Three-dimensional nonlinear prediction of tooth movement from the force system and root morphology.

OBJECTIVES: To determine the different impact of moment-to-force ratio (M:F) variation for each tooth and spatial plane and to develop a mathematical model to predict the orthodontic movement for every tooth. MATERIALS AND METHODS: Two full sets of teeth were obtained combining cone-beam computed tomography (CBCT) and optical scans for two patients. Subsequently, a finite element analysis was performed for 510 different force systems for each tooth to evaluate the centers of rotation. RESULTS: The center of CROT locations were analyzed, showing that the M:F effect was related to the spatial plane on which the moment was applied, to the force direction, and to the tooth morphology. The tooth dimensions on each plane were mathematically used to derive their influence on the tooth movement. CONCLUSION: This study established the basis for an orthodontist to determine how the teeth move and their axes of resistance, depending on their morphology alone. The movement is controlled by a parameter (k), which depends on tooth dimensions and force system features. The k for a tooth can be calculated using a CBCT and a specific set of covariates.

Nonlinear dependency of tooth movement on force system directions.

INTRODUCTION: Moment-to-force ratios (M:F) define the type of tooth movement. Typically, the relationship between M:F and tooth movement has been analyzed in a single plane. Here, to improve the 3-dimensional tooth movement theory, we tested the hypothesis that the mathematical relationships between M:F and tooth movement are distinct, depending on force system directions. METHODS: A finite element model of a maxillary first premolar, scaled to average tooth dimensions, was constructed based on a cone-beam computed tomography scan. We conducted finite element analyses of the M:F and tooth movement relationships, represented by the projected axis of rotation in each plane, for 510 different loads. RESULTS: We confirmed that a hyperbolic equation relates the distance and M:F; however, the constant of proportionality ("k") varied nonlinearly with the force direction. With a force applied parallel to the tooth's long axis, "k" was 12 times higher than with a force parallel to the mesiodistal direction and 7 times higher than with a force parallel to the buccolingual direction. CONCLUSIONS: The M:F influence on tooth movement depends on load directions. It is an incomplete parameter to describe the quality of an orthodontic load system if it is not associated with force and moment directions.

Biology of biomechanics: Finite element analysis of a statically determinate system to rotate the occlusal plane for correction of a skeletal Class III open-bite malocclusion.

INTRODUCTION: In the absence of adequate animal or in-vitro models, the biomechanics of human malocclusion must be studied indirectly. Finite element analysis (FEA) is emerging as a clinical technology to assist in diagnosis, treatment planning, and retrospective analysis. The hypothesis tested is that instantaneous FEA can retrospectively simulate long-term mandibular arch retraction and occlusal plane rotation for the correction of a skeletal Class III malocclusion. METHODS: Seventeen published case reports were selected of patients treated with statically determinate mechanics using posterior mandible or infrazygomatic crest bone screw anchorage to retract the mandibular arch. Two-dimensional measurements were made for incisor and molar movements, mandibular arch rotation, and retraction relative to the maxillary arch. A patient with cone-beam computed tomography imaging was selected for a retrospective FEA. RESULTS: The mean age for the sample was 23.3 ± 3.3 years; there were 7 men and 10 women. Mean incisor movements were 3.35 ± 1.55 mm of retraction and 2.18 ± 2.51 mm of extrusion. Corresponding molar movements were retractions of 4.85 ± 1.78 mm and intrusions of 0.85 ± 2.22 mm. Retraction of the mandibular arch relative to the maxillary arch was 4.88 ± 1.41 mm. Mean posterior rotation of the mandibular arch was -5.76° ± 4.77° (counterclockwise). The mean treatment time (n = 16) was 36.2 ± 15.3 months. Bone screws in the posterior mandibular region were more efficient for intruding molars and decreasing the vertical dimension of the occlusion to close an open bite. The full-cusp, skeletal Class III patient selected for FEA was treated to an American Board of Orthodontics Cast-Radiograph Evaluation score of 24 points in about 36 months by en-masse retraction and posterior rotation of the mandibular arch: the bilateral load on the mandibular segment was about 200 cN. The mandibular arch was retracted by about 5 mm, posterior rotation was about 16.5°, and molar intrusion was about 3 mm. There was a 4° decrease in the mandibular plane angle to close the skeletal open bite. Retrospective sequential iterations (FEA animation) simulated the clinical response, as documented with longitudinal cephalometrics. The level of periodontal ligament stress was relatively uniform (<5 kPa) for all teeth in the mandibular arch segment. CONCLUSIONS: En-masse retraction of the mandibular arch is efficient for conservatively treating a skeletal Class III malocclusion. Posterior mandibular anchorage causes intrusion of the molars to close the vertical dimension of the occlusion and the mandibular plane angle. Instantaneous FEA as modeled here could be used to reasonably predict the clinical results of an applied load.

