Journal of Biomechanics 45 (2012) 1092–1097
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Journal of Biomechanics
journal homepage: www.elsevier.com/locate/jbiomech
www.JBiomech.com
Young Scientist Pre-doctoral Award 2010
Limitations to maximum sprinting speed imposed by muscle
mechanical properties
Ross H. Miller n, Brian R. Umberger, Graham E. Caldwell
Department of Kinesiology, University of Massachusetts, Amherst, MA, USA
a r t i c l e i n f o
a b s t r a c t
Article history:
Accepted 12 April 2011
It has been suggested that the force–velocity relationship of skeletal muscle plays a critical limiting role
in the maximum speed at which humans can sprint. However, this theory has not been tested directly,
and it is possible that other muscle mechanical properties play limiting roles as well. In this study,
forward dynamics simulations of human sprinting were generated using a 2D musculoskeletal model
actuated by Hill muscle models. The initial simulation results compared favorably to kinetic, kinematic,
and electromyographic data recorded from sprinting humans. Muscle mechanical properties were then
removed in isolation to quantify their effect on maximum sprinting speed. Removal of the force–
velocity, excitation–activation, and force–length relationships increased the maximum speed by 15, 8,
and 4%, respectively. Removal of the series elastic force–extension relationship decreased the
maximum speed by 26%. Each relationship affected both stride length and stride frequency except
for the force–length relationship, which mainly affected stride length. Removal of all muscular
properties entirely (optimized joint torques) increased speed (þ 22%) to a greater extent than the
removal of any single contractile property. The results indicate that the force–velocity relationship is
indeed the most important contractile property of muscle regarding limits to maximum sprinting
speed, but that other muscular properties also play important roles. Interactions between the various
muscular properties should be considered when explaining limits to maximal human performance.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Sprinting
Speed
Limits
Muscle
Force–velocity
1. Introduction
Sprinting has a long history of research in biomechanics and
human movement science (Furusawa et al., 1927a,b; Best and
Partridge, 1928; Hill, 1928). Sprinting abilities vary widely among
individuals, but all runners have a maximum sprinting speed that
they cannot exceed. Here we define the maximum sprinting
speed as the fastest steady speed attained during the ‘‘plateau’’
phase of a sprint, when the average speed over one stride is
relatively constant from stride to stride.
Several investigators have studied mechanical factors that
influence the maximum sprinting speed. While both stride length
and stride frequency increase as speed increases from a slow jog
to a maximum sprint, the stride length plateaus at a moderate
speed, and the stride frequency increases up until the maximum
speed (Luhtanen and Komi, 1978). Greater stride frequencies
require the legs to move through the stride cycle at faster rates,
and the muscles to shorten and lengthen more rapidly. This
finding has directed attention on the well-known force–velocity
n
Correspondence to: Department of Mechanical & Materials Engineering, 319
McLaughlin Hall, Queen’s University, Kingston, ON, K7L 3N6, Canada.
Tel.: þ1 613 549 6666; fax: þ 1 613 549 2529.
E-mail address: rosshm@gmail.com (R.H. Miller).
0021-9290/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jbiomech.2011.04.040
relationship of skeletal muscle (Hill, 1938; Katz, 1939) as the
potential ‘‘bottleneck’’ that ultimately limits maximum sprinting
speed (Chapman and Caldwell, 1983; Weyand et al., 2010).
Measurements or estimates of force–velocity parameters in vivo
correlate with training-induced improvements in sprint performance (Cormie et al., 2010), distinguish between the muscles of
sprint athletes and other individuals (Thorstensson et al., 1977),
and are better predictors of sprint performance than measures of
isometric strength (Kukolj et al., 1999).
These findings motivate the question of how fast humans
could run if there was no force–velocity relationship. An isolated
property of skeletal muscle cannot be adjusted or removed in vivo,
but computer simulations can be used to investigate these effects.
In forward dynamics computer simulations, the force–velocity
relationship and other muscle mechanical properties have been
shown to influence the performance of walking (Gerritsen et al.,
1998), running (Sellers et al., 2010), pedaling (Van Soest and
Casius, 2000), and jumping (Van Soest and Bobbert, 1993).
