A three-dimensional musculoskeletal model of the chimpanzee (Pan troglodytes) pelvis and hind limb

Musculoskeletal models have become important tools for studying the mechanics, energetics and control of locomotion. Several human models have been developed (e.g. Delp et al., 1990; Hatze, 1977; Modenese et al., 2011), permitting predictions of muscle–tendon kinematics and moment arms across a range of three-dimensional (3D) joint positions. When combined with detailed muscle architecture measurements, both active and passive muscle–tendon forces and muscle fascicle length change can be determined (e.g. Arnold et al., 2010; Arnold and Delp, 2011). Such models may also be used in forward dynamics simulations to evaluate performance-based optimization criteria, which have provided considerable insight into the physiological bases of walking, running and jumping (e.g. Anderson and Pandy, 2001; Miller et al., 2012; Pandy et al., 1990; Umberger, 2010). Understanding how muscular, skeletal and neural traits interact to influence locomotor performance is an important goal not only for improving human health, but also for gaining new insights into the evolution of human locomotor behavior. Despite the considerable value of musculoskeletal modeling and simulation, no comparable model exists for any of our living, great ape relatives. Such a model is essential for providing an evolutionary context for human performance studies.

The common chimpanzee is both a facultative biped and our closest living relative (Goodman, 1999; Mikkelsen et al., 2005). As such, chimpanzees have long been used to gain insight into the origins and evolution of bipedal locomotion in humans (e.g. Elftman and Manter, 1935a; Elftman, 1944; Jenkins, 1972; Sockol et al., 2007; Stern and Susman, 1981; Taylor and Rowntree, 1973; Tardieu, 1992). The similarities and differences that exist in locomotor mechanics and energetics between chimpanzees and humans have provided a number of important insights into our physiological and anatomical evolution (see Aiello and Dean, 1990; Schmitt, 2003; Stern, 2000). For example, chimpanzees use approximately 75% more energy per distance walked than do humans (Rodman and McHenry, 1980; Sockol et al., 2007), suggesting that an important element in human locomotor evolution was reducing locomotor costs relative to our ape-like common ancestor. Musculoskeletal modeling permits decomposition of the mechanical and metabolic costs of a movement task among individual muscle–tendon units (Neptune et al., 2009; Umberger and Rubenson, 2011), and can provide insight into how these costs are influenced by the differences in skeletal structure, muscle properties and neural control between species.

Previous laboratory-based experimental measurements have included ground forces, joint movements and muscle activation patterns of common chimpanzees during bipedal walking (e.g. Ishida et al., 1985; Jenkins, 1972; Jungers et al., 1993; Kimura, 1996; Kumakura, 1989; Sockol et al., 2007; Stern and Larson, 1993; Stern and Susman, 1981; Tardieu, 1992; Thorpe et al., 2004; Tuttle et al., 1979). These data have been important for addressing a number of specific issues related to the evolution of human locomotion; however, these data have generally been considered in isolation and can be quite difficult to integrate. Yet, this is a necessary step for understanding how the neuromusculoskeletal system produces locomotor movements (Zajac et al., 2002). A comprehensive model that permits simultaneous integration of kinematic, kinetic and muscular aspects of locomotion would substantially facilitate this task.

Here, we describe a 3D musculoskeletal model of the chimpanzee pelvis and hind limb. The model provides accurate representations of muscle moment arms and muscle–tendon lengths within the defined joint ranges of motion, and allows detailed examination of 3D moment-generating capabilities about the hip, knee, ankle, first tarsometatarsal and second–fifth metatarsophalangeal joints. It also permits the calculation of individual muscle–tendon unit force and moment-generating capabilities, which are subjected to a detailed sensitivity analysis. In addition, we compared predictions from the chimpanzee model with similar results obtained from human modeling and experimental studies to gain insights into the functional consequences of the morphological differences between the two species. A user-extensible version of this model will be made available at www.simtk.org, which can be viewed, manipulated and analyzed using the freely available biomechanical simulation software OpenSim (Delp et al., 2007).

The disarticulated skeleton of a 55 kg wild-shot adult male chimpanzee [Pan troglodytes (Blumenbach 1775)] housed at the American Museum of Natural History was used to reconstruct the 3D pelvis and hind limb model. The pelvis (innominates and sacrum) as well as the femur, patella, tibia, fibula, tarsals, metatarsals and phalanges of the right limb were CT scanned using a GE Lightspeed CVT scanner (GE Healthcare, Waukesha, WI, USA). Serial scans of each bone were collected at a slice thickness of 0.625–1.0 mm (voltage: 120 kV; current 70 mA). The CT scanner output a stack of DICOM image files for each bone, and these were imported into Avizo (Visualization Sciences Group, Burlington, MA, USA) for 3D reconstruction. The initial wire-frame renderings of each element were typically quite large. Therefore, the number of points and triangle polygons representing each element were down-sampled to a more manageable size (range: 1000 points for patella to 12,000 points for femur), without a noticeable loss of quality in the depiction of surface geometry. A final, watertight shell (i.e. a closed polygonal mesh surface) of the reconstructed element was created in Geomagic (Geomagic Engineering, Morrisville, NC, USA) for export into Software for Interactive Musculoskeletal Modeling (SIMM; Musculographics, Santa Rosa, CA, USA).