Load system of segmental T-loops for canine retraction.

INTRODUCTION: The orthodontic load system, especially ideal moment-to-force ratios, is the commonly used design parameter of segmental T-loops for canine retraction. However, the load system, including moment-to-force ratios, can be affected by the changes in canine angulations and interbracket distances. We hypothesized that clinical changes in canine position and angulation during canine retraction will significantly affect the load system delivered to the tooth. METHODS: The load systems of 2 T-loop groups, one for translation and the other for controlled tipping, from 9 bilateral canine retraction patients were made to the targeted values obtained from finite element analyses and validated. Each loop was tested on the corresponding maxillary dental cast obtained in the clinic. The casts were made before and after each treatment interval so that both initial and residual load systems could be obtained. The pretreatment and posttreatment interbracket distances were recorded for calculating interbracket distance changes. RESULTS: As the interbracket distances decreased, the average retraction-force drop per interbracket distance reduction was 36 cN/mm, a 30% drop per 1 mm of interbracket distance decrease. The average antitipping-moment drops per interbracket distance reductions were 0.02 N-mm per millimeter for controlled tipping and 1.4 N-mm per millimeter for translation, about 0.6% and 17% drops per 1 mm of interbracket decrease, respectively. Consequently, the average moment-to-force ratio increases per 1 mm of interbracket distance reduction were 1.24 mm per millimeter for controlled tipping and 6.34 mm per millimeter for translation. There was a significant residual load, which could continue to move the tooth if the patient missed the next-scheduled appointment. CONCLUSIONS: Clinical changes in canine position and angulation during canine retraction significantly affect the load system. The initial planned moment-to-force ratio needs to be lower to reach the expected average ideal value. Patients should be required to follow the office visit schedule closely to prevent negative effects because of significant moment-to-force ratios increases with time.

Axes of resistance for tooth movement: does the center of resistance exist in 3-dimensional space?

INTRODUCTION: The center of resistance is considered the most important reference point for tooth movement. It is often stated that forces through this point will result in tooth translation. The purpose of this article is to report the results of numeric experiments testing the hypothesis that centers of resistance do not exist in space as 3-dimensional points, primarily because of the geometric asymmetry of the periodontal ligament. As an alternative theory, we propose that, for an arbitrary tooth, translation references can be determined by 2-dimensional projection intersections of 3-dimensional axes of resistance. METHODS: Finite element analyses were conducted on a maxillary first molar model to determine the position of the axes of rotation generated by 3-dimensional couples. Translation tests were performed to compare tooth movement by using different combinations of axes of resistance as references. RESULTS: The couple-generated axes of rotation did not intersect in 3 dimensions; therefore, they do not determine a 3-dimensional center of resistance. Translation was obtained by using projection intersections of the 2 axes of resistance perpendicular to the force direction. CONCLUSIONS: Three-dimensional axes of resistance, or their 2-dimensional projection intersections, should be used to plan movement of an arbitrary tooth. Clinical approximations to a small 3-dimensional "center of resistance volume" might be adequate in nearly symmetric periodontal ligament cases.

Effects of stress relaxation in beta-titanium orthodontic loops.

INTRODUCTION: This study evaluates the changes in the force system of the beta-titanium T-loop spring (TLS) caused by stress relaxation. METHODS: Ninety TLSs with dimensions of 6 × 10 mm, made of 0.017 × 0.025-in beta-titanium alloy and preactivated by concentrated bends, were randomly distributed into 9 groups according to the time point of evaluation. Group 1 was tested immediately after spring preactivation and stress relief by trial activation. The other 8 groups were tested after 24, 48, and 72 hours, and 1, 2, 4, 8, and 12 weeks. By using a moment transducer coupled to a digital extensometer indicator adapted to a universal testing machine, the amounts of horizontal forces and moments and the moment-to-force ratios were recorded at every 0.5 mm of deactivation from 5 mm of the initial activation in an interbracket distance of 23 mm. RESULTS: The horizontal forces and moments were higher (P <0.001) for group 1 compared with the other 8 groups, which were not different among themselves. All groups produced similar moment-to-force ratios (P = 0.600), with no influence of time. CONCLUSIONS: The TLSs preactivated by concentrated bends had progressive load decreases over time, and this effect is critical in the first 24 hours.

Orthodontic mechanotransduction and the role of the P2X7 receptor.