Because sprinting speed is the product of both stride length and
stride frequency, the maximum speed is ultimately limited by
both variables. It is unknown how the mechanical properties of
muscle, such as the force–length and force–velocity relationships,
influence the maximum stride length and frequency, and therefore the maximum sprinting speed.
R.H. Miller et al. / Journal of Biomechanics 45 (2012) 1092–1097
Therefore, the purpose of the study was to determine the
extent to which individual muscle mechanical properties affect
the maximum sprinting speed and the stride length/frequency
combination used to run at that speed. The approach taken was to
remove specific mechanical properties from the dynamics of
neuromuscular force production and observe the resulting effects
on the simulated sprint performance. We hypothesized that
removing the force–velocity relationship would have the greatest
effect on maximum sprinting speed compared to other muscular
properties.
2. Methods
2.1. Experimental data
Twelve adult females (mean 7 sd: age¼ 27 76 years, height ¼1.66 7 0.05 m,
mass¼ 61.07 4.7 kg) participated as subjects after providing informed consent to
protocols approved by the university institutional review board. Each subject
performed five trials of maximal effort sprinting along a 40-m runway while
reflective marker positions, ground reaction forces (GRF), and surface electromyograms (EMG) were sampled from one stride near the middle of the runway.
Marker positions were lowpass filtered (Butterworth, 4th order recursive) at 12 Hz
and used to calculate joint and segment angles. GRF were lowpass filtered at
75 Hz. EMG were bandpass filtered (20–300 Hz), detrended, rectified, then lowpass filtered at 5 Hz to calculate linear envelopes. Linear envelopes were scaled by
their maximum values recorded during maximum isometric contractions. Data
were averaged over trials, then over subjects.
2.2. Computer model
Sprinting was simulated using a 2D bipedal model of the human musculoskeletal system (Fig. 1). The skeleton had nine rigid segments and 18 Hill-based
muscle models (nine per leg). Muscle activation and contractile dynamics, along
with details on muscle-specific parameters, are presented in the Electronic
1093
Supplementary Material. Each muscle was controlled by an excitation signal that
was a periodic, piecewise linear function of 21 nodal values spaced evenly over the
time for one stride. Excitations to the right and left sides were identical but phaseshifted by 50%. Passive restoring torques restricted the joints to realistic ranges of
motion (Riener and Edrich, 1999). Ground contact elements beneath the heel, toe,
and metatarsophalangeal (MTP) joint of each foot generated the vertical (Fgc y)
and horizontal (Fgc x) ground reaction forces:
3
F gcy ¼ Agc 9ygc 9 exp ðBgc y_ gc Þ
F gcx ¼ mF gcy tanh
x_ gc
g
ð1Þ
ð2Þ
where xgc and ygc are the horizontal and vertical positions of the contact element
(ygc ¼0 at the ground). The contact model parameters were Agc ¼ 2.5E þ 08 N m 3,
Bgc ¼ 0.85 N s m 1, m ¼ 0.8, and g ¼ 0.1 m s 1. The supportive effect of arm swing
(Hinrichs et al., 1987) was modeled as a vertical force Farms at the shoulder:
4pt
farms
ð3Þ
F arms ¼ Aarms sin
tf
where Farms and farms are, respectively, the amplitude and phase shift of the arm
swing force, tf is the stride duration, and t is time.
2.3. Simulations
Simulations of one step were generated by optimizing the model’s control
variables to maximize the objective function J:
J¼
DxCoM
tf
ð0:01ey þ 0:0001eo þ 0:001epas þ egrf Þ
ð4Þ
where DxCoM is the change in horizontal position of the center of mass and the
bracketed terms are penalties. The penalty weighting coefficients were explored in
preliminary simulations and set to the smallest values that produced nearly
periodic strides. ey and eo were the sums of the squared differences between the
initial and final segment angular positions and velocities, respectively, and
encouraged periodic kinematic states. epas was the sum of the integrals of the
squared passive joint moments, and discouraged joint hyperextension. egrf was the
ratio between the absolute braking and propelling impulses of the horizontal GRF
component, with the larger impulse in the numerator, minus one to make the
optimal value zero, and encouraged steady speeds. Data for one complete stride
were reconstructed from one step by assuming bilateral symmetry (Anderson and
Pandy, 2001). Optimizations were performed using a parallel simulated annealing
algorithm (Higginson et al., 2005) that required about 3,000,000 function calls and
1.5 days to generate each simulation.
