AMERICAN JOURNAL OF PHYSIOLOGY
Vol. 213, No. 6, Ihcember
1967.
Printed
in U.S.A.
Series elastic component
of mammalian
skeletal muscle1
ALAN
S. BAHLER2
Department of Biomedical
Baltimore, Maryland
balance
nonlinear
technique
stress-strain
The Johns Hopkins
University,
series elastic component
of the frog sartorius
muscle
using fast constant ve!ocity releases and has found the
maximum
extension of the series elastic component
to be
about 3 ‘% of rest length with maximum
isometric
tension. The damping
of this element is less than 200 dyne/
cm per sec. The second (9) quicklv
changes the force
across the muscle from full isometric tension to a given
isotonic level and measures the instantaneous
shortening
that occurs. During
release there is an instantaneous
shortening
of the lightly damped series elastic component
and negligible
shortening
of the slower contractile
component. This method applied to the frog sartorius muscle
yields a series elastic component
that stretches up to 4 %
of rest length at maximum
isometric
tension ( 10). The
third (6, 7) assumes that during an isometric contraction
the contractile
component
is stretching
the series elastic
component
and thus the time derivative
of developed
force is
BAHLER,
ALAN S. Series elastic component of mammalian
skeletal
1967.-A
method
muscle. Am. J. Physiol.
213(6):
1560-1564.
of modifying
the Wilkie
quick-release
technique
of determining
the compliance
of the series elastic
component
of mammalian
muscle
is described.
This modification
eliminates
the effect of
muscle
and lever system mass by adding
an additional
factor
related
to the shortening
of the contractile
component.
The
compliance
of the series elastic
component
of the rat gracilis
anticus
muscle
varies from
1.1 X lo+
cm/dyne
at maximum
isometric
tetanic
tension
to 17.6 X lOA cm/dyne
at zero tension. The extension
of this component
was 0.07 rest length
at
maximum
isometric
tetanic
tension.
The damping
of the series
elastic
component
was calculated
from
an energy
balance
technique
to be typically
300 dyne/cm
per sec. The results of
this study are compared
with those obtained
from other striated
muscles.
method ; energy
modified
quick-release
lightly
damped
series elastic component;
relationship
Engineering,
;
6P
dP
d7
=
6L’
dL’
l
6P
-$+G-
T
BEHAVIOR
of mammalian
skeletal
can be explained
by postulating
the presence of
three functionally
different
components
of the muscle
(2, 3). This three-component
model of muscle consists
of a nonlinear
elastic element in series with a contractile
component
both bridged
by another
nonlinear
elastic
element. The active element of this model, the contractile
component,
is a function
of length, external load, temperature,
and real time. For most parallel-fibered
mammalian skeletal muscles, the parallel
elastic component
can be neglected
at lengths less than or equal to rest
length, L, (length at which maximum
isometric tetanic
tension, PO, is devel oped).
The characteristics
of the series elastic component
are
commonly
determined
by three different methods. The
f rst (4) has mapped out the force-length
diagram
of the
HE
MECHANICAL
where P is the isometric tension, L’ is the length of the
contractile
component,
and t is the time. If 6P/6t is zero,
then
muscle
Received
for publication
9 June
1967.
1 This
work
was supported
by Public
5-F3-GM-23,
697-02,
5-Tl -GM-576,
and
2 Special
Fellow,
Public
Health
Service.
of Electrical
Engineering
and
Bioengineering
University,
Houston,
Texas
77001.
Health
AM-05524.
Present
Service
address
Laboratory,
dP
-dL’
dP
- dt
dt
l
dL’
Therefore,
dP/dL’,
the compliance
of the series elastic
component,
is determined
from the experimental
data
(dP/dt
and l/dL’/dt)
obtained
when the muscle is allowed to shorten after an initial period of isometric contraction.
This technique
has shown the frog sartorius
muscle to have a series elastic component
extension
of
from 3 to 5 % of rest length (6) and the cat papillary
muscle to have a series elastic component
extension
of
greater than 10 %) of rest length ( 7).
