Center Resistencia 3d

ORIGINAL ARTICLE

Axes of resistance for tooth movement:Does the center of resistance exist in3-dimensional space?

a b c
Rodrigo F. Viecilli, Amanda Budiman, and Charles J. BurstoneNew York, NY, and Farmington, Conn
aAssispartmbGradcProfeThe aucts oReprinDentiNY 10Subm0889-Copyrhttp:/

Introduction: The center of resistance is considered themost important reference point for tooth movement. It isoften stated that forces through this point will result in tooth translation. The purpose of this article is to report theresults of numeric experiments testing the hypothesis that centers of resistance do not exist in space as3-dimensional points, primarily because of the geometric asymmetry of the periodontal ligament. As analternative theory, we propose that, for an arbitrary tooth, translation references can be determined by2-dimensional projection intersections of 3-dimensional axes of resistance. Methods: Finite element analyseswere conducted on a maxillary first molar model to determine the position of the axes of rotation generated by3-dimensional couples. Translation tests were performed to compare tooth movement by using differentcombinations of axes of resistance as references. Results: The couple-generated axes of rotation did notintersect in 3 dimensions; therefore, they do not determine a 3-dimensional center of resistance. Translationwas 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, shouldbe used to plan movement of an arbitrary tooth. Clinical approximations to a small 3-dimensional “center ofresistance volume” might be adequate in nearly symmetric periodontal ligament cases. (Am J OrthodDentofacial Orthop 2013;143:163-72)

The concept of the center of resistance of a tooth isanalogous to the concept of the center of mass ofa free body.1 The center of resistance, initially

idealized in 2 dimensions, can be determined by the it-erative trial application of a force until it results in trans-lation, or by the application of a couple. In 2 dimensions,the center of resistance coincides with the center of ro-tation when a couple is applied because the resultantforce is zero, and the center of resistance does not trans-late in absence of a resultant force. Tooth movement istypically described taking into account the position ofthe center of resistance or the moment-to-force ratiocurves that are available in the literature.1-7

tant professor and director of the Biomechanics and mCT Laboratories, De-ent of Orthodontics, College of Dentistry, New York University, New York.uate student, College of Dentistry, New York University, New York.ssor emeritus, University of Connecticut Health Center, Farmington, Conn.uthors report no commercial, proprietary, or financial interest in the prod-r companies described in this article.t requests to: Rodrigo F. Viecilli, Department of Orthodontics, College ofstry, New York University, 345 E 24 St, 6th floor, Clinic 6W, New York,010; e-mail, [email protected], April 2012; revised and accepted, September 2012.5406/$36.00ight � 2013 by the American Association of Orthodontists./dx.doi.org/10.1016/j.ajodo.2012.09.010

The assumptions involved in the determination of thecenter of resistance have not been formalized. When de-termining it, there are problems related to material prop-erties, 2-dimensional (2D) vs 3-dimensional (3D) clinicalvalidity, and inaccuracies related to the methods of de-termination. As orthodontics shifts into 3D technologyand tooth movement planning, it is crucial to revisitthis concept to develop a 3D scientific theory of toothmovement.

Here, we hypothesize that the center of resistancedoes not exist as a point in 3D space primarily becauseof general periodontal ligament (PDL) and tooth asym-metry. Thus, its definition should, in this case, be limitedto 2D projections and a specific loading direction. Wewill perform numeric experiments on a maxillary firstmolar to test these hypotheses. After disproving thata general theory of the 3D center of resistance is truewith a counter-example, we will provide and test an al-ternative 3D concept, the axes of resistance. Then, wewill demonstrate how to link the 2D projected centersof resistance and 3D axes of resistance concepts.

The first problem in the center of resistance concept isthe choice of assumption of rigidity of the dentoalveolarunits. Analytic studies incorporated this as a necessarysimplification to mathematically calculate root centroids,

