The Effective Mechanical Advantage of A.L. 129-1a for Knee Extension

Authors


Abstract

The functional significance of shape differences between modern human and australopithecine distal femora remains unclear. Here, we examine the morphological component of the effective mechanical advantage (EMA) of the quadriceps muscle group in a sample of hominins that includes the fossil A.L. 129-1a (Australopithecus afarensis) and modern humans. Quadriceps muscle moment arms were calculated from three-dimensional computer models of specimens through a range of knee flexion. All hominins were compared using the same limb positions to allow us to examine, in isolation, the morphological component of the lengths of the pertinent moment arms. After taking into account the differences in bicondylar angle, the morphological component of the EMA was calculated as the ratio of the quadriceps muscle and ground reaction force moment arms. Our analyses reveal that A.L. 129-1a would have possessed a morphological component of the quadriceps muscle EMA expected for a hominin of its body mass. Anat Rec, 2011. © 2011 Wiley-Liss, Inc.

INTRODUCTION

Although it is nearly universally agreed that australopiths were bipeds, the exact nature and the energetic efficiency of their bipedalism continue to be debated (see Ward (2002) for a review). The australopithecine knee has been central to this discussion because, although clearly that of a biped, it deviates notably from modern human anatomy. Fossils from Hadar, attributed to Australopithecus afarensis (A. afarensis), have figured prominently in the ongoing controversy because of their antiquity (3.4–3.0 million years old; Kimbel and Delezene, 2009) and relative abundance and because the functional significance of their morphology remains unclear. Some researchers have categorized morphological differences between A. afarensis and modern humans as small and insignificant, and that the knee (along with the rest of the anatomy) betrays a striding biped with a nearly, if not exactly, modern kinematic profile and energetic requirement (Latimer et al., 1987; Lovejoy, 1988; Latimer and Lovejoy 1989, Latimer, 1991; Crompton et al., 1998). At the other end of the spectrum are those that interpret the differences in morphology as revealing a unique form of bipedalism (e.g., Rak, 1991), one often characterized as being inefficient and compromised relative to modern human locomotion (Stern and Susman, 1983; Susman et al., 1984).

Early descriptions of distal femora from Hadar emphasized human-like features, including the bicondylar angle, elliptical profile of the lateral condyle, and a deep patellar groove (Johanson et al., 1976). Of these traits, the bicondylar angle is likely the one undisputed feature of the australopithecine knee and its link to bipedalism is considered unequivocal.

The characterization and implications of other features, including the depth of the patellar groove and the shape of the condyles, remain controversial. The lateral lip of the patellar groove has been described by some as weakly developed in australopiths, and thus less capable of preventing patellar subluxation (Stern and Susman, 1983; Susman et al., 1984; Stern, 2000). Lovejoy (2007) characterized the same fossil material as having a human-like patellar retention morphology. The difference between the two interpretations is the product of how the distal femora were aligned before comparison. Alignment is the placement of all specimens in a common position and orientation, and is a critical step before the analysis of shape. The difficulty achieving proper alignment is a well-understood problem in geometric morphometrics (GM) (Mitteroecker and Gunz, 2009). Consequently, there are several procedures for aligning specimens in GM analyses, although for landmark data Procrustes superimposition (which also removes size) is now the most widespread (Mitteroecker and Gunz, 2009). For functional analyses however, alignments such as Procrustes superimposition, which minimize overall anatomical shape differences, may not be appropriate. This is because to understand functional shape differences, specimens should be in functionally equivalent orientations, that is, orientations that represent how they perform functionally equivalent tasks.

Stern and Susman (1983) aligned the outlines of the distal femora (from an inferior view) using the posterior condylar axis, which is a line connecting the most posterior points on the medial and lateral condyles. Lovejoy (2007) suggested that the proper alignment of distal femora is based on the meniscal axis, which is a line connecting the centers of the meniscal grooves (impressions from the anterior horns of the medial and lateral menisci). When comparing modern human and australopithecine knees, these two orientations result in different apparent anterior projections of the lateral lip of the patellar groove, which may explain the researchers' different conclusions (Lovejoy, 2007).

The shape of the lateral condyle and its functional consequences also remain unresolved. The description of the Hadar material by Johanson et al. (1976) emphasized the elliptical shape of the lateral condyle, but Tardieu's (1979) early assessments suggested that the elliptical shape was only incipiently developed. Tardieu (1998) showed more recently that distal femora of adult A. afarensis material strongly resemble the bony distal femoral anatomy of 10–12-year-old modern humans, which are less elliptical than the same in adult humans. This would suggest that less elliptical lateral condyles are not indicative of compromised bipedalism as children of this age have attained an adult kinematic profile (Ganley and Powers, 2005; Schwartz et al., 2008). Further work by Tardieu (1999) found, however, that fetal distal femoral cartilage shape more strongly resembles the adult bony form than juvenile osteological shape. Thus, subadult bony morphology might not be the best indicator of cartilaginous morphology, and, in turn, subadult function.

An early shape assessment of the Hadar material was made by comparing the maximum mediolateral and anteroposterior dimensions of distal femora as seen from a distal view. Some of the material from Hadar, including A.L. 129-1a, was described as having a more rectangular (pongid-like) shape in which the maximum mediolateral dimension is longer than the maximum anteroposterior dimension (Senut and Tardieu, 1985). Modern humans are described as having a more square distal femur, with the maximum anteroposterior and mediolateral dimensions being nearly equal. This gross shape difference may have important functional consequences as it may be a result of the elongation of the lateral condyle anterior to the meniscal groove. Lovejoy (2007) observed that the length of this portion of the joint surface appears longer (in the anteroposterior dimension) in modern humans when compared with australopiths. Merkl et al. (2007) reached a similar conclusion for A.L. 129-1a via a three-dimensional shape analysis following alignment based on the axis of knee flexion/extension [i.e., transepicondylar axis (TEA)]. As the portion of the lateral condyle anterior to the meniscal groove is also anterior to the axis of knee flexion/extension, this elongated distance in modern humans would provide a greater moment arm for the quadriceps muscle. This may result in a greater effective mechanical advantage (EMA) for modern human knee extension relative to australopiths (Lovejoy, 2007; Merkl et al., 2007). A lower australopithecine EMA would mean that australopiths would have had to generate a greater muscle force to generate a given extension moment about the knee. Knee extension moments are critical during both bipedal walking and running. The main functions of the quadriceps muscle during both gaits is to prevent knee collapse and support the body following initial contact of the foot with the ground and then through stance phase (Perry, 1992; Novacheck, 1998). The large size of the quadriceps muscle in modern humans relative to the knee flexors, as compared with quadrupedal apes, underscores its important role in bipedal locomotion (Thorpe et al., 1999). A lower australopithecine knee extension EMA might indicate increased metabolic cost of locomotion or reduced locomotor performance (e.g., acceleration). Although such conclusions about EMA appear reasonable, this question has not been quantitatively investigated, providing the impetus for this project.

