Jap.J.Physiol.,27,483-499,1977
FORCE-LOAD-VELOCITY INTERNAL
RELATION
LOAD
OF
AND
TETANIZED
CARDIAC
MUSCLE
Hidenobu
MASHIMA
Department of Physiology,School of Medicine,Juntendo Hongo,Tokyo,113 Japan
Abstract
A
frog
alternating containing was
ventricular
current 9mM
made
muscle
stimulation
the
measured.The in
K+-rich
tration.The
and
force
was
solution,or
fully
20V/cm
various
by
reducing was
external
is the
optimal
isometric
muscle
constants.The
values
shortening At
muscle
difference velocity muscle
and
by
lengths
shorter the
increased
the
load,v
constants
maximum is the
were
in
simple shorten-
than
0.92Lm,the load to
both
force
at
velocity,a,b,a'and
for
external
proportion
isometric
internal and the
velocity
for
lengthening
at
load
calculated
20•Ž.
defined load
and
the
b'are
a/Fm=0.51,b=0.75Lm/sec
a'/Fm=0.39,b'=0.75Lm/sec
between
a
for
is
is the of
were stimulus
lengthening,
force,Fm
length,Lm,P
release
Ca2+concen-
described
equation:(P+A)(v+b)=b(F+A),A=(F/Fm)a
F
by
solution
the
ing,and(2F-P+A')(-v+b')=b'(F+A'),A'(F/Fm)a'for where
a
loads
reducing
the
relation
tetanized in
controlled
against varied
by
be
University,
tetanus,the
velocities
force-load-velocity
hyperbolic
could
10Hz
isometric
shortening
isometric
intensity
strip
at
Ca2+.During
and
THE
FROG
the
at
as a
decrease
the
given in
length.
Although many studies on the load-velocity relation of cardiac muscles have been done,some difficulties still remain.It is hard to measure the velocity under different loads at exactly the same length of the contractile component because the active state of cardiac muscle is variable during twitch.Moreover,the relatively high resting tension and the time-dependent onset of contraction in cardiac muscles disturbed the measurement.But most difficulties could be eliminated if the cardiac muscle could be tetanized.The first attempt to tetanize the cardiac muscle was made by HENDERSON et al.(1971)in rat papillary muscles by applying a train of electrical pulses in Ca2+-rich solution.FORMAN et al.(1972)also sucReceived
for publication
May 13,1977
真 島英信 483
484
H.MASHIMA
ceeded in developing tetanic contraction in cat papillary muscles with repetitive electrical stimulation in a solution containing 10mM Ca2+and 10mm caffeine,and studied the effect of muscle length on the force-velocity relationship.The data were fitted with Hill's hyperbolic equation for a constant muscle length but the extrapolated maximum velocity diminished almost in proportion to a decrease in the muscle length. In a previous report(MASHIMA,1977),the optimal conditions for the electrical stimulation to tetanize the frog ventricular muscles were established.The present study was undertaken in order to elucidate the effects of isometric force and Ca2+ concentration as well as muscle length,on the load-velocity relation for tetanized cardiac muscles,and also to examine whether or not the force-load-velocity relation described by MASHIMA et al.(1972)in the skeletal muscle is also found in the cardiac muscle. METHODS
Material used was a small strip prepared from the ventricle of the frog(Rana nigromaculata),about10mm in length and lessthan 0.8mm in diameter.At first,thecircularstripwas dissectedfrom the ventriclenear the coronary sulcus, and
then
it
was
opened
by
were
ligated
with
preparation paratus
was
Whole
almost
similar
apparatus
was
turbance.The
muscle
the
that
described
a table
was
contained
taneously
by
a
whole
transverse
current(AC)was previous was
electric
employed
field
to
developed
by
Ca2+.In
through
a
stimulation
present
of
CO2
gas
One
end
steel
needle
was
conveyed
penetrated
between
the
same
20V/cm
AC
stimulation
in
the
20•Ž
solution
by
was
parallel simul-
alternating a
tetanic solution
was rest
platinum in
contraction.In
maximum
tension containing
effected
between
NaCl,12mM at
experiments.The
the
2min
maintained
of
bath
bath
stimulated
tetanic
and
solution(110mM
pair the
was
dis-
a polystyrol
electrodes.The
that
for
3-4
NaHCO3,2mM
KC1,
heat
bubbled
sec
stimuli.The
a thermoelectric
always
the ap-
mechanical
of
muscle
of the
KUSHIMA(1971).
in
walls
complete
10Hz
and
solution.A
the
stimulator,inserting
Ringer's
the
of
ends of
external
horizontally
confirmed
study,the
CaCl2,pH7.2)was
throughout
at
absorb
opposite
generate was
high-current
temperature 1.8mM
AC
the
MASHIMA
to
Ringer's the
length
study(MASHIMA,1977),it
9mM
of on
Both
arrangement
by
mounted
10ml placed
portion.
threads.The
designed
preparation
muscle.Thus,the
appropriate
nylon
electrodes(7•~1.5cm)were
with
5%
on
an
thin to
set
(3•~7•~1.5cm)which foil
cutting
with
exchanger 95% O2
and
mixture. of
the
preparation
attached to by
connected
tightly
previously
used
at the
to
was tip
anodal
a needle to
the
ligate
needle the
penetrated
the
pin
attached the
of
of
RCA
at by
the
tip
of it
end
the
ligated
lever,by
5734
tieing
muscle
at
isometric
portion
which
tube,and the
with under
the
isotonic the
other
by
binocular
a stainless
muscle
tension
end
lever.The
remaining a
the
was
muscle
ends
of
the
microscope.The
also was
thread
CARDIAC
FORCE-VELOCITY
RELATION
485
length of wooden vertical arm of the isotonic lever was 10cm and that of horizontal arm made of thin aluminium plate was 1.5cm,and the equivalent mass was 320mg as a whole.As the external load was hung from 0.5cm from the pivot,the load on the muscle was 1/20 of the external load.The movement of the lever was detected photoelectrically.The tension and displacement were displayed simultaneously on an ink-writing rectigraph(Nihon Kohden RJG-3022), or sometimes on a cathod-ray oscilloscope.The compliance of whole mechanical system was about 5pm/g.The muscle length was varied with an accuracy of 0.1mm by moving the isometric lever which was mounted on a sliding scale with a vernier. The electromagnetic relay was set to the isotonic lever for quick release and the moving velocity of the released lever was adjusted by a piston-type velocity controller devised by BUCHTHALand KAISER(1951)in order to minimize the oscillation of the lever after the release(Fig.1). The cross sectional area of the preparation was estimated by the method described in a previous paper(MASHIMA,1977). A
B
Fig.1. Measurementofshorteningvelocity(A)andlengtheningvelocity(8).Muscle was tetanized isometrically and recording paper started to run at S.Controlled was made at R.Upper curve,tension;lower curve,displacement;A,isometric 3.1g,load:2.5g,v:1.4mm/sec;B,isometric tension:2.8g,load:4g,v:-5.3mm/sec.
release tension:
RESULTS
1.
The load-velocity relations at various isometric forces for a constant initial length In a previous study(MASHIMA,1977),it was confirmed that frog cardiac muscle could be fully tetanized by AC stimulation at 10Hz and 20V/cm in the presence of 9mm Ca2+and that various steady forces less than the maximum force could be obtained in K+-rich solution by reducing the stimulus intensity.
486
H.MASHIMA
In the present study,the load-velocity relations for a constant muscle length were determined at various tetanic forces.Usually the initial length was maintained at 0.9Lm,where Lmis the optimum length at which the maximum force,Fm,was developed because the resting tension at 0.9Lm was sufficiently small(less than 5%)compared with the developed tension after overstretching the muscle once to 1.1-1.2Lm(MASHIMA,1977).The controlled release was applied during the plateau of isometric contraction and the shortening velocity against the load was measured at the linear part of the shortening curve immediately after the release (Fig.1A).The velocity of released lever was always controlled a little higher than the maximum shortening velocity,in order to minimize the overshort of displacement after the release.One of the load-velocity relations thus obtained is shown in Fig.2,curve 1.This load-velocity curve for the isometric force,F1(2.8g), generated by the maximum stimulus,was fitted with Hill's hyperbolic equation, (P+a1)(v+b1)=b1(F1+a1),
(1)
Fig.2. Load-velocity curves at various isometric forces for a constant initial length(10mm =0.9Lm).Isometric force,F1:2.8g,F2:2.4g,F3:2.0g,F4:1.55g,F3:1.1g,F6:0.6g. All curves(1-6)are drawn,using the same dynamic constants of a/Fm=0.32 and b= 0.52cm/sec.
