Introduction: A novel approach to characterizing orthodontic spring-generated force and moment systems has been developed. This method allows simultaneous measurement of all 6 force and moment components acting on a tooth. Methods: A continuous full archwire space-closure technique was simulated, and the complete force and moment systems acting on the teeth adjacent to the extraction space were measured. Results and Conclusions: The data showed that, in addition to the intended forces and moments, there are nontrivial activation-dependent interactions with the other load components, and these complex relationships are affected by the position of the triangular loop.
Crown and root positioning are essential for achieving esthetic, functional, and stable orthodontic results. The necessary tooth movement and control are derived from the clinician’s ability to manipulate the force-moment systems produced by orthodontic appliances. However, the traction generated by actual clinical appliances has never been quantified. Thus, in practice, control is achieved empirically and qualitatively.
Experimental and computational studies, including finite element analyses, have demonstrated that load control is affected by wire material or cross section, loop shape and size, gable bends, interbracket positioning, and attachment/ligation method. Unfortunately, the applicability of these studies is partly compromised. For example, in many experimental and most analytical studies, rigid attachments (clamping) are used. In contrast, in clinical situations, archwires are secured firmly, not rigidly, with stainless steel or elastomeric ligature ties. This has an effect on the generated load system. Furthermore, with few exceptions, previous studies focused primarily on 2-dimensional (2D) (plane) segmental springs usually located between 2 isolated teeth, But, because clinical appliances are often attached to groups of teeth via noncoplanar brackets, activation inevitably creates unintended and (most likely) undesirable load components.
Our objectives were to demonstrate the instrument by quantifying the effects of triangular loop activation and location on the generated load system. There are 4 important departures from typical studies: (1) rather than a 2D plane simulation, an actual 3-dimensional (3D) clinical strap-up is simulated; (2) all 6 load components (vs only 2 force components and 1 moment component in plane simulations) are measured, (3) simultaneously (vs independently); and (4) all previous elements were incorporated into 1 experiment.
Material and methods
For each tooth, the buccolingual, mesiodistal, and occlusogingival axes were the local x, y, and z axes, respectively ( Table and Fig 1 ). Thus, the distal and mesial brackets adjacent to the extraction space each has a unique coordinate system aligned so that the x-axes (buccal) are perpendicular to the plane of their bracket bases, and the z-axes are in the occlusal direction. Therefore, the paths of space closure are aligned with their respective +y and −y axes, approximately 15° apart (only the z-axes of the 2 teeth are parallel).
|Mx (second order)|
|My (third order)|
|Mz (first order)|
|+Mz||Distal crown out|
|−Mz||Distal crown in|
An instrument ( Fig 2 ) based on a microscope frame was designed and built to measure all 6 load components ( Fig 3 ) acting on the tooth of interest (TOI) in a typodont (US patent 6,120,287). It consists of a load cell attached to the frame, an x-y-z adjustment mechanism (microscope stage), and a 3D coordinate measuring system (not shown in Fig 2 ). The TOI was separated from the typodont. Then, by using a full-arch impression to align it to the original occlusion, the TOI was cemented to the adapter. The state-of-the-art gamma load cell (ATI Industrial Automation, Apex, NC) can simultaneously measure, with minute deflections, 3 force (0-65 ± 0.2 N) and 3 moment (0-5 ± 0.0009 Newton-meter [N-m]) components. By using the measuring system to determine the relative positions of the load cell and the TOI bracket, the instrument was calibrated to report moment and force values at the bracket. These readings were displayed simultaneously on a computer with an ISA bus interface.
For illustrative purposes, a clinical case was replicated on the maxillary arch of a typodont (model D85SDP-INU.1, Kilgore International, Coldwater, Mich). The maxillary canines were removed to simulate incisor segment closure. The typodont base was fastened to the microscope stage, and the premolar (TOI) was attached to the load cell ( Fig 2 ).
Stainless steel triangular closing loops were fabricated from 0.016 × 0.022-in maxillary small size preformed archwire (Ormco, Glendora, Calif). Duplicating previous studies, loop geometry consisted of an equilateral triangle with 8-mm sides ( Fig. 3 ). All springs were preactivated with gable bends at the base of the triangle. To minimize residual stresses, the gables were overbent and then adjusted to the desired 15°. The wires were inspected for unwanted third-order bends, and a template was used to assess symmetry. With a modified protocol, the springs were heated to 700°F for 10 minutes and then bench cooled.
The teeth were bonded with 0.018-in Ormco brackets or banded with molar bands. To enable a passive buccolingual fit of the wire into the brackets, composite offsets were built onto the buccal surface of teeth as needed. Elastomeric ties were used to hold the wire in the brackets. Four loop locations in the 12.7-mm interbracket distance were considered ( Fig 4 ).
