SCIENTIFICSECTION

Optimization of unilateral molarrotation correction by a trans-palatalbar: a three-dimensional analysis usingthe finite element method

Allahyar Geramy and Tahura EtezadiOrthodontics Department, Tehran University of Medical Sciences, Tehran, Iran

Objective: The main goal of this study was to optimize unilateral molar rotation correction by modifying a trans-palatal arch

(TPA) design using the finite element method.

Design: Three-dimensional analysis of different TPA designs was carried out using the finite element method.

Setting: Department of Orthodontics, Tehran University of Medical Sciences, Iran.

Material and methods: For this investigation, 13 three-dimensional finite element models were produced for different TPA

designs without pre-activation bends. Each model contained a palatal bar and two tubes. Optimizing unilateral molar

rotations was achieved by five separate different paths: incorporating U-loop(s), ‘R’ loop(s) or helix/helices, a reverse action of

the helix/helices and adding a straight wire to the design. The mesial part of the left side tube was displaced 0.1, 0.25, 0.5 and

1 mm, successively towards the midline, simulating palatal bar tab engagement in a mesio-palatal rotated maxillary left molar.

The mesio-distal force, moment and energy produced in the normal side (right) molar were recorded for each of the models.

Results: Findings showed that in all designs, the associated mesializing force was lower than that seen in the traditional design

and the moment showed an increasing pattern when compared with a simple palatal bar. Regarding energy levels, the same

increasing pattern was observed in the designs between activations of 0.1 and 1.0 mm.

Conclusion: According to our optimized system, the TPA design with the highest energy and moment, but the lowest

mesializing force associated with derotating a maxillary molar tooth was a parallel wire II design (i.e. adding a straight wire).

Key words: Asymmetric activation, finite element method, rotation, transpalatal bar, unilateral

Received 11 November 2012; accepted 18 February 2013

Introduction

A trans-palatal arch (TPA) is a lingual arch which connects

upper right and left molars and passes a few millimetres

away from the palate.1 Different treatment objectives are

considered to be achievable with a TPA and its various

shape modifications.2 The original design (also known as a

trans-palatal bar) included a straight bar extending across

the palate, connecting the maxillary first permanent

molars.1,3 An alternative design is the Goshgarian TPA,

which has an additional U-loop oriented either mesially or

distally.1–5 Although TPAs may be constructed from TMA

wires,4 they are usually made of a 0.8–0.9 mm (0.32 or 0.36-

inch) stainless steel wire, adapted to the contour of the

palate, maximizing comfort for the patient and minimising

interference with speech and soft tissue irritation.1,2,6

A TPA can be used for anchorage reinforcement or as

a space maintainer in passive form. When used in these

circumstances it is preferable to solder the TPA directly

onto the bands. For active tooth movement, prefabri-

cated lingual attachments are welded to the molar

bands, allowing insertion of the TPA, which can

subsequently be removed for adjustment.1,2,7 This form

of TPA can be used to institute a variety of tooth

movements, changing or stabilizing maxillary molar

position in three dimensions (3D),8 producing first-,

second- and third-order molar adjustments, correction

of rotations unilaterally or bilaterally,1,4 correction of

unilateral cross-bites, expansion, constriction, distaliza-

tion, buccal root torque, intrusion and correcting mesio-

distal asymmetries.1–5,7,9,10 The criteria for use of a TPA

are dictated by the biomechanical anchorage needs in

Journal of Orthodontics, Vol. 40, 2013, 197–205

Address for correspondence: A. Geramy, Orthodontics

Department, Tehran University of Medical Sciences, Tehran, Iran.

Email: gueramya@yahoo.com# 2013 British Orthodontic Society DOI 10.1179/1465313313Y.0000000050

