Varying coronary
volume will vary vascular resistance and thereby have an effect
on coronary hemodynamics. Six ventricular septa were isolated from
anesthetized dogs, dispersed in a biaxial stretch apparatus at
diastolic stress, and perfused artificially with an oxygenated
perfluorochemical emulsion at maximal vasodilation. Flow and thickness
were measured continuously by an electromagnetic flow probe and
sonomicrometer. Pressure was varied sinusoidally around
30, 50, and 70 mmHg with an amplitude of 7.5 mmHg; frequencies ranged
between 0.015 and 7 Hz. Bode plots of admittance (flow/pressure) and
capacitance (scaled thickness/pressure) were constructed. A
two-compartment model was used in which the resistances vary with
volume. Realistic values of microvascular compliance (~0.3 ml · mmHg
1 · 100 g1) were found. Values 10 times higher were then
found when resistances were forced to be constant. We concluded that
volume dependence of resistances have to be taken into account when
dynamic or static pressure-flow relations are studied and conceal the
effect of a large intramyocardial compliance on arterial hemodynamics.
intramyocardial compliance; model; transfer function; admittance; pressure-dependent resistance
 |
INTRODUCTION |
CHARACTERIZATION of the dynamic interrelationship among
coronary arterial pressure, flow, and volume is essential to our
understanding of coronary physiology not only in relation to coronary
resistance and capacitance but also to myocardial oxygen consumption
and control of coronary flow (3, 11, 12). Resistance and capacitance are the two physical parameters relating pressure and flow and are
clearly dependent on variations of intravascular volume (16). However,
because it is difficult to accurately measure intravascular volume, the
results of many studies are implicitly or explicitly interpreted as if
resistance and/or capacitance were constant, at least in certain
working conditions (7). When coronary pressure is changed, diameters of
vessels of all size vary with pressure, indicating that their
resistances vary as well (21). The assumption of constant resistance
during variations in arterial pressure may be responsible for some of
the controversy in this field (4, 13, 22, 23, 27, 28).
Although some of the determinants of intravascular volume such as
perfusion pressure (25), contraction (15, 18, 31), and vasomotor tone
(3, 12) are recognized in a general sense, little information is
available about the dynamics of the volume changes. The available data
consist primarily of either direct or indirect estimates of the time
constant of volume change after a perturbation. For example, based on
the time course of arterial pressure change after coronary occlusion, a
time constant of ~1.8 s was estimated (29). Similar values were
obtained from comparison of the differences of the time integrals of
the coronary arterial inflow and venous outflow (9, 20, 31). A
recent study using digital subtraction angiography of an intravascular
contrast agent in isolated dog interventricular septa (19) directly
measured intravascular volume change after a step change in pressure
and found a time constant of ~3-4 s. The only detailed studies
on the dynamics of the coronary vasculature examined input impedance (7, 30) over the frequency range of 2-10 Hz. On the basis of the
time constant estimates, however, this frequency range is likely to be
too high to provide useful information on volume responses
because the volume changes of the microcirculation occur over a much
longer time frame.
The hypothesis of this study is that dynamic coronary flow responses to
changes in perfusion pressure are affected by the continuous change of
resistance as dictated by its volume dependence. Hence, an average
resistance value at a certain mean perfusion pressure would not
suffice; one has to take into account the resistance variations as
induced by the pressure variations around that mean as well.
| |
METHODS |
Specimen Preparation
We used the canine isolated, perfused interventricular septal
preparation previously described (26). Briefly, mongrel dogs of either
sex weighing 18-22 kg were anesthetized with intravenous pentobarbital sodium (35 mg/kg). The animals were intubated and ventilated, and the heart was exposed via a midline sternotomy. Heparin
sodium (5,000 U) was infused to minimize thrombi formation. Each dog
was systemically cooled to 28°C, at which time the heart was
arrested by rapid injection into the ascending aorta of cold (4°C)
cardioplegic solution with the composition (in mM) 120 Na+,
16.0 K+, 16 Mg2+, 1.2 Ca2+, 160.4 Cl, 10 HCO3, and
1.0 adenosine. The heart, which usually fibrillated and then became
asystolic within 1-2 min, was removed, and the septal artery was
quickly cannulated and connected to a reservoir so that the specimen
was continually perfused with cold cardioplegic solution for the
duration of the preparatory time. The perfusion pressure was kept <30
mmHg throughout the preparatory steps to minimize tissue edema.
The left anterior descending, left circumflex, and right coronary
arteries were individually cannulated and filled (by injection via a
syringe) with dental rubber to which a few drops of catalyst had been
added. Once the dental rubber filled the smallest visible arteries on
the surface of the heart wall, the left and right ventricular free
walls were removed by cutting within the perfusion boundary of these
embolized arteries. We have verified that this methodology results in
an isolated, perfused septal bed with no shunts or leaks (26). Four
edges of the septum, roughly defining a rectangle, were connected by a
series of threads in a trampoline-like fashion to the carriages of a
biaxial stretching apparatus so that the specimen could be stretched in
the base-to-apex and circumferential directions [see Fig. 1 in
Resar et al. (26)]. The forces in each direction were measured by
force transducers (model sf-10; Interface) mounted on the carriages.
Septal thickness was measured with a pair of sonomicrometer crystals
(Triton Technology, La Jolla, CA) glued onto the left and right
ventricular surfaces of the septum. The crystals were placed in an area
adjacent to the area demarcated by the markers. Two platinum pacing
wires connected to a stimulator (model S88; Grass Instruments, Quincy, MA) were sewn onto the outer edges of the specimen. After the septum
was mounted onto the stretching apparatus, the cannula was connected to
a pressurized reservoir with stiff tubing. The air pressure in the
reservoir was controlled by an electrically controlled pneumatic needle
valve connected to a signal generator. Flow at the entrance to the
septum was measured by means of a 1-mm-inner diameter cannulating flow
probe (model 774-100-2.0-1.0; Skalar Medical, Delft, The Netherlands)
connected to a flowmeter (model 1401, Skalar Medical). The inlet
pressure in a plastic T piece connecting the cannula and tubing was
measured with a micromanometer (model PC-450; Millar Instruments,
Houston, TX).
