Vol. 282, Issue 1, H244-H255, January 2002
Comparative analysis of aortic impedance and wave reflection
in ferrets and dogs
R.
Burattini1,2 and
K.
B.
Campbell2
1 Department of Electronics and Automatics, University of
Ancona, 60131 Ancona, Italy; and 2 Department of Veterinary and
Comparative Anatomy, Pharmacology and Physiology, Washington State
University, Pullman, Washington 99164-6520
 |
ABSTRACT |
Our modified version of the T-tube arterial model
(consisting of two parallel, loss-free transmission paths terminating
in lumped loads of complex and frequency-dependent nature) was applied to experimental measurements of ascending aortic pressure and of
ascending and descending aortic flows taken from dogs and ferrets. Our
aim was to provide quantitative evaluation of the aortic pressure and
flow pulse wave components as they relate to the distribution of
arterial properties and relate to wave travel and reflection in
mammalians of consistently different size and shape. Estimated effective lengths (distances to effective reflection sites) of the
head-end (dh) and body-end
(db) transmission paths were ~12 and 30 cm,
respectively, in the dog and 6.5 and 13 cm, respectively, in the
ferret. These lengths and distributions of estimated arterial properties were consistent with the difference in the body size and
with the more central location of the heart in the ferret's body than
it is in the dog's body. In both animal species the ascending aortic
pressure and flow waves could be interpreted in terms of forward and
reflected components arising from the two distinct effective reflection
sites, although the higher
dh/db ratio in the ferret
determined the presence of one broad, indistinct minimum in the modulus
of ascending aortic impedance in the frequency range from 0 to 10 Hz,
rather than two distinct minima as observed in the dog.
arterial pulse wave propagation; arterial effective reflection
sites; arterial compliance; T-tube arterial model
| |
INTRODUCTION |
ARTERIAL WAVE
REFLECTION is generally recognized as an important phenomenon
affecting pressure and flow pulse contour from the ejecting ventricle
(14, 16, 18-22, 27, 29). Differences in aortic
pressure waveforms and input impedance characteristics among various
animal species have been attributed to differences in wave reflection
as they relate to animal size and shape (1, 17, 18).
Specifically, the size of the animal determines transmission path
length. Because there is little variation in wave velocity among
animals, size (through its effect on transmission path length) also
determines the time of arrival of reflected waves at the aortic root.
An essential aspect of arterial wave reflection, as seen from the
aortic root, is whether reflected waves appear to arise from one or two
functionally discrete reflecting sites. This issue has long been a
matter of discussion (3, 6, 7, 9, 12, 15, 18, 24). The
presence of two arterial effective reflecting sites (one being the
result of all arterial terminations in the upper part of the body, and
the other the result of all terminations in the lower part of the body)
was first suggested and conceptually explained in terms of an
asymmetric T-tube model by McDonald (16, 18) and
has been used by O'Rourke and co-workers (1,
17-21) as a conceptual basis for explaining arterial
impedance patterns and pressure and flow wave shapes in a variety of
animals and in humans. Recently, it has been shown that a modified
T-tube formulation with loss-free transmission paths terminating in
lumped loads of complex and frequency-dependent nature is necessary for identifying the parameters of physiological interest, for interpreting wave travel and reflection along the descending thoracic aorta and,
eventually, for discriminating between proximal and distal mechanical
properties of descending aortic circulation (3-6, 9, 11,
25). However, these reported applications of modified T-tube
model were performed on dogs. To ascertain the applicability of this
model within the mammalian kingdom, a comparative study in dogs and
ferrets is performed in the present study.
The ferret has a decidedly different body size and shape than the dog.
The typical 1.6-kg male ferret is a long slender animal with short
legs, whereas a typical 25-kg dog is much bulkier and longer limbed.
Additionally, the ferret differs from the dog in having a heart more
centrally placed along the long axis of the body. Thus our T-tube
model-based comparison of arterial hemodynamics in dogs and ferrets was
expected to provide quantitative evaluation of the aortic pressure and
flow pulse wave components as they relate to distribution of arterial
properties and wave travel and reflection in mammalians of consistently
different size and shape.
