Force-Velocity Relationship of Cat Cardiac
Muscle, Studied by Isotonic and
Quick-Release Techniques
By Mark I. M. Noble, Ph.D., B.Sc, M.B., B.S., T. Earle Bowen, Ph.D., B.Sc,
and Lloyd L. Hefner, M.D.
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ABSTRACT
Contractions of isolated cat papillary muscle were studied using a lever
system with an electromagnetic load which allowed an on-line computer to
control the experiment and to process all the data. Isotonic force-velocity curves
were determined in 17 cat papillary muscles; the curves were not hyperbolic.
Force-velocity curves at constant time in the contraction and constant
contractile element length were obtained with a systolic quick-release
technique in 9 muscles. The velocity of shortening after release to low force
was almost always less than the maximum recorded following release to
slightly higher force. When quick-release force-velocity curves determined at
different times in the contraction were compared, the maximum velocity
occurred at approximately 60% of the time to peak isometric force. The fall in
velocity at lower forces was more marked later in the contraction. The shape of
the quick-release force-velocity curves was found to depend on muscle length.
At a constant time of release, and ignoring the low force end of the curves, the
quick-release force-velocity relationships were not hyperbolic at muscle lengths
appreciably below optimum, but near the optimal length the curves were
hyperbolic. When these quick-release force-velocity curves were corrected for
the presence of an elastic element in parallel with the contractile and series
elastic elements, it was found that none of the contractile element force-velocity
curves was hyperbolic.
ADDITIONAL KEY WORDS
myocardium
force-velocity relation
length-force curve
systolic quick release
muscle models
A. V. Hill's equation
active state of cardiac muscle
• There is conflicting evidence (1-3) about
whether force-velocity curves of heart muscle,
determined from measurements of isotonic
contractions, are hyperbolic; the hyperbolic
curves are usually described by Hill's equation
From the Departments of Medicine and Physiology,
University of Alabama Medical Center, Birmingham,
Alabama 35233, and the Cardiovascular Research
Institute, University of California San Francisco
Medical Center, San Francisco, California 94122.
This work was supported in part by U. S. Public
Health Service Program Project Grants HE 11310 and
HE 06285 from the National Heart Institute and
Grant FR-00145 (Computer Research Laboratory).
Dr. Noble was a Senior Fellow of the San Francisco
Bay Area Heart Research Committee.
Dr. Noble's present address is Department of
Medicine, Charing Cross Hospital Medical School, St.
Dunstan's Road, London, W. 6, England.
Received October 7, 1968. Accepted for publication
April 8, 1969.
Circulation Research, Vol. XXIV, June 1969
(4). This method of determining the force-velocity relationship has been criticized because
the measurements are made at varying times
during the contraction and at varying contractile element lengths (5, 2, 6). The quickrelease technique (2, 6-9) allows the time
during the contraction and the contractile
element length at which measurements are
made to be constant for each force-velocity
curve; this method would therefore be expected to give hyperbolic force-velocity curves.
However, while this has been found to be
true by some workers (6, 10), Brady has
found that quick-release force-velocity curves
are not hyperbolic (2). The purpose of the
present investigation was to obtain fresh
evidence on these questions.
An understanding of force-velocity relations
821
822
NOBLE, BOWEN, HEFNER
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behave as if in parallel with the CE and with
the whole (model I), part of (model III), or
none (model II) of the SE (Fig. 1).
Intensity of the Active State.—The activity
of the CE (a function of time). Measured as
the force developed by the CE when it is
prevented from shortening or lengthening.
Vmax.—The velocity of shortening of the
CE when there is no force opposing its
shortening.
Isometric Contraction.—Contraction at constant muscle length. The contractile element
shortens and stretches the SE.
Isotonic Contraction.—Contraction begins
isometrically until a predetermined force is
reached; the muscle then shortens at constant
force.
Quick Release.—Contraction begins isometrically (the CE shortens and stretches the
SE) until a predetermined time; the muscle
then recoils rapidly to a lower force (which
remains constant for the remainder of the
contraction).
Isotonic Eorce-Velocity Curve.—The relationship between peak velocity of muscle
shortening and force during isotonic contractions (as defined above).
is an essential part of the characterization of
the mechanical behavior of the muscle. The
hyperbolic force-velocity curves obtained in
skeletal muscle (4) are considered to be the
most characteristic feature of muscle mechanics. It is therefore considered necessary that
theories, which attempt to explain the mechanism of muscle contraction, predict a hyperbolic force-velocity relationship. If hyperbolic
force-velocity curves are not invariably present in heart muscle at constant time during
the contraction and constant contractile element length, the discrepancy may have to be
considered in terms of different theories of
contraction for skeletal and heart muscle.
