royalsocietypublishing.org/journal/rspb
Research
Cite this article: Bohm S, Mersmann F,
Santuz A, Arampatzis A. 2019 The force–
length–velocity potential of the human soleus
muscle is related to the energetic cost of
running. Proc. R. Soc. B 286: 20192560.
http://dx.doi.org/10.1098/rspb.2019.2560
Received: 5 November 2019
Accepted: 26 November 2019
Subject Category:
Morphology and biomechanics
Subject Areas:
biomechanics, physiology
Keywords:
biomechanics, muscle-tendon unit,
force–length–velocity relationships, gear ratio,
running economy
Author for correspondence:
Sebastian Bohm
e-mail: sebastian.bohm@hu-berlin.de
Electronic supplementary material is available
online at https://doi.org/10.6084/m9.figshare.
c.4767113.
The force–length–velocity potential of
the human soleus muscle is related to the
energetic cost of running
Sebastian Bohm1,2, Falk Mersmann1,2, Alessandro Santuz1,2
and Adamantios Arampatzis1,2
1Department of Training and Movement Sciences, and 2Berlin School of Movement Sciences, Humboldt-
Universität zu Berlin, Berlin, Germany
SB, 0000-0002-5720-3672; FM, 0000-0001-7180-7109; AS, 0000-0002-6577-5101;
AA, 0000-0002-4985-0335
According to the force–length–velocity relationships, the muscle force
potential is determined by the operating length and velocity, which affects
the energetic cost of contraction. During running, the human soleus
muscle produces mechanical work through active shortening and provides
the majority of propulsion. The trade-off between work production and
alterations of the force–length and force–velocity potentials (i.e. fraction
of maximum force according to the force–length–velocity curves) might
mediate the energetic cost of running. By mapping the operating length
and velocity of the soleus fascicles onto the experimentally assessed force–
length and force–velocity curves, we investigated the association between
the energetic cost and the force–length–velocity potentials during running.
The fascicles operated close to optimal length (0.90 ± 0.10 L0) with moderate
velocity (0.118 ± 0.039 Vmax [maximum shortening velocity]) and, thus,
with a force–length potential of 0.92 ± 0.07 and a force–velocity potential
of 0.63 ± 0.09. The overall force–length–velocity potential was inversely
related (r = −0.52, p = 0.02) to the energetic cost, mainly determined by a
reduced
shortening
velocity.
Lower
shortening
velocity
was
largely
explained ( p < 0.001, R2 = 0.928) by greater tendon gearing, shorter Achilles
tendon lever arm, greater muscle belly gearing and smaller ankle angle vel-
ocity. Here, we provide the first experimental evidence that lower shortening
velocities of the soleus muscle improve running economy.
1. Background
Humans are capable runners compared with most other mammals and it has
been suggested that the endurance performance has been a crucial aspect for
human evolution [1]. Running economy is an important physiological factor
for endurance performance [2] and is defined as the mass-specific rate of
oxygen uptake or metabolic energy consumption at a given speed [3,4]. The
main determinant of the metabolic energy consumption during running is
the muscular force needed to support and accelerate the body mass [5]. The
level of muscle activation necessary to generate the required force is dictated
by the force–length and force–velocity potential of the muscle. The force–
length and force–velocity potential express the operating length and velocity
of the muscle fibres with respect to the force–length [6] and force–velocity
relationships [7] (i.e. fraction of maximum force according to the force–
length–velocity curves) [8,9]. When fibres operate at lower shortening velocities
and close to the optimal length, the required active muscle volume for a given
force diminishes, together with the metabolic energy expenditure [4,10]. Besides
the operating length and velocity as the main determinants, the history depen-
dence of force generation (i.e. increased force after active muscle lengthening
© 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original
author and source are credited.
[11] and decreased force after active shortening [12]) may
additionally influence the force potential. Thus, it is reason-
able to argue that the fibre dynamics of the large lower
limb muscles during running are explanatory factors of the
energetic cost and thereby endurance performance.
During human running, the soleus actively shortens [13]
and is the most important muscle for propulsion [14,15]. How-
ever, during active shortening, increased length excursion and
shortening velocity reduce the force–length–velocity potential
of muscle fibres. Due to the steep slope of the hyperbolic
force–velocity curve at low to moderate shortening velocities,
the force–velocity potential might be particularly sensitive to
changes in shortening velocity. Yet the association between
energetic cost and operating fibre dynamics as the force–
length and force–velocity potential during human running
has not been experimentally investigated thus far.