| |
GLOSSARY |
| AV-M |
Aortic valve to mandible distance
|
| AV-T |
Aortic valve to base of tail distance
|
| ci |
Compliance per unit length of head-end (i = h) and body-end
(i = b) transmission tube
|
| cidi |
Head-end (i = h) and body-end (i = b) tube
compliance
|
| CLi |
Head-end (i = h) and body-end (i = b) terminal
load compliance obtained from viscoelastic windkessel representation
|
| Ct |
Total T-tube model compliance
|
| Ci |
Overall head-end (i = h) and body-end (i
= b) compliance
|
| di |
Length of head-end (i = h) and body-end (i
= b) transmission tube
|
| HR |
Heart rate
|
| li |
Inertance per unit length of head-end (i = h) and body-end
(i = b) transmission tube
|
| lidi |
Head-end (i = h) and body-end (i = b) tube
inertance
|
| n |
Integer number that, in accordance with Fourier analysis of pressure
and flow waves, varies from 0 to 15-20
|
 |
Mean pressure in the ascending aorta
|
 |
Mean flow in the ascending aorta
|
| b/ |
Ratio of descending to ascending aortic mean flow
|
| Rdi |
Viscous resistance of viscoelastic windkessel model assumed to
represent head-end (i = h) and body-end (i =
b) terminal load
|
| Rp |
Total peripheral resistance
|
| Rpi |
Zero-frequency impedance of head-end (i = h) and body-end
(i = b) terminal load
|
| T |
Heart period
|
| Ts |
Left ventricular ejection period
|
| W |
Body weight
|
| Zci |
Characteristic impedance of head-end (i = h) and body-end
(i = b) transmission tube
|
Zi(jn ) |
Head-end (i = h) and body-end (i = b) input
impedance
|
| ZLi(jn) |
Head-end (i = h) and body-end (i = b) terminal
load impedance
|
 Li(jn) |
Head-end (i = h) and body-end (i = b) terminal
load reflection coefficient
|
| v |
Descending aortic pulse wave velocity
|
 i |
One-way wave transit time across the head-end (i = h) and
body-end (i = b) transmission tube
|
| zi,pi |
Time constants of head-end (i = h) and body-end (i
= b) terminal load
|
| |
Heart pulsation
|
| |
METHODS |
Experimental preparation.
The investigation conforms to the Guide for the Care and Use of
the Laboratory Animals (National Institutes of Health Publication No. 85-23, Revised 1985).
Six adult male ferrets were anesthetized with an intramuscular
injection of ketamine and rompun (20 and 4 mg/kg, respectively). A
tracheotomy was performed and positive-pressure ventilation instituted
using a Bird Mark 10 respirator and a nonrebreathing inhalation
anesthetic system. A surgical level of anesthesia was maintained using
supplemental methoxyflurane, and the chest was opened with a midsternotomy.
In ferrets 1-5, flow probes (Transonic Systems;
Ithaca, NY) were placed snugly around the ascending aorta and the
descending thoracic aorta between the branching of the left subclavian
artery and the first intercostal arteries. Ascending aortic pressure was measured with catheter-tipped pressure transducers (Millar; Houston, TX) inserted through the left ventricular (LV) apex and positioned beyond the aortic valve at the location of the ascending aorta flow probe. In ferret 5, an additional simultaneous
measurement of pulsatile pressure was taken in proximity of
trifurcation by way of a catheter-tipped transducer inserted in the
left femoral artery. In ferret 5, the distance between sites
of pressure measurements was taken at the end of the experiment after
euthanasia (by administrating an overdose of barbiturate anesthesia)
but while the aorta remained longitudinally tethered to the position it
held in vivo. The distance between the proximal measurement site and
the terminal aortic region at the level of renal arteries was also
taken. In ferret 6, measurements of flow and pressure were
taken from the high descending thoracic aorta 2 cm below the left
subclavian artery.
Pressure and flow measurements were taken with the respirator turned
off. Digital data were collected online using a Hewlett-Packard 1000 computing system. Ten contiguous beats of pressure and flow were
aligned and normalized to mean period (i.e., mean number of samples per
cycle), with the aid of a computerized procedure. The ensemble averages
were then taken to reduce noise (8). The resulting mean
cycles of terminal aortic pressure and flow were submitted to the
analysis procedure.
In ferrets 1-5, our asymmetric T-tube model of the
arterial system was fitted to the full cycle of ascending and
descending aorta flow waveforms to estimate arterial parameters and to
evaluate the impact of wave reflection in the morphology of ascending
aortic and head-end and body-end directed pressure pulses. In
ferret 6, the body-end arm of our T-tube model was fitted to
descending aortic flow to estimate parameters of descending aortic
circulation and to infer wave travel, and reflection in this portion of
the circulation supplied by the ascending aorta.
Results from these ferrets were compared with the results obtained from
fitting our asymmetric T-tube model of the arterial system to the
ascending and descending aorta flow waveforms obtained from five dogs
as described in detail in a previous report (5).
