Vol. 279, Issue 2, R539-R547, August 2000
Coupling of left ventricular and aortic volume elasticity in
the rabbit
T.
Wronski,
P. B.
Persson,
E.
Seeliger,
A.
Harnath, and
B.
Flemming
Johannes-Müller-Institut für Physiologie, Humboldt
Universität (Charité), D-10117 Berlin, Germany
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ABSTRACT |
Changes in volume elasticity (VE) of the left ventricle
and aorta could be important for blood flow. A procedure is presented to rapidly assess VE of the left ventricle and aorta by analyzing changes in the eigenfrequency. Six control rabbits and 11 rabbits with
atheromatosis (12 wk of high-cholesterol feeding) were studied. In
control rabbits, during the first half of the systole, left ventricular
VE continuously increased to +43% (P < 0.05). Then VE
gradually declined to an end-diastolic minimum (20% of the average
systolic levels, P < 0.05). Aortic VE changes were in the opposite direction to the ventricle. Aortic VE continuously decreased throughout the systole; the last value was 20% lower than at
the beginning of the systole (P < 0.05). Conversely,
diastolic VE of the aorta took on greater values. This inverse time
course between ventricle and aorta may reduce energy requirements for conveying blood. High cholesterol-fed rabbits did not reveal the inverse behavior of ventricular and aortic VE, e.g., aortic VE increased during the systole (119%, P < 0.05).
heart; aorta; compliance; left ventricular-arterial coupling; eigenfrequency; cholesterol; atherosclerosis
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INTRODUCTION |
THE LEFT VENTRICLE CAN
BE described as an elastic chamber, the mechanical properties of
which change during the heart cycle. Similarly, compliance of the
arterial system is not constant, although often treated as such, but
varies because it is a function of distending pressure (1,
32). The coupling between the mechanical properties of the
left ventricle and aorta is a decisive element in reducing energy
requirements (5, 14, 21). From this energetic point of view, the cardiovascular system may aim at
optimizing the relationship between flow and pressure along these compartments.
Volume-pressure relationships are commonly described in terms of
compliance, elasticity, volume elasticity (VE), stiffness, or
impedance, and several recent studies have assessed such
volume-pressure relationships with regard to the cardiovascular system
(4, 16, 17, 19,
20, 27, 32). These previous
studies have significantly enhanced our understanding in this field;
however, they have not yet fully resolved the dynamic coupling between the heart and vasculature. This is due to the technically demanding requirements that have led to a rigid spatial and temporal focus. For
instance, some investigators analyze the left ventricle or the arterial
vessels (6, 13, 25), whereas
others study certain episodes, such as the telesystole
(29) or the isovolumic contraction period (6,
13).
In this study, we present a technique that allows determination of
several values of VE, which equal change in pressure over volume
(mmHg/ml). Measurements were made simultaneously for both compartments, i.e., the left ventricle and aorta. The steep pressure changes in left ventricular and aortic pressure induced oscillations of
a modified catheter-transducer system (CTS). The frequency and
dampening of these oscillations depend, on the one hand, on the
eigenfrequency, i.e., on the intrinsic dynamic characteristics of the
system and dampening of the measuring system. On the other hand, they
also rely on the properties of the heart and aorta. The shift in
oscillation frequency and dampening were used for calculation of VE. By
subdividing the dampened oscillation process into smaller segments, it
was possible to determine a corresponding amount of VE estimates. The
experiments were performed under closed-chest conditions in the
anesthetized rabbit.
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MATERIAL AND METHODS |
Experimental groups.
Rabbits (4 mo old) of both genders were used. They had access to food
and water ad libitum. The control group (n = 6) was fed
a standard pellet diet. The experimental group (n = 11)
received a daily 120-g ration consisting of a standard pellet mixture
with added 20 ml vegetable oil and 2 g of cholesterol. Cholesterol feeding was maintained for 12 wk and was terminated 2 wk before the experiment.
Surgical procedures.
Rabbits were anesthetized with intravenous 
-chloralose (60 mg/kg)
and urethan (200 mg/kg). Rectal temperature was maintained between
37°C and 39°C by a thermostat table. The trachea was cannulated via
tracheotomy; each animal breathed spontaneously throughout the
experiments. After surgery, the animal was placed in the supine position. A constant-flow infusion pump was connected to a cannula placed into the right brachial vein. An infusion of a mixture of 180 mmol/l of glucose and 180 mmol/l mannitol at 0.12 ml/min was used to
maintain constant plasma volume and osmolality. A catheter with a
length of 15 cm and an outer diameter of 2.5 mm was inserted into the
left ventricle via the carotid artery. An aortic catheter with similar
dimensions was positioned via the brachial artery.
Transducer.
Left ventricular and aortic pressures were measured with pressure
transducers (Siemens Elema, Sweden), as depicted in Fig. 1, A and B.
