ORIGINAL ARTICLE

Optimum conditions for parallel translation ofmaxillary anterior teeth under retraction forcedetermined with the finite element method

Teasoo Kim,a Joungsik Suh,b Naksoo Kim,c and Moonkyu Leed

Seoul, Korea

Introduction: In this study, we used the finite element method to examine the optimum conditions for paralleltranslation of the anterior teeth under a retraction force. Methods: Finite element models of the 6 maxillaryanterior teeth and the supporting structures (periodontal ligament and alveolar bone) were generated as a stan-dard model based on a dental model (Nissin Dental Products, Kyoto, Japan). After designating the positionand length of the power arm as variables, the initial displacement of each tooth was measured with finite el-ement simulation, and the rotation angle of each tooth was calculated. Results: The relationship between theposition and length of the power arm was analyzed, and model equations for this relationship were proposed.As a result, the length of the power arm was either 4.987 or 8.218 mm when it was located either between thelateral incisor and the canine or between the canine and the first premolar, respectively. Conclusions: Thelength of the power arm increased as its position was moved from the lateral incisor to the premolar. Thiswas because the length of the power arm must be increased to be in equilibrium mechanically. Overall, it isexpected that the efficient positions and lengths of the new dental models can be calculated if these totalprocedures are established as a methodology and applied to new dental models. Moreover, the parallel trans-lation of the maxillary anterior teeth can be generated more effectively. (Am J Orthod Dentofacial Orthop2010;137:639-47)

Orthodontic treatment maintains the occlusion byrearranging the teeth to a desirable position byapplication of an external force to the teeth

and supporting structures.1 The demand for adult ortho-dontic treatment has increased. In addition to morepatients seeking treatment to have their malocclusioncorrected, there has also been an increase in the numberof orthodontic treatments for bimaxillary prognathism inKorea. This suggests that a protruded facial contour iscorrected by retracting the anterior teeth with teeth innormal occlusion.

For the best orthodontic treatment, tooth movementcan be estimated and controlled without harming thestructures surrounding the teeth by application of an op-

aDDS, UNI Dental Clinic, Seoul, KoreabStudent, Department of Mechanical Engineering, Sogang University, Seoul,

Korea.cProfessor, Department of Mechanical Engineering, Sogang University, Seoul,

Korea.dPost-doctoral researcher, Department of Mechanical Engineering, Sogang Uni-

versity, Seoul, Korea.

The authors report no commercial, proprietary, or financial interest in the

products or companies described in this article.

Reprint requests to: Naksoo Kim, Department of Mechanical Engineering,

Room 601, Adam Schall Hall, Sogang University, 1, Shinsoo-Dong, Mapo-

Gu, Seoul 121-742, South Korea; e-mail, nskim@sogang.ac.kr.

Submitted, March 2008; revised and accepted, May 2008.

0889-5406/$36.00

Copyright � 2010 by the American Association of Orthodontists.

doi:10.1016/j.ajodo.2008.05.016

timal external force.1 Accordingly, in the orthodonticprocess, it is essential to examine the biometric phe-nomena both physiologically and histologically. There-fore, biomechanical analysis for the applied orthodontictools should be carried out before the procedure. Thisbiomechanical study is an important way to predictand explain the complicated reactions in orthodontics.1

Previous studies have used the photoelasticitymethod,2 the strain gauge method,3 laser holography,4

and the finite element method (FEM) to examine thebiomechanical characteristics of orthodontic toothmovement. Among them, the FEM can reconstruct thesize and shape of a model similar to the actual object.The FEM allows evaluation of the stress distributionsand 3-dimensional (3D) displacements in various sys-tems with an irregular geometry and nonhomogeneousphysical properties.5

Sliding mechanics to eliminate the loop from thearchwire are used mainly to produce effective orthodon-tic tooth movement.6-8 Although the anchorage might bedestroyed by friction between the bracket and acrhwire,this problem can be solved by using miniscrews as ortho-dontic anchorage.9 There have been attempts to increasethe length of the power arm for applying a retractionforce to the position close to the center of resistance(CR) of the anterior teeth.10 Therefore, previous studieshave examined the possibility of translating the anterior

639

Fig 1. A, FE model; B, FE model of the right side and base coordinates.

Fig 2. Dimensions of the central and lateral incisors and the canine.

