What determines the metabolic cost of human running across a wide range of velocities?

Zuntz (1897) was the first to quantify the metabolic cost of running in dogs, horses and humans. He noted that each species showed an increasing linear relationship between metabolic rate (in W kg⁻¹) and velocity. In the ensuing years, nearly all terrestrial vertebrates studied have also exhibited a linear relationship between metabolic rate and running velocity. Additionally, previous studies have found that running is metabolically more expensive for smaller compared with larger animals per kg body mass (Taylor et al., 1970, 1982; Heglund et al., 1982).

To explain the size-dependent metabolic cost of running, Taylor et al. (1980) proposed the ‘cost of generating force’ hypothesis. They found that when running animals carried extra weight, their rates of oxygen consumption (V̇_O2) increased in almost direct proportion to the total weight supported. The vertical force exerted on the ground averaged over an entire stride period increased proportionally with the load but generating one newton of force on the ground was much more costly for smaller animals, such as mice and rats, than for larger animals, such as horses. Taylor (1985) reasoned that this was because, compared with larger animals, smaller animals (e.g. a mouse) must take quicker strides using faster, less economical muscle fibers.

Pontzer and colleagues have since developed alternative derivations of the cost of generating force hypothesis. Initially, Pontzer (2005) proposed and tested a model (LiMb) for the metabolic cost of running that included a cost for swinging the limbs (assumed to be zero by Kram and Taylor, 1990) and it also fractionated the cost of generating force on the ground into vertical and horizontal force components. Kram and Taylor (1990) assumed that vertical force dominated. Most relevant to the present study, Pontzer (2005) collected data for nine humans running at velocities ranging from 1.75 to 3.5 m s⁻¹. Although the LiMb model does not use direct measurements of contact time, it essentially uses the same 1/t_c approach as Kram and Taylor (1990) to estimate the cost of generating force on the ground. The LiMb model estimates limb swing costs based on the force required to accelerate the limbs and Pontzer (2005) determined a cost parameter that best fitted the data. Pontzer (2005) concluded that for human running, taking into account limb swing costs explained more of the variance (R²=0.43) in their data than using 1/t_c alone (R²=0.29). However, Pontzer (2007) reported that the LiMb model could account for 75% of the variance (R²=0.75) in metabolic rate across running velocity in a different group of human runners but did not offer a definitive explanation for the inconsistency.

Recently, Pontzer (2016) proposed a modified version of the LiMb model: the activation–relaxation and cross-bridge cycling model or ARC model. He argued that the metabolic cost of activation–relaxation of muscle rather than cross-bridge cycling dominates the metabolic cost of generating force during level running. However, the fundamental equation in the ARC model remains numerically equivalent to the original Kram and Taylor (1990) formulation (Eqn 1), in that the metabolic rate is proportional to active muscle volume and 1/t_c. As Pontzer (2016) notes, additional in situ and/or in vitro experiments on isolated muscle are needed to determine the relative contributions of activation–relaxation and cross-bridge cycling ATP costs during muscle actions that replicate those of in vivo muscles during locomotion.

The volume of leg extensor muscle activated (V_m) is critical to the Kram and Taylor (1990) and Pontzer (2016) formulations of the cost of generating force approaches. V_m is determined by the architecture of the limbs/muscles, and the muscle forces required to support body weight, and decelerate and accelerate the body's center of mass. The muscle forces needed are determined by how the limbs act as a series of levers. Each of the leg segments (thigh, shank and foot) acts as a lever with a fulcrum at the respective joint center. Over the stance phase, changes in limb posture affect the ground reaction force (GRF) moment arm, R (defined as the perpendicular distance from the resultant GRF vector to the respective joint center). The lever arm (perpendicular distance) of the muscle force vector relative to the joint center defines the internal muscle–tendon moment arm (r). The effective mechanical advantage (EMA) is the ratio of these two moment arms, r/R (Fig. 1) (Biewener, 1989).

Differences in EMA change the muscle force required and thus the amount of active muscle volume required for running. Smaller EMAs require a greater muscle force to exert a specified force on the ground, which necessitates a greater volume of active muscle and thus presumably elicits greater metabolic rates. EMA is smaller when the joints are more flexed and/or less aligned with the resultant GRF. When an animal's limb posture is more upright (straighter legs), the GRF is more aligned with the joint centers and the force that the muscles must exert to support body weight is less compared with that for a bent limb posture.