From a mechanical point of view, the soleus fibre operating
length and velocity are mainly mediated by the decoupling of
the fibre length trajectories from those of the muscle–tendon
unit (MTU), the Achilles tendon lever arm and the excursions
of the ankle joint. The decoupling of the fibre length trajectories
from the MTU is a result of tendon compliance and the variable
pennation of muscle fibres within the series muscle belly and
can be quantified by the so-called MTU gearing (i.e. ratio of
MTU and fibre velocity) [16]. Tendons, due to their compliance,
take over important portions of the length changes within the
MTU, which substantially reduce the length change and vel-
ocity of the series muscle belly. The magnitude of the
decoupling of the muscle belly from the MTU by the tendon
is expressed by the ratio of MTU velocity and belly velocity
and has been termed tendon gearing [16]. Furthermore, the
rotation of the fibres (i.e. changes of pennation angle) during
muscle shortening and concomitant changes in muscle shape
decouple the fibre length change from the length change of
the muscle belly, further decreasing the fibre shortening
length and velocity [17]. The ratio of muscle belly velocity
and fibre velocity defines the effect of the fibre rotation mech-
anism on the shortening velocity, i.e. belly gearing (or
architectural gear ratio) [16,17]. Independent of the gearing
within the soleus MTU, muscle force is transmitted through
the Achilles tendon lever arm (i.e. the distance between the
tendon’s line of action and the centre of rotation of the ankle
joint). It has been shown that shorter lever arms of the Achilles
tendon are correlated with lower rates of energy consumption
during running [18,19]. The lower energy consumption has
been attributed to a greater energy storage and return by the
Achilles tendon due to the higher muscle force required for a
given joint moment at a smaller lever arm [18]. However,
the increased energy storage and release from the tendon is
associated with a higher muscle force, which in turn increases
the metabolic cost, counteracting or even deteriorating the
effects of increased energy storage and release [20]. Yet shorter
lever arms can reduce the fibre length excursions and fibre
shortening velocity of the soleus muscle at a given ankle joint
excursion during the stance phase, which can increase the
muscle force–length–velocity potential. Therefore, besides
the debated benefits in terms of energy storage and release, a
reduction of the fibre operating length changes and velocity
by a shorter lever arm of the Achilles tendon could be an
important mechanism for the improvement in running econ-
omy. Furthermore, the ankle joint excursion during the
stance phase of running may also influence the operating
length and velocity of the soleus fibres [21]. Although gearing
within the MTU contributes to the decoupling of fibre length
and MTU length trajectories, smaller ankle joint excursions
can decrease fibre length changes and velocities. In fact,
Cavagna & Kaneko [22] as well as Williams & Cavanagh [23]
reported reduced ankle joint excursions in runners with
higher running economy than others [22,23].
In the present study, we investigated the operating length
and velocity of the soleus muscle fascicles (i.e. bundles of
fibres) during running as a function of the experimentally
determined force–length and assessed force–velocity relation-
ships (i.e. force–length and force–velocity potential) and their
association to the energetic cost of running. We further
assessed tendon and belly gearing as well as Achilles tendon
lever arm and ankle joint excursions during the stance phase
of running as mediating factors for the fascicle operating
length and velocity. We hypothesized the force–length–
velocity potential to be associated with the energetic cost of
running, mainly due to the sensitivity of the force–velocity
potential to modulations of fascicle velocity. Finally, we
expected that gearing, tendon lever arm and joint excursion
would explain the majority of the fascicle velocity variability
in the soleus muscle during running.
2. Methods
(a) Experimental design
Nineteen healthy (age: 29 ± 6 years, height: 177 ± 9 cm, mass:
69 ± 9 kg, 7 female), ambitious runners who trained at least three
times per week participated in the present study. The ethics com-
mittee of the Humboldt-Universität zu Berlin approved the study
and the participants gave written informed consent in accordance
with the Declaration of Helsinki.
After familiarization, the participants ran on a treadmill at
2.5 m s−1 for 4 min. By integrating ultrasonography, electromyogra-
phy (EMG) and kinematic data, we measured muscle architectural
parameters (fascicle length, pennation angle and thickness) and
EMG activity, and assessed the MTU length of the soleus muscle
as well as the ankle joint angle of the right leg. Energetic cost of
running was determined by expired gas analysis during an
additional 10 min running trial at the same speed. In the second
part of the experiment, the individual force–fascicle length relation-
ship of the soleus was experimentally determined by means of
maximal isometric voluntary plantar flexion contractions (MVC)
of the right leg at different ankle joint angles on a dynamometer
in combination with ultrasound imaging of the soleus muscle fasci-
cles. The force applied to the Achilles tendon was calculated from
the ankle joint moment and the individual tendon lever arm. The
derived optimal fascicle length for force production was further
used to determine the force–velocity relationship of the soleus fas-
cicles. The order of the two parts of the experiments (running
and MVC) was randomized, yet the ultrasound probe and EMG
electrodes remained attached between both ultrasound measure-
ments. Based on the assessed force–length and force–velocity
relationships, it was possible to calculate the force–length and
force–velocity potential of the soleus muscle as a function of the fas-
cicle operating length and velocity during the stance phase of
running. The product of both potentials then gives the overall
force–length–velocity potential.