EMA

The musculoskeletal system is a system of levers in which bones act as moment arms, articulations as fulcra and muscles apply forces (input forces), which are reacted by the ground [ground reaction forces (GRF)]. The EMA of a particular musculoskeletal lever system is quantified as the ratio of the input (or muscle) and output moment arms. For knee extension, the EMA is the ratio of the quadriceps muscle moment arm and the GRF moment arms (Biewener, 1989). EMA is important because it determines the amount of muscular force that must be generated to create a specific output force necessary to support or propel body mass. A greater EMA means that for a given muscular force a larger moment, and hence output force, can be generated. Alternately, a larger EMA requires a lower muscular force to create a specific moment about the joint or output force. Since metabolic cost is related to the volume of activated muscle, a larger EMA would decrease the metabolic cost for a given activity (Kram and Taylor, 1990; Taylor, 1994). The EMA also determines, in part, the maximum moment that can be generated about a joint. Understanding the relative EMA of different animals can, therefore, contribute to our understanding of their relative locomotor performance and energetic requirements. Locomotion plays a pivotal role in a primate's evolutionary success because it provides for access to food, water and mates, but is also a main determinant of energy requirements.

To understand the relative EMA of different animals, it is, as with many aspects of biology, important to consider the effect of body size, and particularly body mass (Schmidt-Neilsen, 1984). EMA does not scale isometrically with body mass across mammals, but rather with positive allometry, with larger animals having higher EMA (Biewener, 1989; Polk, 2002, 2004). Biewener (1989) demonstrated this relationship among mammals and Polk (2002, 2004) among primates. EMA is thought to increase with body mass as it allows larger animals to use proportionally smaller muscular effort for posture and locomotion. The reduced muscular effort allows both the metabolic cost (i.e., volume of activated muscle) and musculoskeletal stresses to remain relatively constant across a wide range of body masses (Biewener, 1989). The increase in EMA that accompanies increasing body mass appears to be largely the result of using extended limb postures (Biewener, 1989; Polk, 2002, 2004), which reduces the moment arm of the GRF, thereby increasing the EMA.

Both Biewener (1989) and Polk (2002, 2004) include not only limb posture information (e.g., kinematic data), but also morphological data (e.g., muscle moment arms), obtained from radiographs and dissections, in their calculations of EMA and they find similar scaling relationships between body mass and EMA. Both of these empirically-derived results fall near theoretical expectations. The theoretical line relating EMA to body mass (EMA α body mass1/3) is based on the following reasoning from Biewener (1989). If animals were to scale isometrically, the cross-sectional area of muscles, which is proportional to maximum force generation, becomes proportionally smaller in larger animals (scaling as body mass2/3) (Biewener, 1989). To keep the mass-specific volume of activated muscle (the presumed determinant of metabolic cost (Kram and Taylor, 1990; Taylor, 1994)) and musculoskeletal stresses constant across body sizes while generating the required moment about joints, the EMA of joints must scale as body mass1/3.

Polk (2004) argues that all terrestrial mammals contend with the same problem, overcoming gravity, and thus must be subject to the same mechanical constraints. Polk (2004) further argues that hominins are not exempt from these constraints and, consequently, hominin limb posture should be affected by body mass in a similar way. We agree that hominins should be subject to the same mechanical constraints and follow the same allometric relationships, but would also point out that mechanics does not dictate that posture is the only avenue for changing EMA with increasing body mass. EMA is also determined by the length of the morphological moment arms, which could be the subject of selective pressures.

EMA is calculated using both morphological (i.e., bone dimensions) and locomotor (i.e., limb kinematics and GRF direction) information. As the locomotion of A. afarensis is unresolved, we use common limb positions to examine the morphological component in isolation. We are not suggesting that A. afarensis used a modern kinematic profile. Instead, this procedure allows us to compare the morphologies of A. afarensis and modern humans, although it prevents us from making statements about positions used during locomotion. Using common limb positions, we had three goals for this project.

  • 1First, we calculated the quadriceps moment arm for the A. afarensis fossil A.L. 129-1a and a sample of representative modern humans through the first 50° of knee flexion.
  • 2Second, using a common set of lower limb positions, we test the null hypothesis that modern humans and A.L. 129-1a have the same quadriceps muscle EMA (morphological component) and consider it against the alternate hypothesis that A.L. 129-1a has an EMA value outside the range of modern humans.
  • 3Finally, we examine experimentally determined and theoretically derived EMA allometric relationships with body mass which may explain differences between A.L. 129-1a and the modern human sample. We examine the residuals of modern humans and A.L. 129-1a relative to regression lines representing known allometric relationships. We consider the null hypothesis that A.L. 129-1a has an EMA expected for its body mass (A.L. 129-1a has residuals within the range of modern humans) and consider it against the alternate hypothesis that body mass does not account for the difference between modern humans and A.L. 129-1a (A.L. 129-1a has residuals outside the range of the modern human sample).

MATERIALS AND METHODS

Calculating the Quadriceps Moment Arm

Human comparative sample

We created a comparative human sample of femora using sex-specific statistical shape atlases of modern human femora (Mahfouz et al., 2007b). The femoral atlases summarize the shape variation in 88 male and 71 female modern human femora that are part of the William M. Bass Skeletal Collection housed at the University of Tennessee, Knoxville. This collection consists of North Americans, largely of European and African ancestry, having been born during the 20th century. All bones are from adult individuals and free of pathology.

