Moment-to-force ratio

READERS’ FORUM

Letters to the editor* To be commercially available or not to be: that is the question I thank Dr Katz for his generosity in naming the fuzzy logic method “Dr Noroozi’s logic” in his recent Letter to the editor (Katz MI. Are we moving in the right direction? Am J Orthod Dentofacial Orthop 2008;133:788-9), but he should know that it is not mine. Fuzzy logic is a universally accepted concept that has been in use for more than 30 years. However, it seems that Dr Katz’s main concern in his letter was not fuzzy logic, but my software (Noroozi H. Orthodontic treatment planning software. Am J Orthod Dentofacial Orthop 2006;129:834-7). I am not going to defend my software here, but I would like to clarify some points. Because something is commercially available, it does not necessarily mean that it is bad. If it did, we would discard 99% of the things we use daily, from the brackets we bond to the pants we wear! Users must decide what is really useful and what is not. A well-known professor in an outstanding orthodontic department in the United States had his residents test the software, so I think that means it is at least worth testing. In the acknowledgment part of my article, I thanked several experts who helped me in developing the software, all without any expectations on their part. Doing things gratis is still alive and well. I agree with Dr Katz that it is bad to be “sandwiched between the shill and the sales guy,” but I do not understand the resentment embedded in his letter. I simply invited my colleagues to look at an important issue—Angle’s classification— from a fuzzy logic point of view. Neither I, nor Dr Katz, nor anyone else can force orthodontists to use a specific method. They are prudent enough to decide for themselves. Hassan Noroozi Tehran, Iran Am J Orthod Dentofacial Orthop 2008;134:176 0889-5406/$34.00 Copyright © 2008 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2008.06.014

Moment-to-force ratio The article “Moment-to-force ratio, center of rotation, and force level: A finite element study predicting interdependency for simulated orthodontic loading regimens” (Cattaneo PM, Dalstra M, Melsen B. Am J Orthod Dentofacial Orthop 2008; 133:681-9) is informative but confusing. The authors are to be commended for clearly stating the limitations of this study and for not making unsubstantiated claims. I agree with their conclusions but for different reasons. The confusion stems from the use of the erroneous concept of moment-to-force (M/F) ratio. According to classic mechanics, the moment of a force is *The viewpoints expressed are solely those of the author(s) and do not reflect those of the editor(s), publisher(s), or Association.

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measured as a linear force times the perpendicular distance to the center of rotation, M ⫽ F ⫻ D. When a moment is divided by a force, the result is a distance. A distance has units (eg, centimeters, inches); a ratio has no units. Therefore, M/F cannot be a ratio. A moment, Mi, is generated when a linear force, Fi, is applied to the crown of a tooth, Mi ⫽ Fi ⫻ Di. The tooth tends to tip around a center of rotation. A countermoment, Mc, must be applied to the tooth if we want to limit tipping or to achieve translation (no rotation). In the ideal situation, when there is no tipping, the countermoment is equal and opposite to the initial moment, ⫺Mc ⫽ Mi. The ratio of the moments is ⫺1 in the ideal situation. The countermoment, Mc, that the orthodontist applies to the tooth can be measured or calculated, but the initial moment, Mi, is indeterminate. The force, Fi, might be known, but the distance, Di, to the center of rotation is not known. Di depends on the centroid of the root and the center of resistance. These are 2 unknowns. This is consistent with the authors’ conclusions that “The material properties of the periodontal ligament, the morphology of the root, and the alveolar bone are patient specific.” If we do not know the initial moment, Mi, then we cannot know the ideal countermoment, Mc. When orthodontists use the expression M/F ratio, they might be unaware that 2 separate and distinct mechanisms are comingled. It is the relationship of the countermoment, Mc, to the initial force, Fi (Mc/Fi). This is a circuitous attempt to determine the ideal countermoment, Mc, which is required to achieve translation of a tooth. Nevertheless, Mc/Fi is still a distance, and M/F ratios are “guesstimates” of the center of rotation at a particular instant. The initial moment, Mi, remains unknown. The article does not explain how forces and moments are generated and applied to the experimental teeth. It is confusing when the authors refer to an M/F ratio of zero with uncontrolled tipping. How can there be a ratio of zero? Either M ⫽ 0 (no moment) or F ⫽ infinity. Perhaps they should have indicated that there was neither a countermoment nor a ratio. This means that the tipping was the result of the initial linear force, F, on the bracket. It is not surprising that the authors were unable to confirm “the classic prescription of the M/F ratio suggested in the literature.” Therefore, the ratios are guidelines. Orthodontists need to, and do, treat patients pragmatically. The article correctly states that “tooth movement should always be monitored and the outcome compared with the expected tooth movement. In the case of discrepancy between the predicted and actual tooth movements, the force system should be adjusted.” This is exactly what most orthodontists do. It seems that M/F calculations are useless. The Emperor has no clothes. It is time for orthodontists to abandon the M/F ratio.