The skeletal elements were associated with six corresponding body segments defined in SIMM. These include, the pelvis, thigh, shank, hind and mid foot, hallux (first digit) and toes (second–fifth digits). Mobile articulations were assigned at the hip, knee, ankle, first tarsometatarsal, and second to fifth metatarsophalangeal joints by defining joint centers and rotational axes. The hip joint was modeled as a spherical articulation between the pelvis and femur. The center of the hip joint was found by fitting spheres to the acetabulum and the head of the femur, and then making the centers of the two spheres coincident. The knee joint had one rotational degree of freedom, which was coupled to translation of the tibia relative to the femur and translation and rotation of the patella relative to the tibia. The center of the knee was taken as the midpoint of a line between the medial and lateral femoral epicondyles. The profile of the chimpanzee femoral condyles has been described as being more circular than in humans (e.g. Heiple and Lovejoy, 1971; Tardieu, 1983); however, modeling the tibiofemoral articulation as a revolute joint led to considerable interpenetration of the tibia and femur. This problem was overcome by introducing translation in the joint that allowed the tibial plateau to track along the contours of the femoral condyles without any bony penetration. The patella was similarly allowed to translate and rotate as a function of the knee flexion angle, such that the patella tracked along the contours of the femoral condyles. The talocrural joint was modeled as a revolute joint between the tibia–fibula and talus, with the rotational axis running between the inferior aspects of the medial and lateral malleoli (Elftman and Manter, 1935b; Latimer et al., 1987). The first tarsometatarsal joint and the metatarsophalangeal joints of the second through fifth digits were also modeled as revolute joints, with the axes determined from congruence of the adjoining skeletal elements.

In order to provide a reference position from which joint angles can be expressed, a neutral posture was defined where the positions of all joint angles were equal to zero. For the hip, the neutral position was set at full extension, with an intermediate position between abduction–adduction and internal–external rotation. The knee joint was also fully extended in the neutral position. The ankle and metatarsophalangeal joints were set between dorsiflexion and plantar flexion, while the tarsometatarsal joint was positioned between abduction and adduction. In the neutral posture, the pelvis and femur were aligned with the vertical. The chimpanzee tibia has a substantial curvature in both the sagittal and frontal planes; thus, the tibia was aligned in its reference frame so as to maximize congruence between the femoral condyles and the tibial plateau. The foot bones were oriented in their respective reference frames following Elftman and Manter (Elftman and Manter, 1935b). The range of motion limits at each of the joints was set based on cadaveric studies (Holowka and O'Neill, in press; Nagano, 2001; Payne et al., 2006; Thorpe et al., 1999). These limits are reported in Table 1.

A total of 35 muscle–tendon units from the pelvis and right hind limb were represented in the model using 44 muscle–tendon paths (Fig. 1). This includes all of the flexors–extensors, abductors–adductors and mediolateral rotators capable of significant force production during movement. A list of these anatomical muscle–tendon units and the muscle–tendon paths used to represent them in the model, is given in Table 2.

Each muscle–tendon path was assigned origin and insertion points based on a combination of skeletal landmarks, published muscle maps (Swindler and Wood, 1982; Uhlmann, 1968; Stern, 1972) and cadaveric dissections (Fig. 1A,B). The muscle–tendon path origin and insertion were positioned as near as possible to the centroid of the muscle attachment area. Some muscles with broad attachments, such as the mm. gluteals and m. adductor magnus, were not well represented using a single muscle–tendon path. In these cases, three or four separate muscle–tendon paths were used to represent the full distribution of the muscle–tendon unit. Using multiple muscle–tendon paths to represent a single muscle–tendon unit is a common modeling practice (e.g. Arnold et al., 2010; Delp et al., 1990; Van der Helm et al., 1992), and is consistent with electromyographic measurements indicating that chimpanzees recruit isolated subunits of some large hind limb muscles during movement (Stern and Susman, 1981). In one case, three small external (lateral) rotators of the hip (m. superior gemellus, m. inferior gemellus and m. obturator internus) were judged to have such similar paths as to be adequately represented by a single muscle–tendon unit. The small, intrinsic muscles of the foot were not included in the current model; however, the extensible nature of our model permits their addition for future studies addressing issues specific to intrinsic foot function in chimpanzees.

Muscle–tendon units were constrained to anatomically realistic paths using a combination of via points (Fig. 1C) and wrapping surfaces (i.e. cylinders and ellipsoids; Fig. 1D). These parameters place boundary conditions on the movement of specific points or components of muscle–tendon paths (Delp et al., 1990; Delp and Loan, 2000; Van der Helm et al., 1992). In vivo, these constraints are present because of bone, retinacula or other muscle–tendon units.

The force-generation capability of each muscle–tendon unit was represented using a generic Hill-type muscle model (Delp et al., 1990; Zajac, 1989). The Hill muscle model consists of a contractile element arranged in parallel with one elastic element, which are both in series with another elastic element (Fig. 2A). While the contractile and parallel elastic elements are generally associated with the properties of the muscle fibers, and the series elastic element is commonly equated to the tendon, it is important to note that these are both modeling approximations. In particular, it is important to be clear that the ‘tendon’ in the muscle model represents the effects of all in-series elasticity in a muscle, including that due to the aponeurosis or to sarcomeric proteins. Thus, even muscles with little or no external tendon, such as the chimpanzee m. gluteus medius or m. adductor magnus, will have non-zero tendon lengths in the muscle model. Given this, a generic Hill model was scaled to characterize specific muscle–tendon units by specifying the values of four parameters: maximum isometric force, optimal fiber length, tendon slack length and pennation angle. In order to minimize confusion, the term fascicle length is used to refer to data derived from experimental anatomical measurement, while fiber length is used to refer to the related muscle model variable. To determine the values of the muscle model parameters, we combined published measurements with geometric scaling and numerical optimization.

The maximum isometric force was calculated from the physiological cross-sectional area (PCSA; m²) and specific tension (N m⁻²) for skeletal muscle. To determine muscle PCSA values for the chimpanzee model, we used muscle mass, fascicle length and pennation angle data from cadaveric dissections of fresh-frozen male chimpanzee specimens [one 6 year old (Thorpe et al., 1999) and one 7 year old (Nagano, 2001)]. The use of young specimens helped mitigate the known underestimation of PCSA in some adult or elderly animals because of age-related muscle atrophy (e.g. Narici et al., 2003).