The purpose of the present study is to define the series
elastic component
of a mammalian
skeletal muscle, the
rat gracilis anticus. The quick release technique
has been
used since this method is the most applicable
with the
Grants
: Dept.
Rice
1560
MAMMALIAN
SERIES
1561
ELASTIC
M
K
SE
tP
‘:
b
SE
‘1
3
bL
FIG. 1. Linearized
equivalent
model
of muscle
quick-release
method.
(Note : since the contractile
cannot change its length instantaneously,
it has been
this approximation.)
for Wilkie’s
component
neglected
in
present
muscle preparation
and lever system. Rat skeletal
muscle shortens rapidly;
therefore,
the quick release
method has been modified
to correct for the combined
mass of the lever system and the muscle. The damping
of the series elastic component
was determined
by applying
the principle
of conservation
of energy to the quick
release.
METHODS
Preparution. The experiments
were performed
on the
right gracilis
anticus muscle from white male Wistar
rats (Carworth
Farms, Type CFN),
approximately
50
days old, 140-165 g body wt, fed a normal balanced rat
diet. The muscles had a mean weight of 60 mg and a
mean rest length of 2.7 cm (fiber length).
The gracilis
anticus, a thin parallel-fibered
muscle
which takes its origin from the posterior half of the pubic
symphysis and is inserted into the upper part of the
crest and medial border of the tibia, was surgically
removed from anesthetized
rats (45 mg/kg sodium pentobarbital
ip). After removal,
the muscle, with a portion
of the tibia and pubis, was placed immediately
in a
1,500 ml bath (17.5 C) containing
oxygenated
(95 %
OL, 5 % COO) bicarbonate-buffered
Krebs-Ringer
solution pH 7.3 (NaCl, 116.8 InM; NaHCO,,
28 mM; CaClp,
2.5 mM;
MgS04,
3.1 mM; KCl, 3.5 mM; KHtPO,,
1.2
mM;
and glucose, 11.1 mM).
The muscle was trimmed
and small stainless steel yokes were attached to the pubis
and tibia bones by means of “00” noncapillary
braided
black-silk
suture. These yokes were then placed between
the lever member and the force transducer.
Lever system. The lever system used in this study (equivalent mass 350 mg) consists of an electromechanical
torque
source, lightweight
magnesium
lever, velocity and force
transducers,
control
circuit,
and low impedance
pulse
generator.
A detailed
description
of this lever has been
presented previously
( 1).
The muscle was supramaximally
stimulated
by two
platinum
multielectrode
assemblies which
set up an
electric field normal
to the long axis of the muscle
(current
density
= 0.08 amp/cm2).
Stimulation
consisted of a train of 32 2-msec pulses with a pulse separation of 10.5 msec. All records were displayed on a Tek-
PIG.
2. Records
showing
the
effect of quick
changes
of load
(from isometric
to a fixed isotonic
level).
In each frame,
a, force
record;
b, length
record.
Base
lines are given for both.
Initial
length is 2.8 cm. Force sensitivity
is 10 g/major
vertical
division.
Sweep
speed is 10 msec/major
horizontal
division.
A: length
sensitivity
is I mm/major
vertical
division;
AP = 29 g. B: length
sensitivity
is 0.4 mm/major
vertical division;
AP = 21.8 g. C.
length
sensitivity
is 0.4 mm/
major
vertical
division;
AP =
15.8 g. D: length
sensitivity
is
0.2 mm/major
vertical
division;
AP = 6.2 g. Muscle wt = 68 mg;
L, = 2.8 cm; P,, = 34 g; bath
temperature
= 17.7 C.
1562
A. S. BAHLER
SION
LerrOth
AU Transients
Omitted
in Massless
AL
.
System
UE
TIME
ta
LENGTH
[Ime
t Release
FIG.
component
4.
