DOI: 10.1002/adfm.200600120 FULL PAPER Relations Between

DOI: 10.1002/adfm.200600120

Relations Between Shape, Materials Properties, and Functionin Biological Materials Using Laser Speckle Interferometry:In situ Tooth Deformation**

By Paul Zaslansky,* Ron Shahar, Asher A. Friesem, and Steve Weiner

1. Introduction

Teeth, like many biological materials that form skeletal com-ponents, function under load. Their structures are often com-plex, being both hierarchical and graded.[1] It is widely assumedthat these intricate structures are needed to allow optimal func-tion. However, understanding the relations between the form,material properties, complex shape, and the load-bearing func-

tions of such biological entities is daunting. Finite-elementmodeling is commonly used to relate local properties to theoverall manner in which they deform in order to obtain insightinto normal or pathological function. This approach is compli-cated by inhomogeneities and structural variations. Bones andteeth for example are known to be hierarchical,[2,3] and hencetheir materials properties vary from one location to another.Thus, real-time measurements of function under load areneeded. We present an experimental approach to measure de-formations of irregular-shaped objects. Samples millimeters tocentimeters in size can be mounted and deformed. Becausemeasurements are carried out under water, the method is di-rectly applicable to the study of a variety of biological samples.We illustrate the advantages of the method and its potentialuse by analyzing the deformation of whole teeth.

Many vertebrate and invertebrate teeth[4] are composed of apliant material capped on the outer surface by a stiff and hardthin layer. In mammalian teeth,[5,6] these materials are termedenamel and dentin, known in humans to be separated by a200–300 lm thick soft dentin[7,8] adjacent to a well-studied in-terface.[9] Teeth are known to have complex morphologies withgraded structures,[3] presumably well adapted to the functionsthey fulfil.[10] Yet details of the structure–function relations arenot fully known. Here, we investigate the manner in which thesurfaces of crowns of whole human teeth deform, when com-pressive loads similar to those encountered during everydaymastication are applied. The deformation is monitored bytracking nanometer-scale displacements of the tooth’s outersurface, using speckles of laser light. A phase-shifting interfer-ometer is used to convert changes in the intensity of the full-field images to phase-relation estimates of the light fields im-

Adv. Funct. Mater. 2006, 16, 1925–1936 © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1925

–[*] Dr. P. Zaslansky,[+] Prof. S. Weiner

Department of Structural BiologyWeizmann Institute of ScienceRehovot 76100 (Israel)E-mail: [email protected]. R. ShaharKoret School of Veterinary MedicineThe Hebrew University of JerusalemRehovot 76100 (Israel)Prof. A. A. FriesemDepartment of Physics of Complex SystemsWeizmann Institute of ScienceRehovot 76100 (Israel)

[+] Current address: Department of Biomaterials, Max Planck Instituteof Colloids and Interfaces, Wissenschaftspark Golm, 14424 Potsdam,Germany. E-mail: [email protected]

[**] Support for this research was provided from grant RO1 DE 006954from the National Institute of Dental and Craniofacial Research toS. Weiner, The Weizmann Institute of Science. We thank L. Addadi,B. Z. Ginzburg, J. D. Currey, and I. Rousso for their helpful com-ments, and Y. Shopen, D. Leibovitch, G. Elazar, and B. Sharon fortheir excellent technical support. S. Weiner is the incumbent of theDr. Walter and Dr. Trude Burchardt, Professorial Chair of StructuralBiology.

The manner in which stiff biological objects, such as whole bones and teeth, deform under load can provide direct insight intotheir in vivo functions, while highlighting the relations between their structure and materials properties. A new approach forstudying the mechanical functions of such objects, using as an example the crowns of human teeth, is developed. Tooth-crowndeformation under a compressive load is determined in water using laser speckle interferometry. The deformation patterns areanalyzed using a novel procedure that reveals the relative magnitudes of 3D displacements of the outer surface. Nanometer-scale deformations of natural teeth were compared to deformations of identical acrylic replicas, in order to differentiate be-tween contributions of the structure–material properties from contributions of morphology. It is shown that human premolarsdeform in a manner that is largely controlled by shape; in natural teeth, the enamel cap appears to displace mainly as a rigidbody, undergoing moderate deformation. These observations contribute to the understanding of whole-tooth performance un-der load. The approach for analyzing the deformation of loaded whole objects is directly applicable to the study of many stiffbiological specimens, including comparisons between normal and altered (repaired or genetically modified) bones.

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pinging on the surface. These estimates are used to determinethe 3D surface displacements of samples[11,12] that are loaded incompression while mounted within a water chamber.[13]

During mastication, teeth are cyclically driven into, orpushed against food and various hard objects such as opposingteeth.[10] The forces that develop during function rarely reachthe 500–2000 N peak forces, commonly referenced in the lit-erature.[14] The lower-range forces (< 100 N) are characteristicof normal function,[15] and produce loads that must be with-stood reliably, many times each day over many years. Small asthey may seem, biting forces result in a complex distribution ofstresses and strains[16] influenced by the irregular tooth mor-phology and the complex internal structure.[3,5,17] This affectsthe shape of the working tooth, in that displacements of tens ofnanometers often occur at different locations of the crown.

We devised a method to mechanically load whole teethmounted firmly about the roots, while measuring sub-microme-ter deformations by speckle interferometry. We use electronicspeckle pattern-correlation interferometry (ESPI),[12,18] whichis based on estimates of decorrelation between images ofspeckled interference fields. Speckles are formed when laserlight is reflected from the optically rough surfaces of objects:teeth in our example. Changes in the intensity and distributionof such speckle fields are analyzed by phase shifting,[19] andphase relations of the interferometer beams across the lightfield are determined. These relations are related to changes inthe optical path of light reflected from the surface.[18] By phaseunwrapping and scaling[19,20] it is possible to measure displace-ments as small as 25 nm along three orthogonal axes inducedby deformation. Measurements of multiple points on the sur-face yield X, Y, and Z displacement maps (see the Experimen-tal[13]).

