Application of an image-based weighted measure of skeletal bending stiffness to great ape mandibles.код для вставкиСкачать
AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 131:243?251 (2006) Application of an Image-Based Weighted Measure of Skeletal Bending Stiffness to Great Ape Mandibles Neel B. Bhatavadekar,1 David J. Daegling,2 and Andrew J. Rapoff3* 1 Department of Periodontology, University of North Carolina, Chapel Hill, North Carolina 27599 Department of Anthropology, University of Florida, Gainesville, Florida 32611 3 Department of Mechanical Engineering, Union College, Schenectady, New York 12308 2 KEY WORDS biomechanics; bone density; computed tomography; image analysis; moment of inertia ABSTRACT Traditional measures of structural stiffness in the primate skeleton do not consider the heterogeneous material stiffness distribution of bone. This assumption of homogeneity introduces an unknown degree of error in estimating stiffness in skeletal elements. Measures of weighted stiffness can be developed by including heterogeneous grayscale variations evident in computed tomographic (CT) images. Since gray scale correlates with material stiffness, the distribution of bone quality and quantity can be simultaneously considered. We developed weighted measures of bending resistance and applied these to CT images at three locations along the mandibular corpus in the hominoids Gorilla, Pongo, and Pan. We calculated the traditional (unweighted) mo- ment of inertia for comparison to our weighted measure, which weighs each pixel by its gray-scale value. This weighing results in assignment of reduced moment of inertia values to sections of reduced density. Our weighted and unweighted moments differ by up to 22%. These differences are not consistent among sections, however, such that they cannot be calculated by simple correction of unweighted moments. The effect of this result is that the rank ordering of individual sections within species changes if weighted moments are considered. These results suggest that the use of weighted moments may spur different interpretations of comparative data sets that rely on stiffness measures as estimates of biomechanical competence. Am J Phys Anthropol 131:243?251, 2006. V 2006 Wiley-Liss, Inc. Traditional measures of structural stiffness in skeletal elements involve consideration of cross-sectional geometry, but ignore variations of material stiffness within that geometry, i.e., sections are considered to be homogeneous in material density. Given that bone is structurally heterogeneous, errors associated with the measurement of structural stiffness are likely when stiffness measures employ the assumption of homogeneity. We developed a technique for incorporating the material stiffness variation of bone into measures of skeletal rigidity. In the context of understanding bone stiffness and strength, structural heterogeneity means that the elastic modulus of bone is not uniform throughout the member under investigation. The effect on stiffness (and ultimately strength) depends on the degree of heterogeneity, such that in some contexts, considerations of inhomogeneities in bone may have trivial consequences, while in others they could profoundly affect interpretation. As an example, consider a sample of rubber tubes of similar and invariant geometry, except that one specimen has a series of steel ?bers embedded longitudinally through the structure. The latter structure is stiffer than the remainder of the sample, but this is not inferred from a traditional calculation of moments of inertia based solely on specimen geometry. In this case, some allowance must be made for the different materials in the member and their mechanical effects. The solution, of course, is to consider the impact of the amount and location of the different materials (and their associated stiffness values) on overall mechanical behavior. How important this correction would be in this hypothetical example would depend on the precise amount and distribution of the two materials. If steel accounted for less than 1% of the material composition of our rubber beam, the effect on stiffness and strength would be negligible, while a beam of 20% steel would behave considerably differently from its all-rubber counterparts. In the osteological context, then, the question becomes, how much does heterogeneity impact measures of skeletal stiffness? The technology of computed tomography (CT) is ideal for comparative biomechanical investigations (which require information on both geometry and material distribution), since it is a nondestructive technique that permits the examination of large comparative samples. The major technological breakthrough of CT is that plane structures, such as mandibular cross sections, can be visualized without the interference of structures on either side of the section (Houns?eld, 1973). The use of CT scans to measure biomechanical properties is preferred over external linear dimensions, because simplifying assumptions about cross-sectional geometry and bone distribution are circumvented (Smith, 1983; Demes et al., 1984; Daegling, 1989). Because accurate images of cortical bone contours may be obtained using CT, this approach provides a more pre- C 2006 V WILEY-LISS, INC. C *Correspondence to: Andrew J. Rapoff, Ph.D., Department of Mechanical Engineering, Union College, Schenectady, NY 12308-3147. E-mail: firstname.lastname@example.org Received 11 March 2005; accepted 9 November 2005. DOI 10.1002/ajpa.20397 Published online 4 April 2006 in Wiley InterScience (www.interscience.wiley.com). 