by orthodontic fames Dr. Burstone

Charles J. Burstone* and Ryszard J. Pryputniewicz** Farm&on, Corm.

A new tool for measuring tooth movement-luser holography-ogler:, on uccutute. noninvasive approach for determining movement in three dimensions. This iu \,iiro study is designed to establish the required force system applied on the crobttl of (I maxillas incisor thut would produce different centers of rotation, LIS in linguul tipping, translation, and root movement. The relationship between monlmt-to-f~)rc,e ratios and centers of rotation is shown. The experimental data are compured to theoretic approaches. With respect to the location of the center of resistance aud centers of rotation, force systems needed to produce different centers of rotcrtion clre given for a central incisor of averuge root length.

Key words: Orthodontics, holography, centers of rotation, forces

T he stresses generated in the periodontal ligament when the crown of a tooth is subjected to a force have important ramifications for the study of orthodontic tooth movement and periodontal disease. In particular, the orthodontist desires to relate the force system applied to the teeth to the center of rotation and the magnitude of tooth displacement. In the study presented here, laser holography, a new technique applied to orthodontics, was used to predict three-dimensional tooth displacements.

Previously, tooth displacements have been studied from a number of approaches: (1) analytical models, (2) physical models, and (3) direct measurement in vivo.

Burstone discussed conditions for optimization of forces used in orthodontic treat- ment and pointed out that more knowledge is needed to determine what force systems produce the various centers of rotation. Miihlemann,2-4 using dial indicators, measured tooth mobility in normal subjects and patients with periodontal disease. Christiansen and Burstone,j also using dial indicators, estimated that centers of rotation were close to the center of the root with single force loading on the crown of a tooth. Synge6* determined analytically the stress distributions for two root shapes a~?ar, two-dimensional wedge and

This study was supported by Research Grant No. DE-03545 from the National Institute of Dental Research, National Institutes of Health, Bethesda, Md. *Professor and Head, Department of Orthodontics, School of Dental Medicine, University of Con- necticut Health Center, Farmington, Conn. **Permanent address: Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, Mass. 01609.

396 C002-9416/80/040396+ 14$01.40/O 0 1980 The C. V. Mosby CO.

Volume II Number 4 Holographic determination of centers of rotation 397

a three-dimensional cone of revolution and also analyzed the problem of the tooth with a root of general conical form and uniform ligament thickness.8 Dyment and Synge deter- mined values for the elastic coefficients of the periodontal ligament. Furthermore, Synge developed a theory of equilibrium for a compressible membrane which was later extended by Hay1-13 to treat thin membranes and stresses in the periodontal ligament. Two- dimensional analytical models have been developed by Burstone, Nikolai,14 and Davi- dian.15 Three-dimensional stress distributions within the periodontal ligament were de- termined theoretically by Haack16 and Haack and Heft.17

The effects of force on the supporting periodontium and, in turn, tooth movement have also been studied by constructing physical models in photoelastic plastic and analyzing photoelastically the stress distribution produced by the applied force.*-*

Unfortunately, the attempts at mathematical modeling by an analytical approach as well as photoelastic techniques have been limited by a number of oversimplifying as- sumptions, such as (1) the anatomy of the root, periodontal ligament, and alveolar bone were represented by idealized geometric forms, (2) the physical characteristics of the supporting structures were assumed to be homogeneous, isotropic, and linear, whereas the structures of interest here are nonhomogeneous, anisotropic, and nonlinear. Furthermore, in most instances the model was two-dimensional.

In addition to the above, previous experimental studies of force-displacement charac- teristics of teeth have yielded low predictive capability because most of these experiments (1) employed forces that produced three-dimensional displacement of the tooth and yet the tooth displacements were measured along one axis only; (2) produced three-dimensional tooth displacements which resulted from an applied force with three components and yet the force, if monitored at all, was measured along one axis only; (3) employed a force system whose magnitude changed with deflection and yet often the change of the mag- nitude of the force was not measured as the tooth deflected; (4) used displacement and force-measuring systems whose accuracy was suspect because of mechanical inertia dur- ing deflection in the measuring instruments themselves; and (5) the experimental appara- tus was invasive and influenced the tooth movement.