An inherent problem in studies of this type is that deforma-tions are usually measured relative to a fixed coordinate sys-tem. Specifically, when loading stiff biological materials, rigid-body translations often occur that are much larger than thenanometer-scale deformations of interest. Thus, the whole fieldof view moves and an absolute static reference framework is oflimited use. By positioning the origin of the coordinate systemon the moving surface, deformation and rotation are detected,while translation of the surface is not.

In this study, we loaded and compared deformation of natu-ral teeth with exact replicas, constructed from an isotropicacrylic material. Teeth and replicas were loaded by graduallypressing a rod against the tip of their main cusp with microme-ter increments. By comparing deformation induced due tosmaller versus larger contact areas, two complementary loadingregimes were employed: loading of only the tip of the toothcusp (depressing it by multiple 2 lm increments, Fig. 1a) toimitate mastication of hard food (high stress at the tip) versusapplication of load distributed over the upper third of the maincusp (depressing it by multiple 1 lm increments, Fig. 1b) toimitate mastication of softer food. Consequently, the crownswere displaced to a similar extent, although dissimilar stressesdeveloped. By qualitatively comparing the differences in the

distribution of the relative displacements over the entirecrowns, we evaluated the characteristic displacements of thetooth surface. Here, we report whole-tooth deformation ob-served with increasing levels of load.

2. Results

The compression of teeth and their replicas monitoredthroughout a series of load increments resulted in independentsimultaneous measurements of displacements along orthogonalX, Y, and Z axes (see Experimental). Vector maps representingthe total cumulative x and y displacements of each tooth at lowversus high loads revealed similar displacement fields, andtheir relative magnitudes increased with load. Examples of dis-placement vector maps of a tooth compressed by cumulativeloads of 35 and 85 N are shown in Figure 2a and b. Detailedanalysis of the full 3D deformation patterns is very difficultusing such vector representations, because of the inherent com-plexity of interpreting 3D displacements based on orthogonalX, Y, and Z measurements. The analysis is further complicatedby the need to select a reference point, relative to which the in-terferometer phase differences are determined (see Experi-mental). Figures 1 and 2 show such reference points (blackcross). Once set, this point defines the origin (zero) of the coor-

1926 www.afm-journal.de © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Adv. Funct. Mater. 2006, 16, 1925–1936

Buccal

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Figure 1. Teeth mounted in a water chamber. Load is transferred onto thetooth working surface by compressing hardened dental composite.a) Tooth loaded at cusp tip to simulate hard-food chewing (uncon-strained, high-stress regime). b) Tooth loaded over upper third of cusp tosimulate soft-food chewing (constrained, low-stress regime). Displace-ments were calculated relative to the point indicated by +.

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P. Zaslansky et al./ In situ Tooth Deformation

dinate system. This therefore becomes a crucial parameter toconsider during interpretation of displacement magnitudes andtheir orientations. We note that both the object geometry andposition of the reference point are bound to vary between dif-ferent objects, so that simple comparisons between teeth arenot possible. We developed an alternative approach which ana-lyzes the relative displacements of the irregular surfaces, inde-pendent of the location of any specific reference point.

2.1. Representation of Relative Rotation and Deformation

While the absolute positions of the ESPI lasers and imagingcharge-coupled device (CCD) are fixed, the position of the ob-ject changes (albeit slightly) during deformation. For a givenseries of increasing-load measurements, the reference point isin exactly the same position in relation to the detector. It thusappears stationary on all displacement maps in the three or-thogonal axes. We exploit this to filter out any rigid bodymovement of the sample in response to load by adhering to anexperiment protocol whereby consecutive deformation incre-ments are small. Thus, the displacements incurred during anysingle load increment do not significantly move the surfacerelative to any of the detectors of the ESPI CCD, no matterwhere the reference point is placed. (Surface points only moveseveral nanometers relative to each other.) This is becauseeach detected speckle is reflected from an area at least20 lm × 20 lm, while the actual surface displacement at anypoint is << 1 lm during any given load increment. The total dis-placements of surface points are obtained by combining the in-termediate values calculated for each step in the series of loadincrements. Under these conditions, the exact position of thereference point is arbitrary as it is used merely to evaluate thedifferences in displacements of all points in each of the orthog-onal maps.

When different reference points are chosen, the magnitudes(and sign) of the relative displacements in each axis changeaccordingly. The resulting vector maps differ significantly, al-though the displacement differences are not affected. We ex-ploit this in the following manner. At first, new maps are creat-ed for the X, Y, and Z axes by moving the reference pointsystematically over each pixel, calculating the displacementmap, and then summing up the relative displacement values ofall vectors to obtain a single (positive or negative) overall valueof integrated displacement. The integrated displacement valuesfor each reference point for every row and every column formnew 2D maps that contain the relative integrated displace-ments for each axis ( �U, �V, and �W) at every load increment.Next, integrated displacements along the X, Y, and Z axesfound at identical coordinates (x,y) are combined in order toobtain absolute relative displacement magnitudes (RDMs).More precisely, these are the scaled square roots of the sum ofthe squares of the integrated displacements along the orthogo-nal axes. They are scalars representing the magnitudes of rela-tive displacement of each subarea on the imaged surface in 3Dspace. A complete set of RDM values for a series of k load in-crements is given in the form of a 3D matrix:

Adv. Funct. Mater. 2006, 16, 1925–1936 © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.afm-journal.de 1927

a 2 mm

b

Figure 2. Displacement vector maps at low and high load. a) Vector fielddenoting in-plane displacements of a natural tooth under low 35 N loadfor the hard-food regime, relative to an arbitrarily chosen reference pointmarked by the black cross found just below the enamel cap. Arrow lengthscorrespond to the extent of displacement in micrometers, magnified × 200so as to be seen on the enlarged tooth silhouette. The vector map revealsa buccal and downward trend of displacements of about 2 lm at the tip ofthe main cusp (see definition of buccal in Fig. 1). b) A vector map of thesame tooth as (a) under a compressive load of 85 N reveals greater dis-placement magnitudes.

