Three-dimensional movement J. Middleton,
analysis of orthodontic
tooth
M.L. Jones* and A.N. Wilson’
Department of Civil Engineering, University of Wales, Swansea; *Dental University of Wales College of Medicine, The Heath, Cardiff; ‘Department Engineering, Unviersity of Wales, Swansea, Wales, UK
School, of Civil
Received April 1989, accepted January 1990
Abstract A three-dimenrionaljinite element model was used to investigate the biomechanical response of an upper canine tooth. The physical model was developedfiom ceramic replicas and X-rays, and consisted of cancellous and cortical bone, the periodontal ligament, dentine and pulp chamber. Horikontal forces were applied at th-e tip of the crown and at the cervical margin and a rotationalforce was applied at the cervical margin of the tooth crown. The resulting displacements and stressjeldfor each load case arepresented with particular emphasis being placed on the response of the periodontal ligament. K5.e investigation shows that quantitative information on initial tooth movement can be accurately predicted and used to evaluate the response of orthodontic treatment. Keywords: Numerical modelling, tooth movement, 3D finite element model. orthodontic treatment
INTRODUCTION Numerical modelling applied to dental mechanics commenced in the mid 1970s and was initially restricted to the study of two-dimensional problems’-“. Axisymmetric problems were subsequently treated by Wright and Yettram and Williams and Watsons and full threedimensional analysis of orthodontic tooth movement has also been reported-. The development of these techniques is particularly attractive to clinicians since they provide an excellent pictorial representation of the displacement and stress fields that occur in the periodontal membrane and surrounding tissue. In this study a three-dimensional finite element model of an upper canine tooth was developed using 1500 eight-noded brick elements. Point loads were applied at the tip and base of the tooth in the distal direction and the displacement fields and principal stresses were determined in the periodontal ligament and surrounding tissues. The model developed is of greater accuracy than previous studies due to the use of a finer mesh. Also the pre- and post-processing capabilities of the package are particularly powerful in that stress fields and displacement plots can be readily produced, which provides the practising orthodontist with important quantitative information on the trend of the prescribed treatment. FINITE
ELEMENT
MODELLING
The finite element method (FEM) of analysis is now well established and has been successfully applied to predict displacements and the resulting stress fields of Correspondence to: Mr J. Middleton, Department of Civil Engineering, University of Wales, Swansea Sk? 8PP, Wales, UK 0 1990 Butterworth-Heinemann 0141-5425/90/004319-09
many linear and non-linear continuum systems?“‘. Recent publications and ongoing research reveal a growing interest in the application of the FEM to the biomechanical study of teeth subjected to othodontic treatment1’-‘3. In particular, the FEM is now clearly the dominant numerical technique for solving structural or displacement-related problems, and is therefore a natural contender for biomechanical applications. The problem of othodontic tooth movement involves the solution of the system displacement, 5, and the resulting stresses, cr, under a known set of applied forces F. The continuum mechanics problem can be solved by using an approximation method for modelling the real biomechanical response by resolving a set of algebraic equations which idealize the system into a finite number of variables. To solve this problem the following conditions must be satisfied.
Stress-strain relationship U=DE where D is a symmetrical matrix which is described in terms of the material elastic modulus, E, and Poissons ratio, v.
Compatibility 1. Continuity and differentiability of a 6 throughout the domain. 2. Satisfaction of boundary displacement conditions. 3. Strain-displacement law, E = L6, where L is a linear operator.
for BES J. Biomed. Eng. 1990, Vol. 12, July
319
30 analysis ofothodontic tooth movement:J
Middkon
et al.
Equilibrium LTcr+F=o where F are body forces per unit volume. In applying the finite element method the unknowns cr, 6 and E are replaced by discretized variables and through the adoption of trial functions; the values of these unknowns can be interpolated throughout the continuum under investigation. In equation form the numerical derivation can be written as: BrDBdu{S}
- {F} = 0
where the B matrix relates strain the continuum to the discretized term JBrDBdv = K gives rise taneous equations which can be K(6)
(1) at any oint within variab Pes {S}. The to a set of simulwritten as: (2)
= IF)
where K is the transpose matrix and T denotes the transpose operator. Equation (2) can be solved directly to provide the set of displacements {a}, which through the application of the trial or shape functions can be interpolated throughout the discretized continuum. This technique has the attraction that complicated geometrical forms with varying material properties can be accurately modelled. Furthermore, arbitrary loading conditions can be applied - an important factor if orthodontic tooth movement is to be accurate1 predicted. techMany investigators have use dy numerical niques to study tooth movement and these have often been restricted to two-dimensional models 14115. In a full three-dimensional finite this application element model has been used to discretize an upper canine tooth and the surrounding tissue. The element chosen for discretization is the well tried eight-noded quadrilateral element with n-i-linear shape functions.
