The role of the periodontal ligament in bone modeling: The initial development of a time-dependent finite element model

The role of the periodontal ligament in bone modeling: The initial development of a time-dependent finite element model

John Middleton, BSc, MSc, a Malcolm Jones, BDS, PhD, D.Orth., FDSRCS, b and Adrian Wilson, BSc, MSc, PhD ° Cardiff, Wales, and Calga~ Canada

Current remodeling theories, as applied to long bones, suggest that such processes are controlled by mechanical strains either within or on the bone surface. In this study, the stresses and strains within the periodontal ligament and surrounding bone, consequent to orthodontic loading of a tooth, were investigated by application of the finite element method. Previously, various authors have applied two and three dimensional instantaneous (essentially static) models to analyze the problems. The study reported in this article describes an initial time-dependent (continuous/dynamic) finite element model for tooth movement that uses newly developed software, the results being cross-referenced against historical data. These early results, from a two-dimensional mathematical model of a loaded canine tooth, suggest that the remodeling process may be controlled by the periodontal ligament rather than the bone. In the finite element model, bone was found to experience a low strain of 1 × 10 -5, whereas the periodontal ligament experienced a strain of 0.1 when the "tooth model" is loaded. Only this latter figure is above the threshold usually reported to be necessary to initiate the remodeling process. Further developments in this rapidly advancing area of biomechanical research should facilitate a greater increase in our knowledge of tissue stress and strain after loading. (AM J ORTHOD DENTOFAC ORTHOP 1996;109:155-62.)

I n recent years, several investigators 1-5 have transposed the problem of finite element modeling in orthodontics from the two-dimensional linear elastic domain 6 to a full three-dimensional continuum. In addition, various authors >5 have included nonlinear material properties with a view to depicting more completely the in vivo behavior of the tooth. The finite element method (FEM) provides the orthodontist with quantitative data that can extend the understanding of the physi- ologic reactions that occur. In particular, such numerical techniques may yield an improved understanding of the reactions and interactions of individual tissues. 1"° Such detailed information on the stresses and strains in tissues is difficult to obtain

Supported by a grant from the Welsh Scheme for the Development of Health and Social Research and in part by the Dr. Nadwen Trust for Humane Research. "Senior Lecturer, Department of Civil Engineering, University College of Wales, Swansea and Welsh Center for Biornechanics. bprofessor in Orthodontics, Department of Child Dental Health, Univer- sity of Wales College of Medicine, Cardiff and Welsh Center for Biome- chanics. CResearch Assistant, Department of Civil Engineering, University of Calgary, Canada. Copyright © 1996 by the American Association of Orthodontists. 0889-5406/96/$5.00 + 0 8/1/61569

and accurately analyze by any other experimental technique because of the interaction of the surrounding tissues, which may then distort the data obtained for any individual material response. 3

Although it has not been applied in this way in orthodontics previously, the FEM, by the application of new techniques, could theoretically predict the longer-term (time-dependent) movement of teeth.

Current remodeling theories have mainly been applied to long bones. Several researchers have developed models that qualitatively predict the overall reaction of such bones to long-term loading. 7-w These theories have all linked the stress or strain existing in the bone or at the bone surface to the remodeling process. The work reported in this article intends to investigate further the particular case of the mechanics of bone remodeling associated with orthodontic tooth movement, since it is has been suggested that such movement is largely a periodontal mediated phenomenon. 1' Previously, FEM work by the authors when using the more traditional instantaneous type of model has pro- vided support for such a contention. Certainly, in orthodontic bone remodeling, where the periodon-

155

156 Middleton, Jones, and Wilson American Journal of Orthodontics and Dentofacial Orthopedics February 1996

Fig. 1. Finite element mesh with load.

F

tal l igament serves as a potent ial source for those cells tradit ionally associated with resorpt ion/deposition, it is theoret ical ly possible for it to play a pr imary role in remodeling.