RDM�y� x� k� �

��Y� �X

m�n� �U�m�n�k� � �U�y� x�k��

� �2

� ��Y� �X

m�n��V�m� n�k� � �V�y� x�k��

� �2

� ��Y� �X

m�n� �W�m� n�k� � �W�y� x� k��

� �2��

�Y � �X �1�

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where �Y and �X represent the overall dimensions of the dis-placement map. Each point on the surface is described as (y,x)where y and x represent the respective row and column coordi-nates, �U, �V, and �W represent matrices of the scaled (phase-measured) displacement values of the X, Y, and Z axes, respec-tively, and m and n are internal summation indices. RDMvalues are non-negative, and they have units of distance in mi-crometers. (For additional details of the calculation see Experi-mental.) By calculating the average RDM for groups of 4 × 4adjacent surface points (thus covering areas of 80 lm × 80 lmon the surface) over the entire tooth-crown surface, maps ofapproximately 100 × 100 RDM values are produced.

2.2. Interpretation of Deformation and Rotation Employingthe RDM Approach

We calculated RDM maps simulating in-plane rotation(Fig. 3a) and simple axial compression (Fig. 3d) of a circulardisc. Figure 3b is a vector map of the combined X and Y dis-placements for rotation with the reference point at the centerof rotation (red dot). Here, the Z component of displacement(w) is zero. In a disc rotating around its center, displacementsof points located further from the center are greater than thedisplacement of points located close to the center (red dots I inFig. 3a and b). This is only clearly visible in vector maps withthe reference point set in the center. When the reference pointis positioned elsewhere very different vector maps result andthey are difficult to interpret (data not shown). This does notoccur for RDM maps that are reference-point independent, asseen in the corresponding pseudo-3D representation (Fig. 3c).It is cone-shaped and the RDM values away from the centersymmetrically and monotonically assume higher values. Thus, aconical-shaped RDM map is indicative of rotation.

When the same disc is simulated to be uniaxially loaded incompression (Fig. 3d), a vector map combining the X and Ydisplacement components relative to the center of the disc re-veals greater magnitudes of displacement near the loaded edge,with no displacements across the center of the disc (Fig. 3e).Again, to simplify the simulation, the w component was as-sumed to be zero, and we ignored edge effects (so-called Saint–Venant effects) and lateral expansion due to Poisson’s ratio.The vector maps vary significantly if the reference point ismoved away from the center of the disc (data not shown).A pseudo-3D representation of the reference-point-indepen-dent RDM map (Fig. 3f) reveals a compression map that isV-shaped. The minimum RDM values are aligned across thedisc, oriented in a direction perpendicular to the a prioriknown (vertical) loading direction. The simulated strain ofabout 0.5 % produces RDM values with magnitudes similar tothose shown with a rotation of about 6° (whole-body rotation).

When simulating concurrent compression and rotation of thedisc, a combination of displacements arises (Fig. 3g). As theRDMs are scalars, the contributions of simultaneous rotationand deformation add up. The RDM map accordingly showsboth deformation (V-shape) and rotation (cone-shaped with adistinct minimum). We emphasize that in the RDM representa-

tion, only the magnitudes of the relative displacements are giv-en, and not their directions. A complete interpretation of thefull 3D displacements requires additional analysis of the dis-placement orientations. However, much of the mechanical de-formation in the samples can be readily interpreted from dis-placement magnitudes alone. This is specifically seen whencomparing objects that are structurally different but are similarin morphology, if they are loaded in the same way. This isshown for teeth in the next section.

2.3. Deformation of Whole Teeth versus Isotropic Replicas

Whole teeth and their morphologically identical replicaswere loaded in a similar manner, by carefully controlling thedisplacements applied to the cusp tip in a series of increasingload steps. By applying the load incrementally with sub-mi-crometer precision, we could ensure that both mounting andloading resulted in excellent reproducibility. As these experi-ments were deformation-controlled and not load-controlled,we were able to compare surface displacements of samplescomposed of different materials. This is important because theelastic moduli of teeth are very different from their replicas.(Teeth are composed of different materials with moduli greaterthan 15 GPa, while acrylic resin has an elastic modulus of lessthan 4.0 GPa.) From the RDM analysis of the 3D displace-ments during intermediate increments we observed a gradualand proportional increase in the magnitudes of surface dis-placements with increasing load in all samples (Fig. 4). Wefound that the most-informative RDM maps were of the final(cumulative) load increment.

The results are shown as color-coded RDM contour maps inwhich identically loaded natural teeth and replicas are com-pared for the soft-food regime (Fig. 5) and hard-food regime(Fig. 6) experiments. The overall patterns of the RDM contourmaps are similar in the teeth and the replicas, but importantdifferences in the distribution and RDMs can be observed.These primarily reflect structural differences between eachtooth and its replica.

