Download Determinants of stroke volume and systolic and

Determinants
of stroke volume and
systolic and diastolic aortic pressure
NIKOS STERGIOPULOS,
JEAN-JACQUES
MEISTER, AND NICO WESTERHOF
Biomedical Engineering
Laboratory,
Swiss Federal Institute of Technology,
1015 Lausanne, Switzerland;
and Laboratory
for Physiology, Institute for
Cardiovascular
Research, Free University, 1081 BT Amsterdam,
The Netherlands
Stergiopulos,
Nikos, Jean- Jacques
Meister,
and Nice
Westerhof.
Determinants
of stroke volume and systolic and
diastolic
aortic pressure. Am. J. PhysioZ. 270 (Heart Circ.
PhysioZ. 39): H2050-H2059,
1996.-We
investigated
how
parameters
describing
the heart and the arterial
system
contribute
to the systolic and diastolic pressures (Ps and Pd,
respectively)
and stroke volume (SV). We have described the
heart by the varying-elastance
model with six parameters
and the systemic arterial
tree by the three-element
windkesse1 model, leading to a total of nine parameters.
Application
of
dimensional
analysis
led to a total of six dimensionless
parameters
describing
dimensionless
Ps and Pd, i.e., pressures with respect to venous pressure (P&
and P&).
SV
was normalized
with respect to unloaded ventricular
volume
(Vd). Sensitivity
analysis showed that PJP”, P&,
and SVNd
could be accurately
described
by four, three, and three
dimensionless
parameters,
respectively.
With this limited
number of parameters,
it was then possible to obtain empirical analytical
expressions
for PJPv, Pd/P”, and SVN+
The
analytic predictions
were tested against the model values and
found to be as follows: Ps predicted = (1.0007 5 0.0062) Ps, r =
0.987; Pd predicted
= (1.016 ? 0.0085) Pd, r = 0.992; and SV
predicted = (0.9987 2 0.0028) SV, r = 0.996. We conclude that
aortic Ps, Pd, and SV can be accurately
described by a limited
number of parameters
and that, for any condition of the heart
and the arterial
system, Ps, Pd, and SV can be presented
in
analytical
form.
varying-elastance
model; three-element
dimensional
analysis; coupling of heart
windkessel
model;
and arterial system
PRESSURE
AND FLOW
in the ascending aorta arise
from the interaction between the heart and the arterial
system; i.e., both heart and arterial load determine
aortic pressure and cardiac output. The determinants
of aortic systolic and diastolic pressures (Ps and Pd,
respectively) and stroke volume (SV) are not quantitatively known. A number of studies have been performed
to obtain insight into the effects of changes in parameters such as peripheral
resistance
(RJ and total
arterial compliance (C) on aortic pressure and flow (3,
8). However, a general and quantitative
approach to
obtain all determinants
of Ps, Pd, and SV has, to the
best of our knowledge, not been carried out. This is in
part due to the fact that a model of both the heart and
the arterial system could not be formulated
in a quantitative way with a limited number of parameters. Such a
quantitative
description
is achieved by use of the
varying-elastance
model for the heart and the threeelement windkessel model for the arterial system, both
of which contain a minimal number of essential parameters. The three-element
windkessel model is a generally accepted and rather accurate description
of the
arterial system (9, 16). The varying-elastance
model of
the heart has also been shown to be an acceptable
limited-parameter
model (10, 14). Thus heart and
arterial system can be described with a limited number
of independent
parameters,
and heart and load can be
coupled to yield aortic pressure and flow. From these,
Ps, Pd, and SV are determined
and used to quantify
their sensitivity to cardiac and arterial parameters. We
will use dimensional analysis to reduce the number of
independent parameters by combining them into dimensionless groups (7). Subsequently,
we will study the
effects of these dimensionless
groups of variables on
pressures and SV. This analysis will allow us to decide
which parameters
are important
determinants
of Ps,
Pd, and SV This information
will be used to derive
empirical formulas for Ps, Pd, and SV.
