Material parameters and stress profiles within the periodontal ligament Kim L. Andersen, MSc, Erik H. Pedersen, DDS, and Birte Melsen, DDS, Dr. Odont. Aarltus, Denmark Levels and profiles of initial stress in the periodontal ligament after application of various force systems were studied. Two finite-element models, based on sections of human autopsy material, were developed to simulate one full and one partial mandible. The validity of the finite-element model was improved by identiiication of material parameters; the mechanical properties of the tissue were described by means of strain-gauge measurements of initial tooth movements in human autopsy material. The multiple modeling technique, in which data from a coarse global model are transferred to a more detailed one, was used to identify bone structure and boundary conditions. Parameters known to influence the results were varied to establish the validity of the finite-element model. Iterative calculation methods were used to gain stable results. However, optimizing features of the bone ,structure and boundary conditions did not influence the results significantly. The elastic stiffness of the periodontal ligament was determined to 0.07 MPa and 'r = 0.49 ('r being the Poisson's ratio). Stress profiles were obtained for various force systems--as in tipping, translation, and root movement. As we expected, there was a marked variation in the stress distribution from cervix to apex when tipping forces were applied. Bodily movement of the tooth produced an almost uniform stress distribution; root movement produced stress patterns opposite to those observed during tipping; and masticatory forces alone produced stress patterns almost identical to those achieved by masticatory force in combination with orthodontic forces. (AMJ ORTHOD DENTOFA(~ ORTHOP 1991 ;99:427-40.)

The application of orthodontic force in periodontal-tissue remodeling has been the subject of extensive research. * Although the relationship between orthodontic force and bone remodeling is still not fully understood, it is believed that the stress and strain on the periodontal ligament produced by application of force is a determining factor. 2-s A precondition for further clarification of the relationship between stress/strain and bone remodeling is, however, availability of the stress/strain profiles of the tissues involved. The purpose of the present study was (1) to evaluate the validity of the finite-element method for determining stress/strain conditions in the periodontal tissues after application of force and (2) to use the method for determining initial stress levels and profiles in the periodontal ligament. BACKGROUND The finite-element method The finite-element method was first used in medical orthopedics but was later introduced in other

From the Royal Dental College of Aarhus, Denmark. Supported by The Danish Research Council. Funding provided by ColgatePalmolive AIS and The Dannin Foundation. 8/1/20725

specialties, such as orthodontics. The first finiteelement models described the tooth-bone structure twodimensionally, using average geometric relationships and homogeneous and isotropic material models. 9~3 Three-dimensional, finite-element models were first published in 1973. ~4 Since then, increased interest has produced a number of papers on three-dimensional models. ~527Of these, only three2s27 describe the relation between well-known force systems and the distribution of stress/strain in surrounding tissues. These analyses. were based on average morphologic dimensions and models constructed of isotropic materials. Strain-gauge technique The strain-gauge techriique for registration of tooth movement has been demonstrated as a precise and valid method. ~'28 A disadvantage, however, is that the method is invasive. A laser-holographic technique was therefore introduced by Pryputniewicz et al. 29 as a noninvasive alternative for registration of initial tooth displacements. In vivo tooth movements were measured with this technique in human subjects, 3°-32but interpretations of the results were difficult. A laser reflection technqiue and holographic interferometry were used to evaluate tooth displacements in a macerated human skull. 3~3~ The value of such studies is limited by the fact that the mechanical properties of the surrounding 427

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Orthod. Dentofac. Orthop. May 1991

rection perpendicular to the long axis of the tooth to an extension from the crown and one mounted directly to the crown surface, parallel with the long axis of the tooth. The clip gauges were constructed to measure linear displacement with a sensitivity of 0.1 /.tm. The accuracy of the measuring device was high; the coefficient of variation was 7% for the values measured. These values were amplified and transfen:ed to a computer, and the center of rotation and the angular displacement were calculated. The center of resistance-coinciding with the line of action of the translational force, with the angle of rotation approaching z e r o - was established. All collected measurements were stored for subsequent statistical processing.

