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.I. theor. Biol. (1976) 63, 275-309
Optimal Control Aspects of Left Ventricular
Ejection Dynamics
ERIK J. Nor>ws
.-lutomatic Control Laboratory, Unil>ersitJ?
of Ghent, Belgium
(Received 25 June 1975, and in reaisedform 27 October 1975)
A model of the ejecting left ventricle is developedin which ventricular
elastanceas a function of time is optimized with respectto a simpleperformance index selected on an energetic basis. The model correctly
predicts a number of well known experimental findings concerning the
effects of preload and afterload conditions and varying system parameters on left ventricular pressureand elastancewaveforms and on the
ejection period. The resultscharacterize ventricular systolic elastanceas
dependenton both end-diastolicvolume and meanaortic pressure.
1. Introduction
The dynamic performance of the left ventricle during systole has been
analysed in numerous papers. One approach that has frequently been used.
characterizes the ventricle in terms of the elastance function E(t) defined b>
E(t) = /qt)/V(t).
(1)
P(t) and k’(t) representing the instantaneous values of left ventricular pressure
and volume. Several systolic time courses of E(t) have been proposed in the
literature, some being chosen for their simplicity (Warner, 1965; Defares.
Hara, Osborn & McLeod, 1963; Beneken, 1964), others being calculated
from physiological data (Greene, Clark, Mohr & Bourland, 1973; Suga,
1971). Ventricular performance measurementson dogs by Suga (1969, 1970),
result in elastance curves that are essentially unaffected by changes in either
left ventricular end-diastolic volume V,, or arterial loading conditions.
Other studies however (Greene et al.. 1973; Taylor, Cove11& Ross, 1969)
characterize ventricular elastance as dependent on end-diastolic volume.
This paper tries to approach the problem of developing a ventricular
pumping model, and determining E(t), relying on optimization aspects in
biological control theory. The relevance of the optimization concept, and
supporting evidence for it, have been examined in detail by Milsum (1968):
loosely speaking, it is assumedthat a selective survival advantage is gained
I’,13.
IX
17
E.
J.
NOLDUS
by those organisms whose subsystems operate in an optimal fashion on some
energetic and/or informational basis, and that as a result living systems have
evolved towards optimal performance when executing a given task. Although
it is by no means clear what performance criteria are relevant for biological
systems, optimization seems to be a worthwhile viewpoint both for theoretical
analysis, as for suggesting new experiments.
Specifically, in the present problem, our aim is to apply dynamic optimization theory to the mechanism of ventricular ejection, in order to
determine which is the best force-developing strategy to eject a given volume
of blood out of the ventricle. To this end the ventricular pressure P(f) during
ejection, which acts as an external driving force on the system under analysis.
will be optimized on the basis of a reasonable performance index related to
the energy expenditure of the working ventricle. We then wish to examine
how the results of this optimal strategy differ from the established facts
concerning, for example, the responses of the ventricle to different types of
loading conditions. If the obtained results are at least qualitatively consistent with the system’s actual behaviour, then the approach may reveal
some new insights in this behaviour, and in the system’s properties when
performing under varying internal and external circumstances. It will be
shown that computed pressure and elastance curves bear good resemblance
to measured data and that calculated and experimental results display similar
tendencies when certain parameters are changed, such as average aorlic
pressure, end-diastolic volume, and the parameters characterizing load
dynamics. This suggests that the hypotheses formulated in the performance
index may to some extent express basic principles which determine the
operation of the ventricle. In this way an ejection model is obtained, based
on a new, rational fundamental philosophy, which to some extent may
provide a logical explanation why the ventricle operates in a specific way
under given working conditions. A model of this type may predict on a
rational basis how the system will react to various external perturbations,
internal parameter variations, and even to internal structural changes, fat
example as a result of accident or illness.
2. Problem Statement
To analyse the circulatory effects of left ventricular pumping, models of
the ventricle itself, and of the arterial system must be defined. Figure 1 shows
an electrical analogue of a model, similar to the ones proposed by Suga
(1971). The model is quite elementary, but since we are mainly interested in
investigating the system’s qualitative behaviour, and the general shapes and
tendencies of waveforms, it will suit our purpose. The same model. or
LLFI‘
VENTRICULAK
I:JLC
I ION
177
Dk N.(.hlt(‘S
-.. I
E(f)
*--Ventricular
FIG.
‘~l~~~:lhss?cl
elastance
1. Electrical
equivalent
of the pumping
model
under
1013
inj:cstigation.
slightly diRerent versions of it, have been used by many investigators. F‘or :t
discussion, see the review by Noordergraaf (I 969) on the subject. The model’>
dynamic equations are
(?a)
V(t) = 1; - .!,i(0) dU
P(f) = P,(f)-I-ri(t)+L
di(l)
dt
(2bi
I’,(f) reprexenls the aorlic blood pressure, i(t) the blood flow ralc e.jectcd o:~t
of the ventricle, P the aortic valvular resistance. and L the blood inertia. /Z
and C are the peripheral resistance and compliance of a lumped arterial
Windkessel load. In the right hand side of equation (?a), the first term I,,
represents the ventricular volume at the beginning of ejection, while 11x
second term is the blood voiume ejected out of the ventricle during the time
interval (0, t). Equation (2b) states that the ventricular pressure equals the
sum of aortic pressure, the pressure drop across the aortic valve, \vhich is
proportional to the blood flow rate i(l), and the blood accelerating pressure.
which is proportional to di(t)/dt. Finally, equation (2~) expressestha: !hr:
blood flow rate into the arterial system equals the blood flow rate leaking
out of it, plus the blood storage rate in the system. The two latter !erms arc
proportional to, respectively, the aor!ic prea;sureP,,(r) and iis deriiaii\c
dP,,(t)/dt. Following Defarex, Osborn & H;IKI f f91:41. and ;~c’u:~rding10
278
E.