METHODS
MathematicaZ
modeZ. The mathematical
model for the
heart and the arterial
system (14) consists of the varyingelastance
model for the heart (10) and the three-element
windkessel
model (16) representing
the arterial
load. An
electrical analog of the entire heart-arterial
system model is
shown in Fig. 1A.
The varying-elastance
model for the heart consists of a
pressure source (venous pressure, Pv) emptying
into the left
ventricle
with a time-varying
elastance, E(t). Elastance
relates ventricular
pressure, P, to ventricular
volume, V, through
the formula
THE
HZ050
0363-6135/96
$5.00
Copyright
0 1996
where Vd is the unloaded volume defined as the intercept of
the end-systolic
pressure-volume
relationship
with the volume axis (10).
Changes in elastance are due to the contraction
of the
myocardium.
A typical heart cycle of the elastance
of an
isolated cat heart is depicted in Fig. 1B. The minimum
value
of the elastance, Emin, together with Pv, determine
filling and
thus end-diastolic
volume (V&. The maximum
value of the
elastance,
the end-systolic
pressure-volume
relationship
or
E max7 is considered a measure of the contractility
of the heart.
For a given cardiac state (heart rate and contractility
constant), the elastance-time
curve is assumed to remain unchanged and independent
of alterations
on the load (10).
We found that the shape of the elastance curve is best
approximated
by the following periodic “double-Hill”
function
the American
Physiological
Society
DETERMINANTS
mitral
valve
OF
STROm
VOLUME
aortic
valve
RP
E max
E min
‘P
0.2
Time [s]
Fig. 1. A: electrical
representation
of model of heart
and arterial
system.
Pv is venous
pressure
and Rv a (small)
resistor
over which
filling takes place. E(t) is time-varying
elastance.
Three parameters
of windkessel
(Rc, characteristic
resistance;
Rp, peripheral
resistance;
and C, total
arterial
compliance)
are indicated.
B: waveform
of
time-varying
elastance
with its minimum
(Emin), maximum
(J&&,
and peak time (TJ values pertaining
to cat heart.
The first term in the brackets (Hill function)
describes the
ascending part of the E(t) curve and the second (inverted
Hill
function) the descending part. The curve is shown in Fig. l&
where only a single heartbeat
is presented.
Dimensionless
shape parameters,
al and CX~,define the relative appearance
time of each curve within the heart period, T, and exponents,
nl and n2, determine
the steepness of the ascending
and
descending parts, respectively.
The arterial
system is represented
with a linear threeelement windkessel
(16). Thus the whole arterial
system is
conceived as a point load with the characteristic
impedance,
&, which models the inertial
and compliant
effects of the
ascending aorta, connected in series with a parallel combination of the peripheral
resistance,
Rp, and the mean total
arterial compliance,
C. With proper choice of parameters,
the
three-element
windkessel
can model the arterial input impedance well and can thus provide an accurate description
of the
heart load. Nonlinearities
arising from the pressure dependence of C have minor effects on aortic pressure
and flow
waveforms and will thus be neglected in this analysis (11).
For the purposes of this study, we will use as a typical
example the human
circulation.
For the validation
of the
model, we will use data from earlier experiments
on isolated
cat hearts (3). The control parameters
for the human and cat
heart-arterial
system model are given in Table 1.
Circulation
of the four phases of the heart cycle (isovolumic
contraction,
ejection, isovolumic
relaxation,
and filling)
is
done in a straightforward
manner, as explained
in previous
studies (1). Equation
1 is coupled to the equation
of the
three-element
windkessel
only in the ejection phase. In this
phase, a solution
was obtained
numerically
by use of a
fourth-order
Runge-Kutta
integration
scheme. During
ejection, ventricular
and aortic pressures
were assumed equal.
During isovolumic contraction
and relaxation,
aortic flow was
zero (no incisura),
and Eq. 1 applies for ventricular
pressure
while aortic pressure decayed exponentially,
as dictated by
the three-element
windkessel.
In the filling phase, ventricular volume relates to pressure through
the formula dV/dt =
P - P)/..“, which is solved with Eq. 1 by use of a RungeKutta integration
scheme.
