INTRODUCTION

Mechanisms by which appliances affect force/couple systems around the continuous archwire are the subject of considerable research. It has been argued that certain ligation types produce decreased forces and rotational moments.1 Because treatment efficiency and patient comfort are affected by both ligation and tissue mechanics,2 a better understanding of ligation mechanics is required.

Growing literature describes the experimental measurement of forces acting on teeth during simulated orthodontic treatment.3–7 Capabilities of the orthodontic simulator (OSIM)8 in measuring forces and couples acting on all 14 teeth of the maxillary or mandibular arch were previously demonstrated9,10 with a high right canine pilot experiment that involved a limited sample size and a non–clinically representative method of application.

This study will analyze passive ligation mechanics by assessing the effects on forces and couples exerted by the wire during a high canine malocclusion correction. This paper will describe experimental methods with discussion of results, including limitations, for passive self-ligating brackets. Part 211 will examine elastic ligation mechanics and will compare the 2 ligation systems.

MATERIALS AND METHODS

The experimental apparatus and data processing software utilized in this study are very similar to those described by Badawi et al.10 Testing procedures and methods have been altered to ensure more statistically relevant results while maintaining a high degree of accuracy.

Damon series (Ormco, Orange, Calif) copper-nickel-titanium alloy 0.014-inch round wire was tested. Passive ligation brackets were represented by the Damon 3MX series. A sample size of 12 bracket sets was utilized, with 12 new wires per bracket set, for a total of 144 tests. This yielded a standard deviation in the measured force data of less than 0.05 N, which will allow tests for statistical significance when ligation methods are compared.

The six components of the force/couple system acting on a bracket are reported by the OSIM in a coordinate system at the geometric center of the slot. These coordinate systems (indicated with “BR”) are shown in Figure 1a.

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The center of resistance (CR) was chosen as the reference location because a force alone applied at CR will produce pure translation of the tooth. Rotation of the tooth is the result of couples only. This allows independent interpretation of force and couple results in terms of tooth translation and rotation. Figure 1b shows the relation between the 2 systems, in particular the offsets used to locate CR relative to BR, and CR axes aligned parallel to BR axes. A transformation is required to calculate the forces and couples at CR, as shown in Figure 1c, where force and couple data reported in the CR system are indicated by an asterisk.

Testing Procedure

Consistent alignment of the brackets and teeth was accomplished by fitting a straight 0.021 × 0.025-inch stainless steel rectangular wire into the brackets. Elastics were placed around the brackets to ensure that proper bracket prescription is expressed. Bonding between brackets and 5-mm-diameter aluminum pegs, simulating teeth, was done using Loctite E-60HP Hysol Epoxy (Henkel, Düsseldorf, Germany). The straight wire was removed and pegs mounted on the OSIM. Pegs were oriented to minimize forces using a 0.018 × 0.025-inch Damon guide wire ligated in the brackets.

The OSIM was placed in a chamber warmed to 37°C and was allowed to sit for 1.5 hours. A single 0.014-inch test wire was then ligated. The horizontal and vertical positions of each tooth were finely adjusted to ensure minimal preloads in the Y- and Z-axis directions at the neutral plane (NP).

The test wire was then removed from the system, and the right canine (tooth 1-3) was moved superiorly 4 mm from the NP. The wire was reinserted and the high canine tooth was ligated first, followed by 1-2, 1-4, and then the remaining teeth, alternating on each side of the canine. A data set was taken at the maximum canine position. The high canine was then lowered toward NP in increments of 0.2 mm, with data collected at each increment. To ensure that equilibrium was established, OSIM was allowed to sit for a period of 5 seconds before a data set was taken. The summation of all forces and couples about the OSIM global coordinate system is calculated with each data set as a check on the internal data processing and load cell calibration. The process was repeated for all 12 wires per bracket set.

Following the 12 tests, the bracket set was removed from OSIM and the process repeated with a new bracket set and a new set of 12 wires. Numeric processing and graphic visualization of the data were prepared using Microsoft Excel (Microsoft, Redmond, Wash) and Pro/ENGINEER Wildfire (PTC, Needham, Wash).

