Metabolic cost of generating muscular force in human walking: insights from load-carrying and speed experiments

Timothy M. Griffin, Thomas J. Roberts, Rodger Kram

Abstract

We sought to understand how leg muscle function determines the metabolic cost of walking. We first indirectly assessed the metabolic cost of swinging the legs and then examined the cost of generating muscular force during the stance phase. Four men and four women walked at 0.5, 1.0, 1.5, and 2.0 m/s carrying loads equal to 0, 10, 20, and 30% body mass positioned symmetrically about the waist. The net metabolic rate increased in nearly direct proportion to the external mechanical power during moderate-speed (0.5–1.5 m/s) load carrying, suggesting that the cost of swinging the legs is relatively small. The active muscle volume required to generate force on the ground and the rate of generating this force accounted for >85% of the increase in net metabolic rate across moderate speeds and most loading conditions. Although these factors explained less of the increase in metabolic rate between 1.5 and 2.0 m/s (∼50%), the cost of generating force per unit volume of active muscle [i.e., the cost coefficient (k)] was similar across all conditions [k = 0.11 ± 0.03 (SD) J/cm3]. These data indicate that, regardless of the work muscles do, the metabolic cost of walking can be largely explained by the cost of generating muscular force during the stance phase.

  • locomotion
  • oxygen consumption
  • muscle
  • efficiency
  • gait

the metabolic demand of walking is set by muscles that act to perform work on the center of mass, swing the legs relative to the center of mass, and support body weight. Most studies that relate the biomechanics and metabolic cost of walking compare the body's mechanical and metabolic energies (13, 15, 20, 31, 38, 48, 50, 66). Cavagna and Kaneko (13) were among the first to separately calculate the work required to lift and accelerate the center of mass (i.e., external work) and the work required to swing the limbs relative to the center of mass (i.e., internal work). One important discovery from these studies was that much of the work of walking is done not by active muscle contraction but by a passive mechanism of exchange between kinetic energy and gravitational potential energy. This exchange occurs in an inverted pendulum-like way as the center of mass vaults up and over the stiff support leg (14), thereby reducing the muscular work required to lift and accelerate the center of mass. Similarly, the muscular work required to swing the legs is reduced by a pendulum-like exchange of potential and kinetic energy as the leg swings forward (42, 43). When these passive mechanisms were accounted for, it appeared that performing external and internal work each accounted for about half of the metabolic energy used to walk at ∼1.0 m/s (13, 63).

There are several potential problems with these studies. First, they likely underestimated external work by about one-third, because they did not account for the work performed by the individual legs during double support (19). When the legs transition from one step to the next, the trailing and leading legs simultaneously perform positive and negative external work, respectively, to redirect and restore the center-of-mass velocity. A second limitation of these studies is that they did not consider nonwork costs, such as isometric muscle activity for stabilization, which may contribute significantly to the metabolic cost of walking during the stance (24) and swing (36) phases. Third, evidence from other studies suggests that stance leg muscle actions, that is, performing work to redirect and restore the center-of-mass velocity (18) and generating force to support body weight (27), dominate the net metabolic cost of walking. Electromyographic (EMG) recordings indicate that leg muscles are primarily active during the stance phase. EMG activity of the swing leg is nearly absent for the entire swing phase, except at the very beginning and end (6). The net muscle moments at the ankle, knee, and hip joints are also much smaller during the swing phase than during the stance phase, suggesting that muscle forces are greatest during the stance phase (54).

The first goal of this study was to determine how swinging the legs contributes to the cost of walking. The muscle actions during walking can be divided into stance and swing leg actions. Independent manipulation of exclusively stance leg actions can indirectly provide insight into the cost of swinging the legs (57). To accomplish this, we investigated how carrying loads placed symmetrically about the waist affected the mechanics and energetics of walking. At a given speed, stride frequency and leg swing time are nearly constant when humans carry loads on their backs (37, 39). Thus the muscular work or force and, presumably, the metabolic cost required to swing the legs relative to the center of mass do not appreciably change during load carrying. The increased metabolic cost of carrying loads should therefore reflect only an increase in the stance leg costs. If the metabolic cost of swinging the legs is negligible, we would expect the net metabolic rate to increase in direct proportion to the external work rate. In this case, the ratio of external work rate to net metabolic rate, defined here as the net locomotor efficiency, should remain constant with increasing loads.

Although muscles perform various mechanical functions during a step, the metabolic cost of walking may be simply explained by the cost of generating muscular force. Muscles perform a variety of functions by acting as motors, tensile struts, and brakes in different leg muscles and at different periods during the stride in the same leg muscles (9, 17, 24, 25, 29). Thus the energy used by muscles during the stance phase is a mix of energy used to perform work and to generate force isometrically. Despite this complexity, the metabolic cost of walking should be proportional to the magnitude and rate of generating force if muscles operate with consistent relative shortening velocities (v/vmax, where vmax is the theoretical maximal shortening velocity under an unloaded condition) and efficiencies across a range of walking speeds. Muscles require less metabolic energy to generate force when they are active isometrically than when they shorten and perform work (21), but the cost of generating force while performing work should be a constant multiple of the energy used to generate force isometrically (1, see Ref. 3 for a discussion of optimum muscle-tendon function).