163

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_surname

Delta:1_surname

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_surname

Delta:1_surname

Delta:1_given name

Delta:1_given name

Delta:1_given name

Delta:1_given name

mailto:[email protected]

http://dx.doi.org/10.1016/j.ajodo.2012.09.010

164 Viecilli, Budiman, and Burstone

assuming that they are close to the centers of resistance. Incontrast, contemporary studies typically model nonrigidmineral units to determine centers of resistance. The un-certainty about these assumptions is most likely becausethe scientific determination and clinical purpose of thecenter of resistance are somewhat different problems. IfPDL stresses initiate orthodontic mechanotransduction,and bone and tooth bending will be mostly reversed afterload application is removed, bone and tooth nonrigiddisplacements are likely irrelevant to the final toothmove-ments.8-10 On the other hand, the scientific determinationof the center of resistance is limited to instantaneoustooth displacement before bone modeling takes place.The clinical extension of the use of the concepts ofresistance is the planning of long-term tooth movement.This extension requires the often-overlooked assumptionthat the catabolic bone response is directly proportional tothe initial deformation pattern of the PDL. Bonemodelingrelationships with PDL stresses and strains have only re-cently started to become clear with translatable biome-chanical models.8,9 It has been recently observed thatthe critical entity in instantaneous tooth displacement isthe PDL, and bone and teeth could be reasonablyassumed to be rigid to define PDL stresses for toothmovement.11

The second major issue in the definition of center ofresistance is related to its technical determination. Inautopsy specimens, a major weakness was that the toothdisplacement gauge itself could load the tooth, thereforecausing errors.12 This difficulty was overcome by mathe-matical adjustments, with laser holographic interferome-try or magnetic sensors.13-17 The most commonly usedcontemporary method to evaluate tooth displacement isnumeric, using finite element analysis, which offersmuch more flexibility on modeling conditions.18 Nonlin-ear PDL behavior is characterized approximately by a non-linear (bilinear) stress-strain curve: the 2 lines separated bya strain threshold.5,11,18-24 Therefore, linear modeling ofthe PDL for finite element analysis could be consideredfairly accurate if loads are kept sufficiently low tomaintain PDL strains in the first section of thecurve.8,9,11 However, for a general load magnitudemodel, the PDL must be modeled nonlinearly becauseload magnitude can affect the data.6,25-27 Studies onPDL properties have most recently explored anisotropy,hyperelasticity, and inclusion of PDL fibers (materialproperties and orientation are yet mostly undefinedexperimentally).7,28,29 Development of these models isimportant; however, unlike many compound (matrixand fibers modeled together as a single entity) linearand nonlinear models, there are limited experimentaldata on tooth displacement to match the mathematicallevel of sophistication of such complex models.

February 2013 � Vol 143 � Issue 2 American

The final major issue related to the concept of centerof resistance is its actual existence as a point in 3D spacein a nonidealized (asymmetric) tooth and PDL. In 3 di-mensions, a couple causes rotation of a body about anaxis of rotation. Hence, in the general case, applicationof a couple in 3 dimensions is not sufficient to determinea 3D center of resistance. The idea that the center ofresistance exists as a point in 3 dimensions comesfrom previous studies that idealized a 3D tooth withsymmetric roots, calculating the centroid of the rootmodeled as a paraboloid of revolution.14 This assump-tion is a simplification because, in reality, PDL deforma-tions are major determinants of the pattern of toothdisplacement. Moreover, the roots and the surroundingPDL of an arbitrary tooth have asymmetric surfacesthat can result in asymmetric deformation patterns.

Nagerl et al,13 in a pioneering experiment, concludedthat “the centers of resistance depended on the directionof load application.” This statement was repeated inmore recent studies, with 3D models for several typesof teeth and modeling approaches.7,8 However, thesestudies did not address the main theoretical difficulty:in 3 dimensions, the concept of a center of resistanceas a point is not a general realistic case. There is noworking theory to connect 2D models of centers ofresistance to 3D axes of resistance. The concept ofaxes of resistance of a tooth was introduced ina comprehensive numeric study of the mechanicalenvironment of tooth movement.8 In this same study,it was noted that the axes of rotation generated by or-thogonal moments in 3 dimensions (axes of resistance)did not intersect in a 3D center of resistance point.

MATERIAL AND METHODS

The maxillary left buccal segment of a dry skull (fromfirst premolar to second molar) was scanned and recon-structed with a cone-beam computed tomography iCATsystem (Imaging Sciences International, Hatfield, Pa).The voxel resolution of the scan was 0.125 mm.

