16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
Early Ideas about Muscular Contraction
Hippocrates thought that the tendons caused the body to move (he confused
tendons with nerves, and in fact used the same word, neuron, for both).
Aristotle compared the movements of animals to the movements of puppets and
thought that the tendons played the role of the puppet strings, bringing about
motion as they were tightened and released.
Muscles themselves were not credited with the ability to contract until the third
century BC when Erasistratus suggested that the animal spirit flows from the
head through the nerves to the muscle. He thought the nerves were hollow
tubes, through which the muscles could be filled with pneuma, causing them to
expand in breadth but contract in length, thus moving the joints. Actually,
muscles don't increase in volume as they contract!
Jan Swammerdam in the early 1660's showed that muscle contracts without
changing its volume. Using a frog muscle in a sealed air-filled glass and
preserving the length of a nerve, the nerve was stimulated mechanically (pulling
on it with a fine wire). A drop of water in the small tube should have risen if the
muscle volume increased, but it did not. Extended to human muscles by Francis
Glisson in 1677 (arm in a water filled rigid tube, sealed at the elbow).
Most of what has been learned about muscle mechanics is from whole muscles
removed from the animal. Many of the most important experiments being
performed between 1910 and 1950 by A.V. Hill and his collaborators at
University College, London. Isolated from muscle preparation (alive for several
days in an oxygenated solution). When given a stimulus, the mechanical and
thermal activation will not be synchronous at all points because the wave of
electrical excitation is fairly slow (30-40 m/s in amphibian muscle).
The fact that muscle is turned on electrically is very interesting.
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
Figure 1. The experiment of Jan Swammerdam, circa 1663, showing that a
muscle does not increase in volume as it contracts. A frog’s muscle (b) is placed
in an air-filled tube closed at the bottom (a). When the fine wire (c) is pulled, the
nerve is pinched against the support (d), causing the muscle to contract. The
drop of water in the capillary tube (e) does not move up when the muscle
contracts. From Needham (1971).
Mechanical Events: Twitch and Tetanus
The first mechanical event it is possible to measure following stimulation is not
the development of force, but the resistance to an externally imposed stretch.
Even before the electrical action potential is over, about 3-5 msec after the
stimulating shock, the contractile machinery feels stiffer to an external pull than
it does when subjected to a similar pull without first being shocked.
There is latency for about 15 msec following the shock and the muscle produces
no force (if stimulated under isometric = constant length conditions). Finally, the
muscle responds, and if it was given a single stimulus, it produces a single
transient rise in tension = TWITCH.
The strength of the stimulus must be strong enough to depolarize the muscle
membrane - otherwise nothing happens. Over a limited range above the
threshold amplitude, the peak force developed in the twitch rises with the
strength of the stimulus, as more muscle fibers are recruited into the force-
generating enterprise. Once the majority of muscle fibers become active there are
no further increases in force.
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
If a train of stimulations is given, the force has a steady magnitude with a little
ripple at the stimulation frequency = Unfused Tetanus. As the frequency is
raised, mean force rises and the ripple finally reaches a very low level (about 30
shocks per sec). Further increase in freq. produce no further increases in mean
force = TETANIC FUSION (mammalian muscle at body temp. 50-60 shocks per
sec).
Figure 2. Twitch and tetanus. When a series of stimuli is given, muscle force
rises to an uneven plateau (unfused tetanus) which has a ripple at the frequency
of stimulation. As the frequency is increased, the plateau rises and becomes
smoother, reaching a limit as the tetanus becomes fused.
Tension-Length Curves: Passive and Active
Marey knew that somehow the elasticity of muscle must be one of the features
that determine how the separate effects of a sequence of shocks coalesce in a
tetanus. There are 2 separate elements of elastic behavior: one due to PASSIVE
and one due to ACTIVE properties.
Passive properties:
Force is recorded as the muscle is stretched to a number of constant lengths, with
no stimulation. Curve gets progressively steeper with larger stretch, same reason
that a piece of yarn gets stiffer as it's extended - fibrous elements which were
redundant at low extension become tensed at high extension, thereby adding
their spring stiffness in parallel.
The derivative of stress with respect to strain, dσ/dλ, is shown to be a linearly
increasing function of the stress ( σ = F/A)
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
dσ
1
dλ
= α(σ +β)
where λ is the Lagrangian strain,
8
G22
G22
o
, when a muscle of rest length G22
8
o
is
stretched to a new length
88
G22
.
Integrating,
2 σ = μe
αλ
? β
where μ is the constant of integration (rabbit heart muscle, many collagenous
tissues - tendons, skin, resting skeletal muscle obey similar exponential
relationships between stress and strain. No plausible derivation of this form from
first principles has yet been given!
When the muscle is tetanized, the tension at each length is greater than it was
when the muscle was resting (some show a local max.)
