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Curvature length and cross-sectional geometry of the femur and humerus in anthropoid primates.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 127:46 –57 (2005)
Curvature, Length, and Cross-Sectional Geometry of the
Femur and Humerus in Anthropoid Primates
Atsushi Yamanaka,1* Harumoto Gunji,2 and Hidemi Ishida3
1
Department of Oral Anatomy, Kagoshima University Dental School, Kagoshima 890-8544, Japan
Japan Monkey Center, Aichi 484-0081, Japan
3
School of Human Nursing, University of Shiga Prefecture, Shiga 522-8533, Japan
2
KEY WORDS
beam theory; bending strength; 3-D data processing; scaling; long bone;
terrestriality; arboreality; Old World monkeys; New World monkeys; gibbons
ABSTRACT
The aims of this study were to describe
the curvature of anthropoid limb bones quantitatively, to
determine how limb bone curvature scales with body
mass, and to discuss how bone curvature influences static
measures of bone strength. Femora and humeri in six
anthropoid genera of Old World monkeys, New World
monkeys, and gibbons were used. Bone length, curvature,
and cross-sectional properties were incorporated into the
analysis. These variables were obtained by a new method
using three-dimensional morphological data reconstructed from consecutive CT images. This method revealed the patterns of curvature of anthropoid limb bones.
Log-transformed scaling analyses of the characters revealed that bone length and especially bone curvature
strongly reflected taxonomic/locomotor differences. As
compared with Old World monkeys, New World monkeys
and gibbons in particular have a proportionally long and
less curved femur and humerus relative to body mass. It is
Much attention has been paid to relationships
between the bending strength of limb bones and
modes of locomotion because bending stresses predominantly occur in limb bones of terrestrial mammals (Lanyon and Bourn, 1979; Rubin and Lanyon,
1982; Biewener, 1983; Biewener et al., 1983, 1988;
Biewener and Taylor, 1986). Bending stresses occur
in two manners, as illustrated in Figure 1. In the
first case, an external force is exerted from the lateral side of a limb bone (Fig. 1a). Since the distance
l between the point of action and a certain cross
section provides a moment arm, the bending stress
in the bone’s outermost cortex is given by
␴b ⫽
共F t 䡠 l兲X
,
I
(1)
where ␴b ⫽ bending stress, Ft ⫽ applied lateral
force, X ⫽ distance from the neutral plane of bending to the bone surface, and I ⫽ second moment of
inertia of the bone’s cross section. In the second case,
an external axial force is exerted to a curved bone
(Fig. 1b). Since the longitudinal curvature C of the
bone provides a moment arm, the bending stress in
the bone’s outermost cortex is given by
©
2004 WILEY-LISS, INC.
also revealed that the section modulus relative to body
mass varies less between taxonomic/locomotor groups in
anthropoids. Calculation of theoretical bending strengths
implied that Old World monkeys achieve near-constant
bending strength in accordance with the tendency observed in general terrestrial mammals. Relatively shorter
bone length and larger A-P curvature of Old World monkeys largely contribute to this uniformity. Bending
strengths in New World monkeys and gibbons were, however, a little lower under lateral loading and extremely
stronger and more variable under axial loading as compared with Old World monkeys, due to their relative elongated and weakly curved femora and humeri. These results suggest that arboreal locomotion, including
quadrupedalism and suspension, requires functional demands quite dissimilar to those required in terrestrial
quadrupedalism. Am J Phys Anthropol 127:46 –57, 2005.
©
2004 Wiley-Liss, Inc.
␴b ⫽
共F a 䡠 C兲X
,
I
(2)
where ␴b ⫽ bending stress, Fa ⫽ applied axial force,
X ⫽ distance from the neutral plane of bending to
the bone surface, and I ⫽ second moment of inertia
of the bone’s cross section. Bending stress under
axial load will be zero when bones are straight (C ⫽
0), and will increase in proportion to C. Therefore,
the cross-sectional properties of the bone shaft (I
and X), bone length (L), and bone curvature (C) are
Grant sponsor: Cooperation Research Program, Primate Research
Institute, Kyoto University.
*Correspondence to: Atsushi Yamanaka, Department of Oral Anatomy, Kagoshima University Dental School, 8-35-1 Sakuragaoka, Kagoshima 890-8544, Japan.
E-mail: yamanaka@denta.hal.kagoshima-u.ac.jp
Received 5 September 2002; accepted 31 October 2003.
DOI 10.1002/ajpa.10439
Published online 7 October 2004 in Wiley InterScience (www.
interscience.wiley.com).