2.4. Analysis of Reproducibility

Phase-shift data tend to be very noisy when measured inwater.[13] We found, however, that repeated data sets resultedin similar RDM maps. This was quantified in two complemen-tary ways. The standard deviations were calculated for the foursets of measurements performed immediately after mountingand preloading during the hard-food simulation setup. The val-ues, calculated for points on a central line from the left to theright side of the crowns, are plotted beneath each sample inFigure 6. The average standard deviation is about 0.3 lm. Thisis small compared to the total range of displacements withineach specimen which varies between 3 and 14 lm. A morecomprehensive estimate of reproducibility was obtained by de-termining the average RDM differences between sets (differ-ence of volumes beneath the RDM surfaces). These are shown


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I

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d e f

g

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I

[µmm]

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Figure 3. Simulation of rotation and deformation of a planar disc with corresponding RDMs. a) Schematic drawing of a disc rotating around a center axis.Two points I and II are indicated as reference areas close to and far from the center of rotation, respectively. b) Vector map of the combined lateral dis-placements relative to point I at the center of the disc. c) Pseudo-3D representation of an RDM map produced by rotation. Note how areas away fromthe center have higher RDM values (cone shaped). d) Schematic drawing of a disc compressed along one in-plane axis. e) Vector field of the combinedlateral displacements relative to an area at the center of the disc (red dot, (d)). Expansion along the orthogonal axis and edge effects are not shown.f) Pseudo-3D representation of the RDM map, produced by deformation. Points away from the horizontal line bisecting the disc have higher RDM values(V-shaped). g) Pseudo-3D representation of the RDM map for combined rotation and deformation. Color-coded scale bar represents magnitudes ofRDM values in (c), (f), and (g).

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in Table 1 as differences between repeat measurements foreach sample. While replicas generally show a constant averageRDM difference of 0.18 lm, natural teeth show a progressivedecrease in average RDM difference as each data set was col-lected. This unexpected observation may be due to the visco-elastic response of teeth to repeated load cycles. All these dif-ference values are small compared to the overall displacementsmeasured.

3. Discussion

We show that ESPI measurements can be used to map nano-meter-scale surface displacements that occur on the irregularsurfaces of deforming objects under water. Load can be appliedin different ways, by taking advantage of the versatility andadaptability of a polymerizable dental polymer. Furthermore,our RDM representation readily detects deformations(V-shaped forms) and rotations (cone-shaped forms) but is in-sensitive to whole-body translations. As the latter contributesnothing to the interpretation of deformation this is an advan-tage.

The RDM maps of the compressed premolars are all domi-nated by a V-shaped pattern rather than a cone-shaped pattern,implying that most of the displacements are due to deforma-tion rather than rotation. Despite the difference in structureand in properties between natural teeth and the replicas, theorientations of the V shapes are similar. This suggests that thedeformation patterns are determined to a large extent by themorphology or the overall shape of the tooth, and not by its in-ternal structure. This is striking because clearly the tooth has a

complex internal design composed of a thin enamel cap and agraded dentin bulk, whereas the replica is made of a uniformisotropic material. The patterns for the hard- and soft-food re-gimes are quite similar for each tooth, except that in additionto the V-shaped motif, a pronounced cone-shaped motif occursin the hard-food regime, suggesting that substantial rotationoccurred in addition to deformation at higher loads (compare,for example, Fig. 5a to f with Fig. 6a and f). This increased ro-tation for the hard-food regime experiments, might be due tomore off-axis rotational freedom existing when load is only ap-plied to the tip of the main cusp as opposed to over much ofthe cusp in the soft-food experiments. Note in particular themarkedly high RDM values on the unloaded cusp (right side)of the first tooth and its replica (appearing in Fig. 5a and b, andFig. 6a and b). The unloaded cusp is probably rotating, whilethe load is applied to the main buccal cusp and the tooth isprobably bending.

There is a marked difference in the distribution of magni-tudes of the RDM values between the natural teeth and theirreplicas (compare Figs. 5 and 6a vs. b, 6c vs. d, and 6e vs. f).More uniform RDM maps are seen in the natural teeth as com-pared to the replicas. This certainly reflects differences in theload response of the internal structure, as the externalmorphologies (architecture) are identical. Because the RDMmaps from the natural teeth are of the external surface of thecrown, the deformations observed occur in the enamel cap. Wethus attribute the difference in RDM patterns between eachtooth and its replica to the stiffness of this cap. This varies indetail from tooth to tooth, but in general the enamel cap evenlydistributes the load over the entire crown even though loadingis confined to one area of one of the cusps. In this respect, thecap acts as a rigid body. Sub-micrometer structural deforma-tions can be detected even with the relatively low loads thatwere used (< 100 N) implying that the natural cap deforms un-der mastication forces that are quite low. Note that RDM val-ues (see soft-food experiments Fig. 5a, c, and e) do not appearto increase monotonically at greater distances from the toothcenter, as is expected for an object that is only rotating aroundits center (see Section 2.2). The dominant deformation of apremolar is therefore due to compression. This may not neces-sarily be the case for other teeth.


0 1 2 3 4 5 6

5 µm 10 µm 15 µm 20 µm

25 µm 30 µm 35 µm

40 µm

RDM[µm]

Figure 4. RDM contour plot maps of intermediate load increments (example of hard-food experiment compressing a natural tooth). Distal aspect of toothis shown. The displacements induced by the compression rod during a typical series of load increments are indicated.

Table 1. Average RDM difference [lm] between four repeated sets of ex-periments.

Tooth 1 Replica 1 Tooth 2 Replica 2 Tooth 3 Replica 3

Err 1–2 [a] 0.4676 0.2755 0.1043 0.0263 0.5815 0.2796

Err 2–3 [a] 0.0142 0.1906 0.0736 0.1555 0.0485 0.0215

Err 3–4 [a] 0.0074 0.2060 0.0120 0.1185 0.0172 0.0141

[a] Err 1–2, etc. refers to the average difference between RDM values insets one and two, and so on.

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Figure 5. Contour plots of RDM maps of distal aspect of natural teeth and replicas loaded in accordance with the soft-food regime. a,c,e) Differentnatural lower premolars and b,d,f) corresponding replica deformation maps. Circle in (a) indicates gingival-buccal region; squares in (c) and (d) indicatelateral positions of minimum RDM value. x and y axes indicate coordinates across tooth surface, with each RDM value obtained from an area of approxi-mately 80 lm × 80 lm. Note lateral millimeter units versus RDM micrometer units.

