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Changes in Spring-Mass Model Parameters and Energy Cost During Track Running to Exhaustion

Slawinski, Jean1,2; Heubert, Richard3; Quievre, Jacques1; Billat, Véronique3; Hannon, Christine1

Journal of Strength and Conditioning Research: May 2008 - Volume 22 - Issue 3 - p 930-936
doi: 10.1519/JSC.0b013e31816a4475
Original Research

The purpose of this study was to determine whether exhaustion modifies the stiffness characteristics, as defined in the spring-mass model, during track running. We also investigated whether stiffer runners are also the most economical. Nine well-trained runners performed an exhaustive exercise over 2000 meters on an indoor track. This exhaustive exercise was preceded by a warm-up and was followed by an active recovery. Throughout all the exercises, the energy cost of running (Cr) was measured. Vertical and leg stiffness was measured with a force plate (Kvert and Kleg, respectively) integrated into the track. The results show that Cr increases significantly after the 2000-meter run (0.192 ± 0.006 to 0.217 ± 0.013 mL·kg1·m1). However, Kvert and Kleg remained constant (32.52 ± 6.42 to 32.59 ± 5.48 and 11.12 ± 2.76 to 11.14 ± 2.48 kN·m1, respectively). An inverse correlation was observed between Cr and Kleg, but only during the 2000-meter exercise (r = −0.67; P ≤ 0.05). During the warm-up or the recovery, Cr and Kleg, were not correlated (r = 0.354; P = 0.82 and r = 0.21; P = 0.59, respectively). On track, exhaustion induced by a 2000-meter run has no effect on Kleg or Kvert. The inverse correlation was only observed between Cr and Kleg during the 2000-meter run and not before or after the exercise, suggesting that the stiffness of the runner may be not associated with the Cr.

1Laboratory of Biomechanics and Physiology, National Institute of Sports (INSEP), Paris, France; 2Team Lagardère, Paris, France; 3Department STAPS, UFR of Fundamental Applied Science, University of Evry, Evry Cedex, France

Address correspondence to Jean Slawinski,

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In running, the repeated high-velocity, short-duration eccentric muscle contractions induce a specific form of fatigue that develops during running races (18,23,24). Characteristics of this fatigue are a failure of the contractile capacity of the exercised muscles with a reduced tolerance to muscle stretch and a delayed transfer from muscle stretch to muscle shortening in the stretch-shortening cycle (SSC). Komi (18) concluded that the deterioration of the muscle function reduces the impact tolerance that leads to a loss of elastic energy potential and a decrease in the system's stiffness. When running, muscles, tendons, and ligaments can all behave as springs, storing elastic energy when they are stretched and returning it when they recoil. This complex system of musculoskeletal springs can be described as a single linear spring. A simple spring-mass model (Figure 1), consisting of a single linear leg spring and a mass equivalent to the runner's mass, has been shown to describe and predict the mechanics of running remarkably well (3,13,21). This spring-mass model has been recently used to investigate the effect of exhaustion during treadmill running (10). These authors have confirmed that the stiffness of the spring-mass model decreases significantly at the end of a running exercise performed at 80% of the maximal oxygen uptake (V̇o2max).

Figure 1

Figure 1

Numerous authors have also shown that an individual variation in stiffness could significantly influence the metabolic cost of running (20). Indeed, a significant inverse correlation has been reported between the runner's stiffness and the energetic cost of running (Cr) (9,17,19,27). These authors argued that stiffer runners gain a greater elastic energy return from musculotendinous structures. Thus, the stiffer runners run more economically than other runners and consequently perform better. However, all these studies measured the stiffness and the Cr during treadmill running. Running on treadmill is very different from a competition situation. Numerous mechanicals differences between treadmill and track running have been observed. Wank et al. (28) have shown that the swing amplitude of the leg, the vertical displacement, and the variance in vertical and horizontal velocity of the center of mass are lower in treadmill running. The subjects reduced their stride length (SL) and contact time (CT) and increased stride rate (SR) in treadmill running. Moreover, in the fatigue state, these mechanical differences may be greater (15). However, no study has been done on the effect of fatigue induced by a supralactic threshold exercise during track running on a runner's stiffness.

The purpose of this study is to determine whether exhaustion, induced by a supralactic threshold running exercise and performed on track, modifies the stiffness characteristics, as defined in the spring-mass model. We also investigated whether stiffer runners are also the most economical.

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Experimental Approach to the Problem

To examine the effect of fatigue on a runner's stiffness, the spring-mass model was used (3). The measurements were done before and after an exhaustive 2000-meter running exercise on an indoor track. Runners were instructed to run as fast as they could for 2000 meters.