The reconstructed voxel data were loaded intoScanIP/FE software (version 4.3; Simpleware, Exeter,United Kingdom) for image segmentation. Gray scalethresholding was used as an initial partition, differenti-ating between the maxillary bone and each posteriortooth. The floodfill tool (an implementation of the re-gion growing algorithm) was used to isolate each toothand bone based on pixel connectivity; this allowed fora separate mask (color coding) of each entity. To createthe PDL, each tooth's mask was duplicated. Then, the di-lation tool was used on this new “accessory”mask to ex-pand about 2 voxels (250 mm) away from the tooth, andthe resulting pixels were subtracted from the total min-eral mask. This method produced fairly accurate and

Journal of Orthodontics and Dentofacial Orthopedics

Fig 1. Finite element model development workflow: A, segmentation of cone-beam computed tomog-raphy images, with bone material properties proportional to the gray scale; B, detailed view of surfacemask rendering of PDL and teeth;C, detailed view of maxillary first molar's highly organic root anatomyon the finite element model;D, 3Dmesh of the posterior segment (note that themesh is denser near thePDL region); E, first molar and PDL nodal view in ANSYS illustrating the coordinate system centerednear the mesiobuccal cusp: x, mesio/distal; y, occluso/apical; and z, linguo/buccal. Only thebone, first molar, and first molar PDL were imported into ANSYS.

Viecilli, Budiman, and Burstone 165

nonuniform PDL thicknesses, varying from approxi-mately 0.15 to 0.5 mm, depending on location, withadapted connections between bone, tooth, and PDL.

The 3D mask surfaces were smoothed with a topol-ogy-preserving algorithm, eliminating minor voxelirregularities while preserving the surface morphology.By using a direct mask-to-mesh technique, a quadratictetrahedrone mesh was produced. The meshingalgorithm was optimally adjusted to provide a minimumof 2 tetrahedron elements in the PDL space betweenbone and tooth, optimizing nodal displacement accu-racy in this material. The resulting mesh had 618,428elements, with less than 0.001% ill-shaped tetrahedrons.The mesh with its material properties was imported intothe APDL software (ANSYS, Canonsburg, Pa) for finite el-ement analyses. The preconditioned conjugate gradientiterative solver at the 10�8 level of accuracy was used.

The maxillary first molar dentoalveolar complexmodel was isolated for this analysis because of its

American Journal of Orthodontics and Dentofacial Orthoped

intricate morphology and had up to about 600,000 ele-ments after postresult refinements. Rigid and nonrigidmineral properties were modeled. In the nonrigid model,the boundary conditions were determined by constrain-ing the bone at its apical and anteroposterior limit sur-faces. The finite element model characteristics anddevelopment process are described in Figures 1 and 2.The tooth-size measurements approximate an averagemaxillary first molar and are shown by the black arrowsin Figure 3.

Table I summarizes all material properties used in thisstudy. Material properties of the bone were modeled asheterogeneous. The cortical bone element materialproperties ranged from approximately 10 to 20 GPa,and trabecular bone properties from 10 GPa to 0.05MPa. These properties were assigned on an individual el-ement basis; ie, Young's modulus of each element wasapproximated as proportional to its gray scale accordingto a mathematical relationship determined previously by

ics February 2013 � Vol 143 � Issue 2

Fig 3. Projections of the coordinate system (blue), tooth, its measurements (black arrows), axes of re-sistance (red), and the 3D distances (red arrows) between the axes of resistance. The measurementsare not to scale. The distances between the axes of resistance represent their directions and approx-imate relative magnitude relationships but are here exaggerated for illustrative and didactic purposes.A, The buccal surface; B, the mesial surface; C, the occlusal surface.

Fig 2. Coordinate system projections illustrating the direction conventions for moments and rota-tions according to the right-hand rule. A, The buccal surface; B, the mesial surface; C, the occlusalsurface.


February 2013 � Vol 143 � Issue 2 American Journal of Orthodontics and Dentofacial Orthopedics

Table I. Material properties (MPa) of the finite ele-ment models

Structure

Nonrigid boneand tooth model

Rigid bone andtooth model

Young'smodulus

Poisson'sratio

Young'smodulus

Poisson'sratio

Enamel 80000 0.3 800000 0.3Dentin 20000 0.3 800000 0.3Bone 0.05-20000 0.325 Constrained 0.325PDL*(\7.5% strain)

0.05 0.3 0.05 0.3

*The PDL was modeled as linear because the experiments were per-formed with loads that lead to PDL principal strains\7.5%: ie, dis-talization and expansion forces up to 1 N and intrusive forces up to0.4 N.


