The premise of comparative investigations of skeletal morphology is that there is an underlying relationship between bone structural and material properties and the load history of the element in question (Carter and Beaupre, 2001). This perspective has informed numerous studies of the functional linkage between masticatory forces and mandibular morphology [reviewed in Hogue (2008)]. Because of an extensive experimental foundation (Hylander, 1977, 1979, 1981, 1984), the biomechanics of primate mastication have been a focal area for understanding general questions of the relationship of bone structure to the physiological loading environment.
A series of investigations (Hylander, 1984, 1985) established that a specific loading regime, lateral transverse bending or “wishboning,” is strongly implicated in the evolution of an ossified mandibular symphysis during primate evolution. In addition, morphological variation in the anterior corpus of anthropoid mandibles has been interpreted specifically within the context of the biomechanics of wishboning (Vinyard and Ravosa, 1998; Daegling, 2001). In terms of patterns of bone strain observed in vivo under physiological and artificial loading conditions, the mandible appears to behave as a curved beam, in which the expected stress and strain on nominally tensile and compressive surfaces are no longer equal. Instead, the lingual face of the symphyseal surface experiences tensile strains that are larger than the corresponding normal compressive strains on the labial surface. The disparity between these strains is a function of both the degree of curvature in the anterior corpus and its thickness. In theory, knowledge of these and other structural parameters are sufficient to specify the nature of this strain disparity (Young and Budynas, 2002); however, given that the morphology of the mandibular symphysis does not meet the geometric and material restrictions required by formulaic solutions, it is widely appreciated that these theoretical solutions are inexact.
Experimental verification of these labiolingual strain disparities using strain gauges has also proven to be elusive. In vivo investigation (Hylander, 1984) has, for anatomical reasons, been restricted to sampling the labial surfaces and the superior aspect of the planum alveolare (the lingual surface of the alveolar process extending to the superior transverse torus). In vitro studies can sample locations more extensively; however, in many specimens the extreme flexion of the anterior corpus precludes successful bonding of strain gauges to the midline on the superior and inferior transverse tori, which are the locations at which wishboning strains are generally assumed to be at their maximum.
Hylander's (1984, 1985) curved beam model was intended to provide a first-order approximation of relative stress and strain disparities at the primate mandibular symphysis. Its necessary utilization of several simplifying assumptions about geometry and structural and material properties provided a heuristic model for studying the comparative anatomy of the anthropoid mandible. Alternatively, these assumptions can be bypassed through application of finite element methods (Korioth et al., 1992; Ichim et al., 2006), but at present these models are not applicable to large comparative samples. There is no question that finite element analysis will provide a more accurate description of the stress and strain field of the mandible than an idealized curved beam model. However, the outstanding issue is whether the principles of curved beam mechanics provide a general (i.e., “good enough”) description of states of stress and strain under wishboning loads that allows for the discernment of biomechanical differences among primate species.
There are three reasons why a more precise assessment of normal strain disparities in wishboning, the focus of this study, is desirable. First, such assessment allows evaluation of the suitability of curved beam formulae for characterizing actual mechanical behavior in the anterior corpus (i.e., model verification). Second, such data permit a test of a dynamic strain similarity hypothesis (Rubin and Lanyon, 1984), which predicts similar stress and strain magnitudes in vivo under wishboning loads (Hylander, 1985; Vinyard and Ravosa, 1998). Finally, a related question is whether wishboning loads impose different functional constraints in different higher taxa among anthropoids; that is, are safety factors under wishboning stress equivalent across the suborder?
MATERIALS AND METHODS
We examined midline symphyseal strains in vitro under simulated wishboning loads in three specimens of colobine monkeys (two adult male Procolobus badius and one adult male Colobus polykomos). The examination of colobine monkeys provides a test of the hypothesis offered by Hylander (1985) that anthropoid species with shorter faces and wider mandibular dental arcades will experience relatively reduced wishboning strains in comparison to the model species of Macaca fascicularis. Using a three-dimensional (3D) DIC technique, we examined full-field strains on the labial face of the symphysis as well as the planum alveolare along the lingual aspect. In contrast to the use of strain gauges, which represent a limited-field technique, 3D DIC permits detailed assessment of surface strain gradients and permits a more complete description of strain disparities of labial and lingual strains.
