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Influence of elastic modulus of periodontal ligament

on the centers of rotation of incisors

: Finite element analysis

Hyesun Kim

The Graduate School

Yonsei University

Department of Dentistry

Influence of elastic modulus of periodontal ligament

on the centers of rotation of incisors

: Finite element analysis

The Dissertation Thesis

Submitted to the Department of Dentistry

and the Graduate School of Yonsei University

in partial fulfillment of the

requirements for the degree of

Doctor of philosophy of Dental Science

Hyesun Kim

June 2016

감사의 글

부족하기만 한 저를 지도해 주신 교수님들의 크신 가르침으로 이 논

문을 완성할 수 있었습니다. 논문 구상부터 완성까지 많은 배려와 세심

한 지도를 베풀어 주신 이기준 지도 교수님께 진심으로 감사 드립니다.

많은 업무로 바쁘신 중에도 귀한 시간을 내 주시어 부족한 논문에 대한

조언을 주신 백형선 교수님, 유형석 교수님, 최광철 원장님, 조영수 박

사님께 깊이 감사 드립니다. 또한, 교정학을 공부할 수 있도록 기회를

주신 박영철 교수님, 황충주 교수님, 김경호 교수님, 정주령 교수님, 차

정렬 교수님, 최윤정 교수님, 최성환 교수님께 깊이 감사 드립니다.

논문이 나오기까지 격려와 조언을 주신 연세대학교 교정과 의국

선생님들과 대학원 선생님들께 깊은 감사의 마음을 전합니다.

항상 크신 지혜와 사랑으로 바른 길로 이끌어 주시는 아버님, 어머님,

늘 끝없는 사랑과 정성으로 응원해 주시는 아버지, 어머니께 깊은

감사의 마음을 올립니다. 언제나 곁에서 조언하고 감싸주어 인생의

든든한 동반자가 되어 주는 남편 윤병선 선생님께 존경과 사랑의

마음을 전하며, 늘 의젓하게 자신의 일을 대견하게 해 나가는 두 딸,

준희와 성은이에게 사랑과 고마움의 마음을 전합니다.

2016 년 6 월

저자 씀

i

Table of Contents

List of figures .............................................................................................................ii

List of tables ............................................................................................................. iii

Abstract (English) .....................................................................................................iv

I. Introduction ............................................................................................................1

II. Materials and methods ..........................................................................................4

A. Construction of finite element models ........................................................4

B. Mechanical variables of the material properties ........................................4

C. Loading condition ..........................................................................................8

D. Calculation of center of rotation / center of resistance ............................9

III. Results ............................................................................................................... 11

IV. Discussion ......................................................................................................... 18

V. Conclusion ........................................................................................................... 21

References ............................................................................................................... 23

Abstract (Korean) ................................................................................................... 30

ii

List of Figures

Figure 1. Three dimensional finite element model ...............................................5

Figure 2. Modeling of the upper incisor .................................................................6

Figure 3. Model elements of upper incisors and surrounding tissue....................7

Figure 4. Force application at different occluso-gingival levels (㎜) ................8

Figure 5. Center of rotation ......................................................................................9

Figure 6. Vertical position of Crot of upper central incisors according to elastic

modulus of PDL at each force application level (㎜) ......................... 11

Figure 7. Change of horizontal & vertical position of Crot of upper central

incisors according to force application level (㎜) (E – 71.5 MPa) ..... 12

Figure 8. Vertical position of Cres(㎜)of upper central incisors according to E

................................................................................................................. 13

Figure 9. Location of the center of rotations in the X-Y plane in accordance

with various E ....................................................................................... 14

Figure 10. Stress distribution pattern in the PDL after application of retraction

force on upper central incisors at Cres ............................................ 15

Figure 11. The characteristics of teeth movement according to E in accordance

with different force application levels. ............................................. 17

iii

List of Tables

Table 1. Variation in the elastic properties of the PDL used in dental finite

element method ...................................................................................... 10

Table 2. Mechanical properties of each materials ............................................. 10

Table 3. Changes in initial tooth displacements in varying degrees of E ........ 16

iv

ABSTRACT

Influence of elastic modulus of periodontal ligament

on the centers of rotation of incisors

: Finite element analysis

Hyesun Kim

Department of Dentistry, Graduate School, Yonsei University

(Directed by Professor Kee Joon Lee, D.D.S., M.S.D., Ph.D.)

Consideration of the physical properties of the periodontium is important in

diagnosis and treatment planning of orthodontic treatment. Moreover,

periodontal ligament (PDL) is the most important tissue in proceeding of tooth

movement under orthodontic loading. The elastic modulus of periodontal

ligament (E) used in previous studies has quite wide range of value from 0.07 to

1,750 MPa.

