Posted at the Institutional Resources for Unique Collection and Academic Archives at Tokyo Dental College,
Available from http://ir.tdc.ac.jp/
Title
Stress distribution in the mini-screw and alveolar
bone during orthodontic treatment : a finite
element study analysis
Author(s) 黒田, 俊太郎
Journal , (): -
URL http://hdl.handle.net/10130/3426
Right
1
Stress distribution in the mini-screw and alveolar bone during
orthodontic treatment: a finite element study analysis
Shuntaro Kuroda
Tokyo Dental College
Department of Orthodontics
2
Abstract:
The purpose of this study was to investigate the reason of the high failure rate of the
mini-screw during orthodontic treatment. We hypothesized that decreased the mini-
screw length ratio outside the bone to inside the bone (outside/inside length ratio),
equivalent to the crown–root ratio of the tooth, would lead to increased stability of the
mini-screw against lateral loads when assessed using the finite element analysis method.
We analyzed stress distribution of the mini-screw in the cortical and trabecular bone and
the von Mises stress level when a 2 N force was applied to the head of four mini-screws
that were 6, 8, 10 and 12 mm in length. The direction of the force was perpendicular to
major axis of the screws. The stress level at the cortical bone increased in proportion to
the length of the mini-screw outside the bone. The length of the mini-screw inside the
bone did not affect the stress level on the cortical bone. We demonstrated that if cortical
bone thickness is 1.2 mm, the von Mises stress will not surpass the yield stress of each
material, regardless of the outside/inside length ratio of the mini-screw.
3
Introduction:
Use of the mini-screw has become increasingly prevalent in the past decade because
they confer a number of advantages when compared with conventional intra- and extra-
oral anchorage reinforcement. Despite the many studies on the mini-screw, few standard
evaluation methods have been established, and reasons for the varying success of their
use remain unclear1. In addition, few reports have been published on the structural
mechanics of the mini-screw in relation to bone. The study we conducted using an
established finite element analysis method contributes to the knowledge base and may
be used to guide optimal clinical usage of the mini-screw in the field of orthodontics.
Most published papers report the success rate for the mini-screw to be approximately
80%, including cases where the mini-screw was mobile but effectively served as an
anchorage during the treatment period2-6. However, the cause of mini-screw failure
often remains unexplained. Mini-screw failure seems to be associated with the
conditions for insertion and, more importantly, the force of insertion in terms of
magnitude and direction. Many studies and clinical reports have proposed that forces
should remain below 200 gf2,7-10. In one report, application of a 400 gf force resulted in
a significant increase in the failure rate11. Regarding the direction, some researchers
state that application of a force in the lateral direction or in a direction that causes
twisting or pulling of the mini-screw should be avoided12. Insertion conditions are
determined by the magnitude and direction of the force and by location of insertion, and
they include the condition of the mini-screw and that of the insertion site within the oral
cavity. The latter depends closely on the treatment plan and the force system. According
to anatomical requirements, one must consider the location and depth of insertion to
avoid injury to neighbouring structures such as the roots of adjacent teeth, nerves, blood
4
vessels and maxillary sinus.
The cortical bone thickness (CBT) at the location of insertion is an important factor for
promoting initial stability when the mini-screw is inserted13. The conditions of the mini-
screw itself primarily relate to the diameter and length of the mini-screw, although they
may also differ in material or design.
Although the impact on success rate is known for many of the parameters associated
with the use of the mini-screw, few reports have been published concerning the stress
distribution of the mini-screw in the bone complex. Moreover, stress distribution in
cortical bone or trabecular bone or at the cortical–trabecular bone interface varies for
individual cases. Stress distribution in the mini-screw also seems to vary depending on
the ratio of the length of the mini-screw outside the bone to that inside the bone
(outside/inside length ratio). The first study on mini-screw load transfer using finite
element analysis was conducted in 200414. They studied stress distribution after
application of a 50-cN load on a mini-screw inserted into bone and reported that the
stress was mostly distributed in the cortical bone.
The present study using finite element analysis was based on the assumption that
the mini-screw would become more stable against lateral load with a decrease in the
outside/inside length ratio.
