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ORIGINAL ARTICLE

Moment-to-force ratio, center of rotation, andforce level: A finite element study predicting

their interdependency for simulated orthodonticloading regimens

Paolo M. Cattaneo,a Michel Dalstra,b and Birte Melsenc

Aarhus, Denmark

Introduction: Changes in the stress/strain distribution in the periodontium after the application of orthodon-

tic forces trigger remodeling processes that make tooth movement possible. The type of orthodontic tooth

movement is linked to the force system applied to the bracket. By combining moments and forces, often

expressed as the moment-to-force (M/F) ratio, it is possible to determine the prescribed type of tooth

movement. According to classical theory, fixed values for M/F are associated with specific tooth movements.

Methods: A segment of a mandible containing the canine and the first premolar obtained from autopsy was

scanned with microcomputed tomography, and a finite element model was generated. In a series of finite

element analyses simulating teeth subjected to various orthodontic loading regimens, the influences of the

M/F ratio and the force magnitude were examined. Results:By applying a range of values of M/F, different

types of tooth movement were generated, although the classic prescription of the M/F ratio suggested in the

literature could not be confirmed. Due to the nonlinear behavior of the periodontal ligament, loading modes

with a constant M/F ratio, yet varying the force magnitude, resulted in different types of tooth movement.

Conclusions: The material properties of the periodontal ligament, the morphology of the root, and the

alveolar bone are patient specific. Therefore, the M/F values generally advocated to obtain orthodontic tooth

movement should be used only as guidelines. To be effective and accurate, the force system selected for a

specific tooth movement must be monitored and the outcome compared with the predicted tooth movement.

(Am J Orthod Dentofacial Orthop 2008;133:681-9)

After an orthodontic loading regimen, the first

reaction is an alteration in the strain-stress

distribution in the periodontal ligament (PDL)

and the surrounding alveolar bone. This leads to intra-

alveolar displacement of the tooth and bending of the

surrounding alveolar bone if the forces are large

enough.

The study of teeth displacement following the

application of orthodontic loading regimes is not trivial

as it depends on several and complex parameters such

as material properties of periodontal tissues, shape and

length of tooth, width of the PDL, and the marginalbone level.1 Moreover, since the decay rate of a force

and the decay of the moment that this force generates

are not equal, the actual force system can vary in

experiment situations.2 As soon as the teeth start

moving, the geometry characterizing the original force

system also changes, and a new force system is gener-

ated, leading to a different displacement. The conse-

quence of the undulating force system makes it nearly

impossible to establish fixed values describing the

tooths displacement over time.3

Based on a mathematical model, Burstone4 estab-

lished the localization of the center of resistance (CR)

in a single-rooted tooth with a parabolic shape to be at40% from the apex to the measured length between the

alveolar crest and the apex of the root. Based on this

assumption, the center of rotation (CRot) could be

calculated with the so-called Burstone formula:

moment

f orce

0.068 h2

y

where h is the distance from the alveolar crest to the

apex, and y is the distance between the CR and the

CRot, assuming the PDL to be idealized as a 2-dimen-

From the Department of Orthodontics, School of Dentistry, University of

Aarhus, Aarhus, Denmark.aAssistant professor.bAssociate professor.cProfessor and chairman.

Reprint requests to: Paolo M. Cattaneo, Department of Orthodontics, School of

Dentistry, University of Aarhus, Vennelyst Boulevard 9, DK-8000 Aarhus C,

Denmark; e-mail, [email protected].

Submitted, November 2005; revised and accepted, May 2006.

0889-5406/$34.00

Copyright 2008 by the American Association of Orthodontists.

doi:10.1016/j.ajodo.2006.05.038

681
mailto:[email protected]:[email protected]

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sional entity with a shape resembling a parabola, the

stress distribution to be uniform, and the stress-strain

ratio to be linear. Later, Christiansen and Burstone5

determined the CRot experimentally by applying dif-

ferent forces on maxillary central incisors, and theyconfirmed that the CRot is related to the moment-to-

force (M/F) ratio, but the CRot seemed to be positioned

more apically than theoretically calculated. Most clin-

ical investigations confirmed these results. The discrep-

ancy between the theoretically calculated and the ex-

perimentally established localization of the CR was

explained by the fact that Burstones formula was

developed on a 2-dimensional model. All consider-

ations regarding the position of the CR were based on

the assumption that the PDL exhibits linear behavior.