640 Kim et al American Journal of Orthodontics and Dentofacial Orthopedics

May 2010

teeth parallel to the desired direction by using a retractionforce. Sia et al11 measured the rotation angle of the cen-tral incisor according to the length of the power arm us-ing a 2-point 3D displacement magnetic sensordeveloped by Yoshida et al.12-14 Sia et al11 concludedthat the CR of the maxillary central incisor is locatedat approximately 77% of the root length from the apex.Burstone and Pryputniewicz4 reported that the CR is lo-cated at 67% of the root length from the apex on a 3Dmodel with parabolic root geometry. Using finite ele-ment (FE) models, Tanne et al15 concluded that the loca-tion of the CR shifted toward the root when the rootlength of the alveolar bone increased. However, previousstudies described only the CR of 1 tooth model or usedthe CR of the maxillary central incisor to represent theCR of all teeth. Few studies have examined the relation-ship between the position and the length of the powerarm because there are few explanations for the tippingphenomenon according to the position of the power arm.

We examined the optimum conditions for paralleltranslation of the maxillary anterior teeth with normalinclination under a retraction force using the FEM.The FE models of the 6 maxillary anterior teeth andthe supporting structures (periodontal ligament [PDL]and alveolar bone) were generated as a standard model,and FE simulations were performed as a horizontal re-traction force was applied bilaterally to the powerarm. The initial tooth movements and degree of rotationwere measured according to the position and length ofthe power arm, and the relationship between the vari-ables was then analyzed and explained mechanically.

MATERIAL AND METHODS

FE models of the 6 maxillary anterior teeth and sup-porting structures (PDL and alveolar bone) were gener-ated as a standard model based on a normal axis of theanterior teeth arranged in a dental model (Nissin DentalProducts, Kyoto, Japan) as shown in Figure 1. Maxillary

Table I. Material properties of the PDL, alveolar bone,bracket, and wire

MaterialYoung’s

modulus (MPa)Poisson’s

ratio

Tooth (rigid body) — —

PDL (average value) 1 0.3

Alveolar bone (average value) 1000 0.3

Stainless steel bracket and wire 200000 0.3

Data from Reimann et al21 and Dorow and Sander22.

American Journal of Orthodontics and Dentofacial Orthopedics Kim et al 641Volume 137, Number 5

first premolars were excluded because they might be ex-tracted. Figure 2 shows the dimensions of the central in-cisor, lateral incisor, and canine. In general, an FEmodel of a dental structure is generated by scanneddata such as computed tomography (CT). We did notuse a CT machine, and we measured the dental modelof the Nissin product manually. Like CT, a tooth con-sists of 5 or 6 fault planes, and each plane was connectedby a Ferguson curve. A PDL was extruded with a thick-ness of 0.25 mm horizontally. This method had many as-sumptions, and this structure might be a little differentfrom a real dental structure. However, since a personhas each dental structure, we did not focus on how tomake it similar to a realistic dental structure, but weconcentrated on the new methodology of parallel trans-lation of the maxillary anterior teeth based on our rea-sonable logic and procedures.

The PDL plays the major role in orthodontic toothmovement, and the stiffness of the tooth is relativelygreater than that of the PDL and the alveolar bone.16

Therefore, the tooth was assumed to be a rigid body,and the PDL and the alveolar bone were assumed tobe isotropic and have homogeneous linear elasticity.This does not perfectly reflect the complex structureand behavior of the PDL. However, in combined exper-imental and numeric studies, this assumption was dem-onstrated to be valid for orthodontic loading andsufficient for describing initial tooth displacement.17-19

ABAQUS (standard version 6.5-1, Dassault Systems,Providence, RI) was used to construct a 3D FE modelof the maxillary anterior teeth and to carry out the FEanalysis. A mesh of the tooth was generated with 3D4-node rigid elements, and those of the other elastic ma-terials consisted of 3D 8-node linear brick elements.

