Biomechanics of cross-sectional size and shape in the hominoid mandibular corpus.код для вставкиСкачать
AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 80:91-106 (1989) Biomechanics of Cross-Sectional Size and Shape in the Hominoid Mandibular Corpus DAVID J. DAEGLING Department of Anthropology, State University of New York at Stony Brook, Stony Brook, New York 11794 KEY WORDS Computed tomography, Mandible ABSTRACT Mandibular cross sections of Pan, Pongo, Gorilla, Homo, and two fossil specimens ofParanthropus were examined by computed tomography (CT) to determine the biomechanical properties of the hominoid mandibular corpus. Images obtained by CT reveal that while the fossil hominids do not differ significantly from extant hominoids in the relative contribution of compact bone to total subperiosteal area, the shape of the Paranthropus corpora indicates that the mechanical design of the robust australopithecine mandible is fundamentally distinct from that of modern hominoids in terms of its ability to resist transverse bending and torsion. It is also apparent that, among the modern hominoids, interspecific and sexual differences in corpus shape are not significant from a biomechanical perspective. While ellipse models have been used previously to describe the size, shape, and subsequent biomechanical properties of the corpus, the present study shows that such models do not predict the biomechanical properties of corpus cross-sectional geometry in an accurate or reliable manner. The traditional “robusticity” index of the mandibular corpus is of limited utility for biomechanical interpretations. The relationship of compact bone distribution in the corpus to dimensions such as mandibular length and arch width may provide a more functionally meaningful definition of mandibular robusticity. There is a concensus that the phenomenon of pronounced mandibular robusticity in the “robust”australopithecines is a direct reflection of their feeding habits. The peculiar gnathic morphology that is characteristic of these hominids presumably reflects their trophic adaptations and, ultimately, their ecological specializations. Yet despite direct approaches to infer the dietary proclivities of these forms (Walker, 1981; Grine, 1981, 1986; Grine and Kay, 1988),the nature of the trophic specializations of Paranthropus remains unresolved. Even a relatively abundant fossil record of Paranthropus mandibles in southern and eastern Af’rica has not served to clarify this issue. While previous biomechanical studies (e.g., DuBrul, 1977; Hylander, 1979a, 1988; Smith, 1983) have succeeded in elucidating certain functionally relevant features of the external design of mandibular morphology @ 1989 ALAN R. LISS, INC. in Paranthropus, nothing is known of the internal design of the “robust” australopithecine mandible. The demonstration that variation in diet significantly affects both the external and internal anatomy of the mandibular corpus (Bouvier and Hylander, 1981) suggests that an approach incorporating information on the design of mandibular cross sections may provide additional insights about mechanical correlates of mandibular robusticity in the hominid fossil record. The success of previous studies in describing the functional consequences of differences in mandibular design (Hylander, 1979a, 1988; Smith, 1983; Bouvier, 1986a,b) indicates that biomechanical approaches may indeed be productive, but only if the appropriate structural model for the manReceived June 14,1988; accepted November 4,1988. 92 D.J. DAEGLING dibular corpus can be discerned through comparative methods. Most contemporary treatments of mandibular mechanics include modelling the corpus as a loaded beam and as a twisted member (Smith, 1978; Walker, 1978; Hylander, 1979a, 1984,1985). It follows, then, that the cross-sectional geometry of the corpus is an essential consideration in the description of its mechanical design (Hylander, 1979a). To date, however, the shape of primate mandibular cross sections has been assumed to be more or less the same, both within and between species, for purposes of comparison (Smith, 1983; Chamberlain and Wood, 1985; Bouvier, 1986b). How well this assumption corresponds to actual gnathic morphology has not been empirically determined. The present study specifically addresses the issues raised by Smith (1983):namely, 1) whether the mandibular corpus is best modelled as a solid elliptical beam or as a hollow elliptical beam, 2) whether idealized geometrical shapes obscure the finer mechanical details of the mandibular corpus to the point that their use is counterproductive, and 3) whether computed tomography (CT) offers a suitable nondestructive methodology for resolving the more subtle mechanical features of the hominoid mandibular corpus through the accurate depiction of internal morphology. The question of the mechanical significance of interspecific variation in the size and shape of the hominoid mandibular corpus may be more effectively addressed through a consideration of these points. Current knowledge on the shape and internal form of hominoid mandibular cross sections is limited to only a few empirical studies (e.g., Demes et al., 1984). Data for cross-sectional anatomy in the fossil record are even more limited, being confined to what can be observed from naturally fractured and damaged specimens. Given the potential of CT scanning for the investigation of modern and fossil material (Jungers and Minns, 1979;Ruff and Leo, 1986; Conroy and Vannier, 1987; Conroy, 1988), the mandibles of contemporary great apes and humans, as well as partial mandibular corpora of Paranthropus from Swartkrans, were examined using this methodology. Pongo pygmaeus, Pan troglodytes, and Homo sapiens. For each taxon, five females and five males were represented (the sexual attribution of the modern human mandibles was based on the development of muscle markings and the mental eminence) so that the effects of sexual dimorphism on mandibular mechanical design could be discerned (cf. Wood, 1978; Chamberlain and Wood, 1985; Hylander, 1988). Only mandibles of fully adult individuals, as determined by the eruption of M,, were utilized in this study. One human specimen lacked third molars; however, the second molar was fully erupted and displayed considerable attrition. Any specimen that displayed noticeable alveolar resorption was excluded from consideration, and