The cold cardioplegic perfusate was then changed to a perfluorochemical
emulsion (FC-43; Green Cross, Osaka, Japan) with the following
composition of the aqueous solution (in mM): 137 Na+, 5.2 K+, 2.6 Ca2+, 2.1 Mg2+, 1.54 H2PO4, 119 Cl, 25.8 HCO3, and
11.5 glucose, along with 2 mg/100 ml adenosine and 5.8 g/100 ml
albumin. Adenosine was sufficient to fully dilate the vascular bed. The
emulsion was perfused at room temperature. The perfusate was gently
bubbled with 95% O2-5% CO2 to maintain
pH between 7.4 and 7.5 and to keep oxygen tension >600 mmHg. After a
few minutes the specimen could be electrically stimulated to beat, at
which time the pressure was increased sufficiently to produce diastolic
flows of ~25 ml/min. The specimen was paced at a rate of 0.4 Hz for
~30-45 min to allow recovery and stabilization from the
cardioplegia. Sufficient lidocaine was added to the perfusate so that
no spontaneous contractions ensued if the specimen was not electrically
stimulated. All measurements were done in noncontracting conditions and
at diastolic stretch of the septum.
Protocols and Data Analysis
Tissue thickness as an index of vascular volume.
In six septa we used our previously reported digital subtraction
angiographic method (19) to directly measure intravascular volume while
also measuring tissue thickness during some interventions to verify
that thickness could serve as an index of volume. Briefly, the specimen
was suspended under an X-ray tube and video camera system. An
approximately 1-cm2 region without any visible venous
effluent was identified and demarcated by four lead markers. These lead
markers also served to correct for scatter and veiling glare when the
images were analyzed. Tissue thickness immediately adjacent to this
region was measured by sonomicrometer crystals glued to the right and left ventricular surfaces. A calibration wedge consisting of a chamber
with linearly increasing depth was placed in-line with the perfusion
system and positioned under the X-ray tube next to the specimen. Mask
images were first taken at a nominal input pressure of 60 mmHg, with
the specimen slightly stretched to biaxial forces of 300 g (to prevent
it from sagging). An emulsion of ethiodol (10 ml) containing 475 mg/ml
iodine that had been previously sonicated for 2 min at 100 W (model
W-385; Heat System-Ultrasonics, Farmington, NY) in the presence of 10 ml of physiologically balanced solution with 5 g/100 ml Pluronic F-68
was then added to the perfusate after an equal volume of the original
perfusate was removed. We previously verified that the spheres of
emulsion prepared in this manner (~3 µm in diameter) were too large
to leak out of the vasculature (19). Once the iodine-containing
perfusate equilibrated in both the wedge and specimen, we imposed an
inflow occlusion and/or a constant amplitude sinusoidal input pressure.
For the latter, the input pressure was varied sinusoidally with a
peak-to-peak amplitude of ~40 mmHg around a mean pressure of 60 mmHg
at frequencies of ~0.002, 0.04, and 0.4 Hz. For both interventions,
simultaneous X-ray images and thickness measurements were recorded. The
data from the X-ray images were converted into an equivalent thickness of iodine according to previously reported methods by using the wedge
as a calibration device (19). Dividing this iodine thickness by the
thickness of the specimen yields the vascular volume as a percentage.
Coronary dynamics.
In another six specimens, comprising the main set of studies, the
dynamic responses of flow and thickness (i.e., vascular volume) to
pressure perturbations were examined. With the specimen completely
unloaded, we imposed constant amplitude (15 mmHg peak to peak)
sinusoidal pressure oscillations at frequencies ranging from ~0.015
to 10 Hz. The perturbations were performed at mean pressures
(Pp) of 30, 50, and 70 mmHg. After each change in frequency and mean pressure, the system was allowed to equilibrate for a few
minutes. The order of the mean pressures was chosen arbitrarily. In the
lower frequency range (0.015-0.5 Hz) the data were digitized at a
sampling rate of 10 Hz. The higher frequency (0.75-10 Hz) data
were digitized at a sampling rate of 100 Hz. The amplitude of the
pressure oscillations was chosen to provide sufficient resolution for
the thickness signal and yet still be in the linear response range of
flow and thickness variations as forced by the pressure variations.
Linearity of responses was verified in several specimens by imposing
pressure amplitudes varying from ~4 to 24 mmHg peak to peak at
frequencies of 0.2 and 2 Hz at each mean pressure.
A segment of digitized pressure, flow, and thickness data, consisting
of an integer number of waves, was analyzed with custom software.
First, any baseline drift was removed by applying a linear
interpolation algorithm to the entire data set. The data at each
discrete frequency were then centered about the mean, after which at
least two complete periods from a steady-state condition at each
frequency and pressure were extracted. These extracted files were
further analyzed to obtain the amplitude and phase angle of the
pressure, flow, and thickness by spectral analysis using the MATLAB
fast Fourier algorithm. Amplitude and phase of each variable resulted
in complex expressions of those variables as P(
), Q(), and
Th(), respectively, where P is pressure, Q is flow, and Th is
thickness, and is the frequency in Hz.
To correct for the pressure drop from the reservoir to the tip of the
cannula, we first measured the system impedance with the cannula open
to air. Measured P() [P()measured] was
then corrected for the pressure drop using Q() and the perfusion
system impedance to obtain the real perfusion pressure
[P()]. Unless mentioned otherwise, perfusion pressures
reported are pressures corrected for the cannula influence.
A few times between changes in perfusion pressures, the perfusion line
was occluded for 50 s. The peripheral pressure at the end of the
occlusion was taken as the value for PZF used in some calculations as described below.
After the experiment, the perfusion area was delineated by Evans blue,
cut out of the septum, and weighed. Flow was then normalized to 100 g
of tissue weight.
Descriptive parameters for Bode plots.
