Introduction

The energetic cost of running affects the behavior/performance of animals in nature, humans seeking fitness and athletes in competition. We believe that reasonable biomechanical explanations for the energetic cost of level running have been developed (Alexander & Ker, 1990; Arellano & Kram, 2014; Kram & Taylor, 1990; Minetti & Alexander, 1997; Roberts et al., 1998), but the world is not flat. We all know intuitively that running up even a slight incline is dramatically more exhausting, yet we lack a coherent biomechanical model for the energetic cost of uphill running.

In this paper, we develop and test a new model for the metabolic cost of uphill human running. Historically, several different approaches have been applied to this topic. Most of these approaches result in unrealistically high values for the efficiency of performing vertical work (Asmussen & Bonde-Petersen, 1974; Lloyd & Zacks, 1972; Pugh, 1971), or are only valid for running up steep inclines (Margaria et al., 1963; Margaria, 1968; Minetti et al., 2002) and not for running up inclines more typical of recreational/fitness running. The purpose of the current study was to reexamine the cost of uphill running, based upon our understanding of level running energetics (Kram & Taylor, 1990; Roberts et al., 1998) and ground reaction forces during uphill running (Gottschall & Kram, 2005).

First, we give an overview of how the energetics of uphill running have been approached in the past. Margaria and co-workers (Margaria et al., 1963; Margaria, 1968) calculated net mechanical efficiency of uphill running as: (1) Net mechanical efficiency = vertical mechanical power/net metabolic rate Here, the vertical mechanical power is the rate of performing work to raise the body mass (m) against gravity (g): (2) vertical mechanical power = m ⋅ g ⋅ s i n θ ⋅ v where θ is the incline in degrees and v is the running velocity parallel to the incline. Margaria obtained the net metabolic rate by subtracting the basal metabolic rate from the metabolic rate during running. In level running, at a constant velocity, upon landing the body absorbs mechanical power (performs negative work) and then generates positive power (performs positive work) but no net external mechanical power is required because the negative and positive work quantities are opposite in sign but equal in magnitude (Cavagna, Saibene & Margaria, 1964). Margaria (1968) proposed that the equal and opposite positive and negative external work can be considered to be wasted, since performing this work has a metabolic cost but does not propel the runner forward.

However, in uphill running, net positive external work and power are produced since the center of mass (COM) is raised against gravity. Margaria et al. (1963) hypothesized and demonstrated that on steeper inclines, the wasted external work decreases and the observed net mechanical efficiency approaches the same value as the efficiency of predominantly concentric exercise, such as cycle ergometry (∼25%). It is important to note that this approach only results in such physiologically realistic efficiency values when the energetic cost of running is dominated by the work needed to raise the COM (i.e., at steep inclines) (Minetti et al., 2002). For running up inclines more typical of recreational/fitness running the net mechanical efficiencies calculated are much lower than the values for concentric muscle contractions (Smith, Barclay & Loiselle, 2005).

Another approach is to calculate “vertical efficiency” by dividing the mechanical power needed to lift the COM vertically by the difference in metabolic rates between locomotion on an incline and level locomotion at the same velocity (e.g., Full & Tullis, 1990; Rubenson et al., 2006). Published values for vertical efficiency range from 30% for red kangaroos (Kram & Dawson, 1998) to ∼46% for humans (Asmussen & Bonde-Petersen, 1974; Lloyd & Zacks, 1972; Pugh, 1971), to values near 50% (walking turtles; Zani & Kram, 2008) or even higher (60% for mice and 66% for chimpanzees; Taylor, Caldwell & Rowntree, 1972). In running, these efficiency values, which are much higher than isolated muscle contraction efficiency, have been attributed to elastic energy storage and reutilization in muscle–tendon complexes (Asmussen & Bonde-Petersen, 1974; Lloyd & Zacks, 1972; Cooke et al., 1991). But, as emphasized by Roberts et al. (1997), the increase in potential energy of the body in uphill locomotion can only be done by active concentric muscle work, since passive elastic mechanisms simply return energy stored previously in a step. Thus, these high efficiency values remain enigmatic.