AND
AORTIC
HZ051
PRESSURE
Dimensional
analysis. For a given shape of the elastancetime curve, the above heart-arterial
system contains eight
basic independent
parameters,
three of which (Rc, Rp, and C)
pertain to the arterial system. Four parameters
(z Em=, Emin,
and Vd) pertain to the heart. The eighth parameter
is venous
(filling) pressure, Pv. For a complete description
of the heart,
however, one should include the four parameters
that define
the shape of the E(t) curve, namely, CQ, cxz, nl and n2. For
simplicity, we will assume that the shape of the E(t) curve is
approximately
triangular
and sufficiently
determined
by
specifying its peak time, Tp (see Fig. 1B). The limitations
of
this assumption
will be considered in DISCUSSION.
The system
is now assumed to be well defined, and any other important
determinants
of cardiac or arterial function (such as humoral
or neural factors) are assumed to be external to the system;
i.e., their influence is simply to modify one or more of the nine
basic independent
parameters.
From this assumption,
any
variable resulting
from the heart-arterial
system interaction
will be a single function of the nine basic parameters.
In this
study we will focus on Ps, Pd, and SV, which can be expressed
in general terms as
where fl, f& and fs are the unknown
functions
still to be
defined.
Ideally, one can do parametric
studies in which one of the
independent
parameters
in the right-hand
side of Eqs. 3-5 is
varied while the others are kept constant. This procedure
would then be repeated eight times for the remaining
parameters. The result of this tedious task would be to fill a
nine-dimensional
parameter
space, out of which it would
have been very difficult to extract a global relationship.
The logical way of approaching
this problem is to use the
theory of dimensional
analysis (7). By virtue of the Buckingham II theorem (7), we can rewrite Eqs. 3-5 in dimensionless
form, in which the independent
dimensionless
parameters,
often called II terms, are formed by groups of the original
independent
parameters.
The major advantage
of this approach is that the number of Il terms is equal to the number of
original
independent
parameters
(9) minus the number
of
basic dimensions
(units) of the system (Ref. 3; namely, time,
volume, and pressure).
Thus the number of II terms in our
equations
reduces to six. Therefore,
without
any loss of
Table 1. Basic model parameters
Independent
Cardiac
and Arterial
System
Parameters
Characteristic
impedance
(Rc),
s ml1
Peripheral
resistance
(R&, mmHg
s ml-l
Total arterial
compliance
(C), ml/mmHg
Heart
period (79, s
Maximum
elastance
(&-&,
mmHg/ml
Minimum
elastance
(Emin), mmHg/ml
Unloaded
volume
(Vd), ml
Venous
pressure
(PJ, mmHg
Time to peak elastance
(Q, s
Shape factor q
Shape factor q
Exponent
Q
Exponent
Q
mmHg
l
l
l
l
Human
Cat
0.51
1.05
1.60
1.00
2.31
0.06
20.0
7.50
0.43
0.303
0.508
1.32
21.9
1.50
21.5
0.057
0.39
41.3
2.00
0.75
7.50
0.25
0.25
0.63
2.00
15.0
H2052
DETERMINANTS
generality,
Qs.
3-5 are rewritten
sv
-vd
-@3iP-9-
OF STROKE
VOLUME
as
Rc RpC
1
KIlax
K
Tp
CEmm ’ -Emin ’ EminVd
’ -T I
l P T
AND
A
AORTIC
PRESSURE
120
(81
The choice of II terms in the right-hand
side of Qs. 6-8 is
not unique; however, it was made such that each term refers
to easily identified
ratios of the independent
variables.
The
dimensionless
parameter
III = RJRp is a pure arterial
parameter.
The parameters
II2 = RpC/T and lL13 = l/(C&&
pertain to both heart and arterial
system and are considered
“coupling
parameters.”
The parameters
IId = EmJEmin and
l-I6 = Tp/T are cardiac parameters.
The parameter
IIs =
Pv/(EminVd) gives the coupling
of the heart with the venous
system.
