The production of external power during the acceleration phase of sprinting has been studied in humans for many decades ^{(17)} . Cavagna et al. ^{(7)} studied the power output at each step during sprint running from the start to the maximal running velocity (9.4 m·s^{-1} ). They found that the power generated by the contractile component of the leg muscles increased in parallel with speed up to submaximal values (approximately 5 m·s^{-1} ). They proposed that the elastic component of the leg muscles provide the additional power required to sustain higher speeds up to maximal values (V_{max} ). They finally suggested that a sprint runner uses the work absorbed in his leg muscles (negative work) at high speed to release further positive work and thus increase power output. Cavagna et al. ^{(8)} also recognized that a runner bounces more quickly and stiffly (equivalent to increased elasticity of the support leg) at high running speed than at low running speeds. The role of limb stiffness in maximal running velocity has been clearly outlined in other animals, particularly in lizards ^{(12)} .

Luthanen and Komi (1980) ^{(24)} divided the stance phase during running into two apparent spring constants: the first was labeled the apparent spring constant during the eccentric phase, and the second was labeled the apparent spring constant during the concentric phase. They found that the apparent spring constant during the eccentric phase increased with speed. Mero and Komi ^{(28)} showed an abrupt increase in the apparent spring constant during the eccentric phase, between 90% and 100% of maximal running speed in sprinters. The best sprint runners had the greatest apparent spring constant during the eccentric phase. These findings for trained sprinters may be due to specific characteristics and training-induced adaptations. The question of whether leg power and leg stiffness may act together to determine the sprinting ability of teenage runners has never been verified. We have therefore investigated this relationship.

The studies cited above all used high-speed cameras and long force platforms to measure the external power. However, simpler ergometric tools can be used to measure the power generated during sprinting. Ergometric treadmills ^{(9–11,22,23)} , in which the runner moves a running belt and pulls a strain gauge-equipped rod joining his/her waist to a fixed point at the rear of the treadmill, have been employed to evaluate mechanical power during acceleration. The mean external power output during acceleration, calculated from the speed of the belt and from the pulling force, was 500–900 W for adult women and men ^{(9–11,22)} . Maximal power can also be measured in the laboratory by jumping tests ^{(5)} . Mero et al. ^{(30)} demonstrated functional links between jumping and sprinting performance. The mean power developed during hopping in place can be estimated from the time the foot is in contact with the ground and from the flight time ^{(6)} . The stiffness of the legs can be evaluated with the spring-mass model using these same variables of time, especially for hopping in place ^{(26)} . Hopping in place has basic mechanical features similar to the spring-mass model used during forward hopping or running ^{(13)} . Consequently, the two main components (forward power and leg stiffness) involved in sprinting ^{(7)} can be evaluated using two simple ergometric systems (ergometric treadmill and vertical jumps).

We measured the acceleration and maximal running velocity of a group of teenage runners in a field sprinting test. We then examined the sprint performances of each individual runner with respect to their forward power and the leg stiffness measured in the laboratory. Our working hypothesis is that forward power is the main determinant of acceleration, whereas maximal running velocity is linked to both the forward power and the leg stiffness.

MATERIALS AND METHODS
The subjects were 11 male handball players (age 16 ± 1 yr; height 1.79 ± 0.06 m; body mass 68 ± 7 kg) who were all regular members of a handball team. They are required to accelerate and run fast over short distances during training sessions and in matches. Consequently, although they are not specialist sprinters, they have substantial sprinting capacity. The subjects and their parents were fully informed of the nature of the tests before giving their written consent to participate in the study. Injuries during the experimental period prevented one subject performing all the tests.

Testing procedures.
The subjects executed tests under field (track sprinting test) and laboratory conditions (treadmill sprinting test, jumping test).