INTRODUCTION: The P2X7 receptor plays a crucial role in bone biology and inflammation. Its main function is to promote necrotic tissue metabolism by ensuring a normal acute-phase inflammatory response. We used a mouse model to describe and compare orthodontic mechanotransduction in wild-type and P2X7 knock-out mice. METHODS: By using finite element analysis, mouse orthodontic mechanics were scaled to produce typical human stress levels. External root resorption, bone modeling, and bone remodeling were analyzed with fluorescent bone labels, Masson trichrome stain, and microcomputed tomography. Relationships between the biologic responses and the calculated stresses were statistically tested and compared between mouse types. RESULTS: There were direct relationships between certain stress magnitudes and root resorption and bone formation. Hyalinization and root and bone resorption were different in the 2 types of mice. CONCLUSIONS: Orthodontic responses are related to the principal stress patterns in the periodontal ligament, and the P2X7 receptor plays a significant role in their mechanotransduction.

Force system generated by an adjustable molar root movement mechanism.

INTRODUCTION: Tooth movement simulation is important for planning the optimal force system and appliance design to correct a specific malocclusion. Experimental verification of a 3-dimensional force system is described for a unique molar root movement strategy that can be adapted to many clinical scenarios. METHODS: The force system was measured for molar root movement springs that had adjustable alpha (anterior) and beta (posterior) moments. A 3-dimensional transducer assessed moments and forces in 3 planes during deactivation and simulated molar rotation. Two experimental situations were compared by using 10 springs in each group: spring reactivation was performed to compensate for changes in the force system caused by molar movement, or there was no reactivation. RESULTS: Without reactivation, the force system becomes unfavorable after approximately 5 degrees of molar movement (rotation). With reactivations, a favorable force system through 20 degrees of molar movement is maintained. CONCLUSIONS: Present root-movement appliances require periodic adjustment to achieve optimal tooth movement. Additional studies are needed to design orthodontic appliances for delivering optimal force systems for the entire range of tooth movement.

Three-dimensional mechanical environment of orthodontic tooth movement and root resorption.

INTRODUCTION: The tension-compression theory of bone mechanotransduction is ubiquitous in orthodontics. However, partly due to deficiencies in the characterization of the mechanical environment, there is no consensus on the mechanisms that link stimuli to root resorption and bone response. In this study, we analyzed the predominant directions of tension and compression in the alveolar structures. METHODS: An idealized tooth model was constructed with computer-aided design for finite element stress analysis. The principal stress magnitudes and directions were calculated in tipping and translation. RESULTS: The highest principal stress magnitudes in the root, periodontal ligament (PDL), and alveolar surface occurred predominantly in the longitudinal, radial, and hoop directions, respectively. On the compression side, the only structure consistently in compression in all directions was the PDL; however, magnitudes were different in different directions. CONCLUSIONS: In the same region of root, PDL, and bone, there can be compression in 1 structure and tension in another. At a given point in a structure, compression and tension can coexist in different directions. Magnitudes of compression and tension are typically different in different directions. Because of direction swaps between principal stresses, previously published data of only stress magnitude plots can be confusing and perhaps impossible to understand or correlate with biological responses. To prevent ambiguities, a reference to a principal stress should include not only the structure, but also its predominant direction. Combined stress magnitude and direction results suggest that the PDL is the initiator of mechanotransduction.

Self-corrective T-loop design for differential space closure.

INTRODUCTION: The current approach to measuring T-loop force systems in patients requiring differential anchorage does not consider active unit angulations and steps during space closure. The angulations and steps during movement introduced by rotation can considerably modify the force system acting on the teeth. METHODS: In this study, geometric modifications were determined during controlled tipping of the 6 anterior teeth, where there was no movement of the posterior teeth, thus configuring a type A anchorage situation. RESULTS: An optimal beta-titanium alloy 0.017 x 0.025-in T-loop spring was designed by using a simulation performed with LOOP software (dHAL Orthodontic Software, Athens, Greece) to allow compensation for anterior unit-position effect on the final force system. The force systems produced by this T-loop spring with and without geometric correction of the brackets have significant differences that should be considered in the segmented arch approach to space closure. CONCLUSIONS: The effects of steps, angles, and vertical forces were combined to produce an ideal T-loop design that would provide a more determinate force system. The effects and force systems are estimates based on simplified locations of the centers of resistance, assuming relatively constant behavior of the centers of rotation. These simplifications might differ slightly from what happens in vivo. The finite element method or an accurate spring tester capable of reproducing the geometric corrections should be used to ensure a precise force system.