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Changes in spring-mass characteristics during treadmill running to exhaustion


Medicine & Science in Sports & Exercise: August 2002 - Volume 34 - Issue 8 - pp 1324-1331

DUTTO, D. J., and G. A. SMITH. Changes in spring-mass characteristics during treadmill running to exhaustion. Med. Sci. Sports Exerc., Vol. 34, No. 8, pp. 1324–1331, 2002.

Purpose: To determine whether the stiffness characteristics of the leg change during a treadmill run to voluntary exhaustion.

Methods: Fifteen runners performed a test run at a constant speed that elicited approximately 80% of their V̇O2peak. The run was performed on a treadmill instrumented to measure vertical ground reaction forces; vertical stiffness and leg stiffness were calculated from these forces. Force data were sampled for 15 s every 5 min and immediately before the end of the test. From the force data, average stiffness characteristics were determined for each sample period. An ANOVA with repeated measures (α = 0.01) was performed for the group on both vertical and leg stiffness. A single-subject, case-series analysis was also performed on each subject by using ANOVA (α = 0.01).

Results: Group analysis revealed significant decreases (P < 0.01) in both vertical (23.9 to 23.1 kN·m−1) and leg (9.3 to 9.0 kN·m−1) stiffness over the run. Based on single-subject ANOVA, 14 of the 15 runners experienced significant (P ≤ 0.01) changes in kvert over the run. A significant correlation between changes in stride rate and vertical stiffness was found (r = 0.85). Changes in the stiffness properties of the leg, as determined via the spring-mass model, resulted in changes in vertical displacement of the center of mass and leg length (distance from ankle to hip) during stance, as opposed to changes in peak force during ground contact.

Conclusions: Observed changes in stride rate possibly result from changes in the stiffness characteristics of the leg during a run to fatigue.

Department of Kinesiology and Health Promotion, California State Polytechnic University, Pomona, Pomona, CA; and Biomechanics Laboratory, Norwegian University of Sport and Physical Education, Oslo, NORWAY

Submitted for publication October 2001.

Accepted for publication April 2002.

Reduction in neuromuscular control associated with fatigue may be detrimental to runners. It has been demonstrated that fatigued runners develop a muscle imbalance between ankle dorsiflexors and plantarflexors, increasing the impact acceleration of the tibia (22). Fatigue tends to reduce or alter neuromuscular response and susceptibility of strain injury in fatigued muscles (19). During extended running, the neuromuscular system of the runner must continually respond to altered muscle and/or neural function associated with the exercise. Recent research has examined both the neurological and kinetic behavior of the leg before and after a marathon (1,18). Force generation during a sled jump was reduced after a marathon run resulting in a decreased velocity of push-off, despite increased foot contact time (1). Vertical ground reaction force has been observed to decrease with marathon running (23). It has been suggested that changes in force production were associated with decreased muscular stiffness of the soleus (1). Running kinematics of the hip, knee, and ankle did not change significantly after a marathon run (18). Observed changes in the mechanical behavior of the leg may be associated in altered stiffness of the leg system, as determined from modeling the body as a simple spring-mass system.

The spring-mass model (Fig. 1) has been found to be appropriate to describe various locomotive parameters (3,10,16,20,21). Spring-like behavior of the leg is associated with center of mass displacement, foot contact time, and stride rate. While running at constant speed, stride rate changes proportional to the stiffness of the leg system (10). Anatomical components that determine stiffness include the muscle and connective tissue of the leg. The magnitude and timing of muscle contraction can be adjusted based on peripheral and/or central neural sources, affecting leg kinematics. Consciously changing leg kinematics (e.g., increasing the amount of stance knee flexion) during running can decrease stiffness, as seen in the case of “Groucho” running (21).

Leg kinematics change during running to fatigue. Stride rate has been observed to increase (9), decrease (2,4,24,26,27), or remain constant (8) with fatigue. Changes in stride rate may be metabolically costly to the runner, as shifting from a preferred rate increases oxygen consumption at a given speed (6,7,15). Shifts in stride rate while running at constant speed is possibly the result of a change in stiffness with a concomitant increase in metabolic cost.