The foundation of the DIC technique is the evaluation of image data from digital cameras. The resolution of the DIC technique is a function of the cameras' resolution, the field of view, and calibration procedure. The digital cameras used in this investigation have a resolution of 1600 × 1200 pixels (width × height). The field of view of the cameras is determined by the focal length of the lens used and the length of the lens extensions attached between the camera and the lens. For this investigation the field of view was set to 15 mm by 12 mm using a 50 mm lens with a 5 mm extension tube attached between the camera and the lens. A calibration procedure is used to correlate locations in three-dimensional space on the surface of the object being evaluated with the location on the camera's two-dimensional photo array. Calibrations were performed daily prior to the start of measurements. For the 15 mm by 12 mm field of view, the typical volumetric field of view was 20 mm × 15 mm × 5 mm (width × height × depth). The average calibration accuracy computed from the calibration procedure was 0.02 pixels (approximately 0.2 μm). Additionally, prior to the start of each measurement a floor measurement was taken. A floor measurement takes a snapshot at the no-load condition and then takes a second snapshot at the no-load condition. The floor measurement is used to evaluate the error in the measurements prior to each loading. In instances where the floor measurement is greater than the calibration accuracy, the system is recalibrated prior to specimen loading.
The digital cameras are used to record grayscale images of the object under investigation before deformation and after deformation. The intensity of each pixel in these images is assigned an eight bit number (0–black to 255–white) that represents its light intensity. These images are subdivided into square facets that typically measure 15 pixels by 15 pixels. Each facet is assumed to contain a unique light intensity pattern before deformation that can be correlated to the light intensity pattern of the facet after deformation (Fig. 1). Interpolation methods are used to represent the light intensity as a function for each facet in an image. From the postdeformation light intensity function the coordinates of a deformed facet can be located with respect to the undeformed facet. These coordinates are then used to establish the displacement functions for each facet, which in turn are used to compute strains. The strains at each point are determined using three facets. As part of the interpolation method a five-pixel facet step was specified. The size of the virtual strain gauge for the measurements can be computed from the dimension of each pixel (15 mm/1600 pixels =0.009375 mm/pixel) multiplied by the minimum distance between facet centers (10 pixels). This results in a virtual strain gauge that is 0.094 mm in length.
For this study, we used a two-camera DIC system. The advantage of a two-camera system over a single camera system is that the two-camera system allows for depth of field or 3D resolution of deformation and strain. The 3D DIC system used in this study was the ARAMIS 2M system by GOM mbH (Braunschweig, Germany). The ARAMIS version 6.0 software was used to analyze all of the data in this study.
Each specimen was immersed in water for 24 hr prior to testing; a speckle-coating was spray-painted onto each specimen such that surfaces exhibited a stochastic pattern that provided the unique light intensity patterns for the sampled facets described above. Specimens were embedded bilaterally at the ramus in blocks of epoxy resin, within which screws were affixed for attachment to a mechanical testing machine (Fig. 2). Each specimen was loaded four times in lateral transverse bending to 7 Nm. The four loadings were required as each surface (labial and lingual) was not planar; that is, a single field of view of the cameras was not sufficiently parallel to the surfaces of interest on either labial or lingual aspects. Consequently, the cameras required a repositioning on each side so that reliable measurements could be made of features of interest. Principal and shear strains were calculated for each sampled facet. To permit comparison among specimens, strains are also reported normalized to a baseline strain and with respect to location along the axis coincident with the midsagittal plane (Fig. 3). The most superior midline facet examined on labial and lingual aspects (respectively) provided a baseline shear strain that was assigned a normalized value of 1.0 at normalized position 0.0. Strains at other locations were reported as normalized values. For example given a shear strain of 100 με at the labial alveolar margin (location 0.0), a strain of 250 με sampled 30% along the length of the labial surface would be reported as a normalized strain of 2.5 at location 0.3. The choice of a load magnitude of 7 Nm was not based on physiological criteria (the magnitude of in vivo wishboning moments is unknown for any taxon), but because previous strain gauge work (Daegling and McGraw, 2009) on colobine mandibles utilized this load and thus comparison with DIC data in the current study was facilitated.
Surface strain patterns observed from the DIC data were recruited to examine the validity of the curved beam model used in comparative research of primate masticatory biomechanics. In the anthropological literature, the conventional model of symphyseal stress under wishboning makes several predictions: (1) the entire labial face of the symphysis is predominantly under compression due to the transversely oriented normal strains; (2) the planum alveolare experiences a transition of normal strains along its superoinferior axis, such that compression is more pronounced in proximity to the dentition and tension predominates in the vicinity of superior transverse torus (Hylander, 1985); (3) the colobine mandible will experience, relative to papionin primates (Macaca in particular), a reduced disparity of lingual:labial normal strains owing to their broader mandibular arcade and consequent reduction of curvature at the symphysis; (4) the maximum normal strains will be found in the midsagittal plane; and (5) among the strains in the midsagittal plane, higher magnitudes will be found in association with the most extreme curvatures (i.e., in the vicinity of the superior transverse torus).