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Nine trained runners, 2 women and 7 men, volunteered to take part in this experiment. Subjects were 10,000-meter competitive runners (they all have taken part in a national or national military competition). They trained about 5-6 times per week at 60-100% of V̇o2max (Table 1). Prior to participation, all the subjects were informed of the risks and stress associated with the experimental protocol and gave a written voluntary informed consent and approval was given by the ethics committee in accordance with the guidelines of the Hospital of Paris St. Louis.

Table 1

Table 1

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All subjects performed a maximal exercise for 2000 meters. During this exercise, the runners had to run as fast as they could for 2000 meters on an indoor 340-meter synthetic track in order to run the 2000 meters. This track was equipped with a force plate integrated into the ground (600 × 1200 mm). This system allows the maximal vertical and horizontal ground reaction force to be measured. The sampling frequency of the force plate was fixed at 500 Hz for 0.51 seconds, ensuring that the entire contact phase was recorded. This exhaustive exercise was preceded by a 10-minute warm-up at 3.6 m·s1 and was followed by an active recovery of 6 minutes at the same speed (Figure 2). During the warm-up and recovery, subjects adopted the required velocity using visual marks set at 20-meter intervals along the track with audio signals determining the speed needed to cover 20-meter intervals. Fingertip capillary blood samples were collected before the warm-up, after running the 2000 meters, and after the recovery in order to measure blood lactate concentration (Lactate Pro; Arkray, Kyoto, Japan).

Figure 2

Figure 2

Throughout all the exercises (i.e., warm-up, 2000-meter run, and recovery), the respiratory and pulmonary gas exchange variables were measured using a breath-by-breath portable gas analyzer (Cosmed K4b2, Rome, Italy). Before each test, O2 and CO2 analyzers were calibrated using ambient air and sample gas references. The flowmeter was calibrated with a volume of air of 3-l (Quinton Instruments, Seattle, WA). The accuracy of this system has been tested by numerous researchers and is acceptable for V̇o2 and V̇co2 measurement during supralactic threshold exercise (19).

Once per lap, the speed was monitored using an optical acquisition system (Optojump; Microgate, Bolzano, Italy) over a 10-meter length around the force plate. This system allows measurement of CT, flight time (FT), and SL and then calculation of the SR and speed (V).

During the 2000-m exercise, basically the runners came over the force plate 6 times. However, the SL was greater than during the warm-up and then the runners were not able to hit the force plate 6 times. Each runner hit the force plate 3.8 ± 1.1 times. In order to increase the number of measures for each parameter analyzed with the force plate, we took into account the whole runs over the force plate.

During the warm-up and recovery, the subject ran over a force plate 4-6 times. We also took in account the whole runs over the force plate for the calculation of stiffness.

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Data Analyses


The breath-by-breath oxygen uptake data were fitted to 2 exponential functions using a least-squares fit method: a single-exponential function comprising a delayed linear component (Equation 1) and a double-exponential function comprising 2 exponential terms that start at 2 distinct time delays from the onset of exercise (Equation 2). Fisher's exact test was used to choose the model for which the fit was associated with the highest F value (26).

Where y0 is the baseline V̇o2 (ml·min1), A1 and A2 are the asymptotic amplitudes for the exponential terms (ml·min1), τ1 and τ2 are the time constants (minutes), and TD1 and TD2 are the time delay from the onset of exercise (seconds).

The Cr was evaluated during the steady state of oxygen uptake, 30 seconds before its end. The equivalent energetic of lactate (EEL) was equal to 3 mLO2·kg1·mmol1·L1 (6), this value was added to C in order to estimate the contribution of the anaerobic pathway (C + EEL).

Cr was calculated during the 2000-meter run, the warm-up, and the recovery run. V is the average velocity respectively measured with the Optojump during the 2000-meter run, the warm-up, or the recovery run. M is the body mass.

V̇o2max was defined as V̇o2max obtained during the 2000-meter run. This value was expressed in mL·kg1·min1 and was only used to give information about the physical characteristics of the runners (Table 1).

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Stiffness (Kleg and Kvert.)

From recording the vertical and horizontal forces, the maximal vertical ground reaction force (VGRF) and horizontal ground reaction force (HGRF) were determined. Vertical displacement of the body's center of mass (ΔZ; Figure 1) was also determined from vertical force records. Vertical acceleration during ground contact was first calculated from the following equation:

where Fz(t) is the VGRF during stance, BW is body weight, and M is body mass. Double integration of az(t) with respect to time provided an estimate for the displacement of the center of mass during ground contact (7). It was assumed that the vertical velocity of the center of mass was 0 at the time of peak force. Thus, it was assumed that the peak vertical force coincides with the peak center of mass displacement (10).

