Download Interrelation between Restitution Time

Clinical Science (1970) 39, 625-639.
I N T E R R E L A T I O N BETWEEN RESTITUTION
TIME-CONSTANT A N D A L T E R N A T I N G
M Y O C A R D I A L CONTRACTILITY I N D O G S
Y . MAHLER
AND
S . ROGEL
Cardiac Physiology Laboratory,
Hadassah University Hospital, Jerusalem, Israel
(Received 5 March 1970)
SUMMARY
1. In open chest dogs with complete heart block the ventricles were stimulated
electrically at various steady rates, and a single premature or delayed beat was interposed to alter myocardial tension. This was followed by transient mechanical
alternans. Steady alternans was induced by high stimulation frequencies.
2. It appeared that the recovery of the force of contraction of a beat could be
approximated by an exponential function of the preceding time interval. We observed
a correlation between the time-constant of this exponential curve and the force of
contraction of the previous beat.
3. The results demonstrate that a general rule of mechanical alternans can be
expressed by a single equation. This equation describes adequately (1) the duration
of transient mechanical alternation produced by a delayed beat, (2) the production
of steady alternation at rapid ventricular rates, and (3) the manifestations of postextrasystolic potentiation during mechanical alternans.
4. Curves relating amplitude of one beat to the successive beat, based on experiments performed in in vivo dog hearts, corresponded to the results predicted by the
equation.
The existence of mechanical alternation of the heart beat has been known for a century
(Traube, 1865) but its cause and mechanism of production is still a matter of dispute. It has
been assumed to be related to alternating electrical activity of the ventricles (Badeer, Ryo,
Gassner, Kass, Cavaluzzi, Gilbert & Brooks, 1967; Braveny, 1964; Hoffman & Suckling,
1954; Koch-Weser, 1967; Noble, 1967; Roselle, Crampton & Case, 1966; Trautwein & Oudel,
1954), to partial asystole of the ventricles (Badeer et al., 1967; Benforado, 1958; Gilmore,
Powell, Graham & Clancy, 1967; Mitchell, Sarnoff & Sonnenblick, 1963), to changing filling
pressure (Badeer et al., 1967; Gleason & Braunwald, 1962; Gilmore et al., 1967; Mitchell et al.,
Correspondence: Dr S. Rogel, Cardiac Physiology Laboratory, Hadassah University Hospital, P.O. Box
499, Jerusalem, Israel.
625
626
Y. Mahler and S. Rogel
1963), to changing atrioventricular conduction time (Brooks, Gilbert & Janse, 1964), or to a
failure of an intrinsic regulatory mechanism of cardiac contractility (Braveny, 1964; Green,
1935; Mitchell et al., 1963; Nayler & Robertson, 1965). The phenomenon of mechanical
alternans is known to occur at high frequencies, or at lower rates when myocardial function is
impaired (Brooks et al., 1964; Gilbert, Gassner, Kass, Drogin, Ryo & Brooks, 1966; Green,
1935). It has always been associated with excessive loading or an abnormally responding
myocardium.
Our purpose is to present data indicating an interplay between contractile force of a beat and
the time-constant of the exponential recovery of contractile force in the succeeding beat (the
‘restitution time-constant’). This relationship exists in the normal heart, as well as in diseased
hearts, at normal or rapid rates. Mechanical alternans appears to be a manifestation of such
interrelation.
METHODS
Experimental procedure
Experiments were performed on fifteen dogs, anaesthetized with sodium pentobarbitone
(30 mg/kg). The heart was exposed by midsternal split and artificial respiration was maintained.
A strain gauge arch was sewn to the right ventricular wall, parallel to and near the septum.
The two limbs of this force transducer were stretched when fixed to the myocardium, so that
the length of the myocardial segment between them remained constant. By doing so, isometric
tension could be obtained during an in vivo experiment, similar to the measurement of isometric
tension in isolated papillary muscle preparations.
Changing inflow pressure, end-diastolic pressure or the generated systolic pressure could not,
therefore, directly affect the tension measured in this isometric segment of the myocardium.
