Download Relation Between Mural Force and Pressure in the Left Ventricle of

Relation Between Mural Force and Pressure
in the Left Ventricle of the Dog
By Lloyd L. Hefner, M.D., L. Thomas Sheffield, M.D., Glenn C. Cobbs, B.S.,
and Willem Klip, M.D., Ph.D.
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• The study of strips of cardiac muscle has
provided much valuable information regarding those factors which determine the force
generated during contraction. Interrelationships between force, velocity of shortening,
di&stolic stretch, duration of systole, and contractility have been delineated.1"5 These findings are not, however, directly and siiiaply
applicable to the physiology of the intact
heart because the force generated by the myocardium and the corresponding pressure are
by no means synonymous. In order to relate
the physiology of muscle strips to that of the
intact ventricle, the quantitative relation between force in the wall of the ventricle and
pressure in the cavity must be known.
The importance of such considerations in
cardiac physiology has long been recognized.0"8 The mechanical disadvantage of
cardiac enlargement has been frequently cited.
Excellent discussions of the problem have
been presented by Burch, Ray, and Cronvich,9
Burton,10 and Linzbaeh.11 More recently, Levine and Wagman have discussed the possible
influence of the size and shape of the heart
on myocardial oxygen consumption, emphasizing the important distinction between the
pressure developed by the ventricle and the
tension exerted by the fibers.12
An easy wa}' to visualize the simple physics
involved in the present experiment is as folFrom the Departments of Medicine and Physiology,
University of Alabama Medical Center, Birmingham,
Alabama.
Supported by American Heart Association grant.
Work was done during Dr. Hefner's tenure as an
Established Investigator of the American Heart
Association and Dr. Sheffield's tenure of a TJ. S.
Public Health Service Research Fellowship (HF9775).
Received for publication March 30, 1962.
654
lows: Gravitational effects are ignored since
they are unimportant in this situation. Consider a static left ventricle of any shape and
size containing blood under a given pressure,
as in figure 1. Now visualize an imaginary
plane through this ventricle as shown in the
figure. This imaginary plane divides the ventricle into two parts and passes through a rim
of myocardium and a cross-sectional area of
the cavity. From elementary hydrostatics it
is known that the pressure of the blood in the
cavity creates a force in a direction perpendicular to the imaginary plane exactly equal
to the product of the pressure (which is force
per unit area) times that cross-sectional area
of the cavity included in the imaginary plane.
The shape of this cross-sectional area of the
cavity in our imaginary plane is immaterial,
and the size and shape of the rest of the ventricle are also completely without effect on
this relation. Since we began by postulating
a static ventricle, Newton's laws of motion
require that the force mentioned above tending to separate the two parts of the ventricle
on each side of the plane be exactly balanced
by an equal and opposite force. This equal
and opposite force must exist in the rim of
myocardium included in the imaginary plane,
shown by the stippled area in figure 1. Note
that this force exists in the rim of myocardium
determined by the imaginary plane, its direction is perpendicular to the plane, and its
magnitude is determined only by the pressure
and cross-sectional area of the cavity and is
independent of the thickness of the wall, the
shape or cross-sectional area of the rim, the
size or shape of the remainder of the ventricle, the distribution of forces exerted by
the individual muscle fibers, the orientation
of the various muscle bundles, or the presence
of shearing forces.
Circulation Research, Volume XI, October 1962
VENTRICULAR FORCE AND PRESSURE
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The distribution of the forces or stresses
within the ventricle wall is of interest and
probably is o"f considerable physiological importance, but a consideration of this distribution is much more complicated10'13 than
the present approach which is concerned only
with the net or resultant force generated by
all the fibers of the myocardium included in
an imaginary plane through the ventricle.
Thus far, we have considered a static ventricle. When the ventricle ejects blood, the
force exerted by the rim of myocardium included in the imaginary plane slightly exceeds
the opposing force represented by the product
of the pressure and the cross-sectional area
of the cavity. This slight excess of force is
necessary to accelerate the ventricular wall.