The second goal of this study was to examine whether the metabolic cost of walking reflects the cost of generating muscular force. To do so, we had humans walk across a range of speeds with and without loads. The metabolic cost of generating force is proportional to the volume of active muscle and the rate of generating force (16). We combined kinetic, kinematic, and anatomic data to estimate the volume of muscle active to generate force during the stance phase. The rate of generating force was measured as the inverse of the time available to generate force on the ground. We calculated a “cost coefficient,” which describes the energy used by a unit volume of active muscle. This coefficient (k) should be constant across speed and load if the cost of generating muscular force completely explains the increase in net metabolic rate.

We studied the mechanics and metabolic cost of humans carrying loads of up to 30% of their body mass while walking across a fourfold speed range. We tested two hypotheses: 1) the cost of swinging the legs is small, and 2) the metabolic cost of walking is directly proportional to the volume of muscle that is active to generate force against the ground and the rate of generating this force.

METHODS

Experimental design. We collected metabolic, kinematic, and kinetic data at four loading conditions (0, 10, 20, and 30% of body mass) and four walking speeds (0.5, 1.0, 1.5, and 2.0 m/s). These 16 speed-load trials were conducted in a randomized order. Subjects walked on a motorized treadmill (metabolic data) and over a walkway (kinematic and kinetic data).

Subjects. Eight healthy volunteers (4 men and 4 women; mean ± SD: body mass = 68.7 ± 12.5 kg, leg length = 0.91 ± 0.06 m, age = 26 ± 5 yr) participated in the study after providing their informed consent, as defined by the Committee for the Protection of Human Subjects at the University of California, Berkeley.

Load-carrying design. Subjects carried symmetrical loads: lead strips secured closely around a well-padded hip belt positioned near the body's center of mass. This arrangement minimized load movement and arm swing interference during walking (Fig. 1). By positioning loads symmetrically, we presumably reduced the muscular activity required to balance asymmetrically positioned loads, such as those carried in backpacks (10).

Fig. 1.

Ground-reaction force collection during load carrying. Lead strips were secured closely around a well-padded hip belt so that subjects carried loads symmetrically. During double support (i.e., when both feet are in contact with the ground), trailing and leading limb ground reaction forces (Ftrail and Flead, respectively) were measured simultaneously to calculate external mechanical power performed by each leg to redirect and restore center-of-mass velocity (19).

Metabolic rate measurements. We measured the rates of O2 consumption (V̇o 2) and CO2 production (V̇co 2) by using an open-circuit respirometry system (Physio-Dyne Instrument, Quogue, NY). We collected the metabolic data on two separate days to avoid fatigue effects. Each day began with a quiet standing trial. All trials lasted 8 min, and we calculated the average V̇o2 (ml O2/s) and V̇co2 (ml CO2/s) values during the last 3 min. The respiratory exchange ratios were < 1.0 for all subjects and conditions, indicating that energy was supplied primarily by oxidative metabolism in all test conditions. We calculated the gross metabolic rate for each trial by using the following standard equation (11) Math1 where Ėmetab,gross is gross metabolic rate, W is watts, s is seconds, and V̅̇̅O2 and V̅̇̅CO2 represent mean V̇o2 and V̇co2, respectively. In a previous study using the same loading belt, we found that metabolic rate while standing did not significantly change with loads of up to 50% of body mass (n = 5, P = 0.75, repeated-measures ANOVA) (28). Therefore, in the present study, we subtracted the metabolic rate during unloaded standing from all walking values to calculate the net metabolic rate (Ėmetab).

External mechanical power measurements. We measured the rate at which the individual legs performed external work by using two force platforms (Fig. 1). The platforms (model LG6-4-2000, AMTI, Newton, MA) were mounted in series and near the midpoint of a 17-m walkway. We collected the vertical (Fz), fore-aft (Fy), and mediolateral (Fx) components of the ground reaction force from both platforms at 1 kHz. We then filtered the force data with a fourth-order recursive, zero phase-shift, Butterworth low-pass filter (100-Hz cutoff). Average walking speeds were measured by using two infrared photocells, placed on either side of the force platforms (3.0 m apart). If the speed was not within 0.05 m/s of the desired speed or if the individual feet did not fall cleanly on separate force platforms, we discarded the trial. We saved and analyzed data for three acceptable trials for each subject at each speed-load condition. All calculations for each trial were performed for a step, which we defined as beginning with ground contact of one foot and ending with ground contact of the opposite foot.

We calculated the external mechanical power by using the individual-leg method as described in detail by Donelan et al. (19). Individual-leg power (ẆILM) is equal to the dot product of two vectors: the external force acting on the leg (Mathleg) and the velocity of the body's center of mass (Mathcom) Math2

We determined the center-of-mass velocities from the ground reaction forces and average forward speed measurements (12). We first calculated the accelerations of the center of mass in the vertical, fore-aft, and mediolateral directions from the respective ground reaction force component. The center-of-mass velocities were then calculated from the time integral of these center-of-mass accelerations. We determined the integration constant for vz by requiring the average vertical velocity over the step to be zero. The integration constant for vy was determined by requiring the average fore-aft velocity to be equal to the average walking speed. For vx, we determined the integration constant by requiring that the mediolateral velocities at the beginning and end of a step be equal in magnitude but opposite in sign (19).

