Download Dynamic Properties of Cardiovascular Systems 1. INTRODUCTION

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)
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DYNAMICPROPERTIES
OF CARDIOVASCULAR
SYSTEMS
105
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.
106
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 (380-1<XXJ)
4500 (2250-6750)
I<XXJ(500-1500)
400 (100-2<XXJ)
80 (25-110)
500 (400-800)
8 (1-20)
90 (70-120)
6 (2-20)
45 (19-71)
70 (0-110)
45 (19-71)
mmHg
mmHg/mL 2
16
16
mL
mmHg
mmHg/mL 2
16
mL/mmHg
4,
mmHg ms/mL
mL
mL
0
rnmHg'ms/mL
rnmHg'ms/mL
ms
rnmHg
mmHg
rnmHg
mL/beat
mL
mL/beat
16
4,
4,
4,
,392
333
p. 379
p. 379
p.
p.
4, p. 353
4, p. 353
4, p. 333
4, p. 414
4, p. 415
4, p. 415
10
10
10
*Range used in the study.
then the central venouspressurewill have the value
Pv(Xn)= (V- Xn- Vo)/C.
Let Ps(x) be the mean (in time) ejection (systolic) pressure and lpd(x)
(2)
=
Pd(x - y) be the mean(in time) diastolicpressure.Then we assumefor the
stroke volume Yn and the venousreturn volume un the following approximation formulas:
and
respectively.
The values Ps(xn) and [F>d(Xn)= Pd(Xn
the pressure-volume
-
Yn) can be approximated from
relation of the heart, known as Frank Starling's
DYNAMIC PROPERTIES OF CARDIOVASCULAR
SYSTEMS
107
relationship(cf. [6, Fig. 4]). Let ~,Ls, Vs' and Fd' Ld be parametersthat
characterizethe systolic and diastolic pressure-volumerelationshipsof the
heart, respectively;then
ps(x)
= ~ - Ls(x - vsf
and
~d(X)
= Pd(X-
y) = Fd+ Ld(X- y)2,
Their values are identified using curve fitting of the experimentaldata in
[4].
Substituting(2)-(4) into (1) will lead to
(5)
where
p.= (~, Ls, Vs,C, R, V, Yo, Ra,Fd' Ld, Rm,r),
and let
v +-.!.--+
xnEI= (°,
s
[1+4(CFs+ Vs- V+ Vo)CLs]1/2
2CLs
2CL
s
Remark. Let K be the parameterspacein which p. is allowedto vary.
For a fixed point p. in K, Equation(5) can be relatedto an iteration of the
map f!' of the interval I into itself.
In practice, f dependson the parameterp.. As we will show in the
sequel,the behaviorof this systemdependson p..
3. STABILITY ANALYSIS
A. DYNAMICPROPERTIES
OFEQUATION(5)
Possiblefixed points (or equilibrium values)of x in Equation(5) may be
found algebraically by putting Xn+1= xn = XI and solving the resulting
equation,
xl=f!,(Xl).
108
GILA FRUCHTERAND SHLOMO BEN-HAIM
An equivalent graphical method is to find the points where the curve f ,,( x)
that maps x n into x n + I intersects the diagonal x n + I = Xn. The stability of
the fixed points dependson the slopeof the curve f" ( x) at the fixed points.
As long as this slope lies between + I and - 1 the fixed point will be at
leastlocally stableattracting. Let us denoteby [Iixl the subsetof points p. in
the parameterspaceK for which there are stable fixed points x I. At the
boundaryof [Iixl the stablepoint XI undergoesa flip bifurcation; the fixed
point becomesunstableand a stableorbit of period 2 is born. This orbit of
period 2 may now be written X2 and found either algebraicallyfrom
X2 = f;( X2)
or graphically from the intersection between the map f;(x) and the
diagonal.The stability of this period 2 orbit depends,again, on the slopeof
the cu~e f;(x) at thesepoints. Let us denoteby [fIx2 the subsetof points
.IL in [fIxl = int(K " (J>
xl), for which the periodic points X2 are stable
attracting. Beyond the boundary of (J>
x2 the period 2 points will in turn
becomeunstableand bifurcate to give an initially stableorbit of period 4.