CARDIAC
FORCE-VELOCITY
487
RELATION
where P is the load,v is the shortening velocity,al and b1are the dynamic constants. These constants were determined by a least-equares method from(F1-P)/v vs.P plot as a1/F1=0.32 and b1=0.52cm/sec and the curve 1 was drawn using these constants. When the external K+concentration was raised to 8mm,the tetanic tension decreased to F2(2.4g).By reducing the stimulus intensity,a smaller tetanic tension,F3(2.0g),was developed.For these reduced isometric forces,the shortening velocities against various loads were measured.The results are also shown in Fig.2.Curves 2 and 3 were also drawn by the calculation using the same dynamic constants,namely,a2/F2=a3/F3=a1/F1 and b2=b3=b1.The measured force-velocity points were well fitted with the calculated curves.This fact indicates that the constant b does not alter with the change in the isometric force, F,but the value of a changes in proportion to F.Much smaller isometric tensions,F4(1.55g),F5(1.1g),F6(0.6g),were developed,reducing the stimulus intensity in the external solution containing 12mm K+.The measured force velocity points for each isometric force were also fitted with the calculated curve, by taking an/Fn=a1/F1 and bn=b1(n=2-6).Thus,we can generalize the dynamic; constants as an/Fn=a/Fm=0.32 and bn=b=0.52cm/sec(=0.47Lm/sec)for all isometric forces,where a and b are the constants for the isometric force of Fm at 1.0Lm.The reason will be described in the section 4.Generalizing equation (1),we obtain the following force-load-velocity relation: (2) (P+A)(v+b)=b(F+A),A=(F/Fm)a, or
where
Fv
is the
load),that isotonic F has
1 +
a
/ Fm
difference
is, the
both
already
been
between
a/Fm
presented
are
or
Table
Dynamic
of
1.The
constants
force
independent
frog
and
the
velocity-dependent is obvious
skeletal
the
(3)
isometric the
constants
in the
constants in
Table1.
the
force
equation(3),it
b and
34cases,areshown
=F-P=Fv,
/v+b
viscous-like
dynamic
v
)
shortening.From
because
The
(
F
muscle
constant
that of
by
F.
muscle a/Fm
of frog cardiac
Fv
the
same
to
equation
et al.(1972). at
dispersed
and skeletal
force(= during
is proportional The
MAsHIMA
ventricular
isotonic
force-1oss
20℃,averagesof from
0.3
to
0.6,
muscles.
usually small in summer and large in winter,the constant b also dispersed from 0.4 to 0.9Lm/sec,also small in summer and large in winter.Consequently,the maximum velocity,Vmax(=b/(a/F,,))remained always around 1.5L./sec.
488
H.MASHIMA
2.The velocity of lengthening induced by a load larger than the isometric force After the load-velocity curve for the shortening muscle was determined by the above described procedure,the lengthening velocity of the same muscle was also measured in several preparations.The muscle was isometrically tetanized by the maximum AC stimulation and was then stretched by a load larger than the isometric force,removing the stop for the isotonic lever with controlled velocity. The changes in tension a.nd muscle length during isotonic lengthening were recorded simultaneously as shown in Fig.1B.The lengthening curve consists of an immediate downstroke due to the lengthening of the series elastic component and a slow lengthening of the contractile component.The lengthening velocity,-v, after the immediate down stroke was measured at the linear part of the curve and plotted against the load as shown in Fig.3,curve 1'.Curve 1 is the load vs.
Fig.3. Load-velocity curves in the shortening and lengthening.Curves 1 and 1',F1= 3.1g;curves 2 and 2',11-2.3g;Dynamic constants,a/Fm=0.58 and b=1.03cm/sec in curves 1 and 2;a'/Fm=0.4 and b'=1.03cm/sec in curves 1'and 2';initial length, 10mm(=0.9Lm).
shortening velocity curve for the same muscle at the initial length of 0.9 Lm,and the dynamic constants are a1/F1=0.58,b1=1.03cm/sec and F1-3.1g.Curve 1' was also hyperbolic,obeying the following equation (4) (2F1-P+a1')(-v+b1')=b1'(F1+a1') and dynamic constants were a1'/F1=0.4 and b1'=1.03cm/sec. Then,the external K+ concentration was raised to 8 mm and the stimulus
CARDIAC
FORCE-VELOCITY
489
RELATION
intensity was slightly decreased in order to decrease the isometric force without changing the initial length of the muscle.Curve 2 in Fig.3 is the load-velocity curve at F2=2.3g where the dynamic constants are also a2/F2=allF1=0.58 and b2=bz=1.03cm/sec.Then,the load vs.lengthening velocity curve was also determined as shown in curve 2',where the dynamic constants were a2'/F2=0.4 and b2'=1.03cm/sec.Obviously,the dynamic constants for the lengthening,as well as for the shortening,are not altered by the change in the isometric force. So that,they can be generalized to a,'/F1=a2'/F2=a'/Fm and b1'=b2'=b',and the generalized dynamic equation can be written as follows: (2F-P+A')(-v+b')=b'(F+A'), A'=(F/Fm)a'. The
dynamic
marized
in
constants Table
for
the
lengthening,averages
(5) of
5 cases
at
20•Ž,are
sum-
1.
Fig.4. Load-velocity curves at various external Cat + concentrations.Ca2+concentration, curve 1:9mM,curve 2:3.6mM,curve 3:1.8mM.All curves are drawn,using the same dynamic constants of a/Fm=0.51 and b=0.95cm/sec.Initial length,10.5mm (0.9Lm);resting tension,0.3g;open circle,measured velocity against the external load; filledcircle,velocity against the corrected load.
490
H.MASHIMA
3.Eject of the external Cat+ concentration on the load-velocity relation The isometric force of the tetanized ventricular muscle increased with an increase in the external Cat + concentration up to 9 mM(MASHIMA,1977).In Fig.4,three load-velocity curves obtained for the same initial length of 0.9Lm in the solution containing 9mM,3.6mM and 1.8mM Ca2+are shown.The isometric forces were 2.75g,2.3g and 1.6g,respectively.Three curves are drawn taking common dynamic constants of a/Fm=0.51 and b=0.95cm/sec.The initial length was always 10.5mm,while Lm was 11.5mm,and the resting tension at 10.5mm was 0.3g.Observed force-velocity points(open circles)are almost on the curves,except the velocity at the load close to the isometric force in each curve.The reason of this high velocity against the load close to the isometric force may probably be due to the fairly high resting tension of this preparation, where the shortening of the series elastic component immediately after the controlled release was so small that the real load to the contractile component could be overestimated by the difference between the external load and the resting tension at the instant of velocity measurement.Actually,the corrected points (filledcirclesin Fig.4)fell almost on the curves.The
method
of correction for
the resting tension will be described in the next section. After all,it is concluded that the dynamic constants are not altered by a change in the isometric force as far as the initial length is 0.9Lm and that the equation (2)holds for any isometric forces,which are developed not only by a reduced stimulus intensity but also by a reduced external Ca2+concentration. 4.The load-velocity curve at the optimal length,Lm All experiments mentioned above were done at the initial length of 0.9Lm, where the resting tension was sufficiently small.However,the resting tension increased markedly at Lm,as one of the results is shown in the upper part of Fig.9. Therefore,it was necessary to subtract the resting tension at the instant of velocity measurement from the external load in order to estimate the real load to the contractile component.One of the results is shown in Fig.5 in which Lm was 14.5 mm.Curve 1 is the load-velocity curve at 13mm where the isometric force was 2.1g and the resting tension was almost zero,while curve 2 was obtained at 14mm where the isometric force was 2.8 g and the resting tension was as small as 0.2g. Both curves have common dynamic constants of al/Fi=a2F2=0.41 and b=1.03 cm/sec.When the muscle was stretched to 15mm,the resting tension increased to 1.6g,although the isometric force remained at 2.8g. In order to measure the resting tension accurately at the instant of velocity measurement immediately after the quick shortening of the series elastic component,the X-Y record of the resting tension and the muscle length during the controlled release was displayed on a oscilloscope,as seen in Fig.6B.In Fig.6A, one of the shortening curves,when the load was 3 g,is also shown.The dotted line in Fig.6 was drawn at the level at which the quick shortening of the series
CARDIAC
Fig.5.
Load-velocity sion.Curve
curve 1,initial
at
FORCE-VELOCITY
the
optimal
rected
external
load
force-velocity
length(Lm)with
491
correction
length:13mm(0.9Lm),F-2.1g,no
14mm(0.97Lm),F=2.8g,resting the
RELATION
for
resting
the
tension,0.2g;•~(a-e),force-velocity at
15mm(1.03Lm),F=2.8g,resting point;
the
length
point tension,1.6
of
arrow
indicates
resting
the
amount
ten-
tension;curve
g;filled of
2, against
circle,cor-
correction.
elastic component was finished.The corresponding resting tension after this shortening is easily determined from Fig.6B,as indicated by R(=0 .6g).The observed velocity against the external load of 3g was 1.2mm/sec and plotted in Fig.5,point a.But the real load to the contractile component at the beginning of shortening should be 3-0.6=2.4 g,so that the point a was shifted to the left by 0.6g as indicated by an arrow.This corrected point illustrated by the filled circle is almost on curve 2.Repeating the same procedures,the measured forcevelocity points b,c and d were also corrected by the amount indicated by arrows and it was confirmed that all of them were on the calculated curve 2.No correction was necessary for the point e because the shortening of series elastic component was more than 1mm at that load(0.5g).After all,the load-velocity curve at 15mm where the developed force was 2.8g could be represented by curve 2,which is the load-velocity curve at 14mm where the developed force was also 2.8 g.Namely,the dynamic constants are not altered throughout the length range of 13-15mm(0.9-1.03Lm). As the dynamic constants are not altered between 0.9-1.0Lm,the constant an/F,,obtained at 0.9Lm in the section 1 can be written as a/Fm,where a is the constant obtained at 1.0Lm.The constant b is normalized by L m,i.e.,when the
492
H.MASHIMA
A
B
Fig.6.Estimation of the resting tension at the instant of velocity measurement.A,shortening curve at the controlled release,initial length:15mm(1.03Lm),F:2.8g,load: 3g,v:1.3mm/sec(corresponding to point a in Fig.5).B,length-tension curve during the controlled release in the resting muscle.The resting tension at the instant of velocity measurement is indicated by R(0.6g),which is the resting tension immediately after the quick shortening of the series elastic component.
