The bilateral force deficit (BLD) is defined as the inability of an individual to develop as much maximal voluntary force with one limb during bilateral (two limbs simultaneously (BL)) as during unilateral (each limb separately (UL)) actions (2,27). This phenomenon has been observed during isometric (16,19,22,31), isokinetic (11,12), and ballistic exercises (2,27) involving one muscle group or the entire limb. The mechanisms underlying BLD have been debated, and the following neural mechanisms have been proposed: (i) neural alterations during BL exercises, including a reduced neural drive due to division of attention and/or interhemispheric inhibition (15,25,35), (ii) a change in motor units recruitment (16,19,24,32,35), and (iii) different muscle coordination strategies (27).
During ballistic lower limb extension movements (e.g., horizontal or vertical jump and starting phase of sprint running, henceforth called “push-off”), nonneural factors have also been considered as possible causes of BLD, notably the muscle mechanical properties characterized by the force–velocity (F-v) relation (2,5,27,34). Indeed, BL push-off results in a higher movement (and, in turn, muscle shortening) velocity than that during UL push-off because a higher total external force is developed against the same resistive load (2). Consequently, knowing the inverse muscle F-v relationship (14), the force-generating capability of each lower limb would be lower during BL extension than during UL extension, which in turn would lead to BLD. However, only Bobbert et al. (2) proposed an actual influence of the F-v properties on BLD through a realistic forward simulation model of the human musculoskeletal system, whereas other studies ended up discarding this hypothesis (5,27,34). Consequently, the nonneural contribution of the F-v properties to BLD during ballistic exertions deserves further investigation.
Assuming that BLD in ballistic lower limb push-off is primarily caused by the F-v properties (2), the changes in external force due to the F-v properties (BLDFv, representing the nonneural part in BLD) ought to depend on the change in individual F-v qualities. Indeed, an F-v relation oriented toward high-velocity capabilities is characterized by a high maximum unloaded velocity. This would induce a lower alteration of the force-generating capability in response to the increase in movement velocity occurring between UL and BL conditions and would in turn reduce BLDFv. The model used by Bobbert et al. (2), which had muscle simulation as its only independent input, allowed these authors to estimate an overall nonneural proportion of BLD associated with F-v properties of about 75%. However, they did not identify, for each individual, the fraction of this part in relation to his own F-v mechanical properties. Identifying this would contribute to explaining the inter- and intraindividual variability observed in BLD during dynamic actions (2,5,12,18,33,34). Moreover, because BLD during lower limb extensions has been linked to jumping or sprint start performance (2,4,5), as well as to daily life activities (e.g., sit-to-stand task [6,20]), it could be valuable to quantify the individual part of BLD due to the F-v properties and to relate it to individual F-v qualities.
Therefore, this study aimed at (i) providing additional information to the debated contribution of the F-v properties to BLD in ballistic lower limb push-off, (ii) quantifying this nonneural contribution of F-v properties, and (iii) relating this contribution to individual muscular F-v qualities. To reach these goals, UL and BL inclined maximum push-offs were performed by subjects and analyzed according to a theoretical macroscopic approach based on an empirically determined F-v relation for the entire neuromuscular system of the lower limb. The hypotheses were that BLD in ballistic lower limb push-off is partly explained by a shift along the F-v relationship and that this nonneural contribution was lower in individuals with high-velocity capabilities.
Subjects and experimental protocol
Fourteen subjects (age, 26.3 ± 4.5 yr; body mass, 83.9 ± 18.3 kg; height, 1.81 ± 0.07 m) gave their written informed consent to participate in this study after being informed of the procedures approved by the local ethics committee and in agreement with the Declaration of Helsinki. Eight subjects were rugby players (four of them played in the Italian first league), and the remaining six practiced physical activities including explosive efforts (e.g., basketball and soccer). After a 10-min warm-up and a brief familiarization with the laboratory equipment, each subject performed (i) horizontal maximum BL tests with different resistive forces and (ii) UL inclined push-offs.