The linearity of the force transducer was checked by loading techniques. For sake of simplicity
the magnitude of tension variations was expressed in arbitrary units only. The output of the
force transducer was recorded on a multichannel thermal recorder together with an atrial or
surface electrocardiogram. Under direct vision, after inflow obstruction, complete atrioventricular block was produced by electro-coagulation of the atrio-ventricular nodal region.
Such a procedure makes it possible to study a wide range of heart rates with a single and
identical focus of excitation, from the slowest to the highest frequencies, including steady rates
as well as premature or delayed contractions. The resultant myocardial tension changes may
thus be compared with the different interbeat intervals since there are no variations in the
spread of activation of the regular or irregular beats. The ventricles were driven through
bipolar electrodes applied to the right ventricular wall close to the apex with a Grass S4
stimulator, delivering pulses of 2 ms duration, 5 V intensity, at ratesrangingfrom20to 320/min.
The stimulator was modified in order to permit the alteration of a single R-Rintervalduring the
steady, basic rate of stimulation. In all experiments and at different basic rates, a single extrabeat was applied, premature or delayed, to produce a change in the force of contraction. The
timing ‘of the extra-beats varied between the limits of the refractory period and up to the
appearance of the first idioventricular beat.
In five dogs the potential for total force development following a premature or delayed beat
was measured by the peak tension reached 1-2 s after the abnormal beat. This tension will be
referred to as the tension following full restitution = F, (Kedem, Mahler & Rogel, 1969).
Restitution and mechanical alternans
627
Method of analysis
It has been claimed that the force of contraction of a beat is determined by the recovery of
contractility as characterized by the restitution curve (Braveny & Kruta, 1958). This curve
reaches a long plateau within 1-2 s, and declines slowly after much longer time intervals. For
the purpose of our analysis, only interbeat intervals shorter than 2 s were used and the curves
can be regarded as exponential. The term ‘restituted beat tension’, FR, is used for the maximal
value of tension reached following full restitution (Kedem et al., 1969). On the basis of our
experience the time-constant of restitution is a function of the force of contraction of the
preceding beat, large amplitudes being accompanied by prolonged time-constants (Rogel &
Mahler, 1970).
On the basis of such relationship between the force of contraction and the time-constant of
the restitution curve, the following equation was developed :
where FRi = restituted beat tension after any beat ‘i’
Fi
= force of contraction of the beat ‘i’
F i + l = force of contraction of the beat following beat ‘i’
T(Fi) = time constant of restitution following beat with force F i
tr
= refractory period, defined as the minimal interval before a detectable mechanical
response can be elicited
t
= the time interval between beats ‘i’ and ‘i + 1’
e
= base of natural logarithm.
To prove the validity of this equation, experimental conditions are required in which the
FRiand the beat-to-beat intervals ( t ) remain constant. If under such conditions a change in
the force of contraction of a beat (Fi) is induced, then Fi+ is a function of Fi according to the
exponential factor in the above equation. The production of such a state was achieved in this
study by sustaining constant ventricular driving and inducing the required changes in Fi by a
single delayed beat, followed by immediate return to the constant interval, t. Thus, the T(F,)
was changed while the other parameters in the equation remained unchanged.
The peak myocardial tension of the beats with the constant interval as well as the tension of
the extra-beats and of the post-extra-beats were then measured and related to the force of
contraction of the preceding beat. The resultant experimental amplitude comparison curves
were compared with those calculated from the equation for each basic rate.
RESULTS
If in the course of steady driving of the ventricle a single interval is prolonged beyond the
basic R-R interval, the resulting contraction exhibits increased amplitude of tension. This
continues until full restitution of contractility is accomplished, following which further
increase of a single R-R interval is not accompanied by significant peak tension change within
the time interval used in these experiments. This asymptotic value is the FR for the steady
driving rate (Fig. 1). Similarly, FR following an extrasystole can be measured as peak tension
Y. Mahler and S. Rogel
628
of a beat 1-2 s after the studied beat (Rogel 8c Mahler, 1970). In order to define the condition
in which almost constant F, values can be obtained, F R was measured after various shortened
and delayed intervals and the resultant curve is shown in Fig. 1. It is noteworthy that the F R
curve changes markedly for premature beats while it is almost unchanged after delayed ones.