For all practical purposes we may ignore this
minor discrepancy. It is possible to estimate
the additional force necessary for ejection,
and it has been found that the discrepancy
under the most extreme circumstances will be
less than 2 per cent.
We may say, therefore, that the product of
the pressure and the cross-sectional area of
the cavity gives a close estimate of the net
force perpendicular to the plane generated
by a complete rim of the myocardium, as
illustrated in figure 1. This conclusion seems
obvious but has not previously been verified
experimentally, nor has the time course of
mural force ever been recorded. An experimental approach to the problem requires a
method for measuring the longitudinal force
developed by the myocardial fibers. To measure directly the force exerted by a muscle
fiber, one must insert a force gauge in series
with the fiber. The following method is based
on this principle: A slit is made in the wall
of the dog left ventricle, and the two edges
of the slit are held together by a stiff gauge
which allows a record to be made of the force
developed by the cut fibers. This setup is
tantamount to sampling the force perpendicular to the imaginary plane discussed above.
Methods
la five consecutive dogs, after induction of
pentobarbital anesthesia, 10 to 12 mg./Kg. given
slowly intravenously, and institution of positive
Circulation Research, Volume XI, October 1962
655
IMAGINARY
PLANE
CAVITY
ENDOCARDIUM
EPICARD1UM
FIGURE 1
Diagram illustrating an imaginary plane through
a left ventricle. The plane divides the ventricle
into tioo parts and passes through a rim of
myocardium and a cross-sectional area of the
cavity. The pressure in the cavity times the crosssectional area of the cavity is the net force perpendicular to the imaginary plane which, if unopposed by an equal and opposite net force in
the rim of myocardium included in the plane,
would cause the two parts of the ventricle to
separate.
pressure artificial respiration, a left thoracotomy
was performed, the pericardium incised, and the
cut edges sutured to the chest wall to provide a
cradle for the heart. Aortic and ventricular pressure, the outer circumference of the left ventricle,
the electrocardiogram, and the force required to
hold together the two edges of a slit in the
ventricle were all recorded simultaneously. Left
ventricular pressure was measured by a Statham
P-23d strain gauge manometer connected to a
short, wide-bore, plastic eannula inserted into
the ventricle through the apex, and aortic pressure was obtained from a similar transducer
connected to a copper eannula inserted into the
ascending aorta through a carotid artery. The
outer circumference of the left ventricle was
measured by a variable resistance gauge described
by Rushmer.14 This gauge consists of a rubber
tube, about 1 mm. in outside diameter, filled with
mercury. The electrical resistance of the mercury
column varies directly with its length and is
recorded through a strain gauge carrier amplifier.
The gauge itself encircles the greatest diameter
of the left ventricle parallel to the atrioventricular
groove, being threaded through the thin wall of
the right ventricle just to the right of the septum,
as illustrated in figure 2. The gauge was applied
so that it was under tension even at the smallest
ventricular size obtained, ensuring that the gauge
HEFNER, SHEFFIELD, COBBS, KLIP
656
CIRCUMFERENCE
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FIGURE 2
(A) Diagram of the positions of the circumference gauge and the force gauge on the
anterior surface of the left ventricle. In one of the five dogs, the orientation of
the slit was perpendicular to that shown. (B) Diagrammatic cross-section of the force
gauge sutured to the myocardium with a slit between the legs. The gauge records the
force needed to prevent the two edges of the slit from separating. (C) Dimensional
view of the force gauge.
was well coupled to the ventricular wall and that
it operated only in its linear range. This latter
factor was verified after each experiment. The
characteristics, including the frequency response,
of this gauge have been studied thoroughly by
Lawton and Collins.15 The amplitude response to
increasing frequency is flat to beyond 30 cycles
per second.