Tests were performed on the Explosive Ergometer (EXER; Figure, Supplemental Digital Content 1 for a diagram of the EXER, http://links.lww.com/MSS/A302) consisting of a carriage seat free to move on one rail along a metal frame (for more details, see Ref. ). The total moving mass (seat + carriage) was 31.6 kg. The main frame could be inclined up to a maximum angle of 30° with respect to the horizontal. The subject could therefore accelerate himself and the carriage seat backward, pushing on two force plates (Laumas PA 300; Laumas Elettronica, Parma, Italy) positioned perpendicular to the rail. The velocity of the carriage seat along the direction of motion was continuously recorded by a wire tachometer (Lika SGI; Lika Electronic, Vicenza, Italy). Force and velocity analog outputs were sampled at 1000 Hz using a data acquisition system (MP100; BIOPAC Systems, Inc., Goleta, CA). The instantaneous power was calculated from the product of instantaneous force and velocity values. An electric motor, linked to the seat by a chain, allowed us to impose known braking forces (from 200 to 2300 N) acting along the direction of motion immediately at the onset of the subject’s push.
For each test, the subject was seated on the carriage seat, secured by a safety harness tightened around the shoulders and abdomen, with his arms on handlebars. The starting position was standardized, with both feet placed on the force plates and on 90° knee flexion. This position was set using adjustable blocks positioned on the rail of the EXER, preventing the downward movement of the carriage seat and, thus, any countermovement before push-off.
F-v relations of lower limb multisegmental neuromuscular system
The external dynamic mechanical capabilities of the entire lower limb multisegmental neuromuscular system (henceforth called “lower limb”) can be modeled by an inverse linear F-v relation during UL or BL multijoint extension (e.g., Refs. [1,3,26,36,38]). To determine BL F-v relations, each subject performed horizontal maximum BL push-offs against seven randomized motor braking forces: 0%, 40%, 80%, 120%, 160%, 200%, and 240% of his body weight. The condition without braking force (0% of body weight) was performed with the motor chain disconnected from the carriage seat. For each trial, subjects were asked to extend their lower limbs as fast as possible. Two trials, separated by 2 min of recovery, were completed at each braking force. Mean total force (
) for the best trial of each condition were determined from the averages of instantaneous values over the entire push-off phase. As previously suggested (3,26,28,30,36), F-v and force–power (F-P) relationships were determined by linear least squares and second-degree polynomial regressions, respectively. F-v curves were then extrapolated to obtain
, which correspond to the intercepts of the F-v curve with the force and velocity axes, respectively. Maximum power of the lower limbs (
, then normalized to body + carriage seat mass) was determined from the first mathematical derivation of the F-P regression equations. Moreover, F-v relations were also determined separately for the right and left lower limb during these BL push-offs from the mean force developed on each of the two force plates.
Inclined push-off performance
Following the procedures described previously, each subject performed two right UL and two left UL inclined maximal push-offs. The sledge angle was set at 20° with respect to the horizontal, and the motor chain was disconnected from the carriage seat (see the diagram of the EXER, Supplemental Digital Content 1, http://links.lww.com/MSS/A302). The inclined position was set to test BLD in different testing conditions than those used to determine F-v relations, whose variables were then used as inputs in the biomechanical model. As mentioned previously, the mean force was determined for the better trial of each condition
). The push-off distance (h PO) was determined for each subject by integrating the velocity signal over time during the push-off phase. On the other hand, subjects also performed two BL inclined push-offs (α = 20°) to compare
values obtained from equations with values measured by the force plates. Furthermore, the mean force developed by each lower limb during these BL push-offs was also quantified (
The theoretical macroscopic approach used here to analyze BLD in ballistic lower limb push-off considers the mechanical constraints of movement dynamics, as well as the mechanical properties of lower limbs that limit the external force production during push-off. This approach was proposed, detailed, and used elsewhere (29,30). Briefly, from movement dynamics, the velocity of the body center of mass at take-off (v TO) or averaged over push-off (
) increases with the increase in the mean external force developed throughout the push-off (
with g as the gravitational acceleration (9.81 m·s−2), α the push-off angle with respect to the horizontal (°), and h PO the extension range of the lower limbs determining the distance over which force can be generated (m). This increase in velocity (
or v TO), and in turn, the increase in performance, with increasing
is represented by the thin line in Figure 1 (h PO = 0.4 m and α = 20°). However, the increase in
is limited by the dynamic mechanical capabilities of the force generator, characterized by the lower limb F-v relation (thick line in Fig. 1 for BL action). The maximum
lower limbs can produce during one extension can be expressed as a function of
as the theoretical maximum external force that can be generated over one lower limb extension at a theoretical null velocity (force axis intercept of the F-v curve) and
the theoretical maximum velocity of the lower limb extension in unloaded conditions (velocity axis intercept of the F-v curve).