As a result of these observations, delayed beats were used to initiate transient mechanical
alternans with unaltered F,.
I
I
Restitution
curve
/-e-e--e-e-.0
0
0
Premature
I
Delayed
I
1
I .oo
0.50
-
FR
t
I
1-50
Interval (s)
FIG.1. Restitution of contractility for a basic rate of 120/min measured as peak tension following
premature and delayed beats. Note the approximately exponential shape of this tension-time
relationship (line connecting solid circles). The asymptotic value of tension for long intervals
represents the F R for the basic rate. S on the restitution curve indicates the tension at the steady
rate. The line marked F R (joining open circles) shows that the peak tension after full recovery
changes markedly following premature beats but it becomes almost constant following delayed
beats. Myocardial tension is measured in arbitrary units. The abscissa indicates the R-R intervals
between the normal beat and the extrasystole. The dotted vertical line intersects the curves at the
basic contraction period (in this case 0.5 s, corresponding to a rate of 120 beatslmin).
When the heart was driven at rapid ventricular rates mechanical alternans was produced.
If the rates of recovery of contractility after the strong beat are compared with those after the
weak beat, the resultant restitution curves reach the same maximal tension (same FR) but
their ascent is different. The time-constant (T) following the large beat is longer than T after
the smaller beat. The relationship between the time-constants following the large and small
beats is approximately the same as the relationship between the amplitudes of these contractions. Such a representative experiment is illustrated in Fig. 2.
A series of tracings is shown in Fig. 3 with different delays during a regular basic rate. When
the peak tension amplitude of each beat is plotted against the tension of the previous one,
starting with the delayed beat, the graph on the right of Fig. 3 is produced. It illustrates a
continuous common curve, irrespective of the duration of the delayed cycle. This also shows
629
Restitution and mechanical alternans
30
I
~
b50
I
tr
0560
0.70
0.80
1.00
0.90
Interval (s)
FIG.2. Separate restitution curves following the large beat and following the small beat during
mechanical alternans. Note that both curves reach the same F R but the restitution after the large
beat has a longer time-constant, T(Fl), than following the small beZit of the mechanical alternans,
T(F2). Time-constant is defined as time required to reach (1-e- l)FR.
The solid line indicates the restitution curve following the small beat and the broken line shows
the restitution curve after the larger beat. S1 and S z mark the large and small beat tensions,
respectively. FR = force of restituted beat; f , = refractory period.
-
A
0
C
D
0.0
' 0
I
20
\.
p
.
30
Fi
FIG.3. Transient mechanical alternans during steady, regular rhythm initiated by a single delayed
beat. The basic cycle length is 0.34 s. The delayed intervals are 152 s (tracing A), 058 s (B),
0.48 s (c)and 0.44s (D). Note that the myocardial tension is markedly increased following the
long pause, and returns to control level with mechanical alternation. The number of beats to reach
control level is lower with shorter delays. The myocardial tension of a beat (Fa+1) is plotted
against the tension of the preceding beat (F,), for each beat (following the delayed one) in all four
tracings (A, B, C, D). Note that a common amplitude comparison curve is valid for the transient
alternans initiated by different delayed intervals.
Y. Mahler and S. Rogel
630
that no significant change could have taken place in the contractile state of the myocardium.
Since in our experimental model the R-R intervals between the studied beats are equal, and
the FR is also relatively constant (as shown in Figs 1 and 3), it is concluded that in mechanical
alternation of contractility the amplitude of a beat is a function of the amplitude of the
previous one.
The shape of the amplitude comparison curve at various steady heart rates is different
according to the relationship expressed in equation (1) (as elaborated in the Appendix). This
shape determines however whether the mechanical alternation following a single delayed beat
is short or prolonged, transient or permanent.