Stej II:
length
comparison
changes
EXTENSION,
t
FIG.
elastic
3. 2%~ I:
component
method
for
obtaining
the
total
uncorrected
of massless
a quick
and
actual
contractile
change
of force.
MM
1.6
TIME
RELEASE
during
series
(AL&.
tronix RM561A
oscilloscope
and recorded
on Polaroid
type 107 film.
Experimental
details. Isometric
experiments
were performed
at y<-hr intervals
on all muscles. At a given
length, the tension developed
180 msec after the onset
of stimulation
was the value used in the length-tension
plot. The reproducibility
of the isometric length-tension
curve served as a measure of the viability
of the preparation. Experiments
were terminated
when a 15 % change
was noted between these isometric length-tension
curves.
The preparation
fulfilled this requirement
of reproducibility for over 3 hr of experimentation.
A minimum
of
nine quick releases was performed
on each muscle. These
experiments
consisted
of releasing
a supramaximally
stimulated
muscle from isometric
conditions
to some
fixed isotonic load.
RESULTS
In practice, any quick release cannot be instantaneous
because the lever system to which the muscle is attached
has finite mass. The effect of this mass can be estimated
by analyzing
the linearized
three-component
model
shown in Fig. 1. In Fig. 1, P is the load, A& is the combined equivalent
mass of the lever system and the muscle,
KsE’ is the force coefficient of the linearized
series elastic
component,
bL is the damping
of the lever system, and
bSE is the damping
of the series elastic component.
The
parallel
elastic component
has been neglected
since it
I. 2
0.8
0.4
0
L
0
IO
30
20
FIG. 5. Extension-load
characteristics
of the series
elastic
component
of a typical
rat gracilis
anticus
muscle.
Experimental
points
have
been
corrected
to eliminate
the effects
of lever
system
mass
and compliance.
Muscle
wt = 65 mg; L, = 2.8 cm; PO = 35 g;
bath
temperature
= 17.5 C.
has a negligible
effect on the characteristics
of the rat
gracilis anticus for muscle lengths less than 110 % rest
length, L,; the contractile
component
has been neglected
since it changes its length at a rate much slower than the
lightly damped series elastic component.
If P is changed
instantaneously,
the change of force across the series
elastic component
would follow the relationship
:
f'8~
= Ape
+fidl--z2)wt
where z is the damping
coefhcient
bT is the total damping
( bT = bL +
frequency (~3 = ~&/A&),
and
change in P. The change of force
< <
1
(Z = b,/2 d&M,),
bsE), o is the resonant
AP is the instantaneous
in the series elastic is
MAMMALIAN
SERIES
1563
ELASTIC
of the form of a damped
sinusoid. That this is the case
with the gracilis anticus can be seen by examining
Fig. 2.
The method of estimating
this instantaneous
extension
is obtained
by the following
procedure.
Step I assumes
that the force across the total uncorrected
series elastic
component
changes instantaneously
for a decrement
of
load (Wilkie’s
(9) method).
This change of force causes
an instantaneous
shortening
of the lightly damped total
uncorrected
series elastic component,
A&E,
which
is
estimated
by assuming that the contractile
component
shortens with a constant
velocity
during
the release.
This total series elastic is found graphically
by extending
a line tangential
to the shortening
curve (slope of line
is the contractile
component
velocity).
The intersection
of this line and the axis formed by the time of release
determines
the change in extension
of the total uncorrected series elastic component
(see Fig. 3).
Step II corrects this total uncorrected
series elastic
component
for the combined
mass of the lever system
and the muscle. The change in length determined
in
step I underestimates
the series elastic component
because
the mass of the system prevents the force and therefore
the velocity (u = V(P)) from changing
instantaneously.
Figure 4 indicates the effect this mass has on the performance of the contractile
component.
The correction
can be obtained
by assuming a piecewise
linear forcevelocity relationship
and a sinusoidally
declining
force
in the series elastic component.
Since step I overestimates
the velocity and hence underestimates
the total series
elastic component,
this correction
factor must be added
to the extension found from step I. The magnitude
of this
correction
can be calculated
by the following
analysis.