The mechanical outputs (
) during a maximum ballistic lower limb push-off have to respect both the mechanical constraints imposed by (i) the movement dynamics (v TO and
, equations 1 and 2) and by (ii) the mechanical capabilities of the lower limb neuromuscular system (maximum external force-generating capability decreasing with increasing velocity, equation 3). Graphically (Fig. 1), during a BL push-off, these optimal mechanical outputs correspond to the intercept (white circle) of the F-v relation (thick line) and the movement dynamics curve (thin line). The maximum value of v TO (
) obtained in this push-off condition can be determined by solving the system formed by equations 1–3. Thus,
can be expressed from
, and h PO:
From equations 2–4, the maximum
that can be developed during a given push-off can be estimated from the force generator and movement characteristics (
, h PO, and α). This force value can be computed in UL (
) or BL (
) push-offs, adjusting the corresponding
values (h PO being the same). Note that Yamauchi et al. (38) showed no difference in
between UL and BL lower limb push-off and a lower
during UL than BL exertion (37).
Moreover, this approach allows one to estimate the theoretical mean force (
) that would be developed during UL push-off if each single limb acted independently of the other and presented the same external mechanical capabilities during UL and BL exertions (i.e., the same F-v relation). This purely theoretical situation excludes any neural capability alterations between UL and BL push-offs and attributes to each limb during UL actions the mechanical capabilities that it presents during BL exertions. The dynamic capabilities of a single lower limb during a BL push-off can be described by a linear F-v relation with the same
and a halved
compared with the F-v curve of both limbs (dashed line in Fig. 1). This is supported by both basic mechanical principles (two independent identical force generators in parallel) and previous experimental results showing similar
values during UL than BL actions (37,38). Consequently, the mechanical outputs developed during this theoretical UL push-off (i.e., UL action during which the limb would present the same F-v relation as during BL push-offs) correspond to the intercept (gray circle in Fig. 1) of the F-v relation of one limb during BL action (dashed line) and the movement dynamics curve (thin line in Fig. 1).
, actual BLD and theoretical BLD due only to F-v properties (BLDFv) can be computed as:
Thus, BLDFv represents the change in external force only because of a shift along the F-v relation. It would be null during isometric exercise (i.e., when
= 0) but not null during ballistic lower limb push-offs. The remaining change in external force is due to the well-known (previously mentioned) neural factors. The relative contribution of the F-v properties to BLD (%BLDFv) can be computed as the ratio between the loss of force due to F-v properties and the loss of force associated with actual BLD:
On this basis,
were computed from
, and h PO for each subject using equations 2–4 for a push-off angle of α = 20°. Then, BLD, BLDFv, and %BLDFv were determined for each subject from equations 5–7 after measuring
during inclined UL (left and right) push-offs.
(All these computations are detailed for a typical subject in the Appendix, Supplemental Digital Content 2, http://links.lww.com/MSS/A303.)