5
Materials and methods:
On the computer, simulation models on mini-screw, cortical bone and trabecular bone
were created as shown below.
First, models of the mini-screw, cortical bone and trabecular bone were created using
a three-dimensional (3D) structure analysis program (Finite Element Stress Analysis
System TRI/3D-BON, FEM, Ratoc System Engineering Co., Ltd., Tokyo). Mini-screw
and bone parameters were set as follows (Fig. 1):
1. Cortical bone: X, 12,000 μm; Y, 12,000 μm; Z, 12,000 μm (coordinates X 53-
203, Y 53-203, Z 5-155)
2. Trabecular bone: X, 9,600 μm; Y, 9,600 μm; Z, 9,600 μm (coordinates X 68-
188, Y 68-188, Z 20-140)
3. Cortical bone thickness: 1,200 μm (15 pixel × 80 μm, 1 pixel = 80 μm)15,16
4. Mini-screw diameter: 1,440 μm
5. Mini-screw length (4-8 = 12 mm, 4-4 = 8 mm, 2-8 = 10 mm, 2-4 = 6 mm)
6. Mini-screw shape: tip protruding at the site of insertion
The mini-screw was implanted vertically into the simulated cortical and trabecular bone
model. At that time, the model with a 4 mm portion of the mini-screw exposed within
the bone and an 8 mm portion buried from the bone was tentatively called the “4-8
model.” Four models (4-4, 4-8, 2-4 and 2-8 models) were created in this way, followed
by finite element analysis as described below.
Raw bone material was defined by computer after referring to the data on bones in the
human body. The mini-screw raw material was assumed to be titanium. Young’s
modulus and the Poisson coefficient for each material were set as shown in Table 1.
The quality of the bone tissue was based on that of healthy adult orthodontic patients
6
(18–35 years old).
Borderline conditions were set as shown in Fig. 2 and Table 2, immobilizing planes 1,
2, 3 and 4. Then the load on the mini-screw was programed. Concentrated loading was
performed, with a 2 N load applied in the X direction (+). Under conditions 4-4 and 4-8,
loading was performed at a point with the following coordinates: X, 119; Y, 128; Z, 205.
Under conditions 2-4 and 2-8, loading was performed at a point with the following
coordinates: X, 119; Y, 128; Z, 180.
On a cross-sectional view of each model, a 2N force (tractive force) directed to the right
was applied vertically to the mini-screw at point A of the mini-screw head (Fig. 3). The
resulting stress distribution in the mini-screw, cortical bone and trabecular bone in each
model was subjected to finite element analysis.
The mini-screw form was designed as simple as possible so that the data from this study
could be applied to most of all the commercially available mini-screws. The screw
thread, known to have little impact on cortical bone strain17, was not adopted as a
variable in this finite element study model.
For each stress value point, we used TRI/3D-FEM to calculate the von Mises stress. The
stress level was compared among several points selected.
Fig. 3 shows the location of each point. The figures showing mean principal stress
(figures colored with red and blue; cross-sectional and top surface views) were prepared
in the following way:
1. A contour figure of stress was prepared using the analysis result-presenting
function of TRI/3D-FEM.
2. The cross section and the cortical bone surface were presented using the 3-D
display function of TRI/3D-BON. The color bar for the contour figure
7
proportionally represents stress levels from minimum (gradation 1: blue) to
maximum (gradation 255: red).
A contour figure was prepared with Excel 2007 as follows:
a. From the calculated stress data for the entire region, the stress level in a
selected area was calculated.
b. Von Mises stress was calculated from X, Y, Z and Toct. data.
c. A pivot table was prepared by extracting data on three variables (X, Y and von
Mises stress).
d. A contour figure was prepared from the pivot table with the use of the contour
figure preparing function of Excel 2007.
The stress distribution in the mini-screw was graphically represented in the
following way:
1. Steps (1) and (2) were performed as described above for preparation of a figure
of stress distribution in bone without the mini-screw.
2. A figure of the stress distribution in the mini-screw was prepared individually.
The stress distribution in bone without a mini-screw was graphically represented as
follows:
1. A contour figure of stress was prepared using the analysis result-presenting
function of TRI/3D-FEM.