Definition of Terms
Contractile Element (CE).—That part of
the muscle in which force is actively developed, possibly the regions in the sarcomeres
where actin and myosin filaments overlap.
Series Elastic Component (SE).—That part
of the muscle with elastic properties that
behave as if in series with the contractile
element.
Parallel Elastic Component (PE).—That
part of the muscle with elastic properties that
HI
IE
PE
PE
PE
SE
SE
FIGURE 1
Diagrammatic representation of muscle models. Left: model I in which the parallel elastic
component (PE) is in parallel with all of the series elastic component (SE). The contractile
element (CE) is assumed to be freely extensible in diastole. Diastolic force therefore equals
PE force in both diastole and isometric systole at any given muscle length. Middle:, model II
in which the PE is not in parallel with any of the SE. Diastolic force equals the PE force
and also equals the SE force, but during systole, the PE shortens and its force contribution
diminishes. Right: model III in which the PE is in parallel with part of the SE, i.e., an
intermediate model between I and II. Series and parallel viscosity have been omitted in
these oversimplified models.
Circulation Research, Vol. XXIV, June 1969
823
HEART FORCE-VELOCITY RELATION
TABLE
Quick-Release Experiments
Cat no.
301
302
303
304
305
306
C3
C4
C5
Frequency of
stimulation
(/min)
No. of
quick-release
series
performed
Range of release
times studied
(% time to
peak force)
Range of initial
muscle lengths
studied (mm)
Cross-sectional
area (mm')
20
20
25
24
8
40-120
40-120
40-120
40-100
75
75
75
75
75
2.35-2.50
3.35-3.50
3.80-4.00
2.55-3.55
2.70-3.04
3.53-3.78
3.10-3.45
3.04-3.20
3.40-3.55
1.08
0.6G
0.68
1.13
1.13
1.99
0.65
0.87
0.94
30
30
30
15
15
15
15
15
15
8
8
10
7
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Quick-Release Force-Velocity Curve.1— The
relationship between velocity of muscle shortening and force measured following quick
release of the muscle (as defined above).
Lmax.— The muscle length at which peak
isometric systolic force minus diastolic force
(peak CE force according to model I) is a
maximum.
Methods
Cats were anesthetized with ether and their
hearts rapidly excised and placed in oxygenated
Krebs-Ringer bicarbonate solution containing 145
mM
Na + ,
2+
4.2 mM K + , 2.5 HIM Ca2
+
2
, 1.2 ITIM
Mg , 125.5 mM C1-, 1.2 mM SO4 -, 1.2 mM
H2PO4-, 27 mM HCO : r, and 5.6 mM glucose.
The right ventricle was opened and a thin
papillary muscle was selected. The length and
cross-sectional area of these muscles are listed in
Table 1. Each end of the muscle was tied with
ligatures, and tension was applied to keep the
muscle at approximately the in-vivo length during
excision from the ventricle. Two short sections of
aluminum tubing were threaded over the ligatures
and knots and crimped on to each end of the
muscle with a needle holder, so that the muscle
was gripped at each end by the crimped metal
with no intervening knots or thread. The muscle
was then mounted in a muscle bath and perfused
with Krebs-Ringer bicarbonate at 24° to 25°C. A
reservoir was used to equilibrate the perfusing
iQuick-release force-velocity curves have been
termed "instantaneous force-velocity curves" (5).
However, the term "instantaneous force-velocity
curves" has also been used to describe curves
constructed from points measured at different times in
the contraction (11).
Circulation Research, Vol. -XXIV, June 1969
solution with 95% O 2 and 5% CO2; the solution
then had a pH of 7.40 and a Po2 greater than 600
mm Hg and was pumped directly from the
reservoir into the muscle bath.
After the muscle had been mounted, it was
stimulated to contract isotonically for 1 to 2
hours. The contractile performance reached a
stable level by this time and continued for 24 to
48 hours. All experiments were done during this
stable period, during which only minor changes in
performance occurred. These changes were not a
problem except during very long experiments
(e.g., isotonic series at four or five muscle lengths
plus quick-release series with quick releases at
five different times of release for four muscle
lengths [see below and Table 1] ). Under these
circumstances there was occasionally a change of
Lmax.