244 N.B. BHATAVADEKAR ET AL. cise assessment of biomechanical function (Ruff and Leo, 1986). Another advantage of CT is the visual display of relative bone density in terms of gray scale: more opaque regions of an image indicate higher mineral density. However, this ability to incorporate material heterogeneity has not been used in comparative biomechanical investigation of the primate skeleton, with a few notable exceptions (Martin and Burr, 1984; Schaf?er et al., 1985; Jungers and Burr, 1994; Rafferty, 1998). The value of understanding patterns of bone material variation for biomechanical inferences has not gone unrecognized. For example, Weiss et al. (1998) used acoustic techniques in combination with geometric parameters such as principal moments of inertia, polar moment of inertia, and the biomechanical shape index for comparisons between cross sections of the femur. A pulse transmission ultrasonic technique was used for detailing variation in elastic properties throughout the mandible, and in doing so, clari?ed the relationship between these properties and mandibular function (Schwartz-Dabney and Dechow, 2003). Mayhew et al. (2004) used densityweighted cross-section moments of inertia derived from peripheral quantitative computed tomography (pQCT) images as a basis for the structural analysis of hip fractures. They used attenuation coef?cients to represent gray-scale values, and incorporated density information into measures of the femoral neck section modulus (a variable derived from moments of inertia) to address questions of fracture risk. Corcoran et al. (1994) made use of mass weighted moments of inertia and bone crosssectional area to evaluate changes in bone cross-sectional area in postmenopausal women. Grayscale variations presented in CT images provide a means to estimate heterogeneity in bone. In addition, spatial information pertaining to this variation (i.e., the precise location of pixels and their gray-scale values within a section) is readily gleaned from CT images. Since gray scale correlates with the material stiffness and strength of bone (Ciarelli et al., 2003; Currey, 1988), as well as its mineral density (Go?tzen et al., 2003), the distribution of bone quality (density) and quantity (area) may be considered in estimating relative stiffness for comparative purposes. We illustrate the application of weighted measures for a sample of mandibular sections of great ape mandibles. Customized software was developed to calculate weighted and unweighted measures of bending resistance at three locations along the mandibular corpus from CT images. In collecting these data, we address two questions, one methodological and one morphological: 1) Does the use of weighted moments of inertia change the distributional pro?les of moments of inertia to the extent that comparative interpretations are altered? 2) Does the use of weighted moments help resolve the question of whether dietary differences among great apes are re?ected in mandibular morphology (cf. Daegling, 1989, 1990, 2001; Taylor, 2002)? MATERIALS AND METHODS Mandible samples and CT scans We used a total of 90 CT images of mandibles from adult males and females of three hominoid species (Gorilla gorilla, Pan troglodytes, and Pongo pygmaeus), obtained from previous studies (Daegling, 1989, 1990; Daegling and Grine, 1991). The images were sampled from symphyseal (S), premolar (P4), and molar (M2) sections. These regions were chosen to represent the gamut of internal stresses present in the mandible during functional loading. The protocol for CT scanning was detailed in Daegling (1989). Scans were performed on a GE CT 9800 machine with 120 kV tube voltage, 170 mA tube current, 4 s scan time, 1.5-mm slice thickness, and in-scan resolutions (pixel dimensions) of either 0.25 mm or 0.49 mm; the bone reconstruction algorithm was used in all scans. The specimens were oriented in each scan such that the x (horizontal)-axis was parallel to the occlusal plane, and the y (vertical)-axis was de?ned perpendicular to that plane. This orientation de?nes axes parallel to the anatomical axes used in calculating moments of inertia. Accuracy of images was assessed empirically for the CT machine used by scanning a sample of human mandibles, sectioning the specimens in the plane of the scan, and verifying the correct level and window width setting for the accurate portrayal of cortical bone contours. The one parameter in which scanning differed among samples is that the Pan sample was scanned in air, while Pongo and Gorilla samples were scanned in water. Calibration using the human specimens established that settings of a �000 Houns?eld units (HU) level and 4,000 HU width accurately imaged the air-scanned specimens, while for water, a �500 HU level and 4,000 HU width were accurate for specimens scanned in a water bath. Areal measurement error between the stated settings and the actual contours determined by direct sectioning was less than 2% (air-scanned specimens, Daegling, 1990) and less than 5% (water-scanned specimens, Daegling, 1989). The