Most previous versions of the cost of generating force approach have made the simplifying assumption that EMA does not change with running velocity (Kram and Taylor, 1990; Roberts et al., 1998; Pontzer, 2005, 2007, 2016; Pontzer et al., 2009a,b). Even with that assumption, Roberts et al. (1998) found that the original cost of generating force hypothesis could explain about 70% of the increase in the rate of metabolic energy consumption in humans across running velocities from 2.0 to 4.0 m s⁻¹. However, in humans there is evidence that as velocity increases, EMA decreases, and thus active muscle volume increases (Biewener et al., 2004).

Here, we performed a more systematic analysis of EMA to determine whether we could more completely explain the increase in metabolic rate across the full velocity range that human runners are capable of sustaining aerobically. Specifically, we explored whether the remaining 30% of the increase in metabolic rate can be attributed to changes in EMA (and thus V_m). Both Griffin et al. (2003) and Pontzer et al. (2009a,b) used estimates of V_m to better predict metabolic cost in walking humans. Pontzer et al. (2009a,b) also estimated V_m to predict the metabolic rates of running bipedal dinosaurs.

Given the expected proportionality, we further hypothesized that k would be constant across running velocity, indicating that the rate of force generation and active muscle volume together can better explain the metabolic cost of running.

Ten high-caliber, male human distance runners participated (27.1±2.5 years, 64.7±4.1 kg, 179.2±5.9 cm). All subjects had recently completed a sub-31 min 10 km race at sea level, a sub-32 min 10 km at the local altitude (∼1655 m) or an equivalent performance in a different distance running event. Subjects gave written informed consent that followed the guidelines of the University of Colorado Boulder Institutional Review Board.

Over two visits, subjects performed a series of running trials on a motorized, force-measuring treadmill (Treadmetrix, Park City, UT, USA). During their first visit, subjects habituated to the treadmill and expired-gas equipment, while we verified that they could run sub-maximally at the three fastest velocities we planned to test (14, 16 and 18 km h⁻¹). Subjects performed 5 min running trials at each velocity and rested for 5 min between trials. To assure a primary reliance on oxidative metabolism, we measured blood lactate concentration. To do so, we obtained 50 μl of blood from the subject's finger at rest and at the completion of each 5 min trial. We analyzed blood samples in duplicate with a YSI 2300 lactate analyzer (YSI, Yellow Springs, OH, USA). Subjects who could run all three velocities with a blood lactate level below 4 mmol l⁻¹ (Heck et al., 1985) and a respiratory exchange ratio (RER) <1.0 were deemed capable of running at all velocities sub-maximally.

Following the three 5 min running trials, we placed 40 reflective markers on the subject's legs using a modified Helen Hayes marker set. Markers were placed bilaterally on the ankle, knee and hip joint centers and clusters were placed on each segment. Subjects then ran 2 min trials at 8, 10, 12, 14, 16 and 18 km h⁻¹ in a random order with ad libitum rest in between. We used a 3D motion-capture system (Vicon 512 System, Oxford, UK) to determine the positions of the ankle, knee and hip joints of both legs relative to the force-measuring treadmill. The short duration of these trials for biomechanics measurements prevented marker movement due to sweat.

During a second visit, subjects arrived to the laboratory 2 h post-prandial to mitigate potential effects of diet on metabolic rate. After resting for 5 min, we measured their metabolic rate while they stood quietly. Then, each subject ran at the same six velocities as the first visit in their same randomized order as in visit one. Trials lasted 5 min for each velocity, and subjects took 5 min breaks between trials. During the standing and running trials, subjects breathed through a standard mouthpiece (Hans Rudolph 2700, Kansas City, MO, USA) and wore a nose clip, allowing us to measure their rates of oxygen uptake (V̇_O2) and carbon dioxide production (V̇_CO2) with an open-circuit expired gas analysis system (Parvomedics TrueOne 2400, Sandy, UT, USA). After completing the 6 submaximal trials, subjects recovered for 10 min and then completed a V̇_O2,max test. For the V̇_O2,max test, subjects ran at 16 km h⁻¹ on a level treadmill for 1 min. Then, we increased the grade by 1% each minute until subjects reached voluntary exhaustion (Daniels, 2013).

Using our measured V̇_O2 and V̇_CO2 along with the energetic equivalents (Peronnet and Massicotte, 1991; Kipp et al., 2018), we calculated and averaged metabolic rate (⁠ in watts) from the last 2 min of each trial when metabolic rate had reached steady state. We defined V̇_O2,max as the greatest 30 s mean value obtained. Our criteria for reaching V̇_O2,max were: a plateau in oxygen consumption with an increase in workload (grade) and/or RER over 1.15 (Issekutz et al., 1962).