The objective of this study was to measure the changes of the centers of

rotation (Crot) of upper central incisors and evaluate the influence of physical

v

properties on orthodontic tooth movement, in response to single horizontal force

applied at different vertical levels with regard to various elastic modulus of PDL

using finite element analysis (FEA). When modeling the periodontal ligament in a

FEM study, a broad range of values has been used for the elastic modulus of the

periodontal ligament.

A 3-dimensional finite element model was constructed from a 3D laser scan

of a maxillary anteriors of a Nissin dental model by the sample survey of adults

with normal occlusion. In this study, PDL was designed with 4 layers of fine

tetrahedron solid elements to estimate the influence of the elastic properties of

PDL on orthodontic teeth movement precisely. The variables of the elasticity of

PDL (0.07 to 1,750 MPa) were consulted previous studies.

The conclusions are as follows:

1. The vertical position of Crot of upper central incisors was displaced

apically with increase of E at same vertical force application level.

2. Centers of resistance (Cres) of upper central incisors was changed

according to the elastic modulus of PDL. On non-rigid bone/tooth model,

Cres of upper central incisors relocated apically at E 50.0 and 68.9 MPa

and shifted more incisally from E 171.9 to 1750 MPa. On rigid

bone/tooth model, Cres of upper central incisors moved apically with

increase of E. On non-rigid bone/rigid tooth model, Cres of upper

central incisors tended to move incisally.

vi

3. The 'functional axes' (traces of the measured Crot) of the upper central

incisors were formulized in accordance with various E. The Crot

distributed on a line under given E in relation to the horizontal forces

applied to the various vertical level. The different linear equations

could be evaluated with coefficients in accordance with various E, which

slopes turn counterclockwise direction gradually with increase of E.

The absolute quantity of displacement and the tendency of rotation of teeth

used to decrease with increased E, when compared the characteristics of teeth

movement according to E. Consideration of the physical properties of the PDL

might be an important factor in diagnosis and treatment planning of orthodontic

treatment.

Key words : elastic modulus of PDL(E) , center of rotation , center of resistance,

functional axis, finite element analysis

１

Influence of elastic modulus of periodontal ligament

on the centers of rotation of incisors

: Finite element analysis

Hyesun Kim

Department of Dentistry, Graduate School, Yonsei University

(Directed by Professor Kee Joon Lee, D.D.S., M.S.D., Ph.D.)

I. Introduction

The center of rotation (Crot) is considered as an important point for tooth

movement. The position of Crot determines the type of tooth movement such as

crown tipping, translation, root movement, and gives us a clue for the equivalent

force and torque need for each force system. The location of center of rotation

is changed with the vertical position of force applied to the teeth.

Many models and methods have been developed to study the relationship

between the location of force, the center of resistance (Cres) and the center of

rotation (Crot) (Yettram AL et al., 1977; Williams KR et al., 1986). In

orthodontics, FEM has been used successfully to model the application of forces

to tooth systems (Selna LG et al., 1975; Takahashi N et al., 1980; Atmaram et

２

al., 1981; Williams KR et al., 1984; Kim et al., 1991).

The PDL is soft, richly vascular and cellular connective tissue that joins the

root cementum to the surrounding alveolar bone proper. The ligament is

approximately 0.25 ㎜ ±50% wide and composed principally of collagen fibers

that run in various directions between the root surface and supporting alveolar

bone, functions as a shock absorber during occlusal load. Magnitude and

direction of orthodontic loads should be changed considering individual

difference in elasticity of PDL (Lindhe et al., 1989; Coolidge E., 1937).

Knowledge of the physical properties of the periodontal ligament would be

useful to help elucidate the role of the ligament in absorbing occlusal load and

to increase understanding of tooth movement under orthodontic loading. Little

information is available about the physical properties of the ligament in

isolation due to problems of obtaining pure samples. Most studies have used a

value of 0.4-0.49 for Poisson´s ratio. However, previous studies have

reported quite different values of wide range of elastic modulus of the ligament

from 0.01 to 1,750 MPa ( Picton et al., 1963; Tanne et al., 1987; Rees et

al.,1997) (Table 1).

Choy et al. (2006) investigated the changes in the initial center of rotation

(Crot) of upper anterior teeth in response to a horizontal load, and recorded a

linear functional axis (a trace of the measured Crot), maintained an angle of

14.5 °to the vertical axis of the anterior segment. Choy et al. (2006) discussed

３

factors affecting the angle of the functional axis, including an inclination of the

roots and physical properties of the PDL.

The objective of this study was to measure the changes of the Crot of the

anterior segments, in response to single horizontal force applied at different

occluso-gingival levels with regard to various elastic modulus of PDL (E) using

FEM. When modeling the periodontal ligament in a FEM study, a wide range of

values has been used for the elastic modulus of the PDL (E) (Table 1).