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NOLDUS
Starling’s law, the stroke volume Vswill be calculated using an expression 01’
the form
b; = u+(c-dP,)V”,
(3)
where, as is usual in experimental studies on the subject, the average aortic
pressure during ejection is approximated by
p,, = 4 [P”(O) -tP,k)l
In all results presented below, it has been verified that P,, differs less than
5 “/, from the true average,
Wdj1J,W
0
do,
such that the approximation is quite satisfactory. The time interval [0, t,,]
represents the ejection period of the ventricle. Equation (3) expresses the
well-established facts that stroke volume increases with end-diastolic volume,
and decreases for increasing arterial pressure. A reasonable and sufficiently
simple performance index is
.l&P2(t)+cd’(t)ijt)]
0
dt,
a>0
(4)
There are several arguments to support the choice of J. First, equation (4)
may be interpreted as a weighted sum of the integral square ventricular
pressure, and the mechanical work generated by the ventricle, during ejection.
Hence the criterion tends to minimize the mechanical work required fat
blood flow ejection, while at the same time it penalizes the buildup of high
pressure peaks. For a fixed I,, the first term in equation (4) penalizes the
mean square ventricular pressure, while for a variable t,. and a given mean
square pressure, it penalizes the time interval t, during the cycle, when high
pressure is maintained. Evidently mechanical flow work is not the only
energy factor involved in the pumping process. During each cycle, an amount
of elastic potential energy is stored in tissues and muscles under tension,
which is only partially converted in useful work. Partially it is dissipated in
heat, to overcome forces resisting deformations, and to overcome tissue
viscosity. The latter terms are difficult to calculate. However, the potential
energy of an elastic spring being proportional to its squared tension, the
first term in equation (4) is clearly related to the elastic potential energy 01
a volume storage element, such as ventricular compliance. Hence .I can also
be interpreted as a weighted sum of mean stored potential energy and
mechanical flow work. During the short isovolumic contraction period
preceding ejection, no mechanical flow work is performed. For reasons of
simplicity, the cost of pressure buildup during this period has not been incorporated in the performance index. This may be justified by the short duration
LEFT
VENTRICULAR
EJECTION
DYNAMIC’S
279
of this part of the cycle. Diastolic energy considerations were assumed not
to affect the control strategy during systole.
fn the performance index, the weighting factor o! is, as yet, unknown, and
of course difficult to determine. However, a method to select a numerical
value for CI will be explained below. Also, it will be demonstrated that the
genera1 resuhs are rather insensitive to the precise value of X.
The problem now consists in determining the function P(t), for 0 5 t 5 t,..
such that equation (4) is minimized, for given values of V, and P,, hence of
Vs. In other words, the objective is to determine the control strategy which
minimizes a well-defined measure of the total energy expenditure, required
for ejecting a given volume of blood.
3. Method of Solution
Let V(t) p -yI, i(t) p x2, P,(t) p x3, and the control function P(t) p II.
Then, after differentiating equation (2a), the system’s equations read
21 = -x2
(5a)
1, =+-).xJ
(5b)
I
1
2.3 = -Rc:x3+c.Y2,
(5cJ
while
J = ‘s’(u” -+-wx2) dt.
0
This constitutes a classical linear optimal control problem, the solution to
which is readily available in the literature (see, for example, Kwakernaak &
Sivan, 1972). Analytical expressions for the extremals of the optimal control
problem can be found, by simultaneously integrating the equations (5), and
the corresponding costate equations:
jl+Co
(6a)
1
fi2 = --
aH
ax,
= --“u+p1+ip2--
aH
1’3
=
-ax,
I
=
j-/2
I
+~~P”’
r
1
p3
c
(6b)
(6~)
280
C.
J.
NOLDL!S
where the Hamiltonian
and u is given by
The vector p, with components p,, 11~and y,, is the costate of the system
state s, with components x1, -K~ and x~. The equations (5)-(6) together
constitute a sixth-order differential system, with boundary conditions:
x2(0)= 0
(SC)
x,(t,) = 0
(94
i9e)
1m+P&?)
= 0,
WI
where p3(t) is the costate variable associated with x,(t). The boundary conditions (9a)-(9b) are self-explanatory, while (9c)-(9d) state that blood flow
is zero at the beginning and the end of ejection. Equation (9e) is a mere
restatement of the definition of li,. The condition (9f), involving the costate
variable us, is easily derivable from (se), using basic principles in optimal
control theory. Along the solutions of the system (5)-(6), the Hamiltonian
H
is a constant, whose value depends on the duration of the ejection interval t,.