each patient, but generally, a more rigid material and

design is required for better stability and mechanical

rigidity.2,11 Conventional TPAs made of stainless steel

are best fitted for this goal. 2 Removing the open U-loop

in the palatal portion can also increase rigidity.1 TPAs

have to deliver various, indication-dependent require-

ments. Single tooth movements requires a system with

defined forces and moments;1 therefore, it is important

to create a configuration that produces the expected

force system when activated.6 It also has to be

considered that the initial force system will change as

the teeth move.12

Moments and forces delivered by TPAs of different

sizes and materials, and with different degrees of

activation have been evaluated in laboratory studies.13,14

In symmetrical activation, equal and opposite moments

can be produced,7 although an ideal symmetric force

system cannot be gained by TPAs.13 In asymmetrical

activation, the delivered force system would be different

and create unequal tooth movements.13

The finite element method is a numerical means of

calculating complex 3D structural problems and has a

proven efficiency in many applications. This 3D method

allows optimization of a process in a precise way, which

cannot be achieved with other in vitro experiments.15–18

The main goal of this study was to optimize unilateral

molar rotation correction using finite element analysis of

modified TPA designs. Optimization starts with the

identification of objectives, the gathering of variables

and then finding values of the variables that optimize the

objectives and finally delivering a practical idea to

decrease side effects while enforcing the desired results if

possible.19

Materials and methods

Biomechanics of unilateral molar rotation correction

Treating a unilateral molar rotation with a TPA is

described by the asymmetric V principle. In this case,

rotation correction moments are equilibrated by a

couple of forces acting on the molars. The produced

moment is equal to the force magnitude multiplied by

the distance between the molars (Figure 1). Thus, when

the conditions are kept constant for the moment

produced by bending the TPA tab, wider palates will

undergo less mesializing or distalizing forces and vice

versa. Usually, it is the mesializing force component that

is considered to be undesirable due to its anchorage loss

effects. Reviewing the biomechanics of this treatment

procedure, any attempt to decrease the mesializing force

component or increase the rotation moment would be

considered as a step towards optimization of unilateral

rotation corrections by a TPA.

Sample description

Thirteen 3D finite element models were designed of a

TPA without pre-activation bends. Each model con-

tained a palatal bar and two tubes. The configurations

of the palatal bars were different between models. For

simpler evaluation, these models were categorized into

five groups.

In the first group (Models 1–5), modifications were

accomplished by successively adding one, two, or three

U-loops to the left half of the palatal bar structure.

Model 1 represented a traditional continuous TPA

(Figure 2A). In Model 2, one U-loop was added at the

midline (uni-U pal.bar); in Model 3 (double-U pal.bar),

an additional U-loop was added equidistant between the

midline U-loop and molar; in Model 4, two U-loops

were added between the midline U-loop and first molar

(triple-U pal.bar) (Figure 2B–D). Model 5 had only a

single U-loop placed near the molar tube (uni-U near

tooth pal.bar) (Figure 3).

In the second group (Models 6 and 7), modifications

included the addition of either single unilateral or

double helices adjacent to the molar tubes (uni-helix

pal.bar and double-helix pal.bar, respectively) to the

TPA (Figure 4).

In the third group (Models 8 and 9), straight parallel

wires were adapted to the palate [parallel wire I and II]

(Figure 5).

Figure 1 M5F6d (where M5moment, F5force and d5distance)

198 Geramy and Etezadi Scientific Section JO September 2013

In the fourth group (Models 10 and 11), one or two

rectangular loops were incorporated into the traditional

TPA (uni-R-loop pal.bar and double R-loop pal.bar)

(Figure 6).

In the fifth group (Models 12 and 13), helices were

used in a reverse action (uni-helix rev.act. pal.bar and

double helix rev.act.pal.bar) (Figure 7).

The treatment of a mesio-palatal rotation of the upper

first left molar was the movement to be optimized. The

cross-section of the stainless steel wire was 0.8 mm.

SolidWorks 2010 (Concord, MA 01742, USA) was

selected for the modelling phase. The models were then

transferred to the ANSYS Workbench Ver. 11.0

(ANSYS Inc., Cononsburg, PA, USA) for analysis.

Young’s modulus (2e5MPa) and Poisson’s ratio (0.3)

were applied. Models were meshed with 24787 nodes

and 6491 elements. The lateral wall of the right molar

tube was restrained so that all rigid body motions were

prevented, simulating the welded attachment on the

palatal side of the right molar band. When conducting a

static analysis, we were calculating the effects at the time

of force system application (palatal bar insertion in this

case). When the palatal bar tab was inserted in the

sheath of the normal side molar band, we felt enough

stability in this side for rotating the affected side palatal

bar tab.

The mesial part of the left side tube was displaced 0.1,

0.25, 0.5 and 1 mm, successively towards the midline,

simulating the palatal bar tab engagement in a mesio-

palatal rotated left molar. The force, moment and

energy produced in the normal side (right) molar were

recorded in different models.