Analog-to-digital conversion was performed at a rate of 1,000 Hz and an
accuracy of 12 bits.
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Fig. 1.
A
schematic of the experimental setup. A: catheter-transducer
system (CTS) coupled to a hollow body (M is the effective
mass of the catheter-filling liquid, ET is the
volume elasticity of the transducer, Z is the friction
resistance of the catheter filling liquid, E'is the volume
elasticity of the hollow body, q is the oscillatory flow in
the catheter). B: electrical analog model of A. C: CTS connected to the additional parallel chamber for
decreasing volume elasticity, (EC is the volume
elasticity of the chamber). D: electrical analog model of
the latter setup is specified. E: an electrical analog model
of a CTS coupled to a hollow body with cylindrical opening. This setup
was used for modeling the ventricle during the ejection phase and
filling phase. Thus there is an additional mass component m (the blood
column in the aorta) and a friction resistance r.
R refers to the viscous resistance of the ventricular wall.
F: the analog model for E, after summarizing
r and R to a resulting r'.
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A chamber for decreasing the VE of the transducer was attached to this
setup (Fig. 1, C and D). This attached
compartment consisted of a short tube of a transparent material, to
allow bubbleless filling. Both sides of the tube were sealed with steel membranes, which, together with the wall of the tube, determine the
elastic properties of the transducer chamber (Fig. 1C). The resulting VE coefficient of the measuring device of ventricular pressure with the attached chamber was 27,950 mmHg/ml. The
corresponding VE for the measuring device of aortic pressure with the
attached chamber was [VE of the hollow body (E'v)]
29,400 mmHg/ml. VE of the transducer system was determined statically
by applying pressure and measuring the volume change. Moreover, a
dynamic VE measurement of the setup was made by applying external
pressure oscillations. The relative standard error for repeated
controls of measuring system VE was 3%.
Figure 2 shows a typical left ventricular
pressure curve with the oscillating component. The pressure induced by
the heart is superimposed by damped oscillations during ejection and
filling (Fig. 2A). The heart itself creates these
oscillations.
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Fig. 2.
A: original recording of left ventricular
pressure superimposed by eigenoscillations. B: left
ventricular pressure after separation from eigenoscillations (for
details see text). C: separated eigenoscillations of the
CTS.
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Data analysis.
For a detailed description of the theoretical background see the
APPENDIX. Left ventricular pressure was separated from the superimposed eigenoscillations as depicted in Fig. 2. To this end, the
transducer chamber was repeatedly disconnected from the measuring
setup, thus providing "pure" blood pressure traces without superimposed oscillations. A reliable reconstruction of the real blood
pressure time course was achieved by a combination of the pure
recordings and the original time series after smoothing. A second-order
Butterworth filter with a corner frequency of 18 Hz eliminated the
superimposed oscillations, however, at the cost of blunting the steep
pressure changes that occur during the isovolumic phases of the heart
cycle. The intersection points of the filtered and unfiltered pressure
recordings during ejection and filling agree well with the pure
pressure traces. Pressure at these intersections stems from left
ventricular pressure only. The time course of left ventricular pressure
was obtained by third-order spline interpolation of the intersection
values and two additional values before and thereafter. Subtraction of
left ventricular pressure from the original pressure signal provides
the damped pressure oscillation time course, as shown in Fig.
2C. Because each pressure is the product of the flow time
integral and the VE coefficient of the transducer (
terms in
Eqs. A2 and A4), the damped oscillation pressure
is divided by the VE value of the transducer and the first derivative
is calculated. This is then the experimentally determined flow
(f), which was used for calculation of
E'. The coefficients E', r'and the
initial value of the q2 were determined by a
least-square fit between experimentally determined f and the
calculated flow q1 (see Eqs. A4a and A4b). The starting values were E'= 300 mmHg/ml,
r'= 1 mmHg s/ml, initial value
q1 = f1 (first value
of f), initial value
dq1/dt = numeric derivative at
point f1, initial value
q2 = initial value
q1, and initial value
dq2/dt = initial value
dq1/dt. Fitting was performed by the
Levenberg-Marquardt algorithm (23). When this procedure
converged and the relative changes in all fit parameters became
<10
4, calculation was terminated and the determined
parameters were stored.
A time interval of 40 ms was chosen for fitting in which averaged
values for E'and r'were obtained. The step rate
was 10 ms.
The effective mass (M coefficient) cannot be fitted, because
various combinations of E'and M will yield the
same angular frequency. Thus the differential Eqs. A4a and A4b cannot be resolved unequivocally. Therefore,
M was determined by the aortic and atrial geometry. It must
be considered, however, that during the ejection phase, axial and
radial distension of the aorta occurs. The axial component tapers off
along the aorta. Thus only a short aortic segment can be taken for
estimating the mass. Aortic diameter was around 3 mm in the animals; 2 mm were assumed for the cylinder length. According to Eq. A1, the resulting mass is M = 0.0025 mmHg
s2/ml = 18.75 · 106
kg/m4.