We calculated the traditional cost coefficient (c) for each velocity by multiplying the mean metabolic rate (⁠⁠) normalized to body weight (W_b) by t_c as proposed by Kram and Taylor (1990) (Eqn 1). Furthermore, we calculated the cost coefficient (k) by dividing metabolic rate in watts by V_m and multiplying by t_c (Eqn 2). For Eqn 1, Kram and Taylor (1990) and Roberts et al. (1998), calculated net by subtracting the y-intercept of the linear regression. However, for several reasons, we used gross ⁠, which includes all of the metabolic energy expended during running. First, baseline metabolism may or may not remain constant when running (Poole et al., 1992; Stainbsy et al., 1980). Further, the linear intercept for human runners is quite close to zero and some authors report a curvilinear relationship between metabolic rate and velocity for human runners over a wide velocity range (up to 5.14 m s⁻¹) (Steudel-Numbers and Wall-Scheffler, 2009; Tam et al., 2012; Batliner et al., 2018; Black et al., 2017). However, we provide the average metabolic rate during standing so that others may calculate net metabolic rate if desired (Table 1).

We collected GRF data at 1000 Hz and kinematics data at 200 Hz during the last 30 s of each biomechanics measurement trial (Vicon 512 System) and analyzed 10 strides (20 steps). We used a Butterworth low-pass filter (14 Hz) to process both GRF and target-marker data (Visual 3D software, C-Motion Inc., Germantown, MD, USA) (Bisseling and Hof, 2006). We determined the touchdown and toe-off times from the vertical GRF recordings using a 30 N threshold. This allowed us to calculate stride frequency and t_c. We calculated internal joint moments using the Visual 3D software. Because measurements of the point of force application (center of pressure) are very noisy during the beginning and end of the foot–ground contact phase, we followed the methods of Griffin et al. (2003), Biewener et al. (2004) and Pontzer et al. (2009a,b) and only included joint moment (M_net joint) values in subsequent analysis when they exceeded 25% of their maximum value.

The only unknown quantities in these three equations are F_m,ankle, F_m,knee and F_m,hip. We calculated F_m,ankle directly from Eqn 4. Eqns 5 and 6 contain two unknowns, F_m,knee and F_m,hip, so we solved them simultaneously.

We calculated means±s.d. for all tested variables and tested for normality using the Shapiro–Wilk normality test. We performed a linear regression analysis on the cost coefficients c and k to determine whether the slopes differed significantly from zero. Additionally, we analyzed the cost coefficient k with a one-way repeated measures ANOVA at each of the velocities tested. We fitted individual subject linear and 2nd order curvilinear regressions to the metabolic rate versus velocity values and used R² values for each subject to assess the strength of the two regression methods. We used a paired samples t-test to compare the means of individual R² values for linear and curvilinear fits. We considered results significant at a P<0.05. We performed statistical analyses using RStudio (version 0.99.892, https://www.rstudio.com/) software.

Across the velocity range tested, 86% of the increase in metabolic rate could be accounted for by the rate of force production (1/t_c) alone using the mean c (Fig. 2A). Moreover, 98% of the increase in metabolic rate could be accounted for by using both 1/t_c and active leg muscle volume using the mean k (Fig. 2B). Linear regression t-tests showed that the slope for c was significantly different from zero (P<0.001), while the cost coefficient k was nearly constant across the velocity range (Fig. 3), with a slope not statistically different from zero (P=0.127). In further support, the one-way repeated measures ANOVA revealed that the cost coefficient k was not significantly different between the different velocities (P=0.575).

Every subject's gross metabolic rate increased by more than 2-fold across the velocity range. At 18 km h⁻¹, subjects' rates of oxygen consumption averaged 82.5% of their V̇_O2,max values (average V̇_O2,max=72.7±3.9 ml O₂ kg⁻¹ min⁻¹; range 67.6–80.3 ml O₂ kg⁻¹ min⁻¹) or 81.4% of their value (average =26.3±1.4 W kg⁻¹; range 24.4–29.3 W kg⁻¹) (Beck et al., 2018). Further, at 18 km h⁻¹, average RER was 0.937±0.04, and average blood lactate concentration was 3.51±0.31 mmol l⁻¹. No subject exceeded an RER of 1.0 or a blood lactate value of 4.0 mmol l⁻¹. Together, these variables clearly indicate that the subjects were at a sub-maximal intensity. Standing metabolic rate was 1.70±0.18 W kg⁻¹. Table 1 reports all metabolic variables.