A representative tracing from an experiment with a moderate driving rate (R-R = 400 ms)
is shown in Fig. 4. The amplitude comparison curve for this rate (plotted below) corresponds
ECGA
40r
10
20
30
F,
FIG.4.Transient mechanical alternans induced at a steady rate (R-R 0.40 s). The myocardial
tension from the first augmented beat following the delayed one (1) gradually returns to control
for such a basic rate is shown below.
level in six beats. The amplitude-amplitude curve (F,-Fi+l)
On this curve the tension values of the upper tracings are shown in sequential order as marked by
the numbers 1-6. The line Fi = F,,, intersects the curve at a point S, the tension value for the
steady state at the above rate. ECGA = atrial electrocardiogram.
Restitution and mechanical alternans
63 1
well with that calculated in the Appendix for such a frequency (C = 1.0). Introduction of a
single, delayed beat (R-R = 580 ms) produces a beat with markedly increased tension due to
more complete recovery of contractility (beat No. 1 in Fig. 4). This causes a shift from the steady
state tension (marked ‘S’ on the graph) to a new position (marked ‘1’). A transient, alternating
return to control tension takes place according to the sequence defined by the curve (as shown
by the dotted line and the arrows in Fig. 4). Six beats are required at this rate to return to steady
state tension (S).
At slower heart rates the calculated as well as the observed curves indicate a shorter, transient
alternating phenomenon and a more rapid return to control tension. If, however, the driving
frequency is high (R-R = 330 ms) the amplitude comparison relationship is such as to produce
a curve which is symmetrical in its central portion about the Fi = Fi+ line. In such instance
a steady, permanent mechanical alternans should result.
30
Fi
+
t
/
I
20 -
10-
10
20
30
Fi
FIG.5 . Persistent mechanical alternans at a rapid ventricular rate (R-R 0.33 s) interrupted by a
delayed beat (R-R 0.48 s). The myocardial tension shows the steady alternation ( S 1 - S I ) and the
transient increase in the beat to beat variations following the delayed beat (1-9). The graph drawn
for such a basic rate is symmetrical in its central portion to the F,= Fi+ line. Therefore, a steady
mechanical alternans is maintained at S1-S2 points. Following the delayed beat the tension returns to
S1-Szalong the curve as indicated by the numbers 1-9 and directed by the arrows.
632
Y. Mahler and S. Rogel
Fig. 5 illustrates steady alternating tensions, S,-S,, on the tracing and on the graph. Increase
of a single R-R interval (to 480 ms) displaces the tension to position ‘1’ on the graph. The
return from this position is conducted, according to the function curve for this rate, through
eight beats (the sequence of which is marked by the consecutive numbers), leading back to
S,-S,, i.e., to a steady mechanical alternans.
Theoretical considerations indicate that at very rapid rates an amplitude to amplitude relation is obtained in which a line symmetrical to the Fi = Fi+ line over a wide range of amplitudes
is produced (see equation (3) in the Appendix). In such cases steady mechanical alternation
may persist at different Fi-Fi+l combinations. Fig. 6 illustrates such a case. It can be seen
0.19 s) causes steady mechanical alternans. The
myocardial tension of the small beats are buried in the larger ones in tracings A, B, C , and are
visible only in D. The large beats have markedly different tension values even though the cycle
length is equal in all tracings. At such a rate the amplitude-amplitude curve (FI-FI+I)is symmetrical to the Fi = Ficl line in a wide range; therefore, mechanical alternans can be steadily
maintained at different levels of tension.
that a series of different steady states of myocardial tension can be achieved at the very same
R-R intervals, kept constant at 190 ms throughout this continuous tracing. Only one cycle
length need be prolonged to initiate transition from one steady state to another.