EQUIPMENT
The apparatus is shown diagrammatically in
Figure 2. The essential features of the apparatus
have been described previously (12, 13). The tip
of the lever was pressed into the aluminum tubing
on one end of the muscle to make a rigid
connection. The tip of the force transducer was
attached in a similar manner to the other end of
the muscle. The lever consisted of a light
magnesium-alloy framework with the fulcrum at
the upper end; the equivalent mass was 350 mg.
The lever carried a wire and moved through the
field of a permanent magnet. The force on the
lever (load on the muscle) was thus controlled by
the current in the wire; this current was
controlled by a potentiometer which was altered
by hand or automatically by a PDP-7 computer.
The amount the muscle could lengthen was set
by an adjustable stop mounted on the end of a
micrometer. When sufficient force was applied by
adjusting the current in the wire, the lever was
held against the stop; changes in diastolic length
824
NOBLE, BOWEN, HEFNER
D
ir
- 5 B
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FIGURE 2
Diagrammatic representation of the apparatus. The
muscle (1) was mounted in the bath (2) between the
force transducer (3) and the tip of the lever. Maximum length was controlled by the micrometer (4).
Current was applied between mercury-filled cups
(5 A and 5B) and passed through the wire (6). The
core of the linear differential-transformer length
transducer (7) was attached to the lever (8) which
swung in the plane of the diagram from the fulcrum
(9) between the poles of a permanent magnet (10);
the poles are behind and in front of the plane of
the diagram in the position ABCD.
could then be made accurately by adjusting the
micrometer. A reference value to obtain absolute
values for length was obtained by measuring the
muscle between the metal crimps with calipers.
Changes in length during contraction were
measured from a linear differential transformer,
the core of which was attached to the lever by a
side arm. It was calibrated (in terms of muscle
length) and checked for linearity by inducing
known displacements of the tip of the lever with
the micrometer. This procedure was followed
prior to each experiment. The measurements of
length were accurate to within 0.002 mm at the
usual sensitivity of the recorder.
The force on the muscle was measured with a
second linear differential transformer, the core of
which was attached to the muscle by a stiff
torsion bar. The compliance of the transducer and
lever system was negligible, i.e., when forces were
applied to the system without a muscle present
(the lever and force transducer were connected
by a rigid balsa wood bridge), no movement
could be detected from the length transducer.
The force produced by any given current in the
lever wire was first established directly. When the
muscle was mounted, it was first made to contract
isotonically against these known forces. The
corresponding deflections of the force transducerrecorder combination were thus calibrated and
checked for linearity for each experiment. Zero
was obtained by dropping the current in the lever
wire to zero. The ability to drop the force to zero
in this way enabled us to perform quick-releases
to forces below the resting force and thus to
extend the range over which quick-release forcevelocity curves could be determined below this
resting force; this is not possible with the
conventional preloaded type of apparatus.
The muscle was stimulated with square-wave
pulses from a Grass stimulator. The electrodes
were placed on either side of the muscle bath and
the stimuli discharged through the solution.
Stimulus frequency was 12 to 30/min, duration 5
to 10 msec, and voltage 15% to 20% above
threshold. The stimulus signal and the length and
force signals were recorded on an Electronics for
Medicine DR 8 Recorder.
ISOTONIC FORCE-VELOCITY CURVES
Velocity of shortening (the first time derivative
of length) was obtained by differentiating the
length signal with an RC circuit; calibration was
achieved with a triangular wave input to the
length channel. It was shown by Fourier analysis
that harmonics in the length signal with
frequencies above 5 cps had an amplitude less
than 5% of that of the fundamental. The
differentiating circuit was linear and produced
phase shifts between 87° and 90° at frequencies
up to 5 cps. In nine experiments, the length and
force signals and their calibration factors were
recorded on digital tape by the PDP-7 computer
(see below); the tapes were analyzed on an IBM
7040 computer which was programmed to
differentiate the length signal and print out the
peak velocity of shortening and isotonic force.
Series of isotonic contractions were recorded by
varying the force from the resting diastolic force
up to the force at which the muscle was unable to
shorten and at which isometric force was
recorded. This procedure was controlled by the
PDP-7 computer in nine experiments. Forcevelocity curves were obtained by plotting peak
velocity of shortening against systolic force. This
procedure was repeated for a series of increasing
muscle lengths.