Leg spring stiffness, kleg, was determined by the following equation:



ΔL represents the maximum vertical deformation of the leg spring, L0 is the resting leg length, θ0 is the angle swept by the leg spring during the first half of the stance, and V is the running speed in m·s1 (Figure 1).

The effective vertical stiffness of the leg spring was calculated from the following:

It is important to note that Kleg represents the total stiffness of the runner during stance, whereas Kvert describes the vertical compliance of the running gait (21).

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Statistical Analyses

The effect of fatigue was determined by repeated-measures analysis of variance. Relationships between the Cr and the stiffness were determined by standard linear regression and tested using a Spearman test. The significance level was set at p ≤ 0.05.

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Fatigue Effects on the Stiffness of the Runner

All the mechanical parameters remain constant when they are measured before and just after the exhaustive 2000-meter run (Table 2). These results show that, on track, rather a significant change between the warm-up, the exercise, and the recovery, fatigue had no significant influence on mechanical parameters such as VGRF, HGRF, Kleg, and Kvert.

Table 2

Table 2

Similarly, SR, SL, CT, FT, and the running speed are not modified (Table 3).

Table 3

Table 3

The 2000-meter run was performed to the best capability of each runner, and the mean performance obtained was 6 minutes, 33 seconds ± 33 seconds. Table 3 shows that Cr increases significantly after the 2000-meter run. Indeed, after the exhaustive exercise, Cr is 11% higher than during the warm-up. These changes in Cr are accompanied by a significant increase in respiratory frequency and ventilation (V̇E) (P ≤ 0.05).

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Relationship Between Energy Cost, Stiffness, and Other Mechanical Parameters

Figure 3A shows that during the 2000-meter run, Kleg correlates with Cr (r = −0.67 and P ≤ 0.05). Before and after the run, Kleg does not correlate with Cr (Figure 3B and C) However, Kvert does not correlate with Cr (r = −0.04; P = 0.93).

Figure 3

Figure 3

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The purpose of this study was to measure the effect of fatigue on the spring-mass characteristics. The results obtained show that stiffness remains constant after an exhaustive 2000-meter run. Moreover, leg stiffness was significantly correlated with energetic cost of running but only during the 2000-meter run.

In regard to the literature, only one study has measured the effect of exhaustion on Kleg and Kvert. Dutto and Smith (10) have shown that Kleg and Kvert decrease significantly from 9.3 to 9.0 kN·m1 for Kleg and from 23.9 to 23.1 kN·m1 for Kvert. The main result of the present study shows that Kleg and Kvert remain constant after the exhausting run. These differences can be explained by the methods used. First, these authors measured Kleg and Kvert every 5 minutes throughout an exhaustive treadmill exercise. In the present work, stiffness was measured before and just after the exhaustive run performed on the track. Second, the exercise duration was not the same. In our study, this duration was only 6 minutes ± 33 seconds rather than in the study of Dutto and Smith (10) in which this duration was about 57 ± 19 minutes. Six minutes of exercise may be not long enough and does not induce enough eccentric contraction to modify the elastic properties of the leg muscles for trained runners. Indeed, in their study, Dutto and Smith showed that the modification of stiffness appears at around 14 minutes of exercise. However, they did not make any measure of the stiffness between the beginning and the 14th minute of exercise. Third, during treadmill running compared to overground running, specific mechanical adaptations may occur, especially at the end of an exhaustive exercise (11,15,28). These adaptations may concern the step rate and particularly the amplitude of the vertical displacement of the center of mass (28). On a treadmill, the subjects favored a type of running that provided them with a higher level of security. A variation in these parameters between the treadmill and the track, especially in a state of fatigue, could influence the stiffness of the runner (13). Thus, on a track, the effects of fatigue on stiffness may be different from those observed on a treadmill. Indeed, on a treadmill, studies on the effect of fatigue on the stride mechanical parameters show that fatigue induces an increase in the SL and a decrease in the SR. This increase is associated with an increase in the CTr (1,4,15,25). On a track, studies suggest that few mechanical modifications appear at the end of an exhaustive exercise and that they would allow the intensity of the exercise to be maintained (12,25,26,30). Our results confirm these results, showing that for trained runners, no change due to fatigue in SR, SL, or other mechanical parameters is observed on a track. However, no study has directly measured the differences between treadmill running and track running in a fatigued state.