window width of 4,000 HU allows for a wide range of HU values to be visually displayed (well beyond the grayscale variation for bone), such that variations in bone density are visually captured. Practically speaking, narrower widths limit the range of CT numbers displayed, and can underestimate density variation. In addition, the use of wide window widths is theoretically preferred for two reasons. First, high widths minimize the artifacts that render high-density objects particularly sensitive to centering window (level) adjustments (Joseph, 1981); and second, narrower widths tend to exacerbate statistical noise in the calculation of CT numbers (McCullough, 1977). Given a range of 256 grayscale values, the window width employed suggests that each grayscale interval encompasses a little over 15 HU. Use of a water bath may impact beam attenuation, such that some bias in attenuation coef?cients, albeit minor, may result. Our concern here is whether these effects will substantially affect grayscale values in airvs. water-scanned specimens. Most tangible is the context of partial volume effects, i.e., at periosteal and endosteal margins, there will be voxels sampled that are incompletely ?lled with bone that will return different values depending on the remaining media (water or air). There are relatively few of these in comparison to the voxels ?lled with bone, but the variance in grayscale may nevertheless be affected. To assess this in these data, we plotted CT number pro?les in linear tracings from the periosteal to endosteal margins in M2 sections of Pan and Gorilla to estimate the effect on variance. Figure 1 shows that the contrasting media of water vs. air do not appear to substantially affect CT numbers once the periosteal margin is encountered (i.e., the slopes of the curves are very similar in this region). The maximum CT values encountered in linear tracings of air-scanned specimens of Pan are 2,447, 2,145, and 2,148, while in the Gorilla specimens they are 1,992, IMAGE-BASED SKELETAL STIFFNESS Fig. 1. CT numbers (Houns?eld units, HU) over linear pixel by pixel lateral-medial traverse of lateral cortex at M2 sections in three specimens each of Pan and Gorilla. Gorilla specimens were scanned in a water bath. Plot is arranged so that at sixth pixel, each scan encounters periosteal surface, and by tenth pixel, all scans have encountered endosteal margin. Similarity of pro?les at periosteal margin suggests that different media of air and water do not dramatically in?uence CT numbers in mandibular bone. Disparate pro?les after tenth pixel are due to factors such as differences in cortical thickness, trabecular density, and surrounding media. Note that with exception of second Pan specimen, it is dif?cult to discern which specimens were scanned in which medium from pixels beyond endosteal margin (10th through 21st pixels). In our analyses, extracortical pixels were set to black (0 gray-scale value, or 1,000 HU). 2,224, and 2,222. These values are similar between specimens, despite the 1,000 HU difference in surrounding media. Since different specimens are sampled in the different media, we cannot establish that the media have no effect on CT numbers, only that the effect is minor compared to the different densities of air and water. The variation in pro?les on the right side of Figure 1 is the result of expected differences in bone thickness and cancellous bone density among specimens. Indeed, it is dif?cult to ascertain which specimens were scanned in air or water in this region of the interior of the corpus, despite the obvious infusion of water into this space in CT images. Even so, we employed procedures to eliminate possible bias by setting the space in the interior of the corpus to zero gray scale (discussed next). These results suggest that the different media likely had an insigni?cant effect on grayscale values calculated in this study. Image conversion and editing Conversion from the CT image format and manual image-editing were required prior to submitting the image to the analysis software (MATLAB Image Analysis Toolbox, MathWorks, Natick, MA). First, each CT image was converted into 8-bit grayscale depth (28 � 256 grayscale ??bins,?? such that 0 � black and 255 � white) bitmap format, using commercially available software (TomoVision, Montreal, Quebec, Canada). The image backgrounds consisted of differing and nonzero gray-scale values, because some mandibles were scanned in a water bath, and some in air. If analyzed in this state, the backgrounds would have erroneously contributed to the moments of inertia. To ensure a background of uniform zero grayscale, the cortical bone external margins were 245 Fig. 2. Cross section after cropping of tooth and cancellous bone from image. Shown are reference x y, centroidal xc yc, and principal xp yp axes, centroid location (x, y), principal angle up, pixel area DAi, and its location (xi, yi). Reference x-axis is parallel to occlusal plane. edge-detected, and extracortical pixel grayscales were set to zero. We used the ??edge()?? function intrinsic to MATLAB and the default Sobel method to detect cortical edges. The Sobel method de?nes an edge as where the gray-scale gradient is a maximum. Then, any teeth and supporting roots appearing in a cross section were cropped, since we assume here that teeth