The dynamic responses are reported in terms of transfer functions in a
standard Bode plot format. The pressure-flow (i.e., admittance)
transfer function consists of plots of the ratio of flow to pressure
modulus versus frequency and the difference between the flow and
pressure phase angles versus frequency. Similarly, the pressure-volume
(i.e., capacitance) transfer function consists of the ratio of volume
to pressure modulus versus frequency and the difference between the
volume and pressure phase angles versus frequency. Volume was
calculated from thickness by applying a correction factor derived from
the fitting procedure explained below. To facilitate interspecimen
comparisons, a few key descriptive parameters were defined as shown in
Fig. 1. For the admittance transfer
function we averaged the moduli in the low-frequency (<0.2 Hz) range
and performed a linear regression of moduli versus frequency in the
high-frequency (1-9 Hz) range. The frequency at which these two
lines intersected was denoted as the corner frequency
(Ac). The lines fitted to the data in the high-frequency range for the three pressures intersected at a common frequency, which
we defined as the "crossover" frequency (x; not
shown in Fig. 1). The phase angle responses were described by the
maximum phase, the frequency at which the maximum occurred
(max), and the frequency at which the phase angle
crossed zero after reaching its maximum (0).
View larger version (22K):
[in this window]
[in a new window]
|
Fig. 1.
Typical Bode plots of admittance and relative ratio of thickness to
pressure for canine septum and definitions of parameters for
descriptions of these curves. Admittance at frequency (left) is defined as Q()/P(), where Q is arterial flow
and P is pressure. Modulus of admittance (top left) is rather
constant up to a corner frequency (Ac). For higher
frequencies, the relation between modulus and is approximated by a
straight line, the slope of which is also used to parameterize the
admittance plot. Phase ( ) of admittance (bottom left) starts
out at 0 but increases to a maximum value (max) at a
frequency max. Th()/P() (right), where Th
represents thickness divided by its mean, is equal to capacitance
multiplied by a constant. Above a certain corner frequency
(Cc), with modulus
|Th()/P()| Cc, the log-log plot
of modulus of |Th()/P()| (top
right) decreases linearly with frequency. A linear curve for lower
frequency also was fitted through the data. Modulus at = 0.05 (|Th()/P()|=0.05) was
determined as an additional descriptive parameter. The phase-frequency
plot of the Th()/P() curve (bottom right) was not
parameterized.
|
|
Similarly, the modulus of the capacitance transfer function was
parameterized by fitting the data with linear regressions in the
low-frequency (0.015-0.2 Hz) and high-frequency (0.5-9 Hz)
ranges. The intercept of the low-frequency regression at a frequency of
0.05 Hz, the slope of the low-frequency data, the slope of the
high-frequency data, and the corner frequency (Cc) of
the intersections of these two lines are reported. The phase of the
pressure-volume response could not be simply parameterized, and these
results will be reported implicitly by the parameters reported below.
Fitting of Bode plots by second-order transfer functions.
The Bode plots were fitted using the following second-order transfer
functions, which were chosen because of the structure of the model
outlined below (see Fig. 2)
where
Gq and 
1-4 are
parameters to be determined, GThV is the conversion
factor for expressing volume in terms of thickness, 
is frequency in
radians per second, and j is an imaginary unit. The two
equations were simultaneously fitted to the experimental data by
minimizing the following cost function
The factors e1i-e4i are
relative differences between the theoretical and experimental
values of modulus and phase of the admittance and capacitance at the
different frequencies as detailed in the APPENDIX;
n is no. of frequencies, and i is index. Note
that is expressed in radians/s and that = 2
, where is
the frequency in Hz. The value of "cost" was minimized by varying
the parameters of the admittance and capacitance functions. The
contributions to the cost function for frequencies <1 Hz were
1.5 times the contributions for higher frequencies.
View larger version (9K):
[in this window]
[in a new window]
|
Fig. 2.
Two-compartment model of septal circulation. The division between the 2 compartments is somewhere within the middle resistance
(Rm). Each compartment consists of a
resistance-capacitance-resistance network. Note that
Rm is thought to be divided between the 2 compartments. Q1, Qm, and Q2 are
flows through R1, Rm, and
R2, which are resistances within the model.
C1 reflects the capacitance of the proximal
compartment and is thought to be determined predominantly by the larger
arteries and larger resistance vessels. Capacitance
C2 reflects the compliance of the microcirculation
predominately. Pin and P0 are inlet and outlet
pressures; PC1 and PC2 reflect pressures over
C1 and C2; and QC1
and QC2 are flows over C1 and
C2. R1, Rm,
and R2 are assumed to vary with the load (volume)
of the capacitances but are forced to be constant for the results
presented in Figs. 9 and 10.
|
|
We used the technique of simulated annealing (1) to find the parameters
that optimally fit the data. Using this technique, we first defined the
range over which each parameter was allowed to vary. Randomly selecting
a value for each parameter within its range defined an initial vector
of parameters. The cost function for this vector was calculated. A
parameter was then randomly selected and its value changed randomly but
within 5% of its original range, resulting in a new vector and a new
value for the cost function. This new parameter vector was accepted as
a new start at a certain probability value (P). Another
parameter was then randomly selected and varied within 5% of its
current value, and so on, until the procedure had been repeated 500 times. From this group of accepted vectors the one with the minimum
cost value was selected as a new starting value for another repetition
of 500 calculations, each with a decrement in P value. When the
cost no longer decreased, the procedure was stopped. The associated parameter vector is called a candidate vector. The whole procedure was
repeated 10 times to generate 10 candidate vectors. From this set of
10, the vector with the lowest cost function value was deemed to be the
best parameter fit to the data. If the cost function had a unique
solution, these 10 parameter vectors would be about equal; otherwise, a
scatter in parameter values would be found.
Two-Compartment Model of Coronary Circulation
We chose a two-compartment model consisting of a network of
resistors and capacitors as depicted in Fig. 2. The capacitances are
intended to reflect those of the proximal and distal portions of the
bed, i.e., the arteries and arterioles and the capillaries and venules, respectively.
To account for the volume dependencies of the resistances and
capacitances during the sinusoidal changes of pressure around a mean,
the resistances and capacitances were assumed to vary linearly around
their nominal working values. That is, around nominal
working values of resistance (R0) and capacitance
(C0), the resistances and capacitances were allowed
to vary linearly according to the formulas R(V) = R0 + KR · 
V and C(V) = C0 + KC · V, where V is the
variation in volume around the working value of volume
(V0). The equations describe the tangent lines at the
R(V) and C(V) curves at the coordinates
(R0,V0) and
(C0,V0), respectively. The K
values are the slopes of these lines and quantify the "sensitivity
constants," KR and KC. Note
that these approximations are only allowed for limited variations in
resistances and capacitances.
