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

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1012141618202224

90 9710

411

111

812

513

213

914

6

153

160

167

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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