Full-field strain patterns under wishboning loads indicate that the most severe curved beam effects in the colobine mandible may be highly localized (Figs. 4 and 5). The planum alveolare is, in general, less strained than much of the labial surface, with the exception that strains increase rapidly as the midline superior transverse torus is approached.
There were certain regions on each specimen that could not be analyzed because of extreme obliquity of the angle between bone surface and the optical axis of the two cameras. In such contexts, the strain values obtained are unreliable. This precluded collection of strain data for all specimens on the inferior portion of the lingual face of the symphysis and the most inferior aspect of the labial face. These unanalyzed regions are shown in Figs. 4 and 5.
Because of differences in the size and morphology of the sampled specimens, comparison of absolute strain magnitudes is not particularly beneficial for understanding the general form of the strain field under wishboning loads. By contrast, the identification of common patterns (if they exist) of strain among specimens is facilitated by normalizing strains and locations in each individual. These normalized values (calculated from engineering shear strains) again reflect the idiosyncratic nature of the strain fields, but also provide an indication of the general effects of wishboning loads at the anthropoid symphysis. At the labial surface, as one samples from superior to inferior, there is an initial decrease in strain (on the order of 10%) followed by an increase of variable magnitude between positions 0.09 and 0.18 (Fig. 6). Another “peak” of strain is observed at about midcorpus (position 0.5–0.6) in two of the three specimens. Proportional changes in strain from initial (position 0.0) values appear to be on the order of 50% at their maximum. Along the inferior portion of the labial symphysis, strains are fairly invariant and slightly above initial values, although data were available for this region from only two of the three specimens.
Along the lingual symphysis the normalized strains follow a more recognizable pattern (Fig. 7). While there are local fluctuations in relative magnitude particular to each specimen, there is a general trend of increased strain as the superior transverse torus is approached (position 0.4) from the alveolar margin (position 0.0). The strain increases by a factor of 2–3 in each case, and this may in fact be an underestimate of the true proportional increase as we were unable to sample strains reliably beyond the superior aspect of the superior transverse torus.
The magnitude of strain disparity between labial and lingual surfaces are reported in Table 1 for various locations. These common positions correspond to transverse anatomical planes that are roughly parallel to the applied bending moment. These disparities provide an approximation of the magnitude of curved beam effects; i.e., under conventional straight beam models, the expected ratio between maximum principal strains (ε1) (lingually) and minimum principal strain (ε2) (labially) is 1.0. The expectation for the primate symphysis, if it behaves as an idealized curved beam, is that the ratio of lingual tensile to labial compressive strain (i.e., |ε1:ε2|) is above unity at opposite surfaces. This is the case in certain regions, but not others. Near the alveolar margins the strain ratio indicates the greater magnitude of labial compressive strain with respect to lingual tensile strain. In contrast, in the vicinity of the midcorpus the ratio assumes the “expected” values that indicate a disparity of lingual:labial normal strain of over 1.0 (specimen 23–9, a Procolobus, trends in this direction prior to loss of data due to camera angle). The Colobus specimen displays a maximum disparity of about 1.7, while the remaining Procolobus shows greater disparity of 2.6. Owing to sampling issues noted above, these disparities are probably underestimating the maximum difference to some degree.
Table 1. Strain disparities between lingual and labial surfaces along the symphyseal midline
21-11 Procolobus badius
23-9 Procolobus badius
22-11 Colobus polykomos
Lingual to labial |ratio|
Lingual to labial |ratio|
Lingual to labial |ratio|
The superior alveolar margin is indicated by a normalized position of zero. The normalized position increases inferiorly as the superior transverse torus is approached. Maximum and minimum principal microstrain magnitudes on the lingual and labial surfaces, respectively, are tabulated for each specimen, as is the absolute value of the ratio of the lingual to labial microstrain. Data are tabulated for the 0.23 position as this is the most inferior location at which reliable data were obtained for specimen 23-9.