If t, is variable, its optimal value can be determined using the condition
H(t,) = 0
(10)
(again see Kwakernaak & Sivan, 1972). The system of equations (5)-(6) has
been integrated in Appendix A. Table 1 displays the resulting curves V(t),
i(t), P,(t) and P(t), for 0 5 t 5 t,, in terms of five integration constants
ar-~1~. The boundary condition (9a) has been taken into account automatically, thus eliminating one integration constant. Now the logical sequence
of calculations is the following. First consider the case where the ejection
time t, is fixed. Then, for given values of the system parameters and the
loading parameters V, and P,, k; is computed using equation (3), such that
the boundary values (9b)-(9f) are known. Expressions for these boundary
values, in terms of the integration constants ~,-a,, are obtained by evaluating
the state variables x,(t), i = 1 . . 3, and the costate variable pj(l) at t = 0
and t = 2,. This is done using the formulas in Table 1. A formula for /am
can easily be found using the results listed in Appendix A. This produces a
x3(0) +x&J
= 2P,
LEFT
VENTRICULAR
EJECTION
TABLE
I)YN.!hllC’S
1
Optimal operation of the model during ejection (0 5
t
5
t,)
282
E. J.
NOLDUS
system of linear algebraic equations in al-u5, which can be solved by
computer in a straightforward manner. Substitution of the results in the
expressions of Table 1 yields the desired solution.
If the ejection time t, is variable, then equation (10) is used to determine
its optimal value. This must be done in a computer algorithm of successive
approximations. N can be obtained by evaluating equation (7) at an arbitrary
time. for example at t = 0 or at t = t,. For t = f, one ha<
If = L?(t,)+
~(u-n,)-R~p,
L
L,<.
,
where
u(t,) = - ;L P2(fJ,
such that equation (10) becomes
1
u2 - 2ux, +x-c
pjx3 = 0,
for t = t,.
(11)
Now, for given values of the system and loading parameters, and for a fixed
value of the weighting coefficient CI, the left hand side of equation (11) can
be computed for any guessed value of t,. t, is adjusted by successive approximations until equation (11) holds. Then the calculations proceed as for the
fixed final time problem. However, the relation (10) can also be used in a
different way. Indeed, after selecting a normal value for t,, the corresponding
value for CI can be obtained from equation (lo), thus conjecturing that the
normal t, is optimal for the considered performance index. In this way an
estimate is produced for the unknown weighting factor in the performance
index. Again this is done in an algorithm of successive approximations, in
which CIis adjusted until equation (11) holds, for a given t,. When doing so
for normal values of all system parameters, a-values are obtained in the
range of c( = 1 mm Hg/(ml/s) to c1= 5 mm Hg/(ml/s).
The heart rate n, corresponding with a given set of parameter values can
be obtained as follows: During the diastolic interval of duration t,, following
the ejection period under consideration, and during a short isovolumic
contraction of duration r, following diastole, the aortic valve is closed. Then
the aortic pressure P,(t) decays exponentially with time constant RC. This
can be verified in the circuit of Fig. I, where the capacitor C unloads over
the resistor R while the valve is closed [i(t) E 01. In periodic operation, the
aortic pressure at the end of isovolumic contraction, that is, at the end of the
exponential decay, equals the aortic pressure at the starting point of ejection.
This yields the relation
PO(O) = P,,(t,,) cup [ - (fd + r)/RC],
LEFT
or,
VENTRICULAR
EJECTION
DYNAMICS
283
t,+z = RC In [Pa(fJ/Po(0)]
from which the heart rate
n = 60/(t,+~+t,)
can be computed.
4. General Results
Figure 2(a) curve A shows the time course of ventricular systolic pressure,
for a set of parameter values for the human system, taken from Defares et ul.
(1964). The origin of the time axis has been shifted to the starting point of
the systolic phase, of duration t, = 0.2 s, which consists of the isovolumic
contraction of duration z = 0.02 s and the ejection period of duration
t, = 0.18 s. According to equation (10) the corresponding value for r) has
been calculated to be a = 3.78 mm Hg/(ml/s). During the isovolumic contraction period, P(t) rises from its initial value, approximately zero, to the
computed initial value for the ejection phase. Since z is a small fraction 01
the cardiac period, the pressure rise during this time interval can be approximated by a linear curve. The computed curve starts at the beginning of
ejection. As a result, the systolic time course of P(t) exhibits two characteristic peaks, which have been observed in numerous experiments (Greene
et al., 1973; Suga, 1969; Clark, Kane & Bourland, 1973). Although in
representative data the primary peak is not always very pronounced, and
may to some extent depend on the type of transducer used, the existence of
two-peak pressure waves seems to be very consistent in experimental
recordings. In Fig. 2(a) curve A the primary peak is very sharp as a result of
the slope discontinuity at the beginning of ejection. This stems from the
fact that the pressure curve has only been calculated during ejection, and
hence a sharp primary peak is inherent to the computational
procedure
followed. The sharpness of the peak is of little importance however, the
general form of the curve being mainly determined by the relative height of
the first and the second peak, and the times at which they occur during
systole. As will be shown in the following sections, the sharpness and relative
heights of the peaks change with various system parameters and with loading
conditions. For the nominal values of Fig. 2(a) curve A, the second peak
occurs at I = 0,145 s, and both peaks have about the same height of 115 to
120 mm Hg. This corresponds approximately with data in Greene (1973).