Calibration of the method.
There is no "gold standard" for measuring elastic properties of
objects that geometry vary time dependently. The acquisition device,
mathematical modeling, and the software were therefore tested in vitro.
Various artificial hollow bodies with known VE were analyzed. An air
bubble of a defined volume was injected. VE of an air bubble can be
determined statically or it can be calculated by the laws of
thermodynamics. An air bubble of 1 ml has a VE coefficient
E'= 760 mmHg/ml, provided atmospheric pressure is 760 mmHg.
This value is valid only for isothermic compression or decompression
with small amplitudes. Because of the rapid oscillations of our
measuring system, this condition is not fulfilled. In fact, the value
for E'approaches 1,064 mmHg/ml, as under adiabatic
conditions. This factor of 1.4 results from the quotient
cp/cv (specific heat at constant
pressure/specific heat at constant volume) for a two atomic gas mixture.
The calibration with external hollow bodies required a piezoelectric
stimulation of the system. Thus a piezoelectric disk was fixed onto one
of the membranes of the attached transducer chambers
(Ec), as depicted in Fig. 1C. The
calibration chamber with the air bubble was then sealed. The
calculation of E'was based on Eq. A2.
To model a branched system, such as the heart attached to the aorta, a
calibration chamber was used containing a cylindrical borehole. One
wall of the chamber consisted of an elastic rubber membrane, whereas
the other wall was stiff. The calculations were based on Eqs.
A4a and A4b.
Statistical analysis.
The heart beats from each animal were analyzed during at least one
complete respiratory cycle. All data of a heart cycle were normalized
to yield a standardized heart beat with the relative duration of one.
These normalized heart beats of each animal during the complete
respiratory cycle were summarized to one resulting representative heart
beat for each group (see Figs. 5 and 6). The tick interval of these
figures is 0.05. For an average heart rate of 300 beats/min, this value
corresponds to 10 ms. The fitting procedure provided moving
average values every 10 ms; the window size was 40 ms.
All values are given as means ± SE. Significance was tested using
the Wilcoxon test for paired or unpaired observations. Differences in
values exceeding P < 0.05 were considered significant.
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RESULTS |
Effects of cholesterol feeding.
Table 1 shows control data for the
anesthetized rabbits under a normal (control) or cholesterol-enriched
diet (cholesterol). Heart rate and mean arterial pressure are
not statistically different between the normal and the atherosclerotic
rabbits.
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Table 1.
HR, MAP, body weight, plasma cholesterol, sodium, potassium, and
calcium in the control and cholesterol-fed rabbits
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None of the control animals shows signs of atherosclerosis. Conversely,
the aortas and carotid arteries from the cholesterol-fed rabbits
consistently reveal atherosclerotic lesions (data not shown). Moreover,
cholesterol plasma concentrations of these animals are drastically
elevated (Table 1).
Validation procedure.
Figure 3 shows the effect of air bubbles
of 0.5, 1, and 2 ml on the angular frequency and the resulting VE. For
this validation experiment, the electrical stimulation of the
piezoelectric disk generated the damped oscillation (Fig.
3A). Reducing the volume of air bubbles from 2 to 0.5 ml
causes an increase of angular frequency from 158 to 164.2 Hz (Fig.
3B). The precision of the calculation diminishes
with decreasing oscillation amplitude, i.e., the variations become
larger over time. Figure 3C shows the three different time
courses of the VE coefficient E'as obtained by Eq. A2. The time courses and absolute levels of angular frequency and
E'are practically identical. In these feasibility
experiments, the frequency alone contains all information of VE,
because an air bubble has no viscous properties and the damping
coefficient does not change.
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Fig. 3.
Validation of the volume elasticity (VE) estimation.
A: damped oscillations of a CTS coupled to an artificial
hollow body were induced by piezoelectric stimulation. Air bubbles of
different volumes were used as artificial hollow bodies. B:
angular frequencies of the damped oscillations from these different
bodies. C: the estimated VEs are depicted. D: the
correlation of the VE estimates vs. calculated VE.
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Figure 3D depicts the calculated values of E'vs.
the theoretical values for air bubbles. Also indicated is the result of
calculating a calibration chamber with rubber membrane and parallel
connection of a mass. The statically measured value of the rubber
membrane is 307 mmHg/ml; the calculated value from Eq. A4 is
300 ± 7.2 mmHg/ml. The correlation coefficient of all values in
Fig. 3D is 0.9997 and the gain is 0.91.
Time course of VE and viscous resistance.
Figure 4A depicts a time
course of left ventricular pressure with eigenfrequency oscillations
for 80 consecutive heart beats. The corresponding values for VE are
shown in Fig. 4B. Synchronous changes to the respiratory
cycle are apparent in both panels. Arrhythmic beats occur at beat
numbers 34 and 48, which also influences VE.
