2002 Schneider

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The use of numeric finite element (FE) calcula-

tions by orthodontic researchers has increased in

recent years. One of their aims is to provide

more information on the behavior of tooth movement

under orthodontic rehabilitation.

Tanne et al1-6 used FE simulations to show that ini-

tial tooth movement depended on the geometry of the

tooth and the alveolus. Middleton et al7 used a

3-dimensional (3-D) FE model to investigate the initial

biomechanical response of a canine. On the basis of FE

calculations, Geramy8 also demonstrated how the ini-

tial mobility of a tooth changes when bone loss occurs.

Bourauel et al9,10 discussed different bone remodeling

theories for the simulation of long-term tooth move-

ment in comparison with experimental investigations.

In addition, some experimental investigations are

found in the literature.11-15 Burstone and Pryput-

niewicz16 examined the position of the center of rota-

tion dependent on orthodontic force systems by means

of holography.

The active processes during tooth movement are

biologic. In orthodontics, mechanical loads can influ-

ence cellular mechanisms. Bone tissue consists of 4 dif-

ferent types of bone cells. In vitro cell culture investi-

gations show that the bone cells respond to different

mechanical loads by increasing their division or activ-

ity.17-19 The complete implementation of these biologic

and biochemical processes into a mechanical calcula-

tion model is almost impossible. So far, different bone-

remodeling theories have been introduced. The inclu-

sion of cell biologic aspects is described by Hart et al20

in a closed automatic control loop. Huiskes,21 Huiskes

et al,22 and Weinans et al23 described the bone remodel-

ing by a change in bone density and an adjustment of

the anisotropy, and Cowin et al24,25 described bone

remodeling in a thermodynamic manner. These versa-

tile investigations were executed with theoretical as

well as experimental methods. In this study, the numer-

ical realization of a bone-remodeling algorithm based

on strains in the alveolar tissue is done in a finite ele-

ment method (FEM) code. The FEM allows for varia-

tions in geometry, in the material parameters of the bio-

logic tissue involved, and in the parameters of the

bone-remodeling function. In this study, mechanical

loads were analyzed in the initial phase, and a resultant

biologic stimulus was calculated. With this stimulus, the

conformational change of bone under tooth movement

is solved in a calculation program. With this method, itwill be possible to study not only the initial tooth mobil-

ity, but also the long-time orthodontic movement of

teeth. With these FE simulations, we can test different

orthodontic force systems by evaluating all components

of movement throughout the treatment.

Parameter studies, characterizing the movement of an

anterior tooth, were carried out. The variation of the

moment/force (M/F) ratios of tipping and rotation is dis-

cussed on the basis of the resulting tooth movement. Vari-

From the Department of Orthodontics, University of Ulm, Germany.aResearch assistant.bDepartment head and professor.

Reprint requests to: Martin Geiger, University of Ulm, Department of Ortho-

dontics, ZMK4, 89081 Ulm, Germany; e-mail, martin.geiger@medizin.

uni-ulm.de.

Submitted, December 2000; revised and accepted, August 2001.

Copyright © 2002 by the American Association of Orthodontists.

0889-5406/2002/$35.00 + 0 8/1/121007

doi:10.1067/mod.2002.121007

257

ORIGINAL ARTICLE

Numerical experiments on long-timeorthodontic tooth movement

Jürgen Schneider, PhD, Dipl-Phys,a Martin Geiger, Dipl-Phys,a and Franz-Günter Sander, DDS, PhDb

Ulm, Germany

In orthodontic treatment, teeth are moved by the use of specific force systems. The force system used

depends on the patient’s orthodontic situation characterized by the geometry of the tooth and the surrounding

alveolar bone, which defines the position of the center of resistance. Therefore, the simulation of bone

remodeling could be helpful for the treatment strategy. In this study, the optimal force system for bodily

movement of a single-root tooth, with an orthodontic bracket attached, was determined. This was achieved by

the use of the numerical finite element method, including a distinct mechanical bone-remodeling algorithm.

