Three types of brackets for the upper right canine were selected; self-ligating bracket (Figure 1C) (Shinye, Hangzhou, China), MLF bracket (Figure 1D) (Shinye), and conventional bracket (Figure 1E) (Shinye). All of the slots were 0.022 × 0.028 inches in size. Four nickel-titanium (NiTi) archwires (0.012, 0.014, 0.016, 0.018-inch diameter, superelastic NiTi; Shinye) and two steel ligatures (0.20 and 0.25 mm) were used in this study.

The RS values were evaluated by a custom-built testing apparatus that was fixed on a universal mechanical machine (Instron3367; Instron Corp., Kawasaki, Japan). A computer-controlled motor was connected to the apparatus. A universal bevel protractor was used as the carrier for bracket bonding and angle control. The testing brackets were bonded to the upper side of the universal bevel protractor and the mesial-distal direction of the slot was supposed to be parallel to the plump line direction (Figure 2A). With changes in the angulation, the component that held the brackets was moved horizontally backward to avoid deviations caused by the bonding height on angulation expression. The displacement of the horizontal movement was calculated by a trigonometric function (Figure 2B).

The upper clamp of the Instron machine was clamped on the upper side of the archwire and the other side was permanently connected with a section of the spring in order to allow the archwire to rise freely along the direction of the slot. Two tie-wires (0.20 and 0.25 mm) were coupled with MLF brackets, and one tie-wire (0.25 mm) was used for ligation with the conventional brackets.

The drawing force was measured using the load cell of the Instron machine with the rotational angle set in the following order: 0°, 2°, 4°, 6°, 8°, 10°, 15°, and 20°. For each combination of bracket, archwire, and ligature, eight drawing forces were measured. The drawing force of the spring was measured in advance to eliminate the effect of the force generated by the spring. Thus, a total of 144 drawing forces with 16 brackets, 16 archwires, and 16 springs were measured in the study. For each assessment, the sliding velocity of the archwire was fixed at 5 mm/minute for a displacement of 7 mm. All measurements were obtained in the dry state at room temperature.

The findings for each combination of bracket, archwire, ligature at a particular angle were recorded as descriptive statistics with means and standard deviations of the RS value calculated by software (IBM SPSS Statistics ver. 23; IBM Corp., Armonk, NY, USA). The obtained statistics were ranked, and the one-way ANOVA test was used to study the difference between different brackets with the same archwire.

The RS values of different bracket–archwire–ligature combinations with MLF brackets, self-ligating brackets, and conventional brackets at each first-order angulation are shown in Tables 1, 2, and 3. Barring some minor fluctuations, the RS value for the same archwire increased with the angulation when no clearance existed.

The RS values of MLF brackets with 0.012-inch archwires showed a flat curve (Figure 3A) even with changes in angulation. With an increase in the dimensions of the archwires and the ligatures, the turning points of the RS values appeared earlier. The RS values also showed a slightly higher increase when coupled with 0.25-mm ligatures for the same archwire (Table 1).

The RS values of self-ligating brackets nearly always remained low (Table 2) when coupled with 0.012- and 0.014-inch archwires, even with an increase in the angulation. The RS values tended to increase significantly when coupled with 0.016- and 0.018-inch archwires at 10° angulation (Figure 3B).

The RS values of the conventional brackets increased with angulation. The values were obviously greater than those of the other two brackets (Table 3). The curve of the RS values displayed a noticeable tendency to increase with an increase in the angulation (Figure 3C).

When coupled with 0.012-inch NiTi archwires, the MLF and self-ligating brackets showed RS values with minor variations, except for some fluctuations. The RS values tended to increase rapidly with 15° angulation for MLF brackets combined with two ligatures while the RS values of self-ligating brackets with 0.014-inch NiTi archwires remained stable. With 0.016-inch NiTi archwires, RS values for both MLF and self-ligating brackets exhibited a tendency to increase at 10° to 15° angulation. When coupled with 0.018-inch NiTi archwires, the RS values of MLF brackets increased at the beginning of the angulation changes. Meanwhile, the RS values of self-ligating brackets showed a more distinct increasing trend compared to those with the previous three archwires, and the turning point of the curve was between 5° and 10°. Compared to the RS values of MLF and self-ligating brackets, the RS values of conventional brackets tended to increase obviously as the angulation gradually increased (Figure 4).