The statistical shape atlases take advantage of standard GM techniques for landmark data. The difference between this approach and standard GM analyses is that instead of representing each bone with as many as several dozen, or even hundreds of, landmarks, femora in the atlases are represented by 7502 landmarks distributed across the surface of the bone. This is accomplished by subjecting a single “template” bone to a series of linear transformations and a local deformation process [termed mutual correspondence warping (Mahfouz et al., 2007b)] until the template bone achieves the shape of a “target” sample bone. The result is two femora that have the same number of landmarks and these points share the same local spatial relationship, thus making GM techniques feasible. Shape variation is summarized by performing principal component analysis, via singular value decomposition, on the coordinate data.

Using sex-specific atlases, we created an average male and an average female femur. We also created bones representing plus and minus two standard deviations along the first two shape components. The first component, which accounts for ∼ 93.1% (male) and 95.0% (female) of the total shape variation, describes size and size-correlated shape. The second component, which accounts for 2.2% (male) and 1.2% (female) of shape variation, describes bone robusticity (Fig. 1).

Figure 1.

The modern human reference sample created using the statistical shape atlas. The five femora on the left are female femora, and the five on the right are male femora. Middle femora in each sex set is the mean individual. Femora to the left of the mean bones are variants along the second principal component, and femora to the right are variants along the first principal component.

Automated procedures were developed previously to extract a host of anthropmetric measurements from femora in the atlas (Mafhouz et al., 2007a). Similar procedures were used to extract the measurements of patellae from sex-specific atlases of patellae (Mahfouz et al., 2007a). We used linear regression with a stepwise variable selection procedure to determine the best sex-specific predictor of the mediolateral width of patellae from the extracted femoral anthropometrics using 55 male and 40 female femur/patella pairs (Male ML Patella: Best predictor = TEA, R = 0.653, SEE = 0.28 cm, %SEE = 6.17%; Female ML Patella: Best Predictor = TEA, R = 0.626, SEE = 0.20 cm, and %SEE = 4.94%). We then used the same procedure to predict anteroposterior and superioinferior dimensions of the patellae from both femoral measurements and patellar mediolateral width (Male AP Patella: Best predictor = Male ML Patella, R = 0.649, SEE = 0.21 cm, %SEE = 8.0%; Female AP Patella: Best Predictor = Female ML Patella, R = 0.501, SEE = 0.14 cm, %SEE = 6.0%; Male SI Patella: Best Predictor = Male ML Patella, R = 0.575, SEE = 0.24 cm, %SEE = 5.3%; Female SI Patella: Best predictor = Female ML Patella, R = 0.760, SEE = 0.17 cm, %SEE = 4.5%). We predicted mediolateral width first for the patellae (as opposed to other dimensions) because the width of the patella must fit with the width of the patellar surface of the femur. To ensure this was the case we measured the width of the patellar surface on all human models and compared them with the predicted mediolateral width of the patellae. In all cases, patellar width closely matched the width of the patellar surface of the femur.

Reconstructing A.L. 129-1a

To anatomically orient A.L. 129-1a, we needed to reconstruct the entire femur. Previously, we performed a virtual reconstruction of A.L. 288-1ap (Lucy's femur) using an isometrically scaled (96.5% of original size) and mirrored version of A.L. 129-1a to rebuild the distal end and condyles (Sylvester et al., 2008b). Our reconstruction of A.L. 288-1ap resulted in a femur with a maximum length 277 ± 5 mm, which includes the original value of 280 mm estimated by Johanson et al., (1982) and the frequently cited 281 mm reported by Jungers (1982). To create our reconstruction of A.L. 129-1a, we took our reconstruction of A.L. 288-1ap and reflected it to make it a right femur and then applied an isometric scaling (0.965−1) to return the A.L. 129-1a virtual model to its original size. The same scaling was then applied to the rest of the A.L. 288-1ap model. This final reconstruction of A.L. 129-1a has a maximum length of 287 mm. We used this value for the length of A.L. 129-la as well as ± 1.92% as recommended by Sylvester et al. (2008b), thus giving minimum and maximum values of 281 and 293 mm.

To create a patella for the A.L. 129-1a we used sex-specific isometrically scaled versions of the average male and female human patellae. We scaled each patella so that the anterioposterior dimension of the patella was the same percentage of the total quadriceps moment arm in A.L. 129-1a as in the average male and female at 0° of knee flexion. We confirmed that the modeled patellae had the appropriate width given the width of the patellar surface of the A.L. 129-1a distal femur by comparing these measures (all measuring ∼ 22 mm). We found that there was a less than 1% difference in calculation of the quadriceps moment arm when using the male and female patella. As a result, we report only the results from the analyses using the male patella. We selected the male patella because Merkl et al., (2007) demonstrated that A.L. 129-1a is closer in three-dimensional shape to the male human average than the female human average.

Using an isometrically scaled version of the average male human patella makes the assumption that the australopithecine patella would have made up the same percentage of the total quadriceps muscle moment arm as it does in modern humans. Ward et al. (1995) demonstrated, however, that modern humans have anteroposteriorly thick patellae (relative to superoinferior and mediolateral dimensions) as compared with chimpanzees, suggesting the possibility of a thin australopithecine patella. Susman (1988; 1989), however, described a patella from Swartkrans (SKX 1084) attributed to Paranthropus as being shaped more like modern humans than it is like apes, providing some confidence that early hominins had relatively thick patellae. The right patella from Dmanisi (D 3418) is described as slightly larger in mediolateral width than SKX 1084, but with a larger medial than lateral articular surface (Lordkipanidze et al., 2007). A patella from Laetoli (EP 2038/03) was recovered in 2003 (Harrison and Kweba, 2011), but remains to be described. Because no patellae from A. afarensis are available against which to check our reconstruction, we conducted parallel analyses with and without the inclusion of patellae. Inclusions of patellae did not change the results of the allometry analyses described below and as a result we only report the analyses which include patellae. If patellae are discovered that show that the patella of A. afarensis was shaped very differently, new analyses would need to be conducted. All subsequent analyses were carried out in custom programs written for Matlab (v 7.3; Mathworks).