Results

A gradual increase in the mesio-distal force, moment

and energy was seen to be induced on the normal side

molar in all designs, which produced similar patterns

(Figures 8–10). Numeric data are shown in Tables 1 and

2. In the relevant sections below, all obtained results

were compared with the normal palatal bar.

Figure 2 TPA designs (A) Conventional TPA (Model 1); (B) uni-U palatal bar (Model 2); (C) double-U palatal bar (Model 3); (D)

triple-U palatal bar (Model 4)

JO September 2013 Scientific Section Unilateral molar rotation correction by palatalbar 199

Force

Adding U-loop(s). In this phase, adding loops

decreased the lowest force finding (0.1 mm of

displacement) from 0.0806 to 0.046 N and for the

highest finding (1 mm of displacement) from 0.806 to

0.46 N. However, for a uni-U-loop palatal bar, the

results were essentially equitable, regardless of the

position of the loop (near tooth versus near mid palatal).

Adding a helix/helices. Incorporating one or two

helices in the palatal bar resulted in a reduction of

force from 0.0806 to 0.0593 N in the lowest finding and

from 0.806 to 0.59 N in the highest finding.

Increasing the straight wire. Palatal bar designs with

added wire showed a decrease in force from 0.0806 N at

0.1 mm of displacement in the traditional design to

0.0403 N in the parallel wire I and 0.037 N in the

parallel wire II. The highest findings were 0.4 N in the

parallel wire I and 0.37 N in the parallel wire II

compared to the traditional design, which was 0.806 N.

Adding ‘R’ loop(s). These results indicated a decrease

in force from 0.0806 to 0.035 N at 0.1 mm of

displacement. The highest obtained value reduced

from 0.806 to 0.358 N.

Helices in reverse action. A decrease in the force was

also shown in this design, from 0.0806 to 0.064 N for the

lowest findings and from 0.806 to 0.641 N for the

highest.

Moments

All designs showed an increasing pattern when com-

pared with the simple palatal bar. The moment findings

for the simple palatal bar are the base for all

comparisons.

Adding U-loop(s). An increase in the obtained

moment occurred with these designs from 0.0914 to

0.409 N mm for the lowest finding and from 0.91434 to

0.409 N mm for the highest. For a uni-U-loop palatal

bar, the near-tooth loop moment increase was more

than the near-mid palatal loop.

Adding helices. The findings showed an increase from

0.0914 to 0.426 N mm at 0.1 mm of displacement.

The highest moment increased from 0.91434 to

4.2647 N mm.

Increasing the straight wire. In these designs, the

moment increased from 0.0914 N mm for the

traditional design to 0.669 N mm for the parallel wire

I and 0.782 N mm for the parallel wire II at the lowest

findings. The highest moment raised from 0.91434 N mm

in the traditional design to 6.6869 N mm in the parallel

wire I and 7.8159 N mm in the parallel wire II.

Figure 3 Near-tooth uni-U palatal bar (Model 5)

Figure 4 (A) Uni-helix palatal bar (Model 6); (B) double-helix palatal bar (Model 7)

200 Geramy and Etezadi Scientific Section JO September 2013

Adding ‘R’ loop(s). In this phase, adding loops increased

the lowest moment from 0.0914 to 0.544 N mm and the

highest from 0.91434 to 5.44 N mm.

Helix/helices in reverse action. Incorporating one or

two helix (helices) in a palatal bar resulted in a raising of

the moment from 0.0914 to 0.85 N mm at the 0.1 mm

displacement and from 0.91434 to 8.5039 N mm at

1 mm displacement.

Energy

The same increasing pattern was shown in a design

between 0.1 mm of and 1.0 mm of activation.

Adding U-loop(s). Adding loops decreased the energy

of the system. The findings were between 6.1361027 mJ

(triple-U) and 2.6361026 mJ (uni-U) at 0.1 mm of

displacement and reached 6.1461025 mJ (triple-U) and

2.6361024 mJ (uni-U) at 1 mm of displacement.

Adding helices. Incorporating one or two helices in a

palatal bar resulted in a reduction in energy from4.4561027 to 5.6661027 mJ at the lowest value and

from 4.4561025 to 5.6761025 mJ at the highest.

Increasing straight wire. The findings of two designs

were between 6.1861026 and 6.161024 mJ in the parallel

wire I and between 6.6e61026 and 6.6861024 mJ in the

parallel wire II. These findings were between 4.961026

and 4.961024 mJ in the traditional design.