Track sprinting test.
Each subject ran 40 m with maximal acceleration from a standing start. The maximal running velocity was recorded with a radar Stalker ATS system^{TM} (Radar Sales, Minneapolis, MN). The reliability of the radar system in our working conditions was checked by measuring given speeds of moving subjects (1–7 m·s^{-1} ) or rolling balls (8–22 m·s^{-1} ) by the radar (v_{r} ) and checking them over a given distance (3 m) using photoelectric cells (v_{pc} ). The linear relationship between the two estimates of the speed (v_{r} = 0.99 × v_{pc} + 0.22) showed that they were well correlated (r = 0.999). The sprinting tests were monitored by placing the radar on a tripod behind the subject at the standing start. The radar was connected to a computer to obtain the raw speed–time curve (Fig. 1 ). Some irregularities, probably due both to segmental movements and acceleration–deceleration phases of the body while running, were obtained in each trial. The data of the speed–time curve corresponding to the leveling-off were identified. Their average value was calculated and defined as the maximal running velocity (v_{max} ) for each trial (Fig. 1 ). The highest value of the three trials was taken as the individual’s maximum running velocity. The acceleration phase was obtained by fitting the speed–time curve from the standing start to the maximal velocity to the following exponential equation:

FIGURE 1: Raw speed-time curve obtained by the radar system (solid curve ) during the 40-m sprint. The curve was fitted by an iterative nonlinear regression technique (broken curve ), allowing us to calculate the initial acceleration.

equation

where τ is the time constant of the relationship. The initial acceleration of each sprint was calculated as the ratio V_{max} /τ. Fitting was done by an iterative nonlinear regression (KaleidaGraph^{TM} ). The capacity of the exponential formula to describe the speed-time curve during the initial acceleration and maximal velocity phases was verified in previous studies ^{(20,32)} .

Treadmill sprinting test.
The subjects accelerated maximally for 8 s on a treadmill designed to measure force and speed and hence power (Sprint Club^{TM} , Médical Développement, Andrézieux-Bouthéon, France, Fig. 2 ). The treadmill has been described in recent studies ^{(11,22)} . The subject was attached to the rear of the treadmill and had to maximally accelerate the treadmill belt from a standing position. This was done by pushing the legs backward in successive strides very similar to track sprinting. The treadmill had a constant-torque motor to modulate the friction due to the subject’s weight. The linear relationship between known loads and torque was established at a treadmill speed of 0.15 m·s^{-1} and was used to select the torque for each subject. The treadmill was equipped with a rod connected to a strain gauge and attached at the rear of the treadmill. An optical encoder at the point of attachment measured the angle of the rod from the horizontal. The subject was attached to the rod by a wide hip-belt. Two signals were obtained, the horizontal force (F _{h} ), calculated from the strain gauge and the optical encoder signals (correction relative to the horizontal force), and the speed of the treadmill belt (v) (Fig. 3 ). The peak velocity was determined as the maximal average value of the oscillating speed–time curve (Fig. 3 ). We did not possess the software needed to fit the speed–time curve obtained during the treadmill test and thus could not calculate the corresponding time constant. Instantaneous treadmill power was calculated as the product (F _{h } ·v); it gave an oscillating curve (Fig. 3 ) that represented power output at each stride. The mean power from the start until the peak velocity was calculated from this signal. The integrated force at each stride (positive force during the pulling impulse and negative force during the braking impulse) was obtained by calculating each positive and negative area under the force–time curve.

FIGURE 2: Diagram of the treadmill ergometer.

FIGURE 3: Computer displays of the treadmill belt velocity, horizontal force, and power output during the treadmill test. The average power was (horizontal dotted line ) calculated from the start to the maximal belt velocity (vertical broken line ).

Jumping test.
Optimal vertical rebounds (i.e., maximal elevation at each jump) were executed from a standing position at 2 Hz (indicated by a metronome) for 10 s. This test is called hopping. The subjects did not use other muscles (arms) by keeping their hands on their hips throughout the hopping test. All subjects performed familiarization trials before doing an experimental trial.

As previously described ^{(6)} , the jump performance was quantified by calculating both the elevation of the center of gravity above the ground level and the average mechanical power of the positive work before take off. The basis of this calculation is described in Bosco et al. ^{(6)} :

The total work performed during a vertical jump is:

equation

where m = mass of the subject, g = the acceleration due to gravity, and h = the total displacement of the center of gravity (CG).