In order to maintain proportionality among different hind limb muscles, we preferred to base our muscle mass and fascicle length values on the near-complete data set of Thorpe et al. (Thorpe et al., 1999). However, for the m. iliacus, the mm. psoas (major and minor), the mm. adductors (adductor magnus, longus and brevis) and the m. tibialis anterior, muscle masses and fascicle lengths were drawn from Nagano (Nagano, 2001). This was a necessary step, as Thorpe et al. (Thorpe et al., 1999) did not report data for the mm. iliopsoas, and their adductor fascicle lengths were typically longer than the total muscle–tendon unit lengths in our model. In addition, the dorsiflexor-to-plantar flexor PCSA ratio in Thorpe et al. (Thorpe et al., 1999) was quite low compared with other chimpanzee data (Holowka and O'Neill, in press). Using the Nagano (Nagano, 2001) data for the m. tibialis anterior yielded a muscle fiber length similar to that in the Thorpe et al. (Thorpe et al., 1999) data set, with less of an imbalance between dorsiflexion and plantar flexion strength. Nagano (Nagano, 2001) did not report raw muscle masses and fascicle lengths; however, these values could be recovered by reversing the scaling procedures performed on his data. Pennation angles were also not reported in Thorpe et al. (Thorpe et al., 1999), and therefore were taken from Nagano (Nagano, 2001) as well. For six muscles, pennation angles were unavailable from any source, and as a result those angles were set at zero (Table 2).

For muscle–tendon units that were represented using multiple muscle–tendon paths, the total PCSA was equally divided among the individual paths. The muscle–tendon path for the mm. gluteus maximus was an exception to this approach, as its superior component (m. gluteus maximus proprius) is substantially smaller than its inferior component (m. gluteus maximus ischiofemoralis) (Stern, 1972). In this case, the m. gluteus maximus ischiofemoralis was assigned 70% of the total PCSA, while the three muscle–tendon paths representing the m. gluteus maximus proprius split the remaining 30%. PCSA represents the total cross-section of muscle available for producing force (m²), which together with specific tension (N m⁻²) determines the force-generating capacity of muscle. The specific tension of vertebrate skeletal muscle is typically reported to be 150,000–300,000 N m⁻² (Josephson, 1993). While the force–length and force–velocity properties of chimpanzee muscles are unknown, at least one recent study has proposed that chimpanzees have larger than average muscle force-producing capabilities (Scholz et al., 2006). Therefore, in the absence of direct measurements, an estimate of specific tension (315,000 N m⁻²) that is slightly above these values, but within the measured range for mammalian skeletal muscle (cf. Brown et al., 1998), was used. This value is similar, but still a bit higher than measurements from other primates, including humans (Maganaris et al., 2001) and macaques (Fitts et al., 1998). The maximum isometric force for each muscle model was obtained by multiplying the scaled PCSA values by the estimated specific tension.

The optimal fiber length for each modeled pelvis and hind limb muscle was taken to be equivalent to its scaled fascicle length. Combining our optimal fiber lengths and the muscle–tendon unit lengths from the musculoskeletal model, we estimated the tendon slack length for each muscle using a numerical optimization procedure (Manal and Buchanan, 2004). In implementing the tendon slack length optimization, the restrictions in the original algorithm that muscle fibers only operate on the ascending limb of the active force–length curve and tendons only function on the linear portion of their stress–strain curve were removed. Optimizations were performed using the constrained nonlinear optimization function (fmincon) in the MATLAB Optimization Toolbox (The MathWorks, Natick, MA, USA).

As an initial evaluation of the robustness of the model, predicted flexion–extension moment arms for hip, knee and ankle muscles were compared with values derived from experiments performed on cadaveric chimpanzee specimens (Holowka and O'Neill, in press; Payne et al., 2006; Thorpe et al., 1999). The femur, tibia and foot lengths of Thorpe et al. (Thorpe et al., 1999), Payne et al. (Payne et al., 2006) and Holowka and O'Neill (Holowka and O'Neill, in press) are, coincidentally, quite similar to the model. As such, the use of limb segment lengths to scale the muscle moment arms had no significant impact on their relative magnitudes. Thus, for ease of comparison, the model (predicted) and chimp (observed) muscle moment arms are presented in dimensional form.

The model can also be used to estimate the moment-producing capability of an individual muscle–tendon unit at a given 3D joint position, by calculating its moment arm length and tendon force. Tendon force is equal to the sum of the active muscle fiber force and the passive force of the parallel elastic element, multiplied by the cosine of the pennation angle. Active (isometric) force generation depends on the muscle force–length curve and muscle activation, which can range from 0 (no activation) to 1 (full activation). In passive force generation, muscle fibers are stretched past their optimal length on the force–length curve and force is developed in the parallel elastic element. In order to calculate maximum isometric joint moments, individual muscles were fully activated and forces were computed at lengths corresponding to the full joint range of motion. Each model joint was positioned in its approximate standing or ‘mid-stance’ posture (following Jenkins, 1972) while the hip, knee and ankle were moved through their full joint ranges of motion.

In developing the model, several decisions about its structure were made and numerous model parameters were specified. Given this, understanding the sensitivity of model outputs to variations in the model structure and parameter values is essential. To this end, the sensitivity of muscle–tendon moment arms and maximum isometric joint moments to variations in muscle path parameters and muscle architecture parameters were evaluated.