One-way ANOVA tests were used to compare the differences in RS values among combinations of different brackets with the same archwire (Table 4). The post-hoc test demonstrated that when coupled with 0.012-inch NiTi archwires, the RS values of MLF, self-ligating, and conventional brackets differed from each other (p < 0.05), while the MLF brackets showed similar RS values even with different ligatures (p > 0.05). When coupled with 0.014- and 0.016-inch NiTi archwires, self-ligating and MLF brackets displayed similar RS values whereas the conventional brackets showed significantly larger RS values (p < 0.05). When coupled with 0.018-inch NiTi archwires, the RS values of self-ligating brackets and MLF brackets with 0.20-mm ligatures reveal similar values. The conventional brackets showed the largest values (p < 0.05). The results indicated that when combined with the same archwire, the RS values of these three brackets were in the following order; conventional brackets > MLF brackets > self-ligating brackets.

The ligation acting on the archwire generates a couple to accomplish rotational control.16 Therefore, although the binding is hardly affected by the ligation method in second-order angulation,1718 ligation was assessed in this study as one of the vital factors affecting the RS in first-order angulation.

Based on the design of the MLF bracket noted above, the assumption that a thinner ligature could be chosen to decrease the RS while a thicker one could be used to strengthen rotational control was proposed. Nevertheless, in this study, the mean comparison indicated minor differences between the 0.20- and 0.25-mm steel ties (p > 0.05). Even so, the variations in the RS values between the two ligature groups were worth discussing.

The difference between the two steel wires coupled with 0.012, 0.014, and 0.016-inch NiTi archwires was 2.625 gF, 3.000 gF, and 2.625 gF, respectively. The difference increased to 6.625 gF when the steel wires were coupled with the 0.018-inch NiTi archwire. This could be because archwire sizes up to 0.018-inch coupled with a 0.25-mm ligature will only eliminate the clearance.19 This aspect might have generated the minor but different biological responses in clinical settings. Combining the data shown in Tables 1, 2, 3, the RS values of different steel ties coupled with 0.012-, 0.014-, and 0.016-inch NiTi archwires produced minimal difference from 0° to 15° rotation, while the RS values of different steel ties coupled with the 0.018-inch NiTi archwire produced distinctive values throughout 0° to 20°. If the formula for the critical contact angle in second-order angulation was analogously applied to the first-order angulation, one may speculate that the RS value would be similar when the angulation is below the critical contact angle, and that the 0.25-mm steel ties will allow the system to more easily achieve or exceed the critical binding angle. In addition, the apparent distinction that appeared in the 0.018-inch group was consistent with the findings of a previous study.19

Many studies have concluded that the elastomeric ligature results in greater friction than steel tying.16172021 Based on the ligation characteristics of MLF brackets, it could be inferred that steel tying should be considered as a subjective ligation technique according to various tooth movement targets. Different ligations such as the round tie, figure 8 tie, and diagonal tie also affect the friction.2021 The single ligation could serve to decrease the RS value by increasing the inter-bracket distance (Figure 5A). Likewise, light or tight tying with thinner or thicker ties, to some extent, may indicate the dominance of the passive or active configuration. Thus, since the dental arch represents a system where the components mutually influence each other, even in combination with an initial thin archwire, a thicker ligation could decrease the clearance in some positions. Similarly, thinner ligation could be used to decrease the RS in some positions even when thicker archwires are placed (Figure 5B). A thin ligature and archwire combination with MLF brackets is just like a “loosely tied stainless steel ligature” with a conventional twin wings bracket and produces negligible or the least friction.1822

The analysis of RS in first-order angulation is inspired by studies based on the second-order theory.2324 Similar to the second-order angulation, we believe that the critical contact angle also exists in first-order angulation (α_c). Thus, α_c might correlate with the size of an archwire, the width of a bracket, and the depth of the slot. Moreover, the depth of the slot may be determined by the geometric parameters of the slot and the size of the ligature (Figure 1).