(b) Assessment of the soleus force–length and
force–velocity relationship
The participants were placed in prone position on the bench
of an isokinetic dynamometer (Biodex Medical, Syst. 3, Inc.,
Shirley, NY) with the knee in fixed flexed position (approx. 120°)
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Proc. R. Soc. B 286: 20192560
2
to restrict the contribution of the bi-articular m. gastrocnemius to
the plantar flexion moment [24] (figure 1a). Following a standard-
ized warm-up, MVCs were performed with the right leg in eight
different joint angles, including a plateau of around 3 s. The
angles ranged from 10° plantar flexion to the individual
maximum dorsiflexion angle, set in random order in uniformly dis-
tributed intervals. The moments at the ankle joint were calculated
taking into account the effects of gravitational and passive
moments and any misalignment between ankle joint axis and
dynamometer axis by means of an established inverse dynamics
approach [25] as well as the contribution of the antagonistic
muscles by means of electromyography (description in electronic
supplementary material; figure 1a). The force applied to the
Achilles tendon during the plantar flexion MVCs was calculated
as quotient of the joint moment and the individual tendon lever
arm (description in electronic supplementary material). Soleus
fascicle behaviour during the MVCs was synchronously captured
at 30 Hz by B-mode ultrasonography (Aloka Prosound Alpha 7,
Hitachi, Tokyo, Japan) with a 6 cm linear array probe (UST-
5713T, 13.3 MHz). The probe was mounted on the shank over the
medial aspect of the soleus muscle belly by means of a custom
made antiskid neoprene/plastic cast (figure 1a). The fascicle
length was post-processed from the ultrasound images (figure 1a)
using a self-developed semi-automatic tracking algorithm [26],
described in more detail in the electronic supplementary material.
Accordingly, an individual force–fascicle length relationship
was calculated for each participant based on a second-order poly-
nomial fit (figure 1b) and the maximum muscle force applied to the
tendon (Fmax) and optimal fascicle length for force generation (L0)
was derived, respectively. Furthermore, we assessed the force–
velocity relationship of the soleus using the classical Hill equation
[7], and the muscle-specific maximum fascicle shortening velocity
(Vmax) and constants of arel and brel. Vmax was derived from the
study of Luden et al. [27], which showed Vmax values for type 1
fibres of 0.77 L0 s−1 and 2.91 L0 s−1 for type 2 fibres of the human
soleus muscle measured in vitro at 15°C [27]. Considering the temp-
erature coefficient [28], Vmax can be predicted as 4.4 L0 s−1 for type
1 fibres and 16.8 L0 s−1 for type 2 fibres under physiological temp-
erature conditions (37°C). Using an average fibre type distribution
(type 1 fibres: 81%, type 2: 19%) of the human soleus muscle
reported in literature [27,29–31], Vmax can be calculated as
6.77 L0 s−1. arel was calculated as 0.1 + 0.4FT, where FT is the fast
twitch fibre type percentage (see above), which then equals to
0.175 [32,33]. The product of arel and Vmax then gives brel as 1.182
[34]. After rearrangement of the Hill equation and extension to
the eccentric component, the operating velocity normalized
to Vmax was used to calculate the individual force potential accord-
ing to the force–velocity relationship.
(c) Assessment of joint kinematics, muscle architecture
and electromyographic activity during running
During running on a treadmill (h/p cosmos mercury, Isny,
Germany, 2.5 m s−1), kinematic data of the right leg were captured
on the basis of anatomically-referenced reflective markers (greater
trochanter, lateral femoral epicondyle and malleolus, fifth meta-
tarsal and calcaneus) by a
Vicon motion capture system
(250 Hz). A 2 min warm-up and familiarization phase on the tread-
mill preceded the captured interval. The touchdown of the foot
and toe off were defined by the kinematic data as the first and
second peak in knee extension, respectively [35]. Ultrasonic
images of the soleus were obtained synchronously with a capture
frequency of 146 Hz and soleus fascicle length was measured as
mentioned above. At least nine steps (11.1 ± 1.5) were analysed
for each participant and averaged [8]. Pennation angle was calcu-
lated based on the angle between the deeper aponeurosis and
the reference fascicle and thickness as distance between both apo-
neuroses. The corresponding length changes of the soleus muscle
belly was calculated as the product of fascicle length and the
respective cosine of the pennation angle [36]. Note that this gives
not the length of the entire soleus muscle belly but the projection
of the instant fascicle length to the plane of the MTU, which can
be used to calculate the changes of the belly length. The length
change of the soleus MTU was calculated as the product of kin-
ematic data-based ankle angle changes and the individual
Achilles tendon lever arm [37], while the initial soleus MTU
length was determined at neutral ankle joint angle based on the
regression equation provided by Hawkins & Hull [38]. The vel-
ocities of MTU, fascicles and muscle belly were calculated as the
first derivative of the MTU, fascicle and belly lengths over the
time. From these data we calculated the MTU gearing (VMTU/
VFascicle [16]), tendon gearing (VMTU/VBelly [16]) and belly gearing
(VBelly/VFascicle [16,17]), where V is the stance phase-averaged vel-
ocity of the soleus MTU, fascicles and muscle belly in absolute (i.e.
positive) values. While belly gearing expresses the effects of fasci-
cle rotation, tendon gearing expresses the effects of tendon
compliance and MTU gearing is an overall expression of the effects
of both components on the fascicle velocity [16].