４

II. Materials and methods

A. Construction of finite element models

A finite element model was constructed from a 3D laser scan of the maxillary

anteriors of a Nissin dental model by the sample survey of adults with normal

occlusion (Model-i12D-400G, Nissin Dental Products, Kyoto, Japan). The

thickness of the PDL was assumed to be uniform (0.25 ㎜) and even around the

root surface. The morphology of the alveolar bone were modeled 1㎜ above

cemento-enamel junction following curvature of the CEJ. Maxillary anteriors are

joined as rigid unit, and force vector was applied to a vertical extension

perpendicular to the occlusal plane in the midsagittal plane between both upper

central incisors.

Tooth length of upper central incisor measured from incisal edge to root apex,

was 24.2 ㎜ : The crown was 11㎜ from incisal edge to labial CEJ and the root

13.2mm. The long axis of each incisor model was inclined 60° facially from the

occlusal plane. A standard coordination system was constructed with the x-axis,

y-axis, and z-axis (Figure 1, 2). The teeth, alveolar bone, and PDL were all

constructed using fine tetrahedron solid elements, and were all assumed to be

isoparametric and homogeneous linear elastic bodies. In this study, PDL was

constructed with 4 layers of elements with fine tetrahedron solid nodes to

simulate the elasticity of PDL (Figure 3).

B. Mechanical variables of the material properties.

The variables of the elasticity of PDL were consulted previous studies (Young’

s modulus and Poisson’s ratio)(Table 1)(Andersen et al., 1991; Tanne et al.,

1987; Farah et al., 1989; Wilson, 1991; Cook et al., 1982; Thresher et al., 1973;

５

Goel et al., 1992; Rees et al., 1997). As the elastic modulus of the PDL(E) is

lower than the alveolar bone, and the tooth, the tooth behave like a rigid body,

and this assumption has been validated in our study.

This study included a non-rigid bone/tooth model, rigid bone/tooth model, and

non rigid bone/rigid tooth model to compare errors by bending phenomenon of

bone and tooth. Material properties of rigid bone/tooth properties were chosen

from the literature (Viecilli et al., 2013) (Table 2).

Figure 1. Three dimensional finite element model.

A. Frontal view; B. Lateral view.

X-axis : (+) superior, (-) inferior,

Y-axis : (+) mesial, (-) distal,

Z-axis : (+) anterior, (-) posterior.

６

Figure 2. Modeling of the upper incisor.

A. Tooth length and bracket position(㎜);

B. Normal inclination of the incisor, 60°to occlusal plane.

７

Figure 3. Model elements of upper incisors and surrounding tissue.

A. The elements of PDL constructed with 4 fine tetrahedron solid nodes;

B. The elements of PDL and surrounding bone;

C. The elements of upper incisors and surrounding tissue.

８

C. Loading condition

A single force vector (200 g) was applied to the vertical extension of the

splint in the midsagittal plane of both upper central incisors at different vertical

levels. It was assumed that there was no play between the teeth and the vertical

extension of the splint. A horizon force was given from the 0.0 ㎜ position(level

of incisal tip of the central incisors) to the 25.0㎜ position with an interval of 5.0

㎜. Around predicted point as center of resistance, force was applied with an

interval of 0.01 ㎜. The midpoints of incisal edges{(X1 , Y1 ), (X1´,Y1´)} and

root apices{(X2 ,Y2 ),(X2´,Y2´)} of central incisors were used as landmarks

for the assessment of center of rotation, using the finite element analysis

program(ANSYS Ver. 12.1, Swanson Analysis System, Canonburg, PA ) (Fig.4).

Figure 4. Force application at different occluso-gingival levels (㎜)

: A horizon force was given from the 0.0 ㎜ position(level of incisal

tip of the central incisors) to the 25.0 ㎜ position with an interval

of 5.0 ㎜.

㎜

９

D. Calculation of center of rotation / center of resistance

Centers of rotation were calculated using following formulae.

Then, centers of resistance could be estimated

by the translation tests and error analyses.

X Crot `=((Y1´-Y1)×(Y2´-Y2)×((Y2´+Y2)-(Y1´+Y1))+(X2´+X2)×(X2´-X2)×(Y1´-Y1)-(X1´+X1)×(X1´-X1)×(Y2´-Y2))

2×((X2´-X2)×(Y1´-Y1)-(X1´-X1)×(Y2´-Y2))

Y Crot =

((X1´-X1)×(X2´-X2)×((X1´+X1)-(X2´+X2))+(X2´-X2)×(Y1´-Y1)×(Y1´+Y1)-(X1´-X1)×(Y2´+Y2)×(Y2´-Y2))

2×((X2´-X2)×(Y1´-Y1)-(X1´-X1)×(Y2´-Y2))

Figure 5. Center of rotation

１０

Table 1. Variation in the elastic properties of the PDL used in dental

finite element method (Rees et al., 1997)

Author(s)Young´s modulus

(MPa)Poisson´s ratio

Andersen et al. 0.07 0.49

Tanne and Sakuda 0.70 0.49

Farah et al. 6.9 0.45

Wilson 50.0 0.49

Cook et al. 68.9 0.49

Atmaram and Mohammed 171.6 0.45

Thresher and Saito 1379.0 0.45

Goel et al. 1750.0 0.49

Table 2. Mechanical properties of each materials (Cattaneo et al., 2005;

Viecilli et al., 2013).