This in turn gives way to a orbit of period 8, and thenceto a hierarchy of
bifurcating stableorbits of periods 16,32,.. . ,2 n. In eachcase,the way in
which a stableorbit of period k becomesunstable,simultaneouslybifurcating to produce a new and initially stable orbit of period 2 k, is basically
similar to the processfor k = 1. Let k = 1,2,...; then
f1Xk
= {ILeK:
3xeI suchthatf "k(x) = x, fj( x) * x, 0 < j< k,
Our investigation on the dynamic properties of Equation (5) when p.
varies, that is, of finding the bifurcation sets and the sets (1ixk,k ~ 1, is
madenumerically. Typical picturesof the bifurcation phenomenonwhen we
fix 11 of the parametersare depictedin Figure 2. A regular behavior of
period doublingcan be seenin Figures2b and 2f. In all other figures we can
see that this period doubling, with creation of smooth periodic branches,
continuesall the time as the periodic points are in the subintervalof I [see
Eq. (5b)], where f!' is smooth. This phenomenonmay occur becauseat
points where f!' is not smooth the implicit function theorem does not
necessaryensuresmooth periodic branches(cf. [6, Figs. 7a and 8a». The
lower and upper bounds of each horizontal axis in Figures 2a-2g are
physiologicalbounds.
DYNAMIC
PROPERTIES OF CARDIOVASCULAR
SYSTEMS
109
(a)
(b)
FIG. 2. The stable fixed points, x, of various periods that can arise by bifurcation
processesin Equation (5) as a function of (a) Fs (mrnHg), (b) Ls (mrnHg/mL 2), (c) R m
(mrnHgoms/mL), (d)R (mrnHg'ms/mL), (e) Ra (mrnHgoms/mL), (t) T (ms), (g) V
(mL).
110
GILA FRUCHTER AND SHLOMO BEN-HAIM
180.00 ~
t~.OO
~.OO
OB.OO
0.00
ie.GO
.
77.eo
, .,
130:00
182.eo
_.00
DYNAMIC
PROPERTIES OF CARDIOVASCULAR
(f)
FIG. 2. (Continued)
SYSTEMS
111
112
GILA FRUCHTER AND SHLOMO BEN-HAIM
(g)
FIG. 2. (Continued)
B. THEROLEOF BIFURCATIONSIN A BEATINGHEART
A normal beating heart can be characterizedby a time-invariant stroke
volume that is asymptotically stable. By asymptotic stability of a timeinvariant stroke volume we mean that if the stroke volume is perturbed
slightly from its time-invariant value, then all subsequentstroke volumes
remainin a correspondingsmall neighborhoodand tendto it underiteration.
As in the caseof a constantsequence(time-invariant)of strokevolumes,by
asymptotic stability of a periodic sequenceof stroke volumes we mean
that the sequenceof the stroke volumes has a neighborhoodof stroke
volumesthat tend to it under iteration.
Consider the system defined in Section 2. Let fjJ~ be the subsetof
points p. in K for which the heartejectstime-invariantasymptoticallystable
stroke volumes Yn' that is, the subregionin the parameterspacefor which
we have a normal beating heart. And, let fjJ~, k > 1, be the subsetsof
points p. in K for which the heart ejects time-variant, k-periodic, k > 1,
asymptotically stable stroke volumes Yn' that is, the subregionsin the
parameterspacefor which we have an abnormal beating heart. Let k =
1,2,...; then
fjJ~
= {p.EK:
3Yn
= yn(p.), n = 0, 1,2,...,
which is asymptotically
stable, suchthat Yn+k = Yn and Yn+j *' Yn, j < k}.
DYNAMICPROPERTIES
OF CARDIOVASCULAR
SYSTEMS
Note that the casek
113
= 2 correspondsto mechanicalaltemans.Our objective
is to find thesesets fJI~ , k = 1,2,. . .. Achievingthis objectivewill
provide physicianswith quantitativeinformation regardingthe location of
parametersthat correspondto the normal and abnormalbeatingheart. Such
information can provide one with the factors that maintain the normal
beatingheart and thosethat causeabnormalbeating.