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Fig. 4.
Time course of ventricular pressure (A) and
ventricular VE (B). The values in B are smoothed
by a 40-ms averaging window. The time axis refers to 80 normalized
heart beats.
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Figure 5A shows the time
series of a normalized beat, averaged over one respiratory cycle, of
left ventricular and aortic pressures. In Fig. 5B, the left
ventricular VE is depicted. In the control group, there is a marked
increase in VE during the first half of the systole, which later tapers
off. The diastolic VE values for both groups are significantly lower
than those during the systole. Remarkably, VE continuously decreases
during the diastole and finally reaches a baseline level that amounts
to <20% of the average systolic levels.
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Fig. 5.
Time course of ventricular and aortic pressure
(A), ventricular VE (B), and aortic VE
(C). The values in B and C are
smoothed by a 40-ms averaging window. The time axis is normalized for
one heart beat. Open symbols indicate control animals
(n = 6), and closed symbols refer to the group
maintained on high-cholesterol diet (n = 11). + Significant changes vs. the initial value (P < 0.05).
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The cholesterol-fed animals reach maximum systolic VE earlier than the
controls. Furthermore, the changes in VE are not pronounced as in the
atherosclerotic group.
The changes in aortic VE are less than those of the left ventricle.
Furthermore, the changes seen during the first half of the systole are
in the opposite direction as for left ventricular VE. As depicted in
Fig. 5C, the control animals exhibit steadily declining
aortic VE values during the systole. Greater VE is observed during the
later one-half of the diastole. This dynamic response is perturbed in
the cholesterol-fed group. In fact, these animals reveal a slight
increase in VE during the systole. Diastolic changes are in the same
direction in both groups; however, the magnitude of changes is somewhat
less in the atherosclerotic animals.
The time course of left ventricular viscous resistance is similar to
the changes in VE (data not shown). In control animals at the onset of
the systole, viscous resistance is 0.16 mmHg s/ml. A maximum of 0.23 mmHg s/ml is reached at 0.35 of the normalized beat. There are no major
changes in systolic viscous resistance for the cholesterol-fed animals.
During the diastole, viscous resistance reaches lower values for both
groups, in resemblance to the VE time course. No consistent changes of
aortic viscous resistance are observed during the systole. Within the
diastole there was an increase to twofold levels; however, the values
had a greater stray than for VE.
In Fig. 6, all values are normalized by
considering the earliest systolic value as 1. After this procedure, the
pressure time courses of both groups overlap. Remarkably, on the other
hand, the VE data maintain their typical pattern. Basically, the same differences are observed as for the absolute data, i.e., the animals kept on an atherogenic diet have a blunted response in left ventricular VE. The differences in the aortic VE time course is also apparent. As
shown in Fig. 6C, the control animals decrease aortic VE
continuously during the systole. The cholesterol-fed animals, in
contrast, increase aortic VE significantly.
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Fig. 6.
Time course of normalized ventricular and aortic pressure
(A), normalized ventricular VE (B), and
normalized aortic VE (C). The values in B and
C are smoothed by a 40-ms averaging window. Open symbols
indicate control animals (n = 6) and closed symbols
refer to the group maintained on high-cholesterol diet
(n = 11). + Significant changes vs. the initial
value (P < 0.05). * Significant differences between
the groups (P < 0.05). rel, Relationship.
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DISCUSSION |
In the present study, analysis of eigenfrequency detuning proved
to be a feasible method to determine VE. The procedure was validated ex
vivo in an artificial setup. As depicted in Fig. 3D, the
estimations correlated very well with the theoretical values
(r = 0.9997, gain 0.91). The reason for having a gain
slightly less than the line of identity is due to the fact that strict adiabatic conditions cannot be achieved. In other words, there is a
temperature exchange between the tested hollow body and the surrounding
environment. Thus the experimental estimates are somewhat below the
theoretical values. The relative error of the experimental method was
around 3% for time invariant systems.
Considerable efforts have been made to model and determine mechanical
properties of the heart and aorta (1, 6,
13, 14, 18, 20,
24, 25, 27, 32).
The VE measure relies on several factors, such as wall stiffness, wall
thickness, heart chamber volume, and finally on the complex geometry of
the heart. VE is particularly important for hemodynamics because it
constitutes an immediate link between changes in pressure and volume.
The energy requirements for the propulsion of blood can in theory be
minimized if VE is adjusted appropriately.
The advantages of the current procedure to determine VE are that the
measurements are obtained in situ, that several determinations can be
made within one cardiac cycle, and that the values are assessed
simultaneously for the left ventricle and aorta.