do not contribute signi?cantly to the stiffness of the mandible (Daegling et al., 1992; Marinescu et al., 2005). This was done by outlining the periodontal ligament space and removing the tooth crowns and roots from the image. Finally, any trabecular bone appearing in a cross section was cropped as well. We assume here that trabecular bone is unlikely to contribute signi?cantly to the bending stiffness of skeletal elements (Ruff, 1983) such as the mandible, and the exclusion of trabecular bone follows precedent with other CT-based studies of mandibular stiffness, to facilitate comparison of our data with previously published studies (Daegling, 1989, 1992, 1993, 2001; Schwartz and Conroy, 1996). The remaining pixels after tooth and/or cancellous bone cropping were set to black. Pixels corresponding to the remaining cortical bone were assigned grayscale values that could range theoretically from 0 (corresponding to a void in the cortex) to 255 (a pure white pixel). Examination of histograms of pixel grayscale values from individual specimens revealed different ranges of grayscale values. For example, most specimens had no pixels assigned a value of 255, and those that did typically had only 1 to 3 pixels with this value. Weighted measure of bending stiffness The development of our weighted bending stiffness measure began by considering the schematic of a mandibular cross section (Fig. 2). The cross section can be 246 N.B. BHATAVADEKAR ET AL. viewed as being discretized into differential areas DAi of identical size, such that the differential areas become pixels in the computerized image analysis. The unweighted area (second) moments of inertia of the cross section about the arbitrary axes x and y are given by the usual formula " 饀� 饀� 饀� 絀xx ; Iyy ; Ixy � n X y2i DAi ; n X i� x2i DAi ; i� n X # xi yi DAi i� where superscript (u) denotes that these are unweighted moments, in that they include no information regarding grayscale variation. We formed our weighted stiffness measure by weighting (multiplying) each pixel area by its grayscale value Gi raised to some power r, so that " 絀xx ; Iyy ; Ixy � n X y2i Cri DAi ; n X i� x2i Cri DAi ; n X i� # xi yi Cri DAi i� where no superscript denotes a weighted moment (determination of the value of r is discussed below). The distances x and y from the weighted centroidal axes xc and yc to the arbitrary axes x and y were determined by modifying the formula for determining centroids of areas, as in 2P n xi Cri DAi 6i��x; y � 6 n 4 P i� Cri DAi n P yi Cri DAi ; i� P i� Cri DAi 3 7 7: 5 The weighted moments of inertia about the centroidal axes were then determined from the parallel axis theorem " n X Cri DAi ; Iyy Ixc xc ; Iyc yc ; Ixc yc � Ixx y2 i� x2 n X Cri DAi ; Ixy xy i� n X Image analyses Each image was analyzed in an unweighted and weighted manner. In the unweighted state, each pixel of cortical bone was assigned a white grayscale value of 255; in the weighted state, a grayscale value was determined for each pixel. For premolar and molar sections, the unweighted moments of inertia were also computed about axes corresponding to the weighted principal axes, so that that both unweighted and weighted moments were referred to the same coordinate axes for some comparison purposes. In any event, the principal axes for unweighted and weighted moments were nearly identical to the centroidal anatomical axes for the premolar and molar sections, and not signi?cantly different from each other. For symphyseal sections, unweighted and weighted moments were transformed into identical weighted centroidal anatomical coordinate axes (vertical and horizontal axes through the weighted centroid in the sagittal plane). Therefore, unweighted Ix(u) and weighted Ixp xp for pxp premolar and molar sections, and unweighted Ix(u) cxc and weighted Ixc xc for symphyseal sections, re?ect resistance to bending in the parasagittal and coronal planes, respectively. Similarly, unweighted Iy(u) and weighted Iyp yp for premopyp (u) lar and molar sections, and unweighted Iycy and weighted c Iycyc for symphyseal sections, re?ect resistance to bending in a transverse plane (i.e., ??wishboning?? loads identi?ed by Hylander, 1984). # Cri DAi : i� The principal weighted moments of inertia are given by the usual formula for determining principal moments of inertia 2 s???????????????????????????????????????????? h i Ixc xc � Iyc yc Ixc xc Iyc yc 2 2 4 Ixp xp ; Iyp yp � � 蘒xc xc ; 2 2 ?3 s??????????????????????????????????????????? Ixc xc � Iyc yc Ixc xc Iyc yc 2 2 5 蘒xc xc 2 2 where Ixpyp : 0. The orientation up of the weighted principal axis xp with respect to the weighted centroidal axis xc is up � calculating the unweighted and weighted centroid locations, principal moments, and principal orientations. All pixels in the image, including the pixels comprising the background, are included in the analysis. We validated our program by comparing program outputs with manually calculated values for numerous simple and composite cross-sectional shapes with homogeneous and heterogeneous variations in gray scale. 