Cory et al.30 The PDL material properties were assignedbased on the experimental tooth displacement data ofPoppe et al.24

For this study, we restricted our load ranges to pro-duce strains within the lower tail of the PDL's nonlinearcurve, allowing for assuming a linear isotropic behaviorof 0.05 MPa. We chose to limit our study to small PDLloads because the PDL stiffening that occurs in higherloads could, by itself, affect the results of the axes of re-sistance. Therefore, it would be difficult to mathemati-cally separate the effect of geometric asymmetry fromthe effect of localized PDL stiffening on the locationof the axes of resistance.6 These modeling decisions war-ranted a more confident result on the specific effect ofgeometric asymmetries on the location of the axes of re-sistance, which are the focus of our hypothesis.

The number, displacement, and coordinates of thenode that defines the axis of rotation were recordedwith an initial couple load of 3 Nmm. The moment vec-tor determined the direction of each axis of resistance.Peak PDL principal strains were under 7.5% in all direc-tions with this load, thus justifying PDL linear propertiesat E 5 0.05 MPa. Cross-product matrix calculationswere performed by inputting nodal coordinates andthe final moment value, to output the necessary nodalforces to obtain a moment vector parallel to each coor-dinate system axis to the tooth (Figs 1, E, and 2). Amoment of a couple parallel to the x direction, forexample, generates a rotation axis that generatesnodal displacement in the y and z directions. Thiscouple-derived axis of rotation (x0) was then tested asone of the 3 axes of resistance. The same procedurewas performed for the other axes. We further approxi-mated the coordinates of the node that defined eachaxis by iterative refinement of the mesh around thenode with the lowest displacements in the plane of


movement, until the position varied within 60.05 mm.The displacement error generated by this error in theposition of the axes was then calculated by finding thedisplacements caused by moments generated by transla-tion forces as far as 60.05 mm from the axes. This ispossible by application of the superposition principle,a convenience of the linear model: the displacement inall directions of the translation tests, and the “error dis-placement” caused by the residual moment error can besummed to estimate a total displacement error range.The error was estimated in worst-case scenarios whenmaximum and minimum displacement errors wereadded or subtracted from the translation test displace-ments near the root apex and the crown. Hence, thesmaller the differences between total displacements,the closer the tooth is to translation. This method to de-termine the axes of rotation, which must be associatedwith an error analysis, was necessary because analyticcalculations of the axes of rotation yield different resultsdepending on which nodes are chosen in the case ofa nonrigid model.

We produced a rigid tooth and bone model to esti-mate the often-neglected error caused by bone andtooth bending (and their associated asymmetric bendingbehavior in different directions) on the determination ofthe axes of resistance. Bone rigidity was modeled by sim-ply assigning zero displacements in all directions asboundary conditions for all nodes. To establish the idealdegree of tooth rigidity within the limits of our model'smesh density, first, a couple was applied in each of the 3spatial directions. Analyses were then run iteratively atprogressively larger tooth elastic moduli (this time, ho-mogeneous for enamel and dentin) until the node num-ber of the lowest displacement node of the tooth nolonger changed for our load of interest (ie, the effectof tooth bending no longer affected the result). It wassufficient to increase the elastic modulus of the toothto 800 GPa to simulate rigid behavior for our establishedaxes of resistance determination error.

After the axes of resistance were obtained in the rigidand nonrigid models, further analyses were limited tothe rigid bone and tooth model only. To ascertain thecorrect combination of axes to obtain translation, weapplied 0.4 N (1y 5 intrusion) and 1.0 N (1z 5expansion;1x5 distalization) forces at 2 planar centersof resistance defined from the planar orthogonal projec-tions of the axes of rotation determined with the cou-ples. This procedure was repeated according to the 2possible center of resistance definitions for each plane(xz, yz, xy), for a total of 3 3 2 loading scenarios: (1)along each hypothetical axis of resistance point projec-tion (eg, the y0 point in the xz plane), and (2) at the in-tersection of the other axes of resistance (eg, the x0z0


Table II. Three-dimensional location of the axes of re-sistance (the orientation of the axis of resistanceshould be used as a reference for translation witha force in a perpendicular direction to the axis)

Axis coordinatefrom mesiobuccalcusp origin

Nonrigid bone andtooth model

Rigid bone andtooth model

x y z x y zx0 axis 11.00 �3.07 11.15 �3.20y0 axis 2.75 �3.59 2.74 �3.15z0 axis 3.07 10.85 3.07 11.08Distance �0.32 0.15 0.52 �0.33 0.07 0.04


point on the xz plane), resulting in a hypothetical 2Dprojected center of resistance.

