Download Periodicities of cardiac mechanics

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SHLOMO A. BEN-HAIM, GILA FRUCHTER, GAL HAYAM, AND YEOUDA EDOUTE
Cardiovascular Research Group, Rappaport Family Institute for Research in the Medical Sciences
and Department of Physiology and Biophysics, Faculty of Medicine, Tt."'hnion- Israel
Institute of Technology, 31096 Haifa, Israel
nonlinear dynamic;mathematicalmodel;chaos
for the "yolution of these mechanical periodicities have
beenproposedby physiologists and clinicians. Somehave
suggestedthat the observedgrossoscillations of volume
and pressure within the heart are actually the result of
microscopic oscillations of one or severalbasicproperties
of the contracting myocardium, such as the calcium flux
(2, 3), whereasothers have suggestedthat oscillations of
the venous return or the afterload are responsible for
these mechanical periodicities (5,6,12,14,17,26).
Because the mechanical action of the heart can be
describedby its pressure-volumerelations as a nonlinear
dynamic system, we propose in the present work that
mechanical periodicities of the heart action may also
result from its nonlinear nature.
In the following sections we first describe a comprehensive model suitable for quantitative description of
ergodicbehavior of cardiac mechanics,and then we apply
stability analysis and determine the factors that will
drive a normal beat-invariable beating heart to an abnormally beating regimen. The theoretical analysis presented in the first part of t.i1epaper is then compared
with experimental results of the isolated working rat
heart.
THE RECENT RECOGNITION of the importance of nonlin-
THEORY
BEN-HAIM,
SHLOMO A., GILA FRUCHTER, GAL HAYAM, AND
YEOUDA EDOUTE. Periodicities of cardiac mechanics. Am. J.
Physiol. 261 (Heart Circ. Physiol. 30): H424-H433, 1991.Using a finite-difference equation to model cardiac mechanics,
we simulated the stable action of the left ventricle. This model
describes the left ventricular end-diastolic volume as a function
of the previous end-diastolic volume and several physiological
parameters describing the mechanical properties and hemodynamic loading conditions of the heart. Our theoretical simulations demonstrated that transitions (bifurcations) can occur
between different modes of dynamic organization of the isolated
working heart as parameters are changed. Different regions in
the parameter space are characterized by different stable limit
cycle periodicities. Experimental studies carried out in an isolated working rat heart model verified the model predictions.
The experimental studies showed that stable periodicities were
invoked by changing the parameter values in the direction
suggestedby the theoretical analysis. We propose in the present
work that mechanical periodicities of the heart action are an
inherent part of its nonlinear nature. The model predictions
and experimental results are compatible with previous experimental data but may contradict several hypotheses suggested
to explain the phenomenon of cardiac periodicities.
ear dynamics to the understandingof periodicbehavior
?f dynamic systemshascauseda proliferation of research
m both the mathematicsand physicsof nonlinearsystems (8, 10, 14, 18,25). Other investigatorshavetreated
the rhythms or periodicities common in physiologyas
nonlinear oscillations of dynamicsystems(22,28-30).
A normally beating heart under steady-stateconditions ejects a constant beat-invariant stroke volume.
Beat-varyingstroke volumeat steadystateis a pathological sign commonto a variety of diseasestates.The most
frequently encounteredtime-varyingstroke-volumepat-
tern,
known
as "mechanical
alternans"
(MA),
corre-
Finite-DifferenceEquationsModel
Th~ left ven~r!cle.(LY) is conceptualized
as.a single
c?nt~mer,recei~mgits Inflow from the.left atnum and
ejectingpa~ of its contentst.othe aortic outflow.Both
the left atnum and the aortic outlet are connectedto
constant-pressure
sourceand sink, respectively.
Let Xn be the end-diastolicvolume(EDV) of the nth
beat.The massbalancefor that beatcanbedescribedas
X =X + U - Y
(1)
n+l
n
n
n
spondsto a stroke volumewith a cycleof period2. During
the MA beating regimen, a large stroke volumeis followed by a small stroke volume, and so on. Clinically,
MA is perceived as a sign of heart failure (14, 17),
hypertrophic cardiomyopathy(4, 13),and valvularheart
disease(7, 9, 16) and is correlatedwith poor outcome.
More complex patterns of beat-varying activity of the
heart at a constant rate havenot reachedclinical recog-
where Un is the blood volumeenteringthe LV during
diastole, Yn is the blood volume ejectedfrom the LV
during systole (stroke volume), and Xn+l denotesthe
EDV of the followingbeat.