2Ixc yc 1 tan1 : 2 Ixc xc Iyc yc We developed a MATLAB-based computer program to analyze the bitmap-converted CT scan images. The program accounts for the grayscale value of each pixel when Allometric compensation The mandibles from which images were derived were obviously of different sizes and shapes, given the genera sampled. Thus, the ?nding of absolutely larger moments in Gorilla vs. Pan is not, in itself, terribly interesting from the standpoint of comparative biomechanics. Therefore, the moments were scaled by appropriate linear dimensions of the mandible, raised to the fourth power so as to form dimensionless indices. The appropriate reference variables depend on the cross section of interest and the mode of bending under consideration (Daegling, 1992; Hylander, 1979). For each premolar and molar section, the moment arm for parasagittal bending ds was de?ned as the distance from the infradentale to the cross section of interest (Daegling, 1992, 1993). For each symphyseal section, the moment arm for wishboning dw was de?ned as the distance from the posterior ramal border to the infradentale (i.e., the length of the mandible; Hylander 1984, 1985). The scaled weighted moments are de?ned as and denoted by ~Ixx Ixp xp =d4s for premolar and molar sections, and ~Iyy Iyc yc =d4w for symphyseal sections (the moment arm for resisting coronal bending at the symphysis is not easily de?ned from mandibular dimensions, because this load is caused by unsymmetric bilat- 247 IMAGE-BASED SKELETAL STIFFNESS eral twisting of the postcanine mandibular corpora; thus, an associated moment (I?xx) is not de?ned for comparisons). The scaled unweighted moments are de?ned similarly 饀� 饀� as and denoted by ~Ixx Ixp xp =d4s for molar and premolar 饀� 饀� 4 ~ sections, and Iyy Iyc yc =ds for symphyseal sections. TABLE 1. Comparison of unweighted and weighted unscaled moments of inertia and principal angles Difference between unweighted and weighted values Unscaled moments1 Effect of grayscale power The magnitude of the weighted moments of inertia and weighted principal axis orientations depend on the power r to which pixel grayscale values are raised. For cortical bone, radiographic grayscale is a power-law function of wet apparent density q over the range of physiologic apparent densities, or C � aqb � v ) q � 1 餋 v� a 1=b where a, b, and v are constants. Our unpublished laboratory data indicate that b 2.4. We arrived at this estimate by preparing small rectangular specimens of bovine cortical bone, and subjected each to various immersion times in a decalcifying agent. We weighed and measured (to determine volume) each specimen, and computed the wet apparent density of each specimen. We microradiographed each specimen, imaged each microradiograph under a light microscope under identical lamp intensities, and captured each image with a digital camera. Finally, we determined average grayscale values throughout the image of each microradiograph, and correlated the average grayscale values with wet apparent densities for all specimens. For cortical bone, the modulus is also a power-law function of apparent density E � aqb � a 餋 v辀=b ab=b where a and b are constants, and the previous expression for density in terms of grayscale is substituted to obtain the right hand side. Therefore, the modulus is proportional to grayscale to the power of r � b/b, or E / Cr : We estimated b 3.4 using the data of Currey (1988), and others (Carter and Hayes, 1977; Rho et al., 1993) estimated 2 9 b 9 3. Therefore, we estimate r to be in the range 0.8 9 r 9 1.3. In our formulation, each pixel grayscale can be raised to any desired power, although powers on the order of 100 result in an over?ow condition during the computerized analysis, because the numerical values of the weighted moments become very large. We report our main results using r � 1, such that the grayscale value serves as a direct surrogate for the modulus. However, we report below the effect on the weighted moments of other values of r on a subset of our data. We varied the exponent over two logarithmic decades (from 0.1 to 10), and computed the weighted and scaled principal moments of inertia and principal angles for the 14 Gorilla molar sections. We used simple regression to examine the effect on the natural log-transformed weighted and scaled moments and the principal angles of varying r, as well as noting the rank order of the moments of each section as r was varied. 饀� 饀� Iyp yp Iyp yp Ixp xp Ixp xp 饀� Taxon Section Pongo Symphyseal Premolar Molar Symphyseal Premolar Molar Symphyseal Premolar Molar Gorilla Pan 1 Iyp yp 12% 12% 22% 15% 16% 19% 18% 18% 20% 12% 11% 21% 15% 15% 19% 20% 17% 20% Ixc xc Ixc xc 饀� Ix c x c 饀� Ix p x p 饀� up up 饀� up 0.18 0.98 0.68 0.18 1.18 0.18 0.28 0.28 0.18 饀� 饀� Differences between physeal sections. Principal angles and Iyc yc Iyc yc 饀� Iyc yc are listed for sym- Statistical analyses We used a model II regression to determine the correlation between moments of inertia and the moment arm dimensions used to calculate the scaled moments of inertia discussed above. We compared mean values of scaled unweighted and weighted moments for each section (premolar, molar, and symphyseal), using a full-effects, repeated-measures analysis of variance (ANOVA) in which the main effects were taxon and sex. If the interaction was not signi?cant (P 0.05), the ANOVA was performed again with the sex effect removed. When a significant ANOVA resulted, individual comparisons were made with a post hoc Fisher?s protected least signi?cant difference (PLSD) test. We further