Non-rigid bone/tooth

modelRigid bone/tooth model Rigid tooth model

Young´s

modulus(MPa)

Poisson´s

ratio

Young´s

modulus(MPa)

Poisson´s

ratio

Young´s

modulus(MPa)

Poisson´s

ratio

Alveolar

bone2.0E+03 0.30 8.0E+05 0.325 2.0E+03 0.30

Teeth 2.0E+04 0.30 8.0E+05 0.325 8.0E+05 0.325

Stainless

steel2.0E+05 0.30 2.0E+05 0.30 2.0E+05 0.30

１１

III . Results

1. The vertical position of Crot of upper central incisors was displaced apically

with increase of E at same vertical force application level.

The vertical position of Crot became more apical with increased E at same

force application level (Figure 6). On same E condition, the vertical position

of Crot was relocated apically when force application level was from occlusal

to Cres. The positions of Crot are distributed on a graph of a national

function. The Crot was positioned on a horizontal tangent at incisal edge and

extended to infinity around Cres. Then the Crot was positioned at infinity

and maintained tangency with force of further away from Cres (Figure 7).

Figure 6. Vertical position of Crot of upper central incisors according to

elastic modulus of PDL(E) at each force application level(㎜).

１２

Figure 7. Change of horizontal & vertical position of Crot of upper central

incisors according to force application level(㎜)(E – 71.5 MPa).

１３

2. The vertical position Cres of upper central incisors is changed according to

the EML.

In this study, the data with a rigid bone / tooth model, a non-rigid bone /

tooth model, and non-rigid bone/rigid tooth model was compared to exclude

bending effect of bone and tooth, clarifying the influence of physical

properties on PDL. On non-rigid bone/tooth model, Cres of upper central

incisors relocated apically at E 50.0 and 68.9 MPa and shifted more incisally

from E 171.9 to 1750 MPa. On rigid bone/tooth model, Cres of upper central

incisors moved apically with increase of E. On non-rigid bone/rigid tooth

model, Cres of upper central incisors tended to move incisally.

Figure 8. Vertical position of Cres(㎜)of upper central incisors according

to E.

１４

3. The ‘functional axes’ of the anterior segment were formulized in accordance

with various E.

The Crot distributed on a line under given E in relation to the horizontal

forces applied to the various vertical level. The different linear equations

could be evaluated with coefficients in accordance with various E, which

slopes turn counterclockwise direction gradually with increase of E.

Figure 9. Location of the center of rotations in the X-Y plane in accordance

with various E.

１５

4. Minimum principal stress distribution is observed in the PDL of upper central

incisors with horizontal retraction force through the Cres.

There is uniformly increased stress on lingual periodontal membrane of

both central incisors, and uniformly decreased stress on labial, which mean

the close movement to translation of upper central incisors to lingual

direction.

Figure 10. Stress distribution pattern in the PDL after application of

retraction force on upper central incisors at Cres ( E - 68.9 MPa ,

vertical force application level - 10.05㎜ from occlusal plane).

A. Frontal view; B. Lingual view; C. Occlusal view.

A

BA

C

１６

5. The characteristic of teeth movement in varying degrees of E between force

application levels was evaluated.

The absolute quantity of displacement and the tendency of rotation of teeth

used to decrease with increased E, when compared the characteristics of

teeth movement according to E.

Table 3. Change in initial tooth displacements in varying degrees of E.

Force

application

level

Initial change of

upper central incisor

E

0.07 0.7 6.9 50 68.9 171.6 1379 1750

incisal

edge

incisal

edge

horizontal change -0.368566 -0.041218 -0.007787 -0.003606 -0.003293 -0.002633 -0.001943 -0.001910

vertical change -0.220470 -0.025089 -0.005083 -0.002476 -0.002271 -0.001834 -0.001371 -0.001350

apex

horizontal change 0.101978 0.010365 0.001084 0.000103 0.000055 -0.000020 -0.000052 -0.000052

vertical change 0.134234 0.013737 0.001539 0.000252 0.000185 0.000072 0.000003 0.000001

25mm

from

incisal

edge

incisal

edge

horizontal change 0.189026 0.061601 0.010630 0.004812 0.004440 0.003704 0.002989 0.002952

vertical change 0.186852 0.050088 0.008406 0.003713 0.003422 0.002855 0.002319 0.002292

apex

horizontal change -0.159390 -0.034068 -0.003690 -0.000646 -0.000507 -0.000287 -0.000160 -0.000157

vertical change -0.075878 -0.022044 -0.002356 -0.000361 -0.000268 -0.000119 -0.000037 -0.000035

１７

Figure 11. The characteristics of teeth movement according to E

in accordance with different force application levels.