2. The mini-screw, cortical bone and trabecular bone were isolated with the use of
inter-image calculation and 3D-display functions of TRI/3D-BON.
3. A figure of stress distribution was prepared by combining data other than those
related to the mini-screw.
8
Results:
Table 3 summarizes the von Mises stress levels at the main points in the cross-sections.
In models 4-4, 4-8, 2-4, and 2-8, von Mises stress A was the largest. Also, the stress
both tensile side and compression side in the mini-screw are distributed symmetrically.
In models 4-4 and 2-4, stress is distributed over the entire mini-screw, without a single
point of excessive stress level. When stress in DS and DB are compared, we see that the
stress in the cortical bone is decreased to 50%–60% of the stress in the mini-screw.
Similarly, stress in GB has decreased to 4% of the stress in GS.
In Figure 4A and Figure 6, we can see that stress is distributed along the interface
between cortical bone and trabecular bone, as well as along parts distal from the mini-
screw. Figure 5 shows that at the cortical bone surface level, the rate of stress decreasing
on the inside of the mini-screw is more lenient compared with that inside the bone. In
addition, the von Mises stress at point E is lowest through all the models.
Fig. 7 illustrates the stress distribution on the contact surface between the mini-screw
and the bone. In models 4-8 and 4-4, more areas are coloured red, indicating greater
stress distribution within the mini-screw. The red areas were distributed inside the
cortical bone and above the level of the surface, whereas only a small amount of stress
was observed in the trabecular bone. Notably, in model 4-4, high stress was distributed
across the mini-screw.
Fig. 8 illustrates the stress distribution on the contact surface between the bone and the
mini-screw. Stress was distributed more extensively at the cortical bone surface level in
models 4-8 and 4-4 than in models 2-4 and 2-8. The number of red-coloured areas was
greater in the marginal area in models 4-8 and 4-4 than in the marginal area in models
2-4 and 2-8. However, the distribution of yellow-coloured areas in the bone wall was
9
more extensive in model 4-4 than in the other models.
10
Discussion:
Creekmore et al.18 (1983) proposed the concept of skeletal anchorage, following
which temporary anchorage devices (TADs) of various designs were developed for
anchorage in adult orthodontic treatment. Following a report by Kanomi19 (1997), TADs
began to be used extensively in orthodontic treatment. TADs known as mini-screws are
advantageous in that they do not require complex manipulation or surgery and are easy
to remove once their purpose is fulfilled. In addition, loading is possible immediately
after surgical insertion of a mini-screw.
The present study makes some assumptions; it is based on adult orthodontics wherein
the first premolar has been extracted and a mini-screw is embedded in the alveolar bone
between the maxillary second premolar and first molar for using as anchorage during
canine distal drive and anterior retraction. We then performed a finite element analysis
of the stress distribution in the mini-screw, the alveolar bone (cortical bone and
trabecular bone) and mini-screw/bone complex. When the models for analysis were
designed, care was taken so that the advantages of finite element analysis could be
utilized adequately, while avoiding loss of general versatility of the models, which
might be caused by excess concern with the details. The number of variables adopted
was minimized. Therefore, the model was a simple cubic shape, and the surface of the
model resembled that of a cortical bone. The models were reproduced assuming that the
alveolar bone, which had the alveolar socket (cortical part), and the mini-screw were
embedded between the root of the tooth.
As previously mentioned, we set 4 conditions for the finite element analysis model (4-4,
4-8, 2-4, and 2-8) designed to assess the stress distribution in the bone and mini-screw
over 4 different ratios. These corresponded to ratios of the length of mini-screw outside
11
the bone to that inside the bone (outside/inside ratio) of 1:1, 1:2, 1:2 (changing the
length outside the bone), and 1:4. If the length outside the bone was increased, the stress
obviously concentrated on the neck of the mini-screw. However, it is clinically valid to
analyze the consequences of changing the length inside the bone. By looking at the
stress level of each analyzed point in this study, there is no significant difference in
stress between each point. Therefore, it is reasonable to expect that the results will be
similar even if the standard of length outside or inside the bone is increased.