Aligning the femora

All femora were oriented by aligning their axis of flexion/extension with the x-coordinate axis. The axis of knee flexion/extension moves within the distal femur during flexion and extension. Knee flexion/extension can be described by two separate axes within the distal femur, one each before and after ∼ 25° of flexion (Iwaki et al., 2000; Freeman and Pinskerova, 2005). It has also been shown, however, that the surgical TEA approximates the instantaneous axis of knee rotation rather well throughout the range of knee motion (Yoshioka et al., 1987; Berger et al., 1993; Churchill et al., 1998; Asano et al., 2005). The surgical TEA is a line connecting the most projecting point on the lateral epicondylar eminence to the sulcus of the medical epicondyle (Yoshioka et al., 1987; Berger et al., 1993; Churchill et al., 1998; Asano et al., 2005). We used the surgical TEA because it required a single, easily and anatomically determined axis that is known to describe knee flexion/extension well. On the human models, these points were located automatically using a custom program written for Matlab. On A.L. 129-1a, the points were more difficult to locate as a result of abrasion on the sides of the specimen. We located the homologous points on the fossil by referencing a cast and selected corresponding points on the virtual model. Next, each femur was rotated around this axis until the center of its femoral head was directly above the surgical TEA. The center of the femoral head was determined as the center of a sphere that minimized the geometric error between the surface of the sphere and the vertices representing the articular surface of the femoral head.

Calculating the quadriceps moment arm (r)

The quadriceps moment arm was calculated as the perpendicular distance from the TEA out to the quadriceps muscle force vector (Fig. 2). The quadriceps force vector was anchored to the top of the patella, a position calculated using contributions from the distal femur and the patella. The femoral contribution was calculated as the minimum distance normal to the surgical TEA to a line that was both tangent to the patellar surface at its deepest point (i.e., closest to the surgical TEA) and parallel to the long axis of the tibia (i.e., having the same angle relative to the coordinate system) (Fig. 3). The patella contribution was then calculated using half the anteroposterior thickness and half the superoinferior height. The patella was positioned such that its anteroposterior axis was aligned and parallel with the line representing the femoral contribution to the moment arm. This also makes the superoinferior axis of the patella parallel to the line tangent to patellar surface at the point of contact. The muscle force vector was anchored to the top of the patella and then directed parallel to the shaft of the femur. The quadriceps muscle moment arm was calculated as the perpendicular distance from the axis of rotation (i.e., TEA) to the force vector (i.e., quadriceps muscle vector) (Brinckmann et al., 2002). The total quadriceps moment arm was multiplied by the cosine of the angle between the shaft of the femur and a line normal to the TEA (Fig. 4). We did this because moments are calculated as the product of the magnitude of the force, the length of the moment arm and the sine of the angle between them. This is because only the force component normal to both the moment arm and the axis of rotation can generate a moment. We used the cosine of the angle between the shaft (quadriceps force vector) and a line normal to TEA (axis of rotation) because it is equivalent to the sine between the shaft and the TEA.

Figure 2.

Sagittal outline of a distal femur and patella used to calculate the quadriceps moment arm (r). Upper arrow represents the quadriceps muscle force vector. The lower horizontal arrow represents the femoral contribution to the moment arm as depicted in Figure 3. The patella (crossed diamond) is positioned such that the anteroposterior axis is aligned and parallel to the femoral contribution line. The total moment arm is measured as the perpendicular distance from the TEA to the muscle force vector.

Figure 3.

Distal femur showing TEA and distance measured for quadriceps moment arm for the vertical tibia analysis. Coordinate axes are: x-coordinate, arrow points medially; y-coordinate, arrow points anteriorly; z-coordinate, arrow points superiorly. Solid lines represents TEA and arrow line represents measured quadriceps moment arm as shortest distance perpendicular to TEA out to the patellar surface.

Figure 4.

Angle between femoral shaft and line normal to TEA.

Our estimates for the modern human quadriceps moment arms are similar to published values. Wilson and Sheehan (2009) measured the in vivo moment arm of the individual quadriceps muscles in 22 individuals using magnetic resonance imaging, and our calculated moment arms are within the range of their data. (See Discussion for more details.) In addition, we examined the contribution that the patella makes to the total moment arm (as a percentage), and found that our results are similar to experimental work (Kaufer, 1971). Given these comparisons, it seems that our virtual method of calculating the quadriceps moment arm gives results that are comparable to experimental techniques. Thus, we find it appropriate to extend our virtual method to A.L. 129-1a.

Quadriceps Muscle EMA Allometry

Estimating body mass

To examine the allometric relationship of EMA with body mass in hominins, we estimated body mass for all the human specimens, based on femoral head diameter (calculated from the virtual models) using formulae provided in Ruff et al. (1991) including the recommended correction factor. A range of body mass estimates for A.L. 129-1a was derived from McHenry and Berger (1998). This range includes values calculated using two regression formulae, one derived using a modern human reference sample and the other using an ape reference sample. Using the human formula, McHenry and Berger (1998) calculate the body mass of A.L. 129-1a to be 28 kg, and 36 kg using the ape regression formula. We used both of these values as well as the average of these two (32 kg).

Knee position and GRF moment arm (R)

To calculate the GRF moment arms, we followed a procedure similar to that outlined by Trinkaus and Rhoads (1999) (Fig. 5). This procedure uses the length of the femur and tibia and the angle of knee flexion. Although Trinkaus and Rhoads (1999) consider the angle between the two segments, we follow the convention in biomechanics where 0° is full extension (femur and tibia axes are aligned when viewed from a lateral perspective), negative values hyperextension, and positive values are flexion (Brinckmann et al., 2002). To calculate the GRF moment arm in this manner, we estimated a tibia length for each specimen. To do this we used crural indices to calculate tibial length based on femoral length. We used a crural index of 83.5 to calculate the tibial length for A.L. 129-1a following Wang et al. (2004). For the human specimens we used a crural index of 82 which is the average crural index of globally distributed human populations reported by Sylvester et al. (2008a). We examined the EMA through the range of motion as the ratio of the calculated quadriceps muscle and GRF moment arms, comparing A.L. 129-1a to the modern human sample.