QUICK-RELEASE FORCE-VELOCITY CURVES
In nine muscles, series of quick-releases were
Circulation Research, Vol. XXIV, June 1969
HEART FORCE-VELOCITY RELATION
825
.028
.056
.064
.112
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§ •• 4 0
AA
AAA
AAAAA
A
A
iBBB A
BBAA
AA
A
ABBBA
AA ABBB
AA
AAA
BABB
AA
AAAAA
A
BBBBAA
AA
AA88
A
A
A BB8BAA
BABB
AA AA
AAAA
ABBBB
AA
AA BAAAAA
168
.196
7.3
14.5
21.8
29.0 36.3
TIME (msec)
43.6
50.8
FIGURE 3
Computer output illustrating the method of determining velocity of shortening after quick
release. A: muscle length during systolic quick release. B: line fitted by the method of least
squared deviations to the final 250 samples of length.
recorded by connecting the apparatus on line to a
Digital PDP-7 computer; this computer also had
control of the systolic force. The computer was
programmed to allow isometric contraction and,
at fixed time during systole, to rapidly release the
force to a series of values which were constant for
the remainder of the contraction. In four
experiments, the time of release was progressively
increased from 40% to 120% of the time to peak
force (in increments of 20%). In the other five
experiments, all releases were timed to occur at
75% of the time to peak force. Diastolic quick
releases were also recorded. Initial muscle length
was varied as before (details for each muscle are
presented in Table 1). The PDP-7 sampled the
signals during release at a rate of 6,000/sec; 300
samples and their calibrations were recorded on
digital magnetic tape.
During the quick-release, there was a very
rapid (5 msec) shortening due to elastic recoil.
The quick-release velocity of shortening was the
Circulation Research, Vol. XXIV, June 1969
velocity with which the muscle shortened after
this very rapid shortening at the moment of
release. Analysis of the tracings was complicated
by the fact that oscillations occur on the length
tracing immediately after the rapid shortening of
release; these oscillations were considerably
greater on the velocity tracings. The difficulty was
resolved in the analysis of the digital tape
recordings which were performed on an IBM
7040 computer. A straight line was fitted (by the
method of least squared deviations) to the
ordinates of length after the first 50 following
release (Fig. 3). The slope of this line was taken
as the quick-release velocity of shortening. This
procedure gave the average velocity of shortening
over a period of 42 msec and ignored the first 8
msec after release. The computer was programmed to print out this velocity of shortening
after quick release and the corresponding force
and to plot the quick-release force-velocity curve
on the printer.
NOBLE, BOWEN, HEFNER
826
TEST FOR THE APPLICABILITY OF HILL'S EQUATION
For the purposes of describing the forcevelocity curves obtained in the present study, we
wished to use an objective method for deciding
whether a given curve could properly be called
hyperbolic. For this purpose, we used a linear
form of Hill's equation (1, 2, 14):
v
0.6
4
|o,4
(1)
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
where P is the load or systolic force, V the
velocity of shortening, Po the peak isometric
force, and a and b are constants. A plot of (Po —
P)/V against P should therefore give a straight
line with a positive slope of 1/b and a positive
intercept of a/b. For each of the isotonic forcevelocity curves and quick-release force-velocity
curves, (Po —P)/V was calculated. The linear
regression of (Po — P)/V was then obtained. If
the slope of the regression line was positive, the
significance of the difference from zero slope was
determined by Student's t-test.
The regression line was not used to calculate
the constants of Hill's equation because the data
points are not weighted equally by this method.
To fit the curves (determined to be hyperbolic by
the above criterion) to Hill's equation, a direct
method was therefore used which minimized the
squared deviations on the velocity axis. The
calculations were performed on the 7040 computer using an iteration process. Hill's constants a
and b were determined, together with Vmax (V
when P = 0) and Po (P when V = 0).
CALCULATION OF CONTRACTILE ELEMENT FORCE
AND VELOCITY ACCORDING TO MODELS I AND II
In each muscle, the quick-release force and
length were recorded and plotted, together with
the diastolic force-length curve (13). The forcestretch relationships of the PE and SE for models
I and II were then determined by the method
previously described (13). These data were fitted
to exponential functions by an iteration process.
The exponential functions were used in the
equations derived in a previous paper (13), and
CE force and velocity for models I and II were
calculated for each experimental force-velocity
point. All calculations were performed on the
7040 computer which was programmed to print
out CE force and velocity and to plot the CE
force-velocity curves for models I and II. At the
present time, no method exists for calculation of
CE force and velocity according to model III.