Over the last decades, numeric methods to calculate

the stress and strain fields in the periodontium have

been extensively used, and the finite element (FE)

method has frequently been the method of choice. The

complexity of shape and tissue composition of the

dental region does, however, set limitations on the

validity of the results of some previous analyses.6-8

Although many attempts to relate the force system to

tooth displacement and the reaction of the surrounding

tissues have been made, a clear concept still has not

been presented. One reason might be that the methods

applied have suffered from deficiencies in the methods

and the properties of the tissues analyzed.

The FE method was introduced into dental biome-

chanical research in 19739 and since then has beenapplied to analyze the stress and strain fields in the

alveolar support structures.8,10-18 The load-transfer

mechanism from the tooth through the PDL to the

alveolar bone depends on the physical properties and

the morphology of the periodontium. The choice of

these parameters determines the outcome of tge FE

analysis and, in orthodontics, the type of tooth move-

ment generated when a specific force system is applied

to the bracket.

Even though the PDL is known to be a nonlinear

viscoelastic material, most previous FE models incor-

porated homogeneous, linear elastic, isotropic, andcontinuous PDL properties. At the same time, the

morphology of the alveolar structures has not been

assigned any specific value in relation to the load-

transfer mechanism.

Our aim in this study was to demonstrate by FE

analyses the influence of the material properties of the

PDL on the type of tooth movement. Moreover, the

influence of the applied force level on the type of tooth

movement, with a fixed M/F ratio, was evaluated and

the results interpreted in the light of existing prescrip-

tions for orthodontic tooth movement.

MATERIAL AND METHODS

Using a microcomputed tomography scanner

(model 40, Scanco Medical, Bassersdorf, Switzerland),

a 3-dimensional (3D) data set of the left segment of a

human mandible including a canine and a first premolarobtained from autopsy was acquired. The 3D data set

had a voxel dimension of 37 m. By using a procedure

previously described,19 a 3D FE model comprising the

mandibular segment, the PDL, the canine, and the first

premolar was generated. The model consisted of

197,186 ten-node tetrahedral elements, 253,309 nodes,

and 754,347 degrees of freedom (Fig 1).

The material properties of bone were assigned to

each element individually based on the true morphol-

ogy of the bone as obtained from the scans by using a

slightlymodified version of the procedure of Cattaneo

et al.20

Three Youngs moduli were considered torepresent full cortical bone (17,500 MPa, Poissons

ratio of 0.3), partly cortical bone (5000 MPa, Poissons

ratio of 0.3), and bone marrow (200 MPa, Poissons

ratio of 0.3), respectively. A nonlinear and nonsymmet-

ric approach was used to describe the material proper-

ties of the PDL when the behavior in compression and

tension were not the same. In compression, the stiffness

of the PDL was substantially lower than in tension. In

compression, a Youngs modulus of 0.005 MPa up to

the 93% strain level was used; then a Youngs modulus

of 8.5 MPa was adopted to simulate precontact between

the roots and the alveolar bone. In tension, the Youngsmoduli was gradually increased from 0.044 MPa at

zero strain level to 0.44 MPa at about 50% strain; then

a smaller Youngs modulus of 0.032 MPa was used.19

A Poissons ratio of 0.3 was used.

The elements representing the 2 teeth were assigned

a Youngs modulus of 20,000 MPa and a Poissons

ratio of 0.3.10,14,21

Different loading regimens were applied to the

model to simulate distinct types of orthodontic tooth

movements. In accordance with clinical practice, the

various loading regimens were defined by the M/F ratio

applied at the bracket.In the first series of analyses, the force applied at

the bracket was kept constant while the M/F ratio was

varied. The force on the premolar acted in the bucco-

lingual direction and in the opposite direction for the

canine22; it had a magnitude of 100 cN for both teeth.

This value was chosen because it corresponded to the

force level used in clinical practice.2,5,22,23 The M/F

ratios were varied from uncontrolled tipping (M/F 0)

to 4, 6, 8, 9, and 10 for the premolar and from 0 to 5,

8, 10, 11, and 12 for the canine.

In the second series of analyses, the force level was

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gradually increased from 0.5 to 200 cN, with the M/F

ratios kept constant at 10 for the premolar and at 12 for

the canine.

For each analysis, the CRot was determined, and its

location was compared with that of the CR.For the boundary conditions, movement was sup-

pressed in all directions for the nodes on the bottom

edge of the bone segment.