MATERIALS AND METHOD

To investigate the role of the periodontal ligament in the bone remodeling process, an initial two-dimensional nonlinear finite element model was developed to study the strains in the periodontal ligament and surrounding tissues. Throughout this article, to avoid confusion, whenever results from the computer modeled tissues are being referred to, the tissue concerned is printed in italics. The viscoelastic behavior of the ligament was incorporated with the use of an overlay model, as has been described previously. '2 All other materials were assigned with linear elastic material constants. The basic material properties used in the analysis can be seen in Table I, their selection has been described in detail previously. 13 The viscoelastic parameters were obtained by back calculations 5 by using the experimental results of Ross? 4

The original FEM model, constructed for the current study, contained 2000 eight-noded isoparametric quadri- lateral elements, and its basic form is illustrated in Fig. 1. The model was developed from measurements of a repre- sentative maxillary canine tooth to provide an accurate

geometry. Although these dimensions were only taken from one tooth, it was of a typical geometry and size. To improve the accuracy of the solution in this region and to overcome numerical instabilities that occurred as a result of the disparity in the stiffness of the materials, the mesh shown in Fig. 1 was developed so that it had a high density of elements in the periodontal ligament, the area of main interest in this investigation. The bone and dentin were assigned linear elastic properties and hence did not exhibit time-dependent behavior. To investigate the strains oc- curring in the model, the results were sampled at the Gauss points (Fig. 2) and reduced 2 by 2 integration was used for the numerical analysis.

In orthodontic treatment, the crown loading condi- tions under which such a tooth may be successfully moved is reasonably well established. Despite this, the complicated nature of oral loading due to additional factors, which might include, for example, the processes of mastication, swallowing, or bruxism, means that the exact value of force exerted at the root surface remains in doubt. The load chosen in the current study was intended to represent a relatively simple force system associated with a typical canine retraction spring. For reasons discussed in previous work, 13 a typical load of 1 N or approximately 100 gm was chosen to be applied to the tooth crown, as shown in Fig. 1.

RESULTS

It should be stated at the outset that in the presenta t ion o f the results, the terms bone, tooth, and periodontal ligament all refer to mathemat ica l representa t ions ra ther than t rue physical entities.

Fig. 3 shows the strains as sampled across five sections of the model of the tooth, bone, and ligament. In the region represent ing the periodontal ligament, it is apparen t that the strain peaks dra- matically, whereas in the o ther sampled areas rep- resenting bone and tooth, the value remains close to zero. Figs. 4 and 5 show quantitat ively the strain in the bone and the periodontal ligament on the palatal (Fig. 4) and buccal (Fig. 5) sides. The strain in the ligament is shown at t ime equals 0 seconds (t = 0), after the l inear elastic response, and time equals 45 seconds (t = 45) at the end of the viscoelastic response. The strain in the bone is shown at time equals 45 seconds, a l though it does not exhibit any t ime-dependen t behavior. F r o m these results, it is shown that the magni tude of strain in the bone is of the order of 1 × 10 -5, which can be c o m p a r e d with the strain in the ligament of 0.1. The ligament itself shows the highest strain close to the cervical mar- gin, the region in which remodel ing has been shown previously to be most prevalent. ~'I6

For strains viewed by Car tes ian analysis (Figs. 6

American Journal of Orthodontics and DentofiTcial Orthopedics Middleton, Jones, and Wilson 157 Volume 109, No. 2

/ : ] . . . . . .

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Lines on which the Gauss Points lie

Peridontal ligament . . . . . . . . . . . . Gauss points in bone (Buccal) . . . . . Gauss points in dentine (Buccal)

Gauss points in dentine (Palatal) Gauss points in bone (Palatal)

Fig. 2. Position of Gauss points adjacent to ligament.

Table I. Average values for the material properties

Young's moduhts Material (E) N/mm 2 Poisson ~ ratio (v) Viscosity (u )

Enamel 84100 0.20 Dentine 18600 0.31 Cortical bone 13800 0.26 Cancellous bone 345 0.30 Pulp 2 0.45

Periodontal [E1]1.5 0.45 Ligament [E2]0.75 0.45 2.21

and 7), it is apparent that strains of larger magnitude exist in the periodontal ligament in comparison to the local bone, the strain in the x-direction of the ligament being particularly large because of the direction of the load. It is worth noting that the shear strain is also contributing to the overall strain state and could therefore have 3n additional effect on the overall remodeling process.