value of b is 1.03cm/sec at 1.45cm(=Lm),it is expressed as 0.71Lm/sec.The value of Vmaxis also calculated by taking the normalized value of b,i.e.,Vmax= b/(alFm).Averaged values of these normalized dynamic constants,as well as those of skeletal muscle of the same species(MASHIMA et al.,1972),are summarized in Table 1.It is noticed that in the cardiac muscle a/F.is twice larger and b is 1/2.5 smaller than those of skeletal muscle,assuming Q10of b as about 2. As a result,the cardiac Vm,x is about one-fifth of that of skeletal muscle.Cardiac Fm is also about one-fourth of that of skeletal muscle,but it is hard to estimate the amount of intercellular connective tissues in the cardiac preparation. 5.The load-velocity relation at the initial length shorter than Lm When the muscle length was shortened below 0.9 Lm,the observed forcevelocity points did not fit well with the calculated force-velocity curve.One of a group of load-velocity curves at 0.8Lm(10mm)is shown in Fig.7.In this case Lm was 12.5mm.Curve 1 was obtained at 12mm after the correction of the resting tension where the isometric force was 3.7g and the resting tension was 0.6g.The dynamic constants of curve 1 are a/Fm=0.47 and b=1.02cm/sec (=0.82Lm/sec).Curves 2,3 and 4 are calculated using the same dynamic constants at the isometric forces of 2.45 g,2.0g and 1.1g,respectively.These isometric forces were obtained by reducing the intensity of AC stimulation in the 8mm K+solution,keeping the initial length always at 10mm.The load-velocity curve at each isometric force is shown by dotted curve in Fig.7.Obviously,the observed curves differ systematically from the calculated curves.And the maxi-
CARDIAC
FORCE-VELOCITY
493
RELATION
mum velocity decreased as the isometric force decreased.It is assumed that this slower velocity than that expected is resulted from an increase in the internal load which is induced by some structural change in the muscle fiber at shorter muscle lengths.In the present study,the the calculated and external loads internal
load
measured
from
internal load is defined as the difference between at a given shortening velocity.In Fig.8B,the
Fig.7
is plotted
against
the velocity.Clearly,the
Fig.7. Load-velocity curves at various isometric forces for a constant initial length of 0.8Lm.Isometric force,curve 1:3.7g,curve 2:2.45g,curve 3:2.0g,curve 4:1.1g. All curves are calculated by using the same dynamic constants of a/Fm=0.47 and b= 1.02cm/sec.Observed load-velocity relations illustrated by dotted curves are no on the calculated curves. internal
load
at 10mm
isometric forces.The the velocity in Fig.8,A
is directly
proportional
internal loads and C.The
to the velocity,regardless
of the
at 9mm and 11mm are also plotted against results are similar to that of Fig.8B,and the
only difference is the slope of linearity,which increases as the muscle length comes shorter.From these results the following equation can be introduced: Pi=γ where internal plotted increases
Pi
is
the
internal
load.In against almost
load,ƒÒ
Fig.9,the the
the
developed
relative
linearly
is
muscle with
(6)
υ
shortening isometric
length,taking
a decrease
be-
in
velocity tension Lm
the
and ƒÁ
muscle
as
and
is
the
1.0.It
was
length
between
the
facto
value
of
found
of r
are
that ƒÁ
0.7-0.9Lm
494
H.MASHIMA
A
B
C
Fig.8. Relation between the internal load and the shortening velocity at different isometric forces.A,at 9mm(0.72Lm);B,at 10mm(0.8Lm),obtained from data in Fig.7;C, at 11mm(0.88Lm).Inserted figures indicate the isometric forces.
and
that ƒÁ
is
zero
at
lengths
longer
γ=c(d-L/Lm)
than
0.92Lm.This
(L≦0.92Lm)
can
be
expressed
by
(7)
where L is the muscle length,c and d are constants.In the case of Fig.9,norAverages of 5 malized values of c=0.41 (g/Fm)/(cm/sec) and d=0.92L/Lm. and d=0.92L/Lm. It was concluded that the cases are c=0.64 (g/Fm)/(crn/sec)
internal load defined as the difference between the calculated and external loads is a linear function of the shortening velocity and that the internal load at a given velocity increases also linearly by 6.4%Fm with a decrease of 10%Lm in the muscle length at least between 0.7-0.92Lm,although no internal load is observed at lengths longer than 0.92Lm. DISCUSSION
As described by BRADY(1968),the study of cardiac muscle mechanics has fallen far behind that of skeletal muscle despite the similar basic contractile mechanism in the two type of muscles.The difficulties accompanying with the study of cardiac muscle,however,could be overcome in this study by tetanizing the cardiac
CARDIAC
Fig.9.
Relation length
curve
scale.DT,developed
between of
the
the muscle
FORCE-VELOCITY
factor used
of for
tension;RT,resting
internal Figs.7,8
load(ƒÁ)and and
495
RELATION
9 is also
tension;Fm,maximum
the
muscle
shown
length.Tensionwith
the
same
length
tension;Lm,optimal
length.
muscle.The maximum steady force was measured at the plateau of tetanic contraction,where the full active state was sustained and the time-dependent force change could be avoided.Moreover,the resting tension at 0.9Lm became sufficiently small compared with the developed tension after once overstretching the preparation to 1.1-1.2Lm. The force-velocity curves for isolated cardiac muscles have been determined mostly in a twitch by the afterload method in the cat papillary muscle(ABBOTT and MOMMAERTS,1959;SONNENBLICK,1962)and by the quick release method in the rabbit papillary muscle(EDMANand NILSSON1968;NILSSON,1972),in the frog ventricle(MASHIMAand MATSUMURA,1964;MASHIMA and KUSHIMA,1971)and in the Limulus heart muscle(PARMLEYet al.,1970).Most of them are hyperbolic, although some of them are rather curvilinear,especially when the isometric peak tension is depressed by high temperatures(MASHIMA and MATSUMURA,1964)or decreased muscle length(MASHIMA and KUSHIMA,1971).NOBLE et al.(1969) compared the afterload and quick release methods in cat papillary muscle and concluded that afterload force-velocity curves were not hyperbolic and the shape of quick release force-velocity curves depended on muscle length.MEISS and
496
H.MASHIMA
SONNENBLICK(1972)determined the velocity-force curve in cat papillary muscle by measuring the force during constant-velocity shortening and obtained the non-hyperbolic curve similar to the afterload force-velocity curve.They suggested that the force-velocity curve determined by the quick release method would contain a systemic bias related to the inactivation induced by quick shortening. According to EDMANand NILSSON(1972),however,the force-velocity curve determined by the damped-release technique exhibited a true hyperbolic shape and variability in the active state intensity between individual velocity determinations might account for the non-hyperbolic shape of afterload force-velocity curve. On the other hand,in the tetanic contraction of cardiac muscle,the force-velocity curves obtained by the controlled release technique are hyperbolic both in the present study and in the result of FORMANet al.(1972).During the plateau of tetanic contraction the active state is steady and the initial length of the contractile component can be always kept constant,so that the force-velocity curve for cardiac muscles must be essentially hyperbolic,obeying Hill's equation.However,it was noticed that the velocity at the load close to the isometric force was frequently higher than the value expected from the calculated force-velocity curve, as seen in Fig.2.Similar high velocity at heavy load was observed so frequently as about half out of 34cases,even after the correction for the resting tension was made.EDMAN et al.(1976)observed the non-hyperbolic force-velocity relation at more than 0.78 Po in the single fiber of frog skeletal muscle and suggested that due to the geometric factors not all of the myosin cross-bridges in the overlap zone were properly oriented to allow interaction with nearby actin sites during an isometric response.It is possible that similar geometrical factors may also take place in cardiac muscles. In the present study,the isometric tetanic force,F,was varied by reducing the stimulus intensity or by reducing the external Ca2+ concentration without changing the initial length(0.9Lm),and the isometric force in Hill's equation was treate as a variable.The results is expressed by the generalized Hill's equation(2),where the dynamic constants are written as A=(F/Fm)a and b,instead of Hill's constant a and b.This means that the value of A varies in proportion to F.Only when the isometric force is the maximum(F=Fm),A=a and the equation(3)becomes original Hill's equation.From the equation(3)it is obvious that the viscous-like force,Fv,is not only a function of the velocity but also a linear function of F. Assuming that each cross-bridge,unit force generator,generates a proper force, f,and moves againsta proper viscosity,fv,the totalforceand viscosityshould become F=nf and Fv=nfv,respectively,where n is the number of working crossbridges,because all cross-bridges are in parallel.Therefore,the fact that the viscosity of the striated muscle is force-dependent is not so strange from the viewpoint of the sliding-filament concept,as already pointed out by MASHIMAet al. (1972)in the skeletal muscle. The resting tension of cardiac muscle is known to be greater than that of the
CARDIAC
skeletal
muscle.But
tetanic
in
contraction
This
greater
was
FORCE-VELOCITY
frog
ventricular
usually
twice
developed
tension
tension.Moreover,when resting
Thus,the
resting
oped tension
study,that
is,the
fact
noncontractile and more
intercellular
for
the
Hill's
shortening.In muscle
load.If
the
difficult
the
extremely of
thin
the
cording
to
GORDON filaments
overlap
internal
other
proportion of
no
load et
of
cat
papillary
moval
of
the
resting
velocity relation
was
was
at
muscle
decrease
be in
at
was
the
during
constant
independent
the
lengths
between of
in
the
initial
Z
simple
study,it
is
with
at
the
muscle
Lm,on from
is
shorter
than
shortenboth
amount
of than
bands,so internal
sides overlap
1.65ƒÊm
that
between
another load
in-
0.92-0.7Lm,
0.92Lm. shortening
technique and
1.0-0.88Lm muscle
the
velocity.