On a cross-sectional view, stress inside the mini-screw was approximately symmetrical
on both sides. In all models tested, the margin close to the cortical bone surface was the
densest red area. The contour figure (Fig. 5) shows that this area was exposed to the
highest stress. Stress was extensively distributed in the part of the mini-screw inserted
within the cortical bone. The part inserted within the trabecular bone is shown as a dull
yellow area that indicates a rapid decrease in stress. These findings indicate that the
mini-screw is supported by cortical bone. The stress distributed in the portion of the
mini-screw outside the bone was higher than that in the portion of the mini-screw within
the trabecular bone. This difference may be because part of the mini-screw outside the
bone is closer to the action point or because the elastic modulus of the trabecular bone is
lower than that of the cortical bone, thus failing to show a high stress level. Furthermore,
when distribution of the color within the mini-screw was analyzed, red-colored areas
were more in models 4-8 and 4-4 than in models 2-4 and 2-8. This finding suggests that
the stress within the mini-screw increases with an increase in the length of the mini-
screw outside the bone. These findings also apply to stress distribution within the mini-
screw, which is graphically represented in Fig. 7. In Fig. 8 for simulation of bone
excluding the mini-screw, stress decreased sharply as the distance from the margin
12
increased. In model 4-4, the stress in the bone was distributed in a manner surrounding
the mini-screw; however, the stress level was low and did not seem to be associated
with any major problem in the bone or mini-screw.
On a cross-sectional view, stress was distributed in a considerably wide fashion along
the interface between the cortical bone and trabecular bone. While stress distribution to
the trabecular bone is low, this finding indicates that trabecular bone is indispensable in
the analysis of stress distribution.
At the margin of the mini-screw closer to the cortical bone surface level, the mini-screw
side was represented by point DS and the bone side was represented by point DB. With
an increase in the length of the mini-screw outside the bone, von Mises stress at point
DS increased. This change was determined only by the length of the mini-screw outside
the bone. Sufficient mechanical interlocking between the mini-screw and cortical bone
has the greatest effect on mini-screw stability7,20,21. However, insertion into thick
cortical bone or high-density hard tissue can also lead to displacement, while the
drilling before insertion can cause overheating8,12.
The von Mises stress at point DB seems to reflect the load on the surrounding bone, as
determined by the length of the mini-screw outside the bone or the outside/inside length
ratio. With increasing distance from the mini-screw, the percentage decrease in stress
within the cortical bone became considerably larger relative to that within the mini-
screw (Fig. 5). This finding indicates that the length of the mini-screw outside the bone
should be controlled for appropriate stress distribution in the mini-screw and that the
outside/inside length ratio should be controlled for appropriate stress distribution in the
bone.
In the present study, the mini-screw width and CBT were kept constant. Under these
13
conditions, the von Mises stress inside the mini-screw decreased and the mini-screw
became more stable as the length of the mini-screw outside the bone decreased. When
the length outside the bone was kept constant, the mini-screw was more stable with a
longer length inside the bone. Some studies suggest that a depth of insertion of at least
5–6 mm is required, and deeper placement is required at sites with poor bone
quality9,10,18.
The cross-sectional view of stress distribution at point E indicates that von Mises stress
is lower if the length of the mini-screw outside the bone decreased. Furthermore, as
indicated by Fig. 5, von Mises stress is lowest at the cortical bone surface level. This
stable point may be viewed as the center of rotation of the mini-screw.
According to Fig. 4A and Fig. 5, the stress distribution at point F and in the surrounding
region is approximately symmetrical to that at point D and in the surrounding region.
Point G serves as the point of contact between the mini-screw and the cortical–
trabecular bone interface. At this point, three different materials are combined, resulting
in stress distribution different from that at other points. At point G, the side closer to the
mini-screw is termed point GS. At point GS, von Mises stress is greater when the length
of the mini-screw outside the bone is 4 mm than when the length is 2 mm. Von Mises
stress at point GS also correlated with the stress at the margin of the mini-screw (point
D) closer to the cortical bone surface. If the stress at this point is high, it will be
distributed, probably resulting in high stress at point GS. If the length of the mini-screw
outside the bone is kept constant, von Mises stress at point GS increases with an
increase in the length of the mini-screw inside the bone. Therefore, it seems likely that
with an increase in the length inside the bone, there is a relative increase in the
outside/inside length ratio, thus resulting in greater von Mises stress at point GS von
14
Mises stress at point GB is only approximately 4% of the stress at point GS (Table 3,
Fig. 4B). This indicates that there is barely any stress distributed on the bone side at this
point.