Surface EMG of soleus was measured by means of the wireless
EMG system according to the procedure described above (proces-
sing description in electronic supplementary material) and
normalized to the maximum processed EMG value obtained
from all the individual MVCs (EMGmax). All parameters were
averaged over the same steps as for the muscle fascicle assessment.
(d) Energetic cost of running
After detaching the ultrasound probe, the participants continued
with a 10 min running trial at the same speed (2.5 m s−1).
(a)
(b)
upper 
aponeur.
deeper 
aponeur.
1 cm
F
10
4500
500
1000
1500
2000
2500
3000
3500
4000
20
30
40
50
60
70
80
fascicle length (mm)
force (N)
q
Figure 1. Experimental set-up for the determination of the soleus force–fas-
cicle length relationship. (a) Maximum isometric plantar flexions (MVC) in
eight different joint angles were performed on a dynamometer. During
the MVCs, the soleus muscle fascicle length (F), pennation angle (Θ) and
muscle thickness were measured based on ultrasound images. (b) Exemplary
force–fascicle length relationship of the soleus muscle by the MVCs (squares)
and the respective second-order polynomial fit (dashed line).
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Proc. R. Soc. B 286: 20192560
3
A breath-by-breath cardio pulmonary exercise testing system
(MetaLyzer 3B – R2, Cortex Biophysik GmbH, Leipzig, Germany)
was used to record the percentage of concentration of both
oxygen and carbon dioxide expired and rate of oxygen consump-
tion ( _VO2) and carbon dioxide production ( _VCO2) was calculated
as average of the last 3 min. Running economy was expressed in
units of energy by
energetic cost ¼ 16:89  _VO2 þ 4:84  _VCO2,
ð2:1Þ
where the energetic cost is expressed in [W kg−1] and _VO2 and
_VCO2 in [ml s−1 kg−1] [3,39].
(e) Statistics
Differences between soleus MTU and soleus fascicle length
changes (absolute and normalized to L0) and velocities as well as
between belly gearing and tendon gearing were tested by means
of a paired t-test for dependent samples. The Pearson correlation
coefficient was calculated in order to assess the relationship of
the energetic cost of running and the force–velocity potential,
force–length–velocity potential and fascicle velocity during the
stance phase. As normality was not given for the force–length
potential, we used the Spearman correlation coefficient to assess
its relationship to the energetic cost. A Pearson correlation coeffi-
cient was also used to analyse the relationship of EMG activity
(mean and maximum) and force–length–velocity potential. We
further conducted a multiple regression analysis to assess the mag-
nitude of the effect of the four independent variables of stance
phase-averaged tendon gearing, belly gearing, angular velocity
of the ankle joint as well as Achilles tendon lever arm on the absol-
ute soleus fascicle velocity. The statistics were performed using
SPSS Statistics (IBM Corp., Version 20.0, Armonk, USA) and the
level of significance was set to α = 0.05. All values are reported as
means and standard deviations.
3. Results
The experimentally assessed L0 was on average 41.3 ± 5.2 mm
and corresponding Fmax was 2887.1 ± 724.2 N. The assessed
Vmax based on the values of arel = 0.175 and brel = 1.182 s−1
was 279.0 ± 34.9 mm s−1. Achilles tendon lever arm showed
an average length of 56.7 ± 7.4 mm.
The averaged stance and swing times during running
were 304 ± 23 ms and 439 ± 26 ms, respectively. During the
stance phase, the ankle joint showed angles between 17.0 ±
3.8° dorsiflexion and 14.5 ± 6.0° plantarflexion (figure 2),
and rotated with an average angular velocity of 164 ± 12°/s.
The average activation of soleus normalized to EMGmax
throughout the stance phase was 0.32 ± 0.19 EMGmax and
the maximum activation was 0.52 ± 0.18 EMGmax at 40 ± 6%
of the stance phase (figure 2). While the MTU showed a
lengthening–shortening behaviour during the stance phase,
the muscle fascicles shortened continuously with signifi-
cantly less length changes as the MTU ( p < 0.001; figure 2,
table 1). The pennation angle increased coincidentally with
fascicle
shortening
while
thickness
remained
almost
unchanged (figure 2, table 1). Operating range (i.e. minimum
to maximum) of the fascicles throughout the stance phase
covered the top of the ascending limb of the force–length
curve (0.75 ± 0.09 L0 to 1.01 ± 0.12 L0; figure 3) with a mean
fascicle operating length close to the optimal length (i.e.
0.90 ± 0.10 L0). Accordingly, the averaged force–length poten-
tial of the soleus fascicles was high; (i.e. 0.92 ± 0.07; figure 3).