We used the ground reaction force from each leg, together with the center-of-mass velocity, to calculate the individual-leg external work. During the step-to-step transition (i.e., during double support), the trailing and leading legs perform work to redirect and restore the center-of-mass velocity. Therefore, during double support, we calculated the power generated by the trailing and leading legs separately. The individual-leg external work is equal to the cumulative time integral of the leg mechanical power. We calculated the positive leg work separately for the trailing and leading legs during double support and for the stance leg during single support by restricting the integration of the positive leg power to the appropriate time intervals. The total positive leg external work (WILM) was calculated by summing the positive trailing and leading leg work during double support with the stance leg work during single support. We doubled this individual-leg work value to account for the work performed by both legs during a stride. We then divided this value by the stride time to calculate the external mechanical power performed by the individual legs (ẆILM).

We calculated the net locomotor efficiency as ẆILMmetab to facilitate comparing the external mechanical power and net metabolic rate. Had we included the negative external power in this calculation, the net locomotor efficiency would have been slightly greater, but it would not have affected our conclusion, because the negative external power was always approximately equal in magnitude but opposite in sign to ẆILM. The net locomotor efficiency is not equivalent to muscle efficiency, as discussed in detail elsewhere (60).

We used foot-ground contact data from force platform measurements to calculate stride frequency and duty factor. Stride frequency (Hz) was calculated as the inverse of stride time (tstride), and duty factor was calculated as the ratio of ground-contact time for a single foot (tc) to stride time: tc/tstride.

Active muscle volume measurements. The metabolic cost of generating muscular force is proportional to the volume of active muscle and the rate at which this volume uses energy (51). Accounting for the active muscle volume from both legs (2Vact,leg), we related the net metabolic cost of walking to the cost of generating ground force during the stance phase as follows Math3 where k is a cost coefficient (J/cm3). According to Eq. 3, k will be constant at different speed and load conditions if the net metabolic rate is directly proportional to the volume of active muscle in the legs and the rate of generating this force. The present calculation of this cost coefficient, k, is different from the cost coefficient, c, calculated previously by Kram and Taylor (35) and Roberts et al. (52). The present value accounts for muscle forces produced to generate vertical and fore-aft ground forces as well as any changes in the muscle mechanical advantage; it is the same as “k” presented in Roberts et al. (51) and “k1” described by Wright and Weyand (65).

We quantified the active volume of muscle to generate force during stance (Vact,leg) by multiplying the mean active fascicle length by the active cross-sectional area for composite ankle, knee, and hip extensor muscles. We calculated the active cross-sectional area by assuming equal muscle stress among extensors. Extensor muscle forces were determined by calculating the net extensor muscle moments at each of the joints, and we used anatomic measurements from cadaver leg muscles to calculate composite muscle dimensions. These methods are similar to those that have been described in detail by Roberts et al. (51).

Anatomic measurements were taken from five cadaver legs. Muscle fascicle length, pennation angle, moment arm, and mass were measured for muscles that act primarily as extensors of the ankle, knee, and hip joints (Table 1). These measurements were made by using methods described by Alexander (2) and Roberts et al. (51). The mean muscle fascicle length and moment arm at each joint were calculated, because the relative contribution of individual muscles to the net muscle moment at a joint cannot be determined from external measurements. These mean composite fascicle lengths () and moment arms () were weighted by each muscle's physiological cross-sectional area (Am) for that joint (7, 51). Thus these mean values were weighted according to each muscle's capacity to generate force. The mean values describe a composite muscle at each joint with a characteristic mean fascicle length and muscle moment arm. To account for differences in body size among the subjects, we divided the composite fascicle length and moment arm values by the respective cadaver's leg length (Lleg; Table 1). We then multiplied these mean normalized values by each subject's leg length for all calculations of active muscle force and volume.

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Table 1.

Normalized fascicle lengths and moment arms for composite joint extensor muscles

We calculated the net ankle, knee, and hip muscle moments in the sagittal plane by using an inverse dynamics solution (20). High-speed (200 Hz) video (JC Labs, Mountain View, CA) was recorded as we collected ground reaction force data. Retroreflective markers were placed on the fifth metatarsophalangeal joint, lateral malleolus, lateral epicondyle of the femur, greater trochanter, and acromion process. Force platform and kinematic data were synchronized by using a simple circuit that simultaneously lit a light-emitting diode and sent a voltage signal to the computer analog-to-digital converter. We used automatic point-tracking software (Peak Motus 2000) to digitize the movements of the markers during each trial. The marker position data were conditioned by using quintic spline processing (Peak Motus 2000). Marker positions were then used to calculate linear velocities and accelerations of the foot, shank, and thigh segments as well as joint angles, segment angles, and segment angular accelerations. We calculated segment masses, center-of-mass locations, and moments of inertia from anthropometric measurements (68). Using a rigid linked-segment model, we calculated the net muscle moments at the ankle, knee, and hip joints by applying Newton-Euler equations of angular and translational motion to each segment, starting distally and moving proximally (20). A net muscle moment includes the moments produced by all the muscles, tendons, ligaments, and contact forces at the joint, although the moments produced by the muscles usually dominate within the normal range of motion (23, 64).