All data are presented as mean ± SD. After checking distributions normality (Shapiro–Wilk test) and variance homogeneity (F test), differences between mean values of
were analyzed with a one-way ANOVA for repeated measures and Newman–Keuls post hoc tests. Following the same statistical procedure, the characteristics of F-v relations (
) obtained during horizontal BL push-offs were compared between right, left, and both lower limbs. The Pearson correlation coefficient was used to assess the associations between BLDFv and F-v mechanical properties (
). Furthermore, in case of correlations between these variables, correlation analyses were performed for each variable after adjusting the remaining variables as covariates; each correlation between BLDFv and one F-v mechanical property was tested, controlling the other properties to remove their potential effect as confounding factors. For all statistical analyses, a P value of 0.05 was accepted as the level of significance.
Individual F-v and F-P relations were accurately fitted by linear (r 2 = 0.75–0.99, P ≤ 0.012) and second-degree polynomial (r 2 = 0.70–1.00, P ≤ 0.024) regressions, respectively (data of a typical subject in Fig. 2). ANOVA did not show a difference in
between considering one or two lower limbs during BL push-offs (P = 0.838), but they demonstrated differences in
between these conditions (P < 0.001), with no difference between left and right lower limbs (P = 0.91) (Fig. 2). Mean values ± SD of
, and h PO characterizing BL F-v relations were 24.2 ± 2.97 N·kg−1, 2.78 ± 0.63 m·s−1, 16.34 ± 2.26 W·kg−1, and 0.39 ± 0.04 m, respectively.
Mean values ± SD of
are presented in Figure 3. All these values were significantly different from each other (P < 0.001), with the exception of right/left comparisons (
, P = 0.86;
, P = 0.87). The external force developed by each lower limb during UL actions was significantly higher than during BL actions (P < 0.001). The external force produced during theoretical UL push-off with BL mechanical capabilities was lower than the force produced during actual UL push-offs but higher than during BL actions (P < 0.001). Consequently, the external force developed by both limbs during BL actions was lower than the sum of forces that each limb can produce during actual UL contractions (the difference representing BLD) and lower than the sum of forces that each limb can produce during theoretical UL push-offs with BL mechanical capabilities (the difference being BLDFv). The resulting BLD, BLDFv, and %BLDFv values were 36.7% ± 5.7%, 19.9% ± 3.6%, and 43.5% ± 9.1%, respectively. The relative contribution of the F-v properties to BLD (%BLDFv) amounted to 43.5%, which was significantly lower than 50% (95% confidence interval for the population mean, 38.2%–48.8%). BLDFv was significantly correlated to
(Table 1). These relations were significant even after controlling for
(Table 1, Fig. 4). Contrastingly, neither
retained its association with BLDFv once
had been statistically controlled. Furthermore, BLD tended to be correlated to
(r = −0.47, P = 0.09) but not to
(r = 0.27, P = 0.35). The values of the
measured during inclined push-off and predicted from equations 2–4 were 1228 ± 192 and 1154 ± 218 N, respectively, with their difference amounting to −73.5 ± 96.4 N, which supports the validity of our computational approach.
The primary findings of this investigation were that (i) the F-v properties do indeed play an important role in BLD during ballistic lower limb push-off, (ii) a method based on a macroscopic biomechanical model and experimental measurements could be used to individually quantify the contribution of F-v properties to BLD, and (iii) this contribution was related to individual mechanical F-v qualities.
This study was based on theoretical developments aiming to describe the mechanical outputs that result from the action of the entire lower limb neuromuscular system and not to model the musculoskeletal structures at the origin of these outputs. The main limit of this approach could be the macroscopic level from which the multisegmental neuromuscular system is considered, inducing (i) the description of its mechanical external capabilities by the empirically determined F-v relations and (ii) the application of principles of dynamics to a whole body considered as a system. (These points have been discussed in Ref. (29).) However, the slight difference (approximately 6.2%) in
between the experimental measurements and the equation predictions is in line with the accuracy recently reported for this approach (30), which confirms its validity. The F-v linear model used to characterize the dynamic external capabilities of the lower limb neuromuscular system has previously been reinforced and discussed during multijoint lower limb extensions (1,3,26,30,36,38). These overall dynamic external capabilities are the final result of the different mechanisms involved in limb extension and characterize the mechanical limits of the entire in vivo neuromuscular function. They encompass individual muscle mechanical properties (e.g., intrinsic F-v and length–tension relation and rate of force development), some morphological factors (e.g., cross-sectional area, fascicle length, pennation angle, and tendon properties), neural mechanisms (e.g., motor unit recruitment, firing frequency, motor unit synchronization, and intermuscular coordination), and segmental dynamics (1,7–9,38). Consequently, the present macroscopic approach of the F-v relation includes neural mechanisms, contrary to previous studies (2,5,34). However, these neural mechanisms are not directly related to the difficulty in maximally driving both limbs simultaneously. Consequently, the proportion of BLD resulting from the F-v properties can refer to the nonneural component of BLD because it was computed theoretically on the explicit assumption that the entire neuromuscular system of each limb would present the same external mechanical capabilities during UL and BL exertions and, in turn, that no alterations in neural capabilities would occur between these two conditions (equation 6).