Figure 5.

Calculation of the GRF moment arm. Adapted from Trinkaus and Rhoads (1999). GRF moment arm is labeled R.

Although EMA is the product of both morphology (i.e., bone and soft tissue dimensions) and locomotor information (i.e., limb kinematics and GRF direction), the precise nature of australopithecine locomotion remains unresolved. This prevents us from including locomotor information in EMA calculation. When australopithecine kinematics is resolved, the contribution of locomotor information to EMA can be incorporated with the morphological component which is calculated here.

EMA allometry

As it has been shown that EMA scales with positive allometry with body mass in both mammals generally (Biewener, 1989) and primates specifically (Polk, 2004), it is important to account for body mass when comparing the EMA of A.L. 129-1a to modern humans. We examined the EMA at 10° of knee flexion for both A.L. 129-1a and modern humans, as this approximates the position of modern human midstance. This procedure allows us to control for limb position so that we can explore the possibility that body mass explains some of the variation in the morphological component of quadriceps muscle EMA.

To account for the effect of body mass on the quadriceps muscle EMA of hominins, we made the following assumption. We assumed that the quadriceps muscle EMA of hominins scales in a fashion similar to the knees of mammals, primates and theoretical expectations (all of which scale in a similar fashion). We made this assumption based on Polk (2004) who argued that the limbs of all terrestrial mammals are subject to the same biomechanical constraints because they perform the same function (i.e., overcoming gravity to support and propel the body mass). As a result, Polk (2004) argued that limb posture should be affected in the same way by changes in body mass. Polk's argument is greatly strengthened by two empirical findings, both of which we find compelling. First, the slope of the primate regression line relating quadriceps muscle EMA to body mass is very similar to that determine for a larger taxonomic range of mammals by Biewener (1989). This suggests that relationship determined by Biewener is not exclusive to “mouse-to-elephant” relationships but also characterizes smaller taxonomic units. Second, the regression lines from Polk (2004) and Biewener (1989) both conform to theoretical expectations that predict that EMA should scale as body mass raised to the 1/3 power. Thus, the power in these relationship stem from their general agreement and adherence to theoretical expectations.

Although we agree with Polk that terrestrial animals are under the same biomechanical constraints, we point out (as noted earlier) that the important parameter is not joint posture per se, but EMA as this determines the amount of muscular effort required to overcome gravity. Mechanics does not dictate that joint posture is the only avenue to change EMA as it is the quotient of both muscular and GRF moment arms. Thus we assume that EMA is the pertinent variable, and that quadriceps muscle EMA scales with body mass in hominins in a manner similar to other terrestrial mammals as described by Biewener (1989) and Polk (2004).

To explore the effect of body mass, we used the following procedure. We plotted our data with the full regression lines relating knee (extensor) EMA and body mass from Biewener (1989) and Polk (2004), and calculated residuals of the modern humans and A.L. 129-1a relative to both lines. Biewener (1989) examined a host of mammals that ranged in size from 0.092–275 kg. Polk (2004) looked exclusively at primates (cercopithecines and humans). Our logic being that if A.L. 129-1a falls no further from the regression lines than do the modern humans, then these allometric relationships may explain the variation in morphological component of the quadriceps muscle EMA among hominins.

We then examined the residuals in a different way by forcing three regression lines (the lines from Biewener (1989) and Polk (2004), and a theoretical line) through the modern human sex-combined mean by changing only the intercept of each line, thus leaving the allometry coefficient unchanged (i.e., leaving the slope unchanged). The theoretical line relating EMA to body mass (EMA α body mass1/3) is based on the following reasoning from Biewener (1989). If animals were to scale isometrically, the cross-sectional area of muscles, which is proportional to maximum force generation, becomes proportionally smaller in larger animals (scaling as body mass2/3) (Biewener, 1989). To keep the mass-specific volume of activated muscle (the presumed determinant of metabolic cost (Kram and Taylor, 1990; Taylor, 1994)) and musculoskeletal stresses constant across body masses while generating the required moment about joints, the EMA of joints must scale as body mass1/3. We then examined the residuals of the A.L. 129-1a reconstructions relative to the residuals of the modern humans to determine if A.L. 129-1a is further from the regression lines.

RESULTS

Body mass estimates, femoral lengths, bicondylar widths and patellar dimensions for the modern human reference sample are provided in Table 1 along with body mass estimates for A.L. 129-1a from McHenry and Berger (1998) and recreated patellar dimensions for the fossil. Quadriceps muscle moment arms for the modern humans and for A.L. 129-1a are provided in Figure 6. The quadriceps moment arm ranges from 4.3 to 5.1 cm for the human males, from 3.9 to 4.7 cm for the human females, and from 2.2 to 2.3 cm for A.L. 129-1a. As a percentage of the total moment arm, the patella makes up between 25 and 40% of the total quadriceps moment arm through the range of motion explored here (Fig. 7). This is true both for the human samples and A.L. 129-1a which follows a similar angle related pattern for this parameter. The change in the femoral component of the moment arm (expressed as a percentage of the value at 0° of knee flexion) is presented in Figure 8. The femoral contribution to the total moment arm decreases rapidly in A.L. 129-1a as a function of knee angle, but less so in the final 10° explored here. This is in contrast to the modern human pattern that changes very slowly in the early portion of knee flexion, and then steadily decreases after ∼ 20°.

Figure 6.

Quadriceps moment arms through first 50° of knee flexion. Solid circle = average female femur; Open circles = other female femora; Solid square = average male femur; Open squares = other male femora. Solid triangles = A.L. 129-1a.

Figure 7.

Patellar contribution to the total quadriceps muscle moment arm (proportion) through first 50° of knee flexion. Solid circle = average female; Open circles = other female femora; Solid square = average male femur; Open squares = other male femora; Solid triangles = A.L. 129-1a.

Figure 8.

Change in femoral contribution expressed as a proportion of its initial value (0°). Solid circle = average female; Open circles = other female femora; Solid square = average male femur; Open squares = other male femora; Solid triangles = A.L. 129-1a.