This algorithm works with equilibrium iterations separated in 2 calculation steps. Furthermore, a parametric 3-

dimensional finite element model, which allows modifications in the root length and its diameter, is described.

For different geometries, the ideal moment-by-force ratios that induce a bodily movement were determined.

The knowledge of root geometry is important in defining an optimal force system. (Am J Orthod Dentofacial

Orthop 2002;121:257-65)

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258 Schneider, Geiger, and Sander American Journal of Orthodontics and Dentofacial Orthopedics

March 2002

ations of diameter and length are executed by the use of a

parametric geometry model. The aim is a successfully cal-

culated bodily movement for all different geometric mod-

els. The morphology is relevant for the M/F ratios. After

performing these simulations, qualitative predictions

about these geometric dependencies can be made.

MATERIAL AND METHODS

The FE analysis we carried out solved complex

structure-mechanical equation sets by numerical meth-

ods. In contrast to analytic procedures, nearly all geo-

metric problems can be calculated. In orthodontics,

most published calculation studies have solved the

problem of initial tooth movement to evaluate the dis-

tribution of stress or strain in the jaw bone or the peri-

odontal ligament (PDL). The calculations presented

here, instead, work with a bone-remodeling algorithm

and a parameterized model.

FE model

In this study, a model tooth was built with variable

geometric parameters. The model was divided length-

wise into different segments, with the crown comprising

2 segments and the root comprising 7 segments. The

cross-section area of a segment was described by ellipse

functions, whereby both ellipse diameters are responsi-

ble for the shape of the tooth. Perpendicularly to the

ellipses used, the form was outlined by splines. The

model in Figure 1, called the reference model, needs to

be a realistic figure of the anterior tooth root because its

morphology influences the movement of the tooth.

A schematic bracket built of beam FEs was fixed at

the crown. Beam elements offer the advantage of easy

modification of the bracket’s size, and the force system

has an effect only at 1 point. This point is positioned in

the half height of the crown, with a perpendicular offset

of 2.5 mm to the surface of the crown. It is possible to

apply orthodontic M/F systems directly in the calcula-

tion model. In Figure 1, the reference geometry model,

already composed of FEs, is shown. The bracket

designed in beam elements is clearly visible.

Boundary conditions applied during the simulation

are the acting orthodontic force system at the bracket

and the zero displacements to the exterior border of the

mandible. Simplified considerations describe the alveo-lar bone, because, for a separation into cortical and

spongy tissue, geometric information is missing.

For this reason, modeling of the mandibular bone,

which is not yet differentiated between spongy and cor-

tical bone, was generated with identical linear material

parameters according to cortical bone (Table). Differen-

tiation of bone material would require a more detailed

bone-remodeling algorithm that could realize the con-

nection between internal and external bone remodeling.

Fig 1. Reference FE tooth model consisting of 1576 solid elements. Coordinate system and direc-

tions of forces and moments applied to schematic bracket are shown.

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American Journal of Orthodontics and Dentofacial Orthopedics Schneider, Geiger, and Sander 259Volume 121 , Number 3

The complex structure of the PDL is basically inte-

grated26,27 with a uniform thickness and with linear elastic

material properties (Table). These aspects will not repre-

sent the real behavior of the PDL. Tanne et al28 noticed

that, during bone remodeling, the response of the PDL on

mechanical loading shows a bigger deformation than at the

beginning. In addition, Bourauel et al9 and Hinterkausen et

al29 postulated a bilinear material law of the PDL. In the

future, an enhanced material law will describe the PDL

more functionally, because the strain distribution in the

alveolar bone is influenced by the behavior of this mate-rial.30 The tooth and the bracket used for numerical simu-

lation were modeled in rigid material. All components of

translations and rotations were analyzed in the root apex.

The 3 different parameter variations are presented:

(1) Modification of My /Fx and Mz /Fx ratios is real-

ized with a constant geometry (Figs 1 and 2). This

model also stands as the reference for the following

variations in geometry. The model shows the dimension

of a typical anterior tooth. The root height is 16 mm,

and a value of 10 mm is used as the crown height. The

model tooth profile is like the profile of a real tooth.