Before engaging the wire, the instrument was zeroed. Then, a P1 wire ( Fig 4 ) was ligated into the arch with the loop legs separated by a 0.07-mm shim, and both ends were cinched back 90° against the distal molar tubes ( Fig 2 ). Then, with the shim removed, a set of recordings, corresponding to zero activation, was taken. Force and moment recordings were then made at 1.6 and 3.3 mm activations. Activation was accomplished by removing the ties distal to the loop, shortening the distal leg of the wire by pulling it back, and then placing appropriate spacers (1.6 or 3.3 mm) between the cinch bend and the distal side of the tube. New elastomeric ligature ties were placed, and the forces and moments on the TOI were recorded. Nine additional P1 loops were then tested. Then, 10 springs each in the P2, P3, and P4 groups were measured. Thus, a total of 40 springs was tested.
By using a full-arch impression of the original occlusion, the premolar and its socket were reattached with resin into their original positions in the typodont. The lateral incisor’s socket was sawed off, and the incisor assumed the role of the TOI. The sequence described above was then repeated with a new set of 40 springs, 10 each in the P1, P2, P3, and P4 groups.
For each tooth type, an analysis of variance (ANOVA) model with fixed effects for loop location, activation distance, and the 2-way interaction was used to model the absolute value of each force. A random effect for each tooth was included in the model to allow for the correlation induced by repeated sampling of the same tooth. Residual plots were examined for violation of model assumptions of normality and homogeneity of variance. The Sidak adjustment was used for all pairwise comparisons for each force/moment and each interproximal surface to control type 1 statistical error (the probability of seeing a significant difference when, in truth, there is no difference). To compare the magnitudes of forces/moments between premolars and incisors, an ANOVA model with terms for tooth type, loop location, activation distance, the 2-way interactions of these terms, and the 3-way interactions was used to model the absolute value of force. Residual plots were examined to assess possible violation of model assumptions. A Sidak adjustment was used to control for type 1 error.
The 6 force and moment components ( Fig 3 ) as functions of activation for the 4 families of loop position (P1, P2, P3, and P4; Fig 4 ) were obtained for the lateral incisor and the premolar, the teeth adjacent to the extraction space. In general, all 6 components changed with activation and loop location. The forces on the brackets in the buccolingual direction, Fx, are shown in Figure 5 , A (positive Fx is in the buccal direction). Similarly, the mesiodistal forces, Fy, are shown in Figure 5 , B . The positive and negative Fy force components on the premolar and lateral incisor, respectively, move the teeth into the extraction space ( Fig 1 ). Changes in the apical (−Fz, intrusive) and occlusal (+Fz, extrusive) forces are shown in Figure 5 , C .
The concomitant moments about the buccolingual (Mx), mesiodistal (My), and occlusogingival (Mz) directions are shown in Figure 5 , D-F , respectively. The positive directions of Mx, My, and Mz correspond to the buccal, mesial, and occlusal directions according to the right-hand rule convention.
Fy and Mx play the dominant roles in space closure because they produce crown movement in the mesiodistal direction. For all loop locations, |Fy| (the magnitude of Fy) on the incisor ( solid symbols , Fig 5 , B ) is low (<1.0N; ∼0 with the loop at P4) and relatively insensitive to activation and loop position ( Fig 6 , B ). However, the force on the premolar is about twice as high, and it increases with activation. The highest and lowest forces on the premolar occur with the P2 spring and in the most anterior (P1) locations, respectively.
The most mesially located loop (P1) results in an |Mx| that is about 3 times higher on the incisor than on the premolar ( Fig 6 , D ) (at 3.2 mm activation, P <.0001). The most distally located loop (P4) shows the opposite. With a centered loop, P3, the |Mx| on the 2 brackets are nearly equal (at 0, 1.6, and 3.2 mm activation, P = .8538, .0121, and <.0001, respectively). Furthermore, with the spring at P4, Mx on the incisor changes very little with activation ( Fig 6 , D ); it is similar for P1 on the premolar. All x-axis moments on the incisor and premolar are negative and positive, respectively.
Relatively little attention has been focused on forces in the buccolingual direction, Fx. For all loop locations, Fx exists on both teeth, and it is negative (palatally directed) except for location P4, which produces a small buccally directed force on the premolar ( Figs 5 , A , and 6 , A ). |Fx| on the lateral incisor is higher than on the premolar. (For P4, at each activation distance, P <.0001. For P2, at 0 activation, P = .0024; at 1.6 mm activation, P = .9999; at 3.2 mm activation, P = .0138. For P3, the corresponding P values are .0102, <.0001, and <.0001. P <.0001 for all activations of P1.) The largest palatal force on the incisor corresponds to the most distal loop position, P4. Placing the loop mesially reduces the palatal force on the incisor, but, even so, its magnitude remains about the same as the closing force (Fy) on the premolar, and substantially higher than the closing force on the incisor itself ( Fig 5 , B ). (On the premolar, P = .9959, .9812, and .0512 for activations 0, 1.6, and 3.2 mm, respectively. On the incisor, the corresponding P values are .0003, <.0001, and <.0001.)
My has a more consistent pattern ( Fig 5 , E ). On the incisor, |My| increases with activation, but loop location has little effect ( Fig 6 , E ). On the premolar, |My| is smaller, and activation has less influence, and there is a direction reversal of the small magnitude My with P1. All moments on the incisor (+My) produce rotation tending to move the crown buccally or the root palatally. The premolar experiences smaller and, except for the P1 loop, opposite rotations.