(h) is the sum of the displacement of the CG during the flight period (h_{f} ) and the contact period (h_{c} ).

(h_{f} ) is calculated using the contact time (t_{c} ) and flight time (t_{f} ) as follows:

equation

(h_{c} ) can be estimated assuming that the vertical velocity from the lowest point of the CG to the release increases linearly ^{(3)} . If the release velocity is v_{v} , the elevation of the CG during the contact period (h_{c} ) is:

equation

The vertical release velocity and impact velocity are equal in a harmonic jump, then we can write the vertical velocity (v_{v} ) at the impact and release phase as follows:

equation

Replacing (v_{v} ) in equation [3] by its formula given in equation [4], we can rewrite

equation

As the total displacement of CG (h) is the sum of (h_{f} ) and (h_{c} ), we can write

equation

where t_{t} is the total time of the jump and equal to the sum of the contact time and the flight time (t_{t} = t_{c} + t_{f} )

Replacing (h) in equation [1] by its formula given in equation [6], the total work performed during a vertical jump is:

equation

As the time of the positive work during contact can be half the total contact time ([3]), the average mechanical power of the positive work before take-off per body mass is:

equation

Hopping was performed on a vertical force platform (FP 350^{TM} , Médical Développement), t_{f} and t_{c} were determined at each hop from the vertical force signal. The spring-mass model can be used to identify the elastic (or stiffness) properties of the legs under the effect of the body mass during hopping (see reviews ^{(4,25)} ).

The stiffness of the legs during the vertical hopping (K_{vert} ), corresponding to the force change/length change ratio, as in a spring, was calculated as indicated by McMahon and Cheng (1990) ^{(25)} and McMahon et al. (1987) ^{(26)} :

equation

where ω_{0} is the forced oscillation of the hopping body. Equation [9] was resolved by progressive approximation with ω_{0} to be determined, t_{c} and t_{f} as variables satisfying the identity between each semiequation [tang (π − (ω_{0} ·t_{c} /2))] and [ω_{0} ·t_{f} ].

The subjects rebounded on both legs in the hopping test, so that the calculated K_{vert} is the sum of the stiffness of the two legs.

Anthropometric measurements.
Circumferences and skin-fold thickness at different levels of the thigh and the calf, the length of the leg, and the wideness of the condyles of the knee were measured to estimate the leg muscle volume, as described by Shephard et al. ^{(31)} . The leg muscle volume was estimated as follows ^{(31)} :

equation

The total limb volume was estimated as the volume of a cylinder determined by: the height (H) corresponding to the distance from the trochanter major to the external malleolar of the ankle; the basal area corresponding to the mean area of five circumferences around the limb (maximal of the thigh, midthigh, just below the patella, maximal of the calf, and just above the ankle). This leads to the simplified formula:

equation

where ΣC^{2} is the sum of the squares of the five circumferences.

equation

where ΣS is the sum of four skinfolds (front to the midthigh, back of to the midthigh, back of calf, and outside of calf) determined with a Holtain skin-fold caliper and where n is the number of skinfolds

equation

where D is the femoral intercondylar diameter and F is a geometrical factor (equal to 0.235 for the leg, i.e., average bone radius = 23.5% of the femoral intercondylar diameter).

Statistics.
Conventional statistical methods were used, including means, standard deviation (SD), and Pearson correlation coefficient. P < 0.05 was taken as the limit of the significance in all statistical tests. The number of subjects for each relationship was 10 or 11 due to injury (see above).

RESULTS
The results obtained during track running and treadmill running are shown in Table 1 . There was a significant correlation between the maximal track running velocity and treadmill velocity (r = 0.84, P < 0.01) or the average treadmill power (r = 0.73, P < 0.05, Table 2 , Fig. 4 ). The leg muscle volume (7636 ± 686 mL) was also significantly correlated (Table 2 ) with the maximal running velocity (r = 0.62, P < 0.05) and the average treadmill power (r = 0.75, P < 0.05). The initial track acceleration was significantly correlated (r = 0.80, P < 0.01) with the average treadmill power, expressed relative to body mass (Fig. 5 , Table 2 ). The integrated force during the pulling impulses of the treadmill test was 600 ± 40 N·s, and was–10 ± 9 N·s during the braking impulses.