The above shortcomings in the studies on the prediction of the tooth movement, under the influence of external forces, can be overcome by the noninvasive techniques of laser holography. Unfortunately, there has been a limited application of the modem holo- graphic methods in dental research. Only a handful of investigators have used holographic techniques of laser holography. Unfortunately, there has been a limited application of the modem holographic methods in dental research. Only a handful of investigators have used holographic techniques in their studies. Wictorin and associates23 investigated elastic deformations of dental joints, Wedenal and Bjelkhagen24, 25 measured uniplaner dis- placements of teeth resulting from masticatory forces, whereas Bowley, Burstone, and Koenig26 demonstrated that holography and mathematical modeling can be used to mea- sure and predict tooth displacements. More recently, Pryputniewicz* has developed a technique that, for the first time, allowed tooth displacements to be measured in the three-dimensional space. This technique, based on recent advances in hologram inter- ferometry,28-32 allows noninvasive measurements of tooth displacements with an accu- racy of 0.5 pm. Burstone and Pryputniewicz 33, 34 have successfully applied this technique to the study of in vivo movements of human teeth.

In this article a new, noninvasive method of laser holography, which has been applied

398 Burstorw nrd Pryputnic,\vic,:

Fig. 1. Schematic representation of the 10: 1 model of the maxillary central incisor. The force of 200 grams in the labiolingual direction was applied at various incisal-gingival positions to produce controlled motion of the root.

to the study of the three-dimensional displacements of teeth, is presented. Since the initial tooth movement is small, accurate measurement of the displacements via laser holography avoids errors found in previous studies of this nature. Furthermore, this measuring tech- nique eliminates the influence of mechanical inertia of the apparatus which changes the nature of the periodontal support.

In order to more closely model the incisor and its supporting structures, careful attention was given to (1) the anatomy of the tooth and periodontal ligament and (2) the mechanical properties of the periodontal ligament.

In the study presented here, primary centers of rotation were determined for varying moment-to-force (M/F) rations with respect to a bracket on a maxillary central incisor. (A primary displacement is that movement produced by a force applied to the tooth before resorption and apposition occur. The second biologic phase of tooth movement is referred to as the secondary displacement.) The study was carried out on a 10: 1 three-dimensional model of a maxillary central incisor. By using the model, it was possible to eliminate the biologic variation in a given subject or patient so that base line data could be developed for comparison with in vivo studies. The present three-dimensional model is superior to previous physical models, allows for analytical studies since geometry and constitutive behavior are known, and can be used for correlation with clinical studies using the same methods.

Methods and materials

Experimental apparatus; recording and reconstruction of holograms. In order to carry out the studies presented in this article, a simplified scale model of a tooth-periodontal ligament-alveolar bone was designed and built. The tooth geometry chosen was that of Haacks16 maxillary central incisor: paraboloid of revolution root shape with a uniformly thick periodontal ligament. The model itself was ten times the size of Haacks geometry to

Volume 77 Number 4 Holographic determination of centers of rotation 399

Fig. 2. Schematic representation of the maxillary central incisor showing scaled dimensions of the 10: 1 model depicted in Fig. 1.

allow the periodontal ligament to be 2.29 mm. thick (Fig. l), a workable space for model construction purposes. * For ease of interpretation of the experimental results, the dimen- sions relating to the actual tooth size are also shown in Fig. 2.) The characteristic dimen- sions of the tooth structures used in this study are shown in Fig. 1. The root is machined out of a solid aluminum blank, the alveolar bone was modeled in dental stone, and viscoelastic silicone rubber (GE/RTV-615) was used to represent the periodontal ligament.

We would like to point out at this time that the aluminum, dental stone, and silicone rubber are not compatible with the actual in vivo characteristics of tooth, alveolar bone, and peridontal ligament, respectively. These materials were used in the present study merely for construction of the experimental model.*

This 10 : 1 model of a maxillary central incisor was loaded with a labiolingual force of 200 grams normal to the long axis of the tooth. The point of force application was varied in the occluso-apical direction, as shown in Fig. 1. The magnitude and direction of the applied force were carefully controlled by a pulley and a dead weight system. Each loading condition (that is, application of a force at a given occluso-apical level) was repeated three times, and the results presented herein are arithmetic averages of the corresponding runs.