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Figure 6. Contour RDM representations of natural teeth and replicas loaded in accordance with the hard-food regime. a,c,e) Three different natural lowerpremolar teeth as in Figure 5 with corresponding replicas (b), (d), and (f). Standard deviations (SDs) of the four repeated sets are shown as line graphsbeneath each sample map. SDs were calculated for RDM values on a line across the width of each crown (see dashed line in (a)). The average SD valueis about 0.3 lm.

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An interesting consequence of the deformation that occursin the teeth but not the replicas is the concentration of strainbelow the enamel cap on the buccal side of premolars. In-creased strain appears as marked RDM gradients. This can alsobe directly observed in conventional ex, ey strain maps (see Ex-perimental). It appears that higher strain develops in this re-gion (circle in Fig. 5a), which can be inferred from the differ-ence between the low RDM values on the enamel cap and theappreciably greater RDM values found above this region. Highlevels of strain and stress have indeed been predicted for thisbuccal gum (gingival) area of teeth[21] based on clinical obser-vations[22] and finite-element simulations.[23] Such a concentra-tion of strain may account for the well-known pathology foundin gingival buccal regions of many teeth, often attributed to ag-gressive tooth brushing and frequently termed ‘abfractions’ or‘non-carious cervical lesions’.[24] This experimental approachcould well provide additional insights into other dental pathol-ogies.

It is interesting to note the positions of the smallest RDMvalues of the natural teeth (found by a numerical search for thelowest RDM value of each 2D RDM matrix) in comparisonwith the replicas. These RDM minima are closer to the lingual(tongue) side of the tooth, whereas the minima of the replicasare very close to the geometrical center of the tooth silhouette(as indicated for example by the red square in Fig. 5c and d).For the natural premolars, the RDM minima are near the con-tact area with neighboring teeth. It may well be that naturalteeth are designed to displace less where contact with neigh-boring teeth occurs, and thus minimize potential damage suchas cracking or fracture. Tooth-restoration materials and tech-niques might need to account for the specific mechanical needsof the contact points of teeth.

In order to investigate the loading of teeth, they were rigidlymounted in acrylic polymer. While this allowed us to study theeffect of load on the crowns of each tooth, we note that in thenatural situation teeth are attached to the mandible by a peri-odontal ligament. Thus, a complex interaction exists betweenthe loaded tooth and the surrounding bone. This can be furtherexplored using ESPI by monitoring the time-dependent re-sponse of loaded teeth measured within whole jaw segments.

4. Conclusions and Broader Applications of ThisApproach

Because detailed deformation patterns of small specimens(in the millimeter-size range) can be determined,[25] the possi-bility of studying, for example, bones of genetically alteredmice with a variety of bone abnormalities can now be consid-ered. This could provide information on the mechanisms re-sponsible for altered bone properties. It would also enable thesystematic study of the mechanical functions of many small in-vertebrate mineralized skeletal elements, including jaws andspicules of various kinds. Furthermore, insights into the func-tion of segments of the object under study can be obtained. Anexample of such an application is the study of the interphaseregion between the enamel and dentin of human teeth.[8] The

benefits of using a replica for deconvoluting shape (morphol-ogy) from structure is well illustrated above. There are, how-ever, many more sophisticated techniques for producing 3Dprototypes, and these could be used to probe the structuresfurther.

The experimental configuration used here focused on com-pression. It can readily be reconfigured to enable bending, ten-sion, or cyclic measurements. The use of fast cameras canshorten the acquisition times by a factor of × 100–1000, and,hence, dynamic studies of deformation followed by fracturecould also be mapped, provided that the displacement magni-tudes between successive images are not too large so as to in-duce complete decorrelation (when deforming more than tensof micrometers). Additionally, the versatility of applying theload via a dental polymer and forming a contact area that is tai-lored can be exploited to try and obtain better insight into thein vivo functioning under different loading trajectories. Thisnovel approach to studying the function of irregular-shapedstiff biological specimens opens up new avenues for a betterunderstanding of the basis of their mechanical behavior. Time-dependent properties may be studied by measuring surface dis-placements without changing the level of load in order to studystress relaxation or creep. All this can easily be adapted to thestudy of synthetic biomaterials and artificial prostheses wherethe function of the materials is largely influenced by the mor-phology.

5. Experimental

Whole, intact, fully erupted human lower premolars were used. Allteeth were extracted from young (age < 16) healthy patients undergoingorthodontic treatment. The teeth were stored in water at 4 °C until test-ing. Measurements of surface displacements from eight teeth were ini-tially analyzed under various mounting and loading conditions andthese were used to calibrate the system for elastic loading so that repro-ducible deformation patterns were obtained. Representative results forthree teeth and their replicas, tested under similar mounting conditions,are reported.

To simulate mastication, load was applied onto the load-bearing areaof all teeth by two regimes: in a manner analogous to biting into hardfood (Fig. 1a, applying load to the tip of the main cusp), and in a man-ner analogous to biting into soft food (Fig. 1b, with load distributedover the upper third of the main cusp). In both cases we gradually dis-placed the cusp tip (by 1 or 2 lm, respectively), and as a result cumula-tive loads of about 100 N were applied along the long axis of the natu-ral teeth (about 40 N in the replicas). To induce a similar overall strain(relative to the tip of the cusp) each tooth and replica was loaded by 40successive vertical compression increments, each resulting in load in-crements of 0.8–2 N.