Table 1: Measured and calculated parameters from track and treadmill tests.

Table 2: Correlation matrix for the most striking data.

FIGURE 4: Relationships between track maximal running velocity and either leg power calculated from the treadmill test and [□] (r = 0.73, P < 0.05) or leg stiffness calculated from hopping [•] (r = 0.68, P < 0.05).

FIGURE 5: Relationships between the initial acceleration during the 40m sprint and the average treadmill power [□] (r = 0.80, P < 0.01), or the leg stiffness calculated from the hopping test [•] (r = 0.15, NS).

The leg stiffness and leg power calculated from the hopping test are summarized in Table 3 . The maximal track running velocity was significantly correlated with hopping power (r = 0.66, P < 0.05). The leg stiffness calculated from hopping was significantly correlated with the maximal running velocity (r = 0.68, P < 0.05, Table 2 , Fig. 4 ). There is a significant relationship between leg stiffness and leg power calculated from the hopping test (P < 0.001 Table 2 ).

Table 3: Measured and calculated parameters for the hopping tests.

DISCUSSION
The main results of this study indicate that the initial acceleration and maximal running velocity of an individual sprinting over 40 m are statistically linked to the mean horizontal power calculated from a sprint on a treadmill ergometer. The leg stiffness calculated during hopping is also correlated with the maximal running velocity. We had postulated that either acceleration or maximal velocity was linked to forward power, indicating a functional relationship between muscle power and running performance. The eventual beneficial effect of the leg stiffness on maximal running velocity is a new finding in teenage runners.

Sprinting performance and average treadmill power.
The average net resultant force measured at constant velocity during the concentric phase of the strides is significantly correlated with the maximal running velocity ^{(28)} . However, little information is available on the external power as a determinant of the acceleration and the maximal running velocity during sprinting. Power peaks within 3–4 s during running on an ergometric treadmill ^{(23)} and subsequently decreases substantially. We used a specific treadmill ergometer to calculate the average forward power output from the speed of the running belt and the reaction force developed at each stride. The results are in agreement with those of other studies using the same type of ergometer ^{(9,11,22)} . Cavagna et al. ^{(7)} measured the work done by two sprint runners. From these data we calculated a corresponding power of 750–800 W, which agrees well with our present data. The mean power calculated during the acceleration phase on the treadmill was due almost exclusively to pulling horizontal forces, with very little braking force. Consequently, the power calculated is the result of the propulsive action of the leg muscles. Propulsive forces are clearly dominant during acceleration in track running ^{(29)} , whereas braking forces become noticeable by the first or second contact after leaving the blocks ^{(27)} . Residual friction forces during the foot contact with the treadmill belt during acceleration on the ergometer probably need to be overcome, to give essentially pulling forces. Further studies are needed to analyze the strides during acceleration on the treadmill ergometer. However, the validity of the forward power values measured on the treadmill is supported by their significant relationship with the individual leg muscle volumes. Consequently, the present study is a further example of the usefulness of the treadmill ergometer for evaluating sprint running ^{(11,22,23)} .

Acceleration was significantly correlated with forward power only when expressed per unit of body mass (W·kg_{bm} ^{-1} : specific power), whereas the maximal running velocity was only correlated with the total forward power of the body (W: absolute power). This indicates that the ability to accelerate depends on the specific power, whatever the total weight (or muscle mass) of the runner. But greater absolute power (large muscle mass) is needed to run faster.