no edentulous or partially edentulous individual (in the region P4-M2) was utilized. All great ape specimens were wildshot, while the modern human sample was derived from teaching collections and biological supply houses. The fossil sample consisted of two partial mandibular corpora of Paranthropus robustus that have been described by Grine (1989): SKX 4446, a subadult individual with the third molars completely formed but not erupted into occlusion; and SKX 5013, a fully adult individual. Methods of computed tomography Cross sections were taken by CT on each corpus at the middle of the MI and M2. Images were obtained using two G.E. Model Scanners (8800 and 9800) to establish the reliability of the CT methodology for the accurate depiction of osteological structures in the mandible. Since the mandible functions as a curved beam during mastication (Hylander, 1984, 19851, the selection of a plane of cross section impacts critically on subsequent biomechanical interpretations. Failure to control for any one of three planes of section (i.e., coronal, sagittal, and transverse) may result in an undesirable obliquity of section. Therefore, each specimen was oriented to the X-ray beam of the CT scanner according t o a specific protocol. In the sagittal projection each specimen was oriented so that the plane of section (i.e., the CT beam) passed perpendicular to the occlusal plane. Each section was oriented normal to the long MATERIALS AND METHODS axis of the corpus rather than to the long axis Materials of the tooth row, since a series of sections The sample of modern hominoids con- normal to the cheek tooth row are invariably sisted of 10 specimens each of Gorillagorilla, oblique (Fig. 1A). While the long axis of the CT SCANNING OF HOMINOID MANDIBLES 93 mandibular corpus may change between tooth positions, the change between the first and second molars is minimal. For this reason, CT was performed sequentially at the buccal grooves of M2 and M, without change in orientation. Orientation of each specimen was facilitated by light projections on the scanner platforms and by scout sections (Figs. lA,B). Each CT image was generated from a 1.5 mm thick section. Duration of actual scanning ranged from 4 sec per section on the model 9800 to 10 sec per section on the model 8800. Each specimen image was photographed at 3X magnification; hard copies of images included scale grids for calibration along both the X and Y axes so that the areas reported represent true values. Each specimen was submerged in water to attenuate the X-ray beam. That is, the density of water (Hounsfield Units = 01, in contrast t o that of air (Hounsfield Units = - 1,000), more closely approximates the Hounsfield Unit value for hard tissue (in this case, ca. +1,000 to +2,000), which results in a clearer image a t the outer margins of the specimen. It should be emphasized that the reconstruction algorithm for living bone was used in all instances so that the density values for compact bone would receive the most emphasis in the generation of each image. Calibration of CT images Selection of appropriate width and centering windows for CT images assumes critical importance when quantitative techniques are to be employed (Ruff and Leo, 1986). The apparent distribution and thickness of compact bone changes conspicuously at different window settings. Therefore, it was necessary to verify that window settings reflected actual cross-sectional morphology. Five mandibles from the modern human sample were sectioned with a Buehler-Isomet wafering blade to verify the accuracy of the CT images. Fig. 1. A: Scout section of modern human mandible Window settings that corresponded reason- in occlusal view, indicating proper orientation of speciably well to actual compact bone distribution men along the long axis of the corpus. B: Scout section of troglodytes mandible. Dashed lines indicate planes of differed markedly between the two CT scan- P. section for CT. Sections 1-3 are of the M,, MI, and P4, ners, and this discrepancy was not due to respectively. Because of great interspecific differences in different orientations of the specimens in the the posterior extent of the symphysis, sections of the P4 two CT scanners. In addition, the specimens were not utilized for comparisons. were not cut obliquely with respect to the original CT images, with one exception (see Table 1,specimen 1 on the model 8800).The levels between the two CT scanners would most plausible explanation for the observa- seem to be related to their distinctive image tion of identical images at different window andor film processing operations. 94 D.J. DAEGLING TABLE 1. CT image and sectioned specimen comparisons’ Specimen and section Area CT/9800 Area CT/8800 Cut area 1. Total subperiosteal 390.5 132.4 325.4 131.4 296.9 104.2 235.7 93.6 331.4 134.0 391.6 150.1 325.8 131.7 302.6 107.1 386.4 128.0 324.9 132.7 295.9 99.2 235.9 93.3 334.6 136.5 2. 3. 4. 5. Cortical Total subperiosteal Cortical Total subperiosteal Cortical Total subperiosteal Cortical Total subperiosteal Cortical -2 - Percent difference Cut/9800 Cut/8800 1.05 3.32 0.15 0.99 0.34 4.80 0.08 0.32 0.96 1.87 1.33 14.72 0.28 0.76 2.21 7.38 - ‘All units are in mm’; all sections at M p . ‘Specimens 4 and 5 were not scanned on the model 8800. Window settings of 250L 4000W on the model 8800 and 1500L 4000W on the model 9800 were determined to be reliable for accurate imaging of compact bone distribution. Images originally scanned on the model 8800 were later transferred to the model 9800, and no difference was found to distinguish the original 8800 images from the transferred images (adjusted to 1500L 4000W). However, upgrading the matrix (the number of pixels in an image field) from 3202 to 5122 led to a noticeable distortion of cross-sectional area. Therefore, images that were originally filmed on the model 8800 and subsequently loaded onto the model 9800 were measured using the 320’ pixel matrix of the model 8800. Correspondence of compact bone area in directly sectioned specimens to that obtained from homologous CT images a t the appropriate window settings was verified by digitization (see below), and these results are summarized in Table 1. The window settings used to depict compact bone area in the present study differ substantially from those used by Ruff and Leo (1986) for similar purposes. Since visual display of CT numbers theoretically