Table 1. Estimated body masses
 Mass (kg)Femur Length (mm)Bicondylar Width (mm)Patella SI (mm)Patella AP (mm)
  1. AveM, Average Male; M1+, Male, plus two standard deviations along first shape component; M1-, Male, minus two standard deviations along first shape component; M2+, Male, two standard deviations along second shape component; M2-, Male, two standard deviations along second shape component; F1+, Female, plus two standard deviations along first shape component; F1-, Female, minus two standard deviations along first shape component; F2+, Female, two standard deviations along second shape component; F2-, Female, two standard deviations along second shape component. For A.L. 129-1a, the two extreme values are from McHenry and Berger (1998) using human and ape regression formula, the middle value is the average of these two values.

Ave M68.84708544.924.9
M 1+75.25189045.825.9
M 1-62.34108244.024.0
M 2+74.74739043.623.4
M 2-64.84687946.126.2
Ave F61.04337639.022.8
F 1+65.44797840.323.3
F 1-56.23887237.622.2
F 2+58.14337137.222.1
F 2-65.84338040.923.5
A.L. 129-1a (ape)28  22.012.1
A.L. 129-1a (average)32  22.012.1
A.L. 129-1a (human)36  22.012.1

The EMA for the modern humans and A.L. 129-1a reconstruction (287-mm length) are presented in Figure 9, and for all reconstructions of A.L. 129-1a and modern humans in Table 2 (5° increments). The A.L. 129-1a reconstruction is closest to the female that represents plus two standard deviations along the second shape component and has EMA values that are 82–85% of this individual. The EMA for each A.L. 129-1a reconstruction as a percentage of the average male, female and human EMA are given in Table 3. Throughout the range of motion explored here, the three A.L. 129-1a reconstructions were between 73 and 81% of the average human values.

Figure 9.

EMA for 5–50° of knee flexion. Solid circle = average female femur; Open circles = other female femora; Solid square = average male femur; Open squares = other male femora; Solid triangles = A.L. 129-1a. Note: To make the difference between A.L. 129-1a and the modern humans clearer, the EMA at 5° of knee flexion was not included in the figure.

Table 2. EMA through first 50 degrees of flexion
 Modern humansA.L. 129-1a
AngleAve MM1+M1-M2+M2-Ave FF1+F1-F2+F2-287281293
  1. For modern human reference sample abbreviations are the same as Table 1; A.L. 129-1a (287 mm) = Reconstruction of A.L. 129-1a with maximum length of 287 mm; A.L. 129-1a (293 mm) = Reconstruction of A.L. 129-1a with maximum length of 293mm; A.L. 129-1a (281 mm) = Reconstruction of A.L. 129-1a with maximum length of 281 mm; Angle in degrees.

52.712.652.772.542.832.702.622.792.502.852.102.162.02
101.371.331.411.281.431.371.321.411.261.441.051.081.02
150.920.890.950.860.960.920.880.950.840.970.700.720.68
200.690.670.710.650.720.690.670.720.640.730.520.540.50
250.550.530.570.520.580.550.530.580.510.590.420.430.40
300.460.440.480.430.480.460.440.480.420.490.350.360.34
350.390.380.410.370.410.390.380.410.360.420.300.310.29
400.340.330.360.320.360.340.330.360.310.370.260.270.25
450.300.290.320.280.320.300.290.320.280.320.230.240.23
500.270.260.290.250.290.270.260.280.250.290.210.220.20
Table 3. EMA of A.L. 129-1a as a percentage of modern human sample
 A.L. 129-1a 287 mmA.L. 129-1a 281 mmA.L. 129-1a 293 mm
AngleAveMAveFAveHsAveMAveFAveHsAveMAveFAveHs
  1. AveM, Average modern human male; AveF, Average modern human female; AveHs, Average modern human sex combined.

50.780.780.780.800.800.800.750.750.75
100.770.770.770.790.790.790.750.750.75
150.760.770.770.790.790.790.740.750.74
200.760.750.750.780.780.780.730.720.73
250.760.760.760.780.780.780.730.720.73
300.760.760.760.790.790.790.740.740.74
350.770.770.770.790.790.790.740.740.74
400.760.760.760.790.790.790.730.730.73
450.760.760.760.790.790.790.760.760.76
500.770.780.770.810.810.810.740.740.74

The EMA for the humans and A.L. 129-1a (all three lengths and three mass estimates) at 10° of knee flexion are plotted in Figure 10. Also included in Figure 10 are the full regression lines from Biewener (1989) and Polk (2004) relating EMA of the knee extensors to body mass. As shown in Figure 9, the australopith had a much lower EMA at this angle in absolute terms. The modern human sample and the nine different combinations of A.L. 129-1a lengths and body masses fall above both Polk's (2004) primate regression line and Biewener's (1989) general mammal regression line. That is, using an angle of 10°, A.L. 129-1a and the modern humans have similar positions relative to both regression lines. Residuals from each of the two lines are provided in Table 4. Relative to both lines, only one of the A.L. 129-1a length/body mass combinations has a residual outside of the range of modern humans and this is the 293 mm/36 kg reconstruction relative to the mammalian line from Biewener (1989).

Figure 10.

Log EMA plotted against log Body Mass with full regression equations from Biewener (1989) and Polk (2004). Solid circle = average female femur; Open circles = other female femora; Solid square = average male femur; Open squares = other male femora; Solid triangles = A.L. 129-1a. For A.L. 129-1a the different reconstructions and body mass are as follows. Columns of points are different body masses, from left to right they are 28 kg, 32 kg, and 36 kg. Rows of points are different lengths, from top to bottom, 281 mm, 287 mm, 293 mm. Included in the plot are the full regression lines and equations from Biewener (1989) (dashed) and Polk (2004) (solid).

Table 4. Residuals of specimens to Polk (2004) and Biewener (1989) full regression lines
HomininPolk, 2004Biewener, 1989
  1. For modern human reference sample, abbreviations are the same as Table 1. For A.L. 129-1a values following the specimen name are maximum length and body mass estimate.