-
Root apex
-
Cancellous bone
-
Peridontal
These elements are highly resistant to error under high aspect ratios and severe distortions in geometry, and also lend themselves to simplified coordinate specification and interpretation of results. Only linear elastic analysis under static loading conditions is considered here, although the technique can be extended to consider non-linear material behaviour under both static and dynamic loading conditions.
DISCBETIZATION This investigation considers the problem of a normal upper canine tooth subject to orthodontic loading conditions. Before considering the application of the FEM a detailed investigation of the anatomical layout, biological function and material properties of the canine and surrounding bone are necessary’6’17. In this investigation the numerical model consists of 1500 eight-noded three-dimensional elements, which allows the accurate geometrical details of the roblem to be considered, and also provides a high B egree of accuracy in the solution of displacements and the resulting stress fields. The canine tooth was modelled three-dimensionally and was sub-divided into dentine and pul . A section of the upper jaw was also considered an cpthis consisted of the periodontal ligament and surrounding cortical and cancellous bone. These regions are shown schematically in Figures la and b and form the physical model from which the finite element mesh will be generated. The nodal coordinate dimensions for the mesh were taken from ceramic replica, X-rays and other information given in references 14, 18 and 19. For simplicity the discretization process was commenced from the core of the tooth and developed outwards. The mesh was subdivided into a series of sub-regions, which could be clearly identified both from the viewpoint of geometrical in ut and the allocation of material properties to the B ifferent regions.
Root ligament b
Cervical margin Tooth crown
MESIAL
ASPECT
a
Cortical bone
-
Pulp
-
Dentine
DISTAL
ASPECT
BUCCAL
ASPECT
PALATAL
ASPECT
b
Figure 1 a, Upper canine tooth viewed from the side (buccal aspect). Arrows show positions of loading at cervical margin and tip of tooth crown to cause a distal tipping of the tooth in the line of the dental arch; b, view of left canine tooth from distal aspect (tooth moving towards observer)
320
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30 analysis of othodontic tooth movement:J. MiaWeton et al
Figure
2
Mesh of dental pulp
Figure 4 Periodontal ligament discretized two elements across the thickness
into 160 elements
with
The first stage of the meshing process consisted of modelling the dental pulp, as shown in Figure 2, which was followed by adding the dentine to the pulp to form the canine as shown in Figure 3. The periodontal ligament and surrounding bone were then modelled as shown in Figure 4, 5 and 6, the final form of the section of alveolous being completed by the addition of the limiting cortical plates (to the crest of the ridge, buccally and palatally) as shown in Figure 7. The complete three-dimensional finite element
Figure
3
Mesh of pulp and upper canine
Figure
5
Mesh of tissue surrounding
the periodontal
ligament
computer plot is shown in Figure 8, and Figure 9 shows a typical section through the canine and the jaw. Finally, the boundary conditions were applied by assuming that the edge sections and base of the cancellous tissue were fully fixed, which implies that the displacement at the extremities of the jaw bone are exactly zero.
LOADING CONDlTIONS Jn this study three types of orthodontic
J. Biomed.
loading were
Eng. 1990, Vol. 12, July
321
30 aru 11ysi.sof othodontic tooth movemtnt:J. Middlcton et al. n-------_
Figure 6
Discretized region of cancellous bone showing tooth socket
Figure 8
Completed three-dimensional model of upper canine tooth
322 J. Biomed. Eng. 1990, Vol. 12,July
Sections of the alveolus forming the limiting cortical plates, bucally and palatably
Figure 7
Figure 9
Section through the centre of the upper canine and jaw
30 analysis of othodontic tooth movement:J Table 1
M&&ton
et al.
Physical properties of five materials
Material
Poissons ratio, v
Elastic modulus, E (N mm-“)
Cancellous bone Cortical bone Periodontal ligament fi’P Dentine
0.38 0.26 0.25 0.45 0.31
345 13800 50 2 18600
applied: 1. A 1 N horizontal force applied in a distal direction at the tip of the tooth crown. 2. A 1 N horizontal force applied in a distal direction at the cervical margin of the tooth crown. 3. A horizontal rotational force of 1 N applied at each side of the cervical margin of the tooth crown. The applied loads are shown schematically in Figure 70. Typical single-point loadings applied in orthodontic practice are between 0.5 and 1.5 N force; thus the use of a unit load of 1 N allows simple scaling of the results obtained from the analysis. MATERIAL
a
PROPERTIES
It has been shown that initial movements due to orthodontic treatment are relatively sma1120~21; thus the assumption that the material behaves linearelastically appears to be valid. Therefore, in order to describe the mechanical behaviour of each material, its elastic modulus, E, and Poissons ratio, u, must be defined. The material properties of cancellous and cortical bone are given in references 22 and 23 and limited information is available on the properties of the periodontal ligament from references 19 and 21. From this information the physical properties of each material are defined as in Table 7.
