Dynamic properties of cardiovascular systems

Dynamic Properties of Cardiovascular

Systems

GILA FRUCHTER*ANDSHLOMOBEN-HAIMt CardiovascularResearchGroup, Rappaport Institute for Researchin Medical Sciences,Technion-IsraelInstitute of Technology,Haifa 31096,Israel Received21 July 1991;revised31 October 1991

ABSTRACT A recently derived mathematical model of an isolated heart is extended here to a closed-loop cardiovascular system. Taking the end-diastolic volume as state variable, the authors show that the closed-loop cardiovascular system can be described by a onedimensional nonlinear discrete dynamical system that dependson parametersdescribing the systolic and diastolic properties of the heart, heart rate, total peripheral resistance, and arterial capacitance. Studies of this model show that the system possessesa rich spectrum of dynamical behavior, from stable points through stable cycles to a .. chaotic" behavior. It is shown that such an analysis of dynamic behavior yields those domains in the parameter spacethat correspond to a normal and abnormal beating heart, when the heart ejects time-invariant and time-variant (periodic or aperiodic) stable stroke volumes, respectively. Determination of such domains may lead to better understanding of the specific pathologic mechanism involved in the evolution of an abnormal beating heart.

1. INTRODUCTION The left ventricle (LV) normally ejects,eachheartbeat, a time-invariant stroke volume. This property is maintainedover a wide range of states including rest, exercise, anger, and disease.An illness that disturbs the cardiovascularintegrity may causeejectionof a time-variantstrokevolume, which may be periodic or aperiodic. From clinical experience[1,8,11,17] we know that the changefrom a time-invariantstroke volume (normal) to a time-variant stroke volume (pathologic) occurs, with an initially constant stroke volume becomingperiodic and finally aperiodic. The specialcaseof

*Currently with Mechanical Aerospaceand Nuclear Engineering Department, UCLA, Los Angeles.,California.

tCurrently with CardiovascularDivision, Departmentof Internal Medicine, The Universityof Iowa, Iowa City, Iowa 52242. MATHEMATICAL BIOSCIENCES 110:103-117(1992) @E1sevier SciencePublishingCo., Inc., 1992 655Avenueof theAmericas.New York. NY 10010

103 0025-5564/92/$5.00

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abnormalbeating heart. Someconcluding remarks and the significanceof the resultsare discussedin Section4. 2. MODEL DEVELOPMENT We consider the cardiovascular system-the heart and the vascular system-described in Figure 1. In this model, the heart is reduced,as in [6], to the left ventricle (LV) pump, with the mitral and aortic valves; the vascular system is representedby an elastic chamber (reservoir) with a constantcompliance C and a rigid tube of a constantresistanceR. (See Table 1.) Let un' Xn, and Yn be the venous return volume, the end-diastolic volume, and the stroke volume at the nth beat, respectively.By the mass conservationlaw we obtain that Xn+I=(Xn-Yn)+Un'

(1)

that is, the end-diastolicvolume at the (n + I)th beat is the sum of the end-systolicvolume (the remaining volume) and the venousreturn volume of the nth beat. Let T be the period of one cardiaccycle, and assume,for simplicity, that the ejection time is equal to the filling time. We assumethat the mitral (inlet) valve and the aortic (outlet) valve are normal-that is, the flow is one-directional-and have constantresistancesdenotedby Rm (mitral) and Ra (aortic). We also assumethat the flow (both inflow and outflow) can be approximatedby Poiseuille'slaw. Let V be the total blood volume and V0 the volume of the elastic reservoir (seeFigure I) at atmosphericpressure;

FIG.

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GILA FRUCHTER AND SHLOMO BEN-HAIM

TABLE 1

Diastolic pressure-volume relationship descriptors Vascular compliance Vascular resistance Total blood volume Elastic reservoir volume at atmosphericpressure Aortic resistance Mitral resistance Cardiac cycle Central venous pressure Mean ejection pressure Mean diastolic pressure Venous return volume End-diastolic volume Stroke volume

Fs Ls Vs Fd Ld C R V Vo

Ra Rm T Pv Ps Pd U X Y

120(80-250) 0.1 (0.01-0.12) 100(60-140) 5 (3.5-7.5) 0.01 (0.001-0.1) 0.05 (0.01-0.3) 700 (380-1d(Xn) = Fd+ Ld(Xn

-

(8b)

Yn)2,

and p.= (Fs, Ls, Vs,C,R, V, Vo,Ra,Fd,Ld,Rm,r)