In all four figures (Figs. 4 to 7), it can be seen on the model that tension exists on the representation of the side adjacent to the load (buccal) and that compression is prevalent on the side opposite the load (palatal).

DISCUSSION

It should be restated at the outset of the discussion that the model described is theoretical and that we are at a relatively early stage of modeling orthodontic time dependency response. This is a complex area on the "cutting edge" of hardware and software development. This being the case, we have limited ourselves to a two-dimensional FEM model, while accepting that the solutions obtained will eventually require confirmation in a later three-dimensional model once it has been fully developed and validated. As is usual in finite element stress analysis (FESA) models, the results


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American Journal of Orthodontics and Dentofacial Orthopedics Middleton, Jones, and Wilson 159 Volume 109, No. 2

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obtained are only as good as the initial data used to set the parameters of tissue response. In other words, the computer model is limited by our detailed knowledge of the physical properties of human tissues. Currently, advances in the field of such tissue measurement lag far behind the speed at which the FEM is being developed and applied in biomechanics. It is recognized that direct measurement of the properties of the periodontal ligament is very difficult especially with the superimposed changing nature of tissue response to load with altering blood flow. Nonetheless, such difficulties should not stop the development of this exciting area of biomechanical research. Although the currently described model has recognizable limitations, tentative first steps into the complex area of orthodontic time-dependent modeling are necessary and the results, as far as possible, have been validated against existing historical data of charted tooth movement," while the currently accepted properties for the surrounding tissues are applied. '3 Working within these stated limitations, the results suggest that in this computerized representation of

the tissues a strain of significantly greater magnitude is found in the periodontal ligament in comparison to that found in the bone surrounding the tooth.

Frost, 17 after examination of the bone else- where in the body, has suggested that for bone remodeling to occur, the strain level should exceed a value of 0.02. There is no reason to expect the situation to be any different for alveolar bone. In the current model, the strain in the modeled bone around the tooth does, at no sampled point, reach this required activation threshold. However, in contrast, in the periodontal ligament, a strain of 0.1 is recorded. Thus, if the arguments put forward by Frost ~7 are correct (and applicable to the dentoal- veolus), the bone might not experience sufficient strain during this typical orthodontic loading to begin remodeling, yet this process around the tooth occurs, and at a rapid rate. This process seems likely to be mediated through the vascular periodontal ligament. The role of the ligament has been investigated in this capacity by both Picton 18 and Reitan '5 who have suggested that, if the load is


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Fig. 6. X, Y, and XY strains in modeled PDL and surrounding tissue on palatal side. Load applied in middle of tooth.

sufficiently large to occlude the blood vessels, then the tissues may eventually perish in this region. This leads to other cells undermining the bone from a neighboring position in the periodontal ligament (undermining resorption). It should be recalled that the classic pattern of remodeling in long bones involves bone deposition at the site of compression and resorption from the site of tension. This pattern of behavior would appear to be reversed in the situation of the bony socket surrounding a tooth under load and would indicate that the process of bone remodeling around a tooth may be different to that described classically for long bones. These differences may be related to involvement in the process of the periodontal ligament. The current study that examines loads, and resultant stresses and strains, in addition to previous data we collected, would support this view. 3'434

In addition, previously reported findings agree with the results of this computer simulation: In general, bone around a tooth subject to a load is

removed from the compression side in contrast it is added to the tension face. T M

Other work that uses more advanced numerical analysis techniques is planned in this area to develop a validated three-dimensional model that simulates tooth movement over a longer term. This will facilitate a greater and more detailed understanding of how teeth move while adding to our knowledge of the stresses and strains that act in the surrounding tissues. Eventually, such models will be applied routinely to the evaluation and development of both new orthodontic appliances and materials, as well as to provide a valuable educational tool.