should not vary between different machines, this discrepancy raises concern. One possible explanation relates to the fact that the specimens used in the present study were submerged in water, whereas Ruff and Leo (1986) did not employ this procedure. The differences in surrounding medium density (i.e., water vs. air) may account in part for the different window settings. Another important consideration is that since compact bone thickness in a mandibular cross section is considerably less than in a femoral midshaft, higher window width settings are required to distinguish the precise contours of compact bone (Ruff and Leo, 1986). Since cortical bone area from directly sectioned specimens was compared with hard-copy output of CT examinations, it was apparent that no significant distortion caused by hard-copy processing or image photography was present. Since the accuracy of the CT images was determined empirically in this study, the difference in window settings between this study and that by Ruff and Leo (1986) probably reflect different measurement and procedural techniques. For instance, the settings employed by Ruff and Leo (1986, p. 187)were “specificallydesigned to delete cancellous bone from the image display.” In the present study, the inclusion or deletion of cancellous bone was not considered crucial for determining compact bone contours (Fig. 2) (cf. Burr and Piotrowski, 1982; Ruff, 1983; Hayes, 1986). The choice of an appropriate window setting for fossil material presents special problems, since the absolute density of the mineralized fossil specimen is different from that of modern dried bone. Nevertheless, it can be assumed that the relative densities of material within the fossil specimens are reflective of the original osteological structures. The windows used for scanning the fossils were the same as those for the modern sample. While the window setting that ideally reflects distribution of cortical bone in a fossil cannot be known with certainty because the fossils displayed marked density variations in the same manner as the dried mandibles, it would appear likely that the in vivo pattern of compact bone distribution has been preserved. In fact, CT images of natural fractures in the fossils indicate that the stated window settings are reasonably accurate. Hard copies of these images were CT SCANNING OF HOMINOID MANDIBLES produced at several different window levels, and there was no indication that substantially raising or lowering centering windows from those used for the modern mandibles produced more accurate images. Moreover, the level used (1500L) emphasizes the high CT numbers of both modern and fossil bone, and although the densities of the two materials are certainly different, the wide window width used assured that a sufficiently broad range of CT numbers would be represented in image reconstruction for both. Analytical methods Each cross-sectional image obtained by CT was traced from hard copy onto tracing paper for measurement, which was performed using a Science Accessories Graf-Pen sonic digitizer. Coordinate data from the digitizer were fed directly into the computer program SLICE (Nagurka and Hayes, 19801, which calculated cross-sectional area, second moments of area about the X and Y axes (I, and I,,,,, corresponding to the major and minor morphological axes), and the polar moment of inertia (J) for each CT image. The second moments of area reflect the ability of a cross section to accommodate bending stresses about buccolingual (X) and superoinferior (Y) axes, while the polar moment of inertia provides a measure of the ability of a cross section to resist torsion. Each image tracing was measured five times; the two extreme area measurements were discarded, and the relevant parameters were derived from an 95 average of the three remaining sets of calculations. Two cross-sectional areas were measured: 1) total subperiosteal area, enclosed by the periosteal border to the alveolar margins, with a straight line connecting those margins (i.e., a solid beam); and 2) cortical area, the area of the compact bone joined a t the alveolar margins by a 1 mm thick “cap”(i.e., a hollow beam). The ratio of cortical area to total subperiosteal area (cortical index) was calculated for each cross section. The mean cortical index values for each extant taxon were subjected to an approximate test of equality of means, with the assumption of heterogeneity of variances between samples using the Games and Howell method (Sokal and Rohlf, 1981). This test was also performed for a ratio of I,$, (bending index, for both hollow and solid treatments) to discern any possible sexual differences in mechanical design in the great ape samples that might be related to corpus shape. Conventional analysis of variance was also performed for the entire sample to detect interspecific differences in bending indices. Major axis regression of the three crosssectional moments of inertia against compact bone area was performed for the entire sample so that changes in cortical area could be related to variation in I,, I,, and J. Since cross-sectional moments of inertia are expressed in mm4, the square roots of these variables were regressed against cross-sec- Fig. 2. Comparison of cut specimen with homologous CT section of a human mandible. CT images may overestimateslightly the endosteal marginsdue to the presence of cancellousbone, which is probably of similar density to adjacent compact bone (Hayes, 1986). 96 D.J.DAEGLING tional area so that the null hypothesis of geometric similarity of cross sections is represented by a slope of 1.0. Independence tests (G-tests) of residuals falling above or below a forced isometric line were subsequently performed to detect significant interspecific deviations from geometric similarity for the entire sample. Idealized geometrical configurations of corpus cross sections (solid ellipse and hollow ellipse of uniform cortical wall thickness) were calculated for each specimen from external metrics of the corpus to determine if such models bear any resemblance to actual cross-sectional shape. The formula for the area of a solid ellipse of height d and breadth b is given as A = 1/4[pil (bd), with second moments of area I, and I, calculated as I, = 1/64[pil (bd3) and I, = 1/64[pil(b3d) (Roark, 1965).The polar moment of inertia J is derived as the sum of I, and I For calculations of these variables in a %allow ellipse, the diameters of the internal walls, 6’ and d’, must also be factored into the equations. Thus, for a hollow ellipse A = 1/4[pil (bd - b’d’),I, = 1/64[pil (bd3 - b’dr3), and I = 1/64[pil(b3d - br3d’) (Roark, 1965).