Because the model consists of two compartments, there are two
sensitivities for capacitance (KC1 and
KC2) and two capacitance working points
(C1,0 and C2,0). Similarly,
there are four sensitivity values for resistance
(K1, Km1,
Km2, and K2) and three
resistance working points (R1,0,
Rm,0, and R2,0). Hence, the
middle resistance (Rm) depends on the volume in
both the first and second compartment. Note that when the
sensitivity constants are zero, the model reduces to the more classic
model with constant resistances and capacitances.
The Simulink program within MATLAB was used initially to evaluate the
nonlinear model and to assess the role of the different parameters.
These simulations resulted in several conclusions. 1) We found
that the sensitivities of the capacitances to volume had minimal
effects on the simulated transfer functions at a fixed mean perfusion
pressure. Specifically, moduli and phases of the transfer functions
changed only by tenths of a percentage over a wide range of values of
KC. With larger values of KC,
exceeding the range that seems realistic, the model became unstable.
Consequently, KC1 and KC2 were
assumed to be zero. 2) For a wide range of values of
KR and realistic values of resistances and volumes,
flow and volume variations changed linearly with pressure variations
for amplitudes 
15 mmHg, not withstanding the dependence of resistance on volume. Hence, in terms of signal analysis the responses remained linear, not withstanding our expectation for nonlinear responses based
on the variations in resistance values induced by the pressure variations. 3) Inductance calculated on the basis of
geometrical properties of arteries and the specific mass of blood
affected the transfer functions only for frequencies >7 Hz.
As outlined in the APPENDIX, the model results in more
parameters than can be obtained with the use of the admittance and
capacitance equations provided above. Therefore, additional relations
are needed to obtain an estimation of parameters of the model from the
parameters of the fitting equation. For this purpose, we used the
additional constraint Pmean/Qmean = R1 + Rm + R2. The assumption that the resistance of each of
the compartments depends on volume according to the law of Poiseuille
yields an additional equation. By further assuming that only the radii
of the vessels vary with volume, we may write R =
A/V2, and therefore KR = dR/dV = 
2A/V3 = 2R/V,
where dR/dV is the first derivative of the resistance-volume relationship and A is assumed to be constant and, according
to Poiseulle, is equal to 8
l2, where is viscosity and l is length. Hence, for each
KR an equation is found. Note that absolute volume
now appears in the equations for the model. Therefore, the model
enables one to estimate absolute volumes, whereas this cannot be done
when resistances are assumed constant during the pressure variations
(all K values = 0). Nevertheless, although four additional
equations are defined, two more unknowns (the volumes in each
compartment) are added. The proportion of the intermediate resistance
ascribable to each compartment becomes an important parameter. Hence, a
factor X had to be introduced to define which fraction of
Rm was sensitive to the volume in the proximal
capacitance. Consequently, (1 X) is the fraction of
Rm that is dependent on the volume in the distal
capacitance. Hence, for X = 0, Rm is only
sensitive to the distal capacitance, and for X = 1, Rm is only sensitive to the proximal capacitance.
The resulting set of equations is still overparameterized because
we have a total of 10 equations and 12 unknowns. Therefore, as
discussed later, rather than finding values for each parameter, we
obtain relationships between some parameters. Some additional constraints will then lead to a unique set of parameters coupled to
each set of Bode plots. Because volume will be predicted from the
parameter set, an independent test of the validation of the model is obtained.
Statistical Analysis
Statistical analysis of the correlation between X-ray contrast
agent thickness and tissue thickness was performed using standard linear regression analysis. Analyses of each parameter describing both
the admittance and capacitance transfer functions were performed using
one-way repeated-measures ANOVA. Pairwise comparisons for effects at
the three mean pressures were made using the Student-Newman-Keuls test. Statistical significance was defined to be the P
= 0.05 level.
| |
RESULTS |
Tissue Thickness as an Index of Vascular
Volume
Figure 3, top, shows a
representative recording of the input pressure along with simultaneous
iodine and tissue thicknesses during an inflow occlusion. Note that the
two thicknesses initially fall with the same time course. At about 120 s into the occlusion the iodine thickness plateaus, whereas the tissue
thickness continues to fall. Because the iodine is contained in
the emulsion, which is too large to leave the vasculature, the
continued decrease in tissue thickness in the absence of inflow and any
further change in vascular volume likely represents continued emptying
of the interstitial volume. Figure 3, bottom, shows the iodine
and tissue thickness responses of the same specimen to approximately
sinusoidal pressure inputs with 40-mmHg peak-to-peak amplitudes at a
mean pressure of 50 mmHg at 0.01 and 0.04 Hz. Note the similar
responses for both thicknesses and that the amplitudes of both are
lower at the higher frequency.
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 3.
Typical responses of septal thickness (tissue thickness) as measured
by sonomicrometer and iodine thickness of the septum (iodine
thickness) to occlusion of the perfusion line (top) and
sinusoidal variations in perfusion pressure at 2 different frequencies,
0.01 and 0.05 Hz (bottom). Note that only after 160 s does a
discrepancy occur between the 2 thickness measurements. Iodine
thickness stabilizes, whereas septum thickness keeps decreasing,
probably because of decreasing interstitial volume. Note that with
sinusoidal variation in perfusion pressure, the amplitude of both
thickness signals is lower at higher frequency. Pressure amplitude was
taken rather high to obtain accurate measurements of iodine thickness.
As a result, higher harmonics are present in thickness signals at lower
frequency.
|
|
Figure 4 demonstrates the regression lines
between the iodine and tissue thickness variations for the
interventions shown in Fig. 3. For the inflow occlusion, only
the data before the point at which the tissue and iodine responses
begin to deviate are included. The results of the regression analysis
for all of the specimens subjected to at least one of these
interventions are summarized in Table 1.
There was a uniformly high degree of correlation between the iodine and
the tissue thickness variations. Note that the specimen that was
subjected to both the sinusoidal and occlusion interventions had slopes
of nearly equal value.