b NUMERICAL
SOLUTION
The finite element model was generated using the ANSYS packageg4 and consisted of 1500 eight-noded three-dimensional elements with 7560 degrees of freedom. The solution of the resulting equations was performed on a VAX8700 and for each load case considered the CPU time was 600s. The resulting displacements and stresses were written to a results file in pre aration for post-processing. The or tl! odontist’s primary interest is in predicting the movement, or displacement, of teeth. The significance of this is reflected in the number of elements used in the model. In particular, emphasis has been laced on accurate modelling of the eriodontal Pigament, since its res onse under loa B is known to affect local blood Kow, cell differentiation and tissue remodelling. Most investigations concerning the response of the periodontal ligament and surinformation on rounding tissue have provided principal stresses and give little if any quantitative information of the displaced form. In the results presented here, deformed meshes are provided showing the response of the periodontum under each load case. This type of output provides a powerful means
C Figure the tip crown; margin
10 a, Horizontal point load applied in the distal direction at of the tooth crown and b, at the cervical margin of the tooth c, horizontal rotation force applied at the side of the cervical of the tooth crown
J. Biomed. Eng. 1990, Vol. 12, July
323
30 analysis of othodontic tooth movement:J. Middleton et al.
a
C
b
Figure 11 a, Displaced form of the periodontal ligament subject to a point load at the tip of the tooth crown and b, subject to a point load at the cervical margin of the tooth crown; c, responseof the periodonticalligamentsubject to rotationalloading at the cervical margin
of visualization and provides the othodontist with a reliable and invaluable means of predicting tooth movement under ideal conditions. RESULTS The scaled displaced forms of the periodontal ligament subject to the three loading types considered are shown in Figure 11. Sections shown in Figures 7 la and b are through the plane l-2, which is shown schematically in Figure 12. The response of the periodontal ligament shows that maximum displacement takes place with the unit load applied at the top of the tooth; the compressed region of the periodontium can be seen clearly at the base of the tooth, as can the dilated region which is due to tensile stresses. A parallel reference line has been included in Figure 1la to provide a comparison of the deformed mesh under load. It is noted that the dentine forming the root is sub’ect to bending and does not behave as a purely rigi d body which is often considered to be the case in orthodontic practice25. Figure 1 lc shows the response of the periodontal ligament subject to rotational loading. The displaced form shows that the tooth has twisted in the socket, with rotations at the root apex being slightly less than those at the base of the tooth. Plots of principle stresses for nodes l-l 1 at position (1) (see Figure 12) are shown for the periodontal ligament with load at the top of the tooth crown in Figure 13. Similar plots for load at the cervical margin of the tooth crown are shown in
Figure 14. Plots of principle stresses in the periodontal ligament at position (1) subject to rotation are shown in Figure 15. Figures 16 and 17 show plots of principle stress at nodes 1- 11 at position (3) with the unit load being applied at the top of the tooth crown and the cervical margin, respectively. By comparing the results of
324
J. Biomed. Eng. 1990, Vol. 12, July
Figures 13 and 17 it is shown that stresses produced with the load applied at the cervical margin are considerably less than those with the load a plied at the top of the tooth crown. The results sg own in Figure 18 are principle stresses at position (3), with the tooth subject to rotational load, and, as would be expected, are similar to the results shown in Figure 15. From these results it is shown that load at the tip of the canine crown causes the highest stress and that this occurs around the neck of the tooth. With load at the cervical margin of the tooth crown, the stresses in the periodontal ligament are less than half those with load applied at the tip. Stresses due to rotational load are again maximum at the neck of the tooth and gradually tend to a much reduced value at the root apex.