% this study, b’ and d‘ were determined empirically for each species by measuring compact bone thickness a t the midpoint of corpus height on the buccal and lingual sides and at the most inferior point of the corpus on each cross section. These three measurements were averaged for each species to obtain a single cortical thickness value from which the values of b’ and d’ were determined. Since there is no compact bone along the superior aspect of the corpus, the initial expectation is that the hollow ellipse model will tend to overestimate cross-sectional area and corresponding moments of inertia. This presents no special problems if this discrepancy is consistent within and between species. The percentage difference between the calculated value for an elliptical cross section and the actual digitized area was determined for both MI and M, cross sections in each specimen. Mean percentage differences in each taxon were determined by averaging individual percentage differences, and the mean percentage differences were expressed as real and absolute values (i.e., with and without respect to sign [+, -I). Solid ellipse values were compared with area values for total subperiosteal area (solid beam), and the values determined from a hollow ellipse model were compared with cortical area (hollow beam). RESULTS The utility of idealized models of mandibular cross-sectional geometry depends on their reliability in predicting the area, shape, and subsequent biomechanical features of the corpus. Table 2 summarizes the reliability of the solid and hollow ellipse models for each taxon. The accuracy and reliability of these models are, to a degree, species specific. TABLE 2. Percentage differences of elliptical models from digitized value areas Ixx, Iyy,and J Actual difference’ mean (s.d.) Absolute difference’ (s.d.) I mean I Observed range Hollow section a t Mz Area Pan Pongo Gorilla Homo 14.12 19.67 19.31 13.02 24.56 18.34 24.18 23.72 14.12 11.80 14.13 13.02 +29.38 29.38 15.06 24.78 20.91 21.78 10.28 19.00 18.50 -18.97 to +32.79 -19.31 to +71.06 - 0.77 t o +67.10 +10.34 -16.37 3.32 +15.21 Ixx I,, + 5.86 to +52.90 +24.56 +10.66 +20.71 +23.72 Pan Pongo Gorilla Homo +19.38 +20.76 21.78 18.37 25.02 18.69 Pan Pongo Gorilla Homo +40.98 5.73 +26.67 +36.62 22.25 21.22 18.23 17.36 40.98 17.59 26.67 36.62 22.25 11.95 18.23 17.36 Pan Pongo Gorilla Homo +31.95 4.24 +21.06 f24.79 18.99 16.77 21.71 15.28 31.95 13.94 24.44 24.79 18.99 9.23 17.34 15.28 + 4.26 + J + -13.99 to +39.91 -16.72 to +45.76 4.41 to +52.00 + + 0.11 to +65.53 + to +82.67 to +44.48 t o +56.08 to f62.35 + 6.22 to +64.82 -22.12 to +28.85 -1 1.90 to +59.07 8.65 to +63.29 + 97 CT SCANNING OF HOMINOID MANDIBLES TABLE 2. Percentage differences of elliptical models from digitized value areas Ixx, Iyy, and J Actual difference1 mean (s.d.) Solid section a t Mz Area Pan Pongo Gorilla Homo Absolute difference1 hd.) I mean1 Observed ranee + 7.14 4.05 + 2.62 + 3.09 7.05 6.97 4.79 6.30 8.04 6.68 4.55 5.22 5.88 4.16 2.74 4.48 3.25 to +20.27 -13.49 to 8.00 - 4.59 to 9.82 - 7.63 to +12.99 1.31 -17.39 -11.11 -11.90 13.98 5.79 10.00 10.94 10.25 17.39 12.37 12.83 8.99 5.79 8.18 9.70 -25.97 -29.84 -27.31 -28.42 +23.20 - 4.54 +12.78 +17.91 19.72 25.47 7.95 17.99 23.87 20.69 12.78 18.61 18.81 14.00 7.95 17.17 12.50 7.53 9.30 11.07 10.01 -15.57 - 6.01 - 5.11 15.57 8.34 10.25 6.74 7.53 7.03 5.90 +22.77 5.01 +21.63 +18.39 + 6.32 11.67 10.58 8.37 22.77 10.05 21.63 18.39 6.32 7.19 10.58 8.37 + + Pan Pongo Gorilla Homo +25.34 - 3.58 19.38 1.99 + + 5.44 20.75 12.49 20.21 25.34 16.17 19.38 16.51 5.44 12.43 12.49 10.50 +18.27 to +32.13 -34.75 to +35.69 3.80 to +43.56 -26.37 to 137.55 Pan Pongo Gorilla Homo +15.86 -20.82 9.89 +14.29 + 11.78 20.23 9.05 12.64 15.86 23.08 11.58 14.29 11.78 17.30 6.44 12.64 1.53 to +35.21 -53.17 to 6.62 - 7.03 to +23.00 0.87 to +36.08 + +23.32 - 7.87 +17.38 5.00 6.00 16.28 10.72 17.36 23.32 13.76 17.38 14.83 6.00 11.11 10.72 9.19 +14.25 to -38.16 to 4.61 to -17.18 to + 4.51 - 5.76 + 3.57 + 0.36 3.53 9.01 4.57 3.75 3.06 9.16 4.89 2.91 2.15 4.98 2.91 2.19 3.94 -27.31 -20.53 -23.07 10.13 9.38 9.95 17.47 9.52 27.31 20.53 24.14 4.36 9.38 9.95 15.78 -14.20 -39.96 -37.55 -45.08 4.98 -24.55 - 9.44 1.19 + 8.52 19.08 10.03 10.40 8.14 27.62 10.75 7.41 5.15 13.64 8.45 6.97 -17.50 -47.81 -22.69 -17.26 to +I839 4.26 -27.02 -17.88 - .. -20.99 8.88 10.33 9.49 . .. 14.92 8.73 27.02 17.88 22.67 3.78 10.33 9.49 .. . 11.88 -13.73 -37.23 -33.97 -39.00 1xx Pan Pongo Gor i11a Homo %an Pongo Gorilla Homo J Pan Pongo Gorilla Homo Hollow section at M I Area Pan Pongo Gorilla Homo - - + 0.60 + + + to +20.70 to - 8.46 to 6.33 4.65 to + + 3.37 to +65.23 -30.79 to +45.93 2.24 to +24.22 - 3.52 to +57.96 - + -18.73 -29.18 -20.84 -23.06 to +20.56 to - 4.05 to 8.01 8.91 to + + +14.80 to -18.11 to 1.64 to 9.94 to +35.56 +20.98 +34.81 +34.08 IXX ,I J Pan Pongo Gorilla Homo Solid section a t M1 Area Pan Pongo Gorilla Homo + Ixx ,I Pan Pongo Gorilla Homo - Pan Pongo Gorilla Homo - Pan Pongo Gorilla Homo - J + + + + +30.22 +16.03 +37.86 f35.44 + + + 2.54 to 7.04 -14.62 to +11.86 -10.01 to 3.21 - 3.92 to 7.71 - to +14.74 to - 9.26 to - 6.99 5.34 to + to + 8.33 to f15.35 to + 3.56 + to 9.83 to - 5.63 to - 5.17 to 8.38 + 'Percentage differences between ellipse model predictions and digitized values (predicted - digitized/digitized) are given as actual and absolute percentage differences. Actual percentage differences distinguish between underestimates (-) and overestimates (+) of actual values by the ellipse model; absolute differences are calculated without respect to sign. Means are calculated from the sum of individual percentage differences. 98 D.J. DAEGLING Hollow ellipse model Theoretical considerations notwithstanding, modelling the corpus as a hollow elliptical beam is unreliable. This model overestimated compact bone area by as much as 20% in many cases (Table 2). Not only are the mean percentage differences (determined as the average of percentage differences in individuals) between the actual digitized and model-generated values quite substantial, but the magnitudes of the standard deviations of the means sometimes approach or exceed the mean values themselves. In sum, modelling the corpus as a hollow ellipse of uniform cortical thickness is both inaccurate and unreliable, even when average compact bone thickness has been determined empirically for each taxon. Since cross-sectional moments of inertia are highly interrelated with cortical area, the large discrepancies between actual and model values for area also result in substantial differences in terms of the mechanical properties of a section.' 