Considering the biologic changes during orthodontictreatment, significant error could be generated if thePDL and alveolar bone are not included in the FE anal-ysis of the orthodontic phenomena, in which the toothmoves through the PDL and the alveolar bone.1 Accord-ing to Coolidge,20 the mesh of the PDL consists of 3D 8-node linear brick elements with a 0.25-mm thickness and3 layers. The alveolar bone surrounding the PDL is also

constructed of 3D 8-node linear brick elements with 3layers, and the displacements of the nodes on the bottomsurfaces of the alveolar bone are restricted in the x-, y-,and z-axes. In the 6 anterior teeth, the total numbers ofnodes and elements were 5880 and 5700, respectively.The PDL had 86,400 nodes and 65,880 elements, andthe alveolar bone comprised 18,720 nodes and 20,550elements. A wire with a rectangular cross-section of0.022 3 0.025 in2 was placed by using a bracket witha rectangular cross-section of 0.021 3 0.025 in2. Boththe wire and the bracket were made from stainless steel.The bracket and archwire were also assumed to exhibitlinear elasticity. Table I lists the material properties ofeach object based on the results reported by Reimannet al21 and Dorow and Sander.22 In the standard coordi-nate system, the origin of the geometric coordinate wasthe center point of a horizontal line connecting the inci-sal edges of both central incisors. The 1x-axis was set inthe centrifugal direction, the 1y-axis was set in thelabial direction, and the 1z-axis was set in the apicaldirection.

To determine the reliability of the FE simulation, theresults were compared with those of Sia et al11 after an-alyzing with the same boundary conditions. Accordingto Sia et al, a 150-g horizontal retraction force was ap-plied bilaterally parallel to the archwire after the powerarms had been soldered to both sides of the mesial ca-nine region of the archwire. Each power arm contained6 small hooks with 2 mm between them. In our study,a 150-g horizontal retraction force was also applied bi-laterally parallel to the archwire to simulate the en-masse retraction of the anterior teeth. To be equivalentto the direction of the archwire, the retraction forcewas applied at an angle of 35�, and the components ofthe force at the x- and y-axes were calculated as shownin Figure 3. However, the power arm is not normally lo-cated at the canine but between the lateral incisor andthe canine, or between the canine and the first premolar.Therefore, after examining the reliability of the FE sim-ulation, the 2 above-mentioned cases were analyzed torealistically describe the orthodontic tooth movement.Although the magnitude of the retraction force wasthe same, its degree was changed to correspond to thedirection of the archwire according to the position ofthe power arm. The retraction force was applied at anangle of 23� or 45� if the power arm was located be-tween the lateral incisor and the canine or between thecanine and the first premolar, respectively. Becausethe length of each power arm ranged from 3 to 11 mmat 2-mm intervals, 5 cases were analyzed according tothe position of the power arm.

Mechanical phenomena such as bending, shearing,and sliding were generated by the retraction force

Fig 4. Displacements of the central and lateral incisors and the canine according to the length of thepower arm when it is located at the canine. B, Bottom; M, middle; T, top.

Fig 3. Loading conditions with the position of the power arm.

642 Kim et al American Journal of Orthodontics and Dentofacial Orthopedics

May 2010

to the power arm in the realistic orthodontic tooth move-ment. They were considered for performing the FE sim-ulation to realize the interaction between the bracketand archwire graphically. Accordingly, the friction con-tact between the bracket and the archwire followed thefriction factor as 0.2, which was assumed in this study.In addition, the bracket was tied with the anterior teethto increase the convergence of computation, because thebracket generally sticks to the teeth in realistic ortho-dontic tooth movement, and the interaction betweenthe bracket and teeth has little impact on this work.Since the power arm was connected with the archwire,they were considered 1 object, and it is also possibleto describe the bending of the power arm under a retrac-tion force.

To measure the displacement of the tooth under theretraction force, a virtual line connecting the root apexto the incisal edge of each tooth was drawn based on

the central part of each tooth. The displacement dataof 31 nodes (35 nodes for the canine) passing throughthis virtual line were obtained. The data for the lingualdirection (–y-axis) were only considered to calculate therotation angle. Since the difference in displacement be-tween the root apex and the incisal edge was the numer-ator, and the length of the tooth was the denominator,the rotation angle of each case was determined by usingan arctangent trigonometrical function.

RESULTS

Before comparing the FE simulation with the exper-imental results, it was estimated that the power armsused in the experiment of Sia et al11 were differentfrom those normally used because the power armswere soldered at both sides of the canine. Therefore,in the FE simulation, the power arm and the archwire

Fig 5. Displacements of the central and lateral incisors and the canine according to the length of thepower arm when it is located between the lateral incisor and the canine. B, Bottom; M, middle; T, top.

Fig 6. Displacements of the central and lateral incisors and the canine according to the length of thepower arm when it is located between the canine and the first premolar. B, Bottom; M, middle; T, top.