Force system

Fig. 1. Radiograph of specimen 1.

tissues could differ from those of actual human tissue because of the limitations of the artificial substitute for the periodontal ligament (RTV-615, General Electric, Waterford, N.Y.). 36 Moreover, repeated loading might cause microfractures.

MATERIALS AND METHODS To establish the finite-element model, previously unknown parameters concerning material properties and bone structure were identified from autopsy material.

Autopsy material Mandibular sections that included the first molar and the first and second premolars were obtained at autopsy from the cadavers of a 15-year-old girl, a 20year-old man, and an 80-year-old man. The tissue blocks were removed within 6 hours of death and stored for no more than 4 hours in Ringer's solution, pH 7.4, at room temperature, to minimize the postmortem changes on the mechanical properties of the tissues. 37 The second premolar was extracted to simulate the clinical situation of space closure. A radiograph of specimen 1, from the 15-year-old girl, is shown in Fig. 1. The measurements were taken at room temperature (22 ° C).

Strain-gauge method The experimental method is shown schematically in Fig. 2. Tooth movements were registered by three strain-gauge bridges, two mounted in a horizontal di-

A rigid extension mounted to the crown of the tooth made it possible to vary the point of force application in the occlusoapical direction (Fig. 2). Steel indicators were placed on the bone surface of the specimens in • order to identify the geometric long axis of the tooth on a radiographic exposure. The loading was generated from a pull-and-dead-weight system, with the force vector adjusted constantly perpendicular to this axis. Each force application was repeated four times, and the resuits are presented as mean values. The tooth movements were registered 120 seconds after the application of force. 38

Finite-element analysis To allow analysis of the initial stress in the periodontal ligament after application of orthodontic forces, two finite-element models, based on a full and a p~lrtial mandibular morphology, were produced. Boundary conditions and material structure parameters for the segment model were determined by a technique called "the multiple modeling technique," in which data are transferred from a rough model to a more detailed one. The computer software ADINA (ADINA Engineering, Watertown, N.J.) was used as the numerical tool for determination of stress conditions.

Geometrical modeling Both the mandibular and the segment models were based on sections of biologic structures--the model of the whole mandible on sections of a macerated mandible, and the model of the segment on sections from specimen 1. The model of the mandible consisted of 168 isoparametric elements interconnected in 255 nodes. A general-element configuration of 17 nodes was

Volume 99 Number 5

429

Material parameters and stress profiles within periodontal ligament

I

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x

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[Y

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y

Finite-elementconfigurationsimulating compactand trabecularbone parts of mandible.

adapted to each of the sections, and the final model included compact and trabecular bone components (Fig. 3). The segment was modeled in a similar way. A grid was fitted for each of the 15 horizontal sections of the autopsy material (Fig. 4). The model included five tis-

sues: compact and trabecular bone, periodontal ligament, dentin, and enamel. Lines separating these tissues were identified. The single layers were interconnected, and the final finite-element model, consisting of 1835 nodes and 1386 isoparametric elements, was established (Fig. 5).

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Am. J. Orthod. Dentofac. Orthop. May 1991

X

Fig. 4. Finite-element grid fitted to horizontal section of autopsy material 1.

Material parameters Three different approaches were taken to establish material parameters of the tissues used in the two finiteelement models. 1. As the stiffness of both enamel and dentin is relatively intense, these materials were modeled as isotropic. Measurements of engineering constants such as the modulus of elasticity and Poisson's ratio have been performed by other authors. 39"~2The compact and trabecular bones were modeled as orthotropic for both models. The property values chosen from the literature are shown in Table II. 2. Determination of the directional dependence of the orthotropic material properties for the compact and

trabecular bone assumed that fibers in bones are oriented along the direction of maximum stress, and that nature's criterion for optimizing fiber directions is stiffness, rather than strength, with minimal bending of the mandible. The optimal fiber direction was determined analytically; the optimum structural load was considered to be the swallowing force. The validity of this method was evaluated through comparison with histologic sections. 3. Because the literature showed major variations in values of mechanical properties of the periodontium, we decided to determine them in this study. The elastic stiffness was determined by iterative comparisons of values of tooth displacement in which