Sprinting performance and leg stiffness.
We used the spring-mass model to calculate vertical leg stiffness during hopping ^{(4,13–16,18,19,25)} and to evaluate the links between leg stiffness and the maximal running velocity. Although vertical stiffness (k_{vert} ; vertical displacement of the hip divided by the force) and leg stiffness (k_{leg} ; leg length change divided by the force) are equivalent during hopping in place ^{(1)} , k_{vert} is greater than k_{leg} during running [at a given speed]^{(15,25)} . The calculation proposed by McMahon and Cheng ^{(25)} for their mathematical model predicts that k_{leg} varies little with increasing running speed, whereas k_{vert} increases approximately quadratically with speed. Consequently, the absolute difference between k_{leg} and k_{vert} widens with increasing speed ^{(25)} . Experimental data by Farley et al. ^{(14)} and He et al. ^{(19)} indicate that leg spring stiffness is almost constant with increasing running velocity. These studies used kinetic data and calculated the theoretical length change of the spring–mass model. A recent study ^{(2)} pointed out that k_{leg} determined using kinematic and kinetic data was significantly higher than k_{leg} calculated as proposed by McMahon and Cheng ^{(25)} . This last study ^{(2)} reported that the increase in knee spring stiffness leads to an increase in the leg spring stiffness up to a running velocity of 6.5 m·s^{-1} .

The spring–mass model was not believed to be involved as an harmonic oscillation during running in other studies ^{(21,24,28)} . They evaluated the elasticity of the contact leg using kinematic data to calculate the apparent spring constant. This was done by subdividing the support phase into two harmonic waves each having different apparent spring constants (eccentric and concentric apparent spring constant). The eccentric apparent spring constant increases with increasing running velocity, whereas the concentric apparent spring constant remains constant ^{(24,28)} .

The spring-mass model can be used to calculate leg stiffness during vertical or forward hopping ^{(4,13–15,18,19,25,26)} , taking into account the recent study of Arampatzis et al. ^{(2)} showing that the theoretical leg length change is overestimated compared with its real change measured kinematically. The leg contact elasticity can also be obtained by calculating two apparent spring constants ^{(21,24,28)} during sprinting. We believe that using these two methods leads to a common interpretation—that the elastic properties of the spring leg (either the leg stiffness or the apparent spring constant during the eccentric phase) increase with running velocity.

Our data on leg stiffness during hopping corroborate previous studies on hopping at 2 Hz ^{(18)} and on bouncing gaits at 2 Hz ^{(13)} . Leg stiffness values are statistically linked to maximal running velocity (r = 0.68, P < 0.05), but there appears to be no relationship with the initial acceleration (r = 0.18). Nevertheless, the leg spring stiffness was not measured during sprinting. Further studies using kinetic and kinematic methods as employed by Arampatsis et al. ^{(2)} during running phases up to maximal sprint running velocities may provide additional evidence about the relationship between leg stiffness and maximal velocity.

Mero and Komi ^{(28)} showed that a group of top male sprinters had higher apparent spring constants during the eccentric phase than did less skilled sprinters. We assumed that any subject that produced a stiff rebound during hopping could produce a stiff rebound during running at maximal velocity. Our findings could be in line with those of Mero and Komi ^{(28)} by showing a significant correlation between the individual maximal running velocities and the corresponding k_{vert} during hopping. Our data indicate that the leg stiffness calculated from hopping, which is correlated with its vertical power, is a good indicator of the power absorbed and then restituted during the successive eccentric and concentric phases of the leg impulses (reactive power). This reactive power is involved at each stride to maintain the high running velocity. However, the absence of any correlation between leg stiffness and the acceleration during sprinting agrees with the fact that there is a need for forward power (see above) and less reactive power during acceleration. Work efficiency decreases during running when the effective leg stiffness decreases ^{(26)} . Farley ^{(12)} also suggested that the maximal running speed may be limited by the maximum stiffness of the limb spring–mass system. However, we do not know whether there is any limit in the increase of leg stiffness to favor the sprint performance. Although there is no clear response, running above the maximal velocity by using a towing system causes a decrease in the apparent spring constant during the eccentric phase ^{(28)} . And the forward power, which involves both concentric muscle contraction and a stretch-shortening cycle, needs at least minimal knee and ankle flexion for its production. Consequently, the leg stiffness is probably limited to an optimal value, even at high running speeds.

