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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: [email protected]# 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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