Every orthodontist would like to accurately predict the movements resulting from their mechanics. But this task is far from simple. Regardless of the type of appliance we use (continuous archwires, segmented arches, aligners), most of the time we apply forces and moments at points far from the center of resistance (CR) of the teeth. Therefore, it is not enough for us to know the force systems released by the different accessories. We also need to know how to TRANSFER these forces – applied by brackets, tubes and aligners – to the CR of the tooth, or group of teeth that we want to move.

BE CAREFUL WHEN ANALYZING A SET OF ARROWS

The best way to predict tooth movement is through the analysis of deactivation diagrams, which represent the force systems (forces and moments) perceived by the teeth. But ATTENTION: these diagrams almost always refer to the forces and moments perceived by the ACCESSORIES (e.g. brackets or tubes), and as mentioned, these devices are often far from the CR of the teeth. Therefore, we need to take into account the location of the CR, as well as the distance of this area in relation to the line of force being applied in each situation, in order to have more accurate predictions.

AN EXAMPLE FOR TRAINING THE TRANSFER OF FORCES

The analysis of the transfer of forces from the brackets to the CR is a simple process, which can be carried out in two steps. But before describing these steps, I would like to propose a test (Figure 1):

Figure 1: Can you predict the force system perceived by the teeth?

The figure above represents an interbracket relationship of geometry I, in which the two teeth have the same mesial angulation. The insertion of a straight wire segment in this situation therefore produces a force system also in geometry I. That is, the canine bracket receives an intrusive force and a clockwise moment, while the premolar bracket receives an extrusive force and a clockwise moment. From Charles Burstone’s studies, we know that the magnitude of these moments is the same in this specific situation (references 1 and 2). However, no mathematical calculation will be required to answer this objective test. Let’s see:

If you consider that the force system perceived by the bracket will be the same as perceived by the object as a whole (i.e., by the CR of the tooth+bracket), you would choose option C as an answer. However, this answer is wrong, as it did not take into account the degree of angulation of the teeth. Yeah! The greater the angulation of the teeth, the less chance we have of making a correct prediction if we only think about the forces applied to the brackets.

We are going to transfer forces and moments to the CR, so that this verification of the equivalence of force systems will point us to the correct answer. If you are not familiar with this transfer, just follow these steps:

1) First, you directly transfer the force from the application point (e.g. bracket) to the CR, maintaining the same intensity and direction. In the case of applying a couple, this transfer from the bracket to the CR is also simple and straightforward, since the couples are free vectors (that is, the CR perceives couples regardless of the point of application).

2) In the second step, when the line of action of the force doesn’t go directly through the CR, you must add the MOMENT OF THIS FORCE that will be perceived by the CR, as I will illustrate below.

Figure 2: When transferring the force system from the brackets to the CR, we must add the moments generated by the forces applied far from the CR (dashed lines). In the canine, the moment will be counterclockwise (M=Fxd1), and in the premolar the moment will be clockwise (M=FXd2)

Transferring the forces/moments to the canine CR: the intrusive force and a clockwise couple are transferred to the CR, and we need to add the moment of a force, generated by the intrusive force that passes a certain distance perpendicular to the CR, producing a counterclockwise moment. Note that the two moments cancel each other out, leaving only an intrusive force on the canine CR.* (Figure 2).

Transferring the forces/moments to the premolar CR: the extrusive force and clockwise couple are transferred to the CR, and we need to add the moment of a force, generated by the extrusive force that passes a certain distance perpendicular to the CR, producing a clockwise moment. Note that the two moments add up, producing an extrusive force and a large clockwise moment in the CR of the premolar. (Figure 2).

*Even without calculating these variables, it was possible to choose the correct option (D). Let’s understand why the other alternatives were incorrect: A. represents the activation force system. B. impossible system of forces, as it does not meet the principle of equilibrium. C. would only be correct IF the teeth were upright, which was not the case. In addition, this quiz was based on a scientific experiment that transferred the force systems of the six geometries applied in the brackets to the CR of teeth with varying degrees of angulation (reference 3).

TAKE-HOME MESSAGE

We need to be careful when interpreting scientific articles and force diagrams without considering the context (the main variables) of each specific situation. Even when memorizing the classic force systems of the six geometries, some simplifications can lead to significant mistakes. Let me emphasize, for example, the fact that the term “six geometries” has two meanings: 1) the geometric relationships between the brackets; and 2) the force systems produced by these relationships (reference 2).

In the quiz example, it was clear that when installing a straight wire between two teeth with an interbracket relationship of geometry I, the force system perceived by the centers of resistance of the teeth corresponded to that of a geometry IV. In fact, depending on the degree of initial angulation of the teeth, the force system in the CR may be of a totally different type of geometry than that applied to the bracket (reference 3).

Therefore, we must always transfer the forces and moments of the brackets to the CR of the teeth that we want to move. This step is critical for proper prediction of tooth movement during any biomechanical planning.

 

References:

1- Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J Orthod. 1974;65(3):270–289.
2- Gameiro GH. The six geometries revisited: History, terminology and applications. Seminars in Orthodontics. 2020; 26(3):110-116.
3- Halazonetis DJ. Ideal arch force systems: a center-of-resistance perspective. Am J Orthod Dentofacial Orthop. 1998;114(3):256-64.