'These results are based on comparisons of ellipse modelgenerated values to moments of area about morphological axes. The ellipse predictions could also be compared with moments of area about principal axes. In this study, the morphological axes are so similar to the principal axes that the fit of ellipse predictions changes negligibly. This would not be true of cross sections in other areas of the mandible (i.e., the symphysis) where the distinction between principal and morphological axes assumes critical importance. It should be emphasized, however, that a formula for a symmetrical hollow ellipse was used in this study, while it is likely that a model for an asymmetrical hollow ellipse (i.e., nonuniform wall thickness) would better predict compact bone area, given the variation observed in the CT images (Fig. 3). The use of a n arbitrary 1mm cap to close off sections of compact bone has also contributed to the large discrepancy between digitized and ellipse model-generated values. Solid ellipse model The solid ellipse is somewhat more reliable as a predictor of total subperiosteal area (Table 2). The actual mean percentage differences between the predicted and digitized area values are acceptably low (<6%) in all extant taxa at M, and in Pongo, Gorilla, and Homo at M2.This model appears to be a reasonable predictor of total subperiosteal area; however, examination of the observed ranges of percentage differences (Table 2) suggests that the solid ellipse model is still a poor substitute for empirically determined cross-sectional areas. Despite the fact that the solid ellipse model may, on average, be within 6% of digitized values for total subperiosteal area, subsequent calculations of second and polar moments of inertia based on this model fail to predict accurately and Fig. 3. CT sections at M, of three Gorilla mandibles. Note variations in the distribution and thickness of compact bone throughout the cross sections. Buccal margins are to the left, lingual margins to the right. 99 CT SCANNING OF HOMINOID MANDIBLES TABLE 3. Cortical indices (cortical area/total subperiosteal area) Taxon Pongo Gorilla Sex F M M M M M F F F F F M M M M M F F F F F M M M Index at MI 0.429 0.403 0.426 0.299 0.425 0.4 17 0.353 0.428 0.360 0.384 0.302 0.390 0.404 0.467 0.383 0.346 0.398 0.351 0.353 0.475 0.405 0.479 0.462 0.463 0.418 0.329 0.388 0.411 0.539 0.489 0.387 0.432 0.311 0.459 0.381 0.472 0.384 0.424 0.350 0.411 Index at 0.438 0.364 0.427 0.313 0.474 0.410 0.332 0.532 0.424 0.478 0.256 0.367 0.357 0.407 0.403 0.254 0.348 0.245 0.394 0.414 0.407 0.499 0.373 0.416 0.381 0.330 0.379 0.371 0.574 0.411 0.350 0.398 0.319 0.441 0.351 0.470 0.383 0.404 0.339 0.404 Mg reveals that the observed differences between taxa (Table 3) are not statistically significant. Intraspecific variation in this index is substantial, and although the Paranthropus values appear to indicate a relatively large contribution of compact bone to cross-sectional area, they fall within the observed sample ranges of modern hominoids in most cases. Marked variation in compact bone thickness is observed within individuals as well, and this variation is apparent not only in the uneven distribution of compact bone in any given cross section (Fig. 3) but also in the differences in cortical thickness for sections at M1 and M,. There is no significant sexual dimorphism in the relative amount of compact bone in the great ape sample, and there is no apparent sexual difference in the distribution of cortical bone within cross sections. Scaling of cross-sectional moments of inertia All cross-sectional moments of inertia scale close to isometry with cortical area for all taxa at both tooth positions (Table 4,Fig. M 4).The 95% confidence limits of the slopes M include isometry using either least squares F Homo F or major axis line fitting techniques; thus, F cross-sectional moments of inertia tend to F increase in expected proportion to compact F bone area for the sample as a whole. That is, M M an increase in the amount of compact bone in M a cross section is not accompanied by any M mechanically significant changes in the disM tribution of the compact bone (Fig. 4). IndeParanthroous 0.418 0.397 SKX 4446 pendence tests of residuals approach signifi0.432 0.480 SKX 5013 cance with respect to the distribution of Pongo specimens for the I, on area regresreliably the mechanical properties of a man- sion a t M,. At M,, Pongo also displays a high dibular cross section. second moment of inertia about the horizontal axis given the amount of compact bone in Variation of compact bone cross section (not shown in Fig. 4);in fact, all Analysis of variance for cortical indices the specimens fall above a forced isometric (compactbone aredtotal subperiosteal area) line, which is significantly different from the TABLE 4. Regression summaries-Moments MZ Ixx on area I on area J”& area MI I,, on area I on area f o n area of inertia on cortical areal N SloDe 95% C.L. Y interceot r 42 42 42 1.04 1.05 1.02 0.91-1.18 0.92-1.19 0.92-1.14 -2.81 -3.37 -2.58 0.93 0.93 0.95 42 42 42 0.97 1.06 0.98 0.87-1.08 0.96-1.17 0.90-1.07 -2.46 -3.49 -2.38 0.94 0.96 0.96 ‘Major axis line-fitting technique in natural log space. 