During ballistic lower limb push-offs, the maximum force each lower limb can produce is lower during BL than UL actions, which was confirmed here by the difference between
(Fig. 3). The resulting BLD values (approximately 35%) were in line with previous studies testing dynamic exertions (2,12,13,27). Note that these BLD values were computed here using
values obtained from the equations, and not from those measured, to reduce the effect of measurement errors on the actual interindividual variability in BLDFv. As concluded by Bobbert et al. (2), the origin of the lower force produced by each limb during BL push-off can be divided into two main kinds of mechanisms, neural alterations and the role of F-v properties. The distinction was made here by considering the purely theoretical situation in which each single lower limb presented the same dynamic mechanical capabilities in UL and BL actions (and in turn the same F-v relation, dashed line in Fig. 1), i.e., assuming that no neural capability alterations occurred between UL and BL conditions. Even in this specific theoretical situation, the force each limb could produce during maximum push-offs would be lower during BL than UL actions because of a higher extension velocity. This BLD due to the F-v properties (i.e., BLDFv), brought about by the difference between
, would then reach approximately 20%. This gives a relative contribution of the F-v properties to BLD (%BLDFv) slightly lower than 50%. The remaining part of BLD is therefore explained by an alteration between UL and BL exertions in the neural capability to maximally drive both limbs simultaneously, as previously suggested (13,15,27). The external dynamic capabilities of each single limb are thus actually different between UL and BL actions, which can be characterized by different F-v relations. To sum up, when considering the external F-v relation of the entire neuromuscular system of each lower limb, BLD is due to (i) a change in the F-v relation for each limb because of neural alterations (slightly >50% of the BLD, shift from the black to the gray circle in Fig. 1) and (ii) a shift along the F-v relation because of a change in movement velocity (slightly <50% of the BLD, shift from the gray circle to the white diamond in Fig. 1).
Even if the theoretical and experimental approaches used here were very different, especially by considering the lower limb neuromuscular system at a more macroscopic level than that of Bobbert et al. (2), the present findings support that the F-v properties do play an important role in determining BLD during ballistic lower limb push-offs, which has often been challenged (5,27,34). From a forward simulation model of the musculoskeletal system, these authors suggested that 75% of the BLD was explained by the increase in extension velocity and 25% by a reduced active state of certain extensor muscles. They attributed this latter part primarily to differences in activation dynamics between BL and UL jumping due to different levels in the muscle active state at the start of the jump. This neural origin was reduced in the present experiment because the lower limbs did not support the body weight in the starting position because of the mechanical blocks preventing downward movements on the EXER, thus preventing any stretch-shortening cycle from occurring. In addition, no deficit in maximum neural drive related to a potential difficulty in driving both limbs simultaneously was incorporated in the model simulations reported by Bobbert et al. This can lead to an underestimation of the effect of neural mechanisms on BLD because maximum voluntary activation of muscles was previously found to be affected by interlimb interactions during BL contractions (15,19,35), even if this has not been systematically observed (22,31). In addition, Bobbert et al. noted differences in the push-off distance between UL and BL experimental jumps, which can also explain differences in BLDFv estimations. However, no difference in push-off distance was observed in the present study between UL and BL conditions (comparison of h PO, P = 0.19), mainly resulting from the mechanically fixed starting position and the configuration of the EXER ergometer. All these points can partly explain the difference in %BLDFv prediction.
