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AW\, J:- PhJ 5,0 1!...,j:l I Vol. :<~J) J t/lt) U 4~4- Jf~J3 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. Theor.Bwl. 148:175-192,1991. . 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 aortic valve disease.J. Am. con. Cardiol. 4: 1-7,1984. 17. LEE, Y. C., S. J. SU170N. Pulsusand alternans: echocardiographic e\idence ofAND reduced venous return alternating end-diastolic fiber length as causative factors. Chest 80: 756-759, 1981. 18. MAY, S. A; Ben-Halm, for aortic re~r~tation. Cardiol 52' R. 211-213 1983 10. DE\.AN~Y. L. An introduction to Chaotic S\.~tem,~. ~lenlo Park, CA: Benjamin/Cummings, 1986. . 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 261: 459-467,1976, 19. MCGAUGHEY, M. D., W. L. MAUGHAM,K. SUNAGAWA, ANDK. SAGAWA. Alternating contractility in pulsus alternans studied in the isolated canine heart. Circulation 71: 357-362, 1985. Received 27 September 1990;acceptedin final form 26 February 1991. 20. NOBLE,R. J., AND D, O. NUTfER. The demonstration of alternating contractile state in pulsus alternans. J. Clin. Invest. 49: 11661177,1970. 21. OPIE, L. H., K, R. L. MANS FORD, AND P. OWEN. Effect of increased heart work on glycolysis and adenine nucleotides in the perfused REFERENCES 1. ADLER, D., AND Y. MAHLER. Modeling mechanical alternans in heart of normal and diabetic rats. Biochem. J. 124: 475-490, 1971. 22. PAVLlDIS, T. 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