A. Crown lingual torque of upper central incisors;

B. Root lingual torque of upper central incisors;

C. Crown lingual torque but similar with translation;

D. Root lingual torque but with decreased movement change.

C

BA

D

１８

IV. Discussion

In this study the influence of elastic modulus of PDL was assessed on the

initial tooth movement of upper incisors. Differences in tooth displacement may

come from the change in the amount of displacement and the change of relations

between centers of rotation and centers of resistance (Ericsson et al., 1984;

Khoo et al., 1988; Tanne et al., 1991). The present study showed that

decreased initial tooth displacement and apically relocation of Cres and Crot with

increased resiliency of PDL. It does suggest that physical properties of PDL

should be considered in designing the equivalent force and torque system for

orthodontic teeth movement.

A lot of research has been reported about the location of the Cres.

Burstone(1966) described the Cres in a single rooted tooth at a point 40% of the

root length. Nikolai(1974) and Matsuura(1984) reported the location of Cres as

0.52 for the maxillary central incisor on the basis of a two-dimensional analysis.

Burstone and Pryputniewicz (1980) used a three-dimensional model and

established with a holographic technique the Cres at 0.31 the root length

measured from the alveolar crest. Tanne(1988) suggested the location of the

Cres at 0.24 the root length located farther coronally. The most relevant aspect

in the cited studies was the definition of a common Cres for the upper anterior

segment, having a position between 8.1-14.7 ㎜ apically to the incisal

edge(Vanden Bulcke et al., 1987; Melsen et al., 1990; Pedersen et al., 1991)

Choy et al.(2006) investigated the changes in the initial centers of rotation of

the upper six anterior teeth which roots coated with silicon in response to a

horizontal load, and obtained a linear´functional axis: y=3.861x + 53.153´.

They expected that there may be many factors affecting the angle of the

functional axis, including an inclination of the roots and the physical properties

１９

of the PDL. The ‘functional axes’ of the anterior segment could be formularized

in accordance with various EML. The functional axes have positive slopes

passing from quadrant 1 to quadrant 3 with E 0.07, 0.7, 6.9, 50.0, 68.9, which

suggest that a small intrusive component of upper anterior segment could be

expected with lingual horizontal retraction force. By contrast, the functional

axes have negative slopes with E 171.9, 379.0, 1750.0, which may suggest

that a small extrusive component of upper anterior segment could be expected

with lingual horizontal retraction force. The´functional axis: y=3.861x +

53.153´obtained with silicon PDL by Choy et al.(2006) was similar with that

of E 0.07 and 0.7 .

To obviate confused estimation of results from bone and tooth bending, in this

study various type of model was planned with the rigid tooth / rigid bone model

and the rigid tooth / non-rigid bone model. The material properties of rigid tooth

/ bone model were assigned based on the data of Viecilli et al. with increased

elastic modulus of the tooth and bone to 800 GPa to improve resistance against

bending (Viecilli et al., 2013). On rigid bone/tooth model, Cres of upper central

incisors moved apically with increase of E, which represented the effect of

elastic properties of PDL on teeth movement independently. On non-rigid

bone/rigid tooth model, Cres of upper central incisors tended to move incisally

which implied bone bending covering the teeth movement. On non-rigid

bone/tooth model, Cres of upper central incisors relocated apically at E 50.0 and

68.9 MPa and shifted more incisally from E 171.9 to 1750 MPa with the mixed

result of bone and teeth bending to conceal the influence of E on tooth

movement alone.

When modeling the periodontal ligament in a finite element stress analysis

study, what physical property values to ascribe to this structure. A wide range

of values has been used for the elastic modulus of the ligament (Anderson et al.,

1991; Tanne et al., 1983; Farah et al., 1989; Takahashi et al., 1989, Wilson,

２０

1991; Cook et al., 1982: Ko et al, 1996; Atmaram et al., 1981; Thresher et al.,

1973; Goel et al., 1992; Rees et al., 1997). This model was designed to compare

teeth displacement with variation of EML. Further experimental studies are

expected to reveal the exact characteristics the living PDL, accommodating

visco-elastic, non-linear properties and thickness variation. Also, objective

criteria should be established in diagnosis for physical condition of PDL.

２１

V. Conclusion

From this study we measured Influence of elastic modulus of periodontal

ligament on the centers of rotation of incisors with regard to constant horizontal

force applied at different vertical levels using FEM. When modeling the

periodontal ligament in a FEM study, a wide range of values has been used for

the elastic modulus of the ligament.