If, in fact, the idealized curved beam model utilized in comparative research is fully accurate, absolute principal strain ratios on the labial face should consistently fall below 1.0, while along the lingual face (the planum alveolare specifically), one should observe a transition between ratio values falling below 1.0 at alveolar margins to ratio values above 1.0 near the superior transverse torus. At the positions sampled along the labial face (Fig. 8), the principal strain ratio exceeds unity in the majority of cases, in contradiction to expectation. Along the lingual face (Fig. 9), however, the expected increase in principal strain ratio at progressively inferior locations is observed in all specimens, despite marked differences in the steepness of the trend in different individuals. Direction of principal strains in the sampled specimens was consistent with a transverse bending loading regime; that is, the largest absolute magnitude principal strains were oriented approximately within a transverse plane.
The idiosyncratic strain fields among specimens, in terms of local differences in strain gradients, are not unexpected for biological samples. These differences are primarily attributable to morphological variation and secondarily due to likely differences in the orientation of bending moments relative to anatomical axes. By themselves, these observations do not invalidate the use of idealized models; instead, they serve as reminders that all models are necessarily approximations of actual mechanical behavior.
A model of the mandible as a curved beam under wishboning loads requires a number of assumptions in order for it to be applicable to comparative samples. These include transparently false conditions of isotropy, homogeneity and prismatic, symmetrical geometry. The full-field strain analysis provides a useful accounting of the validity of certain ideas about the effects of wishboning on symphyseal bone (e.g., Hylander, 1985; Ravosa, 1991, 1996, 2000; Vinyard and Ravosa, 1998; Daegling, 2001); specifically, it provides empirical evidence for the disparity of labiolingual strains. Daegling and McGraw (2009) argued that wishboning stress distribution could be productively modeled by considering it to be a case of asymmetric bending; i.e., a loadcase in which the member is bent in a plane that is not coincident to an axis of symmetry. That analysis suggested that a component of bending in wishboning loaded the alveolar process in compression and the symphyseal base in tension in a manner not considered either explicitly or implicitly by the conventional curved beam model used in comparative contexts. Daegling and McGraw (2009) did not, however, analytically incorporate curved beam effects into the asymmetric bending model, and correlation of observed and predicted strain was weak or absent.
Our full-field analysis of symphyseal strain presents an interpretive conundrum: the lingual planum alveolare strain field is consistent with the curved beam and asymmetrical bending predictions, while the data from the labial face would appear to invalidate these models. Specifically, the relative increase in maximum principal strain as the superior transverse torus is approached follows from the expected location of the neutral axis under both models (Hylander, 1985; Daegling and McGraw, 2009). The relatively high strain magnitudes in the vicinity of the superior transverse torus are also explicable with reference to the mechanics of curved beams and asymmetrical bending. Hylander's (1984, 1985) curved beam model predicts that the labial symphyseal surface will be subject to compressive strain such that ε2 > ε1 along its length. Asymmetrical bending effects (Daegling and McGraw, 2009) further stipulate that the relative magnitude of minimum principal strain will decrease from alveolar margins toward the base, perhaps to the point that ε1 > ε2. In fact, neither model prediction is borne out by observation. The most problematic observation with respect to models of wishboning is the predominance of tensile strain along the labial alveolar process. Such an observation could be explained away in the in vivo context because of uncertainty over the superposition of different loading regimes. However, under a controlled loadcase (such as the case here) it is not so easily resolved. A potential explanation is that the load created in vitro produces an instance of anticlastic bending, in which the primary bending of the beam is accompanied by secondary bending in a perpendicular plane, which produces a normal strain of opposite sign (Fig. 10). Under “ideal” conditions (a prismatic, isotropic, and homogenous beam), the magnitude of the anticlastic bending is expected to be the product of normal strain and Poisson's ratio. Given that for bone Poisson's ratio is ≪1.0, the anticlastic bending explanation apparently fails to account for the predominance of tensile strain along the labial alveolar process. Whether one can generalize this idealized anticlastic bending effect to an irregular, anisotropic and heterogeneous structure, as in the present case, is doubtful.