CONCLUSIONS Previous researchers have apElf&&numerical methods to orthodontic problems ’ . In this study advances in software capabilities, pre- and postprocessing and the availability of improved hardware have permitted a more accurate model to be developed. This consisted of over 1500 solid threedimensional elements which were used to model the biomechanical behaviour of five different tissue materials which form the tooth and immediate surrounding tissue. The results obtained for stresses and displacements compare quantitatively with those of Tanne et al ‘l. The highest stresses were observed with a 1 N force at the crown tip of the tooth and this also reduced the maximum localized displacement in ti: . dontium. Overall displacements, such as thosz sEr% in Figure 10, provide useful information on the movement of the tooth under different loading conditions. It is shown from the plots that the dentine forming the tooth bends under load and that a com-
30 analysis of othodontic tooth movement:J Middleton et al. Distal 2
t
Buccal
Palatal
1 Mesial
Figure
12
Schematic representation of upper canine tooth showing sampling points for the finite element results
0
-0.02
N
,;
-0.04
+7+ya22 / . -f I/
I * -
I
E
ii
i
-0.06
-0.08
(J3
--_
(7.7
---
1. I
positions
-0.005
-0.0 IO
-0.0 15
-
-0.0~0
I.
L
I -0.10
Node
Ul
.I
Z
0
.I I
! -0.025
-
-
Figure 13 Principle stresses in the periodontal ligament at position 1 of Figure 12 with the point load at the crown of the tooth
Figure 14 Principle stresses in the periodontal ligament at position 1 of Figure 72 with the point load at the cervical margin
J. Biomed. Eng. 1990, Vol. 12, July
325
30 analysis of othodontictooth movement:J Mia’dkton et al.
0.0 IS
0.020
0.015
0.0 IO
0.010
0.005 _
0.005 N
‘z Z
Node
0
z
z m
‘E Z ii E
positions
-’ 2
4
‘!‘/
-0.005
5
.’
Node
0
-0.005
.N
./
-0.0 15
.
ui
-
u2
---
U?
-*-
-0.0 10
/
/
-0.0 IS
1
I
2
6
8
10
\
.-.
/ u1
-
.
(-J’
I
03
-0-
_/
/a-
.
./
-0.0 10
I
. -
positions
1
--I
i
-0.020 Figure 15 Principle stresses in the periodontal ligament at position of Figure 72 with the tooth subject to rotational loading
1
Figure 16 Principle stresses in the periodontal ligament at position 3 of Figure 12 with the point load at the crown of the tooth
Node
I-
I-
. -
positions
/
/ .
I f /
Figure 17 Principle stresses in the periodontal ligament at position 3 of Figure 12 with the point load at the cervical margin
326
J. Biomed. Eng. 1990, Vol. 12, July
Figure 18 Principle stresses in the periodontal ligament at position 3 of Figure 72 with the tooth subject to rotational loading
30 analysis of othodontictooth movement:J. Miaiileton et al.
plicated form of stress transfer takes place throughout the root. The results from Figures 74 and 16 show that the maximum compressive and tensile stresses occur at the opposite point of load ap lication and on the same side as the point of loa B application, respectively. Stresses are maximum at the neck of the tooth which is probably due to the relative stiffness of the cortical plates. Maximum stresses in tension and compression were of the order of 1000 gem-” and occur with the point load applied at the crown tip. The compressed and dilated regions of the periodontal subject to these stresses can also be clearly seen from Figures Ila and b. It is shown from this stud that three-dimensional numerical modelling of o x odontic treatment can provide the practitioner with an accurate means of predicting both displacements and stress fields. In particular, the analysis provides quantitative information on tooth movement and through the application of graphical post-processing the response to treatment can be clearly visualized.
9. 10. 11.
12.
13.
14.
15.
16.
17.
REFERENCES 1. Tesk JA, Widera 0. Stress distribution in bone arising from loading on endosteal dental implants. JBiomed Muter Res Symp 1973; 4(7): 251-61. 2. Craig RG, Farah JW. Stresses from loading distal-extension removable-partial dentures. J Prosth Dent 1978; 39: 264-77. I3 Takahashi N, Kitagami T, Komori T. Analysis of stress on a fixed partial denture with a blade-vent implant abutment. JProsth Dent 1978; 40: 186-91. 4. Wright KWJ, Yettram AL. Finite element stress and analysis of a Class I amalgam restoration subjected to setting and thermal expansion. J Dent Res 1978; 57: 715-23. 5. Williams KR, Watson CS. Examination of the failure of a Wiptam-post-restored tooth. J Dent 1986; 14: 14-7. 6. Cook SD, Klawitterd, Weinstein AM, Lavemia CJ. The design and evaluation of dental implants with finite element analysis. Biomechunics Symposium New York: American Society of Mechanical Engineers 1980, pp. 169-92. 7. Weinstein AM, Klawitter JJ, Cook SD. Implant-bone interface characteristics of bioglass dental implants. J Biomed Mater Res 1980; 14: 23-29. 8. Takahashi N, Kitagami T, Komori T. Evaluation of
18.