Ave M0.210.12
M1+0.180.09
M1-0.240.14
M2+0.190.10
M2-0.220.12
Ave F0.230.13
F1+0.200.11
F1-0.260.16
F2+0.200.11
F2-0.240.15
A.L. 129-1a 287mm 28 kg0.250.14
A.L. 129-1a 281mm 28 kg0.260.15
A.L. 129-1a 293mm 28 kg0.240.12
A.L. 129-1a 287mm 32 kg0.230.12
A.L. 129-1a 281mm 32 kg0.240.13
A.L. 129-1a 293mm 32 kg0.210.10
A.L. 129-1a 287mm 36 kg0.210.10
A.L. 129-1a 281mm 36 kg0.220.11
A.L. 129-1a 293mm 36 kg0.190.08

To more easily compare A.L. 129-1a with the modern humans, the regression lines from Biewener (1989) and Polk (2004) and the theoretical regression line were forced through the mean human value by changing only the intercept (Fig. 11). This allowed us to use the modern humans as a reference point for hominin evolution (because modern humans are the only living hominin) to evaluate the A.L. 129-1a reconstructions while maintaining the allometric scaling relationships. The residuals of all hominins relative to all three regression lines forced through the sex-combined human mean are presented in Table 5. Most of A.L. 129-1a length/body mass combinations are within the range of residual values of the modern humans. Relative to Polk's regression line only the smallest (281 mm/28 kg) version of A.L. 129-1a has a residual outside the range of the modern humans. Relative to Biewener's and the theoretical regression line, only the largest (293 mm/36 kg) version of A.L. 129-1a has a residual larger outside the range of modern humans. In all cases where the residuals of the A.L. 129-1a reconstruction do exceed those of the modern humans, they do so by only a small amount.

Figure 11.

Log EMA plotted against log Body Mass with regression equations from Biewener (1989) and Polk (2004) and theoretical allometric line forced through mean modern human.

Solid circle = average female femur; Open circles = other female femora; Solid square = average male; Open squares = other male femora. Solid triangles = A.L. 129-1a. For A.L. 129-1a the different reconstructions and body mass are as follows. Columns of points are different body masses, from left to right they are 28 kg, 32 kg, and 36 kg. Rows of points are different lengths, from top to bottom, 281 mm, 287 mm, 293 mm. Included in the plot are the full regression lines and equations from Biewener (1989) (dashed) and Polk (2004) (solid).

Table 5. Residuals of samples allometric regression lines forced through human mean
 Polk, 2004Biewener, 1989Theoretical
  1. For modern human reference sample, abbreviations are the same as Table 1. For A.L. 129-1a values following the specimen name are maximum length and body mass estimate.

Ave M−0.01−0.01−0.01
M1+−0.03-0.03−0.03
M1-0.020.020.02
M2+−0.03−0.03−0.03
M2-0.000.000.00
Ave F0.010.010.01
F1+−0.01−0.01−0.01
F1-0.040.040.04
F2+−0.01−0.02−0.02
F2-0.020.020.02
A.L. 129-1a 287mm 28 kg0.030.010.01
A.L. 129-1a 281mm 28 kg0.050.030.02
A.L. 129-1a 293mm 28 kg0.020.00−0.01
A.L. 129-1a 287mm 32 kg0.01−0.01−0.01
A.L. 129-1a 281mm 32 kg0.020.010.00
A.L. 129-1a 293mm 32 kg0.00−0.02−0.02
A.L. 129-1a 287mm 36 kg−0.01−0.02−0.03
A.L. 129-1a 281mm 36 kg0.00−0.01−0.01
A.L. 129-1a 293mm 36 kg−0.02−0.04−0.04

DISCUSSION

Calculating the Quadriceps Moment Arm

Our estimates for the modern human quadriceps moment arms are very similar to published values and follow very similar angle dependent patterns. Wilson and Sheehan (2009) recently measured the moment arms for the individual quadriceps muscles in 22 humans using dynamic cine phase contrast magnetic resonance imaging, making this the largest sample to date and the only in vivo. These researchers find that the moment arms of the four individual quadriceps muscles range from ∼ 3.5 to 6 cm between the four muscles and the 22 individuals over a range of motion from 0 to 45° of knee flexion. Our own values for a combined quadriceps moment are well within this range of values despite the absence of articular cartilage. Articular cartilage of the femoral patellar surface and of the patella each average about 3 mm (Ateshian et al., 1991; Shepherd and Seedhom, 1999), and thus its inclusion would not substantially change the results. The values from Wilson and Sheehan (2009) are slightly higher than those reported using the tendon excursion method (e.g., Visser et al., 1990; Buford et al., 1997). The tendon excursion method, however, has been questioned because the assumptions required by this method are rarely met in experimental settings, and thus likely do not accurately reflect the actual muscle moment arm (Sheehan, 2007).

We also examined the relative contribution of the patella to the total moment arm of the quadriceps. Such data are rare in the literature. Kaufer (1971), however, examined the contribution (as a percentage) of the patella to the moment arm of the patellar tendon. Although not comparable through the range of motion because we measured the moment arm of the quadriceps muscle and Kaufer (1971) measured the moment arm of the patellar tendon, our data should be roughly similar to Kaufer's (1971) at full knee extension. Kaufer (1971) found that the patella contributes ∼ 30% of the total patellar moment arm at full extension, and our own data suggest it to be between 25 and 28%. Although A.L. 129-1a falls within the range of the modern human values for this metric, the australopith follows a differently shaped pattern. Because the patella of A.L. 129-1a is modeled from a modern human, its contribution to the total quadriceps moment arm follows the same angle dependent pattern. The difference between modern humans and A.L. 129-1a in the relative contributions of the femoral and patellar components is a result of the changes in the femoral component. In A.L. 129-1a, the femoral component decreases more rapidly than to modern humans in the first 30° of knee flexion, and thus the patellar component makes up a proportionally larger part of the total moment arm. In the final 20° of knee flexion explored here, the modern human femoral component continues to decrease at a steady rate, while that of A.L. 129-1a starts to decrease more slowly. This indicates a significant shape difference in the contour of the patellar surface between modern humans and A.L. 129-1a.