We calculated the extensor muscle force (Fm) from the net muscle moment (M) and the mean muscle moment arm (). The knee and hip net muscle moments included a flexion (flex) moment contributed by two-joint muscles that extended one joint but flexed another. We calculated the force from the two-joint muscles [gastrocnemius (gastroc), hamstrings, and rectus femoris (rect.fem)] by assuming that force was distributed equally (by physiological cross-sectional area) across the joint extensor muscles. Therefore, the net muscle moments equal Math4 Math5 Math6 where ank is ankle. The only unknown quantities in these three equations are Fm,ank, Fm,knee, and Fm,hip. We calculated Fm,ank directly from Eq. 4. Equations 5 and 6 contain two unknowns, Fm,knee and Fm,hip, so we solved them simultaneously. The term ankle extension is equivalent to ankle plantarflexion.

We used a single value of for each composite muscle moment arm during stance. Moment arms can vary across ranges of joint motion, particularly for knee and hip extensors (44, 56). For example, the muscle moment arm for hip extensors changes by 10–20% from 0° to 45° of flexion. This potential change should have minimal influence on estimates of relative muscle volume, because the angular excursion of the joints only differs by ∼10° between 0.5 and 2.0 m/s (46) and is unaffected by load carrying (58).

We assumed that any flexor moments caused by one-joint flexor muscles did not appreciably affect our calculations of extensor muscle forces. This assumption is supported by our observation that the extensor force required to balance the flexor moments caused by two-joint muscles was <8% of the total leg extensor force. Furthermore, direct measurements of ankle extensor muscle forces correspond closely to those calculated from force-plate records for hopping kangaroo rats (8) and for humans performing submaximal plantarflexion within normal joint ranges (5).

We calculated the effective mechanical advantage (EMA) of the leg extensor muscles as the ratio of the extensor muscle moment arm to the moment arm of the resultant ground reaction force following Biewener (7). The mean EMA was measured over periods of the stride when the extensor muscle force was >25% of the maximum extensor muscle force. Because we occasionally obtained net extensor forces, even when the ground reaction force acted to extend a joint, we further constrained the EMA measurements to those periods of the stride when the ground reaction force acted to flex the joint.

We quantified the volume of active muscle on the basis of the muscle force generated during stance and the mean resting length () of the fascicles that produce this force. We integrated the extensor muscle force over the time period of support (∫Fm) and divided it by the integrated ground reaction force (∫Fg) (51). This ratio, ∫Fm/∫Fg, facilitates comparisons at different speed and load conditions, because over a stride the legs produce an average force on the ground equal to body weight (wb). Summing the values for each joint gives the total extensor muscle force produced by the leg to generate 1 N of ground force (∫Fleg/∫Fg). If all muscle fibers operate with the same stress, then ∫Fleg/∫Fg is proportional to the cross-sectional area of the active muscle fibers. We calculated the active fascicle length (Lact, in cm) of the extensor muscles by weighting the mean resting fascicle length at each joint according to how much force is produced at that joint (51). This gives an estimate of the mean length of the active muscle fascicles in the leg.

The metabolic cost of generating muscular force is proportional to the rate at which each unit volume of active muscle uses energy (51). We calculated the volume of active muscle per leg (in cm3) as follows Math7 where σ is the force per unit cross-sectional area of active muscle (N/cm2). We used 20 N/cm2 for σ (7). Alternative values for σ would alter the estimate of active muscle volume. However, as long as the active muscle stress does not vary substantially between conditions (speed or load), our estimate of the relative muscle volume required for each condition will be accurate. Although our approach requires a number of assumptions that limit our ability to estimate the absolute volume of active muscle, it provides a reasonable estimate of proportional changes in volume due to speed or load effects.

Statistical analyses. We used a single-factor repeated-measures ANOVA (P < 0.05) to determine statistical differences due to speed or load effects. If we found a statistically significant speed or load effect from the ANOVA, we performed Tukey's honestly significant difference (HSD) post hoc test to determine which speed or load condition produced the significant response (P < 0.05). Unless otherwise noted, all P values are for an ANOVA test.

We performed a post hoc power analysis of the cost coefficient data, and we had an 80% chance of detecting a 29.6% change in the cost coefficient (α = 0.05). Considering that the net metabolic rate changed by 344% across the speeds tested, we had a high probability of detecting differences that were much less than the potential change in the cost coefficient.