RESULTS

The results from the first series of analyses (force

level of 100 cN and M/F ratio varying from 0 to 10 for

the premolar and from 0 to 12 for the canine) are shown

in Figure 2. From these analyses, it can be seen that,

with an M/F ratio of 0, uncontrolled tipping movement

is obtained. With an M/F ratio of 10 for the premolar

and 12 for the canine, the CRot was situated far from

the infinite, where it theoretically should have been

located. M/F ratio was shown to be the best approxi-

mation for pure translation for the premolar, and a

value of 11 for the canine would be the best approxi-

mation. Furthermore, it could be seen that M/F ratios of

6 for the premolar and about 8 for the canine produced

controlled tipping because the CRot coincided with the

apex of the root in that case.

During uncontrolled tipping (M/F 0) at the

bracket level, the premolar moved lingually 0.163 mm,

and the canine moved buccally 0.181 mm (Fig 3). As

expected, when M/F ratios of 9 for the premolar and 11

for the canine are used, the displacements of both teethwere smaller than in the case of uncontrolled tipping.

The crowns of the premolar and the canine were

displaced 0.024 mm lingually and 0.018 mm buccally,

respectively, whereas the apexes of both teeth were

displaced 0.021 mm (lingually for the premolar and

buccally for the canine). This indicates that, although

aiming at translation, moderate tipping occurred for the

premolar whereas the canine exhibited root movement

(Fig 3).

By looking at the displacements of the crown and

the apex (Fig 3), one can notice that the canine had a

peculiar movement: when low forces were applied withan M/F ratio of 12, the crown and the apex started

moving in opposite directions, and, instead of pure

translation, reverse tipping occurred. However, above

30 cN, the movement of the crown tended to be zero,

and root movement occurred.

During uncontrolled tipping (M/F 0) with a force

level at the brackets of 100 cN, regions of compressive

and tensile normal stresses could be identified in the

PDL, although the stress levels in the compression

areas were significantly lower than the tensile stresses.

This was evident at the PDL-bone interface of both the

Fig 1. Exploded view of the FE model of the mandibular

segment with alveolar bone, PDLs, canine (left), and first

premolar (right). The PDL is the space between the

alveolar bone and the roots of the teeth.


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premolar and the canine, particularly in the cervical

areas opposite the direction of the force (Fig 4, top).

The strains in the PDL were in the range of20% to

14%.

Based on the previous results, the stresses in the

PDL were evaluated when translation was simulated by

using a force of 100 cN and M/F ratios of 9 for the

premolar and 11 for the canine. In this case, again the

stress values in the PDL were much higher in the

tension side that in the compression side (Fig 4,

bottom).

When the force level was gradually increased from

0 to 200 cN, the canine and the premolar displayed a

remarkable phenomenon, especially when M/F of 10

for the premolar and M/F of 12 for the canine were

applied: the CRot is not in a fixed position, but it is

displaced after the force level that is applied to the

tooth. To depict these changes, the distance between the

actual CRot and the CR was calculated (Fig 5). This

distance becomes maximal at a force level of about 50

cN for the canine and about 40 cN for the premolar.

After these force levels, the CR-CRot distance starts to

Fig 2. A, Location of the CRot when the force applied at the bracket of the premolar is kept constant

at 100 cN while the M/F ratio is varied from 0 to 10. B, Location of the CRot when the force applied

at the bracket of the canine is kept constant at 100 cN while the M/F ratio is varied from 0 to 12.


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decrease for both teeth, approaching an asymptotic

value. In the case of both the canine and the premolar,

the initial displacement will vary according to the force

level, and this phenomenon is particularly evident in

the range of forces frequently used in daily practice(20-60 cN).

DISCUSSION

The relationship between the type of tooth move-

ment generated by an orthodontic appliance and the

M/F ratio has been based on inadequate methods. In our

study, the importance of the morphology and the

physical properties of the tissues involved was studied.

An FE model was used to determine the relationship

between the M/F ratio applied at the brackets and the

CRot, and the influence of the force level on the CRot.

Although we used correct morphology and approxi-

mated the true physical properties of the PDL and the

surrounding bone, certain limitations of this study

should be mentioned. This study was based on 1 human

sample. The material property of the PDL was assumedto be nonlinear and nonsymmetric, but viscoelasticity

was not considered.19 Moreover, the results were not

directly compared with in-vivo loading, although the

calculated amount of deflection of the teeth reflects the

results of previous experimental studies.5,24

In previously published studies, the material

properties of the PDL were modeled as linear elastic

with elastic moduli ranging from 0.07 to 100

MPa.14,24,25 In other studies, the PDL was modeled

as nonlinear, fiber-reinforced, or with viscoelastic

properties.10,15-17,26-28 In a recent study,19 the PDL

Fig 3. Lingual-buccal displacements of the crowns and the apices of the premolar and the canine

when the force is increased from 0 to 100 cN. Three values of M/F ratio are simulated.