American Journal of Orthodontics and Dentofacial Orthopedics Kim et al 643Volume 137, Number 5

were assumed to be rigid bodies. As shown in Figure 4,it was confirmed that a parallel translation had been gen-erated when the length of the power arm was 7 mm. Par-allel translation means that the anterior teeth move andmaintain the inclination of the axis of each tooth. There-fore, parallel translation occurs because the displace-ments of each node used to measure the displacementare distributed equally. The central and lateral incisorsand the canine moved parallel when the length of thepower arm was 7 mm. However, clockwise rotation,crown-lingual tipping, was generated when the lengthsof the power arm were less than 6 mm. On the other

hand, counterclockwise rotation, crown-labial tipping,occurred when the lengths of the power arm were over8 mm. These results are similar to the rotation angles ac-cording to the length of the power arm reported by Siaet al.11

The effects of the length of the power arm for the 2positions were analyzed because the power arm is lo-cated either between the lateral incisor and the canine,or between the canine and the first premolar in realisticorthodontic tooth movement. Figures 5 and 6 show thedisplacements of each tooth when the power arm was lo-cated between the lateral incisor and the canine, and

Table II. Rotation (�) of each tooth according to positionand length of the power arm

Case PositionLength(mm)

Centralincisor (�)

Lateralincisor (�) Canine (�) Sum (�)

1 1 3 –0.0049 –0.0049 –0.0049 –0.0147

2 1 5 0.0001 0.0001 0.0001 0.0003

3 1 7 0.0049 0.0049 0.0049 0.0148

4 1 9 0.0096 0.0097 0.0097 0.0290

5 1 11 0.0143 0.0143 0.0143 0.0429

6 2 3 –0.0239 –0.0240 –0.0240 –0.0720

7 2 5 –0.0143 –0.0144 –0.0144 –0.0432

8 2 7 –0.0052 –0.0052 –0.0053 –0.0157

9 2 9 0.0037 0.0037 0.0037 0.0111

10 2 11 0.0123 0.0123 0.0122 0.0368

644 Kim et al American Journal of Orthodontics and Dentofacial Orthopedics

May 2010

between the canine and the first premolar, respectively.In the former case, parallel translation occurred whenthe length of the power arm was 5 mm. In the lattercase, it was predicted that the anterior teeth wouldmove parallel when the length of the power arm was be-tween 7 and 9 mm. This means that parallel translationof the anterior teeth will occur when the length of thepower arm is approximately 8 mm.

Based on these results, Table II lists the rotation an-gles of each tooth in 10 cases to express the relationshipbetween the rotation angles and the position and lengthof the power arm as a model equation. The position ofthe power arm was designated as the x-axis, and a valueof 1 indicated that the power arm was located betweenthe lateral incisor and the canine, and a value of 2 indi-cated that the power arm was positioned between the ca-nine and the first premolar. The y-axis means the lengthof the power arm with values of 3 to 11 mm at 2-mm in-tervals. The z-axis indicates the summation of the rota-tion angles of each tooth according to the position andlength of the power arm. As a result, the summationwas described in a 3D view (Fig 7, A), and model equa-tions and the length generating the parallel translationwere calculated (Fig 7, B). It was estimated that thesummation data were also distributed linearly becausethe FE simulation was performed based on the assump-tion that the PDL and the alveolar bone exhibit linearelasticity. Accordingly, the summation data of the rota-tion angles for each position are represented as linearexpressions with the length of the power arm. Thelength of the power arm was 4.987 mm when thex-axis had a value of 1, and the summation of the rota-tion angles was 0�. When the value was 2, the length ofthe power arm was 8.218 mm. The displacements of theteeth for each position were reanalyzed by using theoptimum lengths obtained from the model equation.As shown in Figure 8, the summations of the rotationangles of each tooth were 0.00018� and 0.00040� at

lengths of 4.987 and 8.218 mm, respectively. It wasconfirmed that the parallel translation of the maxillaryanterior teeth increased when the length of the powerarm was 4.987 mm, and it was located between the lat-eral incisor and the canine. In Figure 8, each tooth forthe x-axis with a value of 1 had a relatively uniform dis-placement compared with a value of 2. The uniform dis-placement indicated that the former case can control thedisplacement of each tooth easily. In the former case,the displacement of each tooth was approximately–2.2 mm, but there were some differences between thedisplacements of each tooth in the latter case. Therefore,it was concluded that the parallel translation of the max-illary anterior teeth was more stable in the former casebecause of the uniform displacement of each tooth.