47
LIMB BONE CURVATURE IN ANTHROPOID PRIMATES
TABLE 1. Sample composition
n
Total
Wild
Captive
Total
Wild
Captive
Papio anubis
Macaca
fascicularis
Macaca
nemestrina
Alouatta1
Ateles
(femur)1
Ateles
(humerus)
Lagothrix1
Hylobates1
12
12
4
3
8
9
11
11
3
2
8
9
12
4
8
10
2
8
12
7
5
1
7
6
7
6
0
0
7
6
8
2
6
6
0
6
9
9
4
0
5
9
5
9
0
0
5
9
1
Fig. 1. Bending stresses produced in two manners. a:
Straight beam subjected to lateral loading. b: Curved beam subjected to axial loading. ␴b, bending stress in beam’s outermost
surface; Ft, applied lateral force; Fa, applied axial force; I, second
moment of inertia of cross section; X, distance from neutral plane
of bending to beam’s surface; l, distance between point of action
and cross section; L, beam length; C, beam longitudinal curvature.
the most significant morphological determinants of
bending strength of limb bones.
Limb-bone strength in nonhuman primates was
investigated using cross-sectional geometric properties in terms of locomotor adaptations by many researchers (e.g., Burr et al., 1981, 1989; Schaffler et
al., 1985; Ruff, 1987, 1989, 2002; Demes and
Jungers, 1989; Demes et al., 1991; Kimura, 1991,
1995; Ruff and Runestad, 1992). Several recent
studies adopted the lateral loading model (Fig. 1a),
incorporating both length and cross-sectional properties into the analysis of bone strength (Demes and
Jungers, 1993; Jungers and Burr, 1994; Jungers et
al., 1998; Polk et al., 2000). However, only a few
studies analyzed bone strength considering bone
curvature (Swartz, 1990; Llorens et al., 2001). This
is regrettable, because bone curvature is a nearly
ubiquitous phenomenon. Swartz (1990) investigated
allometric scaling of limb bones’ curvatures on body
mass in anthropoids, and revealed a less curved
humerus in suspensory species. However, no study
has incorporated cross-sectional properties and bone
curvature simultaneously to estimate bone strength
in primates.
Difficulty in quantification may partly explain
why bone curvature has been insufficiently studied
or ignored. As is illustrated in Figure 1b, bone curvature can be defined as the distance between a
bone’s longitudinal axis and neutral axis. It is difficult, however, to determine the two axes a priori
from external morphologies in the case of tubular
bones. Furthermore, even if these two axes are determined, it is technically difficult to obtain cross
sections perpendicular to the longitudinal axis, to
calculate genuine values of I and X, and to measure
n (known body weight)
Species
Genera which contain several different species samples.
the real distance C between the two axes at each
cross-sectional level.
The aims of this study are to describe the curvature of anthropoid limb bones quantitatively, to determine how limb bone curvature scales with body
mass, and to discuss how bone curvature influences
static measures of bone strength. In order to avoid
the above disadvantages when measuring bone curvature, this study utilizes three-dimensional (3-D)
morphological data reconstructed from consecutive
CT images instead of measuring actual bones. Then,
we introduce a new long bone coordinate system,
which is determined from the bone mass distribution. This new method serves to measure curvature,
length, and cross-sectional properties under the biomechanically proper model illustrated in Figure 1,
and can be applied to almost all primate limb bones
regardless of shape differences. All measurements
(curvature, length, and cross-sectional properties)
are taken from 3-D data on computers with this new
technique.
MATERIALS
Specimens used in this study were the femora and
humeri of six anthropoid primate genera given in
Table 1. They came from the Primate Research Institute of Kyoto University and the Japan Monkey
Center (Inuyama, Japan). All specimens were dry
bones. They were all adult, except for a few specimens in which epiphyseal lines had not completely
disappeared. Sexes were pooled. Pathologically affected specimens were excluded.
Specimens were drawn from both wild-caught and
captive individuals. The use of CT equipment
housed in the laboratory restricted our availability
of specimens. It is possible that free-ranging vs.
captive environments will influence bone strength.