The dependence
of P&,
Pd&, and SV/Vd on the lI terms
will be derived. First, a sensitivity
analysis will be performed
to assess which of the II terms are important
in the determination of Ps, Pd, and SV By retaining
only the II terms that
are important,
we can derive an analytic
description
of the
functions, QI, m2, and m3.
RESULTS
Kilidation
of’ modeZ. The model was tested by use of
available data from experiments
on the isolated cat
heart loaded with the three-element
windkessel
(3).
The windkessel parameters
were taken directly from
the original paper. The cardiac parameters
were also
taken from this article. By plotting the relation between mean ventricular
pressure and SV and fitting
this with a parabola
(12), maximal
SV and mean
isovolumic pressure were determined. Maximal SV and
Pv, together with an assumed dead volume of 0.75 ml,
gave the value of Emin. Isovolumic pressure was used to
estimate EmaX. The parameters
of the Hill functions
were chosen such that the ventricular
pressure wave
shape resembled the measured one (3). The resulting
E(t) curve is shown in Fig. 1B. All parameter values are
presented in Table 1.
The cat model produces ventricular
and aortic pressures and aortic flow as given in Fig. 2A. Pressurevolume loops for control, increased contractility
(EmaXX
2), and increased filling (Pv from 7.5 to 8.5 mmHg) are
given in Fig. 2B. In Fig. 3, the changes in Ps, Pd, and SV
for changes in Rp and C in the model and the isolated
cat heart are compared. We conclude that model predictions and experimental
data compare very well in many
aspects.
Human model The model was then converted to the
human cardiovascular
system. The parameters
are
given in Table 1. The dependence of the variables P&,
Pd/P”, and SV& on the Il terms are given in Figs. 4 and
5. We have calculated only a limited number of points
(0.5, 0.75, 1, 1.5, and 2 of control) of the dimensionless
parameters.
More data would have increased the com-
Time
2
I
0
[s]
3
Volume
4
5
6
[ml]
Fig. 2. A: left ventricular
pressure,
aortic pressure,
and aortic flow
for heart-arterial
model
of cat. B: pressure-volume
loops obtained
from heart-arterial
model of cat. Three loops are shown for changes
in
BP (fully drawn
lines). An increase
in filling is given by stippled
line.
Two pressure-volume
loops during
increased
contractility
are given
by dashed lines.
putational
effort considerably
without
large gain in
information.
It may be seen that Ps mainly depends on
III, l&, IIs, and l-Id. Pd depends on l&, l-Is, and IId. SV
depends on l&, IIs, and l-Is. It may also be seen that the
relation between Ps and Pd with II4 is linearly proportional. The same is true for SV in relation to l-Is.
Based on the results, Eqs. 6-8 are rewritten
in the
following simplified form
P
-- S - @ ---1
PV
’ T
Pd
--
PV
sv
-- Vd
-
Emax
’ CEmm’ Emin
@
’
@ P
’
1
P
’ CEma ’ EmiIVd
Equations 9-11 were used to fit the data presented in
Figs. 4 and 5, following a standard procedure described
in the appendix. The analysis was limited to II values in
the range of half to twice the control values, fitting the
data over this range only with arbitrary formulas. The
empirical formulas for Ps, P& and SV resulting from the
DETERMINANTS
OF
STROKE
VOLUME
AND
AORTIC
PRESSURE
H2053
150
sv
--
100
-dI.r.e
vd
50
0
0
50
Systolic
150
B
100
pressure
[mm
150
Hg]
1
I
.’
.: .
,-
100
K
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(14)
min
where the last term in the exponent of Eq. 14 is the product of
I& and IIS. The values of the empirically
determined
coefficients bi (i = l,..., 8), ci (i = l,..., 6), and di (i = l,..., 4) are given
in Table 2.
The predictions
of these analytical
expressions
for Ps, P&
and SV were tested against their actual values (given by
model) in Fig. 6. Ps predicted
= (1.0007 2 0.0062) Ps, r =
0.987; Pd predicted
= (1.016 2 0.0085) P& r = 0.9%;
sv
predicted = (0.9987 2 0.0028) SV, r = 0.996. We conclude that
the formulas are accurate over a wide range of pressures and
svs.