AssumingPoiseuilleflow throughthe a.orticvalveand
assumingfor simplicitythat ejectionaccountsfor half of
the cardiacperiod,we can relatethe strokevolume(Yn)
to the transaorticpressuregradientas
nition; however,
experimentally
these
patterns
have
been
Yn = T .(P. - Pa)/Ra.2
[P. >- P]a
(2)
detected
during preload,
afterload,
and
inotropic
altera\-,
tion perturbations (23). Severalmechanismsresponsible
H424
Yn = 0
0363-6135/91 $1.50 Copyright @ 1991 the American Physiological Society
[P. < Pa]
(3)
PERIODICITIES
TABLE
=
OF CARDIAC ~IECHANICS
H425
1. Model Parameters
-~-
"-'"
Paramftfr
-
-- -
--
==
'-alu.
105cmHzO/ml'
1 ml
10cmH,O
20cmH,O/ml'
23cmH,O
60cmH,O
5,000cmH,O.ms-ml-1
4.000cmH,O-ms-ml-1
200ms
K1
115cmHzO
-0.0645 cmH.O/ms
K..
-L.. systolicpressure-volume
relationship;V.. left ventricular (LV)
volume at peak LV pressure; Fd. unstressedLV diastolic pressure; ~.
L.
V.
Fd
~
P..
P.
R.
R..
T
.
diastolic pressure-volume relationship; P..Ieft atrial pressure; P., aortic
pressure; R.. outflow resistance;R., inflow resistance;T. cardiac period;
Kt. heart rate-independent peak systolic LV pressure; Kz. heart ratedependent peak systolic LV pressure-
0:2
0.4
0.8
0.8
1
1.2
Xn. END-DIASTOLICVOLUME (ML)
FIG. 2. Diastolic pressure-volume relation of an isolated arrested
rat heart used for deriving values of F d and ~.
where P. is the mean systolic pressure within the LV, Pa
is the meanarterial pressureat the aortic outlet (which
equalsthe instantaneousarterial pressure),Ra is the
outflowresistance,
andT is the cardiacperiod.
The P. is relatedto the EDV, Xn, using an approximation of the Frank-Starlingrelation as
P.
=
F.
-
L(Xn
,
-
V.)2
(4)
whereF., L., and V. arethreeparametersapproximating
experimentalsystolicpressure-volume
relations. These
parameterswere used for approximatingexperimental
results into an analytic expressionthat describesthe
interdependence
of systolicpressureand L VEDV.
The effectof the interval betweencontractionson the
steady-statelevel of the maximally developedforce,F.,
is describedwith a linear approximationof the "staircase
effect"
F. = Kl + K2T
j.J.
(5)
H426
PERIODICITIES OF CARDIAC ~IECHA~lrS
where Kj and K2 are two parameters describingthe of R" to fill the LV during diastole with a mean pressure
experimentaldependence
betweenpeakLV pressureand of Pd.The inflow can therefore be described as
the cardiac cycle. The inflow to the heart is driven by
Un = T'(Pv - Pd)/2.R, [Pv~ Pd]
(6)
the left atrial pressure,Pv (which equalsthe meanleft
Un = 0
[P, < Pd]
(7)
atrial pressure),through a mitral valvewith a resistance
H428
PERIODICITIES
OF CARDIAC MECHANICS
A
8
lOG POWER
0
1
2
3
4
Frequency,Hz
c
lOG
0
POWER
0.5
1.5
2
2.5
3
3.5
4
4.5
a
Frequency,Hz
0
1
2
3
4.
5
8
7
8
a
10
Frequency,Hz
E
lOG POWER
FIG. 6. Effect of heart rate on left ventricular (LV) pressureperiodicities: power spectrum of LV pressure during pacing at 460 (A), 310
(B), 230 (C), 110 (D), and 80 (E) ms.
Equation 9 describesthe discrete dynamic systemof
our model.
isovolumic contractions of the experimental preparations
(Fig. 1). Diastolic parameters (Fd and Ld) were identified
from the diastolic pressure-volume relation of the arParameters
rested heart (Fig. 2).
We choseparametersdescribing the behaviorof the
Kl and K2 are the linear regressioncoefficients describisolated working rat heart; their valuesare summarized ing the relation between peak pressure and the heart
in Table 1.
contraction interval of the same experimental preparaValues of Pv and P were identical to the experimental tion.
settings later used. Systolic parameters [Fs(T), Ls, and
Values of Raand Rv were calculated using the pressure
V.J wereidentified from the pressure-volumereiationsof gradient over the valves divided by the cardiac output,
8
~
~~
H429
PERIODICITIES OF CARDIAC MECHA~ICS
'
.