The only effects of activation are with P3 and P4 on incisor intrusion and premolar intrusion/extrusion; all other intrusive/extrusive forces are either small or relatively constant with respect to activation ( Fig 5 , C ). In contrast, the magnitudes and senses (extrusion vs intrusion) of Fz are strikingly sensitive to loop location ( Fig 6 , C ). With the exception of Mz on the incisor with P3 and P4, |Mz| does not change with activation ( Fig 5 , F ). The largest effect of loop position is on the incisor Mz at maximum activation ( Fig 6 , F ).
Studies have traditionally looked at the force-moment systems produced by various loop designs and the effects of appliance material, shape, and position. Those studies generally modeled (experimentally, analytically, or numerically) a loop acting in a plane consisting of 2 isolated teeth adjacent to an extraction space. This study is fundamentally different in 2 ways: it is a highly statically indeterminate system because the loop is part of a continuous full archwire, and arch curvature introduces extremely complex 3D interactions. As a consequence, the results obtained with this model manifest in ways that challenge some accepted notions.
For example, the measured force-moment system experienced by the bracket is not the same as the system generated by the spring per se. This is in contrast to the aforementioned studies in which the loop-generated system is identical to what is applied to the bracket. This critical distinction is due to the archwire connection to the proximal tooth. That is, it is impossible to separate the individual contributions (to the measured force-moment system) of the spring on 1 side of the bracket from that of the wire segment that connects it to the adjacent tooth.
For a specific tooth, the sole determinant of a particular translational and rotational movement direction is the force-moment system that acts on the tooth. For convenience, that load system is generally defined relative to the tooth’s center of resistance. It does not matter how the load is applied to the crown as long as the equivalent force-moment system is produced at the center of resistance. It matters not whether the tooth is free-standing or part of a segment; appliance design is irrelevant; treatment philosophy is immaterial.
Another distinction is that, in this model, space closure requires movement of the lateral incisor in its distal (−y) direction or movement of the premolar in its mesial (+y) direction ( Fig 1 ). These paths are not coincident, and there are out-of-plane forces and moments, so equilibrium principles do not require the spring-generated closing forces on the 2 teeth to be the same magnitude. Furthermore, even if the arch were straightened out into a plane to resemble previous models, the 2 teeth would still not experience the same closing forces because of the above-mentioned load sharing with their respective neighbors. In addition, in plane problems, the closing force, Fy, dominates, and the buccolingual force, Fx, is zero. This is in contrast to our measurements—the buccolingual forces ( Figs 5 , A , and 6 , A ) can exceed the closing force magnitudes ( Figs 5 , B , and 6 , B ).
Further examination of the results shows important contradictions to traditional views on loop mechanics. Activation used to increase the closing force (+Fy) on the premolar ( open symbols in Fig 5 , B ) also changes, to various degrees, the other load components on that tooth and on the incisor ( Fig 5 ). The closing force (−Fy) on the incisor ( solid symbols in Fig 5 , B ) also increases, but not at the same expected rate as on the premolar. It is the concomitant palatal-directed force on the incisor (−Fx, Fig 5 , A ) that increases the most with activation.
Three dimensionality and connections to proximal teeth wreak similar havoc on the measured moments. For example, in the centered loop position (P3) at placement (zero activation), |Mx| on the 2 teeth ( solid squares for the incisor, open squares for the premolar, Fig 5 , D ) are approximately equal (20 N-mm) even though their respective x-axes are not parallel. With activation, however, |Mx| on the premolar increases more rapidly. This can be partly attributed to the loss of symmetry caused by activation displacing the loop distally. In general, on the incisor, |Mx| is about the same as |My| ( solid symbols in Figs 5 , D and E), not |Mx| much greater than |My|, which is near 0, as is the case with traditional 2D models.
Loop position is a means to control the M/F ratio. Often, the goal is to obtain the necessary Mx/Fy for tooth translation. As can be seen in Figure 6 , D , spring location has an influence on Mx, particularly at high activations, but less so on Fy ( Fig 6 , B ). However, the buccal/lingual (Fx, Fig 6 , A ) and the intrusive/extrusive (Fz, Fig 6 , C ) forces on the teeth are more affected.
Thus, load coupling becomes obvious when all 3 force and 3 moment components on a tooth in a 3D simulation are simultaneously measured. It was demonstrated that the extremely complex interrelationships between the load components depend on activation and loop location. These side effects, unintended and difficult to predict or identify, are likely to be detrimental to treatment progress. Interestingly, such load coupling was not detected in segmented triangular springs with various combinations of first- and second-order gable bends.
Although we understand that many factors affect the load system, we focused only on loop location. Other factors were fixed. For example, different results would have been obtained if the loop had been placed in an arch, or in an arch location with a different curvature—ie, with a different angle between the y-axes ( Fig 1 ). Although triangular and T-loops are generally considered interchangeable, it is unlikely that the 2 springs would have similar 3D characteristics. These are obvious questions for further investigations.