If stride rate changes during constant speed running to fatigue, then the stiffness properties of the leg system may become altered as the system fatigues. In fact, decreased stride rate would be the result of a decrease in stiffness. The purpose of this study was to determine if the stiffness characteristics, as defined in the spring-mass model, change during a fatiguing run. It was hypothesized that runners would have reduced stiffness of the leg system with fatigue, resulting in reduced peak ground reaction force, greater displacement of the center of mass during stance, and decreased stride rate.

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Fifteen (4 female, 11 male) healthy, well-trained runners were recruited. All subjects provided informed consent within guidelines established by the University Institutional Review Board. Average descriptive data are given in Table 1. Participants were training a minimum of 40 km·wk−1 for races ranging in distance from 10 to 50 km. Subjects wore their own running shoes and clothing during the test. Additionally, subjects wore a telemetry heart rate monitor (Vantage XL, Polar Electro Inc., Woodbury, NY), so that heart rate could be recorded.

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Test speed determination.

For the test run to exhaustion, a speed that elicited fatigue around 45 min was desired. To achieve this, the speed associated with 80% of peak oxygen consumption was chosen for the test speed (25,28). To determine the approximate speed associated with 80% of peak oxygen consumption, a maximal oxygen uptake test was performed by each subject a week before the fatiguing test run. All subjects performed five continuous, submaximal runs of increasing speed (duration of each stage was 4 min) followed immediately by a graded run to exhaustion. Oxygen consumption and heart rate were monitored during the test. The V̇O2 values for the last 2 min of each submaximal stage were used to determine the oxygen cost for that speed. Oxygen consumption values from the five submaximal speeds were used to develop a linear regression equation with V̇O2 as the predictor of speed, so that the speed associated with 80% of peak oxygen consumption could be determined for each subject.

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Testing apparatus.

A treadmill (Quinton Q55, Quinton Inc., Bothell, WA) instrumented to measure vertical ground reaction force was used for the test run. Vertical ground reaction force was determined via six uniaxial force transducers (PCB Electronics 208A03 and 208AO2, PCB Piezotronics, Depew, NY) spaced evenly beneath the bed of the treadmill. The instrumented treadmill’s has been determined to be both valid and reliable for measuring both static and dynamic loads (13). A microcomputer equipped with an analog-to-digital conversion board (Metrabyte DAS-16, Keithley Instruments Inc., Cleveland, OH) was used for all data sampling. Force data were collected at a sampling rate of 1000 Hz.

Constant running speed was maintained throughout the test. To assure constant speed, treadmill belt speed was monitored periodically by the investigator with a handheld, digital tachometer (Model 21C13, Kernco Instruments Co., El Paso, TX), and adjusted as necessary (few instances of treadmill speed drift were observed during testing).

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Experimental calculations.

Experimental calculations involved determining the spring properties of the leg and the body’s vertical motion. Calculations used in this study are similar to those described elsewhere (10,16,20). Vertical stiffness is estimated as the ratio of the peak vertical force (maximum force during the active phase of ground contact time) and the maximum center of mass displacement during stance. Leg stiffness is a ratio of the peak vertical force and the change in leg length during stance. Double integration in the time domain of the acceleration-time function (acceleration-time curve generated from the ground reaction force) provided an estimate for the displacement of the center of mass during ground contact (5). It was assumed that the vertical velocity of the center of mass was 0 at the time of peak force. Vertical velocity would be zero if the slope of the vertical position-time curve equaled zero, which would occur at the point of peak displacement of the center of mass during ground contact. Thus, it is assumed that the peak center of mass displacement coincides with the peak vertical force. Change in leg length was determine with the equation (20):MATHwhere L0 is the resting leg length and θ was the angle of the leg at initial contact relative to vertical (θ = sin−1 (utc/2 L0), where u is the horizontal velocity and tc is the foot contact time [time from initial contact to toe-off]).