Results
None of the force-velocity curves from
isotonic contractions in 17 muscles were
hyperbolic, the slope of the regression line of
(Po — P)/V against P being negative. The
4.2S
L-3.9S
O
1.0
2.0
FORCE ( 9 )
3.0
FIGURE 4
Isotonic force-velocity curves for four initial muscle
lengths (L in millimeters) from one muscle.
4,0,-
3.0
i
2.0
1.0
1.0
2.0
3.0
FORCE (9)
FIGURE 5
Test for the applicability of Hill's equation to the four
curves illustrated in Figure 4 (see text). If the forcevelocity curves were rectangular hyperbolas, the
curves in this figure would be straight lines with
positive slopes.
results from one muscle are shown in Figures
4 and 5. These findings were consistent in all
the muscles studied; they are similar to the
Circulation Research, Vol. XXIV, June 1969
HEART FORCE-VELOCITY RELATION
827
FRACTION OF TIMS 70
PEAK iSOMETR/C FORCE
A
0.40
V
0-60
•
o.eo
FRACTION OF T/ME TO
PEAK /SOMETRIC FORCE
• — /.oo
f.2O
1.00 "
0.50 -
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
1.5
FORCE (g)
\
O.75
\
\
\
X
/
/,!/
1.00
FRACTION OF TIME TO
PEAK ISOMETRIC F0RC£
O.CO
0.60
O.SO
,00
,-,.
\
}
'.
^ •
\ \
FRACTION OF TIME TO
F>£AK ISOM£TRIC FORCE
O.+O
0.60
\
\
\
\\
o.eo
\
ti
to
s.
0.50 -
••••"
" v
5
\ \
i!'i / r
N.
'!/
\
1.00
1.20
\
^ \
,S"~\
%.
\ \
S. \
'• \ \
\
\
o
^
ill
0.2 5
i /
\
\
\A
\\
1
1.0
FORCE (g)
1.5
FORCE (g)
FIGURE 6
Force-velocity curves in cat 304 (A) and in cat 301 (B). The upper half of A and of B show
the actual points obtained experimentally; the points are omitted in the lower half so that the
shape of the curves can be seen more clearly. Continuous line and closed circles in A = isotonic
force-velocity curve at initial length 2.85 mm (cat 304). Initial length in B = 2.35 mm. Other
lines depict quick-release force-velocity curves for the same initial lengths measured at different
times during the contraction. Note the marked fall-off in velocity after quick release at very
low force in cat 304; the isotonic curve cuts across the quick-release curves. The fall-off in
velocity at low force was minimal in the muscle from cat 301.
results of Brady (2) and Sonnenblick (10)
but differ from the hyperbolic curves obtained
by Sonnenblick (1, 15).
In four quick-release series, force-velocity
Circulation Reiearcb, Vol. XXIV, June 1969
curves were obtained at four or five different
times during the contraction with initial
muscle length held constant. Examples of such
curves in two individual muscles are shown
NOBLE, BOWEN, HEFNER
828
3.0
2.0
s
i-
2.0
5
1.0
1.0
1.0
2.0
FORCE (g)
L-3.IS-
3.0
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
FIGURE 8
Test for the applicability of Hill's equation to the
four curves illustrated in Figure 7. Hill's equation was
satisfied for initial muscle lengths of 3.45 mm and
3.40 mm but not for 3.25 mm and 3.15 mm.
O
1.0
2.0
FORCE ( g )
3.O
FIGURE 7
Quick-release force-velocity curves for four lengths
in one muscle (cat C3 in Tables 1 and 2). Time of
release was held constant at 75% of the time to
peak isometric force. The fall-off in velocity at low
force has been omitted. L = initial muscle length in
millimeters. Note the change in shape from nonhyperbolic to hyperbolic as initial muscle length
increased. The continuous lines for L = 3.25 and
L = 3.25 were drawn by eye. The continuous lines
for L = 3.40 and L = 3.45 were obtained by fitting
the data to Hill's equation (see text). Note that
velocity of shortening measured at the lowest forces
shown here increased with increase in initial muscle
length and that the velocity of shortening at zero
load (Vmax) obtained from solution of the Hill
equation (i.e., the intercept on the velocity axis of
the continuous line) for L = 3.45 is greater than for
L = 3.40.
(Fig. 6). Maximum velocity of shortening
occurred when the force was appreciably
greater than zero in Figure 6, A, and three of
the curves in Figure 6, B. The fall in velocity
at lower forces was more marked in the curves
obtained later in the contraction. The righthand section of each curve showed decreasing
velocity with increasing force, but the relationship was not hyperbolic. The maximum
velocity was recorded at 60% of the time to
peak isometric force except in the muscle
shown in Figure 6, B, where the maximum
was at 40%. The movement of the curves with
time was not symmetrical for the velocity and
force axes. These curves (Fig. 6) were
obtained at muscle lengths less than Lmax.