The following findings were observed.

1. The vertical position of Crot of upper central incisors was displaced

apically with increase of E at same vertical force application level.

2. Cres of upper central incisors was changed according to the elastic

modulus of PDL. On non-rigid bone/tooth model, Cres of upper central

incisors relocated apically at E 50.0 and 68.9 MPa and shifted more

incisally from E 171.9 to 1750 MPa. On rigid bone/tooth model, CRes of

upper central incisors moved apically with increase of E. On non-rigid

bone/rigid tooth model, Cres of upper central incisors tended to move

incisally.

3. The ‘functional axes (traces of the measured Crot)’ of the anterior

segment were formulized in accordance with various E. The Crot

distributed on a line under given E in relation to the horizontal forces

applied to the various vertical level. The different linear equations

could be evaluated with coefficients in accordance with various E, which

slopes turn counterclockwise direction gradually with increase of E.

２２

The absolute quantity of displacement and the tendency of rotation of teeth

used to decrease with increased E, when compared the characteristics of teeth

movement according to E. Consideration of the physical properties of the PDL

might be an important factor in diagnosis and treatment planning of orthodontic

treatment.

２３

Reference

Andersen KL, Mortensen HT, Pendersen EH, Melsen B. Determination of

stress levels and profiles in the periodontal ligament by means of an improved

three dimensional finite element model for various types of orthodontic and

natural force systems. J. Biomed. Eng. 1991; 13: 293-303.

Atmaram HF, Mohammed H. Estimation of physiologic stresses with a Natural

tooth considering fibrous PDL structure. J. Dent. Res. 1981; 60: 873- 77.

Boyd RL, Leggott PJ, Quinn RS, Eakle WS, Chambers D. Periodontal Implications

of orthodontic treatment in adults with reduced or normal periodontal tissues

versus those of adolescents. Am J Orthod Dentofacial Orthop. 1989; 96:191-9.

Burstone CJ. The mechanics of the segmented arch techniques. Angle Orthod.

1966;36:99-120.

Burstone CJ, Pryputniewica RJ. Holographic determination of centers of rotation

produced by orthodontic forces. Am J Orthod Dentofacial Orthop.

1980;77:396-409.

Cattaneo PM, Dalstra M, Melsen B. The finite element method: A tool to study

orthodontic tooth movement. J Dent Res. 2005;84(5):428-33.

Chen-Ying Wang , Ming-Zen Su , Hao-Hueng Chang. Tension-compression

viscoelastic behaviors of the periodontal ligament. J Formos Med Assoc.

2011;111(9):471-81.

２４

Choy KC, Kim KH , Burstone CJ. Initial changes of centres of rotation of the

anterior segment in response to horizontal forces. Eur Journal of Orthod. 2006;

28 : 471-74.

Cook SD, Weinstein AM, Klawitter JJ. A three dimensional finite element

analysis of a porous rooted Co-Cr-Mo alloy dental implant. J. Dent. Res.

1982;61:25-9.

Coolidge E. The thickness of the human periodontal membrane. J Am Dent

Assoc. 1937;24: 1260-7.

Cordes V, Gardemin M, Lupke M, Seifert H, Borchers L, Staszyk C. Finite

element analysis in 3-D models of equine cheek teeth. The Veterinary Journal.

2012b;193:391-96.

Ericsson I, Lindhe J. Lack of significance of increase tooth mobility

inexperiomental periodontitis. J. Periodontol. 1984; 55:447-52.

Farah JW, Craig RG and Meroueh KA. Finite element analysis of three-and

four-unit bridges. J. Oral Rehabil. 1989; 16: 603-11.

Goel VK, Khera SC, Gurusami S, and Chen RCS. Effect of cavity depth on

stresses in a restored tooth. J. Prosthet. Dent. 1992; 67:174-83.

Haack CD. An analysis of stress in a model of the periodontal ligament.

International Journal of Engineering Science 1972;10 :1093-106.

２５

Jeon P, Turley P, Moon H, Ting K. Analysis of stress in the periodontium of the

Maxillary first molar using a three-dimensional finite element model. Am J

Orthod Dentofacial Orthop. 1999;115:267-74.

Jeremy D. Lina, Jihyun Lee, Huseyin Ozcobanb, Gerold A. Schneiderb,

Sunita P. Hoa. Biomechanical adaptation of the bone-periodontal ligament

(PDL)-tooth fibrous joint as a consequence of disease. J Biomech.

2014;47:2102-114.

Khoo KK, Watts TLP. Upper anterior tooth mobility: selected associations in

untreated periodontitis. J. Periodontol. 1988; 59:231-7.

Kim L. Andersen MSc, Erik H. Pedersen, Birte Melsen. Material parameters and

stress profiles within the periodontal ligament. Am J Orthod Dentofacial Orthop.

1991;99: 427-40.