The forces generated maximum principal strains inthe PDL of less than 7.5% and therefore are compatiblewith our experimentally validated linear modeling ap-proach.

RESULTS

The spatial locations of the axes of resistance (x0,y0, and z0) parallel to each coordinate axis are shownin Table II and Figure 3. The locations of the axesof resistance for the nonrigid and rigid models dif-fered by a maximum of 0.44 mm (z coordinate ofthe y0 axis). Bone and tooth bending mostly increasedthe absolute values of the 3D distances between theaxes of resistance when compared with the rigidmodel.

The absolute values of the 3D distances between theaxes of the maxillary first molar ranged between 0.04and 0.33 mm in the rigid model, which is not affectedby bone and tooth bending. Although the 0.04-mm dis-tance between x0 and y0 in the z direction was within ourerror range, the total 3D distances were out of our errorrange, resulting in real differences according to our erroranalysis. The largest distance in the rigid model was�0.33 mm between the y0 and z0 axes of resistance inthe x direction.

The translation tests and error analyses (Figs 4 and 5)showed that each axis of resistance (x0, y0, and z0)determines the possible parallel lines of action of theforces that would cause translation in the directionsperpendicular to the 2 other axes, respectively (y and z,x and z, x and y). For example, a force in the x directionperpendicular to the z0 axis of resistance causestranslation in the x direction, but associated withrotation around the y axis. To obtain translation in thex direction, it was necessary to find the intersection ofthe y0 and z0 axes of resistance projections on the yzplane. Results for other force directions (y and z) wereanalogous. Translation within the error range could not


be obtained by using the axis of resistance in the samedirection of the force.

DISCUSSION

With the results of this study, it is pertinent to discussthe different center of resistance models used in ortho-dontics and how they relate to the current generalized3D model.

On a 2D tooth in 2D space, a rotation is possibleonly within this plane. Therefore, the tooth rotatesabout the center of resistance (a point) when a coupleis applied. This cannot be extrapolated to a 3D asym-metric tooth. In the 2D case, no axes exist, since rota-tion in other directions would imply the existence of 3Dspace and violate the 2D assumption. Naturally, this isa limited approach to model a clinical situation. In 3dimensions, the geometric description of rotation isdifferent; therefore, the interpretation of the modelsshould not be mixed: ie, a 2D tooth model has move-ment characteristics that are fundamentally differentfrom a 2D projection of a 3D tooth. In a 2D projectionof a 3D arbitrary tooth, the center of rotation and thecenter of resistance do not necessarily coincide whena couple is applied.

In the special case of an idealized paraboloid of rev-olution, all axes of resistance intersect in a 3D center ofresistance because the PDL is symmetric about its centralaxis. On a 2D projection, application of a couple vectorperpendicular to this 2D plane will generate a point pro-jection of the axis of resistance that is coincident withthe other 2 axes of resistance, resulting in the specialcase when a 2D projected center of resistance is coinci-dent with the center of rotation. This projected center ofresistance, determined by 2 axes of rotation (or 1 axis ofrotation with an axis of symmetry), can be used as a ref-erence for translation in any direction (x, y, or z). In or-thodontics, this model is typically assumed to be valid toany tooth for didactic and clinical purposes, although itis essentially a myth.

Here, we determined that pairs of 3D axes of resis-tance can result in 2D “projected” centers of resistance.Each axis represents the zero displacement line producedby a couple: ie, physically, forces that would producea moment in this same direction of the couple only pro-duce translation when passing through this axis. There-fore, to prevent rotation about the x and y axessimultaneously, the 2D projected center of resistancewould be represented by the intersection of the x0 andy0 axes of resistance projections in the xy plane for a forcein the z direction. This is because a z direction force canonly produce moments about the x and y axes. Hence,the z0 axis of resistance provides no information aboutthe necessary position of a translational force in that