The force-velocity data from four of the
remaining five quick-release experiments, in
which time of release was held constant,
showed the same features at short muscle
lengths. The curves were hyperbolic (over the
range of force above that at which maximum
velocity occurred) only at muscle lengths near
Lmax (Figs. 7 and 8). In the fifth muscle in
which time of release was held constant, the
range of muscle length near Lmax was not
fully explored. In this muscle and in two of
the muscles in which time of release was
varied (described in the previous paragraph), the curves were hyperbolic at the
maximum length studied.
The nine muscles in which quick-release
experiments were performed are listed in
Table 1. The values obtained by the hyperbolic fitting procedure from those curves which
were hyperbolic are presented in Table 2.
The results of the analysis of CE forceCirculation Research, Vol. XXIV, June 1969
829
HEART FORCE-VELOCITY RELATION
TABLE 2
Constants of Hill's Equation Obtained by Fitting Hyperbolas to Quick-Release Force-Velocity Curves
Correction for Model I)
Cat
no.
TR
303
304
a
(No
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Po/A
(g/mm>)
(g)
a/A
(g/mm*)
b
(mm/sec)
b/Lmax
(mm/sec/mm)
Vmax
(mm/sec)
Vmax/Lmax
(mm/sec/mm)
4.21
6.19
9.60
14.12
3.74
0.94
1.64
0.41
5.43
4.81
4.27
3.78
1.55
0.44
1.80
0.51
3.00
3.02
3.04
2.98
4.24
6.05
6.67
5.74
3.75
5.35
5.90
5.08
18.00
2.35
1.77
12.40
15.93
2.08
1.57
10.97
6.55
0.97
0.73
3.05
2.15
0.32
0.24
1.00
1.62
2.53
2.75
1.42
0.53
0.83
0.90
0.47
1.99
3.73
3.78
5.30
6.17
2.65
3.10
4.31
1.57
2.16
0.79
1.30
0.83
0.35
0.23
1.59
3.36
0.42
0.89
75
75
75
0.65
3.35
3.40
3.45
2.87
2.91
3.15
4.42
4.48
4.85
6.57
8.46
3.94
10.11
13.02
6.06
4.15
5.51
2.75
1.20
1.60
0.80
1.81
1.90
2.20
0.52
0.55
0.64
C4
75
0.87
3.20
4.08
4.69
10.70
12.30
12.29
3.84
4.69
1.47
C5
75
75
75
0.94
3.50
3.55
*3.52
3.90
4.16
3.77
4.15
4.43
4.01
5.43
2.61
8.31
5.78
2.78
8.84
4.39
2.35
6.15
1.2.3
0.66
1.73
3.15
3.75
2.79
0.89
1.06
0.79
A
(mm2)
IML
(mm)
Po
(g)
80
0.68
4.00
60
1.13
3.55
305
75
75
75
75
1.13
306
75
75
C3
Only those curves for which the plot of (Po — P)/V against P had a positive slope are included. Within each
muscle, experiments are listed in the order in which they were done. T R = time of release as percentage of time to
peak isometric force. A = cross-sectional area. IML = initial muscle length. * This series was done approximately
1 hour after those at 3.50 and 3.55 mm.
IN/7/AL
MUSCLE LENGTH
(mm)
2.80
2 90
2.9S
3.00
INIT/AL MUSCLE UNOTH (mm)
2.80
2.90
2.98
3.00
3.0*
FIGURE 9
Left: CE force-velocity curves assuming model II (Fig. 1) calculated from quick-release data
obtained at five initial muscle lengths in one muscle (cat 305 in Tables 1 and 2). These curves
were identical to the experimental data for the whole muscle. Right: CE force-velocity curves
assuming model I (Fig. 1) from the same experimental data as that used to construct the curves
on the left.
Circulation Research, Vol. XXIV, June 1969
830
NOBLE, BOWEN, HEFNER
velocity relationships for models I and II are
shown in Figure 9. The experimental and CE
model II curves were always identical and
were only hyperbolic at muscle lengths near
Lmax. None of the CE model I curves was
hyperbolic (Fig. 9, B).