The change in external force due to the F-v properties (BLDFv), which are variable among subjects, is influenced by individual mechanical properties of the lower limb neuromuscular system. Indeed, BLDFv is highly correlated with maximum force (
), velocity (
), and power output (
) capabilities of the lower limbs (Table 1). As expected, this nonneural contribution to BLD was lower for subjects presenting F-v profiles oriented toward velocity capabilities (i.e., steep F-v curves in Fig. 1), which were characterized by rather low
values (Table 1, Fig. 4). For such F-v profiles, the decrease in force-generating capability with increasing extension velocity (e.g., during BL push-offs) is reduced, which in turn limits BLDFv. The F-v profile, and in turn, the balance between force and velocity capabilities, thus seem to be the main individual mechanical properties of the neuromuscular system affecting BLDFv. The different mechanical properties analyzed (
) being intercorrelated for the present subject sample, partial correlation analyses were performed to remove potential confounding effects and showed that BLDFv was only related to maximum unloaded velocity (
). Given the importance of the balance between force and velocity capabilities (30), one can expect a significant positive relation between BLDFv and
was statistically controlled. The absence of this relation may be explained by the low interindividual variability in
for the sample of subjects tested (coefficient of variation = 9.3% when normalized to total moving mass) as compared with
(coefficient of variation = 22.5%) values. Furthermore, even if it also largely depends on neural mechanisms, it is worth noting that BLD tended to be directly correlated to
, which means that the individuals with low velocity capabilities were those presenting high BLD values and vice versa.
These results contribute to new insights into the overall understanding of the BLD during dynamic movements, and especially on the factors that may affect it. Indeed, BLD has often been explored during isometric contractions, whereas the necessity to develop maximum force using BL contractions often occurs during dynamic actions, either in sports activities (e.g., jumping in basketball and volleyball, the starting phase in sprint running, and the throw-in in soccer ) or during daily-life activities, such as standing up from a sitting position for the elderly or individuals with Parkinson disease (6,17,20). Note that for these individuals, a sit-to-stand task represents a nonisokinetic dynamic maximum BL effort for which reduced strength or an inability to perform rapid muscle contractions can lead to impaired mobility and eventually to institutionalization (6,20). During such dynamic BL actions, the magnitude of the force production varies among individuals, and these differences can change after training or rehabilitation adaptations. First, the neural capability to maximally drive both limbs simultaneously has been shown to increase with training using BL exertions (18,33). As a result, because BLDFv is negatively correlated to
, which experimentally supports the logical theoretical causal link between these two variables (see the first section of this article), BLD could be reduced by improving
. This improvement, corresponding to an increase in the force capability in the high velocity part of the F-v relation, could be achieved through “ballistic” or “power” training using light (e.g., <30% of one repetition maximum [7,8,10,23]) or negative (21) loading.
To summarize, the present study confirmed that the bilateral deficit in external force during lower limb push-offs was partly due to the F-v properties (for approximately 43% of BLD), which had hitherto been advanced only by Bobbert et al. (2) and challenged by many others (5,27,34). The contribution of this nonneural factor to BLD depends on the individual mechanical F-v profile of the neuromuscular system: the more the F-v profile is oriented toward velocity capabilities, the lower the loss of force from UL to BL push-offs because of changes in movement velocity. These original findings were obtained through a macroscopic theoretical approach, allowing the distinction between neural and nonneural parts of BLD during ballistic lower limb push-offs, which could be further used for individual athlete or patient evaluations.
We thank Alberto Botter (Udine Rugby Football Club) for his help in recruiting the subjects tested and the subjects for their “unilateral and bilateral” involvement in the protocol.
We declare that we have no conflict of interest and that no funding was received for this work.
The results of the present study do not constitute endorsement by the American College of Sports Medicine.