Suga (1969) and Buoncristiani, Liedtke, Strong & Urschel (1973). These
papers report experimentally obtained pressure waves, also showing two
peaks of approximately equal height, the first peak occurring at the beginning
of ejection and the second one approximately at the beginning of the lasr
254
100
FIG. 2. (a) Curve A time course of ventricular systolic pressure, for the following coefficients (a, c, d) in expression (3), and for the following parameter values (Defares et nl.,
1964): a= -100 ml; d = OGO148mm Hg-‘; c = 0.981; R .= 1.O mm Hg/(ml/s):
Y = 0.01 mm Hg/(ml/s); C = 1.9 ml!mm Hg; L = 0.003 mm Hg s2/mI; V0 = 200 ml;
P,, = 100 mm Hg; r, = 0.18 s; r = 0.02 s; hence fs = 0.2 s; and tl = 3.78 mm Hg/(ml/s).
Curve (B) experimentally recorded pressure wave, according to Greene et al. (1973), and
Clark et al. (1973). normalized to the same maximum value and systolic period as curve (A).
(b) ventricular systolic volume as a function of time; (c) systolic arterial pressure as a
function of time; (d) curve (A) systolic elastance as a function of time. Curve (B) an experimentally determined elastance curve, according to Greene et al. (1973), normalized to the
same maximum value and systolic period as curve (A); (e) Curve (A) blood flow as a function
of time during ejection: Curve(B) blood flow as a function of time according to Defares ei al.
(1964); (f) ventricular pressure-volume diagram.
150 -
5
=
.
x
100 -
50-
FIG.
2. (b)
and
(c).
286
E. J.
NOLDUS
-.-----~- .--
r-
5-
.5 -
I
FIG.
2. (d).
LI:tT
VENTRICULAR
EJECTION
600
500
/
‘,G 400
-.
I
E
x
> 300
‘.
200.
100
-1
FIG.
2. (e) and (f).
DYNAMIC‘S
‘X7
288
E.
J.
KOLDUS
quarter of the systolic phase. A typical example of these recordings has been
reproduced in Fig. 2(a) curve B normalized to the same maximum value and
systolic period as for the computed curve. The similarity in shape of both
curves is satisfactory. At the end of systole, the computed pressure becomes
slightly negative. Since pressure must recover during diastole, this causesa
negative peak in the pressure wave at the closing time of the aortic valve.
The negative peak is usually not observed in experimental recordings. Houever, its height is very sensitive to parameter variations and may easily
disappear (seebelow).
Figure 2(b) and (c) display ventricular volume and arterial pressure as
functions of time during systole. During the isovolumic contraction period
z, V(t) remains constant, while P,(t) decays exponentially with time constant
XC. The initial and final values of arterial pressureare consistent with those
given by Defares ct al. (1964). The general shape of both curves is little
affected by changes in system parameters and loading conditions, and
matches data in the literature, among others in Suga (197 I ), Robinson ( I96.5),
Taylor et (11. (1969) and Wilcken, Charlier, Hoffman Rr Guz ( 1964).
Figure 2(d) curve A yields the corresponding elastance curve. This curve
displays a striking resemblancein general shape with similar curves reported
in the literature. The rising portion consists of two segments.separated by a
slightly pronounced primary peak, the initial segment being steeper. The
maximum of the curve is in the latter half of the systolic phase.at f = 0. i 5X \.
posterior to the secondary peak in P(r). The maximum falling gradient is
steeper than the maximum rising gradient. Similarly shapedelnstancc curves
have been obtained in investigations by Greene (1973), using :I compuln;ional
procedure, and in experimental work by Suga (1969, 1970. 1971). For cornparison, Fig. 2(d) curve B also displays a typical example ol‘ an elastancc
curve, as computed by Greene on the basis of experimental data.
Figure 2(e) curve A represents blood flow i(t) during ejection. This curve
does not agree with experimentally recorded flow patterns, or with simulations basedon experimental data. These normally place the maximum flow
in the first half of systole, as in the dotted curve in Fig. 2(e) curve B, which
has been taken from Defares (1964). The discrepancy suggeststhat an oversimplification was made by representing the aortic valve by a constant
resistance I’. Since flow patterns are in general very dependent on valve
characteristics, it is plausible that better results could be obtained by
introducing a nonlinear valve resistance in the model. In section 6, dealing
with the effects of load variations in models with variable ejection time,
some evidence has been obtained, supporting the conjecture that atypical
flow patterns such as in Fig. 2(e) curve A must be attributed lo valve nonlinearitics in the real system. Indeed, a nonlinear valve rehistancc /,(,i) can in
I.T:FT VENTRICULAR
li.lFfTlC)N
DYN4%:!<
\
. 5
.3=5mm
Hg/(ml/s)
1
7 \
'I
\I
3
I
/:
i
FIG. 3. Elastance curves for varying values of the weighting coefficient a.
TABLE 2
a [mmHdb-d/s)l
W (joule)
N (watt)
1
0.9257
1.3705
3
0.9228
1.3685
5
0.9218
1.3677
28?
290
E.
J.