.
.
I
Stable fixed points are those that attract adjacent
solutions, whereas unstable fixed points repel adjacent
solutions. The stable fixed points can be further characterizedby the property that the slope at the fixed point
is smaller than one in its absolute value: If' (Xe) I < 1
(11).
In our dynamic system, only in a specific region of the
parameter spacewill the equilibrium point (Xe) be stable.
This region can be identified analytically as well as
numerically and is illustrated in Fig. 3 (inset A). At the
borders of this region, a bifurcation takes place which
creates a stable cycle of period 2. As depicted in Fig. 3
(inset B), the stable EDV (Xe) cycle is dependent on the
parameter value as well as on the initial condition. Further increasing the parameter value will lead to a periodmultiplying behavior in which the stable cycle of period
2 will be replacedby a stable cycle of period 4, and so on.
Generally, the distance betweeneach bifurcation and the
following one becomes shorter as the period increases.
The point of accumulation is known as chaos, as the
order of the EDVs appears random.
The stable cycle of Xn is illustrated in Fig. 3. For a
beat-invariant case (Fig. 3, inset A), the single fixed
point is the cross section of the map and the diagonal;
for a two-periodic case (Fig. 3, inset B), alternans of
small and large EDVs is shown (Xl and X2). The alternating sequenceof EDVs in higher periods is depicted in
Fig. 3, inset C (where a stable cycle of four EDVs, Xl,
X2, X3, and X4, is present).
The precise mathematical derivation of the regions of
the parameter spacescorresponding to each of the behaviors has been described in detail in a previous paper
1.2
1.4
1.&
1.8
2
(11).
A lOG POWER
10'-
.
0
0.2
0
0.4
0.8
0.8
1
1.2
l.
.
.
I
1.8 1.8 2
Frequency,Hz
8 lOG
POWER
lOr-
/
"--
lII
0
0.2
L
~~
l. -
..
0.&
_.~
0.8
1
Frequency,Hz
FIG.7. Effectof inflowresistance
on leftventricular
(LV)pressure Simulation Results
..
periodicities:power spectrumof LV pressureduring steadystatesat
inflowresistance
(R.)of 4,000(A) and6,000(B) cmH2O.ms.ml-1.
Stable EDV was found to be sensItIveto all parameters
as well as to the initial conditions. Within the parametric
assuminglaminar flow. Simulationswerecarried out to region tested, we observed a period-doubling behavior,
determinethe equilibrium state of the model,given 10 as shown in Fig. 4. Increasing the values of Rv, Ls, and
nominal valuesof the parametersandgraduallychanging Fs(i) increased the period of the EDV cycle, whereas
the 11th parametervaluein a physiologicalrangeabout increasing the values of the other parameters (Pa and Ra)
decreased
the period
of X. Increasing
i initially increased
its nominal valtlP
"-EDV
periodicity;
however,
further increments
of i de-
Stability Analysis
We analyzedthe stability of our state variable (Xn)
with respectto the parametersof the model (Table 1).
The stability analysisdescribedbelow is a generalapproach applied for identification of stable points and
characterization of their stability used in one-dimensional nonlinear dynamicsystems.
For a specific casewhere all parametersare known,
the generalphaseplane can be drawn (Fig. 3, insetsAC). For any value of Xn, the following Xn+l can be
obtainedfrom Fig. 3.
The crosssectionsof our phaseplanewith the diagonal
are the fixed points (Xe) of our system.Iterating the
systemstarting with their valueas initial condition will
maintain their value. Specifically,these points correspond to Xn+l = Xn = Xe; in our case,therefore,their
valuecan be obtainedby solvingEq. 9 for this condition.
creasedEDV periodicity.
It is obvious that all parameters, both those describing
the intrinsic inotropic state of the heart as well as those
describing the loading condition of the heart, can induce
beat-varying action of the heart as a solitary cause.
EXPERIMENTAL STUDIES
To assessthe validity of our model predictions, we
conductedexperimentswith isolatedworking rat hearts.
Our model and parametersfor the model simulations
were all drawn from this experimental setup. In this
experimentalsetup, the inflow to the heart is totally
independentof the LV outflow, as modeled.
Methods
AnimaLS'.