As demonstrated by Fig. 3, the procedure accurately determines VE for
time invariant systems. Moreover, the procedure is very reproducible,
as shown in Fig. 4. Hemodynamic changes, such as the
respiratory rhythm or arrhythmic episodes, are mirrored in changes of VE. The VE time courses for the heart and aorta seen in
Figs. 4-6 are smoothed by a moving average procedure (window size
40 ms). Thus any sudden changes are underestimated. The scope of the
present analysis is therefore confined to changes observed among these
40-ms averages. No conclusions can be made regarding potential
instantaneous alterations in VE. However, changes in the mean values
cannot occur without corresponding alterations in the instantaneous
values. Increasing time resolution to <10 ms is possible; however, in
practice, this proved to be less reliable due to erratic variations and
the relatively coarse digitization. Nonetheless, using this setup, we
could demonstrate a specific change of mean values of ventricular and
vascular VE even within the systole and diastole (Figs. 4-6).
Absolute values for VE vary among previous studies (33).
The absolute values in Figs. 4 and 5 depend on the mass, as indicated
schematically in Fig. 1F. This mass was estimated and thus
the absolute values of VE rely on the accuracy of this approximation.
Aortic VE in control animals was found to be around 45 mmHg/ml, which
is slightly above the estimates of previous studies (33).
Comparisons of relative changes have the advantage that they are
independent of mass. The time course of these values, as shown in Fig.
6, underscores the typical pattern of the absolute data (Fig. 5). Peak
ventricular VE is reached during the middle of the systole, and
concomitantly aortic VE decreases. This behavior of VE for both
compartments should facilitate volume transport during the systole. In
the diastole, left ventricular VE decreases to <20% of the systolic levels, which eases the filling of the left ventricle. Conversely, diastolic aortic VE takes on greater values, thereby enhancing blood
flow to the periphery.
These findings support a concept of Vrettos and Gross
(32). They suggest that a decrease in aortic VE during the
systole reduces the energy demands on the heart. It is more economical to convey larger volumes without changing the pressure difference. This
effect is enhanced if VE increases at the source of volume, while VE of
the connected conduit vessels decrease, as observed in the present
study. We cannot, however, make any conclusions as to whether these
changes in cardiovascular VE are brought about actively or whether they
occur passively in consequence of contraction and ejection.
Our data, however, do not totally agree with a study of Berger and Li
(1). In that study, arterial elasticity reaches a minimum
near the start of ventricular ejection. In our hands, there is a
continuous decline in aortic VE throughout the entire systole. This may
seem in contrast to the general notion that vessel walls become stiffer
as transmural pressure increases (15). However, an inverse
relationship between VE and pressure may indeed be explainable by the
particular vessel wall architecture of large conduit arteries.
Joannides et al. (15) propose that changes in midwall
stress directly affect wall VE by unloading stiffer wall components.
Specifically, during the systole, a decrease in aortic VE may be
accomplished by the vascular smooth muscle cells, which unload the
stiff collagen fibers in favor of the more elastic elastin fibers. In
line with this interpretation, i.e., that elasticity can decrease
during vessel contraction, is the finding that arterial elasticity
declines during sympathetic stimulation in isolated vessels
(15). Remarkably, one only finds a strong relationship
between aortic pressure and diameter after chronic sympathetic blockade
(10). Other factors, such as nitric oxide formation, seem
unlikely to account for adaptations below the range of seconds.
Although the formation of nitric oxide is comparatively rapid, it is
still too slow to mediate changes on an intrabeat scale
(22).
To determine possible changes in the cardiovascular coupling of VE in
atherosclerosis, cholesterol feeding was performed in a similar manner
as in other investigations (2, 3). This diet
produces well-defined atherosclerotic lesions, as described in previous
studies by other groups (8, 9,
26) and ourselves (11). Although this feeding
regimen causes widespread lesions in the aorta, blood pressure and
heart rate do not change (Table 1). Nonetheless, cholesterol feeding
significantly blunted alterations in left ventricular VE.
Moreover, the pattern of aortic VE was perturbed (Fig. 6). We no longer
observed an inverse behavior between ventricular and aortic VE. Thus
the energy requirements for blood flow should be higher in these
animals. Several mechanisms can be put forward to explain the altered
response of aortic VE. These include structural and functional changes,
e.g., the aortas of hyperlipidemic rabbits typically exhibit less
lamellar elastin units (7). Functional alterations also
have been reported, such as the medial smooth muscle layer located
beneath the atheroma, which is hyperactive to vasoconstrictor agents
(12, 30). Furthermore, the endothelium reveals an impaired response to nitric oxide (12,
31).
Perspectives
This study employs a novel technique for assessing changes in left
ventricular and aortic VE throughout the systole and diastole. It is
suggested, that VE constitutes a dynamic element of great importance
within the cardiovascular control network. Changes in VE will have
immediate effects on volume propulsion. During the systole, an increase
in VE at the origin of blood flow (in this case the left ventricle)
together with a decrease in VE at its destination (the aorta) occurs.