this
a sarcomere
length
shortening
be
to
muscle
cardiac
fact,the
longer
must
of
accompanies
maximum
unloading the
in
de-
internal
SONNENBLICK,1965).Ac-
with
muscle
with
relation kind
the muscle
also
shortening
an
added.In
the by
the
elucidated
length
collide
b,but
in the
a
is 2.2ƒÊm of
et
decreasing
sacromere
shortening.At
will
detected
be
2.05ƒÊm(0.93Lm)and
the
and
to
in
with
load
and
center
the
formed
forces,it
to
to be
and
increase
a
cell
length
the
sliding
tension
of
a
isometric
internal
structure
filament
a
not
nature
decreased an
assumed
muscle(SPIRO
al.(1971a,b)estimated
twitch
maximum
fine
thick
with
internal
the
towards
to
the
against
BRUTSAERT
length
each
in
could
mem-
lengths
of
constant
proportional
load
in
surface
proportion
by
muscle
sarcomere
slide
proportionally
while
directly
skeletal
which
ends
is
the
that
the of
load
et al.(1966),the
resistance
creased
is
consider
that
with
increases
(0.75Lm)both
the to
thin to
length
load internal
as
of
resides
lengths,FORMAN
different
velocity
different
calculated
plastic
values
explained at
within
six
direct
meaning
the
tension.
the
as
muscle at
almost
was
in
resting
tension
partly
resting
muscles.
constant
in
internal
filaments,as
same
different
change
works
of
suggestive
almost
not
internal
origin
at
maximum
physiological
which
the
Although
skeletal
same
decrease
does
the
friction
length,because
overlap
a
the
theory.The or
begin
of
explain
sliding-filament viscosity
the varied
to
devel-
the
adopted
on
resting
have
the
curves
study,the
and
value
to
with
present
length
in
force-velocity
velocities
the
creasing
the
the
almost
cross-bridges,such
tissues,which than
equation
maximum
of
the
resting
the
for is
bears were
part
the
once
with
model
component
with
of
stretched
compared
Voigt's
relationship all
was
correction
points
most
cardiac
force-velocity
of
extrapolated
the
the
that
hyperbolas
the
parallel
connective in
al.(1972)indicated
ing
that in
of
contraction.
disadvantage
small
correction,simple elastic
twitch
decreased(MASHIMA,1977).
negligibly
force-velocity
suggests
abundant
As
was
parallel
structures
the muscle
markedly
tension
of
at Lm,however,the
the
corrected
curve.This
are
0.9Lm
measurement
that
reduces
0.9Lm
developed
than
ventricular
at
necessary.For
Fortunately,the
branes
at
the
was
present
tension
tension
tension.For
frog
the
more
relatively
the
1.1-1.2Lm,the
muscle or
497
RELATION
which
the and
the
in
permitted
demonstrated and
length
velocity
that
the rethe
force-velocitytime
after
the
498
H.MASHIMA
stimulus.The
results
contraction,also
when
the
29•Ž(BRUTSAERT
at
20•Žand
tween can
it was
attain
1971),while possible SKY
raise
by
aps
in
maximum
it attains to
and
the
velocity
twich
was
the
gradually
value steady the
TEICHHOLZ(1970),the
fiber of frog skeletal muscie
et
low
in
internal
the
the
external
the
during
between
from
2.5mM
tetanus
was
to
force-velocity
was independent
of pCa
31 .7 5mM
result
Ca2+concentration the
active
be-
state and
intensity KUSHIMA
contraction,which
makes to
relation
in
,
1.47Lm/sec
Ca2+concentration.According
relative
1 .0to
present
twitch(MASHIMA
tetanic
tetanic
23.5mm/sec
al.,1973).The of
during
constant
from
Ca2+solution
during
level
was
raised
velocity
a change the
obtained
increased
maximum
altered
1.8-9.0mM.Perph not
the
al.,1971a;BRUTSAERT
extrapolated not
maximum of
were
Ca2+concentration
et the
study,which
the
velocity
external
however,showed
present
that
maximum
mm/sec
the
showed
0.9Lm.The
at
of
the
, it
PODOLskinned
in the range 5.0-6.75.JuLIAN
(1971)also studied the Ca2+effect on the force-velocity relation of ATP contraction in the glycerinated frog skeletal muscle fiber and showed that the relative force-velocity relation and the maximum velocity was the same at pCa 6.09 and 5.49,although the maximum velocity was reduced at higher pCa value.Then,it is reasonable to assume that the internal Ca2+concentration can be raised higher than pCa 6.75 in a few seconds during the tetanic stimulation even in the 1.8mm Ca2+solution at least at 0.9Lm.Actually,it was frequently observed in the present study that the shortening velocity at small load was lower than the calculated value in the 1.8mm Ca2+solution in preparations in which the tension increased so slowly during tetanic stimulation that the complete plateau was hardly attained in several seconds.Probably,the internal Ca2+concentration was not raised sufficientiyin these preparations
as weli as in the twitches.Moreover,as
it was found
in a previous paper(MASFEIMA,1977)that the length-dependent decline of tension was larger in low Ca2+solutions.Further studies are necessary to elucidate whether or not the length-dependent decline of velocity(constant r or c)is larger in low Ca2+solutions and at shorter muscle lengths than 0.92Lm. REFERENCES
ABBOTT,B. C. and MOMMAER1S,W. F. H. M.(1959) A study of inotropic mechanisms in the papillary muscle preparation. J. Gen. Physiol., 42: 533-551. BRADY,A. J.(1968) Active state in cardiac muscle. Physiol. Rev., 48: 570-600 . BRUTSAERT,D. L., CLAES, V. A., and SONNENBLICK, E. H.(1971a) Velocity of shortening of unloaded heart muscle and the length-tension relation. Circ. Res., 29: 63-75. BRUTSAERT,D. L., CLAES,V. A., and SONNENBLICK, E. H.(1971b) Effects of abrupt load alterations on force-velocity-length and time relations during isotonic contractions of heart muscle: load clamping. J. Physiol., 216: 319-330. BRUTSAERT,D. L., CLAES, V. A., and GOETHALS,M. A.(1973) Effect of calcium on forcevelocity-length relations of heart muscle of the cat. Circ. Res., 32: 385-392. BUCHTHAL,F. and KAISER,E.(1951) The rheology of the cross striated muscle fibre. Dan. Biol. Medd., 21: 1-318. EDMAN,K. A. P. and NILSSON,E.(1968) The mechanical parameters of myocardial contrac-
CARDIAC
FORCE-VELOCITY
RELATION
499
tion studied at a constant length of the contractile element. Acta Physiol. Scand., 72: 205219. EDMAN,K. A. P. and NILSSON,E.(1972) Relationships between force and velocity of shortening in rabbit papillary muscle. Acta Physiol. Scand., 85: 488-500. EDMAN, K. A. P., MULIERT,L. A., and SCUBON-MULIERI,B.(1976) Non-hyperbolic forcevelocity relationship in single muscle fibres. Acta Physiol. Scand., 98: 143-156. FORMAN,R., FORD, L. E., and SONNENBLICK, E. H.(1972) Effect of muscle length on the forcevelocity relationship of tetanized cardiac muscle. Circ. Res., 31: 195-206. GORDON, A. M., HUXLEY,A. F., and JULIAN, F. J. (1966) The variation in isometric tension with sarcomere length in vertebrate muscle fibres. J. Physiol., 184: 170-192. HENDERSON,A. H., FORMAN,R., BRUTSAERT,D. L., and SONNENBLICK, E. H.(1971) Tetanic contraction in mammalian cardiac muscle. Cardiovasc. Res., 5 (Suppl. 1): 96-100. JULIAN,F. J.(1971) The effect of calcium on the force-velocity relation of briefly glycerinated frog muscle fibres. J. Physiol., 218: 117-145. MASHIMA,H. and MATSUMURA,M.(1964) The effect of temperature on the mechanical properties and action potential of isolated frog ventricle. Jap. J. Physiol., 14: 422-438. MASHIMA,H. and KUSHIMA,H.(1971) Determination of the active state by the graphical, experimental and instantaneous methods in the frog ventricle. Jap. Heart J., 12: 545-561. MASHIMA,H., AKAZAWA,K., KUSHIMA,H., and FUJII, K.(1972) The force-load-velocity relation and the viscous-like force in the frog skeletal muscle. Jap. J. Physiol., 22: 103-120. MASHIMA,H.(1977) Tetanic contraction and tension-length relation of frog ventricular muscle. Jap. J. Physiol., 27: 321-335. MEISS, R. A. and SONNENBLICK, E. H.(1972) Controlled shortening in heart muscle: velocityforce and active-state properties. Am. J. Physiol., 222: 630-639. NILSSON,E.(1972) Influence of muscle length on the mechanical parameters of myocardial contraction. Acta Physiol. Scand., 85: 1-23. NOBLE, M. I. M., BOWEN, T. E., and HEFNER, L. L.(1969) Force-velocity relationship of cat cardiac muscle, studied by isotonic and quick-release techniques. Circ. Res., 24: 821-833. PARMLEY,W. W., YEATMAN,L. A., and SONNENBLICK, E. H.(1970) Differences between isotonic and isometric force-velocity relations in cardiac and skeletal muscle. Am. J. Physiol., 219: 546-550. PODOLSKY,R. J. and TEICHHOLZ,L. E.(1970) The relation between calcium and contraction kinetics in skinned muscle fibres. J. Physiol., 211: 19-35. SONNENBLICK,E. H.(1962) Force-velocity relations in mammalian heart muscle. Am. J. Physiol., 202: 931-939. SPIRO, D. and SONNENBLICK, E. H.(1965) The structural basis of the contractile process in heart muscle under physiological and pathological conditions. Progr. Cardiovasc. Dis., 7: 295-335.