Concerning the increase in Cr after the exhaustive exercise, this increase may be explained by an increase in V̇E and the regeneration of the store of the high-energy phosphate and the transformation of the blood lactate, accumulated during the exercise, in glycogen, CO2, and H2O (5). However, the mechanical parameters do not explain this increase.

Several studies (8,9,17,19,27,29) have observed a direct relationship between lower extremity flexibility and the aerobic demand of running. These studies suggested that stiffer individuals gain more benefit from passive elastic mechanisms than less stiff individuals and thus incur lower energy cost. An extreme case was highlighted by McMahon et al. (20) when they examined the mechanics and the aerobic demand of a runner with exaggerated hip and knee flexion during stance (i.e., Groucho running). The effective vertical stiffness was reduced and the Cr increased by as much as 50%. Researchers argued that the increase in Cr was due to the fact that the subjects used additional muscle force by deliberately overflexing the knees. Although the stiffness was not artificially decreased in the present study, the inverse relationship found between Kleg and Cr would support the hypothesis of McMahon et al. (20) and other researchers (9,17).

However, this relationship was only significant when Cr and Kleg are measured during the exhausting exercise. Surprisingly, Cr and Kleg do not correlate before or after the exhaustive exercise. This result demonstrated that the inverse relationship between Cr and Kleg may depend on the running velocity. Indeed, the 2000-meter run was performed at a velocity near that associated with the velocity that elicits V̇o2max rather than the velocity of the warm-up and the recovery, which was 30% lower (70% of the average speed of the 2000-meter run). During constant-load cycling or running exercise of supralactic threshold intensity, a slow increase in oxygen uptake (V̇o2) has been shown to appear after the third minute of exercise (2,14). This phenomenon is called the V̇o2 slow component and induces an increase in the Cr within the third minute and the end of the exercise. A number of physiological factors have been postulated as contributing to this increase in Cr. However, biomechanical factors have received less attention in the literature. Recently, Borrani et al. (4) showed that the increase in Cr in running is not due to the change in the external mechanical cost under the effect of fatigue. However, change in global mechanical descriptors (e.g., SR and CT) suggested that lower limb stiffness may be associated with the increase in Cr during a supralactic threshold exercise. In the present work, the inverse relationship between Kleg and Cr reported only during the 2000-meter run gives some support to the Borrani et al. (4) hypothesis. However, in the present study, no increase in Cr was measured between the third and last minute of exercise.

To conclude, the increase in Cr after an exhaustive 2000-meter run without an increase in Kleg or Kvert showed that, during overground running, the increase in Cr did not result partly from a change in the stiffness of the runner. However, the inverse correlation between Cr and Kleg observed during the 2000-meter run and not before or after this exercise suggested that the stiffness of the runner may be not associated with the Cr as previously suggested. The runner's stiffness measured with the spring-mass model cannot be identified as a discriminating parameter of running economy.

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Practical Applications

Running performance over a mile or 3000 meters depends on the capacity of the best athletes to keep a running style unchanged until the end of the race. Indeed, runners who demonstrated stable running styles were able to run longer during an exhaustive exercise performed at the maximal aerobic speed (15). So technical training in a fatigued state could be interesting.

The relationship between the stiffness and Cr during supralactic threshold exercise observed in the present study allows for the hypothesis that an increase in a runner's stiffness can induce an improvement in Cr. A decrease in Cr leads to an improvement of the endurance running performance. From a practical point of view, an increase in a runner's stiffness is associated with two main types of training. The first one is the technique of running and especially in a fatigued state. The technical training can be included during the usual interval training session performed at the anaerobic threshold or at the maximal aerobic speed. For example, in order to increase stiffness, the runner can perform an interval training session with a greater SR (e.g., 3-3.2 Hz) and a lower SL (in order to conserve the same running speed) than usual. The second one is plyometric training (27). These authors have demonstrated that a 6-week plyometric training program led to an increase in lower leg musculotendinous stiffness. This increase generates an improvement in Cr and in 3-km running performance. This training consists of various jumps, bounds, and hops in both horizontal and vertical planes (e.g., squat jump, double leg jump, alternate leg jump, double leg hurdle jump).

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The authors gratefully acknowledge the French Athletics Federation and the French Ministry of Sport for their financial support. They also thank Jean-Michel Levêque for his help in editing the manuscript.

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      fatigue; stiffness; oxygen uptake; biomechanics

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