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Procedures for test run to fatigue.

After a brief warm-up, each subject performed a run to exhaustion at a speed eliciting approximately 80% of peak V̇O2. Continuation of the test run was based on the runner’s evaluation of their capacity to maintain test speed. Each subject attempted to inform the investigator 1–2 min before ending the test to allow for a final data collection. To provide a measure of effort, heart rate and rating of perceived exertion were monitored throughout the run.

Vertical force data were collected for 15 s every 5 min throughout the test and just before test cessation. From each 15 s of force data, 35–40 steps were identified by locating heel-strike and toe-off. For two subjects (nos. 8 and 10), only 12–14 steps were measured due to experimenter error (5-s data collection periods rather than 15 s). For each of the steps, the variables of interest were determined. Right and left leg data were merged in subsequent analysis, as there was no difference in measured variables.

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Data analysis.

Data analyzed for this investigation represent stiffness characteristics from each collection period during the test run. A group analysis was performed on vertical stiffness and leg stiffness to determine whether changes occurred over the test run. Repeated measures ANOVA (α = 0.01) was used to ascertain whether stiffness changed. Because runners fatigued at varying time periods, variables from the relative time points of beginning, 25%, 50%, 75%, and end of each subject’s test were included in this analysis.

In addition to the group analysis, a single-subject, case-series analysis was performed. Given that individual variations and responses tend to be lost in a group design, single-subject, case-series analysis allows individual reactions to be investigated. A single-factor analysis of variance (ANOVA) was used to determine whether vertical stiffness changes over time. An eta squared (η2) was used to determine the strength of association between vertical stiffness changes over time. Pair-wise post hoc analysis (Tukey HSD) was used to determine which time periods significantly changed from time 0. Due to the number of statistical analyses conducted, the test-wise alpha level was set at 0.01 to control for the inflation of the family-wise alpha level.

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Duration of test run.

Run times for the 15 subjects ranged from 31 to 90 min in length (Table 1). Working at 80% of V̇O2peak, exhaustion might be expected to occur between 30 and 60 min, although some runners have been observed to continue up to 90 min or longer at this workload (17,25,28). From the duration of some subjects’ test runs, some subjects might have been running below 80% of V̇O2peak or were better trained to run at higher intensities. Twelve subjects ran to perceived exhaustion, whereas three subjects (nos. 6, 11, and 13) stopped early (these three runners stopped for reasons other than exhaustion, including equipment malfunction [1 runner] and excessive test duration [2 runners]). All three expressed that they were close to exhaustion at the time the decision was made to end the run. Because of this, data for these three subjects were included in this study. It was observed from the heart rate data that runners achieved an average of 95% (90–102%) of their maximum heart rate at the end of the exhaustive test run.

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Temporal characteristics.

Ten of the 15 runners experienced statistically significant changes in stride rate (Table 2). The degree of change from initial stride rate ranged from −3.7 to 4.4%. Though small, these changes represent deviation from preferred stride rate at the test speed. Statistically different foot contact times (FC time) were observed for nine of the subjects (Table 2) at the end of the run. The observed changes ranged from −3.9 to 8.7% of the average foot contact time of the first sample (obtained 5 min into the test run).

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Force-displacement of spring-mass system.

In general, the leg does not behave precisely as might be predicted from a simple spring-mass model. Representative force-displacement curves for the beginning and end of test runs are shown in Figure 2. As can be seen in Figure 2, the relationship of force and center of mass displacement is slightly different during the time from foot contact to peak force than it is after peak force.

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Group analysis of stiffness measures.

As a group, both vertical and leg stiffness decreased over the test run (Table 3). Leg stiffness decreased initially (from the beginning to 25%) but then remained essentially the same. Vertical stiffness continued to decline over the test run.

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Vertical stiffness.

