Download three dimensional numerical modeling of orthodontic tooth movements

European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS 2000
Barcelona, 11-14 September 2000
 ECCOMAS
THREE DIMENSIONAL NUMERICAL MODELING OF
ORTHODONTIC TOOTH MOVEMENTS
Christoph P. Bourauel
Department of Orthodontics
Department of Orthodontics, Rheinische Friedrich-Wilhelms-Universität Bonn
Welschnonnenstr. 17, 53111 Bonn, Germany
e-mail: [email protected], Web page: http://wodan.meb.uni-bonn.de/
Key words: Orthodontics, Tooth Movement, Bone Remodeling, Simulation, FEM.
Abstract. Orthodontic tooth movements are based on the ability of bone to react on
mechanical stresses or strains with apposition and resorption of alveolar bone. It was the aim
of this study to develop a numerical model that is able to describe the alveolar bone
remodeling processes using basic principles of current bone remodeling theories. An
idealized three dimensional Finite Element model of an upper canine in the shape of an
elliptical paraboloid was automatically generated and numerically processed by an FE
programme. The dimensions of the idealized tooth correspond to the diameters of a human
tooth at the alveolar crest and the root length. Stresses and strains generated by an
orthodontic force system in the tooth and tooth supporting structures were determined in
material nonlinear calculations. The full strain tensor first in the PDL, and second in the
alveolar bone was used to calculate a remodeling signal for the nodes of the alveolar bone
elements and to automatically generate an updated FE model. By repeating this process
orthodontic tooth movement was simulated incrementally with individual steps being in the
range of some 10µm. The model was verified using the clinical results of five patients, being
treated with calibrated orthodontic devices made of nickel titanium alloys. Numerical results
could reproduce the clinical tooth movements with deviations of 10-20 per cent if the strains
in the PDL were used to calculate orthodontic bone remodeling. Calculating the remodeling
signal from the strains in the alveolar bone resulted in significantly larger deviations (30 per
cent). Based on this model, the mechanical key stimulus initiating orthodontic tooth movement
seem to be the normal and the shear strains within the periodontal ligament.
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Christoph P. Bourauel
1
INTRODUCTION
In orthodontics teeth are moved within the alveolar bone to correct malocclusions by
means of specialized orthodontic devices. An example for a certain treatment phase, a socalled canine retraction, is shown in Figure 1. Due to the distribution of mechanical stresses
and strains in the periodontal ligament (PDL) and the alveolar bone, a special kind of bone
remodeling process is initiated with bone resorption in the dircetion of tooth movement and
bone apposition behind the tooth (see Figure 1). By this means teeth may be translated by up
to 5 mm or rotated by several 10 degrees. Frequently planning of orthodontic treatments is
rather complex and the orthodontists demand computer aided support for their treatment
planning. The kernel of such a ‘Computer Aided Orthodontics’ system should be a numerical
model for the simulation of orthodontic tooth movements.
Up to now, in orthodontics only analytical and FE analyses of the initial or physiological
tooth mobility were performede.g. 1-3. But from these investigations it is hardly possible to
exactly calculate the orthodontic tooth movement following the bone remodeling processes. It
is recommended to develop a mathematical model of the orthodontic tooth movement in
analogy to the so-called theories of ‘bone remodeling’ presented in the last decade4-6. The
theoretical basis of these numerical models is Wolff’s law7. According to this a direct
relationship exists between mechanical loading of bone and its architecture. On the other
hand, bone shows the ability to adapt to a change in the external loads. Accordingly, bone has
an optimal structure in the case of mechanical equilibrium. In the case of a change in the loads
bone will be able to remodel until an optimal configuration, adapted to the new state of
equilibrium, is achieved again.
In the course of this study it was investigated whether a connection between theories of
bone remodeling and the finite element models of tooth and tooth supporting structures could
be developed to simulate orthodontic tooth movements. Current bone remodeling theories
derive the key stimulus to start remodeling processes (the remodeling law) from strains or
stresses in the bone. Values of 300-3000 µstrain are required to initiate bone remodeling and
adaption8. However in orthodontics these high values are not reached in the bone but merely
in the PDL. Thus it should be investigated whether the orthodontic tooth movement is
controlled predominantly by the mechanical loading of the PDL or by strains in the alveolar
bone.