Application of the force to the model of the maxillary central incisor resulted in three-dimensional displacements. These displacements were recorded and analyzed by the modem, noninvasive techniques of double-exposure hologram interferometry28-32 and the experimental apparatus shown in Fig. 3. This apparatus consisted of a 0.92 by 1.22 m. flat optical table with air suspension. The illumination for recording and reconstruction of holograms was provided by a 15 mW He-Ne laser. The laser, the optical components for steering and shaping of object and reference beams, the tooth model, and the photosensi- tive material used for recording of holograms were rigidly mounted on top of the optical table by means of magnetic bases. The appropriate exposure times needed to record holograms were determined by taking into account viscoelastic properties of silicone rubber,* which was used to model periodontal ligament, and were effectively monitored by an electronic shutter system. All holograms were recorded on 102 by 127 mm. plates with Agfa-Gevaert lOE75 emulsions.

In order to understand the use of laser holography in the determination of tooth

400 Burstonr und Pryputnirwicz 4nr ./. Orrhocf. April I980

Fig. 3. Experimental setup of holographic apparatus. All of the components of the system were rigidly mounted on the air-suspended optical table. The tooth was loaded by means of pulley and dead-weight systems. The exposure of holograms was controlled by the exposure meter and shutter.

displacements, it is necessary to discuss some of the principles of hologram inter- ferometry.

In practice, a hologram is constructed with an experimental setup similar to the one shown in Figs. 3 and 4. The highly coherent and monochromatic light from the laser source is split into two beams by means of a beam splitter (Fig. 4). One of the beams is directed by mirrors, expanded by means of a spatial filter (microscope objective and a pinhole assembly), and is used to illuminate the teeth to be recorded. This beam, referred to as the object beam, is modulated by a reflection from the tooth and carries all the information about the instantaneous condition of the tooths surface. The second beam is known as the reference beam and is not modulated by any intervening object. If both of these beams are allowed to impinge on some kind of a surface, they will produce a set of fringes, on that surface, as a result of their mutual interference. The spacing of the fringes is entirely dependent on the angle between the two beams and the wavelength of the light used; the fringe opacity is related to the intensities of the interfering beams.

The fringe pattern resulting from the superposition of two beams can be recorded in the photographic emulsion (plate in Fig. 4) which, upon photographic processing, be- comes a hologram. The hologram bears no resemblance to the original object, it is quite unintelligible and gives no hint of the image recorded. It is quite unlikely that one could learn to interpret a hologram visually without actually reconstructing the image.

The hologram can be reconstructed with the original system setup used in recording, but now it is illuminated with the reference beam alone (Fig. 5). During the hologram

Volume 71 Number 4

Holographic determination of centers of rotation 401

m REMOTE

CONTROL

Fig. 4. Schematic representation of hologram recording setup. The highly coherent and monochromatic light from laser illuminates the tooth and exposes the photographic plate.

reconstruction, a portion of the laser light is let through the plate undeviated (the so-called zero-order wave) and the remaining light is diffracted into higher orders. Out of the number of diffracted beams, the most important, in holography, are two first-order wave- fronts, one on each side of the zero-order waveform.

One of these diffracted orders consists of waves that produce an image of the original object, as if it were still located behind the plate at the position it occupied during the recording, although the object had since been removed. A camera placed in this beam may be used to photograph this (reconstructed) virtual, sometimes also called true, image. A typical virtual image obtained during reconstruction of a double-exposure hologram is shown in Fig. 6. The actual displacements and rotations the individual teeth have experi- enced during recording of a hologram can easily be determined from resulting interfero- grams by the techniques discussed by Pryputniewicz. z&3* The virtual images have to be viewed through the hologram as if it were a window for this procedure.

Determination of center of resistance

By definition, the center of resistance is found at a point where a single force produces pure translation. In the experiments reported in this article, we have loaded the 10: 1 model of the maxillary central incisor with a force of constant magnitude and with the line of action horizontal and normal to the long axis of the tooth. The point of loading was then varied occluso-apically, as shown in Figs. 1 and 7. By loading the model in this manner, we have produced varying moments with respect to the center of resistance and, therefore, varying amounts of tooth rotation.