19.
20.
21.
22.
23. 24. 25.
thermal change in pulp chamber. J Dent Res 1977; 56: 1480. Desai CS, Abel JF. Introduction to th Finite Element Method. Amsterdam Van Nostrand Reinhold Co., 1972. Zienkiewicz OC. The Finite Element Method 4th edition. New York: McGraw Hill, 1988. Tanne K, Koenig A, Burstone CJ. Moment to force ratios and the centre of rotation. Am J Orthod Dentofac Ortho@ 1988; 94: 426-31. Zhou SM, Ping H, Wang YF. Analysis of stresses and breaking loads for Class 1 cavity preparations in mandibular first molars. Qvintessence Int 1989; 20: 205-10. Tanne K. Stress distributions in the periodontal membrane associated with various moments to force ratios in orthodontic force systems. J Osaka Univ Dent School 1987; 27: 1-9. Williams KR, Edmundsen JT, Morgan G, Jones ML, Richmond S. Orthodontic movement of a canine into an adjoining extraction site. J Biomed Eng 1986; 8: 115-20. Yettram AL, Wright KWJ, Houston WJB. Centre of rotation of a maxillary central incisor under orthodontic loading. Br J Orthod 1977; 4: 23-7. Waters NE. Mechanical properties of teeth. In: % Medical Properties of Biological Mater& Cambridge: Cambridge University Press, 1980; pp. 99-137. Bonfield W, Datta PK. Young’s modulus of compact bone. Biomechanics 1974; 7: 147-9. Steyn CL, Verweld WS, Van Der Merwe EJ, Foarie OL. Calculation of the position of the axis of rotation when single rooted teeth are othodontically tipped. BY J Orthod 1978; 5: 153-7. Wright KWS. On the mechanical behaviour of human tooth structures, an application of the finite element method of stress analysis. Ph D Thesis. Uxbridge: Brunel University. Tanne K, Sakuda M. Initial stress induced in the periodontical tissue at the time of the application of various types of orthodontic force: Three dimensional analysis by means of the finite element method. J Osaka Univ Dent School 1983; 23: 143-71. Tanne K, Sakuda M, Burstone CJ. Three-dimensional finite element analysis for stress in the periodontal tissue by orthodontic forces. Am J Orthod Dentofac Orthop 1987; 92: 499-505. Grenoble DE, Katz JL, Gilmore RS, Nurty KL. The elastic properties of hard tissues and sapatites. JBiomed Mater Res 1972; 6: 221-33. McElhaney JH. Dynamic response of bone and muscle tissue. JAppl Physiol 1966; 21: 1231-6. ANSYS. Houston, Pennsylvania, USA: Swanson Analysis System Inc. Grimm FM. Bone bending, a feature of orthodontic tooth movement. Am J Orthod 1972; 62: 382-93.
J. Biomed. Eng. 1990, Vol. 12,July
327
analysis of orthodontic
tooth
M.L. Jones* and A.N. Wilson’
Department of Civil Engineering, University of Wales, Swansea; *Dental University of Wales College of Medicine, The Heath, Cardiff; ‘Department Engineering, Unviersity of Wales, Swansea, Wales, UK
School, of Civil
Received April 1989, accepted January 1990
Abstract A three-dimenrionaljinite element model was used to investigate the biomechanical response of an upper canine tooth. The physical model was developedfiom ceramic replicas and X-rays, and consisted of cancellous and cortical bone, the periodontal ligament, dentine and pulp chamber. Horikontal forces were applied at th-e tip of the crown and at the cervical margin and a rotationalforce was applied at the cervical margin of the tooth crown. The resulting displacements and stressjeldfor each load case arepresented with particular emphasis being placed on the response of the periodontal ligament. K5.e investigation shows that quantitative information on initial tooth movement can be accurately predicted and used to evaluate the response of orthodontic treatment. Keywords: Numerical modelling, tooth movement, 3D finite element model. orthodontic treatment
INTRODUCTION Numerical modelling applied to dental mechanics commenced in the mid 1970s and was initially restricted to the study of two-dimensional problems’-“. Axisymmetric problems were subsequently treated by Wright and Yettram and Williams and Watsons and full threedimensional analysis of orthodontic tooth movement has also been reported-. The development of these techniques is particularly attractive to clinicians since they provide an excellent pictorial representation of the displacement and stress fields that occur in the periodontal membrane and surrounding tissue. In this study a three-dimensional finite element model of an upper canine tooth was developed using 1500 eight-noded brick elements. Point loads were applied at the tip and base of the tooth in the distal direction and the displacement fields and principal stresses were determined in the periodontal ligament and surrounding tissues. The model developed is of greater accuracy than previous studies due to the use of a finer mesh. Also the pre- and post-processing capabilities of the package are particularly powerful in that stress fields and displacement plots can be readily produced, which provides the practising orthodontist with important quantitative information on the trend of the prescribed treatment. FINITE
ELEMENT
MODELLING