100 D.J. DAEGLING HOLLOW SECTION A T MI ':[ 1 2.9 ;,7 FORCED ISOMETRIC 1.INE -. + 25- 232.1 - * * MAJOR AXIS SLOPE = 0.97 95% C.L.: 0.87-1.05 r-094 19- section. The second moments of area about the Y axis for the Paranthropus specimens are higher than would be expected for their cross-sectional size (Fig. 4). ratio The bending index provides a test of the hypothesis that sexual differences in corpus shape among modern hominoids are biomechanically significant. Statistical tests for the 1.61- ratio reveal no significant difference etween male and female great apes with respect to mandibular shape. Of course, this finding does not preclude possible differences in the absolute mechanical properties of the corpus caused simply by size dimorphism (Table 5). Figure 5 provides an interspecific comparison of bending indices for hollow and solid model treatments. What is immediately obvious is that the mechanical design of the Paranthropus mandible is distinct from that of any extant hominoid, regardless of the choice of a solid or hollow model. In addition, while the relative positions of modern taxa to one another are not influenced by the choice of design, their positions relative t o Paranthropus change conspicuously from solid to hollow treatments. It is also apparent that the magnitude of the bending index increases from a solid to a hollow design; that is, I, increases relative to I,. However, this is at least partially an artifact of the way in which hollow sections were digitized using the SLICE program. " ' " ' ' J Since CT sections were taken at the middle of each molar, they were digitized with a thin (ca. 1 mm) cap along the superior aspect of the corpus between the alveolar margins. In effect, this reduces the magnitude of I, since the average cortical thickness along the buccal and lingual margins will be greater than compact bone thickness along the superior and basal aspects of a cross section. L 4.3 4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1 LN CORTICAL AREA HOLLOW SECTION A T MI 2.7 2.3 / FORCED ISOMETRIC LINE . \p/ 1.7 - /.' 1.5 - 1.3 I l 43 6 " 4.5 MAJOR AXIS SLOPE. 1.06 95% C.L.: 0.96-1.17 r : 0.96 ~ 4.7 .' ' 49 LN ~ 5.1 ~ ~ ~ 5.3 5.5 5.7 ~ 5.9 ~ 6.1 CORTICAL AREA Fig. 4. Regression of second moments of area on compact bone area at M,. +, Paranthropus; A,Gorilla; 0, Pongo; m, Pun; A, I , on cortical area; B, I, on cortical area; *,Homo. For explanation, see text. other taxa at P < 0.01. Other departures from the isometric line are also worth noting. Gorilla tends to have relatively low polar moments of inertia in the MI region, and the Paranthropus specimens fall below the line for the polar moments of inertia in both cases. These results for polar moment of inertia, however, are not reliable because SLICE calculates J by assuming circularity of cross sections; considerable departures from a circular design lead to pronounced overestimates of the magnitude of J (Miller and Purkey, 1980). Thus, estimates of J for the modern hominoids are exaggerated because their corpora are not circular in cross DISCUSSION The results of this study suggest that the use of idealized geometrical models in mechanical studies is inadvisable in the absence of independent empirical support for their use. The solid ellipse model, which is of some utility in predicting cross-sectional area, seems to be quite limited in its ability to predict the biomechanical properties of a corpus cross section. A hollow ellipse design that assumes uniform cortical thickness is neither accurate nor reliable for predicting either area or cross-sectional moments of CT SCANNING OF HOMINOID MANDIBLES 101 TABLE 5. Cross-sectional areas and moments of inertia’ Area Taxon Sex Hollow section at Mz Pan F M F Pongo M F Gorilla M F Homo M Paranthropus SKX 4446 SKX 5013 Solid section at Mz Pan F M Pongo Gorilla Homo F M F M F M Paranthropus SKX 4446 SKX 5013 Hollow section at M 1 Pan F M F Pongo M F Gorilla M ~~ SKX 5013 ,I 1xx J Mean S.D. Mean S.D. Mean S.D. Mean S.D. 114.4 120.6 154.6 207.8 181.3 249.9 98.2 129.6 14.9 17.2 12.5 46.9 29.0 58.5 6.0 9.0 73.1 67.5 169.0 306.7 148.3 326.1 48.6 83.9 17.1 20.3 48.7 54.3 37.2 118.6 11.5 15.7 21.1 24.7 44.4 95.8 60.2 111.0 21.8 30.5 5.6 7.9 13.2 37.1 13.0 35.9 4.2 4.9 94.2 92.2 213.4 402.5 208.5 437.1 70.4 114.4 21.7 26.3 60.1 80.3 45.2 152.7 12.6 19.1 244.1 270.8 - 210.7 210.7 - 158.2 158.3 - 368.9 369.0 - 287.4 281.7 443.7 629.9 436.2 609.9 267.1 326.9 41.7 49.6 91.6 49.2 38.0 103.0 32.8 39.7 158.6 132.1 406.4 700.2 297.1 642.4 103.4 175.6 41.4 38.2 181.0 56.2 63.8 247.7 27.5 50.0 31.2 37.1 72.1 159.7 86.7 158.7 35.4 46.0 16.3 31.7 53.8 18.1 45.5 9.2 9.1 189.8 169.2 478.5 859.9 383.8 801.1 138.8 221.6 48.1 51.3 210.9 80.5 71.3 288.2 33.8 57.3 614.7 563.9 - 387.7 308.0 - 257.2 225.3 - 644.9 533.3 - 115.1 114.1 166.1 237.7 186.8 274.5 101.3 126.6 13.3 3.7 16.8 31.3 19.6 52.4 9.5 8.9 79.9 ... 70.0 180.1 375.4 157.1 361.2 57.7 84.1 17.7 .. 11.0 81.8 74.0 18.4 174.8 7.7 23.5 21.4 22.6 60.6 104.8 55.1 116.7 17.8 27.1 6.4 5.4 36.5 38.9 11.0 42.6 3.1 5.7 101.1 92.6 240.7 480.2 212.2 477.9 75.5 111.2 21.9 14.1 99.5 110.5 28.1 216.9 9.2 27.4 249.2 251.8 - 224.9 195.1 - 146.1 153.5 - 371.0 348.6 - 293.8 295.8 437.5 622.2 432.8 648.0 259.2 314.0 41.6 31.9 99.2 78.0 32.5 138.2 24.6 49.5 164.5 151.5 432.9 800.3 321.7 781.2 118.4 176.6 47.4 34.8 231.2 151.7 40.8 390.1 24.8 58.6 32.6 35.8 71.7 154.3 77.3 167.5 26.0 j39.0 10.1 9.9 29.6 55.8 15.2 66.7 5.0 11.1 197.1 187.3 504.6 954.6 399.0 948.7 144.4 125.6 54.6 40.6 259.9 198.5 52.2 456.5 27.9 68.9 596.5 583.5 - 421.0 366.1 - 202.4 209.3 - 623.4 575.4 - ~ ~~ - - 9.9 ... Solid section a t MI Pan F M Pongo F M Gorilla F M Homo F M Paranthropus SKX 4446 SKX 5013 ‘Area expressed in mm’; I,,, I,,, - and J in mm‘ X lo-’. inertia. The predictive ability of an asymmetrical elliptical model, however, warrants consideration. While the inadvisability of modelling the mandibular corupus as a simple ellipse has been demonstrated, there remains the question of whether the corpus is best conceived as a hollow or solid structure. Unfortunately, the present study provides little insight into the relative merits of either model for depicting in vivo stress patterns in the hominoid mandible, since actual stress distributions within cross sections were not examined. Hylander (1979b) and Smith (1983)have suggested that the corpus might behave as an open and/or hollow section under load. While the corpus appears t o be anatomically “hollow” in that the cortical bone is distributed only at the periphery of the section, the argument may be raised that the corpus does not behave as a hollow beam during mastication, since extensive trabecular bone networks are routinely found within the interior of the corpus (Figs. 2,6). There is a consensus that the function of such trabec- rc 102 D.J.DAEGLING HOLLOW SECTION MI SOLID SECTION MI M pan1 % pongo I i :G , , SKX 5013 , , Gorilla k+4 Homo rn SKX 5013 SKX 4446 SKX 4446 200 300 wo 500 600 roo 800 900 1000 100 200 300 BENDING INDEX A IYy/;x , 0 x I000 SKX 5013 SKX 4446 I00 200 400 500 , , 600 700 ‘YY/tXX 800 900 1000 - : Homo SKX 5013 , , , 800 900 1000 SKXGorilloi 4446 , 100 BENDING INDEX C 700 x I000 Y,l,Y;/ ;I ;- 300 600 SOLID SECTION M, M Homo 500 BENDING INDEX HOLLOW SECTION M, Gorilla1 , 900 x 1000 D 200 300 400 , , 500 600 .. , , 700 800 u 900 1000 BENDING INDEX IYy/l,, x I000 Fig. 5. Interspecific differences in bending indices (I =) for solid and hollow treatments at M, and M,. For explanation, see text. Based on A., co ical area at M,; B., total subperiosteal area at M,; C., cortical area at M,; D., total subperiostal area at M,. x.