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Material parameters and stress profiles within periodontal ligament

~

431

Z

y

Fig. 5. Final finite-element configuration simulating specimen 1.

the elastic-constant values of the periodontal ligament in the finite-element model were constantly changed until analytical results approximated experimental data. Boundary conditions Boundary conditions for the mandible were transferred to the segment by a multiple modeling technique. The mandible model (which included only half of the mandible) was brought into equilibrium by a process involving assumptions about the distribution of muscle force: chewing force, reactive forces at processus condylaris, and forces in the midsagittal plane. 43'44 Equilibrium was established as the sum of forces and moments in all three planes of space. In the mandible model, forces at specific sites that corresponded to the end sections of the segment model were calculated and then transferred to the segment model. RESULTS Tooth movement The tooth displacement resulting from a horizontal force of 100 cN exerted at different occlusoapical levels of the premolar was registered. The center of resistance

for the premolar of autopsy specimen 1 was located 3.7 mm apically from the alveolar crest (Table I). A force of 100 cN applied at this level will produce a translation of 6.6 Ixm. As the magnitude of translation for the premolar was used in the finite-element analysis for identification of unknown parameters, it was important to justify the validity of these values. However, the process of measurement could not be repeated because the autopsy specimen had been cut into slides for construction of the finite-element model. The validity of the values used for the finite-element model was, therefore, evaluated by comparison with measurements obtained from two additional specimens (Table I). The translational displacement for specimens 2 and 3 were 3.9 i.tm and 6.7 I.tm, respectively. The difference between the measured values, taking the biologic variables such as age, root length, morphology and bone level into consideration, seems to justify the measured value of 6.6 Ixm for specimen 1. The constitutive behavior of the PDL is, in general, believed to be nonlinear and anisotropic. To test the linearity of the deformation of the included structures, the tooth displacement was measured for different force

432 Andersen, Pedersen, and Melsen

Am. J. Orthod.Dentofac.Orthop. May 1991

Table I. Location of center of resistance and distance of translation of premolar for specimens 1, 2, and 3

Autopsy specimen no.

Age (yr)

Root length--ah'eblar crest to apex (ram)

I

15 20 82

il.9 16.0 14.7

2 3

Distancefrom center of resistance to ah'eolar crest (mm) I Mean SD 3.7 6.5 7.9

~ e Distance of translation (tun) I an Range

0.1 0.2 0.2

6.6 3.9 6.7

5.4-8.7 2.4-6.2 5.1-9.8

Table II. Material properties proposed by different authors

Material constants Material group Substantia compacta in the whole mandible Substantia compacta in the whole mandibleand the mandibularsegment

Substantia spongiosa in the whole mandible Substantia spongiosa in mandibularsegment Dentine:in mandibular segment Enamel in mandibular segment

Isotropic

Orthotropic

E = 13,700 E~ = E. = Ej = Gi2 = G, = G.u =

13,000 14,400 21,500 4,740 5,850 6,560

MPa "r = "r~2 = a-~3= 'rzj = "r21 = %, = %_, =

0.3 0.37 0.24 0.22 0.42 0.40 0.33

Author 53 54 55

56 E = 1i,000

a- = 0.3

E = 7,930

'r = 0.3

E = 16,135

'r = 0.3

E = 46,085

'r = 0.3

53 Mean value Mean value

magnitudes ranging from 5 to 250 cN, applied at a level 3.8 mm apical to the center of the crown. Tooth displacement was registered at the same level. A high degree of linearity was seen within the measuring range. The mean load/deflection rate calculated was 10.7 cN/mm.