examined the effects of weighting moments of inertia on the rank-ordering of individuals within species using Kendall?s s rank correlation test. Commercially available software (StatView, SAS Institute, Cary, NC, and BIOM-Stat, Applied Biostatistics, Port Jefferson, NY) was used for all statistical analyses. RESULTS Comparisons of unscaled moments: weighting effects on individual values The effects of weighting are variable across individuals and taxa (Table 1), but the general ?nding is that the error introduced by ignoring density variation is on the order of 10 to 22%. Pongo presents the most dramatic difference between unscaled unweighted and weighted moments of inertia, ranging from 12% for premolar sections up to 22% for molar sections (with the weighted moment of inertia always less in value). The moments in these cases were calculated prior to any axis transformation, to illustrate the full effect of ignoring heterogeneity altogether in a comparative context. Molar sections within each taxon demonstrated the greatest difference between unweighted and weighted moments compared to premolar and symphyseal sections. 248 N.B. BHATAVADEKAR ET AL. Comparison of scaled moments: species differences As described, the moments of inertia were scaled for taxa using the parasagittal moment arm or the wishboning arm, as appropriate. In no case was an interaction indicated for the two factors of sex and species. Signi?cant differences between scaled weighted I?yy moments distinguished Pan from Pongo (P � 0.0247) and Pongo from Gorilla (P � 0.0099) at P4 sections. At the symphysis, signi?cant differences distinguished Gorilla from Pongo (P � 0.0062). The scaled weighted I?xx moments did not discriminate between any of the taxa at P4 and M2 sections, and there was no sexual dimorphism in this index. For reasons of brevity, we present only the statistically signi?cant results; the reported P values were obtained from a post hoc ANOVA Fischer?s PLSD. Allometric comparison: scaling of weighted and unweighted moments TABLE 2. Effect of weighting moments of inertia on rank orders within 14 Gorilla molar sections1 Scaled moments of inertia Rank order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Unweighted (u) I?xx Weighted I?xx A B C D E F G H I J K L M N A D C B E G F H J K I M L N 1 For premolar and molar sections, the unscaled unand unscaled weighted Ixpxp moments scaled weighted Ix(u) pxp with their appropriate parasagittal bending moment arm to the fourth power with nearly identical slopes (1.050 unweighted, 1.053 weighted). The different intercepts (2.795 unweighted, 2.898 weighted) were expected due to the weighting procedure, which necessarily reduced the absolute magnitudes of moments for each section. These unweighted and weighted moments scaled with positive allometry with respect to the parasagittal moment arm (coef?cients of determination R2 of 0.886 unweighted and 0.885 weighted). For symphyseal sections, the unscaled unweighted Iy(u) and unscaled cyc weighted Iycyc moments scaled with their appropriate wishboning moment arm to the fourth power with similar slopes (0.644 unweighted, 0.674 weighted), and with the expected difference in intercepts (1.066 unweighted, 1.395 weighted). These moments scaled with negative allometry with respect to the wishboning moment arm (coef?cients of determination R2 of 0.595 unweighted and 0.584 weighted). Comparisons of rank orders We compared scaled weighted and unweighted rank orders within each taxon for all sections. For example, Table 2 demonstrates how the rank order changes for Gorilla molar sections from when they are unweighted to when they are weighted (Kendall?s s � 0.824, P < 0.0001). We observed signi?cant changes in rank orders for symphyseal and premolar sections for all three taxa (Table 3). These changes in rank order indicate that the effect of weighting sections is neither uniform nor stereotypical among different individuals. Gray-scale variation and its relationship to elastic modulus We compared rank-ordering within the 14 Gorilla molar sections for a series of exponents r that relate gray scale to elastic modulus from 0.9 to 1.1, and for values between 0.1 to 10 (Table 4). For I?xx, there is a single rank-order switch of adjacent individuals for 1 r 1.1, and the rank ordering is identical for 0.9 r 1. Order of magnitude changes in the exponent (i.e., 0.1 to 1.0 and 1.0 to 10) results in dramatic and signi?cant alteration in rank ordering. Nine of 14 sections changed their rank order, three pairs swapped ranks (B D, F G, and L M), and one triplet permuted ranks (I J K and J K I). Kendall?s s coef?cient of rank correlation between unweighted and weighted moments is 0.824 (P < 0.0001). DISCUSSION Three observations support the conclusion that weighted measures of bending resistance are more desirable in comparative applications than the unweighted measures that are traditionally employed. First, an assumption of heterogeneity for bone sections is a more realistic re?ection of bone as a material and structure. Second, our data indicate that unweighted measures may overestimate bending rigidity on the order of 10 to 22%. Third, weighting sections has the effect of changing the relative stiffness of individuals within a sample, an indication that the errors incurred in unweighted measures are