Therefore, a diameter of 6 mm at the alveolar crest was

used. With these dimensions, the different My /Fx andMz /Fx ratios were modified (Fig 2). The resulting com-

ponents of movement are represented corresponding to

the M/F ratio in the following graphs.

In addition, geometric variations of length and

diameter of the tooth were used. This analysis repre-

sents the My /Fx relationship required for a bodily move-

ment, depending on different geometric dimensions.

(2) The root length was varied on the basis of a con-

stant crown length. The root length was changed from

Fig 2. Two complete calculation models. Initial reference geometry with coordinate and force systems

is shown (left). Complete model consists of 7509 finite elements. Model after 400 bone remodeling

iterations (right). At My /Fx ratio of -12.2 Nmm, the tooth bodily moved. Corresponding translation and

rotation values are shown in Figs 3 and 4.

Table. Material parameters of involved tissues

Young’s modulus Poisson ratio

Jawbone 1 GPa 0.35

Periodontal ligament 1 MPa 0.45

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March 2002

12 mm to 22 mm for bone-remodeling simulation. The

cross-section areas and the position of the bracket’s

construction with regard to the crown as well as the

crown’s dimensions never change.

(3) For modification of the root diameter, the 2

ellipse axes were proportionally changed, compared

with the reference tooth. These parameter values were

increased or reduced by 30%. Because of the change of

the crown’s diameter, the antirotational torque must be

adapted throughout the calculation. The distances

between the root segments were not changed; thus the

root length remained constant.

Bone-remodeling algorithm

An equilibrium iteration in remodeling here consists

of 2 different calculation steps. In the first calculation,

the impact of an orthodontic force system to the alveo-

lar bone was analyzed. On the basis of these results, the

local function of biomechanical stimuli was calculated.

The bone-remodeling algorithm calculated the biome-

chanical stimuli from the strain field in the mandibular

bone of the initial tooth movement.24,31 The immediate

relationship between the mechanical stimuli of strain

and the stimuli in the FE model defines the evolutionequation. Different functions for the evolution equation

looked appropriate.22 Instead, a bilinear function that

scaled bone apposition and bone deposition differently

was implemented here. For compression (∑i < 0,

i means X-, Y-, and Z-coordinates), the loaded fraction

in the bone tissue is resorbed, and, for extension (∑i >

0), bone tissue is generated. The scale factors for bone

deposition are twice the scale factors for bone apposi-

tion and are the gradients of the bilinear function.

Provatidis32 analytically demonstrated the unrealis-

tic deformation of the alveolus if the stimuli for bone

remodeling do not depend on the orientation. Therefore,

stimuli are computed according to the evolution equa-

tion from the results of the first calculation step—the 3

oriented principal strains.33 In the second computa-

tional step, the stimuli act as components of load for

each node in the bone tissue. In this step, a new tooth

position was calculated that results from the stimuli’s

function. This new geometry was then the new model

for a further initial calculation.

This procedure with 2 successive calculation steps,

whereby the tooth is definably loaded and the move-

ment results of the bone remodelling algorithm, corre-

sponds with the natural behavior of a tooth loaded with

constant displacement. Detailed information of the

function has been published.34

RESULTS

(1) In these calculations, a constant force Fx of 1 N

in X-direction was applied on the tooth. The variation

concerns the M/F conditions with the antitipping My and

the antirotational Mz moments. The tipping moments My

were modified from -14 Nmm to -10 Nmm. For theantirotational moments Mz, associated values from -6

Nmm to -4.9 Nmm were used. If a value of –12.2 Nmm

was used as the antitipping moment, the rotation around

the Y-axis is almost zero (Figs 3 and 4). With an increase

of the tipping moment My to -14 Nmm, the root was

moved forward. In the case of a decrease to -10 Nmm,

the root moved backward. The bodily movement

occurred only by 1 corresponding My /Fx condition. With

the rotation around the Z-axis, the influence of the

Fig 3. Translation of tooth in all components. Major magnitude of translation is in X-direction, same

as direction of acting force. All other components are so small as to be ignored.