Specimen Preparation and Mounting: All natural teeth were im-mersed in water prior to and during experiments. To facilitate mount-ing, the roots of each tooth were placed in cavities within stainless-steelball bearings (∅ 16 mm) and embedded in self-cure dental acrylic resin(JET acrylic, Lang Dental, IL, USA; Fig. 7a). The ball bearings wereused as standard housings to enable the alignment of each tooth withits specific morphology along a longitudinal physiologic orientation, inorder to mimic, to some extent, in vivo loading conditions. We notehowever that in real life the periodontal ligament lines the root. In-traoral loading conditions, therefore, differ somewhat from those pro-duced here. Prior to loading, each sample was attached to a stainless-steel cradle and the entire configuration was positioned in a sealablestainless-steel load chamber (Fig. 7b). Acrylic resin replicas were pro-


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duced for each tooth by taking dental impressions with quadrafunc-tional siloxane dental impression putty (Aquasil, Dentsply DeTrey,Germany). These replicas were mounted and loaded under conditionsvery similar to those of their corresponding natural teeth.

Mechanical Testing: Gradual compression loading was induced byshifting a stainless-steel rod that was attached to a high-precision DCmotor (PI M-235.5DG, Physik Instrumente, Germany) and controller(Mercury II C-862, Physik Instrumente, Germany). The rod was con-nected in series with a submersible load cell (AL311BN-I6, Honeywell-Sensotec, OH, USA; [L] in Fig. 7b). Prior to loading, the compressionedge of the rod was lined by a small amount of dental composite (FiltekZ250, 3M ESPE, St. Paul, MN, USA; [C] in Fig. 7b). When hardened,this composite layer was used to form the contact area with the toothcusp, simulating mastication of hard or soft food. For experiments bythe hard-food regime (Fig. 1a), the composite was placed on the rodand immediately light-cured (Hand-held LITEX 682, DentamericaCA, USA). Lowering of the rod and compressing the hardened dentalcomposite against the tooth reasonably resembled the action of bitinginto something hard. By using cured composite to load the tooth, as ap-posed to direct contact with the stainless-steel rod, we prevented excessstresses from forming at the cutting tip of the tooth. These could leadto inadvertent chipping or cracking of the enamel cap.

To allow comparison between hard- and soft-food experiments, thesoft-food measurements were performed only upon completion of afull set of hard-food experiments. Each tooth was kept in the exactsame position and orientation so that uncured composite could beadded to the compression edge of the rod. To form the required contactarea, fresh composite was placed on the edge of the rod and waspressed gently against the upper third of the main cusp of the tooth.Using in situ light cure, a gap-free contact area for load transfer wasformed between the rod, composite layer and the tooth (Fig. 1b). Ap-plying load to the tooth in this manner was assumed to be reasonablyanalogous to mastication of soft food. We believe this to hold due tothe fact that in this regime, the tooth becomes constrained about thecusp while the load distributes over a larger area of the tooth’s workingsurface.

To allow laser illumination and phase measurements concomitantwith mechanical testing, the stainless-steel chamber was sealed with ahigh-grade glass window (BK7, k/10 grade). In order to maintain a con-stant wet environment around the tooth, the chamber was filled withwater at room temperature.

All teeth and replicas underwent comparable displacement-con-trolled compression: the tooth cusp tip was pressed in the direction ofthe root tip by a total of either 40 or 80 lm during the soft- or hard-foodexperiments, respectively. The load in each experiment was gradually in-duced by a series of micrometer displacements of the compression rod.A conditioning compression load series was first performed (data werediscarded) followed by four compression series experiments whose datawere used to analyze deformation. Prior to commencing each series, asmall preload (< 5 N) was induced and the system was allowed to settleuntil steady force readings were obtained. Each compression series con-sisted of 40 consecutive discrete increments of load. These different in-crements of 1 lm (soft) or 2 lm (hard) were found to produce compar-able loads in the respective regimes, resulting in displacements of thesame order of magnitude. The load and resultant deformation weremeasured after each increment (see following sections). At the end ofevery compression series, the load was immediately released and thesample was allowed to recover for at least 200 s before performing thenext compression experiment. After repeating four hard-food-regimeexperiments the water was removed from the chamber and fresh un-cured composite was added and pressed into contact with the tooth.Subsequent to light-curing, sealing the chamber, and refilling withwater, four additional compression-series experiments were performed.

Optical Displacement Measurements Using ESPI: The displacementsat each point on the distal tooth surface were determined using ESPI(Q300, Dantec-Ettemeyer, Ulm, Germany) from the difference be-tween light-phase measurements obtained before and after each loadincrement. Cumulative measurements of displacement were obtainedby adding the phase measurements of each point from intermediateload increments. By applying the load in small increments we ensuredthat the area that was imaged by each CCD detector was essentially un-changed between consecutive increments, and also that mechanical vi-brations and water motion were minimized. The ESPI system analyzesspeckle images formed by the laser-illuminated surface [26] by meansof three orthogonal interferometers (using a laser with a wavelength ofk = 780 nm) that are activated sequentially along X, Y, and Z axes ofour coordinate system (Y axial, X lateral, and Z out-of-plane [13]). Thesurface of each sample was imaged through the glass window of thewater chamber at an object distance of 270 mm, and images wereformed on a CCD detector array using a 50 mm television lens. Everydetector in the CCD was illuminated by light reflected from an area ofapproximately 20 lm × 20 lm on the viewed surface. With our setupand using a four-frame temporal phase-stepping technique [27], thephase differences between interfering speckle fields were determinedprior to and after each load increment, appearing as 2p wrapped phase-shift maps [19]. In order to perform automatic phase unwrapping, mod-erate filtering was applied by averaging each data value in the phase-shifted maps with its four nearest neighbors. Displacements corre-sponding to the phase differences were determined after phase un-wrapping, as described elsewhere [13]. The analysis software (ISTRA3D v 3.3, Dantec-Ettemeyer, Ulm, Germany) performs phase unwrap-ping and calculation of changes in the optical path measured relative toone data point, which is used as a reference point. The position of thereference point should ideally correspond to a spot on the deformingsurface that undergoes no displacement during the experiments. None-theless, during each compression increment, all areas on the specimensurface undergo some degree of displacement, albeit of sub-microme-ter magnitude. The small magnitudes of these displacements permit usto assume that the same surface was imaged on each CCD detectorboth before and after load and any of the detectors of the CCD couldbe used as a ‘zero displacement’ reference point. We therefore usedevery point to compute all possible surface-displacement maps alongX, Y, and Z axes. The reference point both establishes an origin of thedisplacement coordinate system (for the three axes) and it sets thepoint relative to which all displacements have either positive or nega-tive values. Gradients within the displacement fields (corresponding to