The finite element method (FEM) of analysis is now well established and has been successfully applied to predict displacements and the resulting stress fields of Correspondence to: Mr J. Middleton, Department of Civil Engineering, University of Wales, Swansea Sk? 8PP, Wales, UK 0 1990 Butterworth-Heinemann 0141-5425/90/004319-09
many linear and non-linear continuum systems?“‘. Recent publications and ongoing research reveal a growing interest in the application of the FEM to the biomechanical study of teeth subjected to othodontic treatment1’-‘3. In particular, the FEM is now clearly the dominant numerical technique for solving structural or displacement-related problems, and is therefore a natural contender for biomechanical applications. The problem of othodontic tooth movement involves the solution of the system displacement, 5, and the resulting stresses, cr, under a known set of applied forces F. The continuum mechanics problem can be solved by using an approximation method for modelling the real biomechanical response by resolving a set of algebraic equations which idealize the system into a finite number of variables. To solve this problem the following conditions must be satisfied.
Stress-strain relationship U=DE where D is a symmetrical matrix which is described in terms of the material elastic modulus, E, and Poissons ratio, v.
Compatibility 1. Continuity and differentiability of a 6 throughout the domain. 2. Satisfaction of boundary displacement conditions. 3. Strain-displacement law, E = L6, where L is a linear operator.
for BES J. Biomed. Eng. 1990, Vol. 12, July
319
30 analysis ofothodontic tooth movement:J
Middkon
et al.
Equilibrium LTcr+F=o where F are body forces per unit volume. In applying the finite element method the unknowns cr, 6 and E are replaced by discretized variables and through the adoption of trial functions; the values of these unknowns can be interpolated throughout the continuum under investigation. In equation form the numerical derivation can be written as: BrDBdu{S}
- {F} = 0
where the B matrix relates strain the continuum to the discretized term JBrDBdv = K gives rise taneous equations which can be K(6)
(1) at any oint within variab Pes {S}. The to a set of simulwritten as: (2)
= IF)
where K is the transpose matrix and T denotes the transpose operator. Equation (2) can be solved directly to provide the set of displacements {a}, which through the application of the trial or shape functions can be interpolated throughout the discretized continuum. This technique has the attraction that complicated geometrical forms with varying material properties can be accurately modelled. Furthermore, arbitrary loading conditions can be applied - an important factor if orthodontic tooth movement is to be accurate1 predicted. techMany investigators have use dy numerical niques to study tooth movement and these have often been restricted to two-dimensional models 14115. In a full three-dimensional finite this application element model has been used to discretize an upper canine tooth and the surrounding tissue. The element chosen for discretization is the well tried eight-noded quadrilateral element with n-i-linear shape functions.
-
Root apex
-
Cancellous bone
-
Peridontal
These elements are highly resistant to error under high aspect ratios and severe distortions in geometry, and also lend themselves to simplified coordinate specification and interpretation of results. Only linear elastic analysis under static loading conditions is considered here, although the technique can be extended to consider non-linear material behaviour under both static and dynamic loading conditions.
DISCBETIZATION This investigation considers the problem of a normal upper canine tooth subject to orthodontic loading conditions. Before considering the application of the FEM a detailed investigation of the anatomical layout, biological function and material properties of the canine and surrounding bone are necessary’6’17. In this investigation the numerical model consists of 1500 eight-noded three-dimensional elements, which allows the accurate geometrical details of the roblem to be considered, and also provides a high B egree of accuracy in the solution of displacements and the resulting stress fields. The canine tooth was modelled three-dimensionally and was sub-divided into dentine and pul . A section of the upper jaw was also considered an cpthis consisted of the periodontal ligament and surrounding cortical and cancellous bone. These regions are shown schematically in Figures la and b and form the physical model from which the finite element mesh will be generated. The nodal coordinate dimensions for the mesh were taken from ceramic replica, X-rays and other information given in references 14, 18 and 19. For simplicity the discretization process was commenced from the core of the tooth and developed outwards. The mesh was subdivided into a series of sub-regions, which could be clearly identified both from the viewpoint of geometrical in ut and the allocation of material properties to the B ifferent regions.