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moment selection becomes still larger than it is around

the Y-axis (Fig 5). With a moment of -5.12 Nmm, the

rotation can be ignored, whereas a modification to

-6 Nmm causes a rotation of nearly 6° after 400 itera-tions (Fig 5). In Figure 3, the absolute movements of the

tooth are represented. The X-direction is always the

main direction of tooth movement—the same as the

direction of force. The movements in the other directions

are very small compared with the main direction. In the

main direction, a movement up to 10 mm is calculated,

which is nearly twice the root diameter.

(2) Depending on the variation of the root length,

the ideal My /Fx condition required for the bodily move-

ment is shown in Figure 6. During these calculations,

the cross-sections of the levels remain constant. The

form and the size of the crown also remain unchanged.

In Figure 6, the ideal force system for antitipping isshown in relation to the root length.

(3) With the diameter variation, a constant crown and

root length was used. These modifications were executed

to adapt the diameters of the ellipse. Because of the diam-

eter modifications, the antirotational component Mz /Fxmust also be adapted. This fact will not be discussed in

detail because of the constant relationship between Mz /Fxand the root diameter. In Figure 7, the influence of the

root diameter on the force system is shown.

Fig 4. Behavior of tipping angle in dependency of My /Fx ratio is indicated. Ideal My /Fx ratio for bodily

tooth movement is -12.2 Nmm.

Fig 5. Behavior of rotation angle in dependency of Mz /Fx ratio is shown. No tooth rotation around

Z-axis occurs at Mz /Fx ratio of -5.12 Nmm.

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DISCUSSION

The aim of this research project was to simulate a

long-time tooth movement of a particular treatment caseto assist the orthodontist in selecting the most appropri-

ate appliance. A bone-remodeling algorithm must be

developed to realize this. The first results were pre-

sented here. The significance of these results must be

discussed, before extensive predictions can be made.

These numeric investigations were executed with sim-

ple linear material properties for the tissues involved.

Simplifications were made because of missing experi-

mental data of materials and missing information of

their morphology. The evolution equation describing

the bone remodeling was simplified as a bilinear func-

tion. Another approach could be realized—eg, the mul-tilinear approach of Huiskes et al,22 including a dead

zone. With this assumption, solving all problems in

orthodontics with high accuracy28,30 is not feasible. For

a general quantitative extrapolation, this bone-remodel-

ing algorithm must be carefully evaluated in clinical

studies.

As the calculations in (1) show, the resulting tooth

movement strongly depends on the M/F relationship

applied. This behavior is well known to orthodontists

Fig 6. Perfect My /Fx ratio for performing bodily movement corresponding to different root lengths.

Fig 7. Perfect My /Fx ratio for performing bodily movement by different root diameters. Root diameter

is varied 30% around reference diameter from wide to narrow.

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and has been already demonstrated by simulations of

initial tooth mobility, but not the change of the mor-

phology of the alveolus throughout the remodeling.28,30

As a result of that, predictions about the position of the

center of resistance (CR) are possible only in the begin-

ning or in the physical case of movement. Statements onthe entire mode of movement considering the change of

morphology of the alveolus can only be made with

bone-remodeling simulations.

Bourauel et al10 previously performed numerical

3-D simulations of bone remodeling but could not pre-

sent the bodily movement in their miscellaneous calcu-

lations.

The results, which emulate the bodily movement of

an anterior tooth, correspond to the magnitude used in

practice. Sander35 chose an My /Fx ratio between 10 and

12 mm for the retraction of a canine; this ratio corre-

sponds to the values calculated. These simulations rep-

resent how exactly the moments must be adjusted to aforce to obtain the desired bodily tooth movement. Even

for deviations of 2 mm, compared with the ideal My /Fxrelationship of 12.2 mm, after a movement of 10 mm, a

tipping around the Y-axis up to 10° is calculated (Fig 4).