Bone
resorption
Bone
apposition
Forces,
Torques
Translation Rotation
Figure 1: Initial (left) and final (middle) intraoral situation of a canine retraction. The tooth is moved through the
alveolar bone by bone remodeling processes induced by the application of orthodontic force systems (right).
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Christoph P. Bourauel
2
MATERIAL AND METHODS
It is to be noted that the distinct components of the system orthodontic device - tooth periodontal ligament - alveolar bone have different mechanical properties and that a tooth
with its individual geometry and bearing in the alveolus is a complex three dimensional
system. This complex geometry represents a great problem, if a numerical model should be
generated automatically and fed into a finite element programme. In a first step an appropriate
approximation of a tooth’s root and adjacent structures was determined. In a second step this
idealized model was used to calculate the bone remodeling around the root of the tooth and
the movement of the tooth and the adjacent structures within the alveolar bone.
2.1 Idealization of the tooth geometry and material parameters
The geometry of a tooth was described by aid of its total length and root length with respect
to the tooth’s long axis as well as the oro vestibular and the mesio distal dimensions of crown
and root at the location of the alveolar crest (indicated by the boundary lines in figure 2a).
These parameters were used to find an approximation of the human canine as well as for the
surrounding bone. In figure 2b a FEM model is shown of a human upper canine with a root
length of 19.5 mm as well as an oro vestibular dimension of 7.6 mm and a mesio distal
thickness of 6.7 mm. This three dimensional FE model consisted of 3800 elements in total.
This model was too complex and too intense in calculation time and could not be generated
automatically for the simulation of orthodontic tooth movements. Consequently, an idealized
tooth model was derived using the geometrical parameters of the canine shown in figure 2c.
The model was generated in the shape of an elliptical paraboloid. The height of this
paraboloid coincides with the rooth length from the apex to the alveolar crest and the long and
short axis of the ellipse are equivalent to the oro vestibular and the mesio distal dimensions at
the alveolar crest respectively. The PDL was generated with a uniform thickness of 0.2 mm
and the bone had an elliptical cross section with the long axis in the mesio distal direction.
a
The model is characterized by a reduced number of elements, having in total 1500
Figure 2: Real tooth geometry, realistic FE-model and idealized representation of a canine in the shape of an
elliptical paraboloid.
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Christoph P. Bourauel
isoparametric 8 noded solid elements. The
Material
[GPa]
PDL
orthodontic force system was applied and
20
Bilinear elastic
FE calculations were performed using the Dentine
Enamel
80
E
0.05 [MPa]
1
FEM package COSMOS/M (Ver. 2.0,
Cortical bone
20
E2 0.22 [MPa]
Windows NT).
ε12 7.5 %
Spongious
bone
3
Isotropic, homogeneous and linear
0.3 for all structures
material behaviour had to be assumed for Poisson´s ratio
dentine, enamel, cortical and spongious Table 1: Material parameters used in this study.
bone, whereas a bilinear approximation
was used for Young’s modulus of the PDL. All values used are given in table 1.
2.2 Basic principle of the model
The complete model developed
for simulation of orthodontic tooth
movements with an idealized bone
segment, the PDL, tooth and the
bracket element is shown in figure 3.
The basic principle corresponds to
the remodeling programmes of other
authors. An important difference is to
be seen in the fact that not only the
change in the bone’s morphology is Figure 3: Principle of the simulation of the orthodontic bone
taken into account but the tooth remodeling.
including its alveolus is moving over
a large distance through the bone. The geometry of the tooth’s root and of the alveolus have to
be kept constant during the movement.
The application of an orthodontic force system results in deformations of the PDL and the
alveolar bone of up to 20 per cent and 0.001 per cent, respectively. Taking either the complete
strain tensor in the PDL or in the bone, the orthodontic bone remodeling can be calculated, in
the following way: The simulation starts with the automatic generation of a FE mesh
representing an initial clinical situation. The application of an orthodontic force system results
in strain distributions in the PDL and the alveolar bone, delivering the displacements of the
alveolar nodes and the amount of tooth movement aid of an orthodontic remodeling law. With
this information, an updated FE-mesh is generated and the calculation starts again with the
application of the orthodontic force system. This principle was implemented in an external
loop to the FE package and in this way, orthodontic tooth movements are calculated in
increments until the simulations are stopped at a predefined final position9.