T

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Figure 7. Tooth and stainless-steel configuration mounted for loading.a) A tooth [T] embedded in acrylic resin as seen after insertion into a stain-less-steel ball and fixed to the test cradle. b) Tooth [T] mounted for loadinginside the emptied stainless-steel chamber. For the hard-food simulations,the composite [C] lining the edge of the compression rod [R] was curedprior to contacting the tooth, whereas for the soft-food simulations the un-cured composite was first pressed against the tip of the tooth to form therequired contact area and only then lightly cured (see Fig. 1b). Load wasmeasured with an intrachamber water-resistant load cell [L] which wasused to monitor the incremental compression load series carried out aftersealing and filling the chamber with water. Scale bar above image: 2 cm.

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surface strains) are unaffected by the choice of reference point. AsESPI measurements are based on correlation and not exact measure-ments of differences between pre- and postdeformation states, smallwhole-body translations along any axis do not affect the results. The or-thogonal u, v, and w components of displacement taken from eachCCD detector are however affected by whole-body rotation as well asby surface deformation, because both involve larger relative surfacedisplacements compared to any chosen reference point.

Numerical Analysis of Deformation: The measured u, v and w com-ponents of displacement determined for all detectors on the CCD arrayspanned 400–500 detector rows and columns, depending on sample size.When arranged into 2D matrices of data, these measurements corre-sponded to the displacements of subareas sized 20 lm × 20 lm on theimaged surface. The displacement maps in each of the three compo-nents had the form:

MATu,v,w(i,j,k) = Du,Dv, Dw(i,j,k) (2)

where k is an integer representing the load increment number withinthe series of measurements, with identical points of detection in all dis-placement maps located at identical columns (i) and rows (j) in all kmatrices, and D represents the incremental nature of displacementsalong each axis. Figure 8 shows an example of one such displacementmap derived in accordance with Equation 2 as seen face on (Fig. 8a)and by a pseudo-3D representation (Fig. 8b). The magnitudes of dis-placement are of the order of 200 nm.

A substantial amount of the displacements induced by compressionoccurred in components other than the teeth. Such components includethe load cell, the composite lining of the moving rod, and the acrylicembedding of the tooth root. As a result, only sub-micrometer surfacedisplacements were observed during any given load increment, al-though the rod movement was 1 or 2 lm, as mentioned above. Sub-mi-crometer displacements were detected in all three components of dis-placement in all experiments.

A low-pass 3 × 3 median filter was applied to each displacement ma-trix, before cumulative orthogonal U, V, and W displacements were cal-culated as a stack of k 2D maps:

U�V�W�i� j�k� ��k

n�1

MATu�v�w�i� j�n� �3�

The cumulative displacements of the last load increment contain infor-mation obtained from all 40 intermediate-displacement increments foran entire experiment. While the cumulative displacement maps varyconsiderably between samples and different reference points, the gradi-ents of the U and V displacement fields, specifically ∂U(i,j)/∂i, ∂U(i,j)/∂j, ∂V(i,j)/∂j, and ∂V(i,j)/∂i are unaffected by the choice of the referencepoint. These gradients can be used as first-order approximations forthe axial and shear surface strains [16,28] as shown for example in Fig-ure 9a and b.

The cumulative displacements were transformed into RDM maps(see Results, Section 2). We found this to be more informative thancomparing gradients, because strain calculations typically requiresmoothing and filtering (resulting in a severe reduction of resolution).To this end, strains are sometimes difficult to interpret when comparingbetween samples that differ both structurally and in loading conditions.Strain, being a tensor, has more than one value at any given point,whereas there is only one non-negative RDM value for each point, cor-responding to its 3D magnitude of displacement. For our analysis, com-parison between RDM distributions of different objects provided ameans of interpreting differences in the deformation of the complexstructures.

The RDM calculated using Equation 1 can be shown to be equiva-lent to calculation of the square root of the sum of squares of the differ-ences between the displacement of each surface point along each axisU(i,j,k), V(i,j,k), and W(i,j,k), and the average displacement of eachdisplacement field of the respective axis (U(k), V(k), and W(k)).RDMs can therefore be expressed equivalently as:

RDM�i� j�k� ��U�k� � U�i� j�k� 2� V�k� � V�i� j�k� 2� W�k� � W�i� j�k� 2

��4�

Thus, the RDM of each point may thereby be seen to depict how mucheach surface area displaces relative to the average displacement of theentire surface. This is the reason we believe RDM maps can be used tocompare deformation of whole biological structures and their artificialreplicas, despite obvious structural and loading differences. The RDMsare affected by rotation which must be subtracted in order to depict de-


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Figure 8. Typical displacement map of one of the k load increments. a) Intensity representation of the shifts of points on the surface produced from an ar-ray of measurements on the i × j detectors of the CCD in accordance with Equation 2 in the text. b) Pseudo-3D representation of the displacementsshown in (a).

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formation. This can be done numerically under the assumption that alldisplacements are extremely small compared to the sample dimensions.