Verification of finite-element model The results obtained by direct calculation and iterative calculation were compared, and we concluded that the iterative method yields results with more numerical stability. The model linearity was tested by comparisons among results for forces of 50, 100, and 150 cN in a mesiodistal direction, applied at bracket level. The model was found to be linear. We investigated the influence of modeling the bone structure in great detail by subjecting the bone structure to an optimal load. Although there were small changes in the results, we concluded that it was not

unimportant to include the optimized structures in the models. We also evaluated the effects of the multiple modeling technique for obtaining better boundary conditions for the segment model by excluding these detailed data in some investigations and concluded that it was not important to use these conditions.

Elastic stiffness of the periodontal ligament The elastic stiffness of the periodontal ligament was evaluated by iterative comaprisons of tooth displacement obtained by strain-gauge measurement and finiteelement analysis. Input data for the finite-element analysis such as force system and boundary conditions were similar to those used in the experiments on autopsy material from specimen 1. The elastic stiffness was determined to be 0.07 MPa. The corresponding Poisson's ratio was found to be 0.49 for the autopsy material.

Material parameters and stress profiles withh, periodontal ligament

Volume 9 9 Number 5

433

7.

x

Y

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SECTION LAYER 4 7 6 5 4 3 2 __

m

1

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Stress levels and profiles

The initial reaction to application of force adjacent to the tooth, is manifested as principal stress in the periodontal ligament of the first premolar of specimen 1 autopsy material. The points of stress and the line of application of force in the mesiodistal direction are seen in Fig. 6. Three different force systems are presented. The first was a distal force of I N at the bracket level, central to the buccal surface of the crown (through the long axis of the tooth), in combination with a couple with moment/force (M/F) ratios of 5, 7, and 24 mm. An M / F ratio of 5 resulted in distal tipping of the premolar. For sections 3 and 7, the absolute stress values were almost uniformly distributed and relatively small. The slight irregularity in stress distribution could probably be ascribed to irregularities in the shapes of the roots (Fig. 7). Stresses past the point of uniform distribution reflected translation at an M / F ratio of 7 (Fig. 8). The higher stress at section I was probably caused by a projection at the buccal surface of the root. An M / F ratio of 24 produced a major variation in stress levels from cervix to apex, with the apical part of the root being displaced distally and the gingival

part, displaced mesially. In sections 3 and 7 (Fig. 9), on the distal and mesial surfaces of the root, a nonlinear stress distribution from the cervix to the apex was observed. The nonlinearity could be explained by the fact that the apical apart of the root was bent slightly as a result of the applied force and therefore the stress levels in the apical part of the periodontal ligament were reduced. This bending resulted in less compressive stress on the distal root surface and less tensile stress on the mesial surface at the apical part. When chewing force was added to the orthodontic force system applied above, the stress distribution was entirely dominated by the effect of chewing. No substantial difference in distribution was detected, either for 50 N of chewing force alone or for chewing force in combination with the orthodontic force system (Figs. 10, 11). The absoltue stress values were much higher than those produced by the orthodontic system, and the effect of morphologic irregularity is clearly seen in the uneven stress distribution in the periodontal ligament. DISCUSSION

In studying the remodeling process that leads to orthodontic tooth movement, researchers need to un-

434

Andersen, Pedersen, a n d

Me~sen

Am. J. Orthod. Dentofac. Orthop.