Mechanical alternation of contractility, so far presented, was obtained under experimental
conditions in which the contractility was kept relatively constant, as was required to evaluate
the validity of our equation. However, mechanical alternans can also be produced by interval
Restitution and mechanical alternans
633
changes which are accompanied by an altered level of FR.A simple example of this is a premature contraction with post-extrasystolic potentiation. In such an instance the post-extrasystolic
beats exhibit alternating peak tension values, but contrary to the previously described rules and
AF
0
10
2
-
0 1 2 3 4 5 6
Fi
FIG.7.Comparison between mechanical alternans induced by a delayed beat (R-R 0.44s, right),
and alternans following a premature beat (R-R 0-20 s, lefr)during steady heart rate (R-R0.32 s).
The amplitude-amplitude relationship following the delayed beat is illustrated in the left lower
graph by the solid line. Note that the tension values following the premature beat (marked 1-7)
are higher than those anticipated according t o the F,-F, line for such a basic rate. This increment
(AF) is due to post-extrasystolic potentiation and is shown to decay from beat to beat (right
lower panel).
curves, these beats are of higher tension than expected from the Fi-Fi + curve (Fig. 7). Similar
observations can be obtained when driving rates are suddenly increased from slow rates to
high frequencies.
DISCUSSION
A number of studies have been reported dealing with the possible causes of mechanical
alternation of contractility-other than factors related to the intrinsic properties of the myocardial fibres. Among these are included attempts at relating mechanical alternans to endG
634
Y. Mahler and S. Rogel
diastolic pressure or volume of the ventricles, to velocity of conduction within the conduction
system, or to partial asystole of the ventricles as a result of inequality of refractory period in the
different fibres of the myocardium (Badeer et al., 1967; Benforado, 1958; Braveny, 1964;
Gleason & Braunwald, 1962; Hoffman & Suckling, 1954; Koch-Weser, 1963; Mitchell et al.,
1963; Noble, 1962; Roselle et al., 1966; Trautwein & Oudel, 1954). However, in a recent article
Guntheroth and his associates (Guntheroth, Morgan, McGough & Scher, 1969) have denied
any correlation between left ventricular end-diastolic pressure or diameter, and stroke volume
during mechanical alternans. They claim therefore, that ‘heterometric autoregulation cannot
account for pulsus alternans’.
Straube (1917) indicated that mechanical alternation is a manifestation of incomplete or
impaired recovery of myocardial contractility. Many others have also accepted intrinsic
myocardial factors determining alternation (Braveny, 1964; Green, 1935 ; Guntheroth et al.,
1969; Mitchell et al., 1963; Nayler & Robertson, 1965). The examples cited to emphasize
the role of such intrinsic, perhaps metabolic factors were impaired performance of the failing
heart, anoxic conditions, rapid ventricular rates and the effect of certain drugs. Thus, mechanical alternans is thought to have a bad prognostic significance(Badeer et al., 1967; Benforado,
1958; Brooks et al., 1964; Cohn, Sandler & Hancock, 1967; Friedberg, 1966; Gilbert, Janse,
Lu, Pinkston & Brooks, 1965; Gilmore et al., 1967; Green, 1935; Roselle et al., 1966).
Some authors attempted to relate mechanical alternation to electrical alternation (Ellis, 1960;
Hogancamp, Kardesch, Danforth & Bing, 1959; Lu, Lange & Brooks, 1968; Roselle et al.,
1966). Others have shown that these two phenomena may occur independently (Badeer et al.,
1967; Braveny, 1964). Microelectrode studies also did not reveal any constant relation between
amplitude or duration of the action potential and the force of contraction (Kleinfeld, Steine
& Magin, 1956; Kleinfeld, Stein & Kossman, 1963).
Alternating contractility of the myocardium is known to be a transient phenomenon, as
well as a sustained condition. It can be initiated by rhythm-disturbances such as premature or
delayed beats, sudden rate changes or rapid ventricular rates. The duration of a transient
mechanical alternans is a function of the prevailing frequency, if the precipitating factor is
constant (Gilbert et al., 1965). It has been shown that in diseased hearts steady alternation can
be induced at slower rates than in normal hearts (Cohn, 1967; Green, 1935; Roselle et al., 1966).