Based on Eqs. B-14,
we have derived the sensitivity
of Ps,
P& and SV to cardiac and arterial
parameters.
Sensitivities
are presented
in dimensionless
form around the control
values in Table 2.
DISCUSSION
50
0
0
100
50
Diastolic
pressure
[mm
150
Hg]
I
C
.’
0
.
2
1
Stroke
. .’
. ..
: .-
volume
[ml]
Fig. 3. A: relationship
between
systolic
blood pressure
(Ps) measured
in an isolated
cat heart loaded with 3-element
windkessel
and model
prediction.
Changes
are obtained
through
variations
in &,, increased
to 2.14 and 4.81 of control
and decreases
of total arterial
compliance
(C) to 0.326 and 0.084 of control.
B: as in A but for diastolic
blood
pressure
(P&. C: as in A but for stroke volume
(SV). Dashed
lines are
lines of identity.
fit are
PS
--
PV
-
b1
.-
E max
E min
(b&Em=
.
b6
The main objectives of this study were, first, to obtain
insight into the sensitivity of Ps, Pd, and SV to cardiac
and arterial parameters
and, second, to derive empirical formulas that give an accurate description of Ps, Pd,
and SV Using dimensional
analysis, we identified the
major cardiac and arterial parameters that contribute
to Ps, Pd, and SV We found that all three variables (Ps,
Pd, and SV) could be described by a limited number of
parameters.
The fact that three to four dimensionless
parameters
were sufficient to describe the variables
made it possible to derive empirical formulas for Ps, Pd,
and SV Obviously, with heart rate known, the predictions on SV can be easily extended to predictions of
cardiac output.
Universality of approach and applicability
of empiricaZ formulas. The use of dimensional
analysis for the
derivation
of the empirical
formulas
(Eqs. 12-14),
apart from being an elegant method of reducing the
number of independent variables, offers another significant advantage: the dimensionless II terms, which are
now the independent
parameters, are for control conditions quite similar for all mammals.
For example,
parameters RJRp and RpCIT were shown earlier to be
the same in all mammals
(6, 15). Also, the cardiac
of
parameter
(Em= /Emin) is a ratio that is independent
heart size because it gives the ratio of isovolumic
pressure and Pd, a ratio which is similar in mammals.
The ratio of V,d (PJEmin + Vd) and Vd is found to be the
same in different mammals. Pressures in the aorta and
Pv are also similar and SV and Vd make a volume ratio
likely to be similar in mammals.
The empirical formulas for the Ps, Pd, and SV (Eqs.
12-14) were derived for the human and do not necessarily apply to all mammals. We have checked this by
applying the empirical formulas to the isolated cat
experiments
and significant
differences were found.
This is mainly due to the fact that the elastance
function for the human is quite different from that of
H2054
0
DETERMINANTS
OF STROKE
0.5
1.5
I
VOLUME
2
AND
2.50
AORTIC
PRESSURE
I
0.5
Rc’Rp
0
0.5
1
l/(C
1.5
1.5
2
2.5
RpC/T
2
2.50
0.5
Emax )
1
2
2.5
2
2.5
1.5
Emax /E min
30
20
10
0
0
0.5
1
1.5
2
2.5
0
0.5
between
(filling)
dimensionless
Ps and
pressure;
V& unloaded
Pd and 6 II terms.
Horizontal
volume;
7’, heart period.
the cat (see Table 1). Also, in Figs. 4 and 5, some
dependence on Z’Pthat is seen was not included in the
analysis. We may expect that if appropriate
analysis
was done to include the additional
II terms resulting
from the E(t) shape factors (cx~,a2, nl, and n2), one could
derive global relationships
for pressures and SV Further research, however, is needed to conclude whether
this is feasible.
GeneraZity. It should be stressed that the model
applies to all cardiovascular
conditions. Nervous and
humoral control will affect the cardiac and arterial
parameters,
but when changes in the independent
parameters
are known and accounted for, the correct
pressures and SV will result.