FORCE-LOAD-VELOCITY INTERNAL
RELATION
LOAD
OF
AND
TETANIZED
CARDIAC
MUSCLE
Hidenobu
MASHIMA
Department of Physiology,School of Medicine,Juntendo Hongo,Tokyo,113 Japan
Abstract
A
frog
alternating containing was
ventricular
current 9mM
made
muscle
stimulation
the
measured.The in
K+-rich
tration.The
and
force
was
solution,or
fully
20V/cm
various
by
reducing was
external
is the
optimal
isometric
muscle
constants.The
values
shortening At
muscle
difference velocity muscle
and
by
lengths
shorter the
increased
the
load,v
constants
maximum is the
were
in
simple shorten-
than
0.92Lm,the load to
both
force
at
velocity,a,b,a'and
for
external
proportion
isometric
internal and the
velocity
for
lengthening
at
load
calculated
20•Ž.
defined load
and
the
b'are
a/Fm=0.51,b=0.75Lm/sec
a'/Fm=0.39,b'=0.75Lm/sec
between
a
for
is
is the of
were stimulus
lengthening,
force,Fm
length,Lm,P
release
Ca2+concen-
described
equation:(P+A)(v+b)=b(F+A),A=(F/Fm)a
F
by
solution
the
ing,and(2F-P+A')(-v+b')=b'(F+A'),A'(F/Fm)a'for where
a
loads
reducing
the
relation
tetanized in
controlled
against varied
by
be
University,
tetanus,the
velocities
force-load-velocity
hyperbolic
could
10Hz
isometric
shortening
isometric
intensity
strip
at
Ca2+.During
and
THE
FROG
the
at
as a
decrease
the
given in
length.
Although many studies on the load-velocity relation of cardiac muscles have been done,some difficulties still remain.It is hard to measure the velocity under different loads at exactly the same length of the contractile component because the active state of cardiac muscle is variable during twitch.Moreover,the relatively high resting tension and the time-dependent onset of contraction in cardiac muscles disturbed the measurement.But most difficulties could be eliminated if the cardiac muscle could be tetanized.The first attempt to tetanize the cardiac muscle was made by HENDERSON et al.(1971)in rat papillary muscles by applying a train of electrical pulses in Ca2+-rich solution.FORMAN et al.(1972)also sucReceived
for publication
May 13,1977
真 島英信 483
484
H.MASHIMA
ceeded in developing tetanic contraction in cat papillary muscles with repetitive electrical stimulation in a solution containing 10mM Ca2+and 10mm caffeine,and studied the effect of muscle length on the force-velocity relationship.The data were fitted with Hill's hyperbolic equation for a constant muscle length but the extrapolated maximum velocity diminished almost in proportion to a decrease in the muscle length. In a previous report(MASHIMA,1977),the optimal conditions for the electrical stimulation to tetanize the frog ventricular muscles were established.The present study was undertaken in order to elucidate the effects of isometric force and Ca2+ concentration as well as muscle length,on the load-velocity relation for tetanized cardiac muscles,and also to examine whether or not the force-load-velocity relation described by MASHIMA et al.(1972)in the skeletal muscle is also found in the cardiac muscle. METHODS
Material used was a small strip prepared from the ventricle of the frog(Rana nigromaculata),about10mm in length and lessthan 0.8mm in diameter.At first,thecircularstripwas dissectedfrom the ventriclenear the coronary sulcus, and
then
it
was
opened
by
were
ligated
with
preparation paratus
was
Whole
almost
similar
apparatus
was
turbance.The
muscle
the
that
described
a table
was
contained
taneously
by
a
whole
transverse
current(AC)was previous was
electric
employed
field
to
developed
by
Ca2+.In
through
a
stimulation
present
of
CO2
gas
One
end
steel
needle
was
conveyed
penetrated
between
the
same
20V/cm
AC
stimulation
in
the
20•Ž
solution
by
was
parallel simul-
alternating a
tetanic solution
was rest
platinum in
contraction.In
maximum
tension containing
effected
between
NaCl,12mM at
experiments.The
the
2min
maintained
of
bath
bath
stimulated
tetanic
and
solution(110mM
pair the
was
dis-
a polystyrol
electrodes.The
that
for
3-4
NaHCO3,2mM
KC1,
heat
bubbled
sec
stimuli.The
a thermoelectric
always
the ap-
mechanical
of
muscle
of the
KUSHIMA(1971).
in
walls
complete
10Hz
and
solution.A
the
stimulator,inserting
Ringer's
the
of
ends of
external
horizontally
confirmed
study,the
CaCl2,pH7.2)was
throughout
at
absorb
opposite
generate was
high-current
temperature 1.8mM
AC
the
MASHIMA
to
Ringer's the
length
study(MASHIMA,1977),it
9mM
of on
Both
arrangement
by
mounted
10ml placed
portion.
threads.The
designed
preparation
muscle.Thus,the
appropriate
nylon
electrodes(7•~1.5cm)were
with
5%
on
an
thin to
set
(3•~7•~1.5cm)which foil
cutting
with
exchanger 95% O2
and
mixture. of
the
preparation
attached to by
connected
tightly
previously
used
at the
to
was tip
anodal
a needle to
the
ligate
needle the
penetrated
the
pin
attached the
of
of
RCA
at by
the
tip
of it
end
the
ligated
lever,by
5734
tieing
muscle
at
isometric
portion
which
tube,and the
with under
the
isotonic the
other
by
binocular
a stainless
muscle
tension
end
lever.The
remaining a
the
was
muscle
ends
of
the
microscope.The
also was
thread
CARDIAC
FORCE-VELOCITY
RELATION
485
length of wooden vertical arm of the isotonic lever was 10cm and that of horizontal arm made of thin aluminium plate was 1.5cm,and the equivalent mass was 320mg as a whole.As the external load was hung from 0.5cm from the pivot,the load on the muscle was 1/20 of the external load.The movement of the lever was detected photoelectrically.The tension and displacement were displayed simultaneously on an ink-writing rectigraph(Nihon Kohden RJG-3022), or sometimes on a cathod-ray oscilloscope.The compliance of whole mechanical system was about 5pm/g.The muscle length was varied with an accuracy of 0.1mm by moving the isometric lever which was mounted on a sliding scale with a vernier. The electromagnetic relay was set to the isotonic lever for quick release and the moving velocity of the released lever was adjusted by a piston-type velocity controller devised by BUCHTHALand KAISER(1951)in order to minimize the oscillation of the lever after the release(Fig.1). The cross sectional area of the preparation was estimated by the method described in a previous paper(MASHIMA,1977). A
B
Fig.1. Measurementofshorteningvelocity(A)andlengtheningvelocity(8).Muscle was tetanized isometrically and recording paper started to run at S.Controlled was made at R.Upper curve,tension;lower curve,displacement;A,isometric 3.1g,load:2.5g,v:1.4mm/sec;B,isometric tension:2.8g,load:4g,v:-5.3mm/sec.
release tension:
RESULTS
1.
The load-velocity relations at various isometric forces for a constant initial length In a previous study(MASHIMA,1977),it was confirmed that frog cardiac muscle could be fully tetanized by AC stimulation at 10Hz and 20V/cm in the presence of 9mm Ca2+and that various steady forces less than the maximum force could be obtained in K+-rich solution by reducing the stimulus intensity.