In Fig. 4A, von Mises stress on the cortical bone surface increases as the distance to the
mini-screw decreases. However, at the cortical–trabecular bone interface, stress was
also distributed at points distant from the mini-screw. We assume that point E serves as
the center of mini-screw rotation. Under this assumption, point F and point G are on the
compression side. At points near the mini-screw, the compression stress is offset by the
tensile stress from point D, resulting in distribution of low compression stress. As the
distance from the mini-screw increases, the residual stress from point D is probably
distributed widely along the interface within the cortical bone. When the length of the
mini-screw inside the bone is 4 mm, the yellow area, indicating stress distribution,
spreads to the apex. However, von Mises stress at the apex (point L) was as low as 0.3–
0.5 MPa, which seems to have little impact on the bone and mini-screw.
Our results are consistent with the findings reported by Dalstra et al.14 (2004). When
force was applied perpendicular to the major axis of the mini-screw, the stress was
concentrated on the cortical bone. Von Mises stress in the cortical bone was >10 times
that in the trabecular bone.
Finite element analysis revealed that the factor affecting stress distribution varied
among individual points, that is, some points affected by the length of the mini-screw
outside the bone, some by the length inside the bone and some by the outside/inside
length ratio. However, this study presents with limitations. We hypothesized that a
smaller outside/inside ratio would lead to increased stability of the mini-screw against
lateral loads when assessed using the finite element analysis method. However, we
15
found that mini-screw stability was not affected by the outside/inside ratio. This is
because von Mises stress at each point in the models never exceeded the reported yield
stress, which is 200 MPa for cortical bone22 and 692 MPa for mini-screws made of
titanium alloy23. Therefore, under the setting tested, bone destruction, mini-screw
displacement and deformation may not have occurred during this study. Nevertheless,
we propose that it is not necessary to design the mini-screw such that the part inside the
bone is so long that it risks its placement close to the maxillary sinus, mandibular canal
or tooth root.
16
Conclusion:
1) Stress due to traction force around the mini-screw was concentrated on the cortical
bone (mini-screw neck) and decreased sharply with an increase in distance from the
mini-screw.
2) Stress at the cortical bone surface was approximately double that at the cortical–
trabecular bone interface.
3) Stress was affected largely by the length of the mini-screw outside the bone, with the
stress around the mini-screw increasing in proportion to its length.
4) The length of the mini-screw inside the bone had little influence on stress around the
mini-screw neck.
5) Under these finite element analysis conditions, a decrease in the outside/inside length
ratio of the mini-screw probably does not result in destruction of the bone or
displacement or deformation of the mini-screw; in addition, it may not affect stability.
17
Acknowledgements
The authors would like to thank Enago (www.enago.com) for their English language
review.