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was given a nonlinear stress-strain relationship based

on experimental results.26,27 Its basic shape with a

low-stiffness toe region and a high-stiffness slope

closely resembles both experimentally29 and mathe-

matically determined relationships.28 Although teeth

comprise various tissues such as dentin, enamel, and

cementum, only the dentin was modeled because our

main goal was to study the reaction in the PDL and

the surrounding alveolar bone. The material property

of the dentin was adapted from several values in the

literature.8,10,21,30

Unlike most previous FE analyses, our model

comprised 2 teeth loaded simultaneously; thus, it was

possible to counteract the force applied on the canine

with another force of the same magnitude, but in the

opposite direction, applied on the premolar. This way

of loading limited the force transfer through the con-

strained nodes because the forces applied to the teeth

counteracted each other. The influence of boundary

conditions on the final outcome was therefore smaller,

as opposed to the situation when only 1 tooth would

have been modeled. In addition to this, it also reflects

Fig 4. Bucco-lingual stress at the bone-PDL interface around the root of the premolar (left) and the

canine (right). Negative values represent compression, and positive values represent tension.

Uncontrolled tipping is simulated by applying a force of 100 cN at the bracket level and an M/F ratio

of 0 (top), and translation is obtained by applying again a force of 100 cN plus M/F ratios of 9 for

the premolar and 11 for the canine (bottom).


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the clinical situation better: a tooth or a group of teeth

is used to load the others.We simulated various tooth movements from un-

controlled tipping to pure translation. This was

achieved by using the values of the M/F ratios recom-

mended by Burstone and Pryputniewicz.23

The CRot values of the resulting tooth movements

are located in a straight line passing through the CR.

The actual location of the CRot is determined by the

following:

1. The value of the M/F ratio. This dependency is

evident, yet our results show that the traditionally

assumed M/F ratios for translation do not perfectly

apply. An M/F ratio of 10, which is generally

accepted for translation of incisors and premolars,was too large for the premolar in this study; it

seemed to approach translation with an M/F ratio of

9. The lack of validity of the assumed values of an

M/F ratio of 12 for the canine was also demon-

strated; the correct value seemed to be between 10

and 11. However, these results were based on 1

human sample; therefore, they should be consid-

ered valid only for this canine and this premolar. In

accordance with the report of Burstone and Pryput-

niewicz,23 small changes in the M/F ratio would

produce large changes in the position of the CRot as

Fig 5. Distance between the CRot and the CR when the force level is gradually increased from 0 to

200 cN while keeping the M/F ratios constant at 10 for the premolar and 12 for the canine.


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the M/F ratio required to generate translation is

approached.

2. The level of the applied force. This was initially not

anticipated but could be explained by the nonlinear

behavior of the PDL. At different force levels, theactual load transfer across the PDL differs, leading

to different types of tooth movement. Christiansen

and Burstone4 observed this phenomenon in their

clinical experiments with forces below 50 cN. Yet

they failed to note that it is related to the mechanics

of the PDL, and they instead ascribed the deviation

between the expected behavior and their findings to

inaccuracies in measuring technique and instrumen-

tation. This means that, with a nonlinear model to

describe the PDL, the location of the CRot is not

constant but depends not only on the M/F ratio, but

also on the force level.

In the deflection curves depicted inFigure 3, it can

be seen that the amount of tooth movement is not

linearly correlated to the applied load. In the first part of

the curves, a relatively small increment of the force is

followed by a relatively large movement of the teeth;

on the contrary, after a force level of about 15 cN, the

slope of the curve diminishes as the force magnitude

increases. Therefore, the application of higher forces

will generate only a moderate increment in tooth

movement. This phenomenon could explain why, as

reported by some authors, increasing the force magni-

tude from 10 to 25 cN increases the mean rate of toothmovement,31 whereas the mean rate of tooth movement

is independent of the force magnitude when the force

level is increased from 50 to 200 cN or more.32,33

Our results indicate that the M/F values generally

advocated for various movements are too high, and a

reduction of the M/F ratio could be suggested. How-

ever, tooth movement should always be monitored and

the outcome compared with the expected tooth move-

ment. In the case of discrepancy between the predicted

and actual tooth movements, the force system should be

adjusted.

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