The quadriceps moment arm for A.L. 129-1a was calculated including an isometrically scaled version of the average male human patella. This makes the assumption that the relationship between australopithecine patellae and distal femora was largely similar to that in modern humans. This is a necessary assumption as fossil femora are rare, patellae are rarer and patella/femur pairs are non-existent for any australopith, including A. afarensis. If associated femora and patellae are found that demonstrate that australopithecine patellae are vastly different in shape relative to their femora, new analyses would need to be conducted to incorporate the differently size patellae for A. afarensis.

Body Mass Estimation

A critical step in this analysis was the estimation of body mass for A.L. 129-1a. It should be noted that McHenry and Berger (1998) did not intend the body masses predicted for fossil specimens to be taken as accurate estimates of actual body mass, but rather they produced them as a means to explore upper to lower limb proportions in A. afarensis and A. africanus. Nonetheless, their estimate for A.L. 129-1a (28-36 kg) seems reasonable based on A.L. 288-1 for which body mass has been estimated with greater confidence at ∼ 30 kg (McHenry 1988; Jungers, 1991). McHenry and Berger (1998) also estimated a body mass for A.L. 129-1b, the proximal tibia associated with A.L. 129-1a, of 27-31 kg. Our own analysis of the size the difference between A.L. 129-1a and A.L. 288-1 finds the latter to be 96.5% the size of the former (linear dimensions) (Sylvester et al., 2008b). This scaling (inversed and cubed) can be used to estimate a body mass for A.L. 129-1a, from the A.L. 288-1 body mass, of 33 kg. This is very close to the mean body mass of 32 kg used in our analyses. As a result we prefer the body mass estimate of 32 kg when interpreting the results.

EMA and Knee Range of Motion

In absolute terms, A.L. 129-1a had a lower EMA for the quadriceps muscle throughout the range of motion compared with modern humans. A.L.129-1a was ∼ 75–80% of the average modern human and 82–85% of the modern human with the lowest EMA values. This confirms the visual assessment of the australopithecine knee made by Lovejoy (2007) and the conclusion concerning A.L. 129-1a reached by Merkl et al. (2007), which is that the shortened anteroposterior dimension results in a smaller quadriceps moment arm and lower EMA when placed in similar limb positions. This is similar to the result reached by Jungers (1991) in his analysis of the hip abductor moment arm in A.L. 288-1, having found the australopithecine hip abductor EMA to be substantially lower than modern humans.

It may be expected that australopiths would have had smaller EMA, even if only for the morphological component, given the known positive allometry between EMA and body mass (Biewener, 1989; Polk, 2004). Polk (2004) provides regression formulae relating knee (extensor) EMA to body mass. Polk's (2004) human/cercopithecine formula predicts a knee EMA at midstance for A.L. 129-1a (32 kg) of 0.62, and EMA values of 0.80 and 0.84 for our average female (61 kg) and male (68.8 kg) respectively. Thus, based on predicted values using Polk's (2004) regression formula, the EMA of A.L. 129-1a would have been 73% of the average modern human male and 77% of the average modern human female. Although Polk's analysis (2004) includes both posture and morphology in calculating knee EMA, we find it interesting that our analysis predicts A.L. 129-1a to be a similar percentage of modern human values based on morphology alone.

In our bivariate plots of EMA at 10° of knee flexion against body mass, A.L. 129-1a and the modern human samples occupy similar positions relative to the full regression lines from Biewener (1989) and Polk (2004), a conclusion reinforced by examination of the residuals. When the regression lines from Biewener (1989) and Polk (2004) and the theoretical line are forced through the human average, most of the length/body mass combinations of A.L. 129-1a are again within the range of modern human deviation from the line. Those that do deviate further than modern humans only do so by a small amount.

Direct comparison of the elevation of our data (i.e., the intercept) relative to the full regression lines from Polk (2004) and Biewener (1989) is hindered by the fact that the data collected here and those from Polk (2004) and Biewener (1989) were collected in different ways. Polk (2002, 2004) measured the quadriceps moment arm on radiographs from the contact point between the femur and tibia to the line of action of the muscle. Biewener (1989) measured the quadriceps moment arms from radiographs and anatomical dissections. We measured it from the axis of knee rotation out to the quadriceps muscle vector anchored to the top of the patella. Thus, we expect our measures to be different than both Polk and Biewener in absolute values. If, however, our measurement of the moment arms are scaled by a constant relative to those measured by Polk and Biewener, this would manifest in bivariate log plots as differences in elevations, but would not affect the slope of the line. Therefore, comparing the elevation of our data points relative to the full regression lines from Polk (2004) and Biewener (1989) is of little value because they may differ only by virtue of the different data collection techniques.

Since our goal was to make comparisons only within hominins and not directly to the previous data, the scaling relationship (i.e., the slope) is of considerably greater importance. As A.L. 129-1a does not deviate from the experimentally determined and theoretically derived allometric regression lines any further than modern humans, it appears that A.L. 129-1a had a morphological component to its EMA that would be expected for a hominin of its body mass. When the kinematics of australopithecine locomotion are resolved, the contribution of limb position and GRF direction would need to be incorporate with the morphological analysis conducted here to understand the EMA of A.L. 129-1a during locomotion.

CONCLUSIONS

Previous analyses have noted many morphological differences between modern human femora and australopiths. Among these is a difference in the length of the quadriceps muscle moment arm. We have shown here that the quadriceps muscle moment arm of A.L. 129-1a is shorter in absolute terms compared with modern humans and have quantified this difference. We have also demonstrated that the morphological component of the quadriceps muscle EMA of A.L. 129-1a is also much smaller than in modern humans. After accounting for the influence of body mass, however, we have shown that the morphological component to the quadriceps muscle EMA of A.L. 129-1a appears to be what would be expected for a hominin of its body mass.

Acknowledgements

The authors wish to thank Andrew Kramer for access to the casts of A.L. 129-1a and A.L. 288-1 and Lee Meadows Jantz for access to the William M. Bass Donated Skeletal Collection. We also thank Christopher Ruff and Jason Organ for helpful discussions.

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