Alternatively, Minetti, Ardigò & Saibene (1994) developed a model which assumed that the metabolic cost can be predicted completely based on measures of mechanical work. In their model, internal work (due to the kinetic energy changes of body segments relative to the body COM), positive external work and negative external work were each assumed to be performed with a separate efficiency value. Minetti, Ardigò & Saibene’s model (1994) also estimates the amount elastic energy storage and release, however the costs of muscle force production to generate tension to allow this energy storage and release is not taken into account.

Although we believe that the cost of generating force to support body weight is the major determinant of the metabolic cost of level running (for review, see Arellano & Kram, 2014; Kram, 2000), none of the models for uphill running explicitly include this cost. Briefly, the cost of generating force hypothesis posits that in running the muscles primarily act to generate tension that allows the tendons to store and return elastic energy. Muscles consume energy whenever they generate tension, regardless of whether they perform work. The cost of generating force to support body weight has been found to be inversely proportional to the foot-ground contact time, presumably because generating force more rapidly requires faster and less economical muscle fibers (Roberts et al., 1998).

In this study, we introduce a model for the metabolic cost of uphill running which combines the cost of generating force and the cost of performing mechanical work approaches. Our overall view is that the net metabolic cost of running is comprised of the costs of generating force to support body weight, braking and propelling body mass in the forward (parallel) direction, swinging the legs and arms and maintaining balance (Arellano & Kram, 2011; Arellano & Kram, in press; Arellano & Kram, 2014; Chang & Kram, 1999; Farley & McMahon, 1992; Kram & Taylor, 1990; Modica & Kram, 2005; Moed & Kram, 2005; Roberts et al., 1998; Teunissen, Grabowski & Kram, 2007). For level running, obviously body weight must be dynamically supported in the vertical direction, but for uphill running, we prefer to call this term the cost of “perpendicular bouncing” to emphasize that the metabolic power required to lift the COM vertically is not included in that term (Fig. 1). This approximation introduces only a small error because the perpendicular component is only slightly less than the vertical component, for example, the cosine of 9 degrees equals 0.988. Furthermore, Gottschall & Kram (2005) observed that both the perpendicular active force peaks and the contact times during uphill running (up 3, 6 and 9°) were not significantly different from those during level running. Thus, based on the cost of generating force hypothesis, the cost of perpendicular bouncing should not change with incline. So, in uphill running, the net metabolic rate should be equal to the sum of the rates of metabolic energy consumption for perpendicular bouncing, braking and propelling body mass parallel to the surface, swinging the legs and arms and, of course, raising of the COM vertically. In Fig. 1 parallel running refers to the task of running parallel to the surface whether that surface is level or inclined. The task of parallel running intrinsically requires bouncing perpendicular to the surface and that bouncing incurs a metabolic cost.

Figure 1: The total metabolic cost of running is comprised of several components. Parallel running refers to the task of running parallel to the surface whether that surface is level or inclined. The task of parallel running intrinsically requires bouncing perpendicular to the surface and that bouncing incurs a metabolic cost.

In contrast to the vertical efficiency approach, we propose that at a certain velocity the metabolic rate required for braking and propelling the body mass parallel to the running surface is less during inclined running (compared to level running), because there is less braking (negative external work) and thus less wasted work (Margaria, 1968; Minetti, Ardigò & Saibene, 1994). Gottschall & Kram (2005) quantified how in uphill running the braking Ground Reaction Forces (GRFs) parallel to the running surface decrease with steeper inclines. The propulsive GRFs parallel to the running surface are greater during uphill running, but the majority of the propulsive GRF impulse parallel to the running surface compensates for the gravitational braking impulse parallel to the surface m⋅g⋅sin(θ)⋅t step , where t step is the time between two consecutive foot strikes. During steeper incline running, most of the propulsive parallel GRF impulse is required to overcome the component of the gravitational braking impulse parallel to the surface. Thus, only a small part of the parallel propulsive GRF impulse is compensating for the braking GRF impulse (Fig. 2). Although initially counterintuitive, the metabolic costs of both braking and propelling forces, parallel to the running surface, should decrease during uphill running. By taking that into account, the efficiency of producing mechanical power to lift the COM vertically should be closer to the efficiency of concentric muscle contractions.

Figure 2: Ground reaction forces for different inclines. (A) Idealized parallel ground reaction force versus time traces for running at 3 m/s. (B) parallel component of gravitational impulse for a single step, and (C) schematic representation of the gravity force vector and its component parallel to the running surface. Forces are normalized to body weight.