NOLDUS
general be approximated by a constant value r if blood flow i(t) remains
sufficiently small throughout ejection, the approximation becoming better for
decreasing average and/or peak blood flow rates. In section 6, Fig. 1 I, flow
patterns i(t) have been computed for varying end-diastolic volume V,. Since
the flow curve generally decreases in amplitude for decreasing V,,, hence
tending to eliminate the effects of possible valve nonlinearities, one would
expect that for lower values of V,, the discrepancy between computed and
actual flow curves should disappear. It can readily be observed that this ib
in effect true. For decreasing V, the maximum in the flow curve shifts towards
the middle of the ejection phase, and the flow pattern becomes symmetrical.
completely in agreement with experimental curves (Suga, 1969, 1970), which
also become symmetrical for low amplitude flow patterns.
The possibility of introducing a nonlinear valve model has not been further
investigated, because of the computational difficulties involved in handling
nonlinear systems, and because it is unclear exactly how valve resistance
varies with blood flow. It should be noted that linear ejection models, in
particular showing constant valve resistances during ejection, are widely used
in the literature (Suga, 1971; Greene et al., 1973). Discrepancies in behaviour
with the real system, resulting from nonlinearities in the real system, are a
common characteristic of all these models.
A computed pressure-volume diagram is given in Fig. 2(f). Here the
resemblance with experimental data (Greene, 1973; Robinson, 1965) ib
satisfactory. As already noted, in Fig. 2(a)-(f) the weighting coefficient E was
selected so, that the corresponding optimal ejection period t, takes a normal
value, t, = 0.18 s. However, the sensitivity of waveforms to the precise value
of M is small. Figure 3 presents elastance curves E(t) for varying values of ‘A
and for a constant ejection time f,, all other parameters also remaining
constant. For increasing c(, the maxima in the elastance curves become more
pronounced, since less weight is attached to pressure peaks in the performance index. However, even relatively strong changes in u do not
essentially alter the shape of the waveforms. As could be expected from the
choice of the performance index, stroke work W and mechanical power IV
transferred by the ventricle to the blood, decrease slightly for increasing 7,
according to Table 2.
5. Effects of Load Variations
We shall start with investigating the effects of changing afterload conditions. First it must be observed that in the living system the mean arterial
pressure P,, which constitutes the main afterload parameter, is not an
independent variable. P, is affected by several factors, but is mainly controlled
I.EFT
VENTRICULAR
EJECTION
191
DYNhMI(‘S
by systemic resistance R. Heart rate does not change with changing arterial
pressure. In their study, Defares et al. (1964) (Fig. 12, page 229) have computed the dependence of P, on R, for a detailed model of the complete
circulatory system, which operates with constant heart rate. This curve has
been repeated in dotted lines in Fig. 4.
The circuit of Fig. 1 on the other hand, does not represent a complete
circulatory system, but only an open-loop section, consisting of the left
ventricle and the arterial load. The feedback path, linking the arterial system
with the venous part, and then back to the left ventricle via the right heart
and the pulmonary system, has not been considered. The circuit can be used
r
,
,;,’
/
‘/
/
R [mm Hg/(ml/si]
FIG. 4. Curve (A) pa-R relationshipfor constantheartrate n, and nominalvaluesof
all other pramcter. Curve (B) the same rel;ttionship accordin 2 to Defilres it 01. (1904).
T.ll.
I‘1
292
E.
J.
NOLDUS
to study the uncontrolled, open loop ejection system, in which P, and R can
be changed independently.
When doing so, increasing P, for constant
systemic resistance R results in strongly growing heart rates, hence in drastic
increasesof transferred mechanical power N. More interesting results can be
obtained by endowing the ejection model with an arterial pressurecontroller,
which establishes a relationship between r’, and R, such as exists in the
complete circulatory system. Since the pumping ventricle does not change its
rate with changing arterial pressure, a good model for such a controller is
the one which keeps heart rate ?I unaffected by changes in R and B,. The
behaviour of the ejection model, controlled in this way, may then be compared with available experimental data, which mostly stem from measurements and simulations in which heart rate was kept constant.
Figure 4 shows the computed relationship between P,, and R, for nominal
values of all other parameters, such that 71remains constant. This curve
almost coincides with the Pa--R curve for constant II, obtained by Defares,
although a more complicated loading network was introduced there, and a
detailed model of the complete circulatory system was used. It follows that
if our model is endowed with a PC,-R relationship, as exists in the complete
closed-loop system, then heart rate is independent of mean arterial pressure,
in agreement with the real system. In the present model, the nominal heart
rate equals 12= 89 min-‘. This is higher than normal, due to the simplification, introduced by selecting a linear, first-order arterial loading network.
A better numerical value for II could be obtained at the expense of complicating loading dynamics, for example as suggestedby Defares (1964), or by
slightly adapting the peripheral resistance R. However, the exact value of
heart rate scarcely affects systolic waveforms. Section 7 presents someresults
for a slightly increased resistance R, and normal heart rates. Plots of left
ventricular pressureagainst time, for varying values of arterial afterload, and
for a fixed ejection period t,, are shown in Fig. 5(a) [Figs 5-8 represent
solutions of the fixed final time problem, with t,, = 0.1s s. which is optimal
for the nominal parameter values, used in Fig. 2(a)]. The left ventricular
pressure waveform generally increases in amplitude with increasing I-‘,,,
conform with results in Greene (1973). Suga (1971), Taylor (1969).