The force required to hold together the two
edges of a slit in the ventricle was measured as
follows: A gauge was constructed of spring steel,
as illustrated in figure 2, upon which were bonded
two foil strain gauges for recording the force
applied to the legs. The dimensions of the gauge
are indicated in figure 2. The gauge weighs 600
ing., is linear from 0 to 100 Gin., has a natural
frequency over 100 c.p.s., and at the maximum
force encountered in these experiments yielded
only 0.4 mm. The two legs were pressed onto an
ink pad and then touched to the surface of the
left ventricle in an area with no visible coronary
arteries, leaving two marks for accurate placement of sutures, allowing precise approximation
of the edges of the slit as described below. Two
deep sutures were placed on these marks and the
gauge tied firmly in place. Then between the two
legs of the gauge a slit was made, extending
deeper than the sutures. The length of the slit
exceeded the width of the gauge by a few millimeters on each side. The depth of the slit extended
Circulation Research, Volume XI, October 1962
657
VENTRICULAR FORCE AND PRESSURE
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through approximately two-thirds the thickness
of the ventricular wall. These procedures assured
that no uncut fibers remained between the two
sutures attached to the gauge. The gauge held the
two edges of the slit together and recorded the
force. The function of the ventricle as a whole
was not detectably disturbed by the presence of
the slit, the edges of which were, in effect, sutured
back together by the attachment of the gauge. In
an ideal experiment the slit would be oriented
so as to be parallel to the circumference gauge,
as consideration of figure 1 makes clear, but in
most of our experiments the slit was actually
made almost perpendicular to this gauge, as illustrated in figure 2. This placement was necessary
to avoid injury to major branches of the coronary artery. The implications of the orientation
of the slit are discussed below.
After the gauges were in place, all measurements
were recorded simultaneously on an eight-channel
oseilloscopie recorder.* Records were made as the
dogs were given, stepwise, 15 or more 100-cc.
transfusions until the ventricle was grossly distended. Records obtained at the natural heart rate
were alternated with records obtained at a slow,
constant heart rate by electrical stimulation of the
cut peripheral end of the vagus with simultaneous
electrical pacing of the heart. These procedures
furnished a wide range of pressure, force, and
size for analysis in each dog. Identical procedures
were carried out in each of five consecutive dogs.
At the end of each experiment the circumference gauge was removed and calibrated. The
force gauge was calibrated by using known
weights. In postmortem study the left ventricle
was transected parallel to the circumference gauge,
and the circumference and average wall thickness
were measured as accurately as possible.
Figure 3 shows the type of records obtained.
Simultaneous values for pressure, outer circumference, and force across the slit were determined
at 0.02-second intervals during ejection systole
for one beat from each record made on each clog.
Thus, many hundreds of measurements from each
of the five dogs were analyzed with a wide range
of pressure, force, size, and stroke volume.
For each measured value for the outer circumference, an approximate value for the inner radius
was calculated as follows: By assuming the ventricle to be a thick-walled sphere, the volume
of muscle is calculated from the measurements of
wall thickness and outer circumference made at
postmortem study. Since the total volume of the
ventricle is equal to the sum of the volume of
blood in the cavity and the volume of the muscle,
•Electronics for Medicine, Inc., White Plains,
Xew York.
Circulation Research, Volume XI, October 1962
FIGURE 3
Typical records obtained from dog 1. Figure 3A
is a record made ivhile the ventricle xoas small,
and figure 3B xoas made after transfusion of 2
L. of blood. Note that changes in pressure were
not proportional to changes in force and that
the force declined during most of ejection systole. Left ventricular pressure (P); left ventricular circumference (C); force across the slit
in the wall of the left ventricle (F).
one solves the corresponding equation for the
inner radius and obtains
3K
where A1 is the inner radius, C the outer circumference, and K the volume of the muscle (a
constant for each dog). The accuracy of the
calculation is determined by the extent to which
the dog left ventricle is a thick-walled sphere.
That the approximation is close enough to be
useful is evidenced by the correlations presented
below.