The soleus fascicles operated between −0.078 ± 0.045
Vmax and 0.322 ± 0.071 Vmax with an average velocity of
0
20
40
60
80
100
–30
–20
–10
0
10
20
30
ankle angle (°)
0
20
40
60
80
100
250
300
350
400
MTU length (mm)
20
40
60
80
100
0
10
20
30
40
50
60
fascicle length (mm)
20
40
60
80
100
0
10
20
30
40
50
pennation angle (°)
20
40
60
80
100
0
5
10
15
20
25
30
thickness (mm)
20
40
60
80
100
stance phase (%)
0
0.2
0.4
0.6
0.8
1.0
EMGNorm 
Figure 2. Ankle angle, soleus muscle–tendon unit (MTU) length, muscle
fascicle length, pennation angle, thickness and electromyographic (EMG)
activity (normalized to maximum voluntary isometric contraction) during
the stance phase of running (2.5 m s−1). Individual (n = 19) data are
shown in thin grey lines and group averages in thick black lines.
royalsocietypublishing.org/journal/rspb
Proc. R. Soc. B 286: 20192560
4
0.118 ± 0.039 Vmax throughout the stance phase (figure 3). The
decoupling of MTU and fascicle length trajectories (figure 2)
enabled a significantly lower absolute operating velocity
of the fascicles (40.0 ± 8.2 mm s−1) compared with the MTU
(166.5 ± 27.7 mm s−1, p < 0.001), resulting in a force–velocity
potential of 0.63 ± 0.09 (figure 3). The achieved total force–
length–velocity potential of the soleus muscle during the
stance phase of running was 0.58 ± 0.10. The calculated velocity
gearing ratios were 4.46 ± 0.98 for MTU gearing, 4.03 ± 0.89 for
tendon gearing and 1.11 ± 0.07 for belly gearing. The magni-
tude of tendon gearing was significantly greater (p < 0.001)
than belly gearing (figure 4).
The energetic cost of running in the investigated velocity
of 2.5 m s−1 was in average 10.69 ± 0.96 W kg−1. An inverse
correlation was observed for the energetic cost and the
overall force–length–velocity potential of the soleus muscle
(r = −0.520, p = 0.022; figure 5). Energetic cost and the force–
velocity potential were also inversely correlated (r = −0.565,
p = 0.012; figure 5), and energetic cost and shortening velocity
were positively correlated (r = 0.561, p = 0.012), indicative for
an association of the economy of running and the operating
velocity of the soleus fascicles during running. The force–
length potential did not show any significant correlation to
the energetic cost (rs = −0.076, p = 0.759; figure 5). A significant
inverse correlation was also observed for the force–length–vel-
ocity potential and the mean (r = −0.504, p = 0.028) and the
maximal EMG activation (r = −0.525, p = 0.021).
The multiple regression model for the assessment of the
fascicle velocity during the stance phase showed a significant
explanatory power ( p < 0.001, R2 = 0.928, adjusted R2 = 0.907)
and was expressed by the equation:
Fasicle velocity¼9:788 (tendon gearing) þ 0:716 (lever arm)
 42:097 (belly gearing)
þ 0:209 (ankle angle velocity) þ 51:341:
The four included independent variables were all signifi-
cant predictors ( p < 0.001 for tendon gearing, tendon lever
arm and belly gearing and p = 0.002 for ankle angular vel-
ocity). Considering the standardized coefficients of −1.006
for tendon gearing, 0.638 for lever arm, −0.367 for belly gear-
ing and 0.310 for the ankle angular velocity, the model
showed that tendon gearing and Achilles tendon lever arm
had the greatest effect on the fascicle velocity.
4. Discussion
By mapping the operating length and velocity of the human
soleus muscle during running onto the individual force–
length and force–velocity curves, we investigated the associ-
ation between the energetic cost of locomotion and the soleus
fascicle force–length and force–velocity potential. The findings
showed that the soleus fascicles operated close to the optimal
length and with moderate continuous shortening during the
stance phase. The significant inverse relationship between
the energetic cost and the force–velocity potential provides
first direct experimental evidence that the fascicle shortening
Table 1. Average values (dimension) as well as changes (range) of the
ankle joint angle (minus indicates dorsiﬂexion), soleus muscle–tendon unit
(MTU) length and fascicle length (absolute and normalized to optimal
length), pennation angle and muscle thickness during the stance phase of
running (n = 19).
dimension
range
ankle angle
−6.1 ± 3.6°
31.5 ± 5.2°
MTU
321.4 ± 22.3 mm
32.2 ± 8.2 mm
(79.5 ± 22.9%L0)
fascicles
36.8 ± 4.2 mm
10.6 ± 3.0 mm*
(25.9 ± 7.8%L0*)
pennation angle
24.0 ± 5.1°
8.9 ± 3.1°
thickness
15.0 ± 3.3 mm
1.7 ± 1.0 mm
*Statistically signiﬁcant difference to MTU (p < 0.05).