The tooth loaded with a lingual force of 200 grams, parallel to the Z axis, rotated primarily with respect to the X axis (that is, the mesiodistal axis), while rotations with respect to the remaining axes were negligible and were omitted for the sake of clarity (Fig. 7). Varying the M/F ratio by moving the horizontal force in the vertical direction,

RECONSTRUCTIN

Fig. 5. Schematic representation of holographic reconstruction setup. One of the first-order wavefronts (resulting from a diffraction of the reconstructing beam by the hologram) appears to emanate from the

position in space where the object was during recording of the hologram. The viewer, looking through the hologram as if it were a window and placing himself or herself in the direction of this beam, sees the three-dimensional virtual (true) image of the tooth, although the tooth itself might have been removed from the recording space.

different amounts of rotation were produced. With the force placed at the incisal edge, rotation was large. The magnitude of rotation decreased as the theoretical center of resistance was approached and then increased as the point of force application neared the apex. The point where this curve intersected the vertical axis (that is, where rotation was zero) was, by definition, the experimental center of resistance. The experimental center of resistance was found to be 9.9 mm. apical to the bracket.

The experimentally found center of resistance was compared to the location of two theoretical centers of resistance (Fig. 7). One was based on a simple two-dimensional parabolic model of the tooth with uniform stress distribution and linear properties of periodontal ligament,jq 35 where the centroid was determined at two-fifths of the root length measured apically from the alveolar crest. However, in order to represent an actual tooth more closely, a second theoretical center of resistance was determined for a three- dimensional root geometry. In this approach, the centroid of a paraboloid of revolution was selected, which was found to be at one-third of the root length measured apically to the alveolar crest. Note the close correspondence between the three-dimensional theoreti- cal center of resistance (10.2 mm.) and the experimentally determined center of resistance (9.9 mm.). The theoretical two-dimensional center of resistance lies further apically at 11.0 mm.

The reason that the center of resistance moves occlusally in the three-dimensional model, as compared with the simple two-dimensional one, becomes apparent when one makes numerous thin sections of the root, parallel to the long axis of the tooth (Fig. 8, a). Each of these sections approximates a two-dimensional parabola for which the centroid is

Volume II Number 4 Holographic determination of centers of rotation 403

Fig. 6. Photographs of images of the model of the maxillary central incisor obtained during reconstruc- tion of double-exposure holograms. The angulation and spacing of fringes change as the point of force application is varied from (a) apical to the centroid, to (b) through the centroid, to (c) incisal to the centroid.

TCR - THEORETICAL CENTER OF RESISTANCE -20

/

Ei= ECR - EXPERIMENTAL CENTER EE

OF RESISTANCE ,-ii

1 5 BRACKET i Fig. 7. Holographically determined rotations at the bracket for loading with a lingual force of 200 grams normal to the long axis of the tooth model and at different occlusogingival positions. Note that theoreti- cally the center of resistance is at a point where there is zero rotation.

located at two-fifths of the height of the section as measured from the base toward the apex (Fig. 8, b), which corresponds precisely with the two-dimensional model used previ- 0us1y.~ Plotting the locations of centroids of each of the thin sections of Fig. 8, a, we obtain a curve similar to the one shown in Fig. 8, c. The centroid for the entire three- dimensional root is then found to be located on the long axis, one-third of the root length apical to the alveolar crest.

Calvin Case36 developed an appliance system based on the use of a single force on an extension attached to the band to produce translation. In recent years this concept has been reintroduced. A gingival extension from the bracket on a typical central incisor would have to be approximately 10 mm. in length to translate the incisor. Using the centroid of a

404 Burstone cd Pryputnieuic-

0 - CENTROW OF A THIN SECTION @J - CENTROID OF A PARABOLOID ,

OF REVOLUTION

/ I (b)

33JTROlDS OF THIN SECTIONS

Fig. 8. The centroid for a two-dimensional parabolic section is found at a point at 2h/5 (a and L$; the centroid for a three-dimensional paraboloid of revolution is at h/3 (c).

paraboloid of revolution as an estimate of the center of resistance, one can determine the length of a gingival extension for other teeth. For example, an average maxillary canine has an alveolar-crest-to-apex dimension of 16 mm. Therefore, the canines centroid lies 5.3 mm. from the alveolar crest and is 0.9 mm. more apical than the centroid for the central incisor, which locates at 4.4 mm. A slightly shorter crown and a more gingival placement of the bracket on the canine suggests that the distance from bracket to centroid will be similar for maxillary canines and maxillary incisors.