Demes and Jungers (1993) reported no significant
differences in the rigidity of femora and humeri of
wild-caught and captive Lemur catta, though Burr
et al. (1989) reached contrary results with the femur
and humerus of Macaca nemestrina. Fleagle and
Meldrum (1988) noted that captivity changed the
cross-sectional bone shape of a zoo-reared specimen
48
A. YAMANAKA ET AL.
Fig. 2. CT images of femur at lesser trochanter level. Both were reconstructed from same raw date file. a: 28 (⫽ 256) gray scale
is assigned to CT value interval of interest only, where 8 bits (⫽ 1 byte) per pixel are required. b: Ternary image, where CT values
which correspond to “cortical bone” are converted into “2,” those corresponding to “cancellous bone” into “1,” and those corresponding
to “air” into “0.” Pixels with “2,” “1,” and “0” are represented as white, gray, and black, respectively.
so that it more closely resembled another species
with a different locomotor repertoire in the wild.
Although most of the captive specimens were chosen
from animals reared in the outdoor enclosures, the
results of this study might be influenced by involving captive animals.
The subsample size of specimens with known body
weights is also given in Table 1. All specimens were
used in the analysis of bone curvature normalized by
bone length. Only specimens with known body
weights were used in the scaling analyses on body
mass, and in the calculations of theoretical bone
strengths.
METHODS
Length, curvature, and cross-sectional properties
were quantified using 3-D morphological data of the
femur and humerus reconstructed from consecutive
CT images.
All CT images were obtained using spiral CT
(TSX-002/4I, Toshiba Medical Systems Co., Ltd.) at
the Department of Zoology, Kyoto University. The
scan parameters were 120 kVp of voltage, 100 mA of
tube current, and 2 mm slice thickness. Spiral scanning was performed with a pitch of 1 (i.e., a table
feed of 2 mm per 360° rotation, which equals 2-mm
slice thickness) over the overall length of a long
bone. Spiral CT scanning enabled us to obtain crosssectional images in x-y planes at arbitrary positions
of the z-axis freely and retrospectively by the zinterpolation algorithm (Kalender et al., 1990; Polacin et al., 1992; Kalender, 2000). Image reconstruction increments along the z-axis do not depend upon
slice thickness and pitch value, although image
quality largely depends on them. We reconstructed
CT images in which each pixel was a square of 0.5
mm, with increments of 0.5 mm along the z-axis.
Consecutive CT images were converted into ternary images with given thresholds. CT measures
and computes the spatial distribution of linear attenuation coefficients, which are given as CT values
or CT numbers, in terms of Hounsfield units
(Hounsfield, 1973). There is a one-to-one correspondence between the set of all CT values and the set of
all pixels of an image. In Toshiba’s CT scanners, 216
different values are available per each pixel. The
range of 216 CT values cannot be evaluated or differentiated in a single view. Conventionally, the 28
(⫽ 256) gray scale is assigned to the CT value interval of interest, the so-called window (Fig. 2a). In this
study, CT values which correspond to “cortical bone”
were labeled as “2,” those corresponding to “cancellous bone” as “1,” and those corresponding to “air” as
“0.” All original CT images were converted into ternary images (Fig. 2b). Threshold values of 1,088
Hounsfield units (HU), 596 HU, and ⫺508 HU were
chosen here for the cortical-cancellous, cortical-air,
and cancellous-air interfaces, respectively. These
threshold values were determined as halfway levels
between two tissues (see Spoor et al., 1993).
A 3-D morphological data set, in which each voxel
was a cubic of 0.5 mm, was reconstructed by piling
up consecutive two-dimensional (2-D) ternary images in which each pixel was a square of 0.5 mm,
because the increments along the z-axis were 0.5
mm (Fig. 3). The ternary values represent the voxels
of 3-D data.
Average bone mineral densities of 1,327 mg/ml for
“cortical bone” and 420 mg/ml for “cancellous bone”
(Yamanaka et al., 2001; for technical details, see
Chen and Lam, 1997) were given to the voxels labeled “2” and “1,” respectively. Assuming that each
voxel was a material point with mass (mass of a
material point ⫽ volume of voxel: (0.5 mm)3 ⫻ bone
LIMB BONE CURVATURE IN ANTHROPOID PRIMATES
Fig. 3. Three-dimensional morphological data in which each
voxel is a cubic of 0.5 mm, reconstructed by piling up consecutive
2-D ternary images in which each pixel is a square of 0.5 mm with
increments of 0.5 mm along z-axis. Ternary values represent
voxels of 3-D data.
mineral density), the coordinates of the center of
mass (G) were calculated. For each 3-D datum, the
tensor of inertia is given by the following matrix
expression:
冉
冊
⌺mi共yi2 ⫹ zi2兲 ⫺ ⌺mixiyi
⫺ ⌺mizixi
⫺ ⌺mixiyi
⌺mi共zi2 ⫹ xi2兲 ⫺ ⌺miyizi
⫺ ⌺mizixi
⫺ ⌺miyizi
⌺mi共xi2 ⫹ yi2兲
(3)
where xi, yi, and zi represent the (x, y, z) coordinates
of the ith voxel relative to G, and mi represents the
mass of the ith voxel.