DISCUSSION

The starting point of this study was the results of Siaet al.11 Based on that article, we considered the paralleltranslation of the teeth and the CR. According to Sia etal, the CR of the maxillary central incisor represents theCR of all teeth. There is no research for the relationshipbetween the position and length of the power arm to ex-plain the tipping phenomenon according to the positionof the power arm.

We numerically analyzed the movement of themaxillary anterior teeth with normal inclination undera retraction force using the FEM and proposed a newmethodology. The new methodology represents clearly1 model equation for the relationship between theposition and length of the power arm. Subsequently,we can predict tooth movement quantitatively as shownin Figure 6. There is little research found in the literaturesimilar to our new methodology for all maxillary ante-rior teeth. In addition, this methodology can be usedin the future to obtain optimal orthodontic treatmentconditions of patients from CT data by generalization.Therefore, this research has some significance inorthodontic treatment.

The FE simulation of 1 selected standard model wasperformed, and the results were compared and analyzed.If these total procedures are established as a methodol-ogy for application to new dental models, it is expectedthat the positions and lengths of new dental models canbe calculated, with more effective parallel translation ofthe maxillary anterior teeth generated. Generally, theposition of the power arm is located either betweenthe lateral incisor and the canine, or between the canineand the first premolar. According to this study, the datafor the summation of the rotation angle were extractedat different lengths of the power arm at 2 positions,and the optimum lengths of the 2 positions were

Fig 7. A, Total rotation in degrees of the anterior teeth according to the position and length of thepower arm in a 3D view; B, model equation between rotation (�) and length of the power arm (mm).

Fig 8. Results from the optimum lengths of the powerarm according to its position by using the model equa-tion.

Fig 9. Relationship between the moments between theCR and the retraction force for controlled tipping. Mi

F,the moment generated by F at the position i; Mi

CR, thecounter moment; F, the retraction force; solid arrows,the moment generated by F; dashed arrows, the countermoment.

American Journal of Orthodontics and Dentofacial Orthopedics Kim et al 645Volume 137, Number 5

calculated. A comparison of the slopes of the modelequations for the 2 positions (Fig 7, B) showed thatthe change in rotation angle was gentler when thepower arm was located between the lateral incisorand the canine. The application of a retraction forceat this position indicates more stable movement ofthe anterior teeth than when the power arm is posi-tioned between the canine and the first premolar.This was confirmed by comparing the displacementof each tooth with the optimum length (Fig 8). There-fore, we concluded that the method of analysis, theprocedures, and the results of the standard dentalmodel were appropriate.

Previous studies examined the CR for parallel trans-lation. Sia et al11 stated that the location of the CR of themaxillary central incisor was approximately 77% of theroot length from the apex. Using finite element models,Tanne et al23 determined the location of the CR for themaxillary central incisor to be at 76% of the root lengthfrom the apex. However, there were some differencesbetween the results of previous studies and our study.Because the power arm was located at the canine, andits length was 7 mm, a similar parallel translation ofthe maxillary anterior teeth was generated to that re-ported by Sia et al.11 Moreover, the CR of each toothwas located at approximately 83% of the root length

646 Kim et al American Journal of Orthodontics and Dentofacial Orthopedics

May 2010

from the apex when the CR was designated as the centerpoint of a line parallel to the point of action of the retrac-tion force through the tooth. In addition, the length ofthe power arm increased as its position was movedfrom the lateral incisor to the first premolar. If the CRwas designated as the center point of a line parallel to thepoint of action, producing parallel translation throughthe tooth, there is some discrepancy because the CRof the tooth was changed by the position of the powerarm. Therefore, a proper basic definition of the CR isneeded. In this article, by definition, a force witha line of action passing through the CR of a tooth pro-duces pure translation without tooth rotation.24,25 Infree space, the CR is the center of the tooth, which cor-responds to its center of gravity. The tooth is embeddedin bone when it is in the mouth. The CR now shifts to thecenter of the portion of the tooth embedded in thebone.26 This suggests that the CR varies according tothe amount of combination with the bone. Rotation isthe movement of a tooth around its CR; this meansthat the CR is equal to the center of rotation. Unintendedcrown-lingual tipping or crown-labial tipping can beproduced by uncontrolled tipping during orthodontictreatment. Many methods compensating for uncon-trolled tipping have been examined to control this.One way is to use the power arm under a retractionforce. In this article, based on these definitions and stud-ies, it was estimated that the CR of the standard modelwas fixed to 1 point according to the combination ofthe PDL and the alveolar bone, and it was estimatedto be located in the root between both central incisors.Figure 9 shows the equilibrium relationship betweenthe moment (Mi