It has also been demonstrated that a gradual change in frequency may prevent occurrence of
mechanical alternans at a rate otherwise causing alternation (Braveny, 1964; Brooks et al.,
1964; Gilbert et al., 1965; Koch-Weser & Blinks, 1963). In such a case, however, the steady
state is very labile and a single rhythm disturbance is sufficient to disturb the equilibrium and
initiate sustained alternation. Braveny (1964) has concluded that ‘excessive loading of the
intrinsic control of the heart activity by the rhythm’ causes a ‘periodic change of the entire
functional state of the heart’-as manifested by alternation. This general statement, however,
does not clarify the genesis of mechanical alternans at different rates or rhythm disturbances.
Although Braveny described changing restitution curves during steady alternans in isolated
dog’s heart at 22.5”, he did not consider it as a causative factor in the mechanism of production
of mechanical alternans.
In their recent study Guntheroth et al. (1969) accepted the theory that part of the myocardium does not contract in the small beats during pulsus alternans, but they claimed that the
inotropic effect of excitation remains unutilized until the next beat, which thus becomes potentiated. The present study indicates, however, that following full recovery of contractility the
Restitution and mechanical alternuns
635
maximal tension after a strong beat or a weak beat of mechanical alternans is equal. Our
suggestion is therefore, not that the number of myocardial cells participating in the contraction
is different in the alternating beats but rather the time-course of recovery of the whole myocardium.
Our working hypothesis has been based on the assumption that the rate of restitution of
contractility after a given beat is an inverse function of its amplitude. The contractility recovers
more quickly after a beat with low tension than after a more forceful systole. This can be shown
in the case of a premature beat, in a post-extrasystolic beat and during sudden rate changes.
But in all these conditions the manifested potentiation is superimposed on the changes resulting
from the relationship between restitution time-constant and amplitude of tension. To analyse
this latter dependence, a model was required in which the intervals could be kept constant
except for a single delayed interval, compared to the basic rate, to induce mechanical alternation. In such a condition the potential for total force development after full recovery (FR)
is fairly aonstant (Rogel & Mahler, 1970). Thus true amplitude comparison curves can be
drawn. The equation developed in the present study indicates that the force of contraction of a
beat is approximately described by an exponential function having a time-constant T(F,) and
a maximal value of FR.
Braveny (1964) has shown restitution curves having different slopes and varying onset following beats of alternating amplitude. Analysis of restitution curves in our experiments with steady
mechanical alternans indicates that the changing time-constant is the major determinant of the
resultant myocardial tension. Variations in the onset of the restitution curves (t, in the equation)
which were either absent or undetectable in the experiments under discussion, cannot, however,
be completely excluded. The restitution curves exhibited equal FR levels for each of the steadily
alternating beats.
The various shapes of the amplitude-amplitude curves, as shown in the results, are due to
the different basic frequencies, and they all result from the same exponential equation (see
Appendix).
Several possible theoretical derivations from equation (1) have been shown to occur in the
actual experiments performed. It was shown that during a regular, steady, low rate an equal
tension is maintained in each beat as indicated in the amplitude-amplitude function curve at
its intercept with the Fi = Fi+l line. The constant tension during normal rhythm is therefore
just a special instance which follows the general rule of mechanical alternans as defined by the
equation. It also follows that at rapid basic rates different steady conditions of alternation can
occur since the curve is symmetrical over a wide range to the Fi = Fi+ line. If the frequency is
slowly raised to this level and thus no sudden interval change is induced, alternation may be
prevented, since the transition from one frequency to another takes place along the F, = F, +
line. This condition however is labile and can easily be replaced by maintained alternation if
the amplitude of only one beat is altered. This leads to a different restitution time-constant of the
extra beat, which in turn is followed by alternation along the symmetrical portion of the line.