Table 1 provides the offsets between BR and CR based on values determined by Sia, Koga, and Yoshida12 for single-rooted teeth and by Dermaut, Kleutghen, and De Clerck13 for multirooted teeth. Average clinical bracket placement on the teeth is based on the findings of Armstrong et al.14; tooth dimensions are based on reports by van Beek15 and McLaughlin, Bennett, and Trevisi.16

Table 1 Location of Center of Resistance Relative to Bracket Coordinate System

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RESULTS

Two representations of OSIM data are provided. The first shows the variation of a specific force or couple component at CR (averaged across all bracket sets and wires) for each tooth around the arch. Figure 2a shows the X-component of the forces, where each column subfigure represents the force history of a single tooth. The horizontal axis represents displaced canine position, labeled on only the 1-6 subfigure as 4, 3, 2, 1, and 0 mm. Each subfigure can be considered a single graph that shares a common vertical axis and an identical horizontal axis range. Subfigures for 1-4 through 1-2 are repeated on an expanded scale in Figure 2b, which also shows data standard deviations. Figures 3 and 4 show force distributions in the Y- and Z-directions, respectively. Figures 5 through 7 show the distributions for couples about the X-, Y-, and Z-axes. The sample size allowed for a force and moment value detection of 0.09 N and of 0.4 Nmm, with a standard deviation of 0.02 N and 0.16 Nmm, respectively.

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The second representation, more qualitative but more easily visualized, is shown in Figures 8 and 9. Bracket and tooth shapes shown are generic. These show the combined force/couple data at CR for the condition when 1-3 is in the +3 mm and the +1 mm position, respectively. The force (8a and 9a) and couple (8b and 9b) components are shown separately for clarity, although it is recognized that these occur simultaneously on each tooth (8c and 9c). Components less than the indicated thresholds are not shown. Interactive three-dimensional (3D) model figures are available online.

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DISCUSSION

This experiment does not attempt to simulate tooth movement resulting from biological processes in the periodontal ligament and alveolar complex, but rather measures forces induced by the archwire using a particular form of ligation.

Here, the desired motion of tooth 1-3 is to translate toward the NP while subjected only to a force through CR in the negative Z-direction (coronal). For archwire equilibrium (net force and moment zero), adjacent 1-4 and 1-2 teeth must carry the necessary reaction forces in the positive Z-direction. These are undesired because they would induce upward translation, but they are unavoidable. In terms of desired movement of only the 1-3 tooth for this simple situation, all other forces and moments acting around the arch are technically undesired.

In Figure 1c, note that forces at CR are identical to forces applied to the bracket, and only the moment values of the couples are changed. Here couples measured at the bracket are small. Couples at CR are therefore affected primarily by bracket forces and offsets; force and related couple distributions therefore have similar shape.

Virtually no generation of force or couple components occurs beyond the teeth adjacent to 1-3. Propagation of the disturbance caused by 1-3 is limited to the adjacent teeth (1-2 and 1-4) for F_X*, F_Y*, and M_Z*. The remaining components (F_Z*, M_X*, and M_Y*) propagate one tooth farther around the arch (1-5 and 1-1). No propagation occurs to the left side of the arch.

F_X* components (mesio-distal) represent resistance to sliding (sum of friction, binding, and/or notching).17,18 As the canine descends, the distance between 1-3 and adjacent brackets diminishes, leaving excess wire length between these brackets that is removed by pushing the wire through the adjacent brackets, as indicated by the signs of F_X* being opposite on 1-4 and 1-2 (Figure 8a).

High mesio-distal forces on 1-4 and 1-2 can be attributed to binding because of the angle with which the wire enters the bracket. For this bracket/wire pair, the critical angle for binding16 is approximately 4.1 degrees. Using the bracket separation in OSIM, binding will occur at 1-4 and 1-2 for all values of the canine position above approximately +0.6 mm. Upon examination of detailed views of the F_X* components (Figure 2b), it is observed that FX* disappears on these teeth when 1-3 is brought within about 0.5 mm of the NP.

As seen in Figure 2, with the exception of 1-4 through 1-2, all brackets display a near zero F_X*. With the wire essentially aligned with these slots, it slips with minimal resistance. This lack of friction arises because F_Y* and F_Z* are also essentially zero.

Figure 3 shows that with the exception of teeth 1-4 through 1-2, F_Y* forces are minimal and all are directed buccally, a consequence of wire curvature around the arch. The magnitude of F_Y* is about one-quarter that of F_X* and F_Z*. The force on 1-3 is greatest at about the +3 mm position of the canine, which approximately coincides with the maximum values for F_X*; this is due to binding.

Binding in adjacent brackets again is thought to be the cause of excess wire length built up between adjacent brackets during descent. This excess wire buildup reaches its maximum at +3.2 mm of malocclusion. As the canine continues to descend, binding decreases, allowing the wire to move more freely within the slot and relieving the wire flex.