RESULTS

Effects of loading. Net metabolic rate increased as subjects carried heavier loads (Fig. 2A), and the percent increase in Ėmetab due to loading was similar for all speeds (see appendix). For example, when subjects carried loads equal to 30% of their body mass, Ėmetab increased 47 ± 17% (mean ± SD for all speeds) above their unloaded rate. At 0.5 and 1.0 m/s, the external work rate (ẆILM) increased in proportion to load, so that the net locomotor efficiency was unaffected by load (P = 0.99 and 0.85, respectively; Fig. 2C). At 2.0 m/s, efficiency decreased from 26.7 to 22.1% between the unloaded and 30% load conditions (P < 0.01). This trend was also evident at 1.5 m/s, although the efficiency values were not significantly different (P = 0.15). The net locomotor efficiency decreased with loading at the faster speeds, because loading had less effect on the rate that the legs performed external work. In the 30% load condition, ẆILM increased 50% at 0.5 m/s, 47% at 1.0 m/s, 36% at 1.5 m/s, and 24% at 2.0 m/s compared with the unloaded condition.

Fig. 2.

Loading increased net metabolic rate (Ėmetab, A) and external mechanical power (ẆILM, B) at each test speed. Metabolic and mechanical values increased in proportion so that net locomotor efficiency (ẆILMmetab, C) was not significantly affected by loading between 0.5 and 1.5 m/s. Efficiency decreased with loading at 2.0 m/s. Data are shown for carrying loads equal to 0, 10, 20, and 30% percent body mass (Mb). Values are means ± SE (n = 8). Lines are second-order polynomial curve fits (KaleidaGraph 3.0). Loading had a significant effect on Ėmetab,ẆILM, and net locomotor efficiency at each of the test speeds (P < 0.05), except where designated not significant (NS).

Loading does not appear to have affected the work involved in swinging the limbs relative to the center of mass. At any given speed, stride frequency did not change as subjects carried loads, and duty factor increased only slightly, i.e., <4%, between the unloaded and 30% load conditions (Table 2). Limb swing time decreased by 5.5 ± 1.1% (mean ± SD for all speeds) between the unloaded and 30% load conditions. Using a model-based equation (41), we estimated the effect of loading on the internal work rate. This equation combines each subject's average forward speed, stride frequency, and duty factor to estimate limb velocity, which is then combined with a dimensionless term accounting for limb geometry and fractional mass to calculate kinetic energy fluctuations. These data suggest that the internal work rate changed by <4.5% at a given speed when subjects carried loads.

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Table 2.

Kinematics of walking at a range of speeds and loads

Active muscle volume and joint extensor forces. Ankle extensor muscles accounted for over half of the muscle volume that was active to generate force during the stance phase (Fig. 3). The ankle extensors accounted for such a large fraction of the active muscle volume, because their EMA [0.28 ± 0.03 (SE)] was much smaller than the EMA for the knee (3.42 ± 1.26) and hip (2.49 ± 0.79) extensors. At 1.5 m/s, the ankle extensor force was 4.4 and 5.0 times larger than the knee and hip extensor forces, respectively (Fig. 4). The relative active muscle volumes in Fig. 3 were not directly proportional to the muscle forces in Fig. 4, because the average fascicle lengths () were shortest in the ankle extensors and longest in the hip extensors (Table 1). As subjects increased speed from 0.5 to 2.0 m/s, the muscle force required to generate a unit force on the ground by the ankle and hip extensors did not change (P = 0.26 and 0.67, respectively), while knee forces increased (P < 0.01; Fig. 4). The increase in active muscle volume between 0.5 and 1.5 m/s was therefore primarily caused by an increase in the volume of active knee extensor muscles.

Fig. 3.

Active volume of ankle, knee, and hip extensor muscles as percentage of total active volume required to generate force during the stance phase at a range of speeds. Ankle extensors accounted for over half of the active muscle required to generate force on the ground during stance. Values are means ± SE (n = 8) for unloaded walking.

Fig. 4.

Ankle, knee, and hip extensor muscle forces (Fm) required per unit ground reaction force (Fg) at a range of speeds. Data are calculated by dividing muscle impulse (∫Fm) during the stance phase by the resultant of the vertical and fore-aft ground reaction force impulse (∫Fg). Values are means ± SE (n = 8) for unloaded walking.

Cost of generating muscular force. The energy used by a unit volume of muscle to generate force during the stance phase (i.e., k) did not significantly change between 0.5 and 1.5 m/s [0.09 ± 0.02 (SD) J/cm3; data for unloaded walking; P > 0.05, Tukey's HSD post hoc test; Fig. 5D]. The cost coefficient did not change across these speeds, because the product of the active muscle volume (Fig. 5C) and the rate of activating this volume (Fig. 5B) increased by approximately the same amount as the net metabolic rate (Fig. 5A). The rate of generating force and the active muscle volume increased significantly between 0.5 and 1.5 m/s, although the rate increased much more than the volume (110 vs. 33%). The cost coefficient did increase significantly between 1.5 and 2.0 m/s (P < 0.05, Tukey's HSD post hoc test). Over this speed range, the rate of generating force increased, whereas the volume of active muscle did not.

Fig. 5.

Metabolic and muscular force data for unloaded walking across a range of speeds. A: net metabolic rate. B: rate of force generation, measured as the inverse of foot-ground contact time (1/tc). C: volume of active muscle per leg to generate force on the ground (Vact,leg). D: cost coefficient (k), which represents energy used by a unit volume of muscle that is active to generate force during the stance phase. Values are means ± SE (n = 8). Lines are 2nd-order polynomial curve fits (KaleidaGraph 3.0). *Significantly different (P < 0.05) from previous (slower) speed value as determined by Tukey's honestly significant difference post hoc test. Vact,leg increased significantly with speed, but only values at 1.5 and 2.0 m/s were significantly different from value at 0.5 m/s, so no asterisks are shown.

