Curvature length and cross-sectional geometry of the femur and humerus in anthropoid primates.код для вставкиСкачать
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: firstname.lastname@example.org 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. LITERATURE CITED Adams DJ, Pedersen DR, Brand RA, Rubin CT, Brown TD. 1995. Three-dimensional geometric and structural symmetry of the turkey ulna. J Orthop Res 13:690 – 699. Aiello LC. 1981. The allometry of primate body proportions. Symp Zool Soc Lond 48:331–358. LIMB BONE CURVATURE IN ANTHROPOID PRIMATES Alexander RM, Jayes AS, Maloiy GMO, Wathuta EM. 1979. Allometry of the limb bones of mammals from shrew (Sorex) to elephant (Loxodonta). J Zool Lond 189:305–314. Alpert NM, Brad JF, Kennedy D, Correia JA. 1990. The principal axes transformation—a method for image registration. J Nucl Med 31:1717–1722. Bertram JEA, Biewener AA. 1988. Bone curvature: sacriﬁcing strength for load predictability? J Theor Biol 131:75–92. Biewener AA. 1982. Bone strength in small mammals and bipedal birds: do safety factors change with body size? J Exp Biol 98:289 –301. Biewener AA. 1983. Locomotory stresses in the limb bones of two small mammals: the ground squirrel and chipmunk. J Exp Biol 103:131–154. Biewener AA. 1990. Biomechanics of mammalian terrestrial locomotion. Science 250:1097–1103. Biewener AA, Taylor CR. 1986. Bone strain: a determinant of gait and speed? J Exp Biol 123:383– 400. Biewener AA, Thomason J, Lanyon LE. 1983. Mechanics of locomotion and jumping in the forelimb of the horse (Equus): in vivo stress developed in the radius and metacarpus. J Zool Lond 201:67– 82. Biewener AA, Thomason JJ, Lanyon LE. 1988. Mechanics of locomotion and jumping in the horse (Equus): in vivo stress in the tibia and metatarsus. J Zool Lond 214:547–565. Burr DB, Piotrowski G, Miller GJ. 1981. Structural strength of the macaque femur. Am J Phys Anthropol 54:305–319. Burr DB, Ruff CB, Johnson C. 1989. Structural adaptations of the femur and humerus to arboreal and terrestrial environments in three species of macaque. Am J Phys Anthropol 79:357–367. Chen X, Lam YM. 1997. CT determination of the mineral density of dry bone specimens using the dipotassium phosphate phantom. Am J Phys Anthropol 103:557–560. Demes B. 1998. Use of strain gauges in the study of primate locomotor biomechanics. In: Strasser E, Fleagle J, Rosenberger A, McHenry H, editors. Primate locomotion: recent advances. New York: Plenum Press. p 237–254. Demes B, Jungers WL. 1989. Functional differentiation of long bones in lorises. Folia Primatol (Basel) 52:58 – 69. Demes B, Jungers WL. 1993. Long bone cross-sectional dimensions, locomotor adaptations and body size in prosimian primates. J Hum Evol 25:57–74. Demes B, Jungers WL, Selpien K. 1991. Body size, locomotion, and long bone cross-sectional geometry in indriid primates. Am J Phys Anthropol 86:537–547. Demes B, Stern JT Jr, Hausman MR, Larson SG, Mcleod KJ, Rubin CT. 1998. Patterns of strain in the macaque ulna during functional activity. Am J Phys Anthropol 106:87–100. Demes B, Qin Y, Stern JT Jr, Larson SG, Rubin CT. 2001. Patterns of strain in the macaque tibia during functional activity. Am J Phys Anthropol 116:257–265. Fleagle JG, Meldrum DJ. 1988. Locomotor behavior and skeletal morphology of two sympatric pitheciine monkeys, Pithecia pithecia and Chiropotes satanas. Am J Primatol 16:227–249. Hounsﬁeld GN. 1973. Computerized transverse axial scanning (tomography): part I. Description of system. Br J Radiol 46: 1016 –1022. Jungers WL, Burr DB. 1994. Body size, long bone geometry and locomotion in quadrupedal monkeys. Z Morphol Anthropol 80: 89 –97. Jungers WL, Burr DB, Cole MS. 1998. Body size and scaling of long bone geometry, bone strength, and positional behavior in cercopithecoid primates. In: Strasser E, Fleagle J, Rosenberger A, McHenry H, editors. Primate locomotion: recent advances. New York: Plenum Press. p 309 –330. Kalender WA. 2000. Computed tomography: fundamentals, system technology, image quality, applications. Munich: Publicis MCD Verlag. Kalender WA, Seissler W, Klotz E, Vock P. 1990. Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation. Radiology 176:181–183. 57 Kimura T. 1991. Long and robust limb bones of primates. In: Ehara A, Kimura T, Takenaka O, Iwamoto M, editors. Primatology today. New York: Elsevier. p 495– 498. Kimura T. 1995. Long bone characteristics of primates. Z Morphol Anthropol 80:265–280. Lanyon LE, Bourn S. 1979. The inﬂuence of mechanical function on the development and remodeling of the tibia. An experimental study in sheep. J Bone Joint Surg [Am] 61:263–273. Llorens L, Casinos A, Berge C, Majoral M, Jouffroy FK. 2001. A biomechanical study of the long bones in platyrrhines. Folia Primatol (Basel) 72:201–216. Polacin A, Kalender WA, Marchal G. 1992. Evaluation of section sensitivity proﬁles and image noise in spiral CT. Radiology 185:29 –35. Polk JD, Demes B, Jungers WL, Biknevicius AR, Heinrich RE, Runestad JA. 2000. A comparison of primate, carnivoran and rodent limb bone cross-sectional properties: are primates really unique? J Hum Evol 39:297–325. Rubin CT, Lanyon LE. 1982. Limb mechanics as a function of speed and gait: a study of functional strains in the radius and tibia of horse and dog. J Exp Biol 101:187–211. Ruff CB. 1987. Structural allometry of the femur and tibia in Hominoidea and Macaca. Folia Primatol (Basel) 48:9 – 49. Ruff CB. 1989. New approaches to structural evolution of limb bones in primates. Folia Primatol (Basel) 53:142–159. Ruff CB. 2002. Long bone articular and diaphyseal structure in Old World monkeys and apes. I: locomotor effects. Am J Phys Anthropol 119:305–342. Ruff CB. 2003. Long bone articular and diaphyseal structure in Old World monkeys and apes. II: estimation of body mass. Am J Phys Anthropol 120:16 –37. Ruff CB, Runestad JA. 1992. Primate limb bone structural adaptations. Annu Rev Anthropol 21:407– 433. Ruff CB, Walker A, Teaford MF. 1989. Body mass, sexual dimorphism and femoral proportions of Proconsul from Rusinga and Mfangano Islands, Kenya. J Hum Evol 18:515–536. Rusinek H, Tsui WH, Levy AV, Noz ME, de Leon MJ. 1993. Principal axes and surface ﬁtting methods for three-dimensional image registration. J Nucl Med 34:2019 –2024. Schafﬂer MB, Burr DB, Jungers WL, Ruff CB. 1985. Structural and mechanical indicators of limb specialization in primates. Folia Primatol (Basel) 45:61–75. Schmitt D. 1994. Forelimb mechanics as a function of substrate type during quadrupedalism in two anthropoid primates. J Hum Evol 26:441– 457. Schmitt D. 1998. Forelimb mechanics during arboreal and terrestrial quadrupedalism in Old World monkeys. In: Strasser E, Fleagle J, Rosenberger A, McHenry H, editors. Primate locomotion: recent advances. New York: Plenum Press. p 175–200. Schmitt D. 1999. Compliant walking in primates. J Zool Lond 248:149 –160. Spoor CF, Zonneveld FW, Macho GA. 1993. Linear measurements of cortical bone and dental enamel by computed tomography: applications and problems. Am J Phys Anthropol 91:469 – 484. Swartz SM. 1990. Curvature of the forelimb bones of anthropoid primates: overall allometric patterns and specializations in suspensory species. Am J Phys Anthropol 83:477– 498. Swartz SM, Bertram JEA, Biewener AA. 1989. Telemetered in vivo strain analysis of locomotor mechanics of brachiating gibbons. Nature 342:270 –272. Tsao J, Chiodo CP, Williamson DS, Wilson MG, Kikinis R. 1998a. Computer-assisted quantiﬁcation of periaxial bone rotation from X-ray CT. J Comput Assist Tomogr 22:615– 620. Tsao J, Stundzia A, Ichise M. 1998b. Fully automated establishment of stereotaxic image orientation in six degrees of freedom for Tc-99m-ECD brain SPECT. J Nucl Med 39:503–508. Weber DA, Ivanovic M. 1994. Correlative image registration. Semin Nucl Med 24:311–323. Yamanaka A, Shimizu D, Takemoto H. 2001. New techniques to reconstruct a 3-D skeletal morphology using continuous CT images. Anthropol Sci (Jpn Ser) 108:81–90.