Results
In figure 3 are typical records obtained
from dog 1. Figure 3A was recorded while
the ventricle was small, and figure 3B after
transfusion of 2 L. of blood. The time course
of force in the slit is of special interest. Note
that (1) the force in the slit rises rapidly
during pre-ejection systole but with or shortlj7
after the onset of ejection ceases to rise and
then falls; this fall is more marked in figure
3B where the stroke volume is larger. (2) The
658
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FIGURE 4
The relation between force (ordinate) and pressure (abscissa) in four of the five
dogs. The units are arbitrary except in dog 1, since the absolute value for the force
depends on the size of the slit, which was not the same in all dogs. All measured
points for these four dogs are shown. Dog 5 is not shown, being very similar to dog
2, with a correlation coefficient of 0.618. The relatively large scatter seen in this figure
indicates that a pressure measurement alone is not a good index to the force developed
by the myocardium.
time course of force and the time course of
pressure differ markedly in contour, pressure
rising at the same time force is falling during
much of ejection systole. (3) Comparing figure 3B with figure 3A. shows that the force
has increased proportionately more than the
pressure. Similar findings were predicted in
the paper by Bureh, Ray, and Cronvich.9
Linzbach11 calculated that the force would be
lower at the end of ejection than at the beginning, as is actually seen in figure 3. This
is not invariably true, however, for when the
stroke volume is small, no decline or even a
slight increase in tension may occur.
Figure 4 shows the relationship between
force in the wall and ventricular pressure for
Circulation Retearch, Volume XI. October 196t
VENTRICULAR FORCE AND PRESSURE
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four of five dogs separately. The correlation
coefficients range from 0.618 to 0.936. There
is considerable scatter, so that for any given
ventricular pressure the corresponding force
may vary within wide limits.
If onlj- those points where the size of the
ventricle is a constant value are selected for
graphing, an excellent correlation between
pressure and force in the wall is obtained, as
shown in figure 5. This graph was obtained
by plotting from dog 1 (fig. 4) all those points
where the ventricular circumference was 190
mm. and therefore represented measurements
near the onset of ejection when the end-diastolic size was small and those points near the
end of ejection when the end-diastolic size
was large. The high degree of correlation is
evidence that the primary reason for the scatter observed in figure 4 is the varying size of
the ventricle. The precision of the relationship between pressure and force seen in figure
5 affords a strong indication of the validity
of the method for measuring force.
Figure 6 shows for the same four dogs the
results of comparing the force across the slit
against the product of pressure and the square
of the inner radius. The inner radius is
squared because we want to multiply the
pressure by a number proportional to the area
of the cavity cut by the plane. The lengthmeter is oriented so that the cavity has a
nearly circular cross-section in this particular
plane, and the square of the inner radius gives
a number proportional to this area. As compared to figure 4, the improvement in the correlation is obvious, the correlation coefficients
ranging between 0.913 and 0.978. All measured points for four dogs are plotted both in
this graph and in figure 4. Dog 5 is not
graphed in either figure 4 or figure 6, since
the results were very similar to dog 2. The
correlation coefficient for dog 4 was 0.618 for
force versus pressure and 0.913 for force
versus the product of pressure and the square
of the inner radius.
In figure 7 a single beat from dog 1 is analyzed. In the top graph, force across the slit
is plotted against the pressure. This graph
shows that for any given pressure, force can
Circulation Research, Volume XI, October 19G2
659
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FIGURE 5
The relation in dog 1 between force and pressure
if the size of the ventricle is not a variable.
From dog 1 all points home been selected for
graphing where the circumference of the ventricle
icas 190 mm. The points graphed, therefore, represent measurements made at various times during
ejection—near the onset of ejection when the enddiastolic size was small and near the end of ejection when the end-diastolic size was large.
have more than one value. The middle graph
shows the relation in the same beat between
force and the product of the pressure and the
square of the outer circumference (use of the
outer radius in place of the outer circumference does not change the shape of the graph
since the radius is proportional to the circumference). Again, the force is shown to be no
function of the product of the pressure and
the square of the outer circumference or radius. On the other hand, the bottom graph
shows that force is a linear function of the
product of the pressure and the square of the
inner radius.