Male Wistar rats (250-350g) wereused.The
animals were anesthetizedby intraperitoneal injection
~
H430
PERIODICITIES
OF CARDIAC ~tECHANICS
A
A
/
A
J\
'"'"""
v-"'-'"
.
I
7
8
0
2
3
4
I5
e
7
8
Frequency,Hz
FIG. 80 Effect of outflow resistance on left \Oentricular (LV) pressure
periodicities: power spectrum of LV pressure during steady states at
outflow resistance (R.) of 4,000 (A) 6,000 (B). and 10,000 (C) cmH2O.
ms.ml-l.
FreQuency,Hz
of methohexital sodium (30-40 mgjkg, Brietal Sodium, powerspectrumanalysisof the LV pressuresignalwas
Eli Lilly, Basingstoke,UK).
carriedout using a fast Fourier transformalgorithm.
Heart perfusion.After removal,the heart wasperfused
t ID .
.
by the Langendorfftechniquein a nonrecirculatingmode E
. Dunng
.
xper~men
a es~gn
at a constant pressureof 85 cmH2Ofor 10 mm.
this time, the left atrium wascannulatedto allow atrial
Oncea steady-stateoperationwasachieved(judgedby
perfusion (atrial pressure= 10-30cmH2O),accordingto stabilization of LV pressure and AF signals),a 6O-s
the modifiedworking heart modelof Opieet al. (21).The baselinerecordingwas carried out. Four pacing rates
L V ejectedagainst an adjustableresistor connectedto were applied in random order. Gradualchangeswere
a vertical tube with adjustable height (range 60-200 then madeto the inflow and outflow resistance.Each
cm). Perfusatetemperaturewaskept at 37:t 0.1°C.Perfu- protocol was maintained for 2 min, and the variables
sion medium was Krebs-Henseleitbicarbonate buffer were recordedthroughout the last minute. After each
(KHBB) equilibrated with 95% O2-5%CO2 at atmos- stagethe heart wasleft to recoverfromthe perturbation,
pheric pressure.The compositionof KHBB was as fol- reachinga newsteadystate (usuallywithin 2-3 min).
lows (mM): 143 Na+, 5 K+, 2.6 Ca2+, 1.2 Mi+, 1.2
We includedin the study only recordingsmadeduring
H2PO., 128Cl-, 25 HC03, 1.2 (SO.)2-,and 10 glucose. normal conductionof electrical activationasjudgedby
LV pressurewas recordedwith a Gould P23 ill pres- the epicardialECG tracing.
sure transducerconnectedto the LV through a 23-gauge
.
needleand a thin polyethylenecatheter (0.4 mm). Aortic E
fl ow (AF) wasrecordedby an electromagnetIc
. flowmeter xperlmentaI ResuIts
with a flow probe placed 3 cm above the aortic valve
Resultswerecollectedfrom 10heartsandwerehighly
(Nihon Kohden, MVF-2100). Bipolar silver electrodes reproducible.
were sutured, 1 mm apart, to the left atrium and used
Figure 5 is an exampleof the analogtracingsof LV
for pacing. Analog data from the pressure and flow pressureand aortic flow of heart beatingin a stablecycle
transducersas well asthe electrocardiogram(ECG)were of 1,2, and 4.
digitized using a 12-bit 500-Hz analog-to-digital conFigure6 summarizesthe powerspectrumof LV presverter (LabMaster, Tecmar, Cleveland, OH). Off-line sure during sequentialincrementsof heart period.It is
~
PERIODICITIES
A
H431
OF CARDIAC MECHANICS
B
lOG POWER
lOG POWER
,o~
A
-~
/"",
,
/
'"'-.
\-I
O'
0.2
0.4
0.8
0.8
1
1.2
1.4
1.8
1.8
2
la
I
2
Frequency,Hz
0
.
.
0.2
0.4
. .
0.1
.
. .
0.1
1
'.2
Frequency,Hz
'.4
.
.
I
1.'
1.1
2
c
lOG
POWER
\
I
r
"""",J
. .
:r
0
0.2
0.4
.
.
.
0.8
0.8
1
.