Fig 4. Results of the translation tests. Comparisons are betweenA andB, C andD, and E and F. Eachpair has the same color scale; therefore, the more uniform the color, the closer the movement is totranslation. A, Force in the x direction at the z0 and y0 intersection; B, force in the x direction alongx0; C, force in the z direction at the x0 and y0 intersection; D, force in the z direction along z0; E, forcein the y direction at the x0 and y0 intersection; F, force in the y direction along y0. Numerically, the A,C, and E scenarios demonstrate translation according to our error analysis (Fig 4); B, D, and F donot translate according to the same criterion.


direction. This result implies that, to represent the centerof resistance as a 2D projected point, a proper 2D projec-tion plane perpendicular to the line of action of the forceshould be used. Figure 6 exemplifies this notation fora single planar projection. The force acts through thex0y0 intersection and is perpendicular to the plane ofview.

The addition of an independent rotation around thez0 axis would, for example, result in movement that fol-lows the axes of resistance theory and must be describedby a screw-axis approach in 3 dimensions. In a linear sys-tem, the superposition principle can be applied to informus that rotations along z0 (according to the right-handrule) and translations along z0 are independent, andthe resultant movement is the sum of both. This provides


an initial basis for a screw-axis theory of 3D tooth move-ment based on axes of resistance.

According to our results, typical didactic 2D repre-sentations of the center of resistance are mathemati-cally incorrect, because they are inconsistent with theway a tooth moves in 3 dimensions. Figure 7 showsthe alternative, more complicated notation that clarifiesthe error in the commonly used didactic representationof the center of resistance and the center of rotation.This notation is also technically descriptive but lessuser-friendly because it requires at least 1 additionalprojection view.

Future studies should examine whether it is possibleto use tooth mobility data in the directions of a fixed 3Dcoordinate system to obtain axes of resistance in other


Fig 5. Error analysis for total displacements expected near the maximum displacement point near theapex and near the crown in the worst-case scenarios for each loading direction (A, Fx; B, Fz; C, Fy); F,force. The smaller the difference between the maximum and minimum displacements, the closer themovement is to translation. Note that translation (no difference between maximum and minimum totaldisplacement near the apex and near the crown) is possible within the error range only when the direc-tion of the force is perpendicular to the intersection of the 2 perpendicular axes of resistance.

Fig 6. Suggested didactic representation of the 2D cen-ter of resistance point that is compatible with a 3D theoryof axes of resistance. For the center of resistance to berepresented by a point, translation needs to be perpendic-ular to the plane of view at z0x0. Rotation situated on thisplane (along the z axis) is centered on z0 on this view.For an arbitrary 3D tooth, the projected center of resis-tance and center of rotation do not coincide.


arbitrary directions. Most likely, the position of the axesof resistance will change with different couple direc-tions. Still, little is known about how real teeth ormore complex units with highly asymmetric PDLsmove with arbitrary 3Dmoment and force combinations.

Our results for this tooth suggest that, clinically,a compromise can sometimes be justified in lieu of bio-mechanical accuracy. The distances between the axes of


resistance in the maxillary first molar used in this studyare small enough to be clinically insignificant to planmovement of this tooth. In spite of individual rootPDL asymmetries, the overall PDL behaved with smallasymmetry on this specimen. Therefore, in this specifictooth, a simplified “center” of resistance could be con-structed not as a point, but as a small 3D volume repre-senting the errors and distances related to rigidity andtooth/PDL asymmetry.

Based on previous studies, a more clinically signifi-cant distance in the occluso-apical direction of 2 mmor more between the mesiodistal and buccolingualaxes of resistance is expected in single-rooted teethwith asymmetric anatomy. Segments including a seriesof asymmetric teeth/PDLs will probably demonstratethe greatest distances.

Nonrigid models of bone and tooth can increase ordecrease the 3D distances between the axes of resis-tance. This effect depends on material properties,boundary conditions, point of load application, andthe bone-modeling approach. Moreover, authors whochoose an analytic method to determine the positionof the rotation axis, a method we previously suggestedto be problematic in nonrigid models, can obtain largedifferences depending on the choice of nodes used forthe calculation. In our molar model, we determinedthat nonrigidity implies a difference of up to 0.44 mmin the 3D location of an axis of resistance. Our bonemodeling method is limited by the fact that a proper hu-man phantom and a calibrated cone-beam computedtomography image are needed to obtain more realisticproperties. This limitation would not affect our conclu-sions. To prevent modeling errors, we suggest that


Fig 7. An alternative representation of the 2D projected center of resistance point that is compatiblewith a 3D theory of axes of resistance. For translation in the x direction, the force can be applied atthe z0 axis of resistance in A, but the z coordinate of the force vector must be given (B and C). This in-formation would be missing by using only the A projection.


rigidity of bone and tooth should be a standard to deter-mine the axes of resistance.

