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Application of an image-based weighted measure of skeletal bending stiffness to great ape mandibles.

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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*
Department of Periodontology, University of North Carolina, Chapel Hill, North Carolina 27599
Department of Anthropology, University of Florida, Gainesville, Florida 32611
Department of Mechanical Engineering, Union College, Schenectady, New York 12308
biomechanics; bone density; computed tomography; image analysis;
moment of inertia
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
*Correspondence to: Andrew J. Rapoff, Ph.D., Department of
Mechanical Engineering, Union College, Schenectady, NY 12308-3147.
Received 11 March 2005; accepted 9 November 2005.
DOI 10.1002/ajpa.20397
Published online 4 April 2006 in Wiley InterScience
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)?
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,
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
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
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
饀� 饀� 饀�
; Iyy ; Ixy �
y2i DAi ;
x2i DAi ;
xi yi DAi
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 �
y2i Cri DAi ;
x2i Cri DAi ;
xi yi Cri DAi
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
xi Cri DAi
6i��x; y � 6
4 P
Cri DAi
yi Cri DAi
; i�
Cri DAi
The weighted moments of inertia about the centroidal
axes were then determined from the parallel axis theorem
Cri DAi ; Iyy
Ixc xc ; Iyc yc ; Ixc yc � Ixx y2
Cri DAi ; Ixy xy
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
premolar and molar sections, and unweighted Ix(u)
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
lar and molar sections, and unweighted Iycy
and weighted
Iycyc for symphyseal sections, re?ect resistance to bending
in a transverse plane (i.e., ??wishboning?? loads identi?ed by
Hylander, 1984).
Cri DAi :
The principal weighted moments of inertia are given by the
usual formula for determining principal moments of inertia
Ixc xc � Iyc yc
Ixc xc Iyc yc 2 2
Ixp xp ; Iyp yp �
蘒xc xc ;
Ixc xc � Iyc yc
Ixc xc Iyc yc 2 2 5
蘒xc xc
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
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-
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
as and denoted by ~Ixx Ixp xp =d4s for molar and premolar
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
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 �
餋 v�
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 �
餋 v辀=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.
Iyp yp Iyp yp
Ixp xp Ixp xp
Iyp yp
Ixc xc Ixc xc
Ix c x c
Ix p x p
up up
Differences between
physeal sections.
Iyc yc Iyc yc
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.
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.
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
For premolar and molar sections, the unscaled unand unscaled weighted Ixpxp moments scaled
weighted Ix(u)
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
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).
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
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
TABLE 3. Kendall?s s coef?cients of rank correlation between unweighted and weighted moments
and P-values for each section within each taxon1
Kendall?s s
Kendall?s s
Kendall?s s
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
Rank orders of log (I?xx)
r � 0.9
r � 1.1
r � 10
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-
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.
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.
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base, skeletal, ape, weighted, bending, stiffness, application, image, mandible, measures, great
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