Principal axis transformation was performed with
this matrix to decide the longitudinal axis of 3-D
data. A principal axis transformation is a well-established technique (e.g., Alpert et al., 1990;
Rusinek et al., 1993; Weber and Ivanovic, 1994; Adams et al., 1995; Tsao et al., 1998a,b) that determines a set of three orthogonal axes representing
the major axes of mass distribution of a rigid object.
Each principal axis is associated with a volumetric
moment of inertia (or an eigenvalue) that measures
the variance of mass distribution perpendicular to
the axis. High eigenvalues indicate that the mass of
an object is dispersed perpendicularly to the axis
direction, whereas low eigenvalues indicate that the
mass of an object is dispersed along the axis direction. Among the three rectangular principal axes of
inertia through G, the axis with the lowest eigenvalue was defined as the longitudinal axis (Fig. 4).
The second transformation was done to settle the
frontal and sagittal planes. In the femora, the most
distal points on the surfaces of the medial and lateral condyles were detected (Fig. 5a). In the humeri,
the most distal points on the medial keel of the
trochlea and on the surface of the capitulum were
detected (Fig. 5b). The plane that was parallel to the
line through the two distalmost points, and also
contained the longitudinal axis, was defined as the
49
frontal plane. The plane that was perpendicular to
the frontal plane and also contained the longitudinal
axis was defined as the sagittal plane.
Then the length, curvature, and cross-sectional
properties were measured. 1) Bone length (L) was
defined as the distance between the following two
planes. One plane was perpendicular to the longitudinal axis, and contained the most proximal point on
the surface of the femoral or humeral head. The
other plane was also perpendicular to the longitudinal axis, and contained the midpoint of the two
distalmost points defined above. 2) Cross-sectional
images perpendicular to the longitudinal axis were
obtained with increments of 10% of the bone length
(Fig. 4). We reconstructed these images by the nearest-neighbor algorithm, because we had to cut the
3-D object obliquely in the original xyz-coordinate
axes under the newly transformed coordinate system. Seven cross-sectional images at the 20 – 80%
level from the distal end were analyzed in this
study. Only the region of “cortical bone” (the region
labeled as “2” in ternary images) was used in the
following analyses of diaphyseal cross sections, because cortical bone is a main component of the diaphysis of long bones, and little cancellous bone
exists. The intersections of the frontal plane and
sagittal plane, and each cross section, were defined
as the mediolateral (M-L) axis and the anteroposterior (A-P) axis, respectively. The second moments of
inertia around the M-L and A-P axes (IML and IAP)
were measured, and the section moduli (ZML and
ZAP) were also calculated. 3) Bone curvature was
defined as the distance between the longitudinal
axis and the centroid of each cross section. “Centroid” was used here as a substitute for the intersection of “neutral axis of bending” and each cross section, because the neutral axis is transient and
changing, depending on the magnitude and orientation of applied loads, but it is expected to pass in
close proximity to the centroid of each cross section.
The M-L and A-P components of the curvature (CML
and CAP) were measured on all cross-sectional images.
All the above image processing and measurements were performed by customized software programmed by A.Y. Three-dimensional data were visualized by AVS software (Advanced Visual
Systems, Inc.).
RESULTS
For analysis of bone curvature, all specimens
listed in Table 1 were used. Bone curvatures CML
and CAP were normalized by bone length L. The
means of CML/L and CAP/L in the femur and humerus are shown in Figures 6 and 7, respectively. In
the anthropoid femur and humerus in common, bone
curvatures are less than several percent of bone
length at maximum. Figure 6 indicates that anthropoid femora curve laterally moving proximally, and
are anteriorly convex at their midpoints. Figure 7
indicates that humeri are laterally convex, and pos-
50
A. YAMANAKA ET AL.
Fig. 4. Three-dimensional morphological data visualized by AVS software (Advanced Visual Systems, Inc.). Translucent blue
voxels represent “cortical bone;” yellow voxels represent “cancellous bone.” xyz-coordinate axes are transformed to principal axes.