Ko CC, Chu CS, Chung KH and Lee MC. Effects of posts on dentin stress

distribution in pulpless teeth. J Prosthet Dent. 1992;66:421-27.

Lindhe, and Karring T. The anatomy of the periodontium. In Textbook of

Clinical Periodontology, 2nd ed J. Lindhe. Munksraad, Copenhagen. 1989.

Mathias Poppe, Christoph Bourauel, Andreas Jager. Determination of the

elasticity parameters of the human periodontal ligament and the location of the

center of resistance of single-rooted teeth. J Orofac Orthop. 2002;358-70.

Matsuura T. Mechanical study on initial changes during canine retraction. J Jpn

Orthod Soc. 1984;3:33-52.

２６

Melsen B, Fotis V, Burstone CJ. Vertical force considerations in differential

space closure. J Clinic. Orhod. 1990;24: 678-83.

Moxham BJ, Berkovitz BKB. The effects of external forces on the periodontal

ligament-The response to axial loads. Pergamon Press, Oxford. 1982; 249-68.

Nägerl H, CJ Burstone, Becker B, Kubein-Messesnburg D. Center of

rotation with transverse forces: an experimental study. Am. J. Orhod. Dentofac.

Orthop. 1991; 99: 337-45.

Nikolai RJ. Periodontal ligament reaction and displacements of a maxillary

central incisor subjected to transverse crown loading. Journal of Biomechanics.

1974; 7 : 93- 99.

Pedersen E, Isidor F, Gjessing P, Andersen K. Location of certres of resistance

for maxillary anterior teeth measured on human autopsy material. European J

Orhod. 1991;13: 452-8.

Picton, DCA. Vertical movement of cheek teeth during biting. Arch. Oral Biol.

1963; 8:109-18.

Provatidis CG. A comparative FEM-study of tooth mobility using isotropic and

anisotropic models of the periodontal ligament. Medical engineering & physics.

2000;22: 359-70.

Ralph WJ. Tensile behavior of the periodontal ligament. J Periodont. Res.

1982;17: 423-26.

２７

Rees JS, Jacobsen PH. Elastic modulus of the periodontal ligament. Biomaterials

1997; 18 : 995-9.

Reimann S, Keiling L, Jager A, Bourarel C. Biomechanical finite-Element

investigation of the position of the centre of resistance of the upper incisors. Eur

J Orthod . 2007; 29 : 219-224.

Richard JS, CJ Burstone. Mechanics of tooth movement. Am J Orthod Dentofacial

Orthop. 1984;85(4): 294-307.

Rodrigo FV, Amanda B, Burstone CJ. Axes of resistance for tooth movement:

Does the center of resistance exist in 3-dimensional space? Am J Orthod

Dentofacial Orthop. 2013 ;143: 163-72.

Sadowsky C, DeGole E. Long term effects of orthodontic treatment on

periodontal health.＂Am J Orhod Dentofacial Orthop. 1981;80:156-72.

Schepdael AV, Lies Geris, Jos Vander Sloten. Analytical determination of stress

pattern in the periodontal ligament during orthodontic tooth movement. Medical

Engineering& physics. 2013; 35:403-10.

Selna LG, Shilingburg HT, Kerr P. Finite element analysis of dental structures

– axisymmetric and plane stress idealizations. J Biomed. Master Res. 1975; 9 :

237-52.

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

anterior teeth subjected to retraction forces in sliding mechanics. An in vivo

study. Angle Orhod. 2007; 77: 999-1003.

２８

Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod Dentofacial

Orthop. 1984;85:294-307.

Stephanie R, Toms, AW Eberhardt, A nonlinear finite element analysis of the

periodontal ligament under orthodontic tooth loading. Am J Orthod Dentofacial

Orthodp. 2003;123:657-65.

Sung SJ et al. A comparative evaluation of different compensating curves in the

lingual and labial techniques using 3D FEM. Am J Orthod. Dentofacial Orthop.

2003;123 : 441-50.

Takahashi N, Kitagami T, Komori T. Behaviour of teeth under various loading

conditions with finite element method. J Oral Rehabil. 1980; 4: 23-7.

Tanne K and Sakuda M. Initial stress induced in the periodontal tissue at the

time of the application of various types of orthodontic force: by means of the

Finite Element Method. J. Osaka Univ. Dent. School. 1983;23: 143-71.

Tanne K, Sakuda K, Burstone CJ. Three-dimensional finite element analysis for

stress in the periodontal tissue by orthodontic forces. Am J Orhod. Dentofcial

Orthop. 1987;92 : 499-505.

Tanne K, Sakuda K, 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.

Thresher RW, Saito GE. The stress analysis of human teeth. J Biomech. 1973;

6 : 443-49.