This increases the pressure difference and thus augments flow. Hence,
vessel and chamber elasticity play an important role in optimizing
hemodynamic demands. In the diastole, the aortic compartment becomes
the source of blood flow. Aortic VE increases during this phase, which
again facilitates blood flow to the peripheral segments. Left
ventricular VE, on the other hand, decreases in the diastole, which
supports the filling phase. If ventricular filling were to occur only
passively, then VE should increase exponentially! After
high-cholesterol feeding, the interplay between the left ventricular
and the aortic VE is perturbed, suggesting that the energy requirements
for maintaining blood flow is higher.
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APPENDIX |
Theoretical background of measurements.
As mentioned in MATERIAL AND METHODS, a liquid-filled
oscillating system was coupled to the heart and aorta. This was done to
continuously monitor VE of these compartments. Changes in cardiac and
aortic elasticity also affect the dynamic properties of this oscillating system, i.e., after connecting the oscillating system (with
known eigenfrequency and dampening characteristics) to the heart
chamber, small volume oscillations were induced by the heart. In other
words, by adding the cardiac elastic components to the system, the
oscillating frequency and the dampening of the measuring device change.
What are the characteristics of the liquid-filled system used? As for
all mechanical oscillating systems, there is an elastic component and a
mass. The elastic element of the measuring device consisted of a
chamber with an elastic wall. The VE coefficient (E) is a
characteristic of the chamber. In liquid-filled systems, the so-called
effective mass replaces the physical mass. In contrast to physical
mass, the effective mass relies on the fluid compartment. The effective
mass (M) in a cylinder or a tube is determined by
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(A1)
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The effective mass has the unit kg/m4 equals 0.133 109 mmHg s2/ml, where 
is the density of
the liquid, l is the length of the fluid cylinder, and
A is the cross-section area of the cylinder.
In addition to mass, fluid friction plays a role and can be determined
by Poiseuille's law. The frequency of the volume oscillations should
on the one hand be high enough to induce several cycles during the
heart period, and on the other hand it must not be too high for
initiating oscillations in the system. The wavelength decreases with
increasing frequency. If the wavelength is greater than the dimensions
of the hollow body, then all wall components oscillate with the same
phase and each of the spatially distributed elastic, mass, and friction
components can be considered as concentrated elements. Thus in this
case, the hollow body is represented by only one value for VE, friction
resistance, and mass.
Every catheter-transducer system (CTS) is an oscillating system in the
above-described sense. Figure 1A shows a schematic CTS
coupled to a hollow body. The liquid in the catheter oscillates between
the transducer and the hollow body. Because of incompressibility of
liquid, the small-volume perturbations induce volume oscillations of
the heart. In the thin catheter, the oscillating fluid reaches greater
values of velocity and acceleration, and therefore the mass becomes
effective only here. Figure 1B shows the electrical analog
model of the CTS. The differential Eq. A2 is evident in this
model. The sum of all partial pressure components in a mesh must be
zero, provided the system is allowed to oscillate freely
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(A2)
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with the solution
where ET is the VE of the transducer,
E' is the VE of the hollow body, M is the
effective mass of the catheter, Z is the friction resistance
of the catheter, and q is the oscillatory flow in the catheter.
The solution of this differential equation is a damped oscillation with
two key parameters: 1) the damping 
and 2) the
angular frequency 
. The angular frequency contains the unknown
parameter E', beside the three known constants:
ET, M, and Z. If
ET 
E', which is normally the case
for conventional pressure measurements, then E' has no
influence on . For this study, we wished to determine the
sensitivity of . Thus it was necessary to decrease
ET of the transducer. Figure 1C
depicts this arrangement schematically. A liquid-filled chamber with a
VE coefficient EC lays parallel to the
transducer and decreases the resulting E
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(A3)
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In the presented experiments, the transducer chambers took on
EC values near 30,000 mmHg/ml. This is much less
than for any modern pressure transducer. Thus under these conditions
the E' of the heart chamber is able to influence .
This simple model allows the calculation of VE, provided that the
hollow body is always sealed. In situ measurement of ventricular VE,
however, is more complicated. The heart chamber opens during the
ejection and filling phases. In consequence, the oscillating volume
branches off, thereby eliciting radial oscillations of the hollow body
and axial volume shifting in the adjoining blood vessel. Thus an
additional mass (m) and friction component (r) lay parallel
to the heart (under the premise that the aorta is considered as a tube)
(28). Figure 1E depicts this arrangement schematically. This model contains four unknown elements: m,
r, R, E'. R represents the
viscous properties of the left ventricular wall. Both resistances
r and R can be summarized to a resulting r'. Figure 1F depicts this model with its
remaining three unknown components. As can be derived from Fig.