0.4
0.6
0.8
1.0
1.2
1.4
1.6
LNorm (L/L0)
0
0.2
0.4
0.6
0.8
1.0
1.2
–0.4 –0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
VNorm (V/Vmax)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
FNorm (F/Fmax)
FNorm (F/Fmax)
Figure 3. Operating length and velocity of soleus muscle fascicles during the stance phase of running mapped onto the averaged normalized force–length and
force–velocity curve. White circles indicate the average operating length and velocity of the stance phase of each participant and the black circle the respective group
average with the standard deviation of all participants (n = 19). The grey shaded areas illustrate the operating range (maximum to minimum) of the operating
length and velocity during the stance phase averaged for all participants. Force is normalized to the maximum force during the maximal isometric plantar flexion
contractions, fascicle length to the experimentally determined optimal fascicle length and fascicle velocity to the assessed maximum shortening velocity. Dotted lines
in the left graph indicate the standard deviation of the individually measured force–length relationships. Note that the data points do not lie on the average curves
because the individual force potentials were calculated for each percentage of the stance phase of each step and then averaged step-wise, which makes a difference
to the calculation using the overall subject-based average length or velocity due to the non-linearity of the curves.
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Proc. R. Soc. B 286: 20192560
5
velocity of the soleus muscle is a determining factor for the
economy of human running.
The triceps surae muscle group contributes substantially
to the overall energetic cost of running [20]. The soleus is the
largest muscle of the triceps surae [40] and the main muscle
to lift and accelerate the centre of mass during locomotion
[14,15]. During the stance phase of running, the fascicles of
the soleus muscle shorten when activated, contributing to the
ankle joint mechanical work/power output [41]. The fascicles
operated on the steep-rising part of the hyperbolic-shaped
force–velocity curve, in average at 11.8% of Vmax, where
already small changes in fascicle shortening velocity cause
relevant effects on the muscle force–velocity potential. As dic-
tated by the force–velocity relationship [7], an increase in
fascicle shortening velocity is accompanied by a decrease in
the muscle force potential. The decrease of the muscle force
potential requires an upregulation of the muscle activation to
maintain the same level of force to support and accelerate the
body mass [4,10]. The observed inverse relationship between
force–velocity potential and energetic cost confirmed our
hypothesis that the soleus fascicle shortening velocity is a key
factor for the energetic cost of running. This link may further
be supported by the observed inverse correlation of EMG
activation and force–length–velocity potential, although it
should be considered that active muscle volume cannot be
assessed accurately from EMG activity. The fascicles worked
in a small range on the upper portion of the ascending limb
of the force–length curve with a high force–length potential
of 0.92. An operating range on the ascending limb close to L0
(0.75–1.01 L0) was a quite consistent observation in the investi-
gated runners, despite notable differences in the optimal
fascicle length (L0 ranging from 33 to 51 mm). In our study,
we did not find any relationship between force–length poten-
tial and energetic cost of running. However, this does not
indicate that the force–length potential is not important for
running economy, but rather that the consistently observed
high force–length potential explained less of the detected varia-
bility in the energetic cost. Besides the favourable high force–
length potential for economical force production, operating
close to optimal length may additionally preserved from rela-
tively higher energetic cost that can arise when contracting at
shorter length. In vitro evidence showed that although force
is reduced at shorter sarcomere length, the ATPase rate seems
not to differ from the rate at optimal length, indicating
0
20
40
60
80
100
stance phase (%)
–100
–50
0
velocity (mm s–1)
belly
fascicle
0
20
40
60
80
100
stance phase (%)
–400
–200
0
200
velocity (mm s–1)
MTU
belly
.  .
Figure 4. Operating velocity of the soleus muscle–tendon unit (MTU) and
muscle belly (top) as well as muscle belly and fascicles (bottom) over the
stance phase, illustrating the effect of tendon gearing and belly gearing,
respectively. Grey shadings indicate the standard deviations (n = 19).
7
8
9
10
11
12
13
14
r = –0.56*
force–velocity potential
energetic cost (W kg–1)
energetic cost (W kg–1)
energetic cost (W kg–1)
7
8
9
10
11
12
13
14
rs = –0.08
force–length potential
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
7
8
9
10
11
12
13
14
r = –0.52*
force–length–velocity potential
0.2
0.4
0.6
0.8
1.0
Figure 5. Association of the force–length–velocity potential, force–velocity
potential and force–length potential of the soleus muscle to the energetic
cost of running. *Statistically significant correlation ( p < 0.05).
royalsocietypublishing.org/journal/rspb
Proc. R. Soc. B 286: 20192560
6
comparably higher cost of contraction at shorter length [42,43].
However, this effect seems more pronounced at very short
lengths, a portion of the force–length curve that is probably
not covered by the soleus during running (operating range
0.75–1.01 L0). Furthermore, we showed that the soleus shor-
tened continuously during the stance phase of running,
which reflects a condition for force depression. However,
since a depression of force was shown to be accompanied by
a decrease in the ATPase activity [44], force depression would
have little or no effect on the energetic cost itself.