2.3.1. Control variables
The control variables were 180 independent muscle excitation nodes (10 per
muscle per step), the initial linear and angular velocities, the amplitude and phase
shift of Farms, the angular stiffness of the MTP joint, and the stride duration. The
initial angular velocities were bounded to be within 7 1 between subjects’
standard deviation of the mean values from the experimental data at initial
ground contact of the right foot. The stride duration was also bounded between
7 1 between subjects’ standard deviation. The arm swing force’s amplitude could
not exceed 25% of the model’s weight (Miller et al., 2009) and the MTP stiffness
could not exceed 300 Nm per rad s 1 (Stefanyshyn and Nigg, 1997).
2.3.2. Initial conditions
The initial angular positions were specified as the mean experimental values
from initial ground contact of the right foot. The initial horizontal hip position was
arbitrary, and the initial vertical hip position was set to the lowest value that gave
a vertical GRF of zero. The initial muscle model activations were calculated by
solving the equations for excitation–activation coupling over the time for one
stride. The initial muscle model CC lengths were calculated by assuming the CC
was isometric at the initial muscle length and activation.
Fig. 1. Diagram of the 2D computer model used to simulate sprinting. The model
included nine segments (trunk, thighs, legs, feet, and toes) and nine muscles per
leg (Iliopsoas, glutei, vasti, biceps femoris (short head), tibialis anterior, soleus,
rectus femoris, hamstrings, and gastrocnemius). Farms represented the kinetic
effect of arm swing.
2.3.3. Muscle mechanical properties
The optimization was performed six different times with six different muscle
model definitions. In the first optimization, the full muscle model was used. In the
second, third, and fourth optimizations, the bounds on the initial linear and
angular velocities were increased to 7 3 standard deviations, and the force–
length, force–velocity, and excitation–activation relationships of the CC were,
respectively, removed in isolation (i.e. each simulation was missing one specific
relationship). In the fifth optimization, the SEC force–extension relationship was
removed, while all of the CC relationships were retained. Fig. 2 shows graphical
examples of how the muscle mechanical properties were adjusted; see the
Electronic Supplementary Material for details. In the sixth optimization, the joint
torques were optimized directly, with no consideration for muscular dynamics.
Like the muscle excitations, the joint torque histories were parameterized by 21
nodal values and smoothed by a sixth-order polynomial prior to applying the
Hip (deg)
Horiz. GRF (bw)
Vert. GRF (bw)
R.H. Miller et al. / Journal of Biomechanics 45 (2012) 1092–1097
Knee (deg)
1094
Ankle (deg)
Stride (%)
Stride (%)
Fig. 3. Joint angles and GRF components vs. the stride cycle from the human
subjects (dashed lines) and the initial simulation (solid lines). Shaded area is
72 between subjects’ standard deviations around the mean of the human
subjects. The stride begins and ends at initial contact of the right foot.
Fig. 2. Changes made to the muscle mechanical properties to remove their
influence from the simulations. FV ¼ CC force–velocity relationship; FL¼ CC
force–length relationship; EA ¼CC excitation–activation relationship; FE ¼ SEC
force–extension relationship.
torques to the skeleton. The maximum permitted torques were set by calculating
the products of the muscle maximum isometric forces times their maximum
moment arms, and summing these products over all muscles spanning each joint.
2.4. Evaluation
The primary outcome variables of the simulations were the average horizontal
speed of the center of mass, the stride length, and the stride frequency. We also
examined how forces produced by the muscle models were affected when the
force–velocity relationship was removed.
3. Results
When all muscle mechanical properties were included, the
model sprinted at 6.75 m s 1, which was 5% faster than the
subjects (6.4270.61 m s 1 on average). The model’s stride length
and frequency (3.39 m and 1.99 Hz) were, respectively, 6% longer
and 1% lower than the means of the subjects (3.2170.30 m and
2.0170.24 Hz). The simulated joint angles and GRF usually fell
within two standard deviations of the mean experimental data
(Fig. 3) even though tracking of these data was not part of the
optimization problem. The cross-correlations between the muscle
model activations and linear envelopes (Fig. 4) averaged 0.60
(range 0.19–0.95). For comparison, cross-correlations between the
linear envelopes for the human subjects (i.e. every subject compared to every other subject) averaged 0.65 (range 0.26–0.98).