Theories of ‘excessive load’ or abnormal interval-strength relation or an ‘oscillatory mechanism’ (Badeer et al., 1967; Nayler & Robertson, 1965) do not seem to be needed or adequate to
explain the observed phenomena which, however, are well described by our general rule.
The increased occurrence of alternans in the diseased heart, and the slower rates at which
alternation may begin, can be explained by the prolonged time constant of the recovery of
contractility in diseased states. Mechanical alternans is dependent on the ratio between the
Y. Mahler and S. Rogel
636
interbeat interval and the time-constant rather than on the rate itself. In the deteriorated
myocardium this ratio is small enough to produce steady alternation even at slower frequencies
than in the normal heart with fast recovery.
It is known that transient mechanical alternans may occur following rhythm disturbances
(Badeer et al., 1967; Braveny, 1964; Guntheroth el ul., 1969; Nayler & Robertson, 1965).
It was shown by Gilbert et al. (1965) that such alternation lasts longer as the basic heart rate
increases or if the initiating delay interval is longer. This observation, as shown also in our
results, is expected and explained by the equation. The duration of transient mechanical
alternans is determined by the shape of the curve which, as shown, is concave at high rate and
thus lengthens the period of alternation; at low frequencies it is of a different shape which
shortens alternation. Similarly, the number of beats required to return to the steady tension is a
function of the amplitude of the initiating beat, which, in turn, is a function of its preceding
interval.
The conclusion from the data presented and the equation proposed, is that the timeconstant of restitution of a beat is related to its myocardial tension, at any heart rate. This
assumption, which seems to be proven in the present study, is adequate to explain the different
forms, magnitude and duration of mechanical alternation of contractility, provided that
phenomena of potentiation do not co-exist. If, however, alternation is initiated by a premature
beat causing potentiation, the post-extrasystolic potentiation can be shown to be superimposed
on the mechanical alternans which is anticipated according to the proposed general rule of
mechanical alternation of contractility.
APPENDIX
According to our theoretical considerations, the force of contraction of a beat can be expressed
by the following formula:
= force of contraction of the beat ‘i’ (measured as peak tension)
where Fi
Fi+l = force of contraction of the beat following beat ‘i’
T(Fi) = time constant of restitution following beat with force F,
t,
= refractory period
= the time interval between beats ‘i’ and ‘i+ 1’
t
= base of natural logarithm
e
For the graphic representation of the relationship between Fi-Fi+, it is assumed that the
time constant T(Fi) is a linear function of F,. Defining the force of contraction during steady
state as F,, the force Fi can also be expressed as Fi = F,.= Equation (1) can thus be given as
(
“‘>
Fi+l = F R . 1-e-K.F..z
In an attempt to normalize equation (l), a steady state is described as Fi+, = F, and ( t - tr)
/I(.F, = C. The force of contraction of a beat in a steady state can thus be defined
F, = FR(1 -e-‘)
Fs
resulting in FR = 1 -e-‘
Restitution and mechanical alternans
637
Substituting FRin equation (l), equation (2) is obtained
Fi+l = Fa
(1 - e-E)
(2)
l-e-c
This equation makes possible the construction of a family of curves (Fig. 8) relating Fi,.JFS
to Fi/F, = z, for various values of parameter C.
I
I
1
I
I
2
I
5 /F,
FIG. 8. Normalized, theoretical amplitude-amplitude curves, calculated for different rates
according to equation (2). F,= force of contraction of beat ‘i’; Fi+l= force of contraction of beat
‘i+ 1’; F,= force of contraction in a steady state. For other symbols see Appendix.
This latter parameter can be considered as a measure for the heart rate since F, changes over
a much narrower range than (t - t,) in the various rates.
For very rapid heart rates ( t - t,)/K . Fi is very small, thus according to the approximation
e-’ = 1 -x which is valid for x 1, equation (1) can be expressed
A
Fi+l =
(3)
-=
Fi
where A is a constant.
Equation (3) indicates a symmetrical relationship between Fi and Fi+l. Therefore, at such
high rates a steady mechanical alternans may occur at any F, value.
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