During the stance phase, the MTU showed length changes
of 80% L0 while the fascicles showed significantly lower length
changes (i.e. 26% L0). The regression model provided evidence
that the MTU–fascicle decoupling mechanisms of tendon and
muscle belly gearing together with the Achilles tendon lever
arm and ankle joint angular velocity determine the fascicle vel-
ocity. The R2 for the model was 0.928, demonstrating a high
goodness-of-fit and a high explanation of variance of the fasci-
cle velocity. The overall MTU gearing ratio of the soleus muscle
indicated a 4.5 fold reduction of the fascicle operating velocity
during the stance phase of running. The tendon gearing ratio
of 4.03 was notably greater than the belly gearing ratio of
1.11, resulting in a higher standardized regression coefficient
(−1.006 versus −0.367). The observed gearing ratios indicate
that the soleus fascicle velocity during the stance phase of run-
ning is mainly governed by the compliance of the series elastic
element. The high portion of tendon gearing in the soleus
muscle is the consequence of greater length changes of the
Achilles tendon and aponeurosis in relation to the muscle
belly length changes. The soleus produces mechanical work/
energy for the lift and acceleration of the body throughout
the entire stance phase. In the first half, where the MTU is
elongated, the fascicles actively shorten. This means that a
part of the mechanical energy of the human body is transferred
to the tendon. Also, in this setting the muscle fascicles produce
work under favourable conditions due to the force–length and
force–velocity relationships (both potentials in this phase were
very high) and save work as strain energy in the tendon. In the
second half, the tendon strain energy is returned and at the
same time the fascicles produce work by active shortening at
a reduced force–velocity potential (fascicle shortening velocity
is higher in this phase). The higher shortening velocity is
associated with a reduction in the EMG activity and an increase
in belly gearing. It has been suggested that increased gearing at
fast shortening velocities and lower forces is a mechanism that
allows particularly slower type fascicles to be more effective in
generating forces [16]. This supports the idea that the observed
activation pattern promoted an economical MTU interaction
during running.
Belly gearing (or the fascicle rotation component) reduced
the shorting velocity of the soleus significantly by 11% in
average throughout the stance phase (ratio = 1.11). The main
contribution of the fascicle rotation component was in the
second half of the stance phase. In situ experiments have
shown that belly gearing in pennate muscles is variable
with higher ratios during low muscle force to amplify belly
shortening at lower fascicle shortening velocity and lower
ratios during higher levels of muscle force to facilitate
muscle force transmission to the tendon in concentric contrac-
tions [17]. In accordance, we found an almost constant belly
gearing of ≈1 in the soleus muscle during the first half of
the stance phase were activation and consequently force
was increased. When the soleus activation level decreased
and the MTU shortened in the second half, the pennation
angle increased and enabled a greater contribution of the fas-
cicle
rotation
component
to
the
reduction
of
fascicle
shortening velocity (maximum belly gearing ratio of 1.18).
As proposed by the variable gearing concept, the low fascicle
rotation component shown by the soleus muscle during the
first half of the stance phase where muscle activation is
increased, facilitated the orientation of the line of action of
the fascicles to the line of action of the MTU [17].
Our results provide further evidence that the Achilles
tendon lever arm and ankle angular excursions during the
stance phase were important explanatory factors of the fascicle
shortening velocity. The lever arm is an anthropometric charac-
teristic and the results showed that shorter lever arms
translated into lower fascicle shortening velocities. The associ-
ation of the Achilles tendon lever arm and fascicle shortening
velocity in the current study provides first direct experimental
evidence that shorter lever arms increase the force–velocity
potential of the soleus muscle during running. Thus, the
reduced fascicle shortening velocity due to a smaller lever
arm is—in addition to tendon and belly gearing—a mechanism
that improves running economy. Further, the association of the
angular velocity of the ankle joint and fascicle shortening vel-
ocity during the stance phase shows that greater angular
excursions and velocities and in consequence greater length
changes of the soleus MTU lead to uneconomical higher
fascicle operating velocities.
Although the soleus probably contributes to a great portion
of the overall energetic cost during running, other limb muscles
that were not considered in the present study are involved.
However, the main energy source (positive work) is the ankle
joint (41%) [41] and the soleus is the greatest muscle among
the main plantar flexors with respect to physiological cross-sec-
tional area (soleus 63%, gastroc. med. 25%, gastroc. lat. 12%)
and volume (53%, 31% and 16% [40]). The key role of soleus
is further supported by the modelling study of Hamner and
Delp (2013), which showed that the soleus is by far the biggest
contributor to the vertical acceleration and fore-aft acceleration
of the centre of mass [14]. This function is achieved by active
shortening, which reduces the force–velocity potential and
consequently requires a greater active muscle volume. In con-
trast, the quadriceps muscle group, the main contributor
during early stance, decelerating and supporting body mass
[14,15], features more economical fascicle dynamics. Recently,
we showed that the fascicles of the vastus lateralis muscle as
a representative of the quadriceps muscle group operates
with a high force–length (i.e. 0.91) and force–velocity potential
(i.e. 0.97) during the stance phase of running [8]. Operating at
high force potentials minimizes the cost of this muscle, which
is energetically expensive due to its long fascicle length
(i.e. L0 = 94 mm [8]), by reducing active muscle volume. This
may indicate that the mechanical energy by muscular work
required for steady state running is generated by muscles
that are metabolically less expensive (i.e. due to shorter fascicle
length as the soleus muscle), probably to compensate for the
reduction of the force–velocity potential.