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 4.
Correlations between iodine thickness and tissue thickness, as
measured with sonomicrometer, of data shown in Fig. 3. Freq, frequency.
Data points for time >160 s after occlusion are not
incorporated. Note the good correlations between dynamic signals.
|
|
Coronary Dynamics
Bode plots.
Typical results of the linearity tests for pressure, flow, and
thickness changes at three mean pressures (Pp) are shown in Fig. 5. For pressure amplitudes 10 mmHg
(20 mmHg peak to peak), the amplitudes of flow and thickness were
proportional to pressure amplitude. Hence, the amplitude of 7.5 mmHg
chosen for this study was well within the linear response range.
View larger version (29K):
[in this window]
[in a new window]
|
Fig. 5.
Check of linearity of flow and thickness responses to pressure
variations. Pressure amplitude at 2 frequencies were varied from 2 to
12 mmHg for 3 different perfusion pressures (Pp = 30, 50, and 70 mmHg). Note difference in scaling of thickness axis between the
2 frequencies. For determination of Bode plots, a pressure amplitude of
7.5 mmHg was taken, well within linear range of responses.
|
|
Figure 6 shows representative results in
one specimen of the input pressure and the flow and thickness responses
for three different frequencies at a mean pressure of 50 mmHg.
Note that the magnitude of the thickness response decreases, whereas
the magnitude of the flow response increases, with increasing
frequency.
View larger version (17K):
[in this window]
[in a new window]
|
Fig. 6.
Typical response of flow (Q1) and thickness to pressure
variations around a mean of 50 mmHg with an amplitude of 7.5 mmHg at
= 0.01, 1, and 2 Hz. Note that there was a mean pressure drop over
the cannula of 5 mmHg. With increasing frequency the thickness
amplitude decreased but flow amplitude increased. Note that at 0.01 Hz
there is no phase difference between flow and pressure, but thickness
lags pressure by a few degrees. At 1 Hz, flow leads pressure by 16°
and the lag of thickness toward pressure has increased to 65°.
These phase differences are even higher at 2 Hz.
|
|
The characteristic features of the responses are more completely
illustrated by the admittance and capacitance transfer functions at all
three pressure levels as shown in Fig. 7.
The pressure dependence of the responses is evident. Specifically, the
low-frequency admittance modulus increases with increasing pressure,
but the slopes of the moduli in the high-frequency range decrease with increasing pressure. This results in a crossover frequency at 1 Hz and
an increase of the corner frequency with pressure. The maximum value of
the phase angle and 0 decrease with increasing pressure,
but the frequency of the maximum phase angle is only slightly dependent
on pressure. Note that the phase angle changes sign at a frequency
close to 10 Hz, which is caused by inertial effects playing a role only
for frequencies >7 Hz. Similarly to the admittance transfer function,
there is some pressure dependence of the capacitance transfer function,
particularly in the low-frequency range. The intercept of the
regression for the low-frequency moduli decreases and the slope becomes
less negative as the pressure increases. However, neither the corner
frequency nor the slopes of the moduli in the high-frequency range are
pressure dependent. There is, however, no clear pressure dependence of
the phase angle responses.
View larger version (33K):
[in this window]
[in a new window]
|
Fig. 7.
Pressure dependence of Bode plots on mean arterial pressure. Note
increased modulus of capacitance ( Th/P GThV)
and decreased modulus of admittance ( Q/P ) at low frequency when
inlet pressure is decreased. For frequencies higher than
Cc, there is no effect of pressure on the capacitance
function. For higher frequencies, the phase of the
capacitance-frequency relation decreases with decreasing pressure. Note
that at high frequencies both modulus and phase of admittance increase
with perfusion pressure. Curves shown are best fits with admittance and
capacitance equations. GThV, conversion factor for
thickness to volume.
|
|
Responses similar to those shown in Fig. 7 were observed in all six
specimens. Tables
2-5
summarize the various descriptive parameters obtained from the
admittance and capacitance transfer functions as defined in Fig. 1, at
conditions defined in Table 6.
Typical fits of the experimental data by the admittance and capacitance
equations over the frequency range of 0.01-5 Hz are also
demonstrated in Fig. 7. The sets of 10 candidate parameters for each
specimen generated by the procedure of annealing are reported in Fig.
8. The means with standard deviations are provided. Note
that there is considerable interspecimen variation but that, with some
exceptions, the parameters are defined rather uniquely by the
estimation procedure. For the results derived below, the parameters
obtained with the curve of best fit were used.
View larger version (16K):
[in this window]
[in a new window]
|
Fig. 8.
Parameters of fitting equations for 6 septa as estimated by minimum
annealing technique. Each estimation procedure for 1 pressure resulted
in 10 parameter estimates. Data are expressed as means ± SD for each
parameter (Gq,
1-4). Note that within each septum
the SD of estimated parameters is rather small but that there is a
considerable intraseptal variation. exp, Experiment.
|
|
Parameter estimation assuming that resistances do not vary with
volume.
Figure 9 provides estimates of the parameters of the
model using the fitting parameters of Fig. 8 and assuming that the
sensitivities of the resistances for volume are zero. Note that in all
septa R1 increased and Rm
decreased with increasing perfusion pressure.
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 9.
Coefficients of model in Fig. 2 as derived from parameters of curves of
best fit ( values), with resistance constants (K) taken to
be zero. Note that in all septa the value of first resistance
(R1) increases and Rm decreases
with increasing Pp. Note widespread and high values of
estimates for distal compliance. Resistance and capacitance (or
compliance) values are expressed per 100 g of tissue; lin, linear.
|
|
In each specimen the sum of the resistances decreased as pressure
increased. However, the sum of resistances was smaller than (Pp PZF)/Q, which again was smaller
than Pp/Q. For the average data, the results from
these resistance calculations are provided in Fig. 10
as a function of perfusion pressure.
View larger version (13K):
[in this window]
[in a new window]
|
Fig. 10.