1F, two differential equations for the two meshes can be
formulated
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(A4a)
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(A4b)
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where q1 is the oscillatory flow mesh 1 and q2 is the oscillatory flow mesh 2.
The differential Eqs. A2, A4a, and A4b
have solutions that are damped oscillations symmetrical to zero.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: T. Wronski, Tucholskystr. 2, D-10117 Berlin, Germany (E-mail: thomas.wronski{at}charite.de).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Received 19 July 1999; accepted in final form 25 February 2000.
| |
REFERENCES |
1.
Berger, SD,
and
Li JK-J
Temporal relationship between left ventricular and arterial system elastances.
IEEE Trans Biomed Eng
39:
404-410,
1992[ISI][Medline].
2.
Bocan, TM,
Mueller SB,
Mazur MJ,
Uhlendorf PD,
Brown EQ,
and
Kieft KA.
The relationship between the degree of dietary-induced hypercholesterolemia in the rabbit and atherosclerotic lesion formation.
Atherosclerosis
102:
9-22,
1993[ISI][Medline].
3.
Braun, M,
Hohlfeld T,
Kienbaum P,
Weber AA,
Sarbia M,
and
Schror K.
Antiatherosclerotic effects of oral cicaprost in experimental hypercholesterolemia in rabbits.
Atherosclerosis
103:
93-105,
1993[ISI][Medline].
4.
Burkhoff, D,
Alexander J,
and
Schipke J.
Assessment of windkessel as a model of aortic input impedance.
Am J Physiol Heart Circ Physiol
255:
H742-H753,
1988[Abstract/Free Full Text].
5.
Burkhoff, D,
and
Sagawa K.
Ventricular efficiency predicted by an analytical model.
Am J Physiol Regulatory Integrative Comp Physiol
250:
R1021-R1027,
1986[Abstract/Free Full Text].
6.
Campbell, KB,
Wu Y,
Kirkpatrick RD,
and
Slinker BK.
Left ventricular pressure response to small-amplitude, sinusoidal volume changes in isolated rabbit heart.
Am J Physiol Heart Circ Physiol
273:
H2044-H2061,
1997[Abstract/Free Full Text].
7.
Dalessandri, KM,
Bogren H,
Lantz BM,
Tsukamoto H,
Bjorkerud S,
and
Brock J.
Aortic compliance in hypercholesterolemic Watanabe rabbits compared with normal New Zealand controls.
J Invest Surg
3:
245-251,
1990[Medline].
8.
Daley, SJ,
Herderick EE,
Cornhill JF,
and
Rogers KA.
Cholesterol-fed and casein-fed rabbit models of atherosclerosis. Part 1: differing lesion area and volume despite equal plasma cholesterol levels.
Arterioscler Thromb
14:
95-104,
1994[Abstract/Free Full Text].
9.
Daley, SJ,
Klemp KF,
Guyton JR,
and
Rogers KA.
Cholesterol-fed and casein-fed rabbit models of atherosclerosis. Part 2: differing morphological severity of atherogenesis despite matched plasma cholesterol levels.
Arterioscler Thromb
14:
105-141,
1994[Abstract/Free Full Text].
10.
Federici, A,
Nocera L,
and
Dambrosio M.
Sympathetic control of aortic pulse pressure pattern.
Funct Neurol
7:
41-44,
1992.
11.
Flemming, B,
Encke T,
Roloff D,
Seeliger E,
and
Wronski T.
Respiratory and cardiovascular responses to hypoxia, hyperoxia and perfusion of the carotid arteries in cholesterol-fed rabbits.
In: Chemoreceptors and Chemoreceptor Reflexes, edited by Acker H,
Trzebski A,
and O'Regan RG.. New York: Plenum, 1990, p. 399-407.
12.
Galle, J,
Busse R,
and
Bassenge E.
Hypercholesterolemia and atherosclerosis change vascular reactivity in rabbits by different mechanisms.
Arterioscler Thromb
11:
1712-1718,
1991[Abstract/Free Full Text].
13.
Honda, H,
Koiwa Y,
Takagi K,
Kikichi J,
Hoshi N,
Takishima T,
and
Butler JP.
Noninvasive measurement of left ventricular myocardial elasticity.
Am J Physiol Heart Circ Physiol
266:
H881-H890,
1994[Abstract/Free Full Text].
14.
Hunter, W,
and
Noordergraaf A.
Can impedance characterize the heart?
J Appl Physiol
40:
250-252,
1976[Abstract/Free Full Text].
15.
Joannides, R,
Richard V,
Moore N,
Godin M,
and
Thuillez C.
Influence of sympathetic tone on mechanical properties of muscular arteries in humans.
Am J Physiol Heart Circ Physiol
268:
H794-H801,
1995[Abstract/Free Full Text].
16.
Kubota, T,
Itaya R,
Alexander J,
Todaka K,
Sugimachi M,
and
Sunagawa K.