F) generated by the retraction force(F) and the countermoment (Mi

CR) of the CR. The sum-mation of the moment of the retraction force and the CRshould be 0 for parallel translation of the anterior teethin this equation:

MCRi 1 MF

i 5 0; MCRi 5 MCR

1 ; MCR2 ; MF

i 5 MF1 ; MF

2

Uncontrolled tipping would be produced if equilib-rium between 2 moments is lost and 1 moment becomeslarger. The moment of the CR was small because thedistance between the CR and the power arm was short,since the power arm was located between the lateral in-cisor and the canine. As a result, the length of the powerarm was relatively short compared with the small mo-ment of the retraction force. On the other hand, the dis-tance between the CR and the power arm became longerthan the former case because the position of the powerarm was between the canine and the first premolar.This suggests that the countermoment of the CR becamerelatively larger and the length of the power arm shouldincrease to increase the moment of the retraction force.

Overall, the location of the CR could not be de-scribed precisely. Since the shapes of teeth, PDL, and al-veolar bone have individual differences, a combinationform also changes on a case-by-case basis. Although thetendency of the location of the CR can be understoodfrom previous studies, it was unreasonable to apply itto patients because the location of the CR would changeaccording to the dental model used. Therefore, in thisstudy, the total procedures including FE modeling, anal-ysis methods, and deduction of the results were estab-lished as a methodology for examining orthodontictooth movement. It is expected that the parallel transla-tion of the maxillary anterior teeth would be generatedmore effectively and reasonably if these procedureswere applied to an individual dental model.

CONCLUSIONS

In this study, FE modeling was performed to analyzethe effect of the position and length of the power armin order to determine the factors affecting the paralleltranslation of the maxillary anterior teeth undera retraction force.

1. As the position of the power arm was moved fromthe incisor to the premolars, the length of the powerarm became longer for parallel translation.

2. The CR was changed by the amount of the combi-nation of the cementum, the PDL, and the alveolarbone of the maxillary anterior teeth. The CR doesnot exist in each tooth but exists only for the fullmodel. According to the FE analysis, the lengthof the power arm was changed with the positionof the CR. This did not indicate that the CR variesaccording to the position and length of the powerarm. The length of the power arm was changed ac-cording to its position to maintain equilibrium ofthe moment mechanically for the fixed CR.

3. If the position of the power arm was changed basedon a fixed CR, crown-lingual tipping or crown-la-bial tipping was generated according to its length.Uncontrolled tipping occurred because of loss ofequilibrium of the moment between the CR andthe retraction force.

4. Based on these results, the position and length ofthe power arm were designated as variables, andthe relationship between the rotation angle andthe variables was described as a model equation.As a result, the length of the power arm generatingparallel translation was 4.987 or 8.218 mm when itwas located between the lateral incisor and the ca-nine, or between the canine and the first premolar,respectively.

American Journal of Orthodontics and Dentofacial Orthopedics Kim et al 647Volume 137, Number 5

If these procedures are established as a methodologyand applied to new dental models, it is expected that ef-ficient positions and lengths of new dental models canbe calculated, and the parallel translation of the maxil-lary anterior teeth will be generated more effectively.

REFERENCES

1. Song JH, Huh H, Park HS. Study on the retraction of anterior teeth

for the lingual orthodontics with the three-dimensional finite ele-

ment method. Transaction of Korean Society of Mechanical Engi-

neers (A) 2004;28:1237-44.

2. Brodsky JF, Caputo AA, Frustman LL. Root tipping: a photoela-

sitc histopathologic correlation. Am J Orthod 1975;67:1-10.

3. Noma H. An experimental study of edgewise mechanism—

distribution of orthodontic forces during labial movement of

lateral incisor measurements by strain gauges. Nippon Kyosei

Shika Gakkai Zasshi 1988;47:351-63.

4. Burstone CJ, Pryputniewicz RJ. Holographic determination of

centers of rotation produced by orthodontic forces. Am J Orthod

1980;77:396-409.