Discussion
The force records in figure 3 are of particular interest since they represent a basic physiological measurement not previously available. They are directly comparable to force
or tension curves obtained from muscle strip
preparations, with the advantage that the
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FIGURE 6
For each of the four dogs, force (ordinate) icas plotted against the product of the
pressure and the square of the inner radius. All points seen in figure 4 are also
represented on these graphs. The improvement in the correlation is evident. The units
assigned are arbitrary except in dog 1. Bog 5 is not shown, being similar to dog 2,
with a correlation coefficient of 0.936.
muscle remains in its natural environment
but the disadvantage that the size of the sample is not readily determined.
Figure 3 shows the typical contour of the
force curve. It is perfectly compatible with
what one would expect on the basis of known
facts of muscle physiology. Thus, if a strip
of cardiac muscle contracts isometrically, the
tension curve has a rounded summit, but if
shortening occurs at any stage during the contraction, the tension (or force) developed is
less than in an isometric contraction. In the
force curves in figure 3, the rise in force ceases
abruptly soon after shortening (ejection) begins, and in most records the force declines
during the remainder of systole. If ejection
Circulation Research, Volume XI, October 1962
661
VENTRICULAR FORCE AND PRESSURE
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had not occurred, the familiar rounded summit of an isometric contraction could be expected. Figure 8 demonstrates this fact,
showing a record made during ventricular
bigeminy, where the premature beat failed to
open the aortic valve. In the isometric beats
the summit of the force curve was rounded,
whereas, in the beats in which ejection occurred, an abrupt fall in tension is seen. The
fall in force development is due to two simultaneously operating influences—the amount of
shortening and the rate of shortening.1"3
Such details of the contour of the force
record provide evidence of the validity of
the method of measuring force, since the contour is consistent with that predicted from
known facts.
The correlations seen in figure 6 are a measure of the success of this study in establishing
that the total force generated by the muscle
perpendicular to an imaginary plane through
the ventricle is determined by the product of
the pressure and the cross-sectional area of the
cavity in the same plane. The significance of
the good correlations is enhanced by considering that our measurement of the cross-sectional area is indirect and is only an approximation for a nonspherical ventricle.
Furthermore, the orientation of the slit in
this study was determined principally by the
direction of the coronary arteries, so that'the
most appropriate orientation of the slit,
namely, parallel to the circumference gauge,
was achieved only once (in dog 5). It follows
from the presentation in the introduction that
for a nonspherical ventricle two different
orientations of the slit, determining two different cross-sections of the cavity, would give
force records which were proportional to one
another only if the variations of the two crosssectional areas of the cavity were also proportional. We believe these difficulties make the
correlation seen in figure 6 more significant.
Theoretical considerations and figure 7 both
show that net force in the wall and pressure
are related to one another through the dimensions of the cavity of the ventricle, not
through the external dimensions of the venCirculation Research, Volume XI, October 19C2
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Analysis of a single beat from dog 1. Measurements were made every 0.02 second during ejection. The upper two graphs show that the force
is not a function of either the pressure alone
or of the product of the pressure and the square
of the outer circumference (or outer radius). The
loiver graph shows the force to be proportional
to the product of the pressure and the square
of the inner radius. Product of the pressure and
the square of the outer radius (PC2); the graph
would not be changed in shape if outer radius
had been used instead of outer circumference.
Product of the pressure and the square of the
inner radius (PA').
tricle. A given small change in the outer
dimensions of the heart may correspond to a
much greater change in the dimensions of the
HEFNER, SHEFFIELD, COBBS, KLIP
662
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FIGURE 8
A record made during ventricular bigeminy in
dog 4. The second beat is premature with little
or no ejection occurring; as evidenced by the
aortic pressure pulse and the circumference record.
Note that the force curve is roughly sinusoidal
in the second beat but quite different in the first
beat. The maximum force developed in the premature beat actually exceeds that developed in
the normal beat, although the pressure developed
is much less. Circumference (C); force (F); left
ventricular pressure (P).