Autoregressive analysis of aortic input impedance: comparison with Fourier transform.
Am J Physiol Heart Circ Physiol
260:
H998-H1002,
1991[Abstract/Free Full Text].
17.
Laskey, WK,
Parker HG,
Ferrari VA,
Kussmaul WG,
and
Noordergraaf A.
Estimation of total systemic arterial compliance in humans.
J Appl Physiol
69:
112-119,
1990[Abstract/Free Full Text].
18.
Little, WC,
and
Freeman GL.
Description of LV pressure-volume relations by time-varying elastance and resistance.
Am J Physiol Heart Circ Physiol
253:
H83-H90,
1987[Abstract/Free Full Text].
19.
Liu, Z,
Brin KP,
and
Yin FCP
Estimation of total arterial compliance: an improved method and evaluation of current methods.
Am J Physiol Heart Circ Physiol
251:
H588-H600,
1986[Abstract/Free Full Text].
20.
Mitchell, GF,
Pfeffer MA,
Westerhof N,
and
Pfeffer JM.
Measurement of aortic input impedance in rats.
Am J Physiol Heart Circ Physiol
267:
H1907-H1915,
1994[Abstract/Free Full Text].
21.
O'Rourke, MF.
Steady and pulsatile energy losses in the systemic circulation under normal conditions and in simulated arterial disease.
Cardiovasc Res
1:
313-326,
1967[Abstract/Free Full Text].
22.
Persson, PB.
Modulation of cardiovascular control mechanisms and their interaction.
Physiol Rev
76:
193-244,
1996[Abstract/Free Full Text].
23.
Press, WH,
Flannery BP,
Teukolsky SA,
and
Vetterling WT.
Numerical Recipes in Pascal. New York: Cambridge University Press, 1994, p. 547-598.
24.
Quick, CM,
Berger DS,
and
Noordergraaf A.
Apparent arterial compliance.
Am J Physiol Heart Circ Physiol
274:
H1393-H1403,
1998[Abstract/Free Full Text].
25.
Shishido, T,
Sugimachi M,
Kawaguchi O,
Miyano H,
Kawada T,
Matsuura W,
Ikeda Y,
Sato T,
Alexander J, Jr,
and
Sunagawa K.
A new method to measure the regional myocardial time-varying elastance using minute vibration.
Am J Physiol Heart Circ Physiol
274:
H1404-H1415,
1998[Abstract/Free Full Text].
26.
Simonsen, U,
Ehrnrooth E,
Gerdes LU,
Faergemann O,
Buch J,
Andreasen F,
and
Mulvany MJ.
Functional properties in vitro of systemic small arteries from rabbits fed a cholesterol-rich diet for 12 weeks.
Clin Sci (Colch)
80:
119-129,
1991[Medline].
27.
Stergiopulos, N,
Segers P,
and
Westerhof N.
Use of pulse pressure method for estimating total arterial compliance in vivo.
Am J Physiol Heart Circ Physiol
276:
H424-H428,
1999[Abstract/Free Full Text].
28.
Stergiopulos, N,
Westerhof BE,
and
Westerhof N.
Total arterial inertance as the fourth element of the windkessel model.
Am J Physiol Heart Circ Physiol
276:
H81-H88,
1999[Abstract/Free Full Text].
29.
Sunagawa, K,
Sugimachi M,
Todaka K,
Hayashida K,
Itaya R,
Chishaki A,
and
Takeshita A.
Optimal coupling of the left ventricle with the arterial system.
Basic Res Cardiol
88:
75-90,
1993.
30.
Tagawa, H,
Tomoike H,
Mitsuoka W,
Satoh S,
Kuga T,
Shimokawa H,
Nakamura M,
and
Takeshita A.
Hyperreactivity of aortic smooth muscle to serotonin is related to the presence of atheroma in Watanabe heritable hyperlipidaemic rabbits.
Cardiovasc Res
27:
2164-2169,
1993[Abstract/Free Full Text].
31.
Tagawa, H,
Tomoike H,
and
Nakamura M.
Putative mechanisms of the impairment of endothelium-dependent relaxation of the aorta with atheromatous plaque in heritable hyperlipidemic rabbits.
Circ Res
68:
330-337,
1991[Abstract/Free Full Text].
32.
Vrettos, AM,
and
Gross DR.
Instantaneous changes in arterial compliance reduce energetic load on left ventricle during systole.
Am J Physiol Heart Circ Physiol
267:
H24-H32,
1994[Abstract/Free Full Text].
33.
Westerhof, N,
and
Elzinga G.
Normalized input impedance and arterial decay time over heart period are independent of animal size.
Am J Physiol Regulatory Integrative Comp Physiol
261:
R126-R133,
1991[Abstract/Free Full Text].
Am J Physiol Regul Integr Comp Physiol 279(2):R539-R547
0363-6119/00 $5.00
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