Total septal resistance (Rtot) as calculated by
several methods categorized by Pp in cases in which
K values are taken as zero. In this case
Rtot = R1 + Rm + R2 and is the inverse of
low-frequency admittance. Low-frequency impedance (inverse of
admittance) drops 15% on average from a linear Pp increase
from 30 to 70 mmHg (open symbols). Rtot should also
equal total pressure drop divided by flow. The assumption that outflow
pressure (P0) is zero results in a drop in
Rtot of 40% over this range of Pp
(solid symbols). Assuming that outflow resistance equals the pressure
at the end of a 50-s occlusion (PZF),
Rtot still decreased by 30% (shaded symbols). Note
that for each individual septum (indicated by differently shaped
symbols) at same Pp, the model with K values of
zero predicts the lowest resistance; the resistance based on zero
outflow pressure is higher, and the resistance based on PZF
is highest.
|
|
Parameter estimation assuming that resistances vary with volume.
Figure 11 demonstrates the relationships between
Rm and most of the other parameters for different
values of X at a pressure of 50 mmHg for the same specimen
whose data are shown in Fig. 7. As shown in the APPENDIX,
R1 is independent of Rm and
X and is therefore not plotted. Note that for a certain range
in Rm unrealistic values for parameters are found.
For example, at low values of Rm, the volume of the
first compartment (V1) becomes negative and
K1 and Km1 are positive. These
values are not realistic, and hence this range of
Rm can be rejected. Likewise, very large values of
Rm result in positive values of
Km2 and negative values for volume of the second
compartment (V2), which again is unrealistic. Finally, for
the lower and upper value of Rm, either both
C1 and C2 or
C2 alone becomes infinitely large. With a lack of
additional constraints, the minimum of the
Rm-C2 relationship was used to determine the actual value of Rm. That, along with
arbitrarily choosing X = 0.75, allowed all parameters to be
estimated uniquely. This value of X implies that
Rm is assumed to be more influenced by the proximal
compartment than by the distal compartment. These criteria are in sync
with realistic constraints on the values for V1 and
V2.
View larger version (22K):
[in this window]
[in a new window]
|
Fig. 11.
For a typical Bode plot at a pressure of 50 mmHg, the estimated
dependence of parameters on Rm are provided for
several values of X, the weighing factor of how the
sensitivities of Rm on load in
C1 and C2 are divided. For
X = 0, Rm is only influenced by distal
volume; for X = 1, Rm is only influenced by
proximal volume. Because R1 was independent of
Rm, these curves are not provided.
C1 and C2, capacitances of 1st
and 2nd compartment (in
ml · mmHg1 · 100 g tissue1); V1 and V2,
volumes (in ml/100 g tissue) in capacitances C1 and
C2; K1 and
Km1 (in
mmHg · ml2 · s1 · 100 g2), dependence of R1 and
Rm on volume in C1;
Km2 and K2, dependence of
Rm and R2 on volume in
C2. Because negative volumes are physically not
realistic, there is only a limited range of Rm that
is realistic. To make a unique choice of Rm
objective, the value was estimated at minimal value for
C2. Arbitrarily, X = 0.75 was chosen.
Symbols indicate minima of C2 for different values
of X. Note that V1 would be infinitely high or low
for X = 0 and V2 for X = 1.
|
|
Figure 12 depicts the estimated parameters as a
function of perfusion pressure. In all experiments,
R1 increased and Rm decreased with increasing perfusion pressure. R2 at
Pp = 70 mmHg was always lower than at Pp = 30 mmHg. The capacitance of the first compartment at Pp = 70 mmHg was higher than that at 30 mmHg except for experiment 4.
The capacitance of the second compartment was larger than that of the
first compartment. As shown in Fig. 12B, in most specimens the
dependency of total resistance on volume decreased with increasing perfusion pressure. Total volume, the sum of the volumes of the two
compartments, increased with perfusion pressure, with one experiment
yielding an unrealistic value because it exceeds tissue volume. Note
that the absolute values of K for this experiment at this
pressure are very low. A small absolute error in K therefore results in a very large change in estimated volume.
View larger version (1516K):
[in this window]
[in a new window]
|
Fig. 12.
A: parameter results for model, assuming that resistances
are dependent on volume and have been obtained according to criteria
outlined in text. Note that value for R1 is
independent of these criteria, whereas C1 is hardly
independent; C2 is ~20 times larger than
C1.
B: the dependence of resistances on volume becomes less at
higher Pp. There are 2 experiments that deserve separate
attention. The septum represented by circles resulted in strong
negative values for sensitivities of resistance on volume. This is the
experiment with a higher PZF value (10.3 mmHg).
Notwithstanding the high absolute values of the K constants,
the capacitances (in
ml · mmHg1 · 100 g1) and volumes (in ml/100 g) estimated from
this experiment are not very different from the means. The larger
pressure dependencies of Rm and
R2 are in agreement with these larger values of
K. The experiment represented by hexagons has K
constants that are very small, 0.1, 0.04, 0.3, and
0.6
mmHg · ml2 · s1
for K1, Km1,
Km2, and K2, respectively, for
a pressure of 70 mmHg. These small values result in a very inaccurate
estimate of volume. The estimate of absolute volume by the model is
independent of any volume calibration but follows from the Bode plots
directly. Calibration of volume is indirect from the flow signal. The
range between 5 and 14 ml/100 g is realistic for the intramural
coronary circulation. Resistance is expressed in
mmHg · ml1 · s · 100 g.
|
|
| |
DISCUSSION |
The data clearly demonstrate that the admittance and capacitance
transfer functions are dependent on the level of mean pressure. A
two-compartment model of the coronary bed was employed to help interpret the results. Resistances and capacitances were estimated in
two conditions. With the first condition, resistances were assumed not
to vary with volume; with the second condition, the sensitivities of
resistances were allowed to vary and were then estimated as well. The
values of the first resistance and capacitance were rather similar for
both conditions. However, when resistances were volume dependent, the
middle resistance and outflow resistance were much higher (2 times) and
the distal compliance much lower (10 times) than in the condition in
which resistances were assumed constant at a mean perfusion pressure.
The vascular volume of the septa calculated from the estimated
parameters for which resistances were assumed to be volume dependent
appeared to be pressure dependent and in a realistic range, indicating
that the model assumptions are not unrealistic.