Adding ‘R’ loop(s). Moving from uni-R loop TPA to a

double-R loop TPA dropped the energy findings. The

results were between 8.8261027 and 2.2261027 mJ in

0.1 mm of displacement and between 8.861025 and

2.2261027 mJ at their highest findings.

Helices in reverse action. Increasing the helices caused

a drop in energy in reverse action. Findings were

between 4.8361027 mJ (double-helix rev. act.) and

Figure 5 (A) Parallel wire I palatal bar (Model 8); (B) parallel wire II palatal bar (Model 9)

Figure 6 (A) Uni-R loop palatal bar (Model 10); (B) double R loop palatal bar (Model 11)

JO September 2013 Scientific Section Unilateral molar rotation correction by palatalbar 201

1.0661026 mJ (uni-helix rev. act.) for their lowest

values and between 4.8361025 mJ (double-helix rev.

act.) and 1.0661024 mJ (uni-helix rev. act.) for the

highest.

The parallel wire II and double ‘R’ loop designs were

chosen based on their force graphs. (Figure 8) The

stages were repeated for the moments. (Figure 9)

According to this screening process, the double ‘R’ loop

was chosen followed by a double helix. Reviewing

energy findings was the final stage of assessment in this

system (Figure 10) The highest energy finding was

traced among the various designs in the same process

as for force and moment. The parallel wire II was

associated with the highest energy, followed by the

traditional design.

According to our optimization goal, a design with the

highest energy, highest moment and lowest mesializing

force was required. This was the parallel wire II design

(Figure 5B).

Discussion

Unilateral rotations of the maxillary molar teeth are

often seen in the dental arches before treatment and

reaching an acceptable occlusion is not possible without

their correction. A number of methods exist to achieve

this, including use of the archwire, elastomerics, a

headgear inner-bow, TPA, quad helix or a heavy labialarch.20 A TPA is one of the most practical appliances for

this goal; however; it is most favourable when used in

symmetrical cases. In a unilateral situation, unwanted

mesio/distal forces are produced to counteract the

moment in the system, which will induce a mesial

movement in the molar whilst derotating. This sagittal

Figure 8 Mesio-distal force produced by uni-lateral activation of the TPA

Figure 7 (A) Uni-helix reverse action palatal bar (Model 12); (B) double helix reverse action palatal bar (Model 13)

202 Geramy and Etezadi Scientific Section JO September 2013

movement may not be appropriate for many cases,

especially when anchorage is critical. It is clear, there-

fore, that the presence of such a force component in

treating a unilateral rotated molar is not ignorable and

should be considered when planning treatment

mechanics.

In this way, the present study was designed to optimize

unilateral molar rotational correction when using a

TPA. Optimization is defined as a process or methodol-

ogy as fully perfect, functional or effective as possible,

requiring a complete knowledge of the involved process,

definition of a goal and consideration of the contem-

porary situation and the defined ideal one. Factors for

optimization are usually selected based on preliminary

experiments and prior knowledge from the literature.21

In this way, optimizing unilateral molar rotations was

done with regard to five different and separate paths:

incorporating U-loop(s), ‘R’ loop(s) or helices, con-

sidering a reverse action of helices and adding straight

wire to the design. In each group of analyses, the nearest

design to the desired one was selected for comparison

with a simple palatal bar. The highest energy finding in

Figure 9 Moment produced by uni-lateral activation of the TPA

Figure 10 Energy produced by the uni-lateral activation of the TPA

JO September 2013 Scientific Section Unilateral molar rotation correction by palatalbar 203

conjunction with the highest moment in the system when

combined with the lowest mesializing force was con-

sidered as the optimized design.

Energy is the ability to do work and these two

quantities are considered to be equal to each other; thus,

higher energy shows more ability to do work. Under the

same activation, different designs can show various

energy levels based on their configurations, with the

highest one considered to be favourable. When choosing

the desirable energy level, we continued towards finding

the highest moment and the lowest mesializing force.

Moment is a free vector tending to rotate bodies.22,23

The higher the moment, the higher the ability of

rotation correction will be. This is a desirable action.

In the process of looking for the minimum mesializing

force, the results of each group were compared with the

traditional palatal bar. In all groups, findings were lower

than the traditional design, necessitating an additional

step to determine the lowest finding among all. The

lowest finding at each group was selected for the

comparison and finally the lowest values were selected

as the optimized ones.