Root ligament b
Cervical margin Tooth crown
MESIAL
ASPECT
a
Cortical bone
-
Pulp
-
Dentine
DISTAL
ASPECT
BUCCAL
ASPECT
PALATAL
ASPECT
b
Figure 1 a, Upper canine tooth viewed from the side (buccal aspect). Arrows show positions of loading at cervical margin and tip of tooth crown to cause a distal tipping of the tooth in the line of the dental arch; b, view of left canine tooth from distal aspect (tooth moving towards observer)
320
J. Biomed. Eng. 1990, Vol. 12, July
30 analysis of othodontic tooth movement:J. MiaWeton et al
Figure
2
Mesh of dental pulp
Figure 4 Periodontal ligament discretized two elements across the thickness
into 160 elements
with
The first stage of the meshing process consisted of modelling the dental pulp, as shown in Figure 2, which was followed by adding the dentine to the pulp to form the canine as shown in Figure 3. The periodontal ligament and surrounding bone were then modelled as shown in Figure 4, 5 and 6, the final form of the section of alveolous being completed by the addition of the limiting cortical plates (to the crest of the ridge, buccally and palatally) as shown in Figure 7. The complete three-dimensional finite element
Figure
3
Mesh of pulp and upper canine
Figure
5
Mesh of tissue surrounding
the periodontal
ligament
computer plot is shown in Figure 8, and Figure 9 shows a typical section through the canine and the jaw. Finally, the boundary conditions were applied by assuming that the edge sections and base of the cancellous tissue were fully fixed, which implies that the displacement at the extremities of the jaw bone are exactly zero.
LOADING CONDlTIONS Jn this study three types of orthodontic
J. Biomed.
loading were
Eng. 1990, Vol. 12, July
321
30 aru 11ysi.sof othodontic tooth movemtnt:J. Middlcton et al. n-------_
Figure 6
Discretized region of cancellous bone showing tooth socket
Figure 8
Completed three-dimensional model of upper canine tooth
322 J. Biomed. Eng. 1990, Vol. 12,July
Sections of the alveolus forming the limiting cortical plates, bucally and palatably
Figure 7
Figure 9
Section through the centre of the upper canine and jaw
30 analysis of othodontic tooth movement:J Table 1
M&&ton
et al.
Physical properties of five materials
Material
Poissons ratio, v
Elastic modulus, E (N mm-“)
Cancellous bone Cortical bone Periodontal ligament fi’P Dentine
0.38 0.26 0.25 0.45 0.31
345 13800 50 2 18600
applied: 1. A 1 N horizontal force applied in a distal direction at the tip of the tooth crown. 2. A 1 N horizontal force applied in a distal direction at the cervical margin of the tooth crown. 3. A horizontal rotational force of 1 N applied at each side of the cervical margin of the tooth crown. The applied loads are shown schematically in Figure 70. Typical single-point loadings applied in orthodontic practice are between 0.5 and 1.5 N force; thus the use of a unit load of 1 N allows simple scaling of the results obtained from the analysis. MATERIAL
a
PROPERTIES
It has been shown that initial movements due to orthodontic treatment are relatively sma1120~21; thus the assumption that the material behaves linearelastically appears to be valid. Therefore, in order to describe the mechanical behaviour of each material, its elastic modulus, E, and Poissons ratio, u, must be defined. The material properties of cancellous and cortical bone are given in references 22 and 23 and limited information is available on the properties of the periodontal ligament from references 19 and 21. From this information the physical properties of each material are defined as in Table 7.
b NUMERICAL
SOLUTION
The finite element model was generated using the ANSYS packageg4 and consisted of 1500 eight-noded three-dimensional elements with 7560 degrees of freedom. The solution of the resulting equations was performed on a VAX8700 and for each load case considered the CPU time was 600s. The resulting displacements and stresses were written to a results file in pre aration for post-processing. The or tl! odontist’s primary interest is in predicting the movement, or displacement, of teeth. The significance of this is reflected in the number of elements used in the model. In particular, emphasis has been laced on accurate modelling of the eriodontal Pigament, since its res onse under loa B is known to affect local blood Kow, cell differentiation and tissue remodelling. Most investigations concerning the response of the periodontal ligament and surinformation on rounding tissue have provided principal stresses and give little if any quantitative information of the displaced form. In the results presented here, deformed meshes are provided showing the response of the periodontum under each load case. This type of output provides a powerful means
C Figure the tip crown; margin
10 a, Horizontal point load applied in the distal direction at of the tooth crown and b, at the cervical margin of the tooth c, horizontal rotation force applied at the side of the cervical of the tooth crown
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30 analysis of othodontic tooth movement:J. Middleton et al.