Received: February 6, 2006Final version: June 7, 2006

Published online: September 12, 2006

–[1] R. Lakes, Nature 1993, 361, 511.[2] S. Weiner, H. D. Wagner, Annu. Rev. Mater. Sci. 1998, 28, 271.[3] W. Tesch, N. Eidelman, P. Roschger, F. Goldenberg, K. Klaushofer,

P. Fratzl, Calcif. Tissue Int. 2001, 69, 147.[4] S. Weiner, P. Zaslansky, in Learning From Nature How to Design New

Implantable Bioaterials: From Biomineralization Fundamentals toBiomimetic Materials and Processing Routes, Vol. 171 (Eds: R. L.Reis, S. Weiner), Kluwer, Dordrecht, The Netherlands 2004, p. 3.

[5] A. R. Ten Cate, in Oral Histology: Development, Structure and Func-tion (Ed: A. R. Ten Cate), Mosby, St. Louis, MO 1994.

[6] a) A. Boyde, in Dental Morphology and Evolution (Ed: A. A. Dahl-berg), University of Chicago Press, Chicago 1971. b) W. Gaunt,A. E. W. Miles, in Structure and Chemical Organization of Teeth (Ed:A. E. W. Miles), Academic, New York 1967.

[7] a) R. Craig, F. Peyton, J. Dent. Res. 1958, 37, 710. b) R. Wang, S. Wei-ner, J. Biomech. Eng. 1998, 31, 135.

[8] P. Zaslansky, A. A. Friesem, S. Weiner, J. Struct. Biol. 2006, 153, 188.[9] a) C. P. Lin, W. H. Douglas, S. L. Erlandsen, J. Histochem. Cytochem.

1993, 41, 381. b) V. Imbeni, J. J. Kruzic, G. W. Marshall, S. J. Marshall,R. O. Ritchie, Nat. Mater. 2005, 4, 229.

[10] P. W. Lucas, Dental Functional Morphology: How Teeth Work, Cam-bridge University Press, Cambridge 2004.

[11] C. M. Vest, Holographic Interferometry, Wiley, New York 1979.[12] R. Jones, C. Wykes, in Electronic Speckle Pattern Correlation Inter-

ferometry, 2nd ed. (Eds: P. L. Knight, W. J. Firth, S. D. Smith), Cam-bridge University Press, Cambridge 1989, p. 1.

[13] P. Zaslansky, J. D. Currey, A. A. Friesem, S. Weiner, J. Biomed. Opt.2005, 10, 024 020.

[14] a) K. J. Anusavice, R. W. Phillips, Phillips’ Science of Dental Materi-als, 11th ed., Saunders, St. Louis, MO 2003. b) D. Tortopidis, M. F.Lyons, R. H. Baxendale, W. N. Gilmour, J. Oral Rehabil. 1998, 25, 681.

[15] a) C. H. Gibbs, P. E. Mahan, H. C. Lundeen, K. Brehnan, E. K.Walsh, W. B. Holbrook, J. Prosthet. Dent. 1981, 46, 443. b) R. G.Craig, in Restorative Dental Materials, 9th ed. (Ed: R. G. Craig), Mos-bey, St. Louis, MO 1993, p. 54. c) A. H. Howell, B. Finn, J. Dent. Res.1950, 29, 133.

[16] A. Asundi, in Photomechanics, Vol. 77 (Ed: P. K. Rastogi), Springer,Berlin 2000, p. 33.

[17] a) G. W. Marshall, Quintessence Int. 1993, 24, 606. b) G. W. Marshall,S. Marshall, J. Kinney, M. Balooch, J. Dent. 1997, 25, 441.

[18] P. K. Rastogi, in Digital Speckle Pattern Interferometry and RelatedTechniques (Ed: P. K. Rastogi), Wiley, Chichester 2001.

[19] G. T. Reid, Opt. Lasers Eng. 1986, 7, 37.[20] J. M. Huntley, J. Strain Anal. Eng. Des. 1998, 33, 105.[21] L. A. Litonjua, S. Andreana, A. K. Patra, R. E. Cohen, Biomed. Ma-

ter. Eng. 2004, 14, 311.[22] W. C. Lee, W. S. Eakle, J. Prosthet. Dent. 1996, 75, 487.[23] a) M. Tanaka, T. Naito, M. Yokota, M. Kohno, J. Oral Rehabil. 2003,

30, 60. b) H. E. Lee, C. L. Lin, C. H. Wang, C. H. Cheng, C. H.Chang, J. Dent. 2002, 30, 283.

[24] a) J. O. Grippo, J. Esthet. Dent. 1991, 3, 14. b) J. O. Grippo, M. Sim-ing, S. Schreiner, J. Am. Dent. Assoc., JADA 2004, 135, 1109.

[25] R. Shahar, P. Zaslansky, M. M. Barak, A. A. Friesem, J. Currey,S. Weiner, J. Biomech. 2006, doi:10.1016/j.jbiomech.2006.01.021.

[26] J. W. Goodman, J. Opt. Soc. Am. 1976, 66, 1145.[27] a) K. Creath, Appl. Opt. 1985, 24, 3053. b) J. M. Huntley, in Digital

Speckle Pattern Interferometry and Related Techniques (Ed: P. K. Ras-togi), Wiley, Chichester 2001, p. 59.

[28] Axial deformation: Mechanics of Materials (Ed: R. R. Craig), Wiley,New York 1996, p. 94.


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Figure 9. Strain maps of variations in displacement gradients on a tooth cap. a) Horizontal axial strain: ∂U(i,j)/∂j gradient along horizontal displacementmeasurements, estimated from the trend of 25 adjacent vertical lines, smoothed by interpolation over 15 × 15 points. Note the small horizontal strain val-ues. b) Vertical axial strain: ∂V(i,j)/∂i gradient calculated using a similar method as for the horizontal strain (but interpolating over horizontal lines). Ex-treme values of strain at the edges are artifacts of the gradient calculations. High values of strain are found mainly near the edge of the enamel cap (de-marcated by the black line across both silhouettes), and are pronounced on the buccal (left) lower region of the tooth, below the loaded cusp.

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