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Christoph P. Bourauel
4
RESULTS
4.1 Basic tooth movements
The results of the simulation of an uncontrolled tipping of a canine is shown in figure 4.
The force system constisted of 1 N of distalizing force and 4.5 Nmm of derotating
momentum. The calculations were stopped at a distal tipping of the bracket of 20°. Figure 4b
illustrates the calculated movement of the tooth with its alveolus through the bony structure
with respect to the initial situation (4a). Figure 4c shows a comparison of the models together
with a calculation based on the position of the centre of resistance to illustrate the expected
tooth movement. The model calculating the tooth movement based on the strains in the PDL
predicts the final position in a good accordance with the expected movements, whereas the
model that is based on the strains in the bone delivers significant side effects that are not in
accordance with clinical experience. Similar results were obtained for different further
movements that have been calculated9.
20
PDL
15
Bone
10
CR
5
0
-5
-10
Tx
Ty
Tz
Rx
Ry
Rz
c
Figure 4: Simulation of an uncontrolled tipping. a) initial FE model, b) FE model at termination of calculation,
c) comparison of the models developed.
4.2 Clinical tooth movements
The simulations of the basic tooth movements indicate that the model calculating the bone
remodeling based on the strains in the bone delivers unreasonable results. Consequently, a
validation of the simulation model had to be done according to the results of clinical cases to
decide whether this model assumption had to be rejected. The validation was done as follows:
Five patients have been selected and isolated canine retractions have been performed using
superelastic nickel titanium alloy retraction springs to ensure nearly constant force systems10.
Plaster casts of the initial and final intraoral situation as well as up to three casts of
intermediate situations were measured with a laser scanner and tooth movements and
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Christoph P. Bourauel
positions were determined using a surfacesurface-matching algorithm. The T-loops inital
4 months
used clinically were calibrated with these
informations
in
the
Orthodontic
Measurement and Simulation System
OMSS11. Together with the informations on
the tooth dimensions, simulation of tooth 2 months
6 months
movement was performed with both model
assumptions. As an example figure 5 depicts
the animated result of the simulation for one
patient. Space closure was achieved by a
basically tipping movement. In addition a
large amount of rotation around the tooth’s Figure 5: Animation of a simulated tooth movement.
long axis (RZ) occurred. The calculation
based on the strains in the PDL could reproduce the clinical tooth movement, whereas the
bone-based model gave significant deviations especially in the components TY and RZ (see
table 2). Similar results were obtained for four further patients.
Tooth Movement
Clinically
PDL
Bone
Force system
Tx [mm]
-7.3
-7.3
-7.3
Fx [N]
-1.0
Ty [mm]
-1.0
-0.8
0.5
Fy [N]
0.3
Tz [mm]
1.3
1.7
1.8
Fz [N]
0.1
Rx [°]
6
10
9
Mx [Nmm]
2.0
Ry [°]
-8
-13
-14
My [Nmm]
8.0
Rz [°]
-21
-19
4
Mz [Nmm]
3.0
Table 2: Results for the tooth movements calculated with the different models.
5
DISCUSSION
In total the clinical results of five canine retractions have been used for the verification of
the numerical model, giving a good accordance of numerical with clinical data with errors of
approximately 20 per cent9. Using the strains in the alveolar bone as a key stimulus to
calculate orthodontic bone remodeling results in significantly larger deviations (more than 30
per cent) and in unreasonable side effects. Consequently, the mechanical modeling of
orthodontic tooth movement should be done using the full strain tensor in the PDL rather than
the strains in the alveolar bone.
The simulation models presented allow a prediction of typical kinds of clinical tooth
movements. It is not possible to give a statement about the duration of a certain treatment step
because of individual differences from one patient to the other are too large for this.
Furthermore, nonlinearities have not yet been considered. This especially holds for the
inclusion of threshold values defining a ‘dead zone’ and a regime where strains are so high
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Christoph P. Bourauel
that orthodontic tooth movement may be constant and finally inhibited. Consequently, future
developments have to put an emphasis upon the functional description of the orthodontic
‘remodeling law’.
REFERENCES
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[7] J. Wolff, Das Gesetz der Transformation der Knochen, Hirschwald, (1892).
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[11] C. Bourauel, D. Drescher and M. Thier, "An experimental setup for the simulation of
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