Buoncristiani (1973), Wit&en (1964). The secondary peak in the wavelbrm
increasessteadily as a function of increasing load, until it becomesdominant
over the primary peak. For high values of P,, P(r) exhibits a single maximum
in the latter half of the systolic phase.This increasein the secondary peak with
increasing afterload is frequently observed experimentally, and also appears
in calculated curves by Greene (1973). Increasing 7, also increases endsystolic pressure, which becomes positive for B, = 120 mm Hg. The
secondary peak time remains almost constant, as confirmed by Greene.
I
L -.---I_-
---_L---_.
100
150
P,t,nrn
FIG.
5. (a,
2nd
Hg)
(b).
-----L.
200
294
E.
J.
NOLDUS
,j.6’
t
05 jI.----_~
1-J
150
I
100
&!mm
----
J.
100
-
--
-
--L_~-
5. (a) Ventricular
.-.
150
,Po (mm
FIG.
200
Hg)
-_1--
200
fig)
pressure curves for varying afterload conditions (ejection period
t, and heart rate IZ constant); (b) maximum ventricular pressure P(f),,,,, as a function of
mean arterial pressure Pa: (c) stroke work W as a function of Pa; (d) mechanical power
N as a function of Pa, for constant heart rate and for varying heart rate.
Maximum
pressure P(t),,,, as a function of P, is shown in Fig. 5(b). This
again corresponds
with Greene’s curve, which is also essentially linear. In
Fig. 5(c) and (d), stroke work Wand transferred mechanical power N have
been computed as functions of P,. Figure 5(c) is the typical parabolic curve
which can also be found in Sonnenblick & Downing (1963) and in Suga
(1971). Stroke work and mechanical power are maximum at is, = 160mm Hg.
Figure 5(d) also shows transferred mechanical power in the case where
systemic resistance is kept constant, hence the heart rate II is allowed to vary.
LEFT
VENTRICULAR
EJECTlON
DYNAMICS
(b)
140 .2
E
-i 130F
I-I_--____L_--
160
186
FE.
200
PO (I?!
6. (a) and (0).
220
295
296
I
1
---L-L-
160
180
220
200
V. (ml)
I.7
------.
180
I--~---.L
200
.-~~~
220
V,(rnll
FIG. 6. (a) Ventricular pressure curves for varying end-diastolic volume VO (ejection
period t, constant); (b) maximum ventricular pressure P(r),,,, as a function of enddiastolic volume V,,; (c) stroke work Was a function of VO; (d) mechanical power N as a
function of V,, for constant heart rate and for varying heart rate.
Then N increases monotonically
and strongly with P,, as a result of
increasing II.
Figure 6(a) is the plot of calculated left ventricular pressure against time,
for varying end-diastolic volume (R was kept constant here). The pressure
curve has a general tendency to rise with increasing V,, but the primary peak
is prominent at higher end-diastolic volumes, while for lower end-diastolic
volumes the secondary peak is prominent. These rather subtle effects have
also been recognized by Greene. Figures b(b)-(d) show maximum ventricular
LEFT
VENTRICULAR
EJECT’ION
‘97
DYNAMICS
pressure P(r),,,, stroke work W and mechanical power N, as functions of
V,. The slope discontinuity in the first curve stems from the fact that, at
lower values of V,, the maximum pressure coincides with the secondary
peak, while at higher V,, the maximum coincides with the primary peak. This
curve does not agree with Greene’s recordings, which display a maximum for
some intermediate V,,. The plots of W and N against V,, are linear, and do
not reach an extremum within the normal range of the end-diastolic volume
(cf. Suga, 197 1; Robinson, 1965). Figure 6(d) also shows mechanical power
N versus P’,,. in the case where n is kept constant, by varying V, and R
simultaneously. Now N increases strongly with iJO, as a result of increasing
stroke volume VS.
Figures 7 and 8 display the time course of ventricular systolic elastance
E(t), for varying P, and V,. They indicate that E(r) is dependent on both
0,7!
E
\
I”
E
F 05
G
O-2’
Fro. 7. Ventricular elastance curves
I, and
heart
rate
II constant).
for
varying
afterload
conditions
(ejection
period
298
III
0-02
--
0.05
045
01
i(s)
FIG. 8. Ventricular
t, constant).
elastance curves for varying end-diastolic
volume (ejection period
preload and afterload conditions. Specifically, the E(t)-waveforms increase
in amplitude for increasing H,, and decrease for increasing V,,. The latter
result confirms experimental findings in the paper by Greene, but contradicts
Suga (1969), according to whom elastance curves are essentially independent
of loading conditions. Variations in the time course of v(t), i(t) and P,(t)
with is, and V,, are conformable to data in the literature, and have not been
reproduced here.
6. Caseswith Variable Ejection Time
In this section some results are presented, pertaining to the free final time
problem. For a fixed value of the weighting coefficient ~1,the ejection time f,
is optimized
using
the condition (10). Figure 9 shows two pressure curves for
002
005
01
t(s)
01
Fw. 9. Ventricular pressure curves for nominal and
decreased mean arterial pressure Pa,, and for a variable
ejection period [x : 2.5 mm HgKml,‘s)].