Determination of centers of rotation by varying moment-to-force (M/F) ratios at the bracket

It has been long established that the point of force application is an important deter- minant of the center of rotation of a tooth. Although it is feasible to carry out some orthodontic treatment by applying the force at different points along the surface of the tooth or through an extension, most of the multibanded techniques employ the application of a force and a pure moment at the bracket on a crown of a tooth.

In this study, single forces were used. The point of force application was varied occlusoapically, and the centers of rotation were then determined. It should be noted that any of these single forces can be replaced with an equivalent force and a couple (a pure moment) at the bracket. In Fig. 9, b and c a single force F, has been replaced by an equivalent force system consisting of a moment (M) and a force (Fb) at the bracket. The sign convention is given in Fig. 9 for the force (F = Fb), the moment (M), and the moment-to-force ratio (M/F). Note that the coordinate system used for the maxillary teeth is the left-hand coordinate system with the Y axis pointing in the occlusal direction. The data obtained in this study were presented as a function of moment-to-force ratio at the bracket, since this is the typical mode of force application that is used clinically. The moment-to-force ratio, in reality, represents nothing more than the distance from the

Volume 77 Number 4

Holographic determination of centers of rotation 405

(b)

x

6

2

I Y Fig. 9. Sign convention for force systems; note that the left-hand rectangular coordinate system is used. Moment in the counterclockwise direction with respect to the mesiodistal axis (the X axis) is positive and a force in the linguolabial direction is positive. a, Force at the bracket: equivalent M/F = 0. b, Force incisal to the bracket: equivalent force system at the bracket results in a positive M/F ratio. c, Force gingival to the bracket: equivalent force system at the bracket produces a negative M/F ratio.

bracket to a point from which a single force could produce the same effect. For example, a force applied at the bracket (Fig. 9, a) has an equivalent M/F ratio of zero, as determined at the bracket. Fig. 9, b illustrates the case in which the force is incisal to the bracket. This force can be substituted by an equivalent force system at the bracket, consisting of a negative lingual force (Fb) and a negative moment (M); therefore, the corresponding M/F ratio has a positive value. In a similar way, the force (F,) apical to the bracket (Fig. 9, c) can be substituted by an equivalent force system at the bracket, consisting of a negative force (Fb) and a positive moment (M), thus yielding a negative M/F ratio.

In Fig. 10 the experimental location of the center of rotation (measured in millimeters from the centroid along the long axis of the tooth) was plotted versus the moment-to-force ratio as evaluated at the bracket. In this figure, the centroid was calculated at one-third of the root length apical to the alveolar crest. As the moment-to-force ratio approaches infinity in either a positive or a negative direction, the center of rotation approaches the centroid of the root. Moment-to-force ratios of minus 2.5 or greater will produce centers of rotation very close to the centroid of the root. The same is true of the moment-to-force ratios of minus 17.5 or less. In other words, if a single force is placed 2.5 mm. or more incisal to the bracket, or 17.5 mm. or more apical to the bracket, the tooth will rotate around a point near the centroid.

With the moment-to-force ratio at the bracket of -9.9, the central incisor translates.

INCISAL EWE (AT 14.2 MM1

-30 -25 -20 -I5 -10

- EXPERIMENTAL RESULTS

- - - THEORETICAL RESULTS

I / I -5 0 5 IO

M/F RATIO AT THE BRACKET

Fig. 10. Center of rotation measured from the centroid of the paraboloid of revolution (h/3) as a function of the M/F ratio at the bracket. The center of rotation approaches infinity as the line of action of the applied force approaches the centroid.

At a M/F ratio of zero (that is, a single force at the bracket), the center of rotation lies slightly apical to the centroid. As the values of the M/F ratio at the bracket become more negative, the center of rotation moves apically. When the M/F of -9.9 is reached, the center of rotation is at infinity. Negatively increasing the M/F ratio moves the center of rotation from infinity to centers incisal to the bracket. Further negative increases of the M/F ratio will cause the center of rotation to move from points incisal to the bracket toward the centroid. It should be noted that the moment-to-force ratios for translation, rotation at the apex, and rotation at the incisal edge are found in a very narrow range. The actual moment-to-force ratios are given in Table I.