２９

Vanden Bulcke MM, Burstone CJ, Sachdeva RCL, Dermaut LR. Location of the

centers of resistance for anterior teeth during retraction using the laser

reflection technique. Am J Orthod. Dentofacial Orthop. 1987;91 : 375-84.

Viecilli RF, Budiman A, Burstone CJ. Axes of resistance for tooth movement;

Does the center of resistance exist in 3 – dimensional space? Am J Orhod.

Dentofcial Orthop. 2013;143(2):163-72.

Williams KR, Edmunson JT, Morgan G, Jones ML, Richmond S. Orthodontic

movement of a canine into an adjoining extraction site. J Biomed. Eng.

1986;8:115-20.

Wilson A. Linear and non-linear analysis of orthodontic tooth movement. Am

J Orthod Dentofacial Orthop. 1991; 96:191-9.

Woo JY, Park YC. Experimental study of the vertical location of the centers of

resistance for maxillary anterior teeth during retraction using the laser

reflection technique. Korean J Orthod. 1993;23:375-90.

Yettram AL, Wright KWJ, Houston WJB. Centre of rotation of a maxillary

central incisor under orthodontic loading. Br. J Orhod. 1977; 4:23-7.

３０

국문요약

치주 인대의 탄성도가

상악 중절치의 회전 중심에 미치는 영향

김혜선

연세대학교 대학원 치의학과

(지도교수 이기준)

교정 치료 시 치아와 치주조직의 물성에 따른 치아의 이동 양상을 파악하여 치아

의 회전 중심과 저항 중심 위치를 예측하는 것은 효율적이고 계획된 치아이동에 필수

적이다. 치주 인대의 복잡한 물리적 특징으로 인하여 인체 내에서의 치주 인대의 교

정적 치아 이동에 대한 정확한 연구와 데이터가 부족하다. 또한 선학들의 치주 인대

의 탄성도에 대한 연구는 꽤 광범위한 결과 (0.07 – 1,750 MPa)를 보였다.

본 연구는 유한 요소해석을 통하여 치주인대 탄성도의 변화에 따른 상악 중절치의

초기 변위와 응력분포를 분석하여, 치주인대의 탄성도의 변화에 따른 상악 중절치의

회전 중심과 저항 중심의 변화 양상을 관찰하여 치주인대의 탄성도의 치아이동에 대

한 영향을 알아보고자 하였다. 또한 이를 기반으로 치주 인대의 탄성도에 따른 교정

력에 대한 다양한 치아 이동을 예측하고자 하였다.

치주 인대의 탄성도에 따른 상악 전치부의 회전 중심의 위치는 다음과 같은 영향을

보였다.

1. 치주 인대의 탄성이 증가할수록, 상악 중절치의 회전 중심(Crot)이 치근단 측

으로 이동하였다.

３１

2. 치주 인대의 탄성도의 변화에 따라 상악 중절치의 저항 중심(Cres)의 위치가

변화하였다. 연식 치아/치조골 모델에서 저항 중심(Cres)은 치주 인대의 탄성

도가 50.0에서 68.9 MPa 사이의 수치로 변화하였을 때 치근단측으로 변위하고

탄성도가 171.9에서 1750 MPa 사이의 수치로 증가하면 저항 중심은 다시 절

단연측으로 이동하였다. 강성 치아/치조골 모델에서 치아와 치조골의 굴곡을 배

제하여 치주인대의 탄성도의 단독적인 영향을 관찰한 결과 상악 중절치의 저항

중심(Cres)은 치근단측으로 이동하였다. 강성 치아/ 연식 치조골 모델에서는

치주인대의 탄성도의 증가에 따라 저항 중심(Cres)은 절단연측으로 변위하였다.

3. 주어진 치주 인대의 탄성도를 대입하여 얻어진 상악 중절치의 회전 중심(Crot)

은 한 직선상에 분포하였다. 각각의 치주 인대의 탄성도에 따른 회전 중심

(Crot)의 분포에 대한 ‘함수식’을 구하였다. 치주 인대의 탄성도가 증가할수록

함수식의 기울기는 증가하였다가 치주인대의 탄성도가 171.9 MPa 일 때 얻어

진 함수의 기울기는 음의 값을 갖게 되며 탄성도가 1750 MPa이 되면 기울기

의 절대값이 감소하여, 치주인대의 탄성도가 증가함에 따라 함수식에 따른 직선

은 시계 반대 방향으로 회전하였다.

치주 인대의 탄성도가 증가할수록 치아이동의 절대량과 회전하려는 경향이 감소하

였다. 이상의 결과는 임상적으로 환자의 치주 인대의 탄성도를 고려하여 상악 전치부

를 이동할 때 교정력의 역학계를 결정함에 있어 참고가 될 수 있을 것이다.

핵심되는 말: 치주인대의 탄성도, 회전 중심, 저항 중심, 치아의 회전 중심의 기능축,

3차원 유한요소