The muscle–tendon paths in the model are specified by origin and insertion points, and in many cases include via points and wrapping surfaces. The effects of uncertainty in these parameters on the resulting musculoskeletal geometry were demonstrated in three muscles, the m. gluteus medius (middle part), the m. vastus intermedius and the m. soleus, which reflect a range of complexity in pathway specification. For these muscles, muscle–tendon path parameters were varied individually, and the effects of those changes on the magnitudes of the moment arms were evaluated. For the m. gluteus medius, the origin was moved 1 cm cranially and caudally along the ilium and the insertion was moved 1 cm cranio-medially and caudo-laterally along the contour of the proximal femur. The wrapping surface that prevents the m. gluteus medius from penetrating the ilium with the hip abducted and internally (medially) rotated was also removed. For the m. vastus intermedius, the via point between the origin and the patellar attachment was moved 1 cm proximally and distally along the femur. Also, the wrapping cylinder that prevents penetration through the distal femur with knee flexion was changed in diameter by ±10%. The kinematic function that prescribed how the patella tracked along the contour of the femoral condyles was changed in magnitude by ±10%. This change had the effect of displacing the line of action of the m. vastus intermedius further away from (+10% change) or closer to (−10% change) the axis of rotation of the knee. For the m. soleus, the origin was moved 0.5 cm proximally and distally along the fibula, and the insertion was moved 0.5 cm anteriorly and posteriorly along the calcaneus. Smaller perturbations were used for the m. soleus than for the more proximal muscles to prevent unrealistic situations, such as a muscle attachment being far removed from the corresponding skeletal element. The anatomically realistic ankle joint axis, running between the malleoli, was also replaced with a pure mediolateral axis through the center of the ankle joint.

The maximum isometric moments generated about the joints are a function not only of musculoskeletal geometry, but also of the force-generating capacities of the muscle. The peak isometric forces generated by the muscle, in turn, depend on the muscle model parameter values. To evaluate the effects of uncertainty in muscle model parameter values, we varied maximum isometric force, optimal fiber length, pennation angle and tendon slack length, and assessed the effects on the maximum isometric moments. For each individual model parameter, we varied the nominal values by ±5% and ±10%, while holding all other model parameters constant. We then recalculated the maximal isometric joint moment over the full range of joint motion. Results are presented for several muscles (m. gracilis, m. sartorius, m. adductor magnus proximal, m. psoas, m. rectus femoris, m. vastus medialis, m. vastus intermedius, m. vastus laterialis, m. biceps femoris long head, m. biceps femoris short head, m. semimembranosus, m. gastrocnemius medialis, m. gastrocnemius lateralis and m. tibialis posterior) that demonstrate the range of sensitivities to muscle model parameter values.

The skeletal model includes a spherical hip joint with relatively large ranges of motion about the three primary joint axes. Consequently, the moment arms of the hip muscles about specific joint axes will depend on the values of the joint angles about the other two axes (cf. Arnold and Delp, 2001). Of course, the degree of these cross-sensitivities may vary considerably among muscles. These effects were evaluated by computing the muscle moment arms about one of the hip joint axes, while systematically varying the angles of the other two joint axes for all of the muscles crossing the hip. Results are presented for four muscles [m. gluteus maximus proprius (caudal part), m. gluteus minimus (middle part), m. pectineus and m. adductor longus] that demonstrate the range of these cross-sensitivities.

Following the validation and sensitivity analyses, we compared muscle PCSA, fiber length and mass data from the chimpanzee model with corresponding data from a similar human musculoskeletal model (Arnold et al., 2010). Both of these models are based on the best available muscle architecture data for each species, except that the data underlying the human model are presently more extensive than for the chimpanzee model.

The moment arms predicted by the model were generally in good agreement with the moment arm functions from tendon-excursion experiments. That is, for a given muscle–tendon unit, the model predicted a similar magnitude and direction over the measured joint flexion–extension range.

At the hip (Table 1, Fig. 3A–C), the tendon-excursion data for the m. gluteus maximus in P. troglodytes is limited to a single measurement, given in Payne et al. (Payne et al., 2006). The model prediction for the m. gluteus maximus proprius (cranial part) was similar to this measurement in direction, but with a larger extensor moment arm maintained through the first two-thirds of hip flexion. Moment arms for the ischial path of the m. adductor magnus and for the m. adductor brevis are both similar in direction and magnitude to the P. troglodytes tendon-excursion data in Thorpe et al. (Thorpe et al., 1999), while the model m. adductor longus is smaller, and exhibits an extensor moment arm for most of hip flexion. The model hamstrings (m. semimembranosus, m. semitendinosus and m. biceps femoris, long head) and remaining muscles around the hip (m. rectus femoris, m. gracilis) are all quite similar to the moment arm functions in Thorpe et al. (Thorpe et al., 1999), with the main difference being the presence of higher-order (i.e. nonlinear) phenomena in the model predictions.

For the knee (Table 1, Fig. 3D–F), the model predictions for the quadriceps muscles are quite similar in both direction and magnitude to the tendon-excursion measurements for the m. rectus femoris. Thorpe et al. (Thorpe et al., 1999) indicate that the moment arm function for rectus femoris – rather than their quadriceps muscle function – provides the best characterization of the mm. vasti (m. vastus lateralis, m. vastus intermedius and m. vastus medialis) and the m. rectus femoris as a group. The model hamstrings (m. semimembranosus, m. semitendinosus and m. biceps femoris, long head) are all similar in magnitude and direction to the moment arm functions in Thorpe et al. (Thorpe et al., 1999). The m. gracilis is similar in magnitude in the first half of knee flexion, but predicts a smaller moment arm in the second half of knee flexion than the tendon-excursion experiment indicates. Finally, the model m. gastrocnemius lateralis is almost identical in magnitude and direction to that in Thorpe et al. (Thorpe et al., 1999), while the model m. gastrocnemius medialis predicts a decrease in moment arm magnitude with increasing knee flexion.