When 10: 1 model of the maxillary central incisor was used, it was found experimen- tally that tipping at the apex requires a moment-to-force ratio of -7.1; to translate the tooth, a moment-to-force ratio of -9.9 is needed; rotation at the incisal edge is caused by M/F = - 11.4. These differences are small and suggest some of the clinical problems in trying to control centers of rotation accurately during tooth movement.

On the basis of a simple two-dimensional model, Burstone has suggested that the M/F ratio with respect to the centroid of a root determines the instantaneous center of rotation of a tooth. The formula developed was (M/F) = 0.068.h2/y where h is the root length from the alveolar crest to the apex and y is the distance from the cetroid (deter- mined at two-fifths of root length) of a parabola, representing the morphology of the Foot,

volume 71 Number 4

Holographic determination of centers of rotation 407

Table I. Moment-to-force ratios required for typical centers of rotation

Location of center of rotation

Moment-to-force ratio at bracket

Experimental Theoretical

Infinity -9.90 -11.00 10 mm. apical to apex -9.24 - 10.35 Apex -7.10 -9.52 Bracket - 12.50 -12.06 Incisal edge -11.40 -11.78

to the center of rotation. In this formula, the M/F ratio is determined at the centroid using (two-fifths)=h as a two-dimensional estimate. The center of rotation of the three- dimensional physical model was determined holographically and compared to this two- dimensional model. Since experimental M/F ratios were given with respect to the bracket, the M/F ratios at the centroid, as compared theoretically, were correlated to the equivalent values at the bracket. The locations of the centers of rotation for the theoretical model were also plotted in Fig. 10 as functions of the M/F ratios at the bracket.

The experimental results show that the location of the center of rotation is less sensi- tive to the M/F ratios than the theory indicates. The central portions of the experimentally obtained curves show smaller changes in the location of the center of rotation for a given change in the M/F ratio than those predicted by the (two-dimensional) theory. For exam- ple, the experiments show that in order to shift a center of rotation from the bracket to the incisal edge (Fig. lo), the M/F ratio at the bracket should change from - 12.5 to - 11.4, whereas according to the (two-dimensional) theory, this change in the M/F ratio should be from -12.1 to -11.8.

The decreased sensitivity of the three-dimensional model suggests that activations of appliances may not be as critical as previously thought. However, small changes in the M/F ratios still should be expected to produce large changes in the position of the center of rotation as one approaches the M/F ratio required for translation.

The relationship between the M/F ratio applied at the bracket and the center of rotation has been demonstrated for a three-dimensional model. Certain limitations in using this model should be pointed out. (1) Although the model has a nonlinear periodontal liga- ment, the actual properties in vivo of the periodontal support are no doubt different. (2) Because of nonlinear periodontal support, varying the magnitude of force while maintaining the same M/F ratio could alter the center of rotation somewhat. This has already been demonstrated in the in vivo studies with single forces applied at the bracket.33, 34 In the study reported here, only one magnitude of force (200 grams) was used. (3) Since only one root length was studied, the nonlinear properties of the periodon- tal ligament could influence the center of rotation if variation in the root length is encoun- tered.

Currently, we are studying the in vivo effects of M/F ratios, force magnitudes, and root geometry on centers of rotation. With its limitations, the 10: 1 three-dimensional model gives the best estimation of the locus of the centers of rotation. These estimates will be refined as more biologic data are available from the in vivo studies.

408 Burstone mui Pryputniewic: 4m J Orrhml .4prrl I980

Summary

A new tool based on laser holography was used to study three-dimensional tooth displacements. In this study, 200 gram loads were placed on 10: 1 model of the maxillary central incisor. It was found that the center of resistance was at a point one-third of the distance from the alveolar crest to the apex. The centers of rotation as measured experi- mentally differed from the theoretical estimates based on the two-dimensional model in that they were less sensitive in establishing commonly used centers of rotation. The IO : 1 model offers a very useful adjunct to the in vivo studies employing laser holography in that the geometry of the tooth and the periodontium can be kept constant under different loading conditions. Hence, it can serve as a base line for comparison of biologic data measured in vivo where greater variabilities are encountered in geometry, loading condi- tions, and the constitutive behavior of the periodontal support.