a
C
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Figure 11 a, Displaced form of the periodontal ligament subject to a point load at the tip of the tooth crown and b, subject to a point load at the cervical margin of the tooth crown; c, responseof the periodonticalligamentsubject to rotationalloading at the cervical margin
of visualization and provides the othodontist with a reliable and invaluable means of predicting tooth movement under ideal conditions. RESULTS The scaled displaced forms of the periodontal ligament subject to the three loading types considered are shown in Figure 11. Sections shown in Figures 7 la and b are through the plane l-2, which is shown schematically in Figure 12. The response of the periodontal ligament shows that maximum displacement takes place with the unit load applied at the top of the tooth; the compressed region of the periodontium can be seen clearly at the base of the tooth, as can the dilated region which is due to tensile stresses. A parallel reference line has been included in Figure 1la to provide a comparison of the deformed mesh under load. It is noted that the dentine forming the root is sub’ect to bending and does not behave as a purely rigi d body which is often considered to be the case in orthodontic practice25. Figure 1 lc shows the response of the periodontal ligament subject to rotational loading. The displaced form shows that the tooth has twisted in the socket, with rotations at the root apex being slightly less than those at the base of the tooth. Plots of principle stresses for nodes l-l 1 at position (1) (see Figure 12) are shown for the periodontal ligament with load at the top of the tooth crown in Figure 13. Similar plots for load at the cervical margin of the tooth crown are shown in
Figure 14. Plots of principle stresses in the periodontal ligament at position (1) subject to rotation are shown in Figure 15. Figures 16 and 17 show plots of principle stress at nodes 1- 11 at position (3) with the unit load being applied at the top of the tooth crown and the cervical margin, respectively. By comparing the results of
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Figures 13 and 17 it is shown that stresses produced with the load applied at the cervical margin are considerably less than those with the load a plied at the top of the tooth crown. The results sg own in Figure 18 are principle stresses at position (3), with the tooth subject to rotational load, and, as would be expected, are similar to the results shown in Figure 15. From these results it is shown that load at the tip of the canine crown causes the highest stress and that this occurs around the neck of the tooth. With load at the cervical margin of the tooth crown, the stresses in the periodontal ligament are less than half those with load applied at the tip. Stresses due to rotational load are again maximum at the neck of the tooth and gradually tend to a much reduced value at the root apex.
CONCLUSIONS Previous researchers have apElf&&numerical methods to orthodontic problems ’ . In this study advances in software capabilities, pre- and postprocessing and the availability of improved hardware have permitted a more accurate model to be developed. This consisted of over 1500 solid threedimensional elements which were used to model the biomechanical behaviour of five different tissue materials which form the tooth and immediate surrounding tissue. The results obtained for stresses and displacements compare quantitatively with those of Tanne et al ‘l. The highest stresses were observed with a 1 N force at the crown tip of the tooth and this also reduced the maximum localized displacement in ti: . dontium. Overall displacements, such as thosz sEr% in Figure 10, provide useful information on the movement of the tooth under different loading conditions. It is shown from the plots that the dentine forming the tooth bends under load and that a com-
30 analysis of othodontic tooth movement:J Middleton et al. Distal 2
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Figure 13 Principle stresses in the periodontal ligament at position 1 of Figure 12 with the point load at the crown of the tooth
Figure 14 Principle stresses in the periodontal ligament at position 1 of Figure 72 with the point load at the cervical margin
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30 analysis of othodontictooth movement:J Mia’dkton et al.
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Figure 17 Principle stresses in the periodontal ligament at position 3 of Figure 12 with the point load at the cervical margin
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Figure 18 Principle stresses in the periodontal ligament at position 3 of Figure 72 with the tooth subject to rotational loading
30 analysis of othodontictooth movement:J. Miaiileton et al.
plicated form of stress transfer takes place throughout the root. The results from Figures 74 and 16 show that the maximum compressive and tensile stresses occur at the opposite point of load ap lication and on the same side as the point of loa B application, respectively. Stresses are maximum at the neck of the tooth which is probably due to the relative stiffness of the cortical plates. Maximum stresses in tension and compression were of the order of 1000 gem-” and occur with the point load applied at the crown tip. The compressed and dilated regions of the periodontal subject to these stresses can also be clearly seen from Figures Ila and b. It is shown from this stud that three-dimensional numerical modelling of o x odontic treatment can provide the practitioner with an accurate means of predicting both displacements and stress fields. In particular, the analysis provides quantitative information on tooth movement and through the application of graphical post-processing the response to treatment can be clearly visualized.
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