At the ankle (Table 1, Fig. 3G–I), the model predictions for the m. gastrocnemius medialis, m. gastrocnemius lateralis and m. soleus are similar in both magnitude and direction to the tendon-excursion measurements of Holowka and O'Neill (Holowka and O'Neill, in press). Thorpe et al. (Thorpe et al., 1999) provide a single linear function for these three muscles as a group, and their moment arm function is smaller in plantar flexion, but slightly larger in maximum dorsiflexion than the muscle-specific measurements of Holowka and O'Neill (Holowka and O'Neill, in press). The model predictions for the deep plantar flexors (m. flexor digitorum tibilais, m. flexor digitorum fibularis, m. tibialis posterior, m. peroneus longus and m. peroneus brevis) and dorsiflexors (m. extensor digitorum longus, m. extensor hallicus longus and m. tibialis anterior) are all quite close in direction and magnitude to the measurements of Holowka and O'Neill (Holowka and O'Neill, in press).

On average, the active hip moments are largest in the extensors and adductors, with the internal and external rotators having similar strength (Fig. 4A–C). In contrast, the passive moments generated by the hip muscle–tendon units are quite small throughout abduction–adduction and internal–external rotation, but increase from 5 to 40 N m in the hip extensors when the hip is positioned in 80 to 120 deg of flexion. At the knee, the active moment was approximately two times larger in the extensors than in the flexors, with the passive moment increasing in the quadriceps muscle–tendon units to a maximum of 4 N m at −140 deg of flexion (Fig. 4D). At the ankle, the active moment was more than six times larger in plantar flexion than dorsiflexion. The passive moment in the plantar flexors increased from 1 to 14 N m through the full range of dorsiflexion (Fig. 4E).

The effects of changing muscle path parameters on muscle moment arms varied considerably, both among muscles as well as among pathway parameters (Fig. 5). For the m. gluteus minimus (middle part), neither the flexion–extension nor the abduction moment arms were affected by varying the location of the origin. In contrast, varying the location of the insertion did alter the moment arms, especially with the hip joint flexed and abducted (Fig. 5A,B). The effects of varying the insertion location on the abduction moment arm were relatively straightforward, with a caudo-distal displacement increasing the moment arm and a cranio-medial displacement reducing the moment arm (Fig. 5B). The effects of varying the insertion location on the flexion–extension moment arm were more complex, depending heavily on the angle of hip flexion (Fig. 5A). The iliac wrapping function had almost no effect on the flexion–extension moment arm (Fig. 5A), but did alter the abduction moment arm if the hip was abducted more than 20 deg (Fig. 5B).

Changing the location of the via point between the m. vastus intermedius origin and the patellar attachment had little effect on the knee extension moment arm (Fig. 5C). The m. vastus intermedius engages a wrapping cylinder in the femoral condyles at ~80 deg of knee flexion. With the knee flexed more than 80 deg, the moment arm was affected by changes in the wrapping cylinder, but not by altering the kinematic function describing patellar tracking along the femoral condyles (Fig. 5C). With the knee extended past 80 deg, the opposite was true. The same was generally true for all of the knee extensor muscles in the model.

Varying the location of the m. soleus origin had almost no effect on the plantar flexion moment arm (Fig. 5D). In contrast, moving the insertion had a relatively large effect on the plantar flexion moment arm, especially with the ankle plantar flexed around 10–30 deg. Predictably, moving the insertion posteriorly increased the magnitude of the moment arm, while moving the insertion anteriorly decreased the moment arm. The parameters describing the ankle joint axis of rotation also affected the m. soleus moment arm. When the anatomically realistic axis was replaced with a pure mediolateral axis, as would be the case in a two-dimensional model, the magnitude of the m. soleus moment arm was reduced (Fig. 5D).

The effects of varying muscle model parameters on the predicted isometric joint moments varied among muscles, and among muscle model parameters (Fig. 6). Peak isometric force had a direct, linear effect on the predicted joint moment (Fig. 6A). The effects of varying optimal fiber lengths were similar in magnitude to peak isometric force on average, but could be larger (e.g. m. vastus intermedius) or smaller (e.g. m. sartorius) in individual muscles (Fig. 6B). The effects of tendon slack length varied considerably across muscles, ranging from relatively small (e.g. m. biceps femoris, short head) to relatively large (e.g. m. rectus femoris), depending on the muscle in question (Fig. 6C). Pennation angle was the only muscle model parameter that had a uniformly small effect on the predicted joint moments (Fig. 6D).

The wide range of sensitivities to perturbing tendon slack lengths can be understood in part by considering the ratio of tendon slack length to optimal muscle fiber length (Table 2). For some muscles (e.g. m. tibialis posterior), tendon slack length is several times longer than optimal muscle fiber length, while the ratio for other muscles [e.g. m. adductor magnus (proximal part)] is well below 1.0, and in a few muscles (e.g. m. pectineus) the ratio is close to zero (Table 2). Muscles with a large ratio of tendon length to muscle fiber length, such as the m. tibialis posterior, tended to be more sensitive to changes in tendon slack length than muscles with a low ratio, such as the m. adductor magnus (proximal part) (Fig. 7). While these sensitivity measurements made at a single joint angle are useful for demonstrating the influence of muscle architecture on model performance, it is important to note that the degree of sensitivity actually may vary over the joint range of motion (cf. m. rectus femoris in Fig. 6C).

Also shown in Fig. 7 are the results obtained for the same muscles in the human musculoskeletal model of Arnold et al. (Arnold et al., 2010). For most hind limb muscles (all but eight), the ratio of tendon slack length to muscle fiber length was greater in the human model than the chimpanzee model. Consequently, most of the muscles in the human model were more sensitive to changing tendon slack length. There were exceptions, such as the m. psoas (Fig. 7), where the tendon slack length to muscle fiber length ratio was greater in the chimpanzee model. However, in all of the cases where the ratio was greater in the chimpanzee model, those muscles had relatively low sensitivity to tendon slack length.