The F_Z* (coronal-apical) force curves (Figure 4) show an S-shape. F_Z* on 1-4 through 1-2 is maximum at the initial canine position, drops suddenly, then slowly increases again to reach a second maximum near +2 mm; it then reduces to zero when the canine reaches the NP. This is possibly due to, first, the nonlinear material behavior during unloading of the CuNiTi wire, which requires further investigation, and second, the interaction between the three force components and wire stiffness in bending. The values of F_Z* are initially determined solely by the bending in the wire. This is illustrated in Figure 10a, which is a free body diagram (FBD) of the loaded segment of the wire from 1-5 to 1-1, showing forces in the BR coordinate system XZ plane. Wire curvature is ignored and F_Y forces are considered of lesser significance. Force magnitudes shown in the figure are approximate but correspond closely to the measured data. The initial rapid reduction in F_Z occurs at the same time that F_X forces due to binding are rapidly increasing from zero. As shown in Figure 10b, the presence of binding on 1-4 and 1-2 reduces the magnitude of F_Z on 1-3 required to maintain the bend of the wire. Note that it is conceivable that with sufficient binding force (total lock) on 1-4 and 1-2, the vertical force on 1-3 could be reduced to essentially zero. Clearly, reduced resistance to sliding is advantageous here. As the canine continues to descend and the binding forces and related moments diminish, the observed F_Z is closer to that required to simply maintain the wire offset (Figure 10c). The magnitude of F_Z on 1-3 is not proportional to displacement of the canine because of the combined effects of the CuNiTi material nonlinearity and binding in the adjacent teeth.

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Generally, three contributing terms to M_X* are known (Figure 1c), two of which involve the assumed CR position. The M_X component is assumed to be absent because of the use of round wire. Figure 5 shows the net result of this interaction to generate moments about CR. With F_Y* forces significantly smaller in magnitude than F_Z*, and because offsets ΔY and ΔZ are similar, M_X* data tend to follow the F_Z force trend, shape, and alternating sign on adjacent teeth, as seen in Figure 4.

Y-axis couples M_Y* (Figures 6, 8, and 9) rotate the tooth mesio-distally. The large F_Z force does not contribute because its line of action is perpendicular. M_Y* magnitudes are therefore much less than M_X*.

Z-axis couple M_Z* (Figures 7 through 9) causes rotation around the tooth long axis. Because F_X is the main contributor, it can be seen that the M_Z* curves mimic the behavior of the F_X data. As such, significant M_Z* couples on 1-4 and 1-2 are the result of binding.

The assumed CR location is a point of contention. Despite its importance in revealing the influence of applied force systems on tooth movement, its location is not known with any great precision and is undoubtedly patient-specific. BR-CR offsets have a major influence on computed CR couples, and therefore on the propensity of the teeth to rotate.

In these experiments, the lack of intermittent perturbation to the system, chewing, for example, will likely affect the results. These disturbances are conjectured to produce sporadic release of binding of the wire in the bracket.19 This would have an effect on F_X and F_Y expression on 1-4 and 1-2, and hence on the couples acting at their CR, because only they experience binding.

This experimental setup also does not include several potentially significant factors such as tooth-to-tooth contact, compliance of the periodontal ligament, and saliva. These are anticipated to have an effect on the generation and/or propagation of forces around the arch and are the subjects of continuing work.

Finally, the simulated pure vertical descent of the canine to the NP would not occur clinically under the action of the measured force systems because of the presence of couples at the CR of this tooth. This is primarily M_X* causing labial rotation of the crown. In a clinical situation, this would be compensated by applying an appropriate countermoment M_X to the bracket to reduce M_X* to zero.

Despite its limitations, the current experiment does provide insight into the interaction of passive self-ligating brackets with a round CuNiTi wire. This serves as a baseline experimental model. A companion paper11 reports on similar tests using standard elastic ligation. Part 2 of this study11 focuses on the mechanics of elastic ligation and how it affects force and couple propagation around the arch.

CONCLUSIONS

• Force/couple data were obtained simultaneously for all teeth. By allowing the wire to slide freely through the bracket slot when below the critical angle for binding, binding is evident only at adjacent brackets to the canine, and friction forces are virtually eliminated throughout the arch. Because this malocclusion requires only the vertical descent of the canine, forces and couples acting on the remaining teeth are essentially undesired.

• Under these special test conditions, passive ligation is well suited because minimal propagation of the force/couple system occurs around the arch, even without consideration of other factors.

• Although no definitive conclusion can be reached from these data regarding the general clinical advantages of passive ligation over other systems, it can be seen that reduced resistance to wire sliding within the slots can potentially reduce the incidence of undesired loads. Depending on the clinical requirements, this may be beneficial.