Three rectangular principal axes intersect at center of mass. Longest principal axis is axis with lowest eigenvalue. Cross-sectional
images perpendicular to the axis were reconstructed with increments of 10% of bone length. a: Baboon femur. b: Baboon humerus.
LIMB BONE CURVATURE IN ANTHROPOID PRIMATES
51
Fig. 5. Three-dimensional morphological data visualized by AVS software (Advanced Visual Systems, Inc.). Red voxels represent
most distal 100 voxels along longitudinal axis (a) on surfaces of femoral medial and lateral condyles, and (b) on medial keel of humeral
trochlea and on surface of capitulum. Center of these 100 voxels is defined as most distal point on each surface. Line through two
distalmost points is added.
sess S-shaped curves from proximally to distally in
the A-P direction.
Variations across taxa in femoral CAP/L are more
remarkable than those in CML/L (Fig. 6). Old World
monkeys possess CAP/L ratios twice as large as those
of New World monkeys. These differences are most
notable around the 50% level. Gibbons have still
lower CAP/L ratios than New World monkeys. New
World monkeys and gibbons possess slightly larger
CML/L ratios than Old World monkeys at proximal
levels. However, these differences are less noticeable
than the A-P curvatures.
Variations across taxa in humeral CAP/L are also
larger than those in CML/L (Fig. 7). The general
pattern of variation is similar to that for the femur.
Old World monkeys possess CAP/L ratios twice as
large as those of New World monkeys. These differences are most notable around the 60% level. Gibbons possess still lower CAP/L ratios than New
World monkeys. Variations in CML/L are less noticeable than those in CAP/L.
Figures 8 –10 display allometric scaling of bone
length, curvature, and cross-sectional properties on
body mass (BM). For these analyses, only specimens
with known body weights were used. The leastsquares regression line was calculated for the three
Old World species, in order to evaluate the devia-
tions of New World monkeys and gibbons from the
scaling of Old World monkeys on each measurement.
Bilogarithmic scaling of femoral and humeral L on
BM is shown in Figure 8. New World monkeys fall
above the regression line of Old World monkeys in
Figure 8a, i.e., their femora are longer than those of
comparably sized Old World monkeys. Gibbons fall
still above New World monkeys. Spider monkeys are
the most upwardly displaced of New World monkeys, though they fall under gibbons. The same
trends are observed and are even more apparent in
the scaling of humeral L (Fig. 8b). One of the reasons
why Old World monkeys possess larger CAP/L in
femora (Fig. 6) and humeri (Fig. 7) may be that they
possess shorter L than New World monkeys and
gibbons do.
The scaling relationships of femoral and humeral
CAP on BM were investigated at the femoral 50%
level and humeral 60% level, where bone curvature
is most pronounced (Fig. 9); therefore, bending
stress under an axial loading is maximum at these
levels. Contrary to the relationship between L and
BM, New World monkeys fall below the regression
line of Old World monkeys, i.e., their femora and
humeri are less curved for their body mass. Gibbons
possess even less curved bones than New World
52
A. YAMANAKA ET AL.
Fig. 6. Femoral curvature standardized by its length. Averages are plotted at each cross-sectional level. Error bars indicate
⫾1 standard deviation. ■, Old World monkeys; ‚, New World
monkeys; , gibbons.
monkeys. The femur and humerus of gibbons are
almost straight, and some of their plots are not
represented in Figure 9. Any distinctiveness of spider monkeys among New World monkeys is not apparent in CAP. The trend that Old World monkeys
possess large femoral and humeral curvatures is
supported even when body size is taken into account.
The relationships between the section modulus
about the M-L axis (ZML) and BM were also investigated at the femoral 50% level and at the humeral
60% level (Fig. 10). ZML was assessed here rather
than ZAP, because A-P curvature (i.e., CAP) is more
pronounced, and ZML is an indicator of bending
strength in the anteroposterior direction. The bending stresses ␴ ⫽ (Fa 䡠 CAP)/ZML at these levels are
thought to be critical for bones under an axial loading condition (where Fa is an external axial force).
The correlation coefficients (r2) for femoral and humeral cross sections of Old World monkeys are high
(0.913 and 0.932, respectively). The slopes of regression lines for both sections are almost 1 (0.948 and
1.081, respectively). Because the dimension of sec-
Fig. 7. Humeral curvature standardized by its length. Averages are plotted at each cross-sectional level. Error bars indicate
⫾1 standard deviation. Symbols as in Figure 6.
tion modulus is L3 (i.e., cube of length), the femoral
and humeral ZML of Old World monkeys scale nearly
isometrically on BM. This isometric scaling of Old
World monkeys is similar to the nearly isometric
(geometrically similar) changes of shaft diameter in
terrestrial mammals generally (Alexander et al.,
1979; Biewener, 1982). Strictly arboreal New World
monkeys and gibbons fall almost on the regression
line of Old World monkeys. The congruency of scaling of the section modulus between Old World monkeys, New World monkeys, and gibbons is contradictory to the upward and downward displacements on
scaling of bone length and bone curvature.