-.(I-
Hg
0.05
-I-.
..-~
FIG. 10. Ventricular pressure curves for nominal and
increased end-diastolic volume V,, and for a variable
ejection period [a 7 2.5 mm Hg,Yml/s)].
0’02
300
I:.
J.
NOLDIJS
a = 2.5 mm Hg/(ml/s), and for the nominal and a decreased value of mean
arterial pressure. Analogous results for the nominal and an increased value
of V, are given in Fig. 10. Comparing with Fig. 2(a) reveals that t, decreases
with decreasing a (10% change in t, for more than 25% change in a), and
that t, increases for decreasing P, and for increasing V,. The latter properties
are well known from numerous results in the literature (Greene, 1973; Suga,
1969; Buoncristiani, 1973; Wilcken, 1964), although the present recordings
suggest a stronger dependence of ejection time on loading conditions, than
is usually observed experimentally. Figure 11 shows flow patterns i(t) for
varying V,. As already pointed out, the flow curve becomes symmetrical for
lower values of V,, in accordance with experimental data (Suga, 1969).
600-
FIG. 1 I. Blood flow curves for varying end-diastolic volume, and for a variable ejection
period [LY: 1.5 mm Hg/(ml/s)].
LEFT
VENTRICULAR
EJECTION
DYNAhlIC‘S
301
7. Effects of Other Parameters
Changing the peripheral resistance R mainly affects heart rate and
mechanical power N, but has little effect on stroke work W and on the
pressure and elastance waveforms, as indicated in Table 3, and in Fig. 12(a).
Both were recorded for c( = 2.5 mm Hg/(ml/s), and for nominal values of
the other parameters. The table shows that a normal heart rate can bc
obtained by slightly adapting the peripheral resistance. The elastance curves
of Fig. 12(a) are influenced by R, only at the beginning of ejection, bul
remain otherwise unaffected. Similar elastance curves for variations in
aortic valvular resistance, blood inertia constant and arterial compliance
have been calculated in Fig. 12(b)-(d). Heart rate, stroke work and mechanical power have been found to be almost insensitive to deviations in L and c’.
For increasing r, stroke work and mechanical power increase slightly, since
more energy is consumed in the aortic valve, while heart rate remains
unaffected.
TABLE
3
Effects of changingperipheral resistanceR
R [mm WW/sN
n (min-I)
N (watt)
W (joule)
J’@hn,x (-
Wh,
Hg)
(mm Hg/ml)
1
83
1.46
1.06
120
0.74
1.2
69
1.22
1.06
121
0.74
1.4
59
1.04
1.06
123
0.74
8. Further Applications
In this section two experiments are described with the model, whose
purpose is to shed some additional light on the value of optimization theory
in ventricular dynamics, and to demonstrate the adequacy of prediction,
using models based on this theory. If optimization is accepted as a fundamental concept in the design philosophy of biological systems, then a given
system should not only be optimally controlled by the external forces acting
on it, but it seems natural that such systems should also possess some
capability to adjust their internal structural parameters towards overall
optimality. The existence of such adaptive parameter adjusting capabilities,
in widely differing classes of biological systems, has convincingly been
demonstrated in Milsum’s (1968) review of this subject. In fact, evidence is
available to such an extent, that an adaptive structure of this type may bc
assumed to be a general characteristic OF living systems.
-
__-_--l_
002
c
I
005
0 015 mm Hg/!mi/s
01
t(s)
I___--.
0.15
OM5
J
1
FIG. 12. (a) Elastance curves for varying peripheral resistance R [a :-= 2.5 mm Hg;(ml/s)J; (b) elastance curves for varying aortic valvula
resistance r; (c) elastance curves for varying blood inertia constant L: (d) elastance curves for varying arterial compliance C.
L
O.!
1
I
I
r-(b)
304
L. J. NULDUS
In Fig. 13, it has been assumed that in the blood ejection mechanism, entldiastolic volume l’,, can be adaptively adjusted on the basis of minimizing
energy consumption. Figure 13 shows the mechanical work N,., generated by
the ventricle per unit of ejected blood volume, represented in normalized
form (relative to its minimum value), against I/,,. The plot reveals that the
V. (ml)
FIG. 13. Mechanical work per unit of ejected blood volume (in normalized form, relative
to its minimum vaIue), as a function of V,, for nominal values of all other parameters.
best value for V0 approximately equals the nominal value of 200 ml. It
certainly adds to the consistency of the system’s behaviour, and to the value
of the optimization concept, if in a model that was optimally designed on
energetic considerations, the nominal value of a loading parameter is also
energetically the best one.
The second example describes a case of ventricular ejection under abnormal
operating conditions. The model has been used to simulate aortic stenosis,
which consists of an obstruction of blood flow from the ventricle to the
0.02
0.05
01
015
--- 1 0
f(s)
FIG. 14. Curve (A) computed ventricular pressure wave in the case of simulated aortic
stenosis; curve (B) computed normal ventricular pressure wave; curve (C) experimentally
recorded ventricular pressure wave in a case of aortic stenosis, according to Van der Werf
(1974). normalized to the same maximum val:ie and systolic period as curve (A).