486
H.MASHIMA
In the present study,the load-velocity relations for a constant muscle length were determined at various tetanic forces.Usually the initial length was maintained at 0.9Lm,where Lmis the optimum length at which the maximum force,Fm,was developed because the resting tension at 0.9Lm was sufficiently small(less than 5%)compared with the developed tension after overstretching the muscle once to 1.1-1.2Lm(MASHIMA,1977).The controlled release was applied during the plateau of isometric contraction and the shortening velocity against the load was measured at the linear part of the shortening curve immediately after the release (Fig.1A).The velocity of released lever was always controlled a little higher than the maximum shortening velocity,in order to minimize the overshort of displacement after the release.One of the load-velocity relations thus obtained is shown in Fig.2,curve 1.This load-velocity curve for the isometric force,F1(2.8g), generated by the maximum stimulus,was fitted with Hill's hyperbolic equation, (P+a1)(v+b1)=b1(F1+a1),
(1)
Fig.2. Load-velocity curves at various isometric forces for a constant initial length(10mm =0.9Lm).Isometric force,F1:2.8g,F2:2.4g,F3:2.0g,F4:1.55g,F3:1.1g,F6:0.6g. All curves(1-6)are drawn,using the same dynamic constants of a/Fm=0.32 and b= 0.52cm/sec.
CARDIAC
FORCE-VELOCITY
487
RELATION
where P is the load,v is the shortening velocity,al and b1are the dynamic constants. These constants were determined by a least-equares method from(F1-P)/v vs.P plot as a1/F1=0.32 and b1=0.52cm/sec and the curve 1 was drawn using these constants. When the external K+concentration was raised to 8mm,the tetanic tension decreased to F2(2.4g).By reducing the stimulus intensity,a smaller tetanic tension,F3(2.0g),was developed.For these reduced isometric forces,the shortening velocities against various loads were measured.The results are also shown in Fig.2.Curves 2 and 3 were also drawn by the calculation using the same dynamic constants,namely,a2/F2=a3/F3=a1/F1 and b2=b3=b1.The measured force-velocity points were well fitted with the calculated curves.This fact indicates that the constant b does not alter with the change in the isometric force, F,but the value of a changes in proportion to F.Much smaller isometric tensions,F4(1.55g),F5(1.1g),F6(0.6g),were developed,reducing the stimulus intensity in the external solution containing 12mm K+.The measured force velocity points for each isometric force were also fitted with the calculated curve, by taking an/Fn=a1/F1 and bn=b1(n=2-6).Thus,we can generalize the dynamic; constants as an/Fn=a/Fm=0.32 and bn=b=0.52cm/sec(=0.47Lm/sec)for all isometric forces,where a and b are the constants for the isometric force of Fm at 1.0Lm.The reason will be described in the section 4.Generalizing equation (1),we obtain the following force-load-velocity relation: (2) (P+A)(v+b)=b(F+A),A=(F/Fm)a, or
where
Fv
is the
load),that isotonic F has
1 +
a
/ Fm
difference
is, the
both
already
been
between
a/Fm
presented
are
or
Table
Dynamic
of
1.The
constants
force
independent
frog
and
the
velocity-dependent is obvious
skeletal
the
(3)
isometric the
constants
in the
constants in
Table1.
the
force
equation(3),it
b and
34cases,areshown
=F-P=Fv,
/v+b
viscous-like
dynamic
v
)
shortening.From
because
The
(
F
muscle
constant
that of
by
F.
muscle a/Fm
of frog cardiac
Fv
the
same
to
equation
et al.(1972). at
dispersed
and skeletal
force(= during
is proportional The
MAsHIMA
ventricular
isotonic
force-1oss
20℃,averagesof from
0.3
to
0.6,
muscles.
usually small in summer and large in winter,the constant b also dispersed from 0.4 to 0.9Lm/sec,also small in summer and large in winter.Consequently,the maximum velocity,Vmax(=b/(a/F,,))remained always around 1.5L./sec.
488
H.MASHIMA
2.The velocity of lengthening induced by a load larger than the isometric force After the load-velocity curve for the shortening muscle was determined by the above described procedure,the lengthening velocity of the same muscle was also measured in several preparations.The muscle was isometrically tetanized by the maximum AC stimulation and was then stretched by a load larger than the isometric force,removing the stop for the isotonic lever with controlled velocity. The changes in tension a.nd muscle length during isotonic lengthening were recorded simultaneously as shown in Fig.1B.The lengthening curve consists of an immediate downstroke due to the lengthening of the series elastic component and a slow lengthening of the contractile component.The lengthening velocity,-v, after the immediate down stroke was measured at the linear part of the curve and plotted against the load as shown in Fig.3,curve 1'.Curve 1 is the load vs.
Fig.3. Load-velocity curves in the shortening and lengthening.Curves 1 and 1',F1= 3.1g;curves 2 and 2',11-2.3g;Dynamic constants,a/Fm=0.58 and b=1.03cm/sec in curves 1 and 2;a'/Fm=0.4 and b'=1.03cm/sec in curves 1'and 2';initial length, 10mm(=0.9Lm).
shortening velocity curve for the same muscle at the initial length of 0.9 Lm,and the dynamic constants are a1/F1=0.58,b1=1.03cm/sec and F1-3.1g.Curve 1' was also hyperbolic,obeying the following equation (4) (2F1-P+a1')(-v+b1')=b1'(F1+a1') and dynamic constants were a1'/F1=0.4 and b1'=1.03cm/sec. Then,the external K+ concentration was raised to 8 mm and the stimulus
CARDIAC
FORCE-VELOCITY
489
RELATION
intensity was slightly decreased in order to decrease the isometric force without changing the initial length of the muscle.Curve 2 in Fig.3 is the load-velocity curve at F2=2.3g where the dynamic constants are also a2/F2=allF1=0.58 and b2=bz=1.03cm/sec.Then,the load vs.lengthening velocity curve was also determined as shown in curve 2',where the dynamic constants were a2'/F2=0.4 and b2'=1.03cm/sec.Obviously,the dynamic constants for the lengthening,as well as for the shortening,are not altered by the change in the isometric force. So that,they can be generalized to a,'/F1=a2'/F2=a'/Fm and b1'=b2'=b',and the generalized dynamic equation can be written as follows: (2F-P+A')(-v+b')=b'(F+A'), A'=(F/Fm)a'. The
dynamic
marized
in
constants Table
for
the
lengthening,averages
(5) of
5 cases
at
20•Ž,are
sum-
1.
Fig.4. Load-velocity curves at various external Cat + concentrations.Ca2+concentration, curve 1:9mM,curve 2:3.6mM,curve 3:1.8mM.All curves are drawn,using the same dynamic constants of a/Fm=0.51 and b=0.95cm/sec.Initial length,10.5mm (0.9Lm);resting tension,0.3g;open circle,measured velocity against the external load; filledcircle,velocity against the corrected load.
490
H.MASHIMA
3.Eject of the external Cat+ concentration on the load-velocity relation The isometric force of the tetanized ventricular muscle increased with an increase in the external Cat + concentration up to 9 mM(MASHIMA,1977).In Fig.4,three load-velocity curves obtained for the same initial length of 0.9Lm in the solution containing 9mM,3.6mM and 1.8mM Ca2+are shown.The isometric forces were 2.75g,2.3g and 1.6g,respectively.Three curves are drawn taking common dynamic constants of a/Fm=0.51 and b=0.95cm/sec.The initial length was always 10.5mm,while Lm was 11.5mm,and the resting tension at 10.5mm was 0.3g.Observed force-velocity points(open circles)are almost on the curves,except the velocity at the load close to the isometric force in each curve.The reason of this high velocity against the load close to the isometric force may probably be due to the fairly high resting tension of this preparation, where the shortening of the series elastic component immediately after the controlled release was so small that the real load to the contractile component could be overestimated by the difference between the external load and the resting tension at the instant of velocity measurement.Actually,the corrected points (filledcirclesin Fig.4)fell almost on the curves.The
method
of correction for
the resting tension will be described in the next section. After all,it is concluded that the dynamic constants are not altered by a change in the isometric force as far as the initial length is 0.9Lm and that the equation (2)holds for any isometric forces,which are developed not only by a reduced stimulus intensity but also by a reduced external Ca2+concentration. 4.The load-velocity curve at the optimal length,Lm All experiments mentioned above were done at the initial length of 0.9Lm, where the resting tension was sufficiently small.However,the resting tension increased markedly at Lm,as one of the results is shown in the upper part of Fig.9. Therefore,it was necessary to subtract the resting tension at the instant of velocity measurement from the external load in order to estimate the real load to the contractile component.One of the results is shown in Fig.5 in which Lm was 14.5 mm.Curve 1 is the load-velocity curve at 13mm where the isometric force was 2.1g and the resting tension was almost zero,while curve 2 was obtained at 14mm where the isometric force was 2.8 g and the resting tension was as small as 0.2g. Both curves have common dynamic constants of al/Fi=a2F2=0.41 and b=1.03 cm/sec.When the muscle was stretched to 15mm,the resting tension increased to 1.6g,although the isometric force remained at 2.8g. In order to measure the resting tension accurately at the instant of velocity measurement immediately after the quick shortening of the series elastic component,the X-Y record of the resting tension and the muscle length during the controlled release was displayed on a oscilloscope,as seen in Fig.6B.In Fig.6A, one of the shortening curves,when the load was 3 g,is also shown.The dotted line in Fig.6 was drawn at the level at which the quick shortening of the series
CARDIAC
Fig.5.