(8c) is the vector of all

theparameters.Supposethat theseis a positiveintegerk 1 suchthat Pv(Xn+K.)= Pv(Xn),

n=O,1,2,

(9)

Let v = (Fs, Ls, ~, C, R, V, Vo, Ro, T), and denoteYn= Ip.(xn). a point Xo = xo(p.) such that Yo = Ip,,(Xo) * 0

(10)

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and its orbit,

xn=f;(xo),

n=O,1,2,

andk-periodicjorjl£' k=1,2,..., if and only if Yo and its orbit, the sequenceYn'n=O,1,2,..., are k= periodic, that is, Yn+k= Yn and Yn+j* Yn, Proof.

j~, k=1,2,..., and let Yn(p.),n=0,1,..., be the corresponding asymptotically stable k-periodic sequence.With no loss of generality we can assumethat Yo * O. The asymptoticstability of this sequence,that is, of the orbit of Yo, implies that there is a correspondingorbit of Xo such that 'P,,(xn)= Yn' n = 0, 1,2,..., that is asymptoticallystable,and by Theorem 1 it is k-periodic. Hence fJ>~C £1'~,and this completesour proof. . 4. CONCLUSION AND DISCUSSION The cardiovascularsystemis describedby a one-dimensionalnonlinear discrete-timedynamical systemthat dependson severalparameters.In this model the heart is representedby a single contracting ventricle (the left ventricle) and the vascular system is lumped into two elements, a total complianceand a total resistance.The state variable is the end-diastolic volume (EDV). The mechanicaland hemodynamicpropertiesof the heart

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GILA FRUCHTER AND SHLOMOBEN-HAIM

and the circulation characteristicsof the vascularsystemare representedby time-invariantparameters.Applying stability analysisto our model we find thosedomainsin the parameterspacein which EDV is a stablefixed point andthosein which it is a stableand k-periodic point, where k = 2,4,8, . . . . Furthermore,the stability analysisprovidesus also with the relationship between EDV and all the parametersencompassedin our model. The relationsare evidentfrom Figures2a-2g. More specifically, in the caseof a stablefixed point, EDV decreaseswhen positively inotropic stimulationsare causedby increasingFs and decreasingLs (Figures 2a and 2b) and when we increasethe mitral resistanceRm (Figure 2c). On the other hand, EDV increaseswhen the vascular resistanceR, the aortic resistanceR a' the cardiaccycle T, and the blood volume V increase(Figures2d-2g). A periodic-doubling is created by a sufficiently large increaseof the inotropic parameters~ and L s and the mitral resistanceR m (Figures 2a- 2c) and by a sufficiently large decreaseof the vascularresistanceR, the aortic resistanceR a' the heart rate 1/ T, and the blood volume V (Figures 2d-2g). Part of our theoretical predictions have intersectionswith the experimentalresultsof Ritzenberget al. [17], where mechanicalperiodicities were createdby inotropic and chronotropicsimulations. Nevertheless,part of our predictions are not generally acceptedin the physiologicalliterature. The most prominentexampleis our prediction that MA is anticipatedas a result of slow heart rate. Generally,high heart rates havebeenassociatedwith MA [17]. This discrepancymay be relatedto the multimechanismnature of MA. Becauseintracellular calcium is increased and may actually becomebeat-variableduring high heart rates, the appearanceof MA in this casecannotbe predictedby the presentmodel, aswe can analyzeonly casesin which parametervaluesare beat-invariable.However, our experimentalstudies [5] have shown clear evidencethat MP can be invoked by decreasedheat rate. Our theoretical results on the stability of EDV are extendedto the stability of the stroke volume (SV). We show by Theorems1 and 2 that for every point in the parameterspacethere is a one-to-onecorrespondence betweenfixed and k-periodic stableEDVs and fixed and k-periodic stable SVs, respectively.Therefore, normal and abnormalbehaviorof the beating heart are related to different regions in the parameter space. On the transition between the regions that correspondto normal and abnormal beating heart there is a bifurcation that gives rise to a pair of attracting points of period 2, correspondingsto mechanicalaltemans (MA). This period doubling continuesin the region of the abnormalbeating heart. In this way we obtain the subregionsthat correspondto other mechanical periodicities(MPs). This work was supported partially by the Lady Davis Fellowship.

DYNAMIC

PROPERnES OF CARDIOVASCULAR

SYSTEMS

REFERENCES

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