CONCLUSIONS

The tooth socket is unique because nowhere in the body is bone tissue connected internally by a structure such as the periodontal ligament. The current two-dimensional model would suggest that the tooth and the surrounding tissues are exposed

American Journal of Orthodontics and Dentofacial Orthopedics Middleton, Jones, and Wilson 161 Volume 109, No. 2

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to an unusual state of strain. The ligament appears to experience relatively high strains, whereas the adjacent bone and dentin appear to be in a low strain field. Thus, if strain is accepted as a contribu- tor to the remodeling process, then the findings of this theoretical study support the contention that it is the periodontal ligament that controls the remodeling process rather than the bone itself. To the extent that the current model accurately repre- sents a "real world" alveolus, it would appear unlikely that an "orthodontic force" applied di- rectly to a tooth facilitates changes in the support- ing alveolus through strain in the bone.

R E F E R E N C E S

1. Wright KWS. On the mechanical behaviour of human tooth structures: an application of the finite element method. [Thesis•] London: Brunel University, 1975.

2. 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.

3. McGuinness NJP, Wilson AN, Jones ML, Middleton J, Robertson NRE. Stress induced by edgewise appliances in the periodontal l i gament -a finite element study. Angle Orthod 1991;62:15-21.

4. Middleton J, Jones ML, Wilson AN. Three dimensional analysis of orthodontic tooth movement. J Biomed Engineer 1990;12:319-27.

5. Wilson AN. Linear and non-linear analysis of orthodontic tooth movement. [PhD Thesis.] Cardiff: University of Wales, 1992.

6. Williams KR, Edmundson JT, Morgan G, Jones ML, Rich- mond S. Orthodontic movement of a canine into an adjoin- ing extraction site. J Biomed Engineer 19S;);8:115-20,

7. Cowin SC. Continuum models of the adaption of bone stress. In: Cowin SC, ed. Mechanical properties of bone. New York: American Society of Mechanical Engineering, 1981:193-210.

8. Hart RT. Quantitative response of bone to mechanical stress. [Dissertation.] Cleveland, Ohio: Department of Me- chanical and Aerospace Engineering, Case Western Re- serve University, 1983.

9. Huiskes R. Adaptive bone remodelling theory applied to prosthetic design analysis. J Biomech 1987;20:1135-50.

10. Beaupre GS, Orr TW, Carter DR. An approach for time dependent bone modelling and remodelling• J Orthop Res 1990;88:651-71.

162 Middleton, Jones, and Wilson American Journal of Orthodontics and Dentofacial Orthopedics Febmaty 1996

11. Proffit WR. Contemporary orthodontics. St. Louis: CV Mosby, 1986:228.

12. Pande GN, Owen DRL Zienkiewicz OC. Overlay models in time dependent non-linear material analysis. J Comput Struct 1977;7:435-44.

13. McGuinness NJP, Wilson AN, Jones ML, Middleton J. A stress analysis of the periodontal ligament under various orthodontic loadings. Eur J Orthod 1990;13:231-42.

14. Ross GG, Lear CS, De Lou R. Modelling the lateral movement of teeth. Biomech 1976;9:723-34.

15. Reitan K. Biomechanical principles and reactions. In: Graber TM, ed. Current orthodontic concepts and techniques. Philadelphia: WB Saunders, 1969:56-159.

16. Reitan K. Effects of force, magnitude and direction of tooth movement on different alveolar bone types. Angle Orthod 1964;34:244-55.

17. Frost HM. A determination of bone architecture: the mini- mum effective strain. Clin Orthop Res 1983;175:286-92.

18. Picton DCA. Distribution of collagen fibres in the periodontal ligament and their role in tooth support. Periodontology Today, International Congress, Zurich 1988:14-31.

Reprint requests to: Professor M.L. Jones Orthodontic Division Department of Child Dental Health University of Wales College of Medicine Heath Park, CF4 4XY Wales, United Kingdom

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The role of the periodontal ligament in bone modeling: The initial development of a time-dependent finite element model