Finally, the theoretical A-P bending strengths at
the femoral 50% and humeral 60% levels were calculated using the two models illustrated in Figure 1.
The theoretical bending strength was defined as the
reciprocal of bending stress, i.e., 1/␴b ⫽ ZML/(Ft ⫻ L)
under lateral loading, and ZML/(Fa ⫻ CAP) under
axial loading. Since the external forces Ft and Fa
were unknown, BM1.0 was substituted for these parameters. Therefore, we cannot compare directly the
absolute values for theoretical strengths obtained
from the two models, because their values are rela-
LIMB BONE CURVATURE IN ANTHROPOID PRIMATES
53
Fig. 8. a: Femoral length vs. body mass. b: Humeral length vs. body mass. Least-squares regression line and equation are given
for three Old World species. Curved lines beside regression line represent 95% confidence limit of predicted value. ■, Papio anubis;
solid diamonds, Macaca fascicularis; ●, Macaca nemestrina; ‚, Alouatta; 䊐, Ateles; E, Lagothrix; , Hylobates.
Fig. 9. Anteroposterior curvature vs. body mass (a) at femoral 50% level, and (b) at humeral 60% level. Species (or genus) symbols
as in Figure 8. Least-squares regression line and equation are given for three Old World species. Curved lines beside regression line
represent 95% confidence limit of predicted value.
tive. Figures 11 and 12 illustrate these theoretical
bending strengths as the y coordinates, with BM on
the x axis.
Femoral bending strengths under lateral loading
are given in Figure 11a. Those of Old World monkeys tend to increase at a smaller body mass. The
same trend was recognized in the results of Jungers
et al. (1998) on cercopithecoid monkeys. When New
World monkeys and gibbons are added, however,
this trend becomes somewhat indistinct. The taxa
with the lowest bending strength comprise baboons,
gibbons, and some New World monkeys. All specimens are plotted within about three times of these
lowest members. Femoral bending strengths under
axial loading are given in Figure 11b. The lowest
members are three species of Old World monkeys.
New World monkeys and gibbons are much higher,
and some are not represented in Figure 11b. These
extremely high values of New World monkeys and
gibbons may result from their small femoral CAP
(Fig. 9a).
Humeral bending strengths under lateral loading
are given in Figure 12a. The declining trend of Old
World monkeys is less distinct than that for femoral
strength. Gibbons and some spider monkeys are
plotted below Old World monkeys. This is mainly
54
A. YAMANAKA ET AL.
Fig. 10. Section modulus around M-L axis vs. body mass (a) at femoral 50% level, and (b) at humeral 60% level. Species (or genus)
symbols as in Figure 8. Least-squares regression line and equation are given for three Old World species. Curved lines beside
regression line represent 95% confidence limit of predicted value.
Fig. 11. Theoretical A-P bending strengths at femoral 50%
level (a) by lateral loading model, and (b) by axial loading model.
Species (or genus) symbols as in Figure 8.
Fig. 12. Theoretical A-P bending strengths at humeral 60%
level (a) by lateral loading model, and (b) by axial loading model.
Species (or genus) symbols as in Figure 8.
LIMB BONE CURVATURE IN ANTHROPOID PRIMATES
caused by their longer humeri relative to the other
monkeys (Fig. 8b). Humeral bending strengths under axial loading are given in Figure 12b. The lowest
members are three species of Old World monkeys, as
in the femur. The strengths of New World monkeys
and gibbons also have extremely high values. This
may result from their small humeral CAP (Fig. 9b),
as is the case in their femora.
DISCUSSION
New method for measuring bone curvature in
biomechanical studies
The present study introduced a new method for
measuring limb bone curvature. We believe this
method has two advantages, but also two weak
points compared with traditional osteometry.