During
time, t,, the massless system will shorten vAta
but the system with mass will shorten approximately
EXTENSION,
L/Lo
0.08
0.06
Tfd3LE 1. Comparison of series elastic component
Compliance,
cm/dyne
X
Muscle
Normalized
Compliance,
LPO
x
10’
Normalized
Extension
at PO
102
Cat
L,
P,
Rat
papillary
(7)
= 1.4 cm
= log
anterior
tibialis
(8)
L, = 2.5 cm
P, = 440 g
Rat gracilis
anticus
L, = 2.7 cm
P, = 30 g
Frog illiofibularis
03
L, = 2.5 cm
P, = 48 g
Frog sartorius
(10)
L, = 3 cm
P, = 45 g
assumed
Frog sartorius
(4)
L, = 2.8 cm
P, = 120 g
ta
5 at P,
50 at 0.2 P,
0.25 at P,
1.5 at 0.1 P,
1.1 at P,
17 at 0.1 P,
3.6
36
0.1 L,
4.4
26.4
0.05
L,
1.3
19.5
0.07
L,
1.1 at 0.6 P,
5 at 0.1 P,
2.0
9.6
1 .6 at 0.7 P,
20 at 0.02 P,
2.4
30.0
0.04
L,
0.5 at P,
6 at 0.1 P,
2.0
26
0.04
L,
-
1.03-O
.05 L,
--.
bA
VA cos
-
wt)
dt = VA t, -
VA 2&-&r
s0
must be added to the extension
Therefore
vA( &/r)
determined
by steb I.
Step III takes account of the change in extension of the
lever system. This extension of the lever system must be
substracted
from the total extension determined
in step
II. The compliance
of the lever system was 3.5 X 1O-7
cm/dyne.
Figure 5 shows the effect of applying
this technique
to
a typical rat gracilis anticus muscle. When this procedure
was repeated
on five different
muscles, the data displayed in Fig.L 6 were obtained. These results show several
interesting
features. First, the series elastic component
is
highly nonlinear
(compliance
varying from 1.1 X 1O+
cm/dyne
at 1 .O P0 to 17.6 X 10mF en/dyne
at 0.0 PO).
Second, the extension of the series elastic component
at
maximum
isometric tension is 0.07 rest length. Last, there
is little variability
in the normalized
results when different muscles are compared.
The series elastic component
is lightly damped.
This
can be verified
by the following
analysis. The energy
stored in the series elastic component
is calculated
by use
of the relationship
:
LsE+L,
E SE
0
6. Stress-strain
of five muscles AX 1
0.2
curve
FIG.
SD.
0.4
0.6
0.8
of the series elastic
=
s
PSE
(LSE)
dLSE
La
1.0
component.
Means
where LSE is the variable
length
of the series elastic,
PSE( LICE) is the force across the series elastic, and Es. is
1564
A.
the energy stored in the series elastic when it is extended
from length L, to length L SE --/- L,. This energy can be
calculated
by integrating
the area under the extensionforce relationship
of the series elastic component.
When
the muscle is released, the energy stored in the series
elastic component
is used to accelerate the lever system
and to supply frictional
losses (both damping
of the lever
system and damping
of the series elastic). The energy
used in accelerating
the lever system is calculated
from
the expression
V
E
lever
V
ML
=
s
dv/c.lt
=
0
where v is the velocity.
component
is calculated
E
(>s> ML
v2
26v
dL
*
=
=SE+=,
Iv-
-
vdv
0
The energy lost to the viscous
from the expression
2bv
VisC
ML
s
s
=a
=
L$E
n-
assuming the velocity varies sinusoidally
from 0 to v as
the series elastic component
shortens from J!,, + L$E to
I,,. With this procedure,
the damping
of the series elastic
component
is typically
300 dyne/cm
per sec.