Developed tension (difference between the active (tetanized) and passive curves)
is greatest when the muscle is held at a length close to the length it occupied in
the body. The maximum developed stress is almost a constant, about 2 kg/cm
2
(in mammalian muscles taken from animals of a wide range of body sizes).
Noteworthy, because many other parameters (shortening speed, activity of
enzymes controlling metabolic rate) are very different in animals and even
between muscles. Cross-sectional area of muscle doesn't have a unique meaning
in muscle which tapers down into a tendon on either end, so divide the weight in
grams by the length in centimeters (assuming muscle density ~ 1 g/cm
3
).
Figure 3. Tension-length curves for frog satorius muscle at 0° C. The passive
curve was measured on the resting muscle at a series of different lengths. The
tetanized curve was measured at a series of constant lengths as the muscle was
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
held in isometric contraction. The rest length, l
o
, was the length of the muscle in
the body. From Aubert et al. (1951).
Figure 4. Schematic force-length curves. The pennate-fibered gastrocnemius
(left), with its short fibers and relatively great volume of connective tissue, does
not show a local maximum in the tetanic length-tension curve. By contrast, the
parallel-fibered sartorius (right) does show a maximum.
Conceptual Model of Muscle
Figure 5. (a) Quick-release apparatus. When the catch is withdrawn, the muscle
is exposed to a constant force determined by the weight in the pan. (b) The
muscle is stimulated tetanically. Upon release of the catch, the muscle shortens
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
rapidly by an amount ? x
2
which depends on the difference in force before and
after release. (c) A conceptual model of muscle.
Series elastic component in this diagram is in the tendons, since they must
ultimately transmit the muscle force to bone. The parallel elastic component acts
in parallel with the part of the muscle which generates the force - contractile
component.
Together, parallel and series elastic components account for the passive tension
properties of muscle. This model only represents the gross features of whole
muscle mechanics. (single muscle fibers, without tendons, requires more
advanced models, Ch 5).
Series Elastic (SE) Component
Quick release experiments provide direct evidence of a series elastic component.
The rapid change in length which accompanies the sharp change in load is
consistent with the mechanical definition of a spring, which has a unique length
for every tension but is entirely indifferent to how fast its length is changing.
Series elastic element for both skeletal and cardiac muscle has been shown to fit
the same exponential form [2. σ = μe
αλ
? β (tension=y vs. extension=x) found for
the parallel elastic element].
Force - Velocity Curves
All of this assumes that the contractile component (CC) is damped by some
viscous mechanism and cannot change its length instantaneously. You've
probably all experienced that muscles shorten more rapidly against light loads
than they do against heavy ones. (lift a light or heavy weight from the floor).
Inertia, yes, but main cause it that muscles which are actively shortening can
produce less force than those which contract isometrically.
Suppose that the CC is not capable of instantaneous length change. Then all the
rapid shortening in the quick release experiment is taken up in the SE
component. Further length changes must now be attributed to the CC alone since
the tension, and thus the SE length, is held constant.
Particularly important is the rate at which the CC shortens before it has time to
move very far from its initial length (broken line tangent to the length-time curve
just after the rapid shortening phase, 1.8b). When this initial slope is plotted
against the isotonic afterload, T, a characteristic curve is obtained which shows
an inverse relation between force, T, and shortening velocity, v. A.V. Hill (1938)
proposed an empirical relation which emphasized the hyperbolic form of the
curve,
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
3. (T + a)(v + B) = (T
o
+ a)b
This is a rectangular hyperbola whose asymptotes are not T=0 and v=0 but T=-a
and v=-b. The isometric tension T
o
defines the force against which the muscle
neither shortens nor lengthens, and the speed v
max
=bT
o
/a is the shortening
velocity against no load. Hill's Equation is found to describe nearly all muscles
thus far examined, including cardiac and smooth muscle as well as skeletal
muscle (even contracting actomyosin threads, more on that later, read Chapters).
Only insect flight muscle seems to be an exception, and this muscle is
extraordinary in may other respects, particularly in its very short working stroke.
Hill's equation can be written in normalized form:
4. v′ = (1?′ T / k )T )/ (1+′
where v′ = v / v
max
, T ′ = T / T
o
, k = a / T
o
= b / v
max
. For most vertebrate muscles,
the curve described by Hill's eq. has a similar shape. In fact, k usually lies within
the range 0.15< k < 0.25.
The mechanical power output available from a muscle,
5. Power = Tv =
v(bT
o
? av)
v + b
has a maximum when the force and speed are between a 1/3 and a 1/4 their
maximum values. It is apparent that the speed of shortening controls the rate at
which mechanical energy leaves the muscle. The peak in the curve corresponds
to about 0.1 T
o
v
max
watts. Bicycles have gears so that you can take advantage of
this, by using the gears they can keep muscle shortening velocity close the
maximum-power point.