The first advantage is that it serves to measure
curvature, length, and cross-sectional properties under the biomechanically proper model illustrated in
Figure 1. We are able to obtain a bone’s cross section
perpendicular to the longitudinal axis, and to measure the curvature at that cross-sectional level. The
second is that it can be applied to almost all primate
limb bones regardless of shape differences, without
a researcher’s decision about bone orientation. This
method revealed the patterns of curvature in anthropoid limb bones quantitatively. The femora and
humeri of Old World monkeys are more anteroposteriorly curved than those of New World monkeys
(Figs. 6a, 7a, 9). This relationship is weakly preserved in the humeral mediolateral curvature (Fig.
7b). It was also revealed that gibbons have very
weakly A-P curved femora as well as very weakly
A-P curved humeri (Figs. 6a, 7a, 9).
The first disadvantage is that this method requires
a lot of processes. In particular, this method requires
CT scanning over the overall length of a long bone, and
is not appropriate for fragmentary fossil limb bones.
The second disadvantage also correlates with the utilization of whole-bone geometry and mass distribution
to determine the longitudinal axis. The femora of New
World monkeys and gibbons are more laterally curved
in the proximal regions than those of Old World monkeys (Fig. 6b). This phenomenon probably results from
the fact that the femoral neck of New World monkeys
and gibbons is longer than that of Old World monkeys.
A large-sized femoral head and long femoral neck locate the present longitudinal axis somewhat medially
(about 1% of bone length), because the axis is determined by bone mass distribution. It may largely explain the relatively similar and monotonic increase in
bone lateral curvature (lateral offset from the longitudinal axis) in the proximal femoral diaphysis of all
species (Fig. 6b). Although the present longitudinal
axis may not be appropriate for measurements such as
femoral neck-shaft angle, it does not differ significantly from the longitudinal axis, which most anthropologists generally consider (note that the above femoral lateral curvature is only 1% of the femoral length).
55
Cross-sectional geometry, length, and curvature
The results of the present scaling analyses suggest that cross-sectional properties of limb bones are
more conservative indicators of taxonomic or locomotor differences among anthropoid primates (Fig.
10). These differences among anthropoids are reflected more in bone length and especially in bone
curvature (Figs. 8, 9). Ruff (1987) investigated allometric scaling patterns of cross-sectional properties
and length on body mass in hominoid and macaque
hindlimb bones. He noted that locomotor adaptive
differences were apparent in proportional differences between body mass and bone length, rather
than in those between body mass and cross-sectional
dimensions. Aiello (1981) reached a similar conclusion by allometric analyses of bone length and external measurements of cross sections in anthropoids. The present results are concordant with these
previous studies. This study also draws attention to
the significance of bone curvature for locomotor
functional studies in anthropoids.
These results, however, do not necessarily mean
that cross-sectional properties are unsuitable indicators for locomotor behavioral affinities and differences. It was shown that within prosimians, leapers
tend to have more robust femora (with elongated
A-P dimensions) than general quadrupeds of the
same size (Demes and Jungers, 1989, 1993; Demes
et al., 1991). Kimura (1991, 1995) suggested that the
limb bone cross-sectional strengths of arboreal
mammals are more robust than those of terrestrial
mammals, although Polk et al. (2000) showed that
the notions of Kimura (1991, 1995) were not supported at least within some mammalian orders, including primates. Cross-sectional geometry may reflect such outstanding locomotor specialization as
observed in leapers, or may reflect differences at
higher categorical levels, e.g., the differences between arboreal and terrestrial mammals. More recently, Ruff (2002) found that limb bone section
moduli were very good indicators of locomotor specializations in anthropoids when compared with the
articular size of the same bones, or when compared
between fore- and hindlimb bones. The conservativeness of cross-sectional properties relative to body
mass, on the other hand, will indicate that these
properties are suitable predictors of body mass in
extinct anthropoids (e.g., see Ruff et al., 1989; Ruff
and Runestad, 1992; Ruff, 2003).
Locomotor adaptations in Old and New World
monkeys and gibbons
As summarized in Biewener (1990), the organization and composition of bone in the limb skeleton are
quite similar, possessing fairly uniform material
properties in a diverse range of mammalian species.
Therefore, it seems likely that selection favors similar peak functional stresses in each type of skeletal
support element in all mammalian species, regardless of body size. If the theoretical bending strengths
56
A. YAMANAKA ET AL.
calculated from bone geometrical dimensions (1/
␴b ⫽ ZML/(BM ⫻ L) under lateral loading, and 1/␴b ⫽
ZML/(BM ⫻ CAP) under axial loading) are not similar
in magnitude regardless of BM or between primate
species, other mechanisms than these morphological
dimensions must be presumed to achieve a constant
bending stress. These mechanisms are probably correlated with the positional behavioral differences
between primate taxa.