Removing any of the CC muscle mechanical properties
increased the model’s maximum sprinting speed, while removing
the SEC force–extension relationship reduced speed (Fig. 5).
Removing the force–velocity relationship had the greatest effect
among the three CC properties: speed increased by 15% to
7.76 m s 1, with a 5% increase in stride length and a 9% increase
in stride frequency. Removing the force–length relationship
increased speed by 4%, due almost entirely to an increase in stride
length. Removing the excitation–activation relationship increased
speed by 8%, with a 2% increase in stride length and a 6% increase
in stride frequency. The greatest effect on speed was produced by
removing the SEC force–extension relationship, which decreased
the sprinting speed by 26% to 5.00 m s 1, accompanied by 14%
decrease in stride length and a 13% decrease in stride frequency. In
this simulation, the periodicity penalty terms were three times
larger than any other simulation (average deviations of 111 and
671 s 1, compared to 2.81 and 221 s 1 for the other simulations),
indicating that the allowed bounds on the initial state were
outside the model’s ability to perform a periodic stride. When all
muscle mechanical properties were removed (joint torques optimized), the speed increased by 22% to 8.24 m s 1, with an 8%
increase in stride length and a 13% increase in stride frequency.
The effect of the force–velocity relationship on the abilities of
muscles to produce force is seen in Fig. 6. Removing the force–
velocity relationship increased the average force by at least 130 N
for every muscle but gastrocnemius. The greatest absolute
increases in average force were seen in iliopsoas (802 N), glutei
(644 N), and vasti (543 N) while the smallest were in gastrocnemius (26 N), tibialis anterior (130 N), and rectus femoris (152 N).
The greatest relative increases in average force (scaled by maximum isometric force) were seen in iliopsoas (35%), biceps
femoris (23%), and glutei (16%) while the smallest were seen in
gastrocnemius (1%), soleus (3%), and tibialis anterior (5%). For
muscles that showed a distinct force peak within the stance
phase, the peak occurred later in stance for glutei (4% of the
stride), vasti (7%), biceps femoris (14%), and gastrocnemius (4%).
These results indicate that removal of the force–velocity relationship affected both the magnitude and timing of muscular force
production.
R.H. Miller et al. / Journal of Biomechanics 45 (2012) 1092–1097
1.0
0.5
0
0
25 50 75 100
1.0
0.5
25 50 75 100
SOL
0
0
25 50 75 100
25 50 75 100
0
25 50 75 100
1.0
0.5
0
0
0.5
25 50 75 100
1.0
0.5
25 50 75 100
0
0
1.0
0
1.0
0
0
0.5
0
25 50 75 100
HAM
0.5
0
TA
0
1.0
BF
VAS
0.5
GAS
0
1.0
RF
GLU
ILP
1.0
1095
0.5
0
25 50 75 100
0
Stride (%)
Stride (%)
Stride (%)
EMG linear envelope
Model excitation
Model activation
Fig. 4. Optimized muscle excitations (thin lines), activations (thick lines), and EMG linear envelopes (dashed lines) for the sprinting simulation. ILP ¼iliopsoas;
GLU ¼glutei; VAS ¼ vasti; BF¼ biceps femoris (short head); TA ¼ tibialis anterior; SOL ¼soleus; RF¼ rectus femoris; HAM¼ hamstrings; GAS ¼gastrocnemius. The stride
begins and ends at heel-strike. Vertical dashed lines indicate toe-off. Linear envelopes were scaled by values from maximum isometric contraction. No EMG data were
available for ILP and BF.