“ this concept, even though the apparent density of cancellous bone is less than compact bone (Carter and Hayes, 1976). Trabecular networks are frequently encountered in CT examinations and in sectioned specimens along the endosteal aspect of the compact bone, although the extent of these networks within cross sections is highly variable (Fig. 6). In those cases in which trabeculation is found throughout section (Fig. 6A), the appropriate model would appear to be that of an enclosed beam with a series of holes (for a different interpretation, see Ruff,1983). On the other hand, a section with relatively little trabeculation (Fig. 6B) would best be treated as a hollow section. Thus, neither solid nor hollow beam models on their own are likely to be totally reliable predictors of the mechanical design and behavior of the mandible. Other aspects of mandibular morphology that have been neglected to this point are the A B roles of the teeth and periodontal ligaments Fig. 6. Two cut sections of human mandibles at M!, (PDL) in the distribution of bite forces. It has showing variation in trabecular networks between indi- been argued that alveolar intrusion into the viduals. The extent of this variation illustrates that idealized models of mandibular shape and form are corpus provides a case for the ”open section” unlikely to be reliable indicators of real biomechanical model of the corpus (Hylander, 1979b; properties.A and B are explained in text. Smith, 19831, but how the presence of teeth ular “struts” or “columns”is to counter principal stress regimes in bones (Lanyon, 1974; Currey, 1984). The observation that cancellous bone behaves in a biomechanically similar manner to compact bone lends support to CT SCANNING OF HOMINOID MANDIBLES within the alveoli affect the mechanical behavior of the corpus is a question critical to the evaluation of this model. Unfortunately, consideration of the tooth and PDL further complicates matters (Smith, 1983). One implicit assumption of the present study is that the mandibular cross section is isotropic; Currey (1984)has remarked that this results in an oversimplification of the true mechanical features of bone. Addition of the PDL and tooth to an analysis of cross-sectional design results in the consideration of a structure that is not only anisotropic, but nonhomogeneous as well. Theoretically, this can be addressed by employing a compositebeam or a finite element approach (Gallagher et al., 1982), in which the strength and stiffness of different materials in a cross section are figured into an analysis. It is certain that factors such as the area of PDL attachment, PDL fiber orientation, and displacement of the tooth in the alveolus during mastication impact on load distribution in the mandibular corpus (Lehman, 1968; Glickman et al., 1970; Ralph and Williams, 1975; Melcher and Walker, 1976; Wills et al., 1978). Explanations of the pecularities of internal mandibular design (Fig. 7) require an understanding of these factors. Despite the idealized treatment of the material properties of the mandible in the present study, the methods employed here provide an empirical assessment of the mechanical consequences of size and shape in the hominoid mandibular corpus. Even so, there may be considerable sources of error in the approach pursued here in terms of the ways in which cross-sectional mechanical properties were evaluated. Specifically, the use of a 1 mm cap on hollow sections introduces an arbitrary amount of bone into the analysis, and the magnitude of this error probably varies differentially between taxa. In addition, relative torsional resistance of hominoid mandibles cannot be evaluated using the conventional calculation of J for reasons outlined above. This particular measure of the ability of the corpus to resist torsion appears to be inappropriate for the inference of mechanical design in the mandible (see Popov, 1976; Miller and Purkey, 1980; Burr and Piotrowski, 1982). The relatively high 1 4 , values for the Paranthropus mandibles support the hypothesis that they were particularly wellsuited to resist transverse bending or “wishboning” during the power stroke of mas- 103 Fig. 7. CT section of the M, region of the SKX 5013 mandibleof Purunthropus. The trabecularstruts seen in this specimen must have played a major role in the dissipation of bite forces and should therefore probably be incorporatedinto biomechanical models of the mandible. tication (Hylander, 1988). Since this bending index in the Paranthropus specimens most closely approaches unity in the sample studied (indicating a more circular cross section), the hypothesis that resistance to torsion was critical to the design of the Paranthropus mandible is also supported, since the most efficient design for resisting torsion is a circular cross section. When the mechanical design of the Paranthropus cross sections is evaluated with respect to other mandibular dimensions, the distinctiveness of this early hominid becomes increasingly apparent. The true “robusticity” of the Purunthropus mandibles is best evaluated by the regression of cross-sectional moments of inertia on mandibular length (Hylander, 1979a, 1988). Due to the fragmentary nature of the material this could not be accomplished in the present study; however, it should be made explicit that the ability of a mandibular cross section to resist internal stresses cannot be known by simple reference to the traditional “robusticity” index of breadthheight. This is an index of shape only, from which limited 104 D.J. DAEGLING mechanically relevant information can be readily discerned. Since biomechanical analyses of long bones routinely refer to the robusticity of an element as an index of midshaft dimensions over shaft length, it seems appropriate that mandibular robusticity be defined by the same criterion. Mandibular length, by approximating the moment arms of the forces acting on the corpus, provides a mechanically relevant