Ground contact time (t_c) decreased over the velocity range and thus the rate of force production (1/t_c) increased. Vertical GRF peaks were greater at faster velocities (Table 2). Three of the 10 subjects transitioned from a rearfoot to a midfoot strike classification at faster velocities (14, 16 or 18 km h⁻¹), as indicated by the disappearance of the impact peaks in the vertical GRF trace (Cavanagh and Lafortune, 1980). All other subjects maintained their same foot strike pattern over the entire range of velocities.

Mean net joint moment (M_net joint) increased with velocity at the ankle and hip (Fig. 4). However, at the knee, M_net joint increased up to 14 or 16 km h⁻¹ and then slightly decreased (Table 3). We observed these patterns in every subject. Accordingly, over the complete velocity range, ankle EMA decreased by 19.7±11.1%, knee EMA increased by 11.1±26.9% and hip EMA decreased by 60.8±11.8% (Fig. 5).

Total mean active muscle volume increased by 52.8±10.6% across the velocity range (Fig. 6). Over the velocity range, active muscle volume increased by 60.5±20.3% at the ankle, 27.5±25.3% at the knee and 81.0±19.5% at the hip.

We accept both of our hypotheses. Over the wide range of velocities tested, EMA of the lower limb joints overall decreased, leading to an increased volume of activated muscle. Additionally, the new cost coefficient, k, from Eqn 2 was essentially constant across the velocity range. Together, the rate of force production (1/t_c) and active leg muscle volume (V_m) explained 98% of the increased metabolic rate required to run at faster velocities (Fig. 2). Our average value for k (0.079 J cm⁻³) is similar to the minimum k (0.09 J cm⁻³) reported by Griffin et al. (2003) for human walking. Further, Pontzer (2016) reported a value of 0.06 J cm⁻³ (slope of his fig. 1b) for a diverse assortment of species using both walking and running gaits. Note that Pontzer (2016) used net metabolic rate, while we use gross metabolic rate.

Roberts et al. (1998) found that across a narrow velocity range (2–4 m s⁻¹), 1/t_c could account for 70% of the increase in metabolic rate in humans. Across our wider velocity range (2.2–5.0 m s⁻¹), we found that 1/t_c alone accounted for 86% of the increase in metabolic rate. We attribute this difference to the use of net metabolic rate by Roberts et al. (1998), whereas we used gross metabolic rate in our calculations. If we use our net metabolic rate data, 1/t_c accounts for 65% of the increase in metabolic rate across our full range of velocities, and from 8 to 14 km h⁻¹ (2.2–3.8 m s⁻¹) 1/t_c accounts for 78% of the increase in metabolic rate. Taken together, our findings agree with Roberts et al. (1998).

Our study builds upon previous studies that suggest that changes in the EMA of the legs influence the metabolic cost of running for humans and other species (Biewener, 1989; Full et al., 1990; McMahon et al., 1987). Here, for humans, we found that from 8 to 18 km h⁻¹, ankle EMA decreased by 19.7±11.1%, while hip EMA decreased by 60.8±11.8%. Accordingly, the ankle plantar flexor active muscle volume increased by 60.5% from 652±150 cm³ to 1046±120 cm³, while the hip extensor active muscle volume increased by 81.0% from 517±92.0 cm³ to 935±110 cm³ across the velocity range. Knee EMA increased 11.1±26.9% over the velocity range; however, active muscle volume still increased 27.5±25.3% because of the increase in the peak resultant GRF over the velocity range. Because changes in EMA of the legs influence the metabolic cost of running, it is possible that EMA could give greater insight into inter-individual variations in metabolic rate.

We observed that each subject's gross metabolic rate increased curvilinearly across the velocity range. The R² values for each subject's curvilinear fits for metabolic rate versus velocity (average R²=0.998) were slightly but significantly greater than the R² values for linear fits (average R²=0.980) (P<0.05). A linear fit of metabolic rate versus velocity resulted in a mean intercept of −1.19 W kg⁻¹ (range: −0.21 to −2.04), while a 2nd order polynomial fit of metabolic rate versus velocity resulted in a mean intercept of 7.64 W kg⁻¹ (range: 4.02 to 9.95). Our data concur with previous studies that reported a non-linear increase in metabolic rate for good human runners over a wide range of velocities (covering a velocity span that changes by at least 2.5 m s⁻¹) (Steudel-Numbers and Wall-Scheffler, 2009; Tam et al., 2012; Batliner et al., 2018; Black et al., 2017). Kram and Taylor (1990) and Roberts et al. (1998) calculated net metabolic rate for a variety of small and large species by subtracting the y-intercept of linear regressions. However, we chose not to do this given the negative intercepts from the linear fit of our data. We believe the negative intercept is an artifact resulting from forcing a linear fit to intrinsically curvilinear data. An alternative approach is to subtract the standing metabolic rate. However, subtracting a constant from our measured metabolic rates does not change our conclusions. Overall, our data suggest that when we more fully account for the biomechanics of human running [i.e. changes in limb posture (EMA) and thus active muscle volume], the relationship between metabolic rate and velocity is better explained.