Let
9';
be the subsets in K defined and found above, numerically. In
thefollowingwe will provethat 9~ = 9';, k = 1,2,... .
Note that we found (numerically) only the one-dimensionalsubsetsof
9';, which yield by fixing 11 of the coordinatesof JL.
THEOREM1
Consider the one-dimensional discrete dynamical system defined in
Section 2,
Xn+1
= Xn -
Yn + Un = f,.(xn)
(6)
Here, Yn and un are defined by (3), that is
+ 1)
)( sgn(Ps(Xn).
2 P,,(Xn))
-
Yn -~
- 2 ( Ps(Xn)-PIJ(Xn)
R + R
Q
and
Un
I
= (Pv( Xn);:d( Xn)) (Sgn(pv(Xn)~ [f)d(Xn))+ 1)
and Ps' PI/' ~d are
definedby (2) and (4), that is,
ps( Xn)
= Fs -
Ls( xn - VS)2,
P,,(Xn)= (V- Xn- VO)/C,
[F>d(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)
114
GILA FRUCHTER AND SHLOMO BEN-HAIM
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"
n=O,
,
,2,. .
First we prove sufficiency. From (9) and (8b) it follows that
Xn+k, = Xn,
n=O,1,2,
We want to prove that k 1= k. From (7b) and (8c) it follows that there is a
positive integer p suchthat
Un+pk=
n=O,1,2,...
Un,
(14)
Now, let m be someinteger. From (6), (11), (12), and (14), one can obtain
f"mpk(XO)
= Xmpk = Xmpk-l-
= Xmpk-2 -
Ympk-2 mpk-1
L
= ... = Xo (
+ Umpk-1
Ympk-1 + Umpk-2 + Umpk-1
Ympk-1
mpk-1
L
Y;+
;=0
( k-1
u;
;=0
pk-1
I
= Xo+ m p - L Y, + L u;
;=0
;=0
I
(15)
Hence
Ympk
k-l
(
)
pk-l
P - i~OYi + i~O Ui
= 'P,,(Xmpk) = 'P,I
Again from (12) we obtain
Xo
Ympk = /P,
+ m( p ( - k-l~ y
pk-l
+ L
1=0
;=0
u;) = Yo = tp.( Xo)'
Since(10) ensuresthat the last equationhas at most two solutions,there is
an integer mo suchthat
Xo
=
Xo
+
mO
( P
( -
k-l
:L
Yj )
1=0
+
pk-l
:L
1=0
Uj
DYNAMIC
PROPERnES OF CARDIOVASCULAR
115
SYSTEMS
and in conclusion
P( -
k-l
i~OYi
pk-l
+
L u; = O.
;=0
(16)
Substituting(16) into (15) we obtain, for m = 1,
Xpk = f:k( xo) = xo.
Hence,from (13) we obtain k1 = pk, andby (7a) and (8a) we concludethat
p= 1.
For j< k, we have
Xj=f!(xo)*xo,
becauseotherwise(12) cannotbe satisfied.This completethe proof of our
theoremfor the sufficiency part.
The necessityfollows immediatelyfor j = k. If thereexistsj < k such
that (12) doesnot hold, than by argumentssimilar to the above,we obtain a
contradictionto the assumptionsof this part of the theorem. .
THEOREM 2
~
k=1,2,
£1'~,
£1'~=
Then
above.
With the notations and the assumptionsof Theorem 1, let £1' and
£1'~, k = 1,2,...,
be the subsets in the parameter space K defined
Proof. Let p.E£1'~, k=1,2,...,
and let xo=xo(p.) be a stable
attracting k-periodic point of f/". With no loss/of generalitywe can assume
that 'P,,(xo)= Yo* O. By Theorem 1 the orbit of Yo, that is, the sequence
{Yn}n, is k-periodic [seeEq. (12)]. The asymptoticstability of Yo and its
orbit follows from the continuity of 'P". Hence £1'~C £1'~. Now, let
p.EfJ>~, 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
116
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
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