Discussion
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The principal finding in this study was that
hyperbolic force-velocity curves were rarely
found in cat papillary muscles. When they
were found, the resting force was always high
and therefore it is doubtful whether the forcevelocity relationship of the CE was hyperbolic
even in these experiments.
The failure of the isotonic force-velocity
curves to fit Hill's equation may be explained
by the change in the intensity of the active
state of the muscle during contraction. During
a series of isotonic contractions with progressively increasing systolic force, the muscle
begins to shorten later and later during the
contraction because more and more time is
required for the muscle to develop enough
force to move the load (1). The measurement
of velocity of shortening is thus made at
progressively later times during the contraction up to the time of peak isometric force
(1). Since the Hill equation applies only to
one level of active state (4) and since the
intensity of the active state changes with time
during the contraction (2, 6, 8, 9, 16, 17), the
isotonic force-velocity curves would not be
expected to fit the equation (5). Abbott and
Mommaerts discussed the specific problem of
determining Po which is that peak isometric
force (usually equated with Po) may occur
after the intensity of the active state has
begun to decay (18). The claim that change
in intensity of the active state with time does
not affect isotonic force-velocity curves (19)
contradicts the almost universally accepted
belief that active state has a slow onset (2, 6,
8-10,16,17).
An additional problem arises when forcevelocity points measured at different times in
the contraction are used. Different amounts of
internal shortening of the CE will have
occurred at the time the muscle begins to
shorten (19). Thus, these force-velocity
points, although measured at nearly the same
muscle length, pertain to different CE lengths.
Brutsaert and Sonnenblick (19) believe that
this is the major source of error in isotonic
force-velocity curves.
The curves shown here (Fig. 4) are
appropriate for the CE in model II (Fig. 1)
since the PE in model II was calculated to be
very stiff in these muscles (the method for
calculating the PE force-stretch curve is
described in an earlier paper [13]). We have
calculated CE force-velocity curves according
to model I from the isotonic data, using
equations presented previously2 (13); they
were also not hyperbolic. By intuitive reasoning, we would also conclude that nonhyperbolic curves would be obtained according to
the intermediate model (III) (20).
We used the quick-release technique to
overcome many of the problems outlined
above. These contractions consisted of an
isometric period continuing up to a predetermined time. The force on the muscle was then
released suddenly to a lower force for the
remainder of the contraction. There was a
sudden shortening at the time of release due
to the undamped recoil of the stretched SE,
and then a slower shortening of the CE
against the new force. The force to which the
muscle was released was varied to obtain the
force-velocity curve. When the time of release
was kept constant, the time at which velocity
of shortening was measured was the same for
all forces, and therefore the intensity of the
active state was constant (unless active state
is affected by the quick-release maneuver [see
below]). We also assumed that the CE did
not shorten appreciably during the rapid
recoil of the SE, i.e., the CE length was nearly
the same after release (when force and
velocity were measured) as before release. CE
shortening during the isometric period prior to
2
The principal effect of these corrections is that PE
force (a function of muscle length) is subtracted from
the measured force and the velocity increased to
correct for the transfer of force from the PE to the
CE during shortening. CE shortening velocities at
very low CE forces were obtained (i.e., close to
Vmax) and found to be affected by initial muscle
length.
Circulation Research, Vol. XXIV, June 1969
HEART FORCE-VELOCITY RELATION
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release was assumed to be constant when
time of release was kept constant. Each point
on one quick-release force-velocity curve thus
pertains to the same CE length. The quickrelease force-velocity curves obtained at progressively later times in the contraction (Fig.
6) pertain to progressively shorter CE lengths
because more time (for internal shortening of
the CE) elapsed before release.
Some published quick-release force-velocity
curves have been hyperbolic (6, 10) and
others have not been hyperbolic (2). Our
study shows that different results would be
obtained at muscle lengths below Lmax than
at muscle lengths near Lmax. Sonnenblick
(10) and Edman and Nilsson (6) found
hyperbolic quick-release force-velocity curves,
but since they do not indicate whether their
muscle lengths were near Lmax, it is not
possible to decide whether there is a discrepancy between their results and those described
in this study. We did find hyperbolic curves at
lengths near Lmax. Our results are also
compatible with those of Brady (2) whose
curves were not hyperbolic; we did not find
hyperbolic force-velocity relationships at
lengths appreciably below Lmax.
Another important finding in the present
study was that Vmax obtained by the quickrelease method was markedly length-dependent (Fig. 7). It has previously been concluded on the basis of isotonic data that Vmax
was independent of muscle length (1,15).