1.5
T/T
'v ' tvcj 'mi"l
Fig. 4. Relations
control.
Pv, venous
1
axis
is normalized
with
respect
to
Limitations.
The analysis was performed
with a
model of the heart and arterial system. In principle,
this approach can be applied to the intact animal but in
such a preparation
it is extremely difficult to induce
variations
in certain parameters
while leaving other
parameters unaffected. For instance, a change in C in
the intact animal will result in changes in pressure,
leading to baroreflex control changes in heart rate and
contractility
(8). It will also lead to changes in cardiac
filling. Moreover, the analysis requires a large number
of data points, even if the sensitivity to the parameters
that are of major importance
is investigated.
In general, experiments
would require very long measurement times in a stable preparation.
The present study
DETERMINANTS
5
OF STROKE
I
I
I
I
0.5
I
1.5
2
VOLUME
AND
AORTIC
H2055
PRESSURE
4
u
3
2
>
CD
2
1
I
I
I
0.5
1
1.5
2
2.5
1.5
2
2.5
2
2.5
0
0
vp
2.5 Cl
I
F$C/T
5
2.5 0
0.5
1
E
max ’ ‘min
8
6
0
0.5
I
1.5
2
2.5 0
0.5
1
1.5
Tp/T
p 1 o$ Ii min J
V
Fig.
5. Relations
between
dimensionless
SV and 6 II terms.
Horizontal
gives the functional dependence of Ps, Pd, and SV on the
important
parameters and forms the basis for further
studies in animals and the human.
A triangular-like
wave shape in the E(t) curve was
chosen for the human model. By changing the wave
shape of E(t), we found similar sensitivity
of the
Table 2. Parameter
values of Eqs. 12-14 estimated
axis is normalized
with
respect
to control.
variables Ps and Pd to the independent
parameters,
although the quantitative
relations varied.
By assuming a triangular
wave shape of E(t), we
have limited the shape factors of the E(t) to a single
parameter,
Tp. If the Hill equations were taken into
account, this would have implied four parameters, i.e.,
by least-squares
fit
Parameters
0.337
1.87
0.271
0.651
0.93
0.595
0.60
0.369
0.595
1.09
0.346
0.271
0.866
0.111
1.04
0.0589
0.568
0.389
DETERMINANTS
OF STROKE
VOLUME
Systolic pressure [mm Hg]
I
B
l
.s
.
.
.
.
100
0
200
Diastolic pressure [mm Hg]
C
. .ri
. 0;
,
0
. ci
.
40
80
120
Stroke volume [ml]
Fig. 6. Comparison
between
model prediction
las (Eqs. 12-14) for Ps (A), Pd (B), and SV (C).
and
empirical
formu-
increasing the number of parameters by three. Better
fits, i.e., fits with more parameters,
will improve the
final result and make it possible to accurately account
for changes in the E(t) curve.
The pressure-volume
relations in diastole and systole
were approximated
by straight lines. There is evidence
that these relations are curved with convexity toward
the volume axis in diastole and concavity in systole (2,
13). This second-order effect was not included to keep
relations simple and the number of parameters small.
AND
AORTIC
PRESSURE
The way we derived the empirical formulas was as
follows. We have started from a chosen dimensionless
parameter
and studied its relation to the variable of
interest. Subsequently, we studied the contributions
of
the other parameters on this relation (see APPENDIX).
By
doing so, we have chosen a certain path to arrive at our
results. If we had started the analysis from a different
dimensionless parameter, the formulas would have had
a different form. However, the same parameters would
have appeared in the formulas. We have also fitted the
relations with linear, exponential or power curves. The
choices were made on the basis of the quality of the fit
during the search for a minimal number of coefficients
in the fitted curve, since the dependence of each coefficient on the other parameters needed to be quantified.
We have carried out the analysis for a limited range
of the dimensionless parameters (from 0.5 to 2 of their
control value). Again this was done to limit the amount
of work. Most variations in the parameters
cover the
physiological working range. For instance, during exercise, heart rate increases considerably
(i.e., factor of
3-4) so that T decreases by the same amount, but $,
decreases considerably as well. Thus the dimensionless
parameter
R&/T
does not undergo large changes.