Critique of Experimental Method
Interstitial volume changes were not accounted for with the
interpretation of the data. However, if interstitial volume changed during our interventions, it did so sufficiently slowly so as not to be
dominantly manifested in the thickness signal in the frequency range
applied. However, such fluid exchange may explain why the thickness
responses did not reach steady values for the frequencies of ~0.01
Hz, as predicted by the model results. Moreover, after completing a set
of measurements at one mean pressure, we always repeated some
measurements at several frequencies and verified that, within the error
of the technique, the results were similar. Thus, even if interstitial
volume changed during a set of measurements at a given pressure, it did
not affect the correspondence between thickness and intravascular
volume changes.
If a crystal were to be positioned over a large vessel, some or all of
the thickness variations would manifest as pulsations of the vessel
walls rather than as volume changes in the small vessels. We took care
to avoid these large vessels when placing the crystals. The hallmark of
such a problem is a variation of thickness in phase with perfusion pressure.
The relations between iodine thickness variation and tissue thickness
variation do not have a slope of unity. The major reason is that the
regions of interest for tissue thickness and iodine thickness
measurements were not at identical physical locations on the
specimen. Nevertheless, the high correlations
strongly support our contention that intravascular volume changes are
linearly indexed by the overall tissue thickness changes. Although the thickness changes may not be a one-to-one measure of volume changes, an
exact relationship is not needed for estimates of capacitance and
volume from our data analysis. These values are obtained from the
frequency dependence of both the flow and thickness responses to
sinusoidal perfusion pressure variations. Absolute volume is the result
of integration over time of the calibrated flow signal by the function
of the capacitance.
VanHuis et al. (30) concluded that wave reflections should be taken
into account when admittance for frequencies >7 Hz is interpreted.
Moreover, as described in METHODS, inertial effects were
discernible at >7 Hz. Hence, 7 Hz was chosen as the upper limit
for the frequency range subjected to quantitative analysis.
Critique of Model and Parameter Estimation
The admittance curves alone would be able to deliver all the fitting
parameters because all values are present in this fitting equation.
However, the uncertainty of the distal parameters
(Rm, R2, and
C2) would increase because the more distal they
are, the less influence these factors have on the admittance curves. On the other hand, the capacitance curves do not provide the same information as the admittance curves because the capacitance equation is missing the quadratic term in the nominator. However, the
capacitance curves strongly relate to the distal parameters. Hence,
fitting the two relations in one procedure optimizes the determination of all parameters involved. However, in doing so, a less satisfactory fit of the admittance at some frequencies may be the result of the
attempt to fit the capacitance data. The phase angle of the capacitance
function was the one with the largest contribution to the cost function.
The present model of the coronary circulation is not the first to
assume volume dependence of resistance. Earlier models (2, 6, 8, 24)
predicted well several aspects of hemodynamic behavior of the coronary
circulation. However, because of the specific form of the nonlinear
pressure-volume and volume-resistance relationships assumed, those
models were not well suited for parameter estimations. This was the
reason for our use of the approach of piecewise linearization for the
resistance-volume and capacitance-volume relations around a working
point. The analysis with Simulink then demonstrated that Bode plots
generated by the model were hardly influenced by the sensitivity
constant of capacitance to volume. Therefore, for fitting the
experimental results in the model, only the dependence of resistance on
volume was accounted for.
Note that we substituted the sensitivity of resistance to volume with
dR/dV in the working point according to Poiseuille's law. This
is only allowed when it is assumed that the vessel lengths are not
affected by the pressure variations. In fact, it is this substitution
that allows for the determination of absolute volume in this study. It
should be noted that if the sensitivities of resistances are taken as
zero, it is impossible to estimate absolute volume as a function of
perfusion pressure. Only volume variations around a working point can
then be determined.
A limitation of the present model is that only two compartments are
considered. The distal compartment reflects the capillary and venous
bed, whereas a differentiation between the two probably would be
better. However, more compartments imply more parameters, and without
additional measurements this is not very useful. Moreover, the shapes
of the transfer functions do not suggest that a higher-order model is
applicable. It should be appreciated that in a compartmental model
distributed functions are lumped into single compartments.
It may very well be that the two lumped compartments each reflect a
portion of the coronary circulation that is different at different
pressures. The estimates of the first resistance and, in most cases,
the first capacitance increase as pressure increases. Increasing
perfusion pressure may change not only the caliber of vessels but also
the distribution of resistance and capacitance. An increase in
perfusion pressure decreases the resistance and capacitance per vessel
in the proximal bed, but the pressure effect is transmitted farther
into the bed. Thus, more distal vessels, with greater resistance than
the proximal ones, are incorporated in series into the proximal bed.
This more than compensates for the decrease in resistance caused by
distention of the proximal vessels, so the net effect is an increase in
overall resistance of the proximal bed.
This explanation for the increasing value of R1
with increasing mean perfusion pressure seems to be in conflict with
the assumption of constant vessel length in the application of
Poiseuille's law to define the sensitivity of resistance on
volume. However, we assume that for the variations around
the working points the effect of diameter changes will be much stronger
than the variations in vessel lengths to be considered. However, this
aspect certainly deserves further study in the future.
As explained in METHODS the model has two parameters too
many compared with the experimental information we have. In relation to
Fig. 11, we discussed the rationale for the choice of additional constraints, such as the requirement that volumes have to be positive, allowing the unique determination of all parameters in the model. Obviously, one can think of additional constraints resulting from different experiments. For example, one can find reasonable estimates for the volume of the proximal large and small arteries. This volume
does not exceed 3.5 ml/100 g for the full proximal coronary circulation
including the large epicardial arteries (27) at a perfusion pressure of
100 mmHg. As is clear from Fig. 11, the criteria we chose fulfill this
particular constraint as well.
As discussed below, the total intravascular volume and its dependence
on perfusion pressure do agree with values published independently. We
have tried to perform the parameter analysis by entering the total
volume at a certain perfusion pressure as a known value. However, this
did not provide stable solutions of our equations, probably because the
value taken for volume was too far off that of the actual volume in the
specific experiment. Hence, in the absence of actual measurements of
additional parameters or signals in the same septum from which Bode
plots are constructed, one cannot do better than formulating additional constraints.