denominator for any index of mandibular robusticity. It is apparent from other data (Hylander, 1988; Grine, unpublished) that the magnitude of second moments of area in Paranthropus mandibles becomes even more marked when they are compared with corpus length. The relatively low values of I, observed in the Paranthropus mandibles for regressions of this variable against compact bone area are probably not meaningful in mechanical terms (Fig. 4). In fact, this seemingly “deficient” vertical bending moment, when regressed against mandibular length, is undoubtedly substantially greater than those of extant hominoids (Hylander, 1988). In addition to the short length of the corpus, the relative width of the dental arcade in Paranthropus probably also has significant implications for the distribution of bending and torsional stresses in the mandibles of these hominids. Considerable attention has surrounded the contention of Wood (1978) and Chamberlain and Wood (1985) that the large canine in male primates has the effect of reducing the “robusticity” of the male mandibular corpus (Kimbel and White, 1988; Hylander, 1988). While sexual differences in this index have been documented among higher primates (Picq, 1987), the results of the present study indicate that no significant difference exists in the mechanical properties of male and female hominoid corpora apart from those that are purely a consequence of size. The amount and distribution of compact bone is of considerable consequence to the mechanical behavior of a mandibular corpus cross section (Smith, 1983; Hylander, 1988). Most of the variation observed in the present study cannot be related to any specific mechanical feature. It is interesting to note that those orangutans that tend to deviate markedly above forced isometric lines for moments of inertia (Fig. 4) do not display a substantially greater amount of compact bone relative to total subperiosteal area. This is an example of how the distribution of compact bone may be optimized in a corpus cross section; in this case, cross-sectional moments of inertia are maximized without an increase in cortical area by distributing bone as far as possible from the neutral axis of the cross section. This suggests that the amount of compact bone within the total subperiosteal area is not necessarily a reliable guide to mechanical efficiency of the corpus as measured here. The consideration of variables such as mandibular arch width, symphyseal size and shape, and mandibular length may further elucidate the relationship of compact bone distribution to biomechanical efficiency in the hominoid mandible. The pronounced intraspecific variation and interspecific differences in crosssectional mechanical design, as determined by CT, indicate that the use of simple models based on external mandibular metrics is of questionable value for the identification of the functionally relevant features of the hominoid corpus. Given the limitations of idealized geometrical models for interspecific comparisons of this nature, it should be recognized that studies that utilize such models across higher taxonomic groups, in attempting to establish correlations between external corpus dimensions and mechanical attributes of the mandible, necessarily exclude a substantial amount of potentially relevant information (cf. Smith, 1983; Hylander, 1988).Application of CT t o the study of cross-sectional design promises to be a productive approach for interpreting the more subtle interspecific differences in mandibular design in modern as well as fossil taxa. SUMMARY AND CONCLUSIONS Mandibular cross sections of 10 specimens each of Pan troglodytes, Pongo pygmaeus, G. gorilla, and H. sapiens and two specimens of Paranthropus robustus were examined by CT to determine empirically the mechanical properties of the hominoid mandibular corpus. The accurate depiction of osteological structures by CT reveals that the utility of solid or hollow ellipse models for estimating the mechanical design of the corpus is limited. The pronounced intraspecific and interspecific variation in compact bone distribution precludes the possibility that a simple hollow elliptical model can depict the mechanical properties of the corpus in an accurate or reliable manner. CT SCANNING OF HOMINOID MANDIBLES In the present study, biomechanical correlates of cross-sectional shape indicate that there is no evidence for sexual dimorphism in the mechanical design of the hominoid corpus apart from differences that are purely a consequence of size. In addition, the contribution of compact bone to total subperiosteal area does not differ significantly between taxa. The most distinctive feature of the Paranthropus corpus is that the second moment of area about the vertical axis is extremely large by comparison to other hominoid mandibles. This suggests that there may have been a significant transverse component to the bite force in these early hominids (Hylander, 1988). The traditional “robusticity” index of the mandibular corpus, while useful for taxonomic comparisons, may be of limited utility for functional interpretations of early hominid gnathic morphology. The determination of the functional significance of mandibular robusticity in the hominid fossil record requires further emphasis on mechanically relevant variables rather than continued use of metric criteria of dubious mechanical significance. CT makes possible the determination of mechanically relevant variables in the study of cross-sectional morphology, and a more comprehensive sampling of extant and extinct primates will assist in the development of a more meaningful definition of mandibular robusticity. ACKNOWLEDGMENTS I thank Dr. C.K. Brain of the Transvaal Museum and Drs. G. Musser and I. Tattersall of the American Museum of Natural History for permission to examine the material in their care. Mr. B. Day provided invaluable technical assistance in CT operations. Drs. F.E. Grine, W.L. Hylander, W.L. Jungers, C.B. Ruff, and J.T. Stern commented on earlier drafts of this manuscript, for which I am grateful. I also thank two anonymous reviewers for their comments. This project was supported by a grant-in-aid of research from the Sigma Xi Foundation. LITERATURE CITED Bouvier M (1986a) A biomechanical analysis of mandibular scaling in Old World monkeys. Am. J. Phys. Anthropol. 69:473-482. 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