At represents the inner radius and Ae the outer
radius of a thick-walled sphere if the volume
of the wall is constant. The graph shows that
the inner radius changes more rapidly than the
outer radius, especially at small sizes, as the size
of the sphere increases. The horizontal distance
between the solid and the dotted lines represents
the thickness of the wall.
cavity. Figure 9 illustrates this point. For
a thick-walled sphere the inner radius is
plotted on the ordinate and the outer radius
on the abscissa. The slope of the graph varies
from infinity (when the inner radius is zero)
to one (at large ventricular sizes). Therefore,
when the ventricle is small, the inner radius
changes much more rapidly than the outer
radius. The principle is still valid for a nonspherical ventricle. The clinical implications
of this principle are obvious: Small increases
in the external dimensions of the heart can
greatly increase the force which the muscle
fibers must develop in order to generate a
given ventricular pressure.
It was pointed out earlier that the virtual
identity between net force in the wall and the
product of intracavitary pressure and crosssectional area is not influenced by such factors
as the shape of either the cavity (the area
is important, not the shape) or the rim of
myocardium, fiber direction, shearing forces,
or any other consideration one can suggest.
Such factors are very important, however, in
determining the distribution of forces in the
rim of myocardium. This important problem
we cannot evaluate because of the simplicity
of the mathematics used and because the
method of force measurement gives only the
net force pulling the two edges of the slit
apart and provides no information about the
distribution of stresses in the sample. It is
very likely, though, that the sample of the
myocardium measured in this way is a representative sample, meaning that the force recorded across the slit is a constant fraction
of the total force perpendicular to the plane
of the slit generated by all the fibers of the
ventricle in the imaginary plane determined
by the slit. The chief evidence for this conclusion is as follows: (1) the previously cited
excellence of the correlation between pressure
(which results from the action of all the fibers) and the force recorded across the slit,
so long as the ventricular size is a constant,
shown graphically in figure 4; (2) the size
FIGURE 9
Circulation Research, Volume XI, October 1962
663
VENTRICULAR FORCE AND PRESSURE
of the sample is a considerable fraction, about
1/20, of all the fibers that would be included
in a complete transection of the ventricle.
Summary
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An investigation of the relationship in the
dog between left ventricular pressure, left
ventricular size, and total longitudinal force
developed by the wall of the left ventricle is
reported. This study required the development of a method for recording the force
needed to keep together the two edges of a
slit in the ventricle. Such a method is described, and samples of the force curves are
shown, the general features of which are consistent with predictions from the physiology
of muscle strips. The net force which the muscle fibers develop perpendicular to a given
plane through the ventricle is almost identical
to the product of the intracavitary pressure
and the area of the cavity included in the
plane. This relationship holds true regardless
of the thickness of the wall or the shape of
the ventricle.
References
3. HILL, A. V.: Heat of shortening and dynamic
constants of muscle. Proc. Boy. Soc, London
126: 136, 1939.
4. PODOLSKY, B. J.: Mechanism of muscular eontraction. Am. J. Med. 30: 70S, 1961.
5.
SONNENBLICK, E . H . , AND McCOLLUM, Z. T. :
Active state, force-velocity relationships, and
inotropic mechanisms in mammalian papillary
muscle. Fed. Proc. 20: 26, 1961.
6. WOODS, E. H.: Few applications of physical
theorem to membranes in human body in
state of tension. J. Anat. & Physiol. 26: 362,
1892.
7. WOLF, A. V.: Demonstration concerning pressuretension relations in various organs. Science
115: 243, 1952.
8. EUSHMER, E. F . : Cardiac Diagnosis: Physiologic
Approach. Philadelphia, W. B. Saunders Co.,
1955, p. 105.
9. BTJRCH, G. E., EAT, C. T., AND CRONVICH, J. A.:
Certain mechanical peculiarities of human
cardiac pump in normal and diseased states.
Circulation 5: 504, 1952.
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Erratum
Vol. X, pages 741 and 743, Intereoronnry Reflex. Demonstration by Coronary Angiography, Santiago V. Guzman et al.: illustrations for figures 2 and 4 should he reversed.
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Appl.
Relation Between Mural Force and Pressure in the Left Ventricle of the Dog
Lloyd L. Hefner, L. Thomas Sheffield, Glenn C. Cobbs and Willem Klip
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Circ Res. 1962;11:654-663
doi: 10.1161/01.RES.11.4.654
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