DISCUSSION
Although
Wilkie’s
quick-release
method is applicable
to the frog sartorius
at 0 C, this procedure
had to be
modified
to include
the effect of lever system mass if
meaningful
results were to be obtained
with rat fast
muscle at 17.5 C. A technique
for eliminating
the effect
of lever system mass has been outlined.
Experimental
results typically
showed the uncorrected
Wilkie method
S. BAHLER
to underestimate
the extension of the series elastic component at P, by 1 % of L,. This discrepancy
amounted
to a 13 % error in the extension of this component.
This paper has shown the rat gracilis anticus muscle
to have a series elastic component
which stretches 7 %
of L, when P, is impressed
across it. Furthermore,
the
compliance
of this element varies from 1.1 X 10-F cm/
dyne at P, to 17.6 X lo+ cm,/dyne at 0 P,. A comparison
of the series elastic components
reported
for different
striated muscles is given in Table 1. This table indicates
a basic similarity
among these striated muscles both in
normalized
compliance
and normalized
total extension.
Since this similarity
exists, one may conclude
that observations
made from other striated muscles are applicable to mammalian
fast muscle in general and to the
rat gracilis
anticus muscle in particular.
These observations indicate
that the series elastic component
of
mammalian
fast muscle is lightly
damped
( 1 1 ), independent of muscle length (5), and independent
of muscle
temperature
( 5).
The viscous damping
of the series elastic component
of the rat gracilis anticus is of the order of 300 dyne/cm
per sec. Woledge ( 11) has reported values of damping
of
from 200 to 500 dyne/cm
per set in frog sartorius which
is in general agreement
with the results obtained
for rat
fast muscle. Wells (8) however finds a viscous element of
1.3 X lo4 dyne/cm
per set for an in situ preparation
of
rat anterior tibialis. This high value of damping
obtained
by Wells is probably
due to the high viscosity caused by
the surrounding
fascia inherent
in an in situ muscle
preparation.
It is a pleasure
to thank
Drs. K. L. Zierler
and J. T. Fales
for
their
support
throughout
this work,
and to acknowledge
the careful
preparation
of the line drawings
by Mr.
Larry
Shack.
REFERENCES
1. RAHLER,
A. S., AND J. T. FALES.
A flexible
lever
system
for
quantitative
measurements
of mammalian
muscle
dynamics.
J. A@Z. Physiol.
2 1: 142 l-l 426,1966.
FENN,
W. O., AND B. S. MARSH.
Muscular
force
at different
speeds
of shortening.
J. Physiol.,
London
85: 277-297,
1935.
HILL, A. V. The heat of shortening
and the dynamic
constants
of muscle.
Proc. Roy. Sot., London,
Ser. B 126: 136-195,
1938.
HILL, A. V. The series elastic
component
of muscle.
Proc. Roy.
Sot., London,
Ser. B 137 : 273-280,
1950.
JEWELL, B. R., AND D. R. WILKIE. An analysis
of the mechanical
components
in frog
striated
muscle.
J. Physiol.,
London
143: 515-540,
1958.
6. PENNYCUICK,
C. J. Frog
fast muscle:
II. method
of measuring
internal
series compliance.
J. Exptl.
Biol.
41: 113-l
18, 1964.
7. SONNENBLICK,
heart
muscle:
1330-l
338,
E. Series
elastic
changes
in muscle
and
contractile
length.
Am.
WILKIE,
at various
London
D.
134:
elements
Physiol.
in
207:
1964.
J. B. Comparison
of mechanical
8. WELLS,
slow and fast muscles,
J. Physiol.,
London
9.
J.
R. Measurement
times
during
527-530,
10.
WILKIE,
D. R. The
Med. Bull.
12: 177-182,
11.
WOLEDGE,
in active
of the series
a single
muscle
properties
between
178 : 252-269,
elastic
twitch.
1965.
component
J. Physiol.,
1956.
mechanical
1956.
R. C. The thermoelastic
muscle.
J. Physiol.,
London
properties
of
muscle.
Brit.
effect
of change
of tension
155 : 187-208,
1961.