Significant changes in vertical stiffness were observed in 14 subjects (Table 4). Between 6 and 39% of the variance in vertical stiffness can be explained by the fatiguing run. Twelve runners decreased (11 significantly) vertical stiffness (up to −8.7%) and two runners increased vertical stiffness (up to 6%). The magnitude of the stiffness values and the degree of change in stiffness varied between subjects, as would be expected with the different test speeds (Fig. 3, a and b). The results of the ANOVA indicate significant changes in vertical stiffness for some subjects, though this may not be readily apparent from examining Figure 3, a and b. Nonlinear vertical stiffness data were observed for some subjects, as determined from post hoc analysis. Changes in vertical stiffness were primarily associated with changes in the displacement of the center of mass (r = −0.78, P < 0.01), as opposed to changes in the peak vertical force (r = −0.22, P > 0.01)(Fig. 4), with decreased vertical stiffness related to increased displacement of the center of mass during stance. Changes in vertical stiffness were found to be fairly proportional (r = 0.85, P < 0.01) to changes in stride rate (Fig. 5).

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Leg stiffness.

Leg stiffness was found to change with exhaustion for many of the runners (Fig. 6). Most of the observed changes in leg stiffness were associated (inversely proportional) with altered leg displacement during stance (r = −0.81, P < 0.01) as opposed to differences in peak vertical force (r = 0.43, P > 0.01) (Fig. 7).

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For the spring-mass model to describe the movement characteristics of the body, the relationship of force produced and displacement of the center of mass must be similar to that predicted by a spring-mass model. The force-displacement curves generated for these runners show a somewhat imperfect shape. This is due primarily to hysteresis present, which is not present in basic spring-mass models. Measured force-displacement relationships (Fig. 2) indicate that perhaps the leg behaves in a spring-like (as determined with a spring-mass model) manner during the first part of stance (initial contact to maximum displacement of the center of mass), and some additional element changing the generation of force during the second phase of stance. It is unclear why this relationship was observed, but perhaps it may be due to the interaction of the runner and the treadmill. By calculating vertical stiffness as the ratio of peak force to peak center of mass displacement, the stiffness of the leg during initial stance was determined. Thus, observed results apply primarily to the initial part of stance.

If the premise is accepted that the mechanical behavior of the leg can be modeled as a spring-mass system, stride rate shifts may be related to changes in the modeled stiffness properties of the leg. Stride rate and stiffness have been shown to be linearly related, with a strong relationship between stride rate and vertical stiffness (10). The relationship of stride rate to the stiffness parameters is fairly important, as stride rate is a basic measure of running performance. Across all runners in this study, there is a strong relationship between the percent change in stride rate and percent change in vertical stiffness (Fig. 5).

Vertical stiffness was computed as the ratio of peak vertical force and maximum center of mass displacement during stance. Evidence has been presented indicating that observed changes in vertical stiffness properties of the leg result in changes in displacement of the center of the mass and not necessarily to peak vertical force (10). In this study, a similar relationship between vertical stiffness and displacement of the center of mass was observed. The relationship between change in vertical stiffness and change in peak vertical force is very weak. Observed changes in vertical stiffness and stride rate with exhaustion exhibit a similar relationship to changes in vertical stiffness due to unfatigued stride rate manipulation (10). Changes in stiffness properties of the leg were related primarily to changes in the amount of leg displacement and not to changes in the peak vertical force.

Runners in this study tended to maintain peak vertical force during a fatiguing run, but this has not been observed in previous studies of exercise to fatigue (1,14,23). In these studies, typically the time that force is applied is increased with the decrease in peak force. The result is typically a lower take-off velocity. Runners in this study maintained peak force and slightly (on average) increased time of force application resulting in an increased center of mass displacement during stance. This resulted in decreased modeled stiffness parameters for the leg. This corresponds to observations of Avela and Komi (1), who found that the stiffness in the soleus muscle was reduced after a marathon run. If this were true for other muscles, then the overall stiffness of the leg would be decreased.