306
E.
J.
NOLDUS
aorta, and is usually caused by a striction near the aortic valve. Aortic
pressure is usually below normal. Heart rate remains normal or decreases
slightly, while the systolic period increases. Recorded ventricular pressure
curves for this condition can be found in Van der Werf (1974). They are
characterized by a strongly increased secondary peak. In order to simulate
aortic stenosis, the following parameter values were introduced in the model :
Aortic resistance Y was increased to 0.05 mm Hg/(ml/s), mean arterial
pressure was set at 75 mm Hg, and, according to Fig. 4, systemic resistance at
O-7 mm Hg/(ml/s). The systolic interval was increased to 0.22 s. Figure 14
shows the resulting pressure wave, as compared to the normal one, and also
an experimental curve according to Van der Werf (1974). The increased
secondary peak is correctly reproduced, which shows that the model is not
only capable of predicting the ventricle’s behaviour under normal operating
conditions, but also in abnormal situations.
In addition the simulation provides some evidence that the ventricle’s
optimal control mechanism does not operate on a preprogrammed, or “open
loop” basis, either genetically or learned, but upon a basis of continuing
adaptive control. By this is meant that the control strategy can be continuously and optimally modified upon the occurrence of structural changes
in the controlled system, for example as a result of illness. It is evident that
such adaptive aspects essentially enhance the power and quality of biological
control.
9. Discussion
A model of the left ventricular ejection system has been developed, which
has as its salient feature the optimization of the ventricular elastance curve
E(l) with respect to a performance index, selected on an energetic basis. The
ventricle has been loaded with a lumped arterial Windkessel load. Although
the performance index is very simple, the model is capable of correctly
reproducing many classical experimental results, at least in a qualitative
sense, including the changes in left ventricular pressure curves in response to
varying preload and afterload conditions (changes in end-diastolic volume
V, and mean arterial pressure P,). The model accurately predicts the shape
of the left ventricular pressure wave, as well as subtle modifications in this
shape, as a function of increasing arterial pressure and end-diastolic volume.
Computing
the corresponding ventricular elastance yields a family of
elastance curves, depending on P, and V,. The dependence is qualitatively
in agreement with the results of Greene et al. (1973). It is in contrast however
with those of Suga (1969, 1971), according to whom ventricular elastance is
essentially independent of P, and I’,. When the ejection period is allowed to
LEFT
VENTRICULAR
EJECTION
DYNAMICS
3.);
change, it decreases for decreasing k;, and for increasing I’,, also in agreement with experimental findings. The model is not capable of satisfactorily
reproducing the time course of blood flow during ejection. Possibly this can
be attributed to an oversimplification
in simulating aortic valve characteristics by a constant resistance. Some evidence supporting this conjecture
has been included in the paper. For reduced end-diastolic solumes and the
corresponding
lower amplitude flow patterns, the discrepancy
between
computed and experimental flow curves tends to disappear, probably as ;I
result of the reduced influence of valve nonlinearities. Finally the effects of
changing peripheral resistance, blood inertia constant and arterial compliance have been briefly examined, and the model’s operation in an
:tbnormal condition of the controlled system has been discussed. It has been
thown that the real system’s behaviour is, in a qualitative sense. adequately
predicted under abnormal as well as normal condi:ions of the controlled
system. In addition an example has been provided in which the nominal
value of a loading parameter is successfully matched with the energetically
optimal value for this parameter, thus further supporting the optimization
concept as a fundamental principle in the design of the ejection mechanism.
Complete numerical agreement with experimental data has not alway\
been achieved. This could not be expected however. in view of the simplified
qualitative nature of the dynamical model. upon which the analysis wa\
carried out. Nevertheless the results strongly suggest that the general control
strategy, including shapes of waveforms
and their dependence on system
parameters, and the nominal values of these parameters, are probably
related to energy minimization considerations.
Further research might be
directed towards the analysis of more accurate dynamical ejection models
on a similar basis, possibly including the entire cardiovascular
system, ancl
taking into account a number of homeostatic relationships among parameters. which in the present analysis have been ignored or crudely approximated. Using a more realistic arterial loading network
might also be
attempted. Of course this could only be done at the expense of strongly
increased computational complexity.
The author is grateful to H. Hinssen and E. Van Tngelghem who wrote the
computer program for the present problem.
APPENDIX
Integration of the Hamiltonian
First note that p1 = a,, a constant.
--.I 4 (PJC) +a.~,, and L4
- 4 (p,/C)--KY,.
r 9.
A
System, equations (5)--(b)
Define z, A y2 + aLx,, zz 4 p2 -ah,.
Then. by iwice differentiating z, and
31
308
E. J. NOLDUS
rearranging terms, it is easy to show that
Eliminating z2 from equations (Ala)-(Alb)
yields a fourth order equation in
zI, from which 21= P(r) = -z,/2L can be obtained as given in Table I, in
terms of the integration constants a,-a5. Now the computation of
-6,La,]
e-6z(fe-‘)+[(C(+J.)UJ+S*L~4]
e-d?r
-f-[(~+~.)f~~+K,Li(~]
e-“1’
and the results of Table 1 is straightforward.
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CLARK,
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ond Medic&
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SLIC;~, H.
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IlYhh.MICS
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