American Journal of Orthodontics and Dentofacial Orthopedics Volume 134, Number 2

Readers’ forum 177

Henry I. Nahoum Laguna Hills, Calif Am J Orthod Dentofacial Orthop 2008;134:176-7 0889-5406/$34.00 Copyright © 2008 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2008.06.010

Author’s response Dr Nahoum claims that the definition of the “moment-toforce (M/F) ratio” used in our article (Cattaneo PM, Dalstra M, Melsen B. Moment-to-force ratio, center of rotation, and force level: A finite element study predicting their interdependency for simulated orthodontic loading regimens. Am J Orthod Dentofacial Orthop 2008;133:681-9) is erroneous. However, this is a misconception on his behalf, since we followed the generally accepted definition. In our article, we stated that the center of resistance depends on the actual morphology, anatomy, and individual material properties of the dentoalveolar tissues and, therefore, is tooth specific. In our simulation, the forces are applied, as in reality, at the bracket level, whereas the countermoment is applied as a free vector (by definition). Dr Nahoum raised an issue about the M/F being zero. In this case, the countermoment was assumed to be zero or, in other words, only a pure force was applied to bracket. We don’t agree with the conclusion that “It seems that M/F calculations are useless.” By stating this, Dr Nahoum is negating the validity of the biomechanical approach to orthodontics that was introduced by Dr Burstone. Finally, we are not sure how to interpret the closing sentences, although being from Denmark, we are well aware of “The Emperor’s New Clothes” short story by Hans Christian Andersen! Paolo M. Cattaneo Aarhus, Denmark Am J Orthod Dentofacial Orthop 2008;134:177 0889-5406/$34.00 Copyright © 2008 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2008.06.008

Friction and anchorage loading Dr Halazonetis’s recent letter to editor (Am J Orthod Dentofacial Orthop 2008;133:484-5) entitled “Friction and anchorage loading” certainly made frictional forces seem simple; he referred to “classical mechanics” twice, and I must admit that term is confusing. When I was in engineering school, we studied statics, dynamics, thermodynamics, and so on, but never classical mechanics. I think that part of the difference of opinion in this controversy is that no one has defined friction in real terms and its relationship to resistance to sliding. Dr Halazonetis did not distinguish between the passive and active states of tooth movement. The passive state is the beginning of tooth movement when the wire touches only the sides of the bracket or ligature (classical friction). Once the tooth begins to move down a wire, the tooth immediately tips, and the active state of tooth movement begins. Dr Halazonetis is correct that, if a bracket is in the passive state (much like a laboratory experiment when the

Fig 1. As a tooth is moved down an archwire, the MC is created. The MC is the product of the applied force F and the distance between the forces MC⫽ FW.

Fig 2. The beam shown here has a force F1 applied to it and is supported at both ends. The forces at the supports F2 and F3 are equal to F1, but F2 and F3 are not equal. The MMB in this beam is at F1. This is the part of the beam that will bend the most. Knowing that location of the MBM is important in engineering because it tells us, eg, where a bridge will be most vulnerable, and the engineer can make FWX adjustments. The formula for MMB is: MBM ⫽ . The L MMB depends on the dimensions of W and X. bracket is not allowed to move), the bracket width makes little difference. But in clinical orthodontics, the tooth moves and the bracket tips, and the width could make a difference. Dr Halazonetis said my mechanics were wrong but did not say why. I will explain why the bracket width makes a difference in resistance to sliding (friction, tipping, notching). When a force is placed on a tooth to move it down an archwire, a moment of force (MF) is created. As the tooth moves, a moment of a couple (MC) is created to counter the MF. The MC is equal to the force generated at the corner of the bracket times the width of the bracket (Fig 1). The maximum bending moment (MMB) when a tooth is moved down an archwire is: FWX MMB ⫽ (bending property of a beam) (Fig 2). L To evaluate that effect of bracket geometry (especially width) in regard to tipping, we need to rearrange MC⫽ FW to MC . F⫽ W