The metabolic cost of carrying loads was directly proportional to the active muscle volume and the rate of activating this volume at all but the slowest speed. This is indicated by the nearly constant cost coefficient at the different loading conditions when subjects walked at 1.0, 1.5, or 2.0 m/s (Table 3). At 0.5 m/s, only the cost coefficient for the 10% body mass load condition was greater than the unloaded walking value (P < 0.05, Tukey's HSD post hoc test). Loading did not significantly affect the rate of generating force on the ground (1/tc) at any of the speeds: carrying loads primarily affected the volume of active muscle required to generate force against the ground. When the unloaded and 30% load conditions were compared, Vact,leg increased 29% at 0.5 m/s, 40% at 1.0 m/s, 34% at 1.5 m/s, and 37% at 2.0 m/s.

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Table 3.

Cost coefficient k at a range of speeds and loads

DISCUSSION

Cost of swinging the legs. Is the metabolic cost of swinging the legs during walking substantial? Our loading experiments, which independently manipulated just the stance limb muscle actions, suggest that it is not. When subjects carry loads about the waist, the work or force required to swing the legs relative to the center of mass does not appreciably change. Although the time available to swing the leg may also influence leg swing cost (36), it seems likely that the cost of swinging the legs is nearly constant across loading. Over a range of moderate speeds (0.5–1.5 m/s), the net locomotor rate increases in proportion to the external mechanical power, and the net locomotor efficiency remains nearly constant with loading (Fig. 2). If the cost of swinging the legs was substantial (e.g., half of the net metabolic rate), then, with loading, the net metabolic rate would only increase half as much as the external mechanical power, and the net locomotor efficiency would increase.

Swinging the legs is likely to be metabolically inexpensive, because the motions are primarily passive and require little muscle activity. Researchers have long suggested that the legs swing forward passively, similar to a swinging pendulum (61). Experimental and theoretical studies generally support this idea (40, 42, 43), but some mechanical energy must be added to the leg during swing to attain the kinematics observed in walking humans (55, 62). This mechanical energy could be added directly from muscles performing mechanical work or from elastic energy storage and recovery in joint flexor muscle-tendon units. This latter possibility would serve as another means of reducing muscle activity costs during the swing phase. Kuo's recent study (36) of a simple passive dynamic walking model suggests that the muscle force generated per time to operate a hip spring, rather than the work performed to swing the leg, more accurately predicts the preferred speed-step length relation of walking humans.

If leg swing costs are high, net locomotor efficiency should increase with loading, because external mechanical work rate will increase more than metabolic rate. We can predict how leg swing costs would affect changes in net locomotor efficiency if we assume that 1) the cost of swinging the legs is unaffected by loading, 2) the remainder of the net metabolic rate is completely determined by the external work rate, and 3) external work is performed with a constant muscle efficiency. If leg swing cost were one-half, one-third, or one-fifth of the net unloaded metabolic rate, the net locomotor efficiency would increase by 12, 8, or 5%, respectively, between the unloaded and heaviest load conditions. We find that loading increases the net locomotor efficiency by 0.2% at 0.5 m/s and 5.7% at 1.0 m/s, suggesting that leg swing costs are at most ∼20% of the net metabolic rate. Estimating the cost of leg swing for the fastest speeds is not feasible, because the net locomotor efficiency decreases with loading.

At fast walking speeds, a more flexed leg posture with loading may dissociate the link between external mechanical power and the metabolic cost of load carrying. Loading has a similar effect on the net metabolic rate at all speeds (see appendix), but the net locomotor efficiency decreases with loading at 2.0 m/s, because the external mechanical power is less affected by load at this speed. At 2.0 m/s, the peak magnitude of the ground reaction force (relative to total weight) and the center-of-mass velocity fluctuations are smaller for the 30% load condition than for the unloaded condition. These results are associated with a more flexed leg posture with loading: the maximum and minimum knee angles during the first half of stance decrease by 5.7° and 6.7°, respectively, between the unloaded and 30% load conditions (P < 0.01, 1-tailed paired Student's t-test). One consequence of using a more flexed leg posture is that the knee extensor muscle force increases with loading. At this extreme speed and load condition (2.0 m/s and 30% body mass load), we find that the net metabolic rate increases in direct proportion to the active muscle volume required to generate force on the ground, rather than the external mechanical power. Net locomotor efficiency may decrease with loading at the fastest speeds if changes in limb posture disrupt the relation between external mechanical power and the metabolic cost of walking (67). In these cases, a force-based approach, which accounts for changes in active muscle volume, may provide a more accurate indicator of metabolic cost.

Muscle force generation during walking. Can the metabolic cost of walking be simply explained by the cost of generating muscular force during the stance phase? Muscles perform a variety of tasks during walking by operating as motors, tensile struts, and brakes (9, 17, 24, 25, 29). Work-based approaches to relating the mechanics and energetics of walking ignore the cost of isometric (i.e., strut) muscle activity. Yet, regardless of how muscles change length, they consume energy when they actively generate force. If v/vmax of muscles performing these different tasks remains nearly constant across a range of speeds, the cost of generating muscular force may explain the metabolic cost of walking.