While the strain fields depicted by DIC caution us that there is no “typical” strain profile for a given load in terms of local strain magnitudes and gradients, the method can nevertheless be recruited to address the question of whether colobine mandibles exhibit the same labiolingual stress disparities as cercopithecine monkeys. Conventional wisdom is that the curved beam effects of wishboning are most severe (and perhaps impose the greatest constraints) in cercopithecine mandibles, given their narrow dental arcades and concomitant tightly curved symphyseal regions (Hylander, 1985; Ravosa, 1996, 2000; Vinyard and Ravosa, 1998; Daegling, 2001). Colobines as a group, by the metrics used to evaluate curved beam effects, should exhibit reduced disparity in labiolingual strains compared to cercopithecines. Hylander (1984) reasoned, based on theoretical principles of overall mandibular geometry of the macaque mandible, that a strain disparity between a factor of 2.5 and 6 typified the cercopithecine symphysis. His experimental data suggested a disparity of at least a factor of 3.5. Applying Hylander's (1984, 1985) theoretical analysis to specimens of C. polykomos and P. badius, we estimate that under a wishboning loading regime lingual strains exceed labial strains by a factor of between 1.9 and 2.2. The relevant strain data collected in this study (Table 1) suggests that this figure is generally accurate, but in at least one case it is underestimating the magnitude of the disparity. These findings suggest that colobine jaws subjected to wishboning loads experience reduced disparities of labiolingual strain relative to the mandibles of cercopithecines, in accordance with the predictions of the general comparative model advocated by Hylander (1985).
The DIC data is in general agreement with strain gauge data collected from the same colobine species under similar loading conditions (Daegling and McGraw, 2009). As different specimens were utilized, direct comparison of principal strain magnitudes is not meaningful. The disparity of labial to lingual strains, however, is of interest as the strain gauge study evaluated the disparity of labial strains to strains at the inferior torus, and the present study describes the disparity at the superior torus. Hylander (1984) considered both transverse tori to be important structural features for mitigating lingual symphyseal stress during wishboning. The DIC data suggests that the lingual:labial principal strain disparity is at least 1.7–2.5 at the superior torus, while the strain gauge data suggest disparities on the order of 1.1–1.9 at the inferior torus in colobines. The asymmetric bending model indicates that stress at the superior torus is usually (but not invariably) higher than that along the inferior torus. This observation coupled with the DIC and strain data offer circumstantial support for the hypothesis of greater labiolingual strain disparities in the vicinity of the superior transverse torus.
DIC provides an empirical description of the wishboning strain field that supports the use of a curved beam model to describe the general mechanical behavior of the anthropoid symphysis. By contrast, specific predictions of symphyseal behavior based on an asymmetrical bending model are weakly supported. Strain gradients along the labial face of the symphysis are highly variable from specimen to specimen. Hylander (1984) observed much the same phenomenon among macaques under controlled-loading experimental contexts. It would appear that any iteration of beam theory is inadequate to explain this variation in mechanical behavior, because the differences between individuals involve highly localized differences in strain magnitudes and gradients. Even though this presents future modeling challenges, there is no reason to speculate on the potential biological significance of this variation until similar patterns are documented in vivo. The structural and material variables responsible for the idiosyncratic differences in the strain field observed may be meaningfully explored through future application of finite element analysis, and DIC represents an important tool for validation of such models.
This study confirms the existence of large gradients of strain magnitude along the lingual symphysis in anthropoid primates; of the regions sampled, the highest strains in wishboning are found in the midsagittal plane at the superior transverse torus. While these findings are generally consistent with theory of curved beams, the prediction of relative strain magnitudes elsewhere in the anterior corpus is problematic with reference to idealized models. Application of heterogeneous structural models that account for geometric irregularity (Bhatavadekar et al., 2006; Rapoff et al., 2007) will improve characterization of the stress field in comparative contexts.
The inference of reduced strain disparities in the colobine symphysis relative to cercopithecines has important implications for anthropoid craniofacial biology. These data suggest that safety factors may not be equivalent across anthropoids; specifically, cercopithecine mandibles are likely under more severe biomechanical constraints than those of colobines or hominoids. It would appear that colobine monkeys, despite potential differences from cercopithecines in terms of dietary toughness, are able to afford less structural reinforcement in the anterior mandibular corpus for the purely architectural reason of possessing a wider dental arcade and reduced symphyseal curvature. This has implications for a hypothesis of dynamic strain similarity between colobines and cercopithecines with respect to masticatory biomechanics. In a “global” sense the hypothesis seems unlikely because masticatory loads that produce similar strain magnitudes in the postcanine corpus will produce dissimilar strain values in the anterior corpus given the general validity of curved beam mechanics in describing mandibular behavior.
We thank Jason Organ and Qian Wang for the invitation to participate in this special issue. The comments of two anonymous reviewers improved the manuscript, as did the editorial help from Jason Organ and Valerie DeLeon.