With the rotation around the Z-axis, this connection is

also very clear (Fig 5). Therefore, constant M/F condi-

tions must be applied during a complete treatment ses-

sion.35-37 The distance of translation in these calcula-

tions cannot be coupled so far with the orthodontic

treatment time, because the movement components

achieved depend on the load status. In case of constant

force and larger moments, extended movements are

then also achieved. This is the direct consequence of the

bilinear evolution equation used,34 and this is why it is

impossible to compare the number of iteration steps

directly with treatment time or other biologic factors. In

this state of elaboration, it is possible only to make a

motion study of the tooth treated.

Throughout the geometry variation, the influences

of the root lengths (2) and the diameters (3) are exam-

ined (Figs 6 and 7). One result is that the position of the

CR strongly depends on the tooth geometry,38 and also

the position of the CR receives a shift relative to the

alveolar crest. During the model creation, it is ensured

that there is a linear relationship between the surfacemodification of the root and the root length. Throughout

the diameter modification, there is a very linear depen-

dency between diameter and My /Fx relation, whereas

different gradients during the length variation occur.

These are very large modifications—about 14 to 15

mm. An explanation has not yet been found for this

behavior. Choy et al39 studied a number of variations in

the geometry of roots in 2 dimensions with analytic

methods and found a relationship between the outline of

the root and the position of the CR. Taking into account

the location of the bracket and the geometry of the ref-

erence tooth discussed here, the CR is then located 7.2

mm below the alveolar crest, or at 45% of the root

length. For a shorter root with a length of 12 mm, the

position of the CR is then 4.2 mm below the alveolarcrest, or only at 35% of the root length. The dimensions

are comparable with other results published.3,14,39 Ger-

amy8 studied initial tooth movements on the basis of a

3-D numerical model to calculate miscellaneous

geometries of bone; he also found a high dependency

between geometry and movement.

Overall, it is difficult to compare the presented mod-

els and the performed parametric studies here with

those of articles published in the past because models

are sometimes totally different and mostly designed

only for calculating initial mobility.

Generally, the influence of the geometric parameters

on tooth movement is pointed out here. For the simula-tion of individual tooth movements, individual nonstan-

dard tooth models must be created. Computerized

tomography sessions could provide the data for this

model.

CONCLUSIONS

1. This study showed that it is possible to integrate a

bone-remodeling algorithm in a FEM code and

apply it to a realistic 3-D tooth model.

2. With this FE model, it was possible to simulate the

bodily tooth movement, and almost any tooth

movement, with rotations for a long distance in the

mandible.

3. The first results are very close to reality. With the

variations shown here, dependencies between the

force system and geometrical parameters were ana-

lyzed.

4. It was shown that the optimal My /Fx ratio for a

bodily movement depends strongly on the tooth

geometry.

5. An important consequence is the use of a real

model (computerized tomography data) of the indi-

vidual tooth; this will increase the accuracy of the

simulation of treatment significantly.

Outlook

Furthermore, new methods will be developed that

allow for the checking and the development of these

bone-remodeling algorithms and use the material prop-

erties of the tissues involved. The first step is to measure

the long-term orthodontic tooth movement free of force

in vivo measurements. This will be done with a high

resolution with a photogrammetrical system.40 Experi-

mental evaluation of the orthodontic force system will

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be done with special 6-component force-moment sen-

sors before and after treatment.41 With these require-

ments and a realistic FE model from computerized

tomography data sets created before the treatment, it

will begin to be possible to evaluate these complex

functions in a patient study. Here, the emphasis was onthe biologic stimuli and the evolution equation, and, if

this is successful, then the material properties of the

PDL will be tested with different functions. If all these

steps are successful, a calculation of real treatment sit-

uations will become possible for the first time.

The authors thank the Deutsche Forschungsgemein-

schaft DFG for financial support of project Sa-272/1-2.

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