Load-velocity sion.Curve
curve 1,initial
at
FORCE-VELOCITY
the
optimal
rected
external
load
force-velocity
length(Lm)with
491
correction
length:13mm(0.9Lm),F-2.1g,no
14mm(0.97Lm),F=2.8g,resting the
RELATION
for
resting
the
tension,0.2g;•~(a-e),force-velocity at
15mm(1.03Lm),F=2.8g,resting point;
the
length
point tension,1.6
of
arrow
indicates
resting
the
amount
ten-
tension;curve
g;filled of
2, against
circle,cor-
correction.
elastic component was finished.The corresponding resting tension after this shortening is easily determined from Fig.6B,as indicated by R(=0 .6g).The observed velocity against the external load of 3g was 1.2mm/sec and plotted in Fig.5,point a.But the real load to the contractile component at the beginning of shortening should be 3-0.6=2.4 g,so that the point a was shifted to the left by 0.6g as indicated by an arrow.This corrected point illustrated by the filled circle is almost on curve 2.Repeating the same procedures,the measured forcevelocity points b,c and d were also corrected by the amount indicated by arrows and it was confirmed that all of them were on the calculated curve 2.No correction was necessary for the point e because the shortening of series elastic component was more than 1mm at that load(0.5g).After all,the load-velocity curve at 15mm where the developed force was 2.8g could be represented by curve 2,which is the load-velocity curve at 14mm where the developed force was also 2.8 g.Namely,the dynamic constants are not altered throughout the length range of 13-15mm(0.9-1.03Lm). As the dynamic constants are not altered between 0.9-1.0Lm,the constant an/F,,obtained at 0.9Lm in the section 1 can be written as a/Fm,where a is the constant obtained at 1.0Lm.The constant b is normalized by L m,i.e.,when the
492
H.MASHIMA
A
B
Fig.6.Estimation of the resting tension at the instant of velocity measurement.A,shortening curve at the controlled release,initial length:15mm(1.03Lm),F:2.8g,load: 3g,v:1.3mm/sec(corresponding to point a in Fig.5).B,length-tension curve during the controlled release in the resting muscle.The resting tension at the instant of velocity measurement is indicated by R(0.6g),which is the resting tension immediately after the quick shortening of the series elastic component.
value of b is 1.03cm/sec at 1.45cm(=Lm),it is expressed as 0.71Lm/sec.The value of Vmaxis also calculated by taking the normalized value of b,i.e.,Vmax= b/(alFm).Averaged values of these normalized dynamic constants,as well as those of skeletal muscle of the same species(MASHIMA et al.,1972),are summarized in Table 1.It is noticed that in the cardiac muscle a/F.is twice larger and b is 1/2.5 smaller than those of skeletal muscle,assuming Q10of b as about 2. As a result,the cardiac Vm,x is about one-fifth of that of skeletal muscle.Cardiac Fm is also about one-fourth of that of skeletal muscle,but it is hard to estimate the amount of intercellular connective tissues in the cardiac preparation. 5.The load-velocity relation at the initial length shorter than Lm When the muscle length was shortened below 0.9 Lm,the observed forcevelocity points did not fit well with the calculated force-velocity curve.One of a group of load-velocity curves at 0.8Lm(10mm)is shown in Fig.7.In this case Lm was 12.5mm.Curve 1 was obtained at 12mm after the correction of the resting tension where the isometric force was 3.7g and the resting tension was 0.6g.The dynamic constants of curve 1 are a/Fm=0.47 and b=1.02cm/sec (=0.82Lm/sec).Curves 2,3 and 4 are calculated using the same dynamic constants at the isometric forces of 2.45 g,2.0g and 1.1g,respectively.These isometric forces were obtained by reducing the intensity of AC stimulation in the 8mm K+solution,keeping the initial length always at 10mm.The load-velocity curve at each isometric force is shown by dotted curve in Fig.7.Obviously,the observed curves differ systematically from the calculated curves.And the maxi-
CARDIAC
FORCE-VELOCITY
493
RELATION
mum velocity decreased as the isometric force decreased.It is assumed that this slower velocity than that expected is resulted from an increase in the internal load which is induced by some structural change in the muscle fiber at shorter muscle lengths.In the present study,the the calculated and external loads internal
load
measured
from
internal load is defined as the difference between at a given shortening velocity.In Fig.8B,the
Fig.7
is plotted
against
the velocity.Clearly,the
Fig.7. Load-velocity curves at various isometric forces for a constant initial length of 0.8Lm.Isometric force,curve 1:3.7g,curve 2:2.45g,curve 3:2.0g,curve 4:1.1g. All curves are calculated by using the same dynamic constants of a/Fm=0.47 and b= 1.02cm/sec.Observed load-velocity relations illustrated by dotted curves are no on the calculated curves. internal
load
at 10mm
isometric forces.The the velocity in Fig.8,A
is directly
proportional
internal loads and C.The
to the velocity,regardless
of the
at 9mm and 11mm are also plotted against results are similar to that of Fig.8B,and the
only difference is the slope of linearity,which increases as the muscle length comes shorter.From these results the following equation can be introduced: Pi=γ where internal plotted increases
Pi
is
the
internal
load.In against almost
load,ƒÒ
Fig.9,the the
the
developed
relative
linearly
is
muscle with
(6)
υ
shortening isometric
length,taking
a decrease
be-
in
velocity tension Lm
the
and ƒÁ
muscle
as
and
is
the
1.0.It
was
length
between
the
facto
value
of
found
of r
are
that ƒÁ
0.7-0.9Lm
494
H.MASHIMA
A
B
C
Fig.8. Relation between the internal load and the shortening velocity at different isometric forces.A,at 9mm(0.72Lm);B,at 10mm(0.8Lm),obtained from data in Fig.7;C, at 11mm(0.88Lm).Inserted figures indicate the isometric forces.
and
that ƒÁ
is
zero
at
lengths
longer
γ=c(d-L/Lm)
than
0.92Lm.This
(L≦0.92Lm)
can
be
expressed
by
(7)
where L is the muscle length,c and d are constants.In the case of Fig.9,norAverages of 5 malized values of c=0.41 (g/Fm)/(cm/sec) and d=0.92L/Lm. and d=0.92L/Lm. It was concluded that the cases are c=0.64 (g/Fm)/(crn/sec)
internal load defined as the difference between the calculated and external loads is a linear function of the shortening velocity and that the internal load at a given velocity increases also linearly by 6.4%Fm with a decrease of 10%Lm in the muscle length at least between 0.7-0.92Lm,although no internal load is observed at lengths longer than 0.92Lm. DISCUSSION
As described by BRADY(1968),the study of cardiac muscle mechanics has fallen far behind that of skeletal muscle despite the similar basic contractile mechanism in the two type of muscles.The difficulties accompanying with the study of cardiac muscle,however,could be overcome in this study by tetanizing the cardiac
CARDIAC
Fig.9.
Relation length
curve
scale.DT,developed
between of
the
the muscle
FORCE-VELOCITY
factor used
of for
tension;RT,resting
internal Figs.7,8
load(ƒÁ)and and
495
RELATION
9 is also
tension;Fm,maximum
the
muscle
shown
length.Tensionwith
the
same
length
tension;Lm,optimal
length.