Although the problem of overparameterization is not solved completely,
the ranges of model parameters are narrowed such that any uncertainty
of estimation of a parameter from a single data set due to the
uncertainty in constraints arrives in the 10-20% range. The
consistent application of criteria makes it likely that trends found in
the parameter estimation reflect physiological ones. However, with
the interpretation of absolute values presented in this study, the
limitations left by overparameterization should be taken into account.
Comparison with Literature
There are only a few studies of the dynamic response of the coronary
vasculature with which to compare our results. Canty et al. (7)
measured the input impedance (from 1 to 10 Hz) of the circumflex
arterial bed in intact hearts during prolonged diastole. Over this
frequency range our findings of increasing flow amplitude with
increasing frequency and a larger change at lower mean pressure, as
well as a peak in phase difference around 4 Hz, are all concordant with
their results. Canty et al. concluded that the input impedance could be
described well by a first-order model representing only one compartment
with a characteristic time constant on the order of a few tenths of a
second. That model contained resistances independent of volume. Their
conclusion does not contradict our present results. The authors
concluded that a rather high back pressure had to be assumed for the
model. With constant resistances, our model fit finds a very large
distal compliance, which causes a virtual constant back
pressure for the first compartment in the frequency range studied by
Canty et al. It should be noted that these authors believed that the course of the admittance Bode plot was determined by viscoelastic effects of small arteries (7). However, such effects have not been
found in capacitance measurements on isolated small arteries (14) and
were therefore not accounted for in our model.
Coronary arterial input impedance was also reported by VanHuis et al.
(30), albeit at only one perfusion pressure. Using an impulse response
technique, which is better suited for examining high than low
frequencies, they studied a frequency range 20 Hz and found results
similar to ours and those of Canty et al. (7). Because the impedance
was not different between diastole and systole, they concluded that, at
these frequencies, little information about the deeper portions of the
vasculature could be obtained.
There are no data on the frequency dependence of vascular volume with
which to compare our results. However, the nearly monotonic decrease in
capacitance transfer function with increasing frequency beginning at
~0.1 Hz suggests a time constant for volume changes of several
seconds. This corresponds with previously reported time constants
found, for example, in studies in which intramural blood volume changes
were measured indirectly by integrating the difference between arterial
and venous flows (9, 27, 31). These data are also consistent with the
slow decay of volume after steps in perfusion pressure demonstrated
previously with the use of the same intravascular Roentgen contrast
method used in the present study.
The value estimated for the distal capacitance is in concordance with
other estimates. For example, Bosman et al. (5) demonstrated that
capillary diameter in rabbit tenissimus muscle may change by 5%
between control and arterial occlusion and by 14% between control and
maximal reactive hyperemia. This implies a total change in volume of
38%. The capacitance value to be calculated from this diameter change
depends on the assumed capillary pressure change with this
intervention. With the assumption of a 20-mmHg variation in capillary
pressure between occlusion and peak reactive hyperemia and an average
volume fraction of 10 ml/100 g, microcirculatory capacitance may be
3.8/20 = 0.19 ml · mmHg1 · 100 g tissue1. Moreover, Morgenstern et al. (25)
demonstrated that total coronary volume changed with arterial pressure
by 0.075 ml · mmHg1 · 100 g1.
Because the majority of volume change is within the microcirculation, this change in volume is brought about by a much smaller variation in
microvascular pressure. With a distal resistance in the order of 20%
of total resistance, the pressure variations would be only 20% of the
arterial pressure variations, and hence microvascular compliance would
be ~0.375
ml · mmHg1 · 100 g1. One may therefore conclude that the value for
the microvascular capacitance found in the present study is not
unrealistic, at least when the volume dependence of resistance is taken
into account. Disregarding this volume dependence results in estimates
of distal compliance that are 10 times higher and therefore unrealistic.
The estimates of total intravascular volume range between 4 and 14 vol% depending on perfusion pressure and not including the one data
point predicting a vascular volume exceeding tissue volume. These
values agree well with other data in the literature. In freshly excised
heart, intramural blood volume is ~4 ml · mmHg1 · 100 g1
(10) but can increase up to 14 ml · mmHg1 · 100 g1
at 100 mmHg, depending on perfusion pressure and the degree of autoregulation (27).
Interpretation of Data
In the experiments we could change the amplitude of pressure variations
15 mmHg around a mean Pp of 50 mmHg without finding the
introduction of a higher harmonic with an amplitude higher than that
expected on the basis of the higher harmonics in the pressure signal.
Also, in the Simulink model and with the use of parameters
corresponding with the parameters found from the fits with the model,
pressure amplitudes could be increased up to 15 mmHg and yet higher
harmonics in flow and volume would remain lower than 5% compared with
the base frequency. Because of the volume dependence of resistance in
the model and in the circulation, one would expect a nonlinear relation
between pressure on one hand and predicted flow and volume on the other
hand. However, we have to conclude that a linear relation between the
different variables at play does not necessarily imply that resistances therefore have to be assumed constant in the working range of these
variables. Vice versa, having resistances and capacitances that are
volume dependent does not imply that relationships between variables
have to be nonlinear. In other words, although our model consists of
physical elements that each give rise to nonlinear input-output
relations, the combination of these elements into one system results in
a linear dependence of output signals (flow and volume) on variations
over a wide range of input signals (pressure).
Because there is sufficient evidence that vessels with a significant
resistance in the coronary bed do vary in diameter, and therefore in
their resistance value, it seems justified to require that an analysis
of coronary hemodynamics incorporates this effect.
To gain insight into the dependence of the transfer functions on the
volume sensitivities of resistance, we set some sensitivities equal to
zero while keeping all other model parameters at the same value. A
typical result of such an analysis is shown in Fig. 13,
in which the solid lines represent a fit with the admittance and
capacitance equations. The corresponding model parameters are also
provided in Fig. 13. Consequently, if only K2 is
made equal to zero, i.e., only the outflow resistance is not volume dependent, then the predicted admittances are hardly altered. In
contrast, the amplitude of the capacitance transfer function increases
by a factor of two. Hence, the volume sensitivity of the distal
resistance is of particular importance in estimating the distal
compliance. In addition, if the other K</
TOP
ABSTRACT
INTRODUCTION