Generally, a runner will adopt a stride rate that minimizes or nearly minimizes oxygen cost at a given running speed (6,15). This is the optimal stride rate for the runner because it minimizes the metabolic cost for the runner. Changes in observed stride rate may represent a shift away from optimal, although these changes were small (up to 6%). It may be that that the optimal stride rate at a given speed (in terms of metabolic cost) changes with fatigue. Decreased stiffness was accompanied by a decrease in stride rate. However, during exercise to exhaustion, these shifts may prove to be increasingly significant to the runner. Further testing is needed to confirm this. Changes in stiffness of the leg are also small (up to 8.7% for vertical and 13.1% for leg stiffness). Perhaps it is the inability of the system to maintain leg stiffness that eventually drives exhaustion when running at a constant speed.

Modeled stiffness is probably driven by physiologic factors relating to muscle activation of the lower limb. The stiffness of the leg can be controlled in response to external perturbations. For example, adjustment of leg stiffness based on the running surface has been observed so that the combination of the surface and leg stiffness remains at a constant value (11,12). Under fatigued conditions, the question remains whether leg stiffness is consciously adjusted or affected by the physiologic ramifications of continued exercise.

It has been suggested that runners might consciously change running kinematics with fatigue (24). In particular, consciously increasing stride length (thereby decreasing stride rate for a given speed) was cited as a possible reaction of a runner to maintain running speed. Stride length would increase as a result of decreasing leg stiffness as was observed in this study. If continually working muscles are unable, either from local (metabolic or neural) or central (metabolic or neural) factors, to maintain contraction patterns necessary to maintain leg kinematics, a shift in leg stiffness may occur. Monotonic changes in vertical stiffness (as observed in 8 of the runners in this study) may be indicative of a continually changing physiological environment in the working musculature over the course of the exhaustive run. Whether these shifts are made consciously or are subconsciously driven cannot be determined from the results of this study, but it is plausible that the changes are unconscious in nature. Increases in stride length with fatigue have been consistently observed under a number of conditions (2,24,27). This lends further plausibility to changes in stride length stemming from leg stiffness changes rather than a conscious change by a runner with exhaustion.

Tibial accelerations increase with decreased leg stiffness while performing “Groucho” running at a constant speed (21). It may be that, with constant speed, running while fatigued tibial accelerations may increase from decreased leg stiffness, increasing the possibility injury. This inference may be plausible from the “Groucho” running results in that the tibial to head acceleration ratio decreased despite increased peak tibial accelerations (21). Between the passive (skeleton and connective tissue) and active elements (muscle), the load on the system probably increases due to the effort of attenuating the increased accelerations, increasing the likelihood of injury.

Tibial accelerations have been observed to increase and stride rate decrease with fatigue (26). Runners decreasing stride rate during an exhaustive run may increase tibial accelerations and the shock on the body. However, changes in tibial shank accelerations might be due to the speed constraint imposed by performing on a treadmill (26). This may be true in several other running studies that had runners perform to exhaustion (24,27). Runners performing overground without the speed constraint imposed by a treadmill may be inclined to reduce speed, thus minimizing changes in observed parameters (tibial accelerations (26) and leg stiffness in this study). Changing speed would probably allow the runner to continue for a longer time. Vertical stiffness is related to speed and, in particular, lower vertical stiffness values are associated with lower running speeds (16,20). Perhaps if runners in the current study were permitted to reduce treadmill speed, they may have regained mechanical equilibrium allowing them to increase the duration of the run.

In summary, modeled stiffness parameters of the leg changed over the course of the run to exhaustion for most runners, accompanied by stride rate changes. Although the mechanisms of stiffness change are unknown, the effects of changing stiffness are clear in terms of the accompanying changes in vertical motion of the center of mass and changes in leg length with exhaustion. It remains to be determined if shifts in stiffness are advantageous and part of an optimization process, or are disadvantageous with any relationship to injury mechanism in running.

Address for correspondence: Darren J. Dutto, Ph.D., Department of Kinesiology and Health Promotion, California State Polytechnic University, Pomona, 3801 W. Temple Ave., Pomona, CA 91768; E-mail:

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