muscle.The maximum steady force was measured at the plateau of tetanic contraction,where the full active state was sustained and the time-dependent force change could be avoided.Moreover,the resting tension at 0.9Lm became sufficiently small compared with the developed tension after once overstretching the preparation to 1.1-1.2Lm. The force-velocity curves for isolated cardiac muscles have been determined mostly in a twitch by the afterload method in the cat papillary muscle(ABBOTT and MOMMAERTS,1959;SONNENBLICK,1962)and by the quick release method in the rabbit papillary muscle(EDMANand NILSSON1968;NILSSON,1972),in the frog ventricle(MASHIMAand MATSUMURA,1964;MASHIMA and KUSHIMA,1971)and in the Limulus heart muscle(PARMLEYet al.,1970).Most of them are hyperbolic, although some of them are rather curvilinear,especially when the isometric peak tension is depressed by high temperatures(MASHIMA and MATSUMURA,1964)or decreased muscle length(MASHIMA and KUSHIMA,1971).NOBLE et al.(1969) compared the afterload and quick release methods in cat papillary muscle and concluded that afterload force-velocity curves were not hyperbolic and the shape of quick release force-velocity curves depended on muscle length.MEISS and
496
H.MASHIMA
SONNENBLICK(1972)determined the velocity-force curve in cat papillary muscle by measuring the force during constant-velocity shortening and obtained the non-hyperbolic curve similar to the afterload force-velocity curve.They suggested that the force-velocity curve determined by the quick release method would contain a systemic bias related to the inactivation induced by quick shortening. According to EDMANand NILSSON(1972),however,the force-velocity curve determined by the damped-release technique exhibited a true hyperbolic shape and variability in the active state intensity between individual velocity determinations might account for the non-hyperbolic shape of afterload force-velocity curve. On the other hand,in the tetanic contraction of cardiac muscle,the force-velocity curves obtained by the controlled release technique are hyperbolic both in the present study and in the result of FORMANet al.(1972).During the plateau of tetanic contraction the active state is steady and the initial length of the contractile component can be always kept constant,so that the force-velocity curve for cardiac muscles must be essentially hyperbolic,obeying Hill's equation.However,it was noticed that the velocity at the load close to the isometric force was frequently higher than the value expected from the calculated force-velocity curve, as seen in Fig.2.Similar high velocity at heavy load was observed so frequently as about half out of 34cases,even after the correction for the resting tension was made.EDMAN et al.(1976)observed the non-hyperbolic force-velocity relation at more than 0.78 Po in the single fiber of frog skeletal muscle and suggested that due to the geometric factors not all of the myosin cross-bridges in the overlap zone were properly oriented to allow interaction with nearby actin sites during an isometric response.It is possible that similar geometrical factors may also take place in cardiac muscles. In the present study,the isometric tetanic force,F,was varied by reducing the stimulus intensity or by reducing the external Ca2+ concentration without changing the initial length(0.9Lm),and the isometric force in Hill's equation was treate as a variable.The results is expressed by the generalized Hill's equation(2),where the dynamic constants are written as A=(F/Fm)a and b,instead of Hill's constant a and b.This means that the value of A varies in proportion to F.Only when the isometric force is the maximum(F=Fm),A=a and the equation(3)becomes original Hill's equation.From the equation(3)it is obvious that the viscous-like force,Fv,is not only a function of the velocity but also a linear function of F. Assuming that each cross-bridge,unit force generator,generates a proper force, f,and moves againsta proper viscosity,fv,the totalforceand viscosityshould become F=nf and Fv=nfv,respectively,where n is the number of working crossbridges,because all cross-bridges are in parallel.Therefore,the fact that the viscosity of the striated muscle is force-dependent is not so strange from the viewpoint of the sliding-filament concept,as already pointed out by MASHIMAet al. (1972)in the skeletal muscle. The resting tension of cardiac muscle is known to be greater than that of the
CARDIAC
skeletal
muscle.But
tetanic
in
contraction
This
greater
was
FORCE-VELOCITY
frog
ventricular
usually
twice
developed
tension
tension.Moreover,when resting
Thus,the
resting
oped tension
study,that
is,the
fact
noncontractile and more
intercellular
for
the
Hill's
shortening.In muscle
load.If
the
difficult
the
extremely of
thin
the
cording
to
GORDON filaments
overlap
internal
other
proportion of
no
load et
of
cat
papillary
moval
of
the
resting
velocity relation
was
was
at
muscle
decrease
be in
at
was
the
during
constant
independent
the
lengths
between of
in
the
initial
Z
simple
study,it
is
with
at
the
muscle
Lm,on from
is
shorter
than
shortenboth
amount
of than
bands,so internal
sides overlap
1.65ƒÊm
that
between
another load
in-
0.92-0.7Lm,
0.92Lm. shortening
technique and
1.0-0.88Lm muscle
the
velocity.
this
a sarcomere
length
shortening
be
to
muscle
cardiac
fact,the
longer
must
of
accompanies
maximum
unloading the
in
de-
internal
SONNENBLICK,1965).Ac-
with
muscle
with
relation kind
the muscle
also
shortening
an
added.In
the by
the
elucidated
length
collide
b,but
in the
a
is 2.2ƒÊm of
et
decreasing
sacromere
shortening.At
will
detected
be
2.05ƒÊm(0.93Lm)and
the
and
to
in
with
load
and
center
the
formed
forces,it
to
to be
and
increase
a
cell
length
the
sliding
tension
of
a
isometric
internal
structure
filament
a
not
nature
decreased an
assumed
muscle(SPIRO
al.(1971a,b)estimated
twitch
maximum
fine
thick
with
internal
the
towards
to
the
against
BRUTSAERT
length
each
in
could
mem-
lengths
of
constant
proportional
load
in
surface
proportion
by
muscle
sarcomere
slide
proportionally
while
directly
skeletal
which
ends
is
the
that
the of
load
et al.(1966),the
resistance
creased
is
consider
that
with
increases
(0.75Lm)both
the to
thin to
length
load internal
as
of
resides
lengths,FORMAN
different
velocity
different
calculated
plastic
values
explained at
within
six
direct
meaning
the
tension.
the
as
muscle at
almost
was
in
resting
tension
partly
resting
muscles.
constant
in
internal
filaments,as
same
different
change
works
of
suggestive
almost
not
internal
origin
at
maximum
physiological
which
the
Although
skeletal
same
decrease
does
the
friction
length,because
overlap
a
the
theory.The or
begin
of
explain
sliding-filament viscosity
the varied
to
devel-
the
adopted
on
resting
have
the
curves
study,the
and
value
to
with
present
length
in
force-velocity
velocities
the
creasing
the
the
almost
cross-bridges,such
tissues,which than
equation
maximum
of
the
resting
the
for is
bears were
part
the
once
with
model
component
with
of
stretched
compared
Voigt's
relationship all
was
correction
points
most
cardiac
force-velocity
of
extrapolated
the
the
that
hyperbolas
the
parallel
connective in
al.(1972)indicated
ing
that in
of
contraction.
disadvantage
small
correction,simple elastic
twitch
decreased(MASHIMA,1977).
negligibly
force-velocity
suggests
abundant
As
was
parallel
structures
the muscle
markedly
tension
of
at Lm,however,the
the
corrected
curve.This
are
0.9Lm
measurement
that
reduces
0.9Lm
developed
than
ventricular
at
necessary.For
Fortunately,the
branes
at
the
was
present
tension
tension
tension.For
frog
the
more
relatively
the
1.1-1.2Lm,the
muscle or
497
RELATION
which
the and
the
in
permitted
demonstrated and
length
velocity
that
the rethe
force-velocitytime
after
the
498
H.MASHIMA
stimulus.The
results
contraction,also
when
the
29•Ž(BRUTSAERT
at
20•Žand
tween can
it was
attain
1971),while possible SKY
raise
by
aps
in
maximum
it attains to
and
the
velocity
twich
was
the
gradually
value steady the
TEICHHOLZ(1970),the
fiber of frog skeletal muscie
et
low
in
internal
the
the
external
the
during
between
from
2.5mM
tetanus
was
to
force-velocity
was independent
of pCa
31 .7 5mM
result
Ca2+concentration the
active
be-
state and
intensity KUSHIMA
contraction,which
makes to
relation
in
,
1.47Lm/sec
Ca2+concentration.According
relative
1 .0to
present
twitch(MASHIMA
tetanic
tetanic
23.5mm/sec
al.,1973).The of
during
constant
from
Ca2+solution
during
level
was
raised
velocity
a change the
obtained
increased
maximum
altered
1.8-9.0mM.Perph not
the
al.,1971a;BRUTSAERT
extrapolated not
maximum of
were
Ca2+concentration
et the
study,which
the
velocity
external
however,showed
present
that
maximum
mm/sec
the
showed
0.9Lm.The
at
of
the
, it
PODOLskinned
in the range 5.0-6.75.JuLIAN
(1971)also studied the Ca2+effect on the force-velocity relation of ATP contraction in the glycerinated frog skeletal muscle fiber and showed that the relative force-velocity relation and the maximum velocity was the same at pCa 6.09 and 5.49,although the maximum velocity was reduced at higher pCa value.Then,it is reasonable to assume that the internal Ca2+concentration can be raised higher than pCa 6.75 in a few seconds during the tetanic stimulation even in the 1.8mm Ca2+solution at least at 0.9Lm.Actually,it was frequently observed in the present study that the shortening velocity at small load was lower than the calculated value in the 1.8mm Ca2+solution in preparations in which the tension increased so slowly during tetanic stimulation that the complete plateau was hardly attained in several seconds.Probably,the internal Ca2+concentration was not raised sufficientiyin these preparations
as weli as in the twitches.Moreover,as
it was found
in a previous paper(MASFEIMA,1977)that the length-dependent decline of tension was larger in low Ca2+solutions.Further studies are necessary to elucidate whether or not the length-dependent decline of velocity(constant r or c)is larger in low Ca2+solutions and at shorter muscle lengths than 0.92Lm. REFERENCES
ABBOTT,B. C. and MOMMAER1S,W. F. H. M.(1959) A study of inotropic mechanisms in the papillary muscle preparation. J. Gen. Physiol., 42: 533-551. BRADY,A. J.(1968) Active state in cardiac muscle. Physiol. Rev., 48: 570-600 . BRUTSAERT,D. L., CLAES, V. A., and SONNENBLICK, E. H.(1971a) Velocity of shortening of unloaded heart muscle and the length-tension relation. Circ. Res., 29: 63-75. BRUTSAERT,D. L., CLAES,V. A., and SONNENBLICK, E. H.(1971b) Effects of abrupt load alterations on force-velocity-length and time relations during isotonic contractions of heart muscle: load clamping. J. Physiol., 216: 319-330. BRUTSAERT,D. L., CLAES, V. A., and GOETHALS,M. A.(1973) Effect of calcium on forcevelocity-length relations of heart muscle of the cat. Circ. Res., 32: 385-392. BUCHTHAL,F. and KAISER,E.(1951) The rheology of the cross striated muscle fibre. Dan. Biol. Medd., 21: 1-318. EDMAN,K. A. P. and NILSSON,E.(1968) The mechanical parameters of myocardial contrac-
CARDIAC
FORCE-VELOCITY
RELATION
499
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