Indeed, we find that across a threefold speed range (0.5–1.5 m/s) the net metabolic rate increases in direct proportion to the active muscle volume required to generate force on the ground and the rate of generating this force, as noted by a constant cost coefficient (k). The cost coefficient is also constant when subjects carry loads of up to 30% of their body mass at 1.0, 1.5, and 2.0 m/s. These data provide general support for our hypothesis that the metabolic cost of walking is directly proportional to the volume of muscle that is active to generate force against the ground and the rate of generating this force. Although we have to make a number of assumptions to estimate the factors that determine muscle force costs, such as the active muscle volume, our results demonstrate that it is possible to explain changes in the metabolic cost of walking from relatively simple measurements. One notable exception to this conclusion is the ∼40% increase in the cost coefficient between 1.5 and 2.0 m/s. This increase could potentially be explained by a relatively greater cost of swinging the limbs. An increase in the volume of muscle that is active to swing the limbs would increase the metabolic cost of walking above that predicted by the volume required to generate force on the ground alone. This possibility, however, seems unlikely, because the net metabolic rate during load carrying at 2.0 m/s increases in direct proportion to the cost of generating force on the ground.

Changes in the cost coefficient might indicate a change in the muscle activity patterns or relative shortening velocities. Muscles operating at faster v/vmax would require activation of a greater cross-sectional area and, thus, a greater volume to generate a given force. Our analysis assumes a constant v/vmax. Therefore, we would not be able to measure a change in active muscle volume if v/vmax actually does change.

Muscle activity assumptions. A critical assumption of our study is that v/vmax is not affected by walking speed or loading. This implies that the force produced per unit cross-sectional area of active muscle and the muscular efficiency remain approximately constant throughout our experimental conditions. In vivo measurements of muscle function in dogs and rats show that strain rate increases with walking speed in some muscles (25, 26). The ordered recruitment pattern of muscle fiber types (32) suggests that these higher strain rates are accomplished by faster muscle fibers, thereby maintaining a constant v/vmax (53), but this has yet to be demonstrated. The active cross-sectional area of muscle required to generate a given force appears to be similar across loading. For example, the cross-sectional area of muscle showing glycogen loss increases in direct proportion to the load carried by running rats (4).

Can the metabolic cost of walking be explained by performing work at a fixed muscle efficiency? If so, then locomotor efficiency should be nearly constant across speed. Our results indicate, however, that the net locomotor efficiency approximately doubles between 0.5 and 1.5 m/s (Fig. 2C). Previous studies that measured internal and external work rates reported similar findings (13, 63) (these calculations of external work did not use the individual-leg method). These authors suggested that a variation in muscle efficiency, according to the force-velocity properties of isolated muscle, accounted for the net locomotor efficiency being greatest at intermediate speeds. However, it seems unlikely that muscle efficiency would change by two-fold over this speed range. Maximum efficiency is similar for fast- and slow-twitch fiber types (30), and leg muscles appear to operate with a similar efficiency across a range of contraction frequencies. Ferguson et al. (22) measured the rate of O2 consumed by the leg during knee extensions performed at 1.00 and 1.67 Hz while controlling for the same total power output. The mechanical efficiency was only slightly less at the higher frequency (28 vs. 24%).

At least two factors may cause the net locomotor efficiency and muscular efficiency to be uncoupled: 1) mechanical work measurements likely underestimate the energy consumed by the muscles, because they do not account for the energy consumed by isometric muscle activity, and 2) mechanical work measurements may overestimate the energy used by muscles, if some of this work is performed passively by elastic tissues in the legs. The observed increase in net locomotor efficiency across speed suggests that mechanical work measurements may underpredict energy consumption at slower speeds and overpredict energy consumption at faster speeds.

Active muscle function and volume. By combining muscle geometry and force measurements, we can begin to estimate which muscles are responsible for setting the metabolic cost of walking. Our results indicate that the ankle extensors generate the greatest forces during walking: approximately four to five times those of the knee and hip extensors. When we account for the active fascicle lengths of the ankle extensors, which are shorter than the knee and hip extensor fascicle lengths, we find that the ankle extensors account for approximately half of the total volume of active leg muscles. Recent in vivo data for muscle length changes in walking humans show that the gastrocnemius muscle is active nearly isometrically during much of the stance phase, while the Achilles tendon stores and then releases elastic strain energy (24). Together, the active muscle volume and muscle function data suggest that the metabolic cost of generating near-isometric muscle force by ankle extensors is a substantial component of the metabolic cost of walking.