To assessthe force–velocity potential we used a biologically
funded value of Vmax, based on in vitro studies on the human
soleus, i.e. 6.77 L0 s−1 (279.0 ± 34.9 mm s−1). However, during
submaximal running in vivo the lower activation level and
selective slow fibre type recruitment may affect the actual
force–velocity potential of the soleus muscle. To evaluate the
effect of the choice of Vmax on the observed inverse correlation
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Proc. R. Soc. B 286: 20192560
7
of force–velocity potential and energetic cost, a sensitivity
analysis was conducted by decreasing and increasing Vmax in
10% intervals and calculating the correlation coefficients,
respectively. The results did not show any substantial effects
on the associations between force–velocity potential and ener-
getic cost until a value of Vmax of <2.0 L0 s−1 (i.e. Vmax+30%:
r = −0.577,
Vmax+20%:
r = −0.574,
Vmax+10%:
r = −0.570,
Vmax−10%:
r = −0.559,
Vmax−20%:
r = −0.552,
Vmax−30%:
r = −0.544,
Vmax−40%:
r = −0.534,
Vmax−50%:
r = −0.522,
Vmax−60%: r = −0.506, Vmax−70%: r = −0.484; p < 0.05), which
confirms and strengthens the overserved association. Further-
more, we assessed the force–length curve during maximal
isometric contractions and used it to calculate the force–
length potential of the soleus muscle during running at
submaximal activation. There is evidence from in vitro studies
that the force–length curve depends on muscle activation
[45,46]. However, in a recent in vivo study by Fontana &
Herzog [47] on the human vastus lateralis muscle, a rightward
shift of optimal length with submaximal activation was not
observed when force was normalized to the maximum EMG
signal. The authors suggested that the shift in optimal length
phenomenon might be related to the in vitro testing set-up
(e.g. non-physiological stimulation frequency range or Ca2+
concentrations). The discrepancy of the in vitro and in vivo evi-
dence clearly warrants future investigation to elucidate the
shifting length phenomenon in the context of in vivo submaxi-
mal locomotion. Given the current human in vivo evidence [47],
we can argue that mapping the submaximal fascicle operating
length onto the force–length curve in the present in vivo study
should not affect the findings.
In the present study we focused on the understanding of
the contribution of the force–length and force–velocity poten-
tial to the energetic cost of running and we showed that the
force–velocity potential is inversely related to the energetic
cost, explaining about one-third of its variance. We argue
that an increase of active muscle volume due to the decreased
force–velocity potential would increase the energetic cost of
running. However, it must be acknowledged that the energetic
cost of muscle contraction is complex and multifactorial.
Independent of active muscle volume, in higher shortening vel-
ocities the rate of cross-bridge cycling is increased and as a
consequence so is the consumed energy. In our study, shorten-
ing velocities of the soleus muscle were on average 0.118 Vmax
throughout the stance phase, a range where the rate of ATP
hydrolysis shows a steep increase [48]. Furthermore, in sub-
maximal intensity contractions as during our investigated
running velocity selective slow fibre type activation might
decrease the energetic cost by reducing the contribution of
energetically more expensive fast twitch fibres.
5. Conclusion
In conclusion, this study provides for the first time experimen-
tal evidence that the energetic cost of running is related to the
force–length–velocity potential of the soleus muscle with lower
shortening velocities of the fascicles as the main influencing
factor (i.e. higher force–velocity potential). The main mechan-
ism for the underlying reduction of the fascicle shortening
velocity during the stance phase was gearing within the
MTU, particularly greater tendon gearing, a shorter Achilles
tendon lever arm as well as, to a minor extent, a lower ankle
angular velocity.
Ethics. The ethics committee of the Humboldt-Universität zu Berlin
approved the study and the participants gave written informed con-
sent in accordance with the Declaration of Helsinki.
Data accessibility. The datasets generated and analysed during the cur-
rent study are available at https://dx.doi.org/10.6084/m9.figshare.
10119320.v1.
Authors’ contributions. S.B., F.M. and A.A. designed research; S.B., F.M.
and A.S. performed research; S.B. analysed data; S.B. and A.A.
drafted the manuscript and F.M. and A.S. made important intellectual
contributions during revision.
Competing interests. We declare we have no competing interests.
Funding. Funding for this research was supplied by the German
Federal Institute of Sport Science (grant no. ZMVI14-070604/17-18).
Acknowledgements. We acknowledge the support by Theresa Domroes,
Antonis Ekizos and Arno Schroll for data analysis.
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The force-length-velocity potential of the human soleus muscle is related to the energetic cost of running.
Bohm, Sebastian,Mersmann, Falk,Santuz, Alessandro,Arampatzis, Adamantios