The present study was done to identify the physiological links between three parameters: power and stiffness, as biomechanical factors, and maximal running velocity (or acceleration), as an indicator of performance. Although power and stiffness are implicit determinants of performance, the significant correlations identified between these factors and performance do not allow us to draw any conclusion whatever about cause and effect relationships. However, the present data indicate that neither power (measured on the treadmill) nor leg stiffness (measured during hopping) is statistically independent with regard to maximal running velocity. But these correlations may be due to a single factor that was not directly measured in the present study. One such factor may be muscle strength, which may direct both muscle power and stiffness toward a significant relation with maximal running velocity. Muscle volume may be taken as an indicator of muscle strength. Muscle volume was significantly correlated with power (r = 0.75, P < 0.01) and not with leg stiffness (r = 0.42, P = 0.22). Power and stiffness are also not clearly dependent on each other (r = 0.54, P = 0.13), which does not favor strength being a linking factor. However, this simple statistical approach does not definitively refute any role of muscle strength in the relationship between maximal running velocity, forward power, and leg stiffness.

Thus, these results obtained with a group of teenage handball players accustomed to sprinting indicate that sprint performance depends on two parameters. One is the muscle power needed to produce initial acceleration so as to reach and maintain a high maximal running velocity. The other is the leg stiffness that may also contribute to the maximal running velocity, although there appears to be no statistical link between this stiffness and initial acceleration. The ability to produce a stiff rebound during the maximal running velocity could be explored using the ability to produce a stiff rebound during a vertical jump. These two characteristics (muscle power and leg stiffness) were explored using relatively simple ergometric devices (ergometric treadmill and contact platforms). The relationship between the individual maximal running velocity and the corresponding vertical stiffness during hopping in place is new and warrants further examination.

The authors thank Profs. A. Belli and T. Busso for their valuable comments and R. Bonnefoy for technical assistance. M. S. Chelly was supported by a grant from the Educational Department of the Tunisian Government.

Address for correspondence: C. Denis, Lab. Physiologie–GIP Exercise, CHU Saint Etienne.

Hôpital de Saint Jean Bonnefonds, Pavillon 12, F 42055 Saint Etienne Cedex 2, France; E-mail: [email protected]

APPENDIX: A summary of the spring-mass model
The spring–mass model during running (^{14,15,19,25} ) consists of a mass and a linear spring (leg spring) (Fig. 6 ). The ratio of the peak vertical force on the spring (F_{max} ) and the peak displacement of the leg spring (ΔL) defines the stiffness of the leg spring (k_{leg} ).

FIGURE 6: Diagram of the spring-mass model during running ^{(14,15)} . L_{0} , ΔL, Δy, and θ represent the initial length of the leg spring, the maximum compression of the initial length of the leg spring, the maximum vertical displacement of the center of mass, and the half of the angle swept by the leg spring during the stance phase, respectively. The spring–mass model drawn with a discontinuous line represents the uncompressed leg spring. The arc drawn with a discontinuous line represents the path of the mass (center of gravity of the subject) during the stance phase.

equation

At the middle of the stance phase, the leg spring is oriented vertically and compressed maximally, then the force is greatest and corresponds to the peak vertical ground reaction force (F_{max} ).

The peak displacement of the leg spring (ΔL) is calculated using the formula in McMahon and Cheng (^{25} ):

equation

where

Δy = maximum vertical displacement of the center of mass
L_{0} = the distance between the foot and the hip when the leg spring is at its initial length (measured as the distance from the ground to the greater trochanter during standing)
θ = half of the angle swept by the leg spring during the stance phase
From the geometric consideration (Fig. 6 ) and assuming that the forward speed (u) is almost constant during the contact time (t_{c} ) of the foot with the ground, the following equation is obtained to calculate θ:

equation

The ratio of the peak vertical force on the spring (F_{max} ) and the peak displacement of the center of mass during the stance phase (Δy) defines the effective vertical stiffness (k_{vert} ).

equation

The relationships between k_{vert} and k_{leg} could be presented using equation [A4]. Defining F_{max} from equation [A1] and Δy from equation [A2] and substituting their corresponding equation in equation [A4], we obtain:

equation

During vertical hopping, the angle θ is null (θ = 0 degree), then by calculating its corresponding cosine and simplifying by ΔL, we obtain k_{vert} = k_{leg} .

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