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Biomechanics of cross-sectional size and shape in the hominoid mandibular corpus.

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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.
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