The effects of changing one joint angle on the muscle moment arm about another joint axis differed among the hip muscles (Fig. 8). Varying the hip internal–external rotation angle had relatively little effect on the m. pectineus flexion–extension moment arm (Fig. 8B). In contrast, varying the hip adduction angle resulted in large changes in the m. gluteus maximus proprius (caudal part) extension moment arm (Fig. 8A). For the m. adductor longus, the flexion–extension moment arm switched sign approximately halfway through the adduction range of motion (Fig. 8C). However, the effect of changing the hip adduction angle was similar to that in the m. gluteus maximus proprius (caudal part), in that abducting the hip tended to decrease the magnitude of the m. adductor longus flexion–extension moment arm, regardless of the sign of the moment arm. There were also cases where changing one joint angle caused the moment arm about another joint axis to change sign. For example, externally rotating the hip, with the hip near full extension, caused the m. gluteus minimus (middle part) to change from a flexor to an extensor (Fig. 8D).

The manner in which PCSA varies with mass and fiber length in the chimpanzee model was almost identical to that in the human model (Fig. 9A,B). That is, PCSA varies directly with muscle mass, with nearly the same slope for both species. The primary difference between species was the presence of two outliers in the human model, the m. gluteus maximus and the m. soleus (Fig. 9B). The m. gluteus maximus had a relatively small PCSA for its mass, while the m. soleus had an especially large PCSA for its mass. PCSA does not vary directly with fiber length for either species (Fig. 9C,D). In muscles with short fibers, PCSA is relatively independent of fiber length, while muscles with long fibers always have small PCSAs. The main difference between species was the presence of more extreme architectures in humans (Fig. 9D), including cases of both large PCSA with short fibers (m. soleus) and small PCSA with long fibers (m. sartorius).

Herein, a 3D musculoskeletal model of the chimpanzee pelvis and hind limb was developed that provides robust predictions of muscle–tendon unit moment arms over a wide range of joint motions (Fig. 1). The model is parameterized in large part based on cadaveric studies of the chimpanzee pelvis and hind limb musculoskeletal structure, allowing prediction of muscle–tendon unit length and force- and moment-generating capacities at the hip, knee and ankle.

The model moment arms corresponded well to the tendon-excursion measurements for the hip, knee and ankle muscles, suggesting that the model provides robust predictions of muscle–tendon unit moment arms over a wide range of joint positions (Fig. 3). Of course, perfect correspondence was not anticipated because some variation in muscle moment arm magnitude and pattern is to be expected within a species, and our chimpanzee model is based on a different individual than the ones used in Thorpe et al. (Thorpe et al., 1999), Payne et al. (Payne et al., 2006) and Holowka and O'Neill (Holowka and O'Neill, in press). The primary discrepancy we observed was the presence of higher-order phenomena in most of the model predictions. The absence of these nonlinear trends in the Thorpe et al. (Thorpe et al., 1999) data set is surprising, because moment arms are expected to vary as trigonometric functions of joint angle for many muscles. It is possible that the measurement technique used in Thorpe et al. (Thorpe et al., 1999), which included moment arm calculations at 10 deg intervals, may account for these patterns. Given this, moment arm studies that include instantaneous, simultaneous measurements of 3D kinematics and muscle–tendon displacements (cf. Delp et al., 1999; McCullough et al., 2011) are needed for chimpanzees. This is of particular importance for muscle–tendon units that are sensitive to joint positions outside the plane of interest, such as the musculature crossing the hip (cf. Fig. 8).

The muscle architecture data set used to parameterize the muscle–tendon units in our model reflects the best available data; however, at present these data are still rather limited. This is due to a range of factors, including the fact that the total number of chimpanzees studied to date is still quite small (Carlson, 2006; Holowka and O'Neill, in press; Myatt et al., 2011; Nagano, 2001; Thorpe et al., 1999) and that measurement techniques for determining optimal fascicle length based on the length of in-series sacromeres have not yet been applied (Lieber et al., 1994; Ward et al., 2009). Because of our assumption that the optimal fiber lengths in the model were equal to empirical fascicle length measurements, scaled to account for body size, some muscles may generate force over a different range of joint angles than predicted here. Clearly, more detailed studies of chimpanzee muscle are needed. Such data would have the potential to improve both active and passive muscle force and moment estimation, as well as provide additional insight into how muscles operate in vivo with respect to their force–length and force–velocity properties.

The ratios of peak strength predicted by the chimpanzee model for the hip and knee joints were generally similar to data from human studies (Anderson et al., 2007). The hip extension-to-flexion strength ratio in the model (1.9) was slightly greater than in humans (1.4), while the knee extension-to-flexion strength ratio in the model (1.6) was slightly lower than in humans (1.9). However, the results for the ankle joint differed considerably between species. The peak ankle plantar flexion moment predicted by the chimpanzee model was 6.6 times greater than the peak ankle dorsiflexion moment. In humans, the plantar flexors are also stronger than the dorsiflexors, but only by 3.0 to 3.5 times (Anderson et al., 2007; Marsh et al., 1981; Sale et al., 1982). The results for the chimpanzee model were because of both relatively weak dorsiflexors and relatively strong plantar flexors (based on PCSA) when compared with data from humans (Ward et al., 2009). Nevertheless, the total maximum isometric joint moments predicted at the hip, knee and ankle all exceed inverse dynamic estimates of the peak joint moments in chimpanzee bipedal and quadrupedal walking (Yamazaki, 1985; Thorpe et al., 2004; see also Sockol et al., 2007), suggesting that the chimpanzee model is sufficiently strong to simulate locomotion. Further, the plateaus of the hip, knee and ankle extension strength curves (Fig. 4) appear to correspond with mid-stance joint angles employed during locomotion in the pygmy chimpanzee, Pan paniscus (D'Août et al., 2002). Corresponding kinematic data for P. troglodytes are not available, at present.