Parallel wire II and double ‘R’ loop designs were

chosen based on their mesio-distal force graphs

(Figure 8). The stages were repeated for the moments

(Figure 9). According to this screening process, the

double ‘R’ loop was chosen, followed by a double helix.

Reviewing energy findings was the final stage of the

assessment of this system (Figure 10). The highest

energy finding was traced among the various designs

using the same process as for force and moment. The

parallel wire II was the highest, followed by the

Table 1 Numeric data for different TPA designs.

Activation Simple pal.bar Uni-U pal.bar Double-U pal.bar Triple-U pal.bar Uni-U near tooth Uni-helix pal.bar

Force (N) 0.1 8.0661022 6.9361022 5.6461022 4.6061022 6.9261022 6.0361022

0.25 0.2016 0.1734 0.14096 0.11497 0.17309 0.15073

0.5 0.4032 0.34679 0.28193 0.22993 0.34618 0.30146

1 0.8064 0.69357 0.56385 0.45987 0.69236 0.60292

Moment (N mm) 0.1 9.1461022 2.6861021 3.5461021 4.0961021 3.3061021 4.2661021

0.25 0.22859 0.6692 0.88606 1.0222 0.82519 1.0662

0.5 0.45717 1.3383 1.7721 2.0445 1.6504 2.1324

1 0.91434 2.6766 3.5442 4.0889 3.3008 4.2647

Energy (mJ) 0.1 4.9061026 2.6361026 1.4461026 6.1361027 2.6361027 5.6661027

0.25 3.0661025 1.6561025 8.9861026 3.8361026 1.6461025 3.5461026

0.5 1.2361024 6.5861025 3.5961025 1.5361025 6.5761025 1.4261025

1 4.9061024 2.6361024 1.4461024 6.1461025 2.6361024 5.6761025

Table 2 Numeric data for different TPA designs.

Activation

Simple

pal.bar

Double

helix

Parallel

wire I

Parallel wire

II (optimized

model) Uni-R loop

Double-

R loop

Uni-helix

rev. act.*

Double helix

rev. act.{

0.1 8.0661022 5.9361022 4.0361022 3.7061022 4.1561022 3.5861022 6.6961022 6.4161022

Force (N) 0.25 0.2016 0.14827 0.10066 9.2661022 0.10478 8.9661022 0.16721 0.16027

0.5 0.4032 0.29654 0.20133 0.18524 0.20756 0.17916 0.33442 0.32055

1 0.8064 0.59308 0.40265 0.37048 0.41513 0.35833 0.66882 0.6411

0.1 9.1461022 3.9661021 6.6961021 7.8261021 5.4461021 2.7061021 8.5061021 6.8261021

Moment (N mm) 0.25 0.22859 0.99946 1.6717 1.956100 1.3608 6.7461021 2.126 1.7046

0.5 0.45717 1.9789 3.3435 3.9079 2.7216 1.3486 4.2519 3.4091

1 0.91434 3.9578 6.6869 7.8159 5.4432 2.6972 8.5039 6.8183

0.1 4.9061026 4.4561027 6.1861026 6.6061026 8.8261027 2.2261027 1.0661026 4.8361027

Energy (mJ) 0.25 3.0661025 2.7861026 3.8661025 4.1361025 5.5161026 1.3961026 6.6461026 3.1661026

0.5 1.2361024 1.1161025 1.5561024 1.6561024 2.2061025 5.5561026 2.6661025 1.2161025

1 4.9061024 4.4561025 6.1861024 6.6061024 8.8261025 2.2261026 1.0661024 4.8361025

*Uni-helix reverse action palatal bar.{Double helix reverse action palatal bar.

204 Geramy and Etezadi Scientific Section JO September 2013

traditional design; therefore, the design with the highest

energy, highest moment and lowest mesio-distal force

was associated with the parallel wire II design

(Figure 5B).The next stage will be to test the new design in

unilateral molar rotation corrections clinically to

observe the results. Experimental studies are used to

decrease the number of possible methods considered in

clinical trial research (which is an expensive time-

consuming procedure). This finite element method study

has provided a useful potential clinical TPA design

modification for unilateral molar rotation correctionthat should be further investigated in a clinical trial.

Conclusion

Based on this finite element method study, unilateralmolar rotation correction was optimized using a parallel

wire II design modification of the TPA through adding

straight wire. The mesial component of the force was

minimized, the moment to correct rotation was max-

imized and the energy of the system was increased.

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