May 1991

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derstand how stress is distributed in the periodontal tissue when force is applied. The distribution of stress around teeth after loading has been studied with finiteelement analysis. 927 The results, however, are dependent on the load, the boundary conditions, and the material parameters. These analyses have been based on average figures for morphologic characteristics of tooth and bone and simplified estimates of material properties. In the present study, the boundary conditions of

the finite-element model were optimized to correspond to the geometry and the structure of the autopsy material that was being analyzed. Determination of the structural parameters for compact and trabecular bone was based on two hypotheses established by Wolff4~ and consolidated by Currey, 46 who concluded that "fibers in bone are oriented in [the] direction of the maximal stress" and "the criteria for optimizing bone in nature is [sic] rather the stiffness than the strength." When the theo-

Volume 99

Material parameters and stress profiles within periodontal ligament

Number 5

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Fig. 8. Stress distribution for distal force of 1 N at bracket and M / F ratio of 7.

retical determination of fiber direction was compared with that of the histologic sections, these assumptions were corroborated. Because the measurements of tooth movement in the autopsy material showed a high degree of linearity, a linear-elastic material was chosen for the finiteelement model. The mechanical properties of the periodontal ligament were obtained by identification of certain parameters. The stiffness of the periodontal liga-

ment was found to be 0.07 MPa, which was very small compared with values cited in the literature. :528 Using three-dimensional finite-element analysis and in vivo measurements of tooth mobility, Tanne et al. 25 found a stiffness of the periodontal ligament of 0.8 MPa, which differed significantly from our results. This difference may be explained by the fact that these authors combined the data from experiments on human beings with data based on standard morphologic and boundary con-

436

Andersen, Pedersen, and Melsen

Am. J. Or:hod. Dentofa¢. Orthop.

May 1991

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ditions for teeth, whereas our experimental data and the input for our finite-element analysis originated from the same material. We evaluated the representativeness of our finiteelement model by comparing data from the model with data from other specimens and previous in vivo experiments. 25"47'.8The results indicated that the amounts of tooth displacement obtained on autopsy material were within the range of what could be expected in vivo. The experimental data in this study were obtained

by the strain-gauge technique, which is invasive, because it adds a minor initial force to the tooth before the actual force is applied. Because the force/displacement relationship was linear, it would not have influenced the results. Another disadvantage of the strain-gauge method--the poor reproducibility of strain-gauge bridges placement, given a new set of starting points--could also be disregarded, since the bridges were reset at the start of each experiment. The present study, having optimized the underlying

Volume99

Material parameters and stress profiles within periodontal ligament

Number 5

LAYER

437

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variables for the finite-element model, made it clear that individualization of the boundary conditions with regard to proportions and bone structure did not significantly alter the results. The material properties, apart from the periodontal ligament, were shown to exert only minor influence. This study thus provides the necessary background for the more simplified models. The findings for the three types of tooth displacement--tipping, translation, and root movement--agree with those for simple analytical

models, 49,5° but they differ somewhat from the threedimensional finite-element results obtained by Tanne et al. 26 in which, rather than being uniform, the stress was found to be concentrated at the center of the root. Periodontal response to orthodontic force is known to depend on magnitude and mode of application-e.g., intermittent or continuous force2 When we compared the distribution of stress from orthodontic force alone with the distribution obtained when orthodontic force was combined with chewing force, we found that

438

Am. J. Orthod. Dentofac. Orthop.

Andersen, Pedersen, attd Melsen

May 1991

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the periodontal ligament, at the levels o f force used in the study, would be subject to intermittent stress. The force exerted by a continuous stainless steel wire is usually considerably higher 51 and may result in a continuous compression of the periodontal ligament, with known consequences, such as local necrosis, hyalinization, indirect resorption, and, eventually, root resorption. The full benefit o f the present study can be obtained

only when analyses of cellular reaction to the different stress levels become available.

REFERENCES

I. Norton LA, Burstone CJ. The biology of tooth movement. Boca Raton, Fla.: CRC Press, 1988. 2. Storey E. The nature of orthodontic tooth movement. A.',I J

OR'roOD 1973;63:292. 3. Norton LA, HaRley KJ, Turkewic J. Bioelectric perturbationof

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6.

7.

8. 9. 10.

I1.

12.

13.

14.

15.

16.

17.

18. 19.

20.

21.

22.

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Material paranteters and stress profiles within periodontal ligament

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