A general finding from the systematic variation of muscle–tendon path parameters was that perturbing the locations of proximal muscle–tendon attachment points (origins and proximal via points) had relatively minor effects on muscle moment arms (Fig. 5). In contrast, muscle moment arms tended to be more sensitive to the locations of muscle–tendon insertion points. These results were due to the insertion points generally lying closer to the axes of rotations than the more proximal points. Therefore, moving an insertion point by a given amount tends to displace the muscle line-of-action more relative to the joint center than does moving an origin or proximal via point by the same amount. This is fortunate from a model development standpoint, as there is often more uncertainty in muscle origin locations than there is for insertions. This is especially true for muscles that have broad proximal attachments but terminate distally via a more focused tendon (e.g. m. gluteus medius and m. gluteus minimus). The greater sensitivity to insertion positions is closely related to the notion that peak muscle moment arms are correlated with the shorter distance between the joint axis and the origin or insertion (Murray et al., 2002). For muscles in the chimpanzee hind limb, the shorter distance will usually be to the insertion, thus the greater sensitivity to insertion placement is to be expected. The main exceptions will be in certain biarticular muscles. For example, the m. gastrocnemius lateralis moment arm at the knee joint will be more sensitive to uncertainty in the origin, rather than the insertion. Fortunately, the major biarticular muscles tend to have well-defined areas of attachment.

It is possible, in some circumstances, to precisely digitize the outline of the area of muscle attachment during a dissection, and then calculate the centroid of the area. Such an approach may be especially useful for muscles with broad proximal attachments. However, the effort involved in such a procedure may be unnecessary in many cases. This is for two reasons. First, our results show that moment arms are not especially sensitive to the location of the point chosen for the origin. Second, the proper origin point for muscles with broad attachments is not actually a fixed point at the centroid of the attachment area, but rather the center of force application, which varies with motor unit recruitment in vivo (Monti et al., 2001).

Consistent with previous studies, model force- and moment-generating capacities were most sensitive to uncertainty in tendon slack lengths (Delp et al., 1990; Out et al., 1996; Redl et al., 2007), although the magnitude of the effect varied across muscles (Figs 6, 7). The different sensitivities were due in large part to differences in the ratio of tendon length to muscle fiber length (Table 2), though model sensitivity may be influenced by other geometric factors, such as the ratio of tendon length or muscle fiber length to moment arm (Delp et al., 1990). Unlike the maximum isometric force, optimal fiber length and pennation angle parameters in the muscle model, which may be based on muscle architecture measurements, tendon slack length does not have an easily measurable analog, as it includes both the external and internal portions of the tendon, as well as other sources of series elasticity. The numerical optimization approach used in the present study (from Manal and Buchanan, 2004) provided a systematic and uniform approach for determining tendon slack lengths that takes into account the operating range and optimal fiber length for each individual muscle. The sensitivities of optimal fiber length and tendon slack length are not completely independent of each other, as the muscle fiber and tendon lie in series, and together with pennation angle determine muscle–tendon length (Fig. 2). The tendon slack length optimization approach has the added effect of partially compensating for uncertainty in optimal fiber length. In Fig. 6B, optimal fiber lengths were perturbed without adjusting tendon slack length. However, we found that if the tendon slack length optimization was re-run after perturbing optimal fiber lengths, the effects on the predicted joint moment were reduced considerably (up to approximately 50%).

The ratios of tendon slack length to muscle fiber length in the chimpanzee model were generally smaller than in the human musculoskeletal model of Arnold et al. (Arnold et al., 2010). In addition to the chimpanzee model being less sensitive to uncertainty in tendon slack length, the smaller tendon to muscle fiber length ratios in the chimpanzee model may also have functional implications. Muscles with long tendons and short fibers are suitable for storage and release of elastic strain energy (Biewener and Roberts, 2000). The smaller tendon length to muscle fiber length ratios predicted in the chimpanzee model implies that chimpanzees have less potential for elastic energy utilization than humans. This may be an important contributing factor to the higher cost of transport in chimpanzees (Sockol et al., 2007), as storage and release of passive elastic energy functions as a metabolic cost-saving mechanism in terrestrial locomotion (e.g. Fukunaga et al., 2001; Umberger, 2010).

The systematic variation of hip joint angles about all three axes revealed that for some muscles, their function with respect to a particular joint axis depended heavily on the angle about one or both of the other hip joint axes (Fig. 8). This was true, for example, in the m. gluteus maximus proprius (caudal part), in that the hip extension moment arm could be varied more than fourfold by abducting the hip. Other muscles, such as the m. pectineus (Fig. 8), were far less sensitive to changes in the other hip joint angles. For both simplicity and consistency, the flexion–extension moment arms reported for the model in Fig. 3 were generated with hip adduction and rotation both set to the neutral position (0 deg). It should be kept in mind then, that for some muscle–tendon units, the degree of agreement with the experimental data could be improved, and for other muscle–tendon units worsened, by manipulating the hip adduction or rotation angles.

Our model provides accurate predictions of muscle moment arms and muscle–tendon unit lengths within the joint ranges of motion established for the model. However, outside of these ranges, its accuracy is undefined and may be reduced. Measurements of chimpanzee hind limb kinematics during walking, climbing and jumping (e.g. Isler, 2005; Scholz et al., 2006; Sockol et al., 2007) indicate that the joint ranges of motion in the model subsume several important muscle-powered movements. As such, the current limits on the joint ranges of motion should be adequate for most locomotor applications.