REFERENCES 1. Burstone, C. J.: Biomechanics of tooth movement. Kraus, B. S., and Riedel, R. A. (editors): Irr, Vistas in

orthodontics, Philadelphia, 1962, Lea & Febiger, pp. 197-213. 2. Miihlemann, H. R.: Periodontometry-A method for measuring tooth mobility, Oral Surg. 4: 1220-1233,

1951. 3. Miihlemann, H. R., and Houglum, M. W.: The determination of the tooth rotation center, Oral Surg. 7:

392-394, 1954. 4. Miihlemann, H. R.: Ten years of tooth-mobility measurements, J. Periodontal. 31: 110-122, 1960. 5. Christiansen, R. L., and Burstone, C. J.: Centers of rotation within the periodontal space, AM. J. ORTHOD.

55: 353-369, 1969. 6. Synge, J. L.: The tightness of the teeth, considered as a problem concerning the equilibrium of a thin

incompressible elastic membrane, Philos. Trans. R. Sot. Lond. 231A: 435-477, 1933. 7. Synge, J. L.: The theory of an incompressible periodontal membrane, INT. J. ORTHOD. 19: 567-573, 1933. 8. Synge, J. L.: The equilibrium of a tooth with a general conical root, Phil. Mag. J. Sci. 15: 969-996, 1933. 9. Dyment, M. L., and Synge, J. L.: The elasticity of the periodontal membrane, Oral Health 25: 105-I 11,

1935. 10. Synge, J. L.: Equilibrium of a thin flat membrane of compressible material, Trans. R. Sot. Can. 31: 57-81,

1937. 11. Hay, G. E.: The equilibrium of a thin compressible membrane, Can, J. Res. 17: 106-121, 1939. 12. Hay, G. E.: The equilibrium of a thin compressible membrane with application to the periodontal mem-

brane, Can. J. Res. 17: 123-140, 1939. 13. Hay, G. E.: Stress in the periodontal membrane, Oral Health 29: 257-261, 1939. 14. Nikolai, R. J.: Periodontal ligament reaction and displacements of a maxillary central incisor subjected to

transverse crown loading, J. B&hem. 7: 93-99, 1974. 15. Davidian, E. J.: Use of a computer model to study aforce distribution on the root of a maxillary central

incisor, AM. J. ORTHOD. 59: 581-588, 1971. 16. Haack, D. C.: An analysis of stresses in a model of the peridontal ligament, Ph.D. dissertation. Kansas

State University, 1968. 17. Haack, D. C., and Heft, E. E.: An analysis of stresses in a model of the periodontal ligament, Int. J. Engr.

Sci. 10: 1093-1106, 1972. 18. Mahler, D. B., and Peyton, F. A.: Photoelasticity as a research technique for analyzing stresses in dental

structures, J. Dent. Res. 34: 831-838, 1955. 19. Lehman, M. L.: Photoelastic tests with ethoxylene resin teeth, J. Dent. Res. 43A: 967, 1964. 20. Lehman, M. L., and Mayer, M. L.: Relationship of dental caries and stress-Concentrations in teeth as

revealed by photoelastic tests, J. Dent. Res. 45: 1706-1714, 1966. 21. Glickman, I., Roebez, F. W., Brian, M., and Pameijer, J. H. N.: Photoelastic analysis of internal stresses in

the periodontium created by occlusal forces, J. Petiodontol. 41: 30-35, 1970. 22. Baeten, L. R.: Canine retraction-A photoelastic study, AM. 1. ORTHOD. 67: 1 l-23, 1975. 23. Wictorin, L., Bjelkhagen, H., and Abramson, N.: Holographic investigation of the elastic deformation of

defective gold solder joints, Acta Ondontol. Stand. 30: 659-670, 1972.

Volume 77 Number 4 Holographic determination of centers of rotation 409

24. Wedendal, P. R., and Bjelkhagen, H. 1.: Holographic interferometry on the elastic deformations of pros- thodontic appliances as simulated by bar elements, Acta Odontol. &and. 32: 189-199, 1974.

25. Wedendal, P. R., and Bjelkhagen, H. I.: Dynamics of human teeth in function by means of double pulsed holography-An experimental investigation, Appl. Opt. 13: 2481-2484, 1974.

26. Bowley, W. W., Burstone, C., Koenig, H., and Siatkowski, R.: Prediction of tooth displacement using laser holography and finite element techniques, Proc. Symp. Comm. V of ISP, Washington, D. C., pp. 241-273, 1974.

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