1.2
1.4
1.8
FreQupnCy,Hz
E
lOG POWER
1~-
/
--~
/\
\
/
FIG. 9. Effect of aortic pressure on left ventricular (LV) pressure
periodicities: power spectrum of LV pressure during steady states at
aortic pressure(P.) of 70 (A). 80 (B). 90 (C). 120(D). and 150(E)
cmH2O.
o' . . . . . . . . . . . . . , I
0 0.2 0.4 0.8 0.8 1 1.2 1.4 1.8 1.8 2 2.2 2.4 2.8 2.8 3
Frequency,Hz
evidentthat Subharmonicsat 1/2 (period 2), 1/3 (period
3), and 1/4 (period 4) becameevident as the heart period
decreased.
When we increasedthe inflow resistance,subharmonics of the basic heart rate evolved at half of the basic
frequency(period 2). These changesin the powerspectrum of the LV pressureare describedin Fig. 7.
Changingthe outflow resistancehad the oppositeeffect on cardiac periodicities. A two-periodic operation
was evident at low outflow resistanceand was replaced
by a fixed-point (cycle 1) operationwhen the outflow
resistancewasincreased(Fig. 8).
Decreasingthe aortic pressurecausedsubharmonics
of the basicheart rate.At high aortic pressures,a fixedpoint operationwaspresent(cycle1). This steadyoperation wasreplacedby a stableperiod2 during operation
at low aortic pressures(Fig. 9).
Similarly, decreasingthe inflow pressureresultedin
PERIODICITIES
change
parameters may invoke
d. in any
. d. of
. the svstem
hd. .
car lac peno IClty on ~ e con Itlon ~ hat other system
parametersare of specIficvalues,ThIs conclusioncontradicts
that
drawn
by Adler,
Mahler,
and
colleagues
(1-3), who have indicated that only intrinsic mechanical parameters can invoke mechanical periodicities.
.
O
1 .
h
d.
ur experiments
support our conc uslon t at car lac
periodicities
may be a form of a stable limit cycle of a
nonlinear dynamic system such as the heart and may
not be caused solely by periodically
alternating
values of
one or
, more of the cardiac
. parameters.
'"
stablesteadystateis replaced
by a stablelimit cyclewith
a period that increases as one or more parameter values
are graduaI Iy changed.
Th ~ auth ors th ank Rut h S.
.
.mger f;or ed'.Itmg t he manuscr~pt,..
This study ,,:a~supported m part by a grant from thE!Chief Scientist
of the
Israel
Address
Mmlst~
for
of Health
reprint
(184-139)...
request.s:
alternan!' after valve replacement
Card~ovascular
Am J.
11. FRUCHTER,
G., ANDS. BEN.HAtM. Stability analy!'i!'of one-dimen!'ional dy.namic system!' applied to an isolated beating heart. J.
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.
12. GA17ULLO, D., G. MILLEStMO, AND G. VaCCA. The role ofl!'olated
extrasystoles in the genesis of pulsus alternans: a preliminarv
report. IRCS Med. Sci. Libr. Compend. 10: 792-793,1982.
-
13. GLEASON, W. L., AND E. BRAUNWALD. Studies on Starling's law
of the heart. IV. Relationsh.ipbetwee,:,left ventri~ular end-dia!'tolic
14. HADA,Y.,C.WOLFE,ANDE.CRAIGE.Pulsusalternansdetermined
by biventricularsimultaneoussystolictime intervals.Circulation
65:617-626,19~2.
.
"
15. HAO,B. L. (Editor). Chaos.SIngapore:World Scientific,1984.
16. HESS, O. M., E, P. SURBER, M. RI17ER,
AND H. P. KRA YENBUEHL.
Pulsusalternans: its influence on systolic
and diastolic function in
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Y. C.,
S. J. SU170N.
Pulsusand
alternans:
echocardiographic
e\idence
ofAND
reduced
venous return
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S. A; Ben-Halm,
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L. An introduction
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vo~ume
and stroke
volume m.man,
~lth 2~:
?bser\.atJons
on the mechamsmofpulsus
alternans.
CLrculatwn
841-848,1962.
,
With our new explanatIon, It IS possIble to reconsIder
cardiac mechanics in a unifying mode. In this context, a
u
H433
OF CARDIAC ~1ECHANICS
~e-
se~rch Group, R,appaport Fam~lyInstitute for Research m the ~edical
ScI~nc~s,T~hnlon-Israel Institute of Technology, PO Box 9691,31096
Haifa, srae.
R. M.
dynamics,
Simple
Nature
mathematical
Lond.
models
with
very
complicated
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M. D., W. L. MAUGHAM,K. SUNAGAWA,
ANDK.
SAGAWA.
Alternating contractility in pulsus alternans studied in
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heart work on glycolysis and adenine nucleotides in the perfused
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