The theoretical bending strengths of Old World
monkeys’ femora and humeri, based on either of two
models, show a gently increasing pattern in the range
of smaller body size (Figs. 11, 12). Jungers et al. (1998)
calculated the theoretical bending strengths under lateral loading in the femur and humerus of Old World
monkeys (10 cercopithecine and 10 colobine species),
and recognized the same trend. Those authors suggested that strength indices with BM2/3 might be more
realistic, if peak forces per unit body mass were somewhat reduced at larger cercopithecoid body sizes. The
present results imply that Old World monkeys achieve
constant bending strength probably by mechanisms
that include better muscular mechanical advantages
and more extended limb posture in larger animals
(Biewener, 1990).
Despite this gradual declining trend in lager animals, the theoretical bending strengths of Old World
monkeys, based on either of two models, fall in a
relatively narrow range when they are compared to
those of New World monkeys and gibbons. The relatively shorter bone length and larger A-P curvature
of Old World monkeys among anthropoids largely
contribute to this convergence.
The theoretical bending strengths under lateral
loading in New World monkeys and gibbons are distributed below the relatively uniform strengths of Old
World monkeys at similar body sizes (Figs. 11a, 12a).
That is probably because New World monkeys and
gibbons have longer femora and humeri than do Old
World monkeys (Fig. 8). Jungers et al. (1998) suggested that cercopithecine long bones should be reinforced in comparison to those of colobines. They also
suggested that the long bones of arboreal colobines
may experience relatively lower peak stresses because
of greater substrate compliance than those of more
terrestrial cercopithecines. In fact, Schmitt (1994,
1998, 1999) documented reductions in peak forces acting on primates moving on branches compared to their
locomotion on the ground. The New World monkeys’
compliant movements along and across arboreal substrates might not require limb bones as strong as those
required by Old World monkeys’ parasagittal movements along substrates or on the ground.
On the other hand, the strengths under axial loading in New World monkeys and gibbons have extremely higher values than the relatively uniform
strengths in Old World monkeys (Figs. 11b, 12b). This
is because they have less curved femora and humeri
(Fig. 9). Swartz (1990) explored quantitatively the curvature of long bones of anthropoid primates, and demonstrated that suspensory species such as gibbons and
spider monkeys have less curved humeri. As Swartz
(1990) discussed, the humerus may be significantly
less curved in brachiators because of its torsional-dominated loading regime (Swartz et al., 1989) and the
greatly increased stress magnitude developed in torsionally loaded curved beams. Alternatively, bending
due to axial loading and bone curvature may be lower
and less important than bending due to lateral loading
and bone length in New World monkeys and gibbons.
The results of this study support the latter interpretation, because of large intrageneric variations in theoretical bending strengths of bones under axial loading,
and the less curved femora of forelimb-suspensory species.
It is difficult to resolve which is the more important factor for bone yield or fracture: bending resulting from bone length, or bending resulting from bone
curvature? It is currently unknown what the direction and magnitude of applied forces are in vivo, and
thus the relative contributions of axial and transverse forces are also unknown. It is also difficult to
answer what the primary adaptive meaning of limb
bone curvature is. Bertram and Biewener (1988)
proposed an explanation of long bone curvature as a
mechanism for improving the predictability of loading direction and, consequently, the pattern of
stresses within the bone, countering losses in structural strength. It is plausible to argue that limb
movements, limb bone orientation with respect to
direction of progression, and overall limb loading are
quite stereotyped in terrestrial locomotion, while
arboreal and suspensory locomotions involve moving in a highly variable way in a three-dimensionally complex fashion. The results of this study are
not contradictory to the hypothesis by Bertram and
Biewener (1988) that long bone curvature acts to
improve loading predictability, countering losses in
structural strength. In any case, a better understanding of the influences of bone length and curvature on bending strengths in primates needs a great
deal of experimental data, especially concerning in
vivo strain (or stress) patterns within bone during
various movements in primate limbs (Swartz et al.,
1989; Demes, 1998; Demes et al., 1998, 2001).
ACKNOWLEDGMENTS
We express our gratitude to M. Nakatsukasa, N.
Ogihara, H. Kumakura, and M. Yamashita for assistance and advice in this study. We are grateful to Y.
Morita and H. Takemoto for technical advice concerning CT scanning and image processing. N. Shigehara
of the Primate Research Institute at Kyoto University
kindly allowed us access to specimens under their
care. We also thank two reviewers for useful comments on an earlier version of this paper.
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