In all situations, the variations in RS values tended to generally increase as the angulation increased without clearance. The differences were observed in the variation range and the critical contact angle values of the different combinations. This study found that the RS values of self-ligating brackets slowly increased. Even when the 0.018-inch NiTi archwires were placed, the variation curve remains flat till the angulation is increased to 15°. Bednar and Gruendeman16 found that the self-ligated Orec SPEED bracket produced a small moment from 2.5° to 15° rotation and a greater moment over 20° rotation. Miles8 and Miles et al.9 compared the Smart-clip and Damon 2 brackets with conventional brackets after initial alignment and found that the self-ligating brackets had a 0.2–0.7-mm irregularity because of the rotational play. Fleming et al.10 used the peer assessment rating (PAR) index to compare the treatment effect of Smart-clip and conventional brackets. Although the findings showed no statistical significance, the PAR score reduction was still lower in the self-ligating bracket group. These similar phenomena might indicate that the clearance of first-order angulation was usually greater in self-ligating brackets than in conventional brackets. In other words, the self-ligating bracket is not sensitive to subtle rotation. The clearance somewhat impedes the rotational control without an active control, so the derotation will not be realized until the rotational angulation is larger than α_c. O'Reilly et al.25 concluded that most clinical situations involve binding and releasing phenomena rather than true friction. Thus, a low RS value is not the only target in orthodontic treatment.

The findings showed one setpoint of angulation after which the curve transformed from comparatively smooth to obviously steep (Figure 6). This setpoint could be considered as the critical contact angle in first-order angulation. Due to the particularity of the first-order direction, each combination of bracket, archwire, and ligation ought to correspond with one setpoint. The range of setpoints for each bracket represents its derotation ability. The turning points of the curves for self-ligating brackets were shown to be mostly concentrated in the range of 10° to 15°. The conventional brackets had few setpoints because the steel ties completely tied the archwires with no passive configuration. The MLF brackets showed the largest range of setpoints under different combinations. This could be because the contact angle for these brackets could always change with ligations and angulations. In addition to 0.20- and 0.25-mm steel ties, other ligature sizes could be chosen to realize different treatment goals. Due to its design, critical contact angle in first-order angulation on each tooth can be adjusted by clinicians even on the same archwire, and theoretically, with countless ligatures, there would be countless critical contact angles as well. Thus, the α could be stabilized at α≈α_c, which would be the most efficient and effective status.26

The limitations of our study are inevitable. First, in contrast to the θ_c in second-order angulation, the calculation formula for α_c in first-order angulation is more difficult to compute due to the complex structure of the irregular bracket cover or the uncertainty of the ligating position. Hence, the theoretical α_c was not employed in this study. The conclusions were based more on empirical results instead of theoretical calculations. Second, the complexity of biomechanics is an inherently unsurmountable problem for in-vitro studies, necessitating further studies or clinical research to improve the understanding of this topic.

In summary, exploration of first-order angulation is more likely to demonstrate that irrespective of the archwire that is inserted in a complete dental arch, in which sliding and rotational control are simultaneously needed, choosing an appropriate ligation method can help orthodontists more efficiently align and level the teeth in the first stage of the treatment. Since the complexity of biomechanical tooth movement could not be completely imitated in an in-vitro study, clinical trial should be carried out to verify the theoretical hypothesis. This process should be more detailed than merely deciding which bracket is the best or advertising for any manufacturer. Indeed, this should involve continuous introspection based on previous clinical and scientific experience so as to more thoroughly meet the orthodontic goal.