Our estimates of the active muscle volume should be interpreted cautiously, however, because the cross-sectional area of muscle that must be activated to generate a given force varies depending on the mechanical function performed by the muscle. Muscles generate the greatest force per cross-sectional area when they are lengthened while active, and they generate the least force per area when they actively shorten at a high strain rate (33). A number of studies suggest that muscle function varies depending on whether a muscle crosses one or two joints (34, 45, 59). Two-joint muscles may be more likely to operate as tensile struts, because power can be transferred from proximal to distal joints during leg extension. If this is the case, muscle stress might vary among the ankle, knee, and hip extensors. We find that two-joint muscles have the capacity to generate 34% of the total ankle extensor moment compared with 15% for the knee and 81% for the hip (moment calculations based on a muscle's cross-sectional area and moment arm). These data indicate that the relative contribution of one- and two-joint muscles to a joint's maximal extensor moment varies greatly, especially between the knee and hip extensors. If the average muscle stress were different between the ankle, knee, and hip extensors, our estimates of active muscle volume would not account for this, because we assumed that muscle stress was equal for all active muscle.

Conclusion. Our results indicate that, for walking at moderate speeds (0.5–1.5 m/s), the swing phase costs are small and that the net metabolic cost of walking can largely be explained by the cost of generating muscular force during the stance phase. Muscles act as motors, brakes, and tensile struts during walking, and an advantage of a force-based approach to relating the mechanics and energetics of walking is that it includes the energy consumed by muscles that are active isometrically as struts. This cost may be substantial when no net work is performed on the environment, as occurs when walking on level, hard surfaces.

APPENDIX

It is important to consider how biomechanical and metabolic data are compared in walking studies, because the gross metabolic rate includes a substantial amount of non-locomotor-related metabolism, which we estimated as the metabolic rate during standing. We calculated the net metabolic rate (i.e., gross-standing) to determine the energy used by the leg muscles to walk. In the present study, we found that the net metabolic rate increased greater than proportional to the load carried. For example, when subjects carried a load equal to 30% of their body mass, the net metabolic rate increased 47% above their unloaded rate. Previous load-carrying studies reported how load affected the gross metabolic rate (37, 47). They found that the gross metabolic rate increases directly or slightly less than proportionally to the load carried for loads less than one-half of body weight.

Our data indicate that the percent increase in gross metabolic rate to carry a given load changes with walking speed (Fig. 6). If it is assumed that non-locomotor-related metabolism does not appreciably change across speed (49), the net metabolic rate is a smaller fraction of the total (i.e., gross) metabolic rate at slower speeds. Therefore, comparing the loaded and unloaded gross metabolic rates will underestimate the increased energy used by the leg muscles to carry the load, especially at slow-to-intermediate walking speeds. By using the net metabolic rate, we find that the percent increase in energy used by the leg muscles during load carrying is similar over a fourfold range in speed. This illustrates the importance of using baseline subtractions to calculate the net metabolic rate when relating the mechanics and energetics of walking. Comparisons of leg and pulmonary V̇o2 across a range of walking speeds, as reported by Poole et al. (49) for cycling, would be of great value to test the validity and values used in baseline subtractions.

Fig. 6.

Ratio of gross (•) and net (○) metabolic rates during loaded (ĖmetabL) and unloaded (ĖmetabU) walking vs. ratio of total mass (body mass + load, Mtotal) to unloaded mass (Mbody). Net metabolic rate is calculated as gross metabolic rate minus metabolic rate during standing. Values are means ± SE (n = 8). Dashed line, direct proportionality of 1 between the 2 ratios. Solid lines, linear least-squares regressions. A: 0.5 m/s, gross (y = 0.22 + 0.76 Mtotal/Mbody, r2 = 0.705, 95% confidence limits of slope = 0.58 and 0.95, P < 0.01) and net (y = - 0.71 + 1.70 Mtotal/Mbody, r2 = 0.701, 95% confidence limits of slope = 1.29 and 2.12, P < 0.01). B: 1.0 m/s, gross (y = 0.19 + 0.79 Mtotal/Mbody, r2 = 0.519, 95% confidence limits of slope = 0.51 and 1.08, P < 0.01) and net (y = - 0.41 + 1.40 Mtotal/Mbody, r2 = 0.574, 95% confidence limits of slope = 0.95 and 1.84, P < 0.01). C: 1.5 m/s, gross (y = 0.00 + 1.00 Mtotal/Mbody, r2 = 0.683, 95% confidence limits of slope = 0.75 and 1.25, P < 0.01) and net (y = - 0.48 + 1.48 Mtotal/Mbody, r2 = 0.655, 95% confidence limits of slope = 1.08 and 1.89, P < 0.01). D: 2.0 m/s, gross (y = -0.24 + 1.24 Mtotal/Mbody, r2 = 0.909, 95% confidence limits of slope = 1.09 and 1.39, P < 0.01) and net (y = - 0.57 + 1.58 Mtotal/Mbody, r2 = 0.888, 95% confidence limits of slope = 1.37 and 1.79, P < 0.01).

Acknowledgments

We are grateful to Max Donelan for critical comments and help with data analysis. We thank Daniel Schmitt for assistance and use of his laboratory to digitize video data, Nitin Moholkar for aiding with data analysis, and Jamie Stern and Stacey Frankovic for help during the preliminary stages of the project.

This work was supported by National Institute of Arthritis and Musculoskeletal and Skin Diseases Grants AR-44688 to R. Kram and AR-46499 to T. J. Roberts.

Footnotes

  • The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

References

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