not uniform in magnitude from specimen to specimen. The implication is that it is unlikely that a weighted moment of inertia is predictable from the unweighted value in any individual. We found up to a 22% difference between the unweighted and weighted moments of inertia (Table 1). The largest differences were consistently found in molar sections in the three taxa. The reason for this ?nding is not obvious. It may be that the relatively thick cortex here indicates that the compact bone takes up more of the subperiosteal space in these sections (Daegling, 1989, 2001), and that more pixels containing bone are sampled here than at more anterior sections, resulting in higher absolute variation in gray-scale values. Partial volume effects (i.e., where a given pixel contains part bone and part air) may be a culprit in producing a wider range of grayscale values that do not re?ect material density accurately. We regard this as unlikely, because partial volume effects should be exacerbated in specimens with thin cortical shells (e.g., in the present study at symphyseal sections), yet these sections display smaller differences between weighted and unweighted moments. We found that scaling of moments of inertia was altered in terms of sample elevation, but not in terms of slope in transforming unweighted moments to weighted ones. The reason behind the shift in elevation is an obvious outcome of the weighting procedure, since the 249 IMAGE-BASED SKELETAL STIFFNESS TABLE 3. Kendall?s s coef?cients of rank correlation between unweighted and weighted moments and P-values for each section within each taxon1 Pan Gorilla Pongo Section Kendall?s s P-value Kendall?s s P-value Kendall?s s P-value Symphyseal Premolar Molar 1.000 0.833 0.667 0.0002 0.0018 0.0123 0.949 0.923 0.824 <0.0001 <0.0001 <0.0001 0.778 0.643 0.786 0.0035 0.0260 0.0065 1 All correlations were signi?cant. TABLE 4. Rank ordering of log-transformed scaled, weighted moments of inertia for 14 Gorilla molar sections as function of varying gray-scale power r1 r � 0.1 A C G B D F E H I J K M L N Rank orders of log (I?xx) r � 0.9 r51 r � 1.1 r � 10 A B C D E F G H I J K L M N A B C E D I F N J H L K L G A B C D E G F H I J K L M N A B C D E F G I H J K L M N 1 Baseline order (A to N) corresponds to log (I?xx) values for r � 1 (in bold). weighted moments must always be less than the unweighted moments. The fact that the slopes remained essentially constant after weighting may indicate that the errors associated with using unweighted measures are mitigated somewhat in an allometric context. Speci?cally, differences in overall size of sections are driving the regression, and the material variations of the included taxa may assume secondary importance. Great ape mandibles also provide an interesting test case of our application, in that there is no consensus on the role that diet plays in producing adult morphology. Gorilla, Pan, and Pongo are all best described as omnivores, although resistant foods typify the diets of Gorilla and Pongo (Doran et al., 2002; MacKinnon, 1974) to the exclusion of Pan (Kuroda, 1992). Both male and female Western lowland gorillas consume herbs, bark, and a minimal diversity of ?brous fruits (Doran et al., 2002). Pongo consumes a variety of hard unripe fruits, seeds, and bark, along with small amounts of insects, eggs, and fungi (MacKinnon, 1974; Ungar, 1995). Undoubtedly the more resistant items in the Pongo diet require large amounts of compressive, crushing components of the bite force (Schwartz, 2000). The diet of the common chimpanzee Pan troglodytes is made up of about 70 to 80% fruit, 5?9% seeds, 12 to 13.5% leaves, 3.5% pith, and 3% ?owers (Kuroda, 1992; Tutin et al., 1997). Pan is regarded as having a softer diet than Gorilla and Pongo, even though Pan is an extremely catholic feeder (Taylor, 2002, 2003). If we accept that these descriptions of diet translate neatly into distinctive biomechanical demands, then the contrast of a hard-diet group (Gorilla and Pongo) to a soft-diet group (Pan) leads to a prediction that gorillas and orangutans should have more functionally robust jaws (i.e., higher moments of inertia given their size). Hard, ?brous foods, given their strength or toughness, likely require both increased bite forces and more masticatory cycles overall (Hylander, 1979; Weijs and de Jongh, 1977). Taylor (2002) argued, based on linear measures of the mandibles of chimpanzees and gorillas, that Gorilla mandibles can be shown to re?ect their heavy reliance on ?brous vegetation relative to Pan. On the other hand, Daegling (1989, 1990, 2001) found little evidence from CT data on corpus sections that the mechanical properties of great ape jaws can be related to their speci?c dietary regimens. In the present context, we hypothesize that Pan cross sections would demonstrate markedly lower biomechanical stiffness relative to Gorilla and Pongo, as re?ected in lower weighted principal moments of inertia, size-adjusted to the appropriate mandibular length measures. The evidence for a biomechanical dichotomy between Gorilla and Pongo on the one hand and Pan on the other is equivocal. In terms of parasagittal bending, the section bearing the brunt of this load among those we sampled is M2. By our weighted measures, the three taxa are indistinguishable once we account for the moment