Active State
The fact that muscle develops its greatest force when the speed of shortening is
zero led AV Hill (1922) to suggest that stimulation always brings about
development of this maximal force, but that some of the force is dissipated in
overcoming an inherent viscous resistance if the muscle is shortening. Thus he
proposed representing the contractile element as a pure force generator in
parallel with a nonlinear dashpot element (defined in a minute, Fig. 6). Pure
force generator="active state" and proposed that it could develop a force T
o
which rose and then fell after a single electrical stimulation. In a tetanus, this
active state force would rise to a constant level numerically equal to the
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
developed isometric tension. The active state force was therefore a function of
the length of the contractile element, x
1
, as was the tetanic developed tension.
Figure 6. (a) Active state muscle model. The active state To(x1,t) is the tension
developed by the force generator in the circle. (b) The dashpot element resists
with a force proportional to the velocity.
Dashpot elements develop 0 force when they are stationary, but resist length
?
changes with a force F = B x1 , where B may be either a constant or a function of
? ?
x1 (Fig. 7b; dot denotes d( )/dt)
Isotonic contractions (i.e., muscle is shortening against a constant load) were first
investigated by Fenn and March (1935). They found (Hill did later, 1938) that the
realtion between developed force and shortening velocity in nonlinear, therefore
that the dashpot element has an acutely velocity-dependent damping.
As the mechanical circuit element suggests, engineering dashpot elements can be
made by fitting a piston into a cylinder with enough clearance to allow fluid to
escape past the piston s it moves. Since muscle contains a lot of water, the
dashpot model suggests that the viscosity of water ultimately determines the
viscous property of active muscle. But water is a Newtonian fluid; its viscosity is
not a function of shear rate, provided laminar flow in maintained. A non-
Newtonian liquid would have to be postulated in order to explain the velocity-
dependent damping in muscle.
By contrast the damping factor B for muscle was shown to be strongly dependent
on shortening speed (Hill curve) and temperature. In order for muscle to suffer
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
such a large change in internal viscosity temperature, it would have to be filled
with a viscous fluid with the properties similar to castor oil.
It may have been these thoughts which led Fenn to doubt that anything as simple
as a dashpot was responsible for the force-velocity behavior of muscle. He
proposed, correctly, that a biochemical reaction controlled the rate of energy
release and therefore the mechanical properties.
Nevertheless, the model (Fig. 6) has proven enormously useful in calculating the
purely mechanical features of skeletal muscle working against a load. If To is
specified as a function of time, and if B(x
1
), K
PE
(x
1
), and K
SE
(x
2
) are given as
empirical relations, then the overall length x and tension T of the muscle can be
calculated.
Muscles Active While Lengthening
In ordinary tasks, such as running, muscle functions to stop the motion of the
body as often as it does to start it. When a load larger than the isometric tetanus
tension T
o
is applied to a muscle in a tetanic state of activation, the muscle
lengthens at a constant speed. The surprise turns out to be that the steady speed
of lengthening is much smaller than would be expected from an extrapolation of
the Hill eqn. to the negative velocity region. In fact, Katz (1939) fount that -
dT/dv, the negative slope of the force-velocity curve, is about six times greater
for slow lengthening than for slow shortening.
Another anomaly is that the muscle "gives" or increases length rapidly, when the
load is raised about a certain threshold, as shown in Fig. 7, this "give" becomes a
very large effect, almost as if the muscle had lost its ability to resist stretching,
when the load is about 1.8 T
o
.
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16.423J/HST515J Space Biomedical Engineering and Life Support Prof. Dava Newman
Figure 7. Hill’s force-velocity curve. The shortening part of the curve was
calculated from eq. (1.4) with k = 0.25. The asymptotes for Hill’s hyperbola
(broken lines) are parallel to the T T 0 axes. Near zero shortening
velocity, the lengthening part of the curve has a negative slope approximately six
times steeper than the shortening part. The externally delivered power was
calculated from the product of tension and shortening velocity.
and v v max
Summary and Conclusion
Introduce the schematic diagram of Fig. 7 whose parameters can be obtained
empirically from mechanical experiments. Force-length of the parallel elastic
spring K
PE
and active force generator T
o
can be found from passive and tetanic
force-length experiments, respectively. Series elastic element K
SE
and the
dashpot element B(x
1
) are determined from the initial (instantaneous) length
change and early slop of the length record in the quick-release experiments.
Limitations: no one has believed in Fig. 6 as a comprehensive representation of
the way muscle actually works since about 1924. Most notable among the failures
of the "viscoelastic" model is its inability to account for the Fenn effect (not
enough time to go into the metabolic properties).
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