Differences in the modeling of the material propertiesof the PDL and bone could affect our results by increas-ing or decreasing the distances between the axes of re-sistance. PDL anisotropy related to fiber orientationand “tension only” behavior, combined with stiffening(nonlinearity) and differences in Poisson's ratio, canact to exacerbate these asymmetries to currently un-known extents, depending on the loading pattern andmagnitude.

The results of the positions of axes of resistance inthis study have restricted clinical applicability tocouples of less than about 3 Nmm and subsequenttranslation forces up to about 0.4 N for intrusionand 1.0 N for expansion or distalization. However,our study shows that the “nonintersecting” natureof the axes of resistance is primarily determined byasymmetric geometric characteristics of a tooth andits associated PDL.


CONCLUSIONS

1. Geometric asymmetry of the tooth and PDL impliesaxes of resistance that do not intersect; hence, theaxes do not define the center of resistance as a 3Dpoint.

2. To obtain pure translation in 1 direction (eg, x), a 2Dprojected center of resistance point can be derivedfrom intersections of the axes of resistance orthog-onal projections on the plane determined by the 2other perpendicular directions (eg, y and z).

3. To be consistent in 3 dimensions, didactic represen-tations of the 2D center of resistance projectionshould be constructed on the plane perpendicularto the force vector.

4. Rotation on a 2D projection plane does not occuraround the 2D projected axes intersections (the 2Dcenters of resistance), when a pure moment isapplied in the perpendicular direction. It occursaround the point that represents the remaining



projected axis of resistance on the same plane. Thecenter of resistance and the center of rotation maynot coincide on a 2D projection of a 3D tooth.

5. For clinical purposes, more studies are necessary todetermine on which teeth or segments the 3D centerof resistance small volume can be a good approxi-mation, and for which the clinician should bemore careful and use the axes of resistance projec-tion combinations to plan tooth movement.

REFERENCES

1. Smith RJ, Burstone CJ. Mechanics of tooth movement. Am JOrthod 1984;85:294-307.

2. Tanne K, Nagataki T, Inoue Y, Sakuda M, Burstone CJ. Patterns ofinitial tooth displacements associated with various root lengthsand alveolar bone heights. Am J Orthod Dentofacial Orthop1991;100:66-71.

3. Halazonetis DJ. Ideal arch force systems: a center-of-resistanceperspective. Am J Orthod Dentofacial Orthop 1998;114:256-64.

4. Viecilli RF. Self-corrective T-loop design for differential spaceclosure. Am J Orthod Dentofacial Orthop 2006;129:48-53.

5. Bourauel C, Keilig L, Rahimi A, Reimann S, Ziegler A, Jager A.Computer-aided analysis of the biomechanics of tooth move-ments. Int J Comput Dent 2007;10:25-40.

6. Cattaneo PM, Dalstra M, Melsen B. Moment-to-force ratio, centerof rotation, and force level: a finite element study predicting theirinterdependency for simulated orthodontic loading regimens. AmJ Orthod Dentofacial Orthop 2008;133:681-9.

7. Meyer BN, Chen J, Katona TR. Does the center of resistance dependon the direction of tooth movement? Am J Orthod DentofacialOrthop 2010;137:354-61.

8. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE.Three-dimensional mechanical environment of orthodontic toothmovement and root resorption. Am J Orthod Dentofacial Orthop2008;133:791.e711-26.

9. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE.Orthodontic mechanotransduction and the role of the P2X7receptor. Am J Orthod Dentofacial Orthop 2009;135:694.e1-16;discussion 694-5.

10. Bourauel C, Freudenreich D, Vollmer D, Kobe D, Drescher D,Jager A. Simulation of orthodontic tooth movements. A compari-son of numerical models. J Orofac Orthop 1999;60:136-51.

11. Hohmann A, Kober C, Young P, Dorow C, Geiger M, Boryor A, et al.Influence of different modeling strategies for the periodontalligament on finite element simulation results. Am J Orthod Dento-facial Orthop 2011;139:775-83.