4. Discussion
Speed (ms-1)
10
8
6
4
2
0
All
No FV
No FL
No EA
No FE
None
All
No FV
No FL
No EA
No FE
None
All
No FV No FL No EA No FE
Muscular Properties
None
4
SL (m)
3
2
1
0
SF (Hz)
3
2
1
0
Fig. 5. (a) Speeds, (b) stride lengths, and (c) stride frequencies from the sprinting
simulations with various muscular properties removed. FV, FL, EA, and FE are the
force–velocity, force–length, excitation–activation, and force–extension relationships, respectively. Horizontal dashed lines indicate values from the initial
simulation that included all muscular properties.
In this study, computer simulations were used to quantify the
effects of muscle mechanical properties on maximum sprinting
speed. Removing CC properties enabled faster speeds, while
removing the SEC force–extension relationship greatly impaired
speed. The changes in speed resulted from modifications in both
stride length and the stride frequency, except for removal of the
force–length relationship, which primarily altered stride length.
Concurrent changes in stride length and stride frequency would
be expected to alter both the timing and magnitude of muscular
force production, as seen in Fig. 6 when the force–velocity
relationship was eliminated.
For the individual muscular properties, the greatest increase in
speed was seen when the force–velocity relationship was
removed ( þ15%). This finding supports the contention made in
several previous studies that the force–velocity relationship is a
critical limiting factor in sprint performance (Chapman and
Caldwell, 1983; Weyand et al., 2000, 2010). However, removal
of the other CC properties also increased the maximum sprinting
speed ( þ8% for excitation–activation; þ 4% for force–length).
Even a 4% difference in maximum speed would have a major
effect on the outcome of a 100- or 200-m race. Further, when all
muscular dynamics were eliminated and the model was actuated
by optimized joint torques, the maximum speed increased to the
fastest speed observed in any simulation ( þ22%). Removal of CC
properties increases speed because they limit the potential kinematic states at which muscles can develop large forces. The
largest forces can be generated only when the CC is near optimal
length in isometric or eccentric conditions. Removing all muscular properties increased speed to a greater extent ( þ22%) than
removing any individual contractile property ( þ4, þ8, and
þ15%), but the increase was less than the sum of these individual
effects ( þ27%). This result suggests that the speed-limiting effects
of CC properties interact with each other. Interactions between
the force–velocity relationship and other muscular properties
R.H. Miller et al. / Journal of Biomechanics 45 (2012) 1092–1097
4000
2000
0
6000
6000
4000
4000
2000
6000
4000
4000
2000
2000
0
0
0 25 50 75 100
6000
4000
2000
0
0 25 50 75 100
6000
4000
4000
SOL
6000
0
0 25 50 75 100
HAM
6000
2000
0
0 25 50 75 100
BF
VAS
2000
0
0 25 50 75 100
TA
RF
GLU
ILP
6000
0 25 50 75 100
6000
GAS
1096
2000
0
4000
2000
0
0 25 50 75 100
0 25 50 75 100
0 25 50 75 100
Stride (%)
Stride (%)
Stride (%)
Fig. 6. Muscle forces (in newtons) during the stride cycle (heel-strike at 0% and 100%; toe-off indicated by vertical dashed lines). Broken lines from sprinting simulation
with all muscle mechanical properties (speed ¼ 6.75 m s 1). Solid lines from sprinting simulation with the force–velocity property removed (speed ¼7.76 m s 1). Muscle
key: ILP ¼ Iliopsoas, GLU ¼glutei, RF¼ rectus femoris, VAS ¼vasti, BF¼ biceps femoris (short head), HAM ¼hamstrings, TA ¼ tibialis anterior, SOL ¼soleus, and
GAS ¼gastrocnemius.
should thus be considered when seeking to explain limits to
maximal performance within the muscular system.
An example of muscular property interaction is seen when the
SEC force–extension relationship is removed. In contrast to the CC
properties, removing the SEC force–extension relationship substantially reduced the model’s maximum sprinting speed ( 26%).