18
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22
Figure legends
Figure 1: Finite element model consisting the mini-screw and alveolar bone
Figure 2: Boundary conditions for the finite element model
Figure 3: Arrangement of points in model 4-8
Figure 4A: Cross-sectional images of stress distribution figures (stress at each point;
von Mises stress, colored stress distribution graph; mean principal stress)
Figure 4B: Stress value for each model
Figure 5: Contour figures for stress distribution at the surface level
Figure 6: Colored images of stress distribution at the cortical–trabecular bone interface,
when viewed from the top
Figure 7: Stress distribution at the mini-screw surface contacting the bone on the tensile
side
Figure 8: Stress distribution at the bone surface contacting the mini-screw on the tensile
side
23
Table 1: Properties of the constituent materials
Table 2: Borderline conditions of finite element study model
Table 3: Von Mises stress at each point
Figure.1 Finite element model consisting of mini-screw and alveolar bone
Figure 2: Name and position of each planes for the finite element model
Arrangement of points in model 4-8
DS: Mini-screw side at point D DB: Bone side at point D FS: Mini-screw side at point F FB: Bone side at point F GS: Mini-screw side at point G GB: Bone side at point G IS: Mini-screw side at point I IB: Bone side at point I Figure 3: Arrangement of points in model 4-8
4-8 4-4
2-4 2-8
Figure 4A: Cross-sectional images of stress distribution figures (stress at each
point; colored stress distribution graph; von Mises stress)
4-8 Von Mises equivalent stress DS: 22-0MPa
DB: 8-71MPa
FS: 22-0MPa
FB: 8-70MPa
GS: 8-28MPa
GB: 0-38MPa
IS: 8-28MPa
IB: 0-38MPa
4-4 Von Mises equivalent stress DS: 21-8MPa
DB: 8-84MPa
FS: 21-8MPa
FB: 8-84MPa
GS: 8-38MPa
GB: 0-37MPa
IS: 8-38MPa
IB: 0-37MPa
2-4 Von Mises equivalent stress DS: 10-80MPa
DB: 4-04MPa
FS: 10-80MPa
FB: 4-04MPa
GS: 4-48MPa
GB: 0-20MPa
IS: 4-48MPa
IB: 0-20MPa
2-8 Von Mises equivalent stress DS: 11-0MPa
DB: 4-87MPa
FS: 10-8MPa
FB: 4-87MPa
GS: 4-10MPa
GB: 0-21MPa
IS: 4-10MPa
IB: 0-21MPa
4-8 4-4
2-4 2-8
Figure 4B: Stress value for each model (von Mises stress)
100
12602468
1012141618202224
90 9610
210
811
412
012
613
213
814
415
015
616
216
8
22-24
20-22
18-20
16-18
14-16
12-14
10-12
8-10
6-8
4-6 100
12602468
1012141618202224
90 9710
411
111
812
513
213
914
6
153
160
167
22-24
20-22
18-20
16-18
14-16
12-14
10-12
8-10
6-8
4-6
2-4
100
1260
2
4
6
8
10
12
90 9710
411
111
812
513
213
914
615
316
0
167
10-12
8-10
6-8
4-6
2-4
0-2
100117
1340
2
4
6
8
10
12
90 9710
411
1
118
125
132
139
146
153
160
167
10-12
8-10
6-8
4-6
2-4
0-2
4-8 4-4
2-4 2-8
Figure 5: Contour figures for stress distribution (von Mises stress) at the surface
level (MPa)
4-8 4-4
2-4 2-8
Figure 6: Colored images of stress distribution at the cortical–trabecular bone
interface, when viewed from the top
4-8 4-4
2-4 2-8
Figure 7: Stress distribution at the mini-screw surface contacting the bone on the
tensile side
4-8 4-4
2-4 2-8
Figure 8: Stress distribution at the bone surface contacting the mini-screw on the
tensile
Material properties of constituent materials: Material Young’s modulus (MPa) Poisson’s ratio
mini-screw 110,000 0.35
cortical bone 14,000 0.30
trabecular bone 3,000 0.30
Table 1: Properties of the constituent materials
Plane X direction Y direction Z direction Xmin Xmax Ymin Ymax Zmin Zmax 1 Immobilized Immobilized Immobilized 53 53 53 203 5 155 2 Immobilized Immobilized Immobilized 53 203 53 53 5 155 3 Immobilized Immobilized Immobilized 203 203 53 203 5 155 4 Immobilized Immobilized Immobilized 53 203 203 203 5 155
Table 2: Borderline conditions of finite element study model
A DS DB FS FB GS GB IS E 4-4 240.52 21.9 8.84 21.9 8.84 8.39 0.37 8.38 1.43 4-8 240.52 22 8.71 22 8.7 9.28 0.38 9.29 1.37 2-4 247.1 10.9 5.04 10.9 5.05 4.58 0.2 4.58 0.76 2-8 247.1 11 4.97 10.9 4.97 5.1 0.21 5.1 0.8 (MPa) Table 3 : Von Mises stress at each points