However, for changes in a single parameter, the formulas apply only for the range 0.5-2 of control.
ImpZications. Figures 7 and 8 show the variations of
Ps, Pd, and SV for changes in Rp and C (arterial) and for
changes in the end-systolic pressure-volume
relation
and heart rate (cardiac). The values derived from
pressure and flow in time are given by the symbols. The
fully drawn lines are from the derived equations. It
may be seen that our empirical formulas, when applied
within the allowed range, predict these variables well.
The sensitivity
of Ps, Pd, and SV to the independent
cardiac and arterial parameters is given in Table 3. It
may be seen from Table 3 that characteristic impedance
has a small effect on Ps only.
Increased Rp decreases SV and increases mean pressure (4). If the pressure wave shape remains the same,
the increase in mean pressure would result in proportional increases in Ps and Pd. However, with an increase
in Rp the arterial pressure in diastole will decrease
more slowly due to increased RpC time, so that Pd
remains higher. This then results in a relatively larger
increase in Pd than Ps. In Table 3 of Elzinga and
Westerhof (3), we indeed see that with an increase in Rp
(by a factor of -2) P d increased by 50% compared with
Ps. This is in good agreement
with the theoretical
predictions given in Table 3.
For a decrease in C, SV increases slightly and mean
pressure increases somewhat. This is again in agreement with experimental
findings in (3). When C decreases, the decay of pressure in diastole will be faster
so that Pd will be lower. Elzinga and Westerhof (3)
showed that a decrease in C (by a factor of -3) resulted
in a decrease in Pd that was about twice as much as the
increase in Ps. Randall et al. (8) studied Ps and Pd in the
intact dog by replacing the aorta with a stiff tube while
keeping Rp and heart rate unchanged.
However, in
DETERMINANTS
OF
STROKE
VOLUME
20
25
AND
AORTIC
H2057
PRESSURE
250
0
IO
5
15
Compliance,
250
I
I
0.2
C [ml/kPa]
I
I
Peripheral
250
I
0.6
0.4
0.8
resistance, F$, [kPa s/ml]
I
I
I
I
1
I
D
200
150
100
50
0 1
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Maximum elastance,
Fig. 7. Effects
of changes
and lines are representations
0
0
E,.,,aX[kPa/ml]
in C (A), $, (B), Emax (C), and heart
of empirical
relationships
given
I
I
I
I
50
100
150
200
250
Heart rate [bpm]
rate (D) on Ps and Pd. Points
by Qs. 12 and 13.
their preparation
in contrast to the isolated heart
studies by Elzinga and Westerhof
(3), end-diastolic
pressure increased.
They found that Ps increased
slightly, whereas Pd decreased. This may be explained
from the fact that end-diastolic
ventricular
pressure
(cardiac filling) increased somewhat and contractility
was also possibly increased.
Comparison
of the sensitivities
to I$., and C shows
that Ps and Pd are about four times less sensitive to
changes in C than to changes in Rp. For SV, the
sensitivity
to C is even smaller compared with the
sensitivity to Rp.
When heart rate decreases (and T increases), we
predict an increase in SV, and a decrease in Ps and Pd.
This is in agreement with the findings on the isolated
feline heart (5). With an increase in T, the pressure in
diastole decreases over a longer period in time so that
Pd is decreased more than Ps. Thus the sensitivity of Pd
to T is greater than the sensitivity of Ps.
With an increase in the end-systolic pressure-volume
relation, interpreted
as an increase in contractility,
both cardiac output (and thus SV) and mean pressure
are predicted to increase, in agreement with (4). Because the duration of diastole and the speed of decay of
Pd are not affected, Ps and Pd changes will be about the
same. Suga et al. (see Fig. 3 in (10)) showed that an
are obtained
from
model
increase in contractility
resulted in roughly the same
changes in SV and Ps, in good agreement with our
prediction
of similar sensitivity
of these parameters
(Table 3).