5. Joo JW, Hoe KH, Cha KS. Finite element modeling and the me-

chanical analysis of orthodontics. Transaction of Korean Society

of Mechanical Engineers (A) 1999;953-8.

6. Knox J, Kralj B, Hubsch P, Middleton J, Jones ML. An evaluation

of the quality of orthodontic attachment offered by single- and

double-mesh bracket bases using the finite element method of

stress analysis. Angle Orthod 2001;71:149-55.

7. Bennett JC, McLaughlin RP. Controlled space closure with a pre-

adjusted appliance system. J Clin Orthod 1990;24:245-60.

8. Vasquez M, Calao E, Becerra F, Ossa J, Enriquez C, Fresneda E.

Initial stress differences between sliding and sectional mechanics

with an endosseous implant as anchorage: a 3-dimensional finite

element analysis. Angle Orthod 2001;71:247-56.

9. Rhee JN, Chun YS, Row J. A comparison between friction and

frictionless mechanics with a new typodont simulation system.

Am J Orthod Dentofacial Orthop 2001;119:292-9.

10. Proffit WR, Fields HW. Mechanical principles in orthodontic

force control. In: Proffit WR, Fields HW, editors. Contemporary

orthodontics. 3rd ed. St Louis: Mosby; 2000. p. 326-61.

11. Sia S, Koga Y, Yoshida N. Determining the center of resistance of

maxillary anterior teeth subjected to retraction forces in sliding

mechanics. Angle Orthod 2007;77:999-1003.

12. Yoshida N, Koga Y, Jost-Brinkmann PG, Abe R, Kobayashi K,

Yamada Y. Study on the location of center of rotation of the tooth

subjected to the load using a magnetic sensing system for three

dimensional displacements. J Jpn Stomatognath Funct 1998;5:

21-30.

13. Yoshida N, Koga Y, Kobayashi K, Yamada Y, Yoneda T. A new

method 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.

14. Yoshida N, Jost-Brinkmann PG, Koga Y, Mimaki F,

Kobayashi K. Experimental evaluation of initial tooth displace-

ment, center of resistance and center of rotation under the influ-

ence of an orthodontic force. Am J Orthod Dentofacial Orthop

2001;120:190-7.

15. Tanne K, Koeing HA, Burstone CJ. Moment to force ratios and the

center of rotation. Am J Orthod Dentofacial Orthop 1988;94:

426-31.

16. Melsen B. Tissue reaction to orthodontic tooth movement—a new

paradigm. Eur J Orthod 2001;23:671-81.

17. Vollmer D, Bourauel C, Maier K, Jager A. Determination of the

centre of resistance in an upper human canine and idealized tooth

model. Eur J Orthod 1999;21:633-48.

18. Poppe M, Bourauel C, Jager A. Determination of the elasticity pa-

rameters of the human periodontal ligament and the location of

the center of resistance of single-rooted teeth. A study of autopsy

specimens and their conversion into finite element models.

J Orofac Orthop 2002;63:358-70.

19. Kawarizadeh A, Bourauel C, Zhang D, Gotz W, Jager A. Correla-

tion of stress and strain profiles and the distribution of osteoclastic

cells induced by orthodontic loading in rat. Eur J Oral Sci 2004;

112:140-7.

20. Coolidge ED. The thickness of the human periodontal membrane.

J Am Dent Assoc 1937;24:1260-70.

21. Reimann S, Keiling L, Jager A, Bourauel C. Biomechanical finite-

element investigation of the position of the centre of resistance of

the upper incisors. Eur J Orthod 2007;29:219-24.

22. Dorow C, Sander FG. Development of a model for the simulation

of orthodontic load on lower first premolars using the finite ele-

ment method. J Orofac Orthop 2005;66:208-18.

23. Tanne K, Nagatani T, Inoue Y, Sakuda M, Burstone CJ. Patterns of

initial tooth displacements associated with various root lengths

and alveolar bone heights. Am J Orthod Dentofacial Orthop

1991;100:66-71.

24. Mulligan TF. Common sense mechanics: 2. Forces and moments.

J Clin Orthod 1979;13:676-83.

25. Smith RJ, Burstone CJ. Mechanics of tooth movement. Am

J Orthod 1984;85:294-307.

26. Haack CD, Weinstein S. Geometry and mechanics as related to

tooth movement studied by means of a two-dimensional model.

J Am Dent Assoc 1963;66:157-64.