Our results using the quick-release technique are similar to those of Brady (2) and
Sonnenblick (9) in that quick-release velocity
of shortening sometimes decreased at very low
forces. The reason for using the quick-release
technique was to eliminate variations in
intensity of the active state between each
point on the force-velocity curve (see above).
However, since the quick-release velocity of
shortening decreases at very low forces (Fig.
6), it would appear that large releases reduce
the intensity of the active state. This effect at
low forces increases with time of release (Fig.
6, in agreement with Brady [2]). A possible
explanation of this phenomenon may be that
quick release leads to buckling of fibers (21),
Circulation Research, Vol. XXIV, June 1969
831
an effect which will be most prominent with
the biggest releases to low forces. The buckled
fibers would then have to straighten before
further shortening of the whole muscle could
occur. However, Brady believes that quick
releases produce a true reduction in active
state because they reduce the ability of the
muscle to redevelop force (22). Quick releases
produce oscillations in length (Fig. 3), and
the muscle is therefore being subjected to
quick stretches during the parts of these
oscillations where length is rapidly increasing.
Quick stretches also appear to reduce the
intensity of the active state (2, 8); this is
therefore another possible explanation for the
reduction in velocity of shortening after large
releases which are followed by greater oscillations than small releases. We have been
unable to detect transient changes in velocity
of shortening after release similar to those
described by Civan and Podolsky (23) in
tetanized skeletal muscle, which these authors
predicted from the model of muscle proposed
by Huxley (24). The occurrence of a similar
effect in cardiac muscle at low forces might
explain the unexpectedly low shortening
velocities.
The results obtained by the quick-release
technique are also affected by the choice of
mechanical model (Fig. 1). The curves in
Figures 6, 7, and 9, A, are appropriate for the
CE in model IP (the PE was very stiff). We
have calculated the force-velocity relations for
the CE appropriate for model I using the
equations described previously (13). None of
these curves was hyperbolic, regardless of
muscle length (Figure 9, B). By intuitive
reasoning, we would conclude that the CE
force-velocity relationship for model III would
be less hyperbolic than for model II (Figure
9, A) but it is possible that some of them, at
lengths near Lmax, would fit A. V. Hill's
equation. At the present time, we have no
3
Most authors prefer model I to model II (5, 13,
17). Model II has been thought to be unlikely (13)
or impossible (5). However, Parmley and Sonnenblick
prefer model II in general, but found it impossible in
one of their six muscles (20). There is no Lmax for
model II, i.e., there is no maximum of peak CE
systolic force when plotted against muscle length.
832
NOBLE, BOWEN, HEFNER
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method of calculating quantitatively the CE
force-velocity relationship for model III.
These simple mechanical models (Fig. 1) do
not deal with complications arising from series
and parallel viscosity. Such considerations,
along with recognition of the nonhomogeneity
of individual sarcomeres in cardiac muscle
(21) with respect to geometrical alignment or
physiological properties, also serve as potential explanations for the nonhyperbolic forcevelocity curves obtained from quick-release
experiments.
At the present time, there is no way of
assessing these factors quantitatively, but this
study strongly suggests that heart muscle
definitely differs from skeletal muscle in not
having a hyperbolic force-velocity relation at
lengths below Lmax. Abbott and Wilkie (25)
and Matsumoto (26) state that Hill's equation
does apply to skeletal muscle at lengths less
than Lmax and that a and b are constant
while Vmax and Po vary. However, Gordon et
al. (27) found that velocity of overlap of actin
and myosin filaments differed from that
predicted by the application of the conclusion
of Abbott and Wilkie (25), and postulated
that a force may be present opposing shortening when the muscle is below its optimum
length. It would seem, therefore, that this
question deserves further study in both
skeletal and heart muscle.
Our study also suggests that, even at lengths
near Lmax, there is a strong possibility that A.
V. Hill's equation does not apply to heart
muscle (when the high resting force is taken
into consideration). This may indicate that
there are greater differences in the mechanism
of contraction between heart and skeletal
muscle than has hitherto been believed.
Acknowledgment
We are grateful to T. C. Donald, B.E.E., for help
with computer programs.
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Force-Velocity Relationship of Cat Cardiac Muscle, Studied by Isotonic and Quick-Release
Techniques
MARK I. NOBLE, T. EARLE BOWEN and LLOYD L. HEFNER
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Circ Res. 1969;24:821-833
doi: 10.1161/01.RES.24.6.821
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Copyright © 1969 American Heart Association, Inc. All rights reserved.
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