We intentionally did not consider the metabolic cost and muscular force required for swinging the legs during running. Taylor's original cost of generating force hypothesis was instigated from his earlier experiments comparing goats, gazelles and cheetahs, which were all about the same overall body mass (28.1–24.2 kg) (Taylor et al., 1974). Despite large differences in limb mass and limb mass distribution, the three species all expended about the same amount of energy to run. Taylor et al. (1974) surmised that at a constant running speed, little metabolic energy is used to accelerate and decelerate the limbs. However, as Modica and Kram (2005) recapitulated in detail, the cost of leg swing is controversial but not zero. Most relevant to the present study, Moed and Kram (2005) found that in humans, across a range of running velocities (2–4 m s⁻¹), the relative cost of leg swing remained nearly the same modest percentage (10%) of the total metabolic rate. Thus, even though we ignored the metabolic cost of leg swing, our results are likely not confounded by an increased percentage of the total metabolic cost due to leg swing across the velocity range.

Kram and Taylor's (1990) original cost of generating force hypothesis interpreted 1/t_c as the rate of generating force within a step and thus reflecting the rate of cross-bridge cycling in the active muscles (see Taylor, 1994). The intrinsic rate of muscle shortening (and hence the rate of cross-bridge cycling) varies widely between muscle fibers within a given muscle (Bottinelli and Reggiani, 2000). For example, within the human vastus lateralis muscle, Bottinelli et al. (1996) found that the maximum shortening velocity spanned a 4- to 9-fold range depending on the method of determining shortening velocity. At the slowest running velocity tested in the present study, average 1/t_c was 3.57 s⁻¹ and at the fastest running velocity tested, average 1/t_c was 5.29 s⁻¹, approximately a 1.5-fold increase. The fastest human sprinters have 1/t_c values of ∼10.6 s⁻¹ (Rabita et al., 2015). Thus, it appears that the factorial range of muscle shortening velocities is at least comparable to the range of 1/t_c across the spectrum of human running velocities.

Pontzer (Pontzer, 2007, 2016; Pontzer et al., 2009a,b) contends that 1/t_c primarily reflects the muscle activation–relaxation costs not cross-bridge cycling costs. Classic (Homsher et al., 1972) and contemporary (Barclay et al., 2008) muscle physiology experiments suggest that activation–deactivation costs comprise only about one-third of the total energetic cost of isometric force production. In stark contrast, other studies report that activation–deactivation comprises 80% of the cost of isometric force production (Zhang et al., 2006). As stated previously, further isolated muscle experiments are needed to resolve these disagreements.

A limitation of our study was the use of muscle moment arm, fascicle length and pennation angle data obtained from cadavers with an average age at death of 78 years. Clearly, muscles atrophy with advanced age. But fortunately, our analysis of how active muscle volume changes as a function of velocity depends only on the fascicle length and pennation angle of the muscles within the limb, which are less affected by age and muscle atrophy (Narici et al., 2003). Another potential limitation is that we assumed a constant muscle moment arm for each joint. In walking, Achilles tendon moment arms have been shown to modestly increase during plantarflexion (Rasske et al., 2017). It is possible that the muscle moment arms during running also vary during stance.

In contrast to bipedal humans, in quadrupedal mammals Biewener (1989) reported that EMA does not vary substantially with velocity or gait changes. Quadrupedal mammals that can switch between multiple gaits have shown linear increases in metabolic rate across running speeds (Taylor et al., 1970, 1982). Given our findings for bipedal humans, further research should seek to establish whether a decrease in EMA at faster velocities occurs with other bipedal species (i.e. birds) that do not switch between walking, trotting or galloping gaits like quadrupeds (Gatesy and Biewener, 1991).

In this study, we asked a simple question: why does metabolic rate increase when humans run faster? We measured the cost coefficient (k) that relates metabolic rate to the rate of force production (1/t_c) and the leg muscle volume activated. After quantifying those variables over a wide range of running velocities, we found that k was nearly constant, indicating that the rate of force production and active leg muscle volume together almost completely account for the metabolic requirements of human running. Our results link the biomechanics and metabolic costs of human running with a simple equation.

We thank Bailey Kowalczyk for assisting with data collection, and Owen Beck and Wouter Hoogkamer for their help with data filtering and processing.