Biomechanics and force system of butterfly arch: A device for three-dimensional controlling of upper molars

Anchorage loss

In a maximum anchorage situation, anchorage loss happens, more or less due to the creation of a reactive force as a result of anterior teeth retraction during space closure. This appears in two distinct forms: (1) in first order because of mesial-in rotation of posterior teeth, especially first molars when the reactive forces apply from buccal surface related to the centers of resistance; therefore, widest part of teeth encroaches the extraction space, and (2) in second order, as the consequence of tip-forward moment that exerts on posterior anchorage units, these teeth encounter “dumping effect” that leads to more anchorage loss. These two effects are main mechanisms of anchorage loss occurrence when mesially directed force tends to loosen the anchorage unit. It should be reminded that the manifestation in the form of either mesial-in rotation or dumping is because the mesially directed force is applied at the buccal side of the anchorage teeth.

The principle of using tensile strength

The best form of resisting against mesially directed force of anterior retraction in anchorage units is to transfer it into the form of axial loading, either compression or tension. Tensile strength of wires that we use in orthodontic practice is too high even in small, round cross-sectional wires. However, when a mesially directed force exerts on one end of a PB, the resultant of intrinsic forces is equal to 0, meaning there is no axial loading. Therefore, connecting left and right sides of the dental arch in molar area do nothing with anchorage reinforcement at all. If using the anchorage units of the other side of the arch is desired, somehow there must be some kind of establishment of “tension” in the connecting wire or bar. In butterfly arch, when a force that tends to displace the teeth of anchorage units applies, tension rises in all cross sections of the connecting oblique wire known as “bracing unit” (red segment in Figure 4). This would create an additional component to counteract mesially directed force. Mathematically, it could be proven by the following equations (Figure 5)

Because cos α < 1 → ΔL2 < ΔL1 →, there is a differential length change (H − H′) which shows tension.

Principle of using tongue function

A wide pentagonal area, which is called here as “tongue trapping area” (often with acrylic coverage), could trap and direct the tongue pressure during functions (swallowing, speech, and chewing). With respect to the center of resistance of upper molar segments, which is closer to upper first molars, the wider part of tongue trapping area is located behind the center of resistance, and therefore, perpendicular tongue pressure against palate creates a high level of force on that area which tends to tip the crowns of upper first and second molars distally (Figure 6). The vertical component of this force, however, could control the vertical dimension very well.

The principle of 3D interactions

• 1. Interaction of vertical and horizontal dimensions.

Figure 7(a) shows the effect of a force on an object that is displaced by a drag force (F) in a distinguished direction (D) with a constant velocity if we eliminate the role of friction. Figure 7(b) illustrates a condition when a perpendicular force (Fv) applies on the moving object. If we need to maintain the original velocity, we have to increase the drag force to the extent that overcomes the additional vertically directed force. The same story is true for the butterfly arch when the tongue presses the tongue trapping area against palate during function. This vertical component of tongue force could affect the inappropriate reactive mesially directed force of anterior retraction and therefore could enhance anchorage in anteroposterior plane.

• 2. Interaction of transverse and horizontal dimensions.

The rigidity of butterfly arch is considerably high. It is made of stiff stainless steel wire, and soldered joints create short segments that increase total rigidity of the framework of the appliance markedly. If this rigid apparatus encounters anchorage loss, the diameters have to be shortened inevitably (Figure 8) and “butterfly wing effect” occurs (Figure 9). This effect happens when a butterfly closes its wings and results in diminution in width. The high rigidity of the arch withstands this phenomenon because a huge force is needed to distort the multiplication sign-like framework.

The principle of creating reactive forces

Regarding causative factors of anchorage loss, there are some components to neutralize mesial-in rotation and mesial tipping of the first- and second orders, respectively, so establishing a mechanism for applying reactive forces in the form of mesial-out and tip-back moments is essential for anchorage control. Mesial-out moment is created by 1-mm preactivation (0.5 mm each side) with opening of omega loop before placing of the appliance (Figures 10 and 11). This expands the inter-molar width slightly and usually remains until removing the butterfly arch in finishing stage. By reaching the settling phase, the inter-molar width would return to its original dimension. As mentioned earlier, the tip-back moment is created by tongue pressure against the tongue trapping area. It does not mean that the axial inclinations of the upper molars essentially change to more upright position. However, preservation of their inclinations mesiodistally, without any bodily movement, is enough to say that controlling of anchorage is quite successful.

The principle of using short segments

The essence of accurate application of butterfly arch is high rigidity for either preservation of transverse dimension or stabilization of the system and prevention of distortion. To preclude long lever arms and huge bending moments at soldering areas due to tongue function during mastication, we must avoid long segments as far as possible. Therefore, soldering points (green arrows in Figure 3) all over the framework are made to increase the total rigidity of the appliance effectively. These soldered joints must be bulky enough not to be broken when mechanical stress exerts.