Sellers et al. (2010) recently reported a similar result. Under
normal conditions, SEC compliance allows the SEC to take up a
portion of the change in total muscle length, which facilitates the
storage and utilization of elastic strain energy (Cavagna, 1977;
Jacobs et al., 1993) and enables the CC to operate at slow
shortening velocities (Alexander, 2002). Without a compliant
SEC, the CC must shorten at faster velocities, which impair force
production. These interactions predict that muscles with long
tendons and short muscle fascicles (e.g. gastrocnemius) should be
able to produce force economically with its fibers remaining
relatively isometric even during fast dynamic movement when
the whole muscle changes length quickly, as others have suggested (Morgan et al., 1978). This state has been observed in
human gastrocnemius during both experimental (e.g. Lichtwark
et al., 2007) and simulation (e.g. Hof et al., 2002) studies of
running, and was also observed in the present study. In simulations with the force–velocity relationship removed, gastrocnemius force production was essentially unaffected (Fig. 6) because
its force–extension relationship allowed the muscle to shorten at
a slow velocity in both cases.
Even when all muscle mechanical properties were removed,
the model did not achieve unrealistically fast speeds. The fastest
steady speed of the model over one stride was 8.24 m s 1. Elite
sprinters can greatly exceed this speed. Florence Griffith-Joyner’s
average speed during her entire world record 100-m dash
performance was 9.53 m s 1. The missing explanatory variable
is muscular strength. In a sensitivity analysis, we found that the
maximum isometric muscle strengths needed to be doubled in
order for the model to achieve world-class speeds. With no
muscular dynamics (joint torque optimization), the model can
achieve super-human speeds if the maximum joint torque bounds
are removed. Here the speed is limited entirely by non-muscular
factors, such as the mass and inertia of the limbs, joint ranges of
motion, and the allowed ranges for the stride duration and initial
conditions.
Although tempting, it is difficult to make conclusions on sprint
training from the present results since we did not manipulate the
muscular properties over a physiological range. However, the
results could be interpreted as an indication that the ability to
generate force at a variety of muscle velocities (power) is more
important for sprinting speed than the ability to generate force at
a variety of muscle lengths (flexibility). It is also important to
consider that the model is a greatly reduced analog of the full
human neuro-musculo-skeletal system, and is therefore subjected
to some limitations. The model was 2D and any dynamics related
to the medial–lateral and transverse planes (and any interactions
between these planes and the sagittal plane) were neglected. The
model lacked explicit arm segments, although the arm swing
force is intended to represent the net kinetic effect of arm swing
on supporting the center of mass (Hinrichs et al., 1987). The role
of the arms in balancing the transverse angular momentum of the
legs (Hinrichs, 1987) is unnecessary in this 2D model. Finally, the
model’s state equations are too complex to derive an analytical
solution, and there is no assurance that the optimization algorithm
located the global maximum of the solution domain. However, the
solution space was searched extensively ( 3 million solutions
evaluated per simulation) and repeated optimization runs with
different random number generator seeds converged to essentially
the same solutions. We are therefore confident that the solutions
presented are very near the global ‘‘maximum speed’’ solutions.
In conclusion, the force–velocity relationship had the greatest
limiting effect on maximum speed among the contractile properties
of muscle. This result supports previous theories that the force–
velocity relationship plays a critical limiting role in the achievement
of maximum sprinting speed. However, other muscular properties,
in particular the force–extension relationship of the SEC, also affect
the maximum speed. Interactions between these properties should
be considered when explaining limits to human muscular performance. Different theories on specifically how and when the force–
velocity relationship limits sprinting speed have been posed in
R.H. Miller et al. / Journal of Biomechanics 45 (2012) 1092–1097
experimental studies (Chapman and Caldwell, 1983; Weyand et al.,
2000). Investigating these theories will be a direction for future
work. Other possible unexplored topics include the removal of CC/
SEC properties from individual muscles, altering property-defining
parameters within a physiological range rather than removing them,
and strengthening individual muscles for training implications.
Electronic Supplementary Material
The Electronic Supplementary Material on the journal’s website includes a description of the muscle model, the algorithm
used to calculate muscle forces and rates of change in activation
and CC length, and the assignment of muscle-specific parameters.
In addition, a description of how the muscle mechanical properties were removed is provided.
Conflict of interest statement
The authors have no personal or financial conflicts of interest
related to publication of the present work.
Acknowledgments
The authors would like to thank the American Society of
Biomechanics for their consideration and for the opportunity to
present our work.
Appendix A. Supplementary materials
Supplementary materials associated with this article can be
found in the online version at doi:10.1016/j.jbiomech.2011.04.040.
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