arm responsible for this load. At the symphysis, the predominant load is lateral transverse bending (wishboning), countered by Iycyc (the moment of inertia about the vertical anatomic axis in the sagittal plane). When scaled against the wishboning moment arm, this moment only signi?cantly differentiates Gorilla from Pongo. Since we take no account of the curved beam effects that are certainly present under this load (Hylander, 1984, 1985), and given that these stress-concentrating effects will be most pronounced in Gorilla jaws (Daegling, 2001), these differences may well be spurious. This is circumstantially supported by our observation that Gorilla is actually weaker than Pongo under lateral transverse bending at premolar sections, where the quali?cations of curved beam mechanics do not apply. These results may also be interpreted as meaning that an improved model for assessing stiffness and strength offers little improvement of discrimination (recalling for the moment that the unweighted CT data do not support functional discrimination of great ape mandibles along dietary lines; Daegling, 1990). That being the case, the cost in resources and time may not be worth the effort, although the cost-bene?t ratio is bound to be contextspeci?c. In the present case, great ape diets may not actually differ in terms of the overall mechanical pro?le of the various foods eaten in terms of toughness, hardness, and strength. The precise relationship of grayscale variation to differences in elastic modulus is incompletely established. The validity of our interpretations is, in part, premised on the assumption that the exponent r describing the relationship of elasticity to gray scale is at or near 1. This involves establishing the relationship of bone den- 250 N.B. BHATAVADEKAR ET AL. sity to gray scale, which we attempted to do experimentally. These procedures in themselves involve certain assumptions, among them, that the grayscale variation sampled in bovine specimens can be directly applied to the CT data reported here. In any case, utilization of bone mineral phantoms in future investigations will mitigate some of the uncertainties regarding grayscale variation that were not controlled here. Conventional CT (the approach used here) is not ideally suited for this type of investigation, and the description of this relationship of grayscale to density (and ultimately modulus) awaits the appropriate application of other radiographic techniques such as quantitative or dual-energy CT to osteological data. While it is a reasonable approximation for illustrating the application of weighted measures to a comparative data set, the precise value of this exponent, estimated by appropriate experimental designs, will provide more reliable estimates of skeletal biomechanical behavior. Our weighted and scaled measures changed little for variations about the best estimate for the exponent r with respect to rank ordering of the sections analyzed, but for larger intervals of this estimate (spanning two orders of magnitude; Table 4), the effect on the rank ordering of specimen stiffness is large. CONCLUSIONS We presented a method for accounting for material variation in bone sections to derive a weighted moment of inertia, using variation in grayscale from CT scans. We applied this method to a sample of great ape mandibles to determine the effects of weighting on comparative biomechanical interpretations. Weighting sections by gray scale leads to stiffness measures that are reduced 10 to 22% from unweighted measures in sex-pooled, taxon-speci?c samples. Our observation of signi?cant changes in rank ordering within species samples between unweighted and weighted measures suggests that the errors induced in ignoring material variation are not uniform in magnitude (although they are by necessity uniform in direction). Unweighted measures, i.e., traditional moments of inertia assuming homogeneity, do not appear to be amenable to simple correction. The weighted-measures protocol introduced here is recommended for skeletal mechanical research, since the unwarranted but simplifying assumption of homogeneity is bypassed. A persistent question in masticatory mechanics is whether the distinct dietary regimens of the great apes are re?ected in jaw morphology. Its solution is tied to the development of appropriate biomechanical measures of function. Our weighted measures do not consistently document meaningful differences among taxa in terms of structural stiffness. This provides support for two alternative hypotheses: 1) the alleged dietary differences among great apes are not accompanied by important differences in masticatory loads, or 2) jaw form in the great apes, analyzed morphometrically, is not re?ective of the mechanical demands of their respective diets. ACKNOWLEDGMENTS Daniel Zahrly and David Pinto contributed to an initial version of the image-analysis code. LITERATURE CITED Carter DR, Hayes WC. 1977. The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg [Am] 59: 954?962. Ciarelli TE, Fyhrie DP, Par?tt AM. 2003. Effects of vertebral bone fragility and bone formation rate on the mineralization levels of cancellous bone from white females. Bone 32:311? 315. Corcoran TA, Sandler RB, Myers ER, Lebowitz HH, Hayes WC. 1994. 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