The Pd-volume relation is a very important
parameter in the determination
in both pressure and SV. The
sensitivities to this parameter are similar for pressures
and flow. A decrease in Emin will result in a larger Ved,
resulting in changes similar to those due to increased
Pv. The large sensitivity shows that, with poor relaxation of the cardiac muscle as in ischemia, the pumping
function of the heart is strongly reduced.
Pv has a large effect on pressures and flow. This is the
Frank-Starling
mechanism. The predictions (Table 3)
are in agreement with animal data (4). In these animal
experiments,
similar changes in SV and pressure were
found for changes in Pv.
We have presented a method to derive the effects of
the major parameters that characterize the heart and
arterial tree on Ps, Pd, and SV. Taking into account all
major determinants,
one can derive to empirical analytical formula for these three variables under all possible
conditions. We have performed a limited analysis that
takes into account the major effects and arrived at
empirical formulas. The effects of changes in single
parameters such as C and Rp or heart rate and contrac-
H2058
DETERMINANTS
100
I
I
A
OF STROKE
I
AND
120
1
cl
cl
m
VOLUME
AORTIC
PRESSURE
BCJ ’
I
I
I
I
I
I
100
cl
80
20
I
IO
I
5
0
0
I
15
Compliance,
I
20
0
25
l-l
60 -
60
40 -
40
20 -
20
I
0.4
Maximum elastance,
Fig.
8. As in Fig.
7 but for SV and lines
based
Table 3. Sensitivity to independent
arterial system parameters
&
9
C
T
E
E
-10
-50
max
min
P”
40
-100
100
I
I
I
I
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
I
0.5
0.6
0 1
0.7
resistance, R,, [kPa s/ml]
I
I
I
I
I
50
100
Emax [kPa/ml]
I
1
I!
I
150
200
250
Heart rate [bpm]
The methodology
used in deriving an appropriate
form of the
function @ was as follows. We first determined
the functional
relationship
between (PJPJ&
and I& for a range of values of
l-I2 from 0.5 to 2. The data were best fitted by a power law
suggesting that Eq. Al can be written as
at Control
sv
0
90
22
-90
32
-100
100
0
I
I
on Eq. 14.
pd
41
%
I
cardiac and
Sensitivity
ps
I
D
80
0.1
I
100
80 -
I
0.3
I
Peripheral
C
I
0.2
I
C [ml/kPa]
100 .
0
1
0
-28
5
28
33
-100
100
= FI( II2 ) ( l-13)F2(U2J
l
Functions
@r and @z were plotted against II2 and could be
well approximated
by an exponential
and a linear function,
respectively,
so that Eq. A2 is written as
Sensitivity
is expressed
as @Y/aX)/(Y/X)
X 100, where Y is systolic
pressure
(P& diastolic
pressure
(Pd), or stroke
volume
(SV) and X is
an independent
parameter.
tility (Emax) can easily be assessed. The results can be
generally applied and hold in conditions of health and
disease.
APPENDIX
Dimensional
analysis (Figs. 5, A-3’) showed that Pd/Pv is a
function
only of I&J, IIs, and II,+ Furthermore,
Pd/Pv is
proportional
to IId which means that the relation simplifies to
with al = 172, a2 = 186, a3 = 0.581, a4 = 0.866, and as = 0.187
obtained
by nonlinear
and linear regression.
Substituting
l-I2 = R&IT, l-I3 = l/(C&&,
and I& = Ema/Emin into ZQ. A3
and normalizing
with respect to the II values of control (see
Table 1), we obtain Eq. 13 in the text. A similar approach was
followed for deriving the other two empirical formulas.
Address
for reprint
requests:
N. Stergiopulos,
Biomedical
Engineering Lab., Swiss Federal
Institute
of Technology,
PSE-Ecublens,
1015
Lausanne,
Switzerland
(Email:
[email protected]).
Received
14 August
1995;
accepted
in final
form
9 November
1995.
DETERMINANTS
OF STROKE
VOLUME
REFERENCES
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