At present, it is not possible to validate the model predictions for abduction–adduction and internal–external rotation at the hip, as the available data are limited to flexion–extension moment arms. Although the magnitude and direction of the cadaveric moment arms at the hip fit well with model predictions for most muscles, it is clear that for some muscles (e.g. gluteal and adductor muscles) the 3D orientations of the pelvis and thigh segments during tendon-excursion experiments can have a significant effect on moment arm metrics. This may account for the discrepancies between the experimental measurements and model predictions for some muscles, such as the m. adductor longus; however, the degree of abduction–adduction and internal–external rotation of the cadaveric specimen of Thorpe et al. (Thorpe et al., 1999) and Payne et al. (Payne et al., 2006) are unknown.

In the chimpanzee model, the muscles with the largest PCSAs were the m. gluteus maximus, m. tibialis posterior, m. soleus and m. vastus lateralis. For humans, the muscles with the largest PCSAs were the m. soleus, m. vastus lateralis, m. gluteus medius and m. gluteus maximus. This substantial overlap reflects the need in both species to generate extensor moments at the hip, knee and ankle to resist gravitational and inertial forces. The muscles with the greatest absolute PCSA in each species, the m. gluteus maximus in chimpanzees and the m. soleus in humans, correspond well with the functional demands of locomotion. Hip extension is the largest moment generated by chimpanzees in quadrupedal locomotion, while ankle plantar flexion is the largest moment generated in human walking (Sockol et al., 2007). The large PCSA for the m. tibialis posterior in chimpanzees may reflect its dual role as a plantar flexor and inverter, which is likely important in arboreal locomotion. The muscles with the longest fibers, the m. sartorius, m. gracilis and m. semitendinosus, were the same in chimpanzees and humans. These muscles are all knee flexors, while also functioning at the hip joint, which necessitates long fibers to accommodate large excursions (Ward et al., 2009).

The manner in which PCSA varies with muscle mass and fiber length in the chimpanzee model was similar to a comparable human model (Arnold et al., 2010). One of the major difference between species is the presence of an extreme muscle architecture in the m. soleus in humans, which favors high PCSA, and hence high force production. This relatively high PCSA is achieved through a moderately large mass and especially short fibers (Ward et al., 2009), as well as a medial expansion onto the tibia. In chimpanzees, the m. soleus has a robust PCSA, but not out of proportion with other lower limb muscles. The differences in m. soleus architecture between humans and chimpanzees seem to reflect the demands placed on this key muscle in different forms of locomotion. Human bipedal locomotion requires substantial force production from the ankle plantar flexors (Komi et al., 1992). However, these high forces are typically exerted through a limited range of motion, which allows muscle fibers to be short. In contrast, locomotion in chimpanzees includes an arboreal component that demands considerable joint excursion, and therefore longer muscle fibers. This effectively limits muscle force capacity unless muscle mass is increased substantially. The architecture exhibited by the m. soleus in humans appears to be an adaptation for meeting the demands of our unique form of bipedalism. While taken to the extreme in the case of the m. soleus, the emphasis on joint excursion in chimpanzees and force production in humans is part of a more general trend noted by Thorpe et al. (Thorpe et al., 1999) and others (Payne et al., 2006; Holowka and O'Neill, in press).

Musculoskeletal models are most powerful when integrated with a rich set of experimental data. Although numerous experimental studies over the past 75 years have investigated chimpanzee locomotion, there is still a great deal to be learned about the basic function of their bones, muscles and tendons during movement.

The 3D musculoskeletal model of the chimpanzee pelvis and hind limb presented here represents an important step in integrating experimental data on the mechanics, energetics and control of locomotor performance. An essential element of the model is that it permits detailed parameterization of the skeletal, muscle–tendon and neural systems. While much simpler models, such as pendulums, mass springs and colliding masses, have provided valuable insights into the fundamental dynamics of various locomotor tasks (e.g. Cavagna et al., 1977; Ruina et al., 2005), more complex models such as ours are needed to investigate the functional roles of specific details of bone and muscle (Delp et al., 1990; Neptune et al., 2009; Pandy, 2003; Umberger and Rubenson, 2011). Indeed, from the standpoint of trait evolution, detailed musculoskeletal models can be of considerable heuristic value, as it is almost impossible to assess the contribution of subtle changes in bone size and shape, muscle–tendon origins and insertions, or muscle architectural parameters on the global performance of the hind limb when that trait is studied in isolation. The extent to which the musculoskeletal morphologies of chimpanzees and humans reflect differences in locomotor capabilities is essential information for comparative studies of hominin fossil remains (e.g. Aiello and Dean, 1990; Lovejoy et al., 2009; Richmond and Jungers, 2008; Robinson, 1972; Stern and Susman, 1983; Zipfel et al., 2011). The use of detailed musculoskeletal models in combination with forward dynamics simulation and numerical optimization procedures provides a reproducible, mechanics-based approach for gaining insights into the relative importance of these different variables for a given muscle-driven task.

The extensible nature of our model allows further development of the underlying musculoskeletal structure (e.g. the addition of trunk, upper limb and head anatomy; intrinsic foot musculature) as well as implementation of inverse or forward dynamics analyses, based on experimental data, numerical optimization or some combination thereof. In this regard, validation of optimization criteria for predicting movement tasks in humans and chimpanzees is crucial if these same criteria are applied to fossil hominins or other species for performance-based prediction when experimental data are unavailable (e.g. Nagano et al., 2005; Sellers et al., 2005). As such, the model can also operate as a hypothesis generator for direct experimentation, and allow estimation of parameter values that are simply not feasible to measure directly.

Thanks to E. Delson and the staff at the American Museum of Natural History for support and assistance in arranging the loan of the chimpanzee skeleton used to build our model. Thanks also to K. Baab, D. Boyer, J. Groenke, B. Patel, J. Sertich and R. Susman for help with CT scanning and digital reconstruction.