12. Christiansen RL, Burstone CJ. Centers of rotation within the peri-odontal space. Am J Orthod 1969;55:353-69.

13. Nagerl H, Burstone CJ, Becker B, Kubein-Messenburg D. Centers ofrotation with transverse forces: an experimental study. Am J Or-thod Dentofacial Orthop 1991;99:337-45.

14. Burstone CJ, Pryputniewicz RJ. Holographic determination of cen-ters of rotation produced by orthodontic forces. Am J Orthod1980;77:396-409.


15. Vanden Bulcke MM, Dermaut LR, Sachdeva RC, Burstone CJ. Thecenter of resistance of anterior teeth during intrusion using thelaser reflection technique and holographic interferometry. Am JOrthod Dentofacial Orthop 1986;90:211-20.

16. Vanden Bulcke MM, Burstone CJ, Sachdeva RC, Dermaut LR.Location of the centers of resistance for anterior teeth duringretraction using the laser reflection technique. Am J Orthod Den-tofacial Orthop 1987;91:375-84.

17. Yoshida N, Koga Y, Kobayashi K, Yamada Y, Yoneda T. A newmethod for qualitative and quantitative evaluation of tooth dis-placement under the application of orthodontic forces using mag-netic sensors. Med Eng Phys 2000;22:293-300.

18. Vollmer D, Bourauel C, Maier K, Jager A. Determination of the cen-tre of resistance in an upper human canine and idealized toothmodel. Eur J Orthod 1999;21:633-48.

19. Dorow C, Krstin N, Sander FG. Experiments to determine the ma-terial properties of the periodontal ligament. J Orofac Orthop2002;63:94-104.

20. Pietrzak G, Curnier A, Botsis J, Scherrer S, Wiskott A, Belser U. Anonlinear elastic model of the periodontal ligament and its numer-ical calibration for the study of tooth mobility. Comput MethodsBiomech Biomed Engin 2002;5:91-100.

21. Kawarizadeh A, Bourauel C, Jager A. Experimental and numericaldetermination of initial tooth mobility and material properties ofthe periodontal ligament in rat molar specimens. Eur J Orthod2003;25:569-78.

22. Dorow C, Krstin N, Sander FG. Determination of the mechanicalproperties of the periodontal ligament in a uniaxial tensional ex-periment. J Orofac Orthop 2003;64:100-7.

23. Ziegler A, Keilig L, Kawarizadeh A, Jager A, Bourauel C. Numericalsimulation of the biomechanical behaviour of multi-rooted teeth.Eur J Orthod 2005;27:333-9.

24. Poppe M, Bourauel C, Jager A. Determination of the elasticity pa-rameters of the human periodontal ligament and the location ofthe center of resistance of single-rooted teeth: a study of autopsyspecimens and their conversion into finite element models. J Oro-fac Orthop 2002;63:358-70.

25. Cattaneo PM, Dalstra M, Melsen B. The finite element method:a tool to study orthodontic tooth movement. J Dent Res 2005;84:428-33.

26. Cattaneo PM, Dalstra M, Melsen B. Strains in periodontal lig-ament and alveolar bone associated with orthodontic toothmovement analyzed by finite element. Orthod Craniofac Res2009;12:120-8.

27. Toms SR, Lemons JE, Bartolucci AA, Eberhardt AW. Nonlinearstress-strain behavior of periodontal ligament under orthodonticloading. Am J Orthod Dentofacial Orthop 2002;122:174-9.

28. Qian H, Chen J, Katona TR. The influence of PDL principal fibers ina 3-dimensional analysis of orthodontic tooth movement. Am JOrthod Dentofacial Orthop 2001;120:272-9.

29. Natali AN, Carniel EL, Pavan PG, Sander FG, Dorow C, Geiger M. Avisco-hyperelastic-damage constitutive model for the analysis ofthe biomechanical response of the periodontal ligament. J Bio-mech Eng 2008;130:031004.

30. Cory E, Nazarian A, Entezari V, Vartanians V, Muller R, Snyder BD.Compressive axial mechanical properties of rat bone as functionsof bone volume fraction, apparent density and micro-CT basedmineral density. J Biomech 2010;43:953-60.


Documents

Center Resistencia 3d