T-tube arterial model.
Our T-tube model of systemic arterial circulation consists of two
uniform, loss-free elastic transmission tubes connected in parallel and
terminating in first-order, low-pass filter loads (Fig.
1). Each of the two arms of the T-tube
model is characterized by the following parameters: the tube inertance
(li) and tube compliance
(ci) per unit length; the tube length
(di); the tube characteristic impedance
(Zci); the one-way wave transit time
across the tube (i); the peripheral resistance (Ri); and the time constants at the
numerator and denominator, respectively, zi
and pi, of the terminal load impedance
[ZLi(jn)].
This terminal load impedance was given the following expression
(5, 6)
|
(1)
|
where i equals h or b depending on whether the
reference is to the head-end or body-end circulations,
respectively.
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Fig. 1.
Modified asymmetric T-tube model of systemic arterial circulation
consisting of two uniform and loss-free, elastic transmission-tubes
(transmission lines in the electrical analogy) connected in parallel
and terminating in first-order low-pass filter loads. Parameters
Zci and i
(i = h and b), represent tube characteristic impedance
and one-way wave transit time across the head-end (subscript h) and the
body-end (subscript b) transmission tubes. Tube lengths
dh and db represent the
distances to two effective reflection sites respectively located in the
head-end and body-end portions of the arterial circulation supplied by
the ascending aorta. ZLh and
ZLb are terminal load impedances the electrical analog
of which incorporates the peripheral resistance
(Rpi, i = h and b) of the
head-end and body-end portions of the arterial system, the overall
static compliance (CLi) and the viscous losses
(Rdi) of wall motion (10). AV-M and
AV-T denote aortic valve to mandible distance and aortic valve to base
of tail distance along the body's long axis, respectively.
|
|
The constraint
|
(2)
|
was added to formalize the assumption that, with increasing
frequency, ZLi(jn)
approximates the tube-characteristic impedance
Zci = 
. A
consequence of this constraint is that no reflections occur at higher frequencies.
Designating the reflection coefficient at the junction between tube and
terminal load [Li(jn)]
|
(3)
|
the ascending aortic input impedance is determined by the
parallel of the head-end (i = h) and the body-end
(i = b) input impedances,
Zi(jn), which are of the
form (23)
|
(4)
|
According to Eqs. 1-4, the model parameters to
be estimated from input pressure and flow measurements (see next
paragraph) are Zci,
i, zi, and
Rpi (i = h and b). The
pi time constants are computed from
zi, Zci,
and Rpi making use of Eq. 2. Values of head-end and body-end tube compliance,
cidi, and tube inertance,
lidi, are given by the following
equations
|
(5)
|
|
(6)
|
Assuming a viscoelastic windkessel (3, 5, 10) as
terminal load configuration (Fig. 1), terminal load compliances, CLi, and resistances,
Rdi, with i = h and b, are obtained from resolving the following system of equations (3, 5, 10)
|
(7)
|
|
(8)
|
Data fit and parameter estimation.
In ferrets 1-5, the parameters
i, Zci, and
zi (i = h and b) were
estimated by minimizing (with a modified Levenberg-Marquardt algorithm,
Minpack, Argonne National Laboratory; Argonne, IL) a cost function
defined as the sum of the squared differences between measured
ascending and descending aortic flows and the corresponding flows
predicted by the model using measured ascending aortic pressure as
model input (5). Total peripheral resistance of systemic
arterial circulation (Rp) was calculated from
the ratio of the mean ascending aortic pressure () to mean
ascending aortic flow (). Head-end and body-end Rpi resistances were computed as
Rpi =
Rp/i, where
i is head-end directed (i = h) or body-end directed (i = b) mean flow.
In the sixth ferret, the parameters, b,
Zcb, and zb were estimated by
minimizing the sum of squared differences between measured descending
aortic flow and flow predicted by the body-end arm of our T-tube model
using measured descending aortic pressure as model input
(6). Total peripheral resistance of descending aortic
circulation (Rpb) was calculated from the ratio
of mean descending aortic pressure to mean descending aortic flow, and pb was calculated from zb,
Zcb, and Rpb making use
of Eq. 2.
An estimate of the pulse wave velocity (v) along the
body-end transmission path was determined in ferret 5 by
dividing the distance between proximal and distal pressure transducers
and the foot-to-foot time delay between the measured pressure waves. The product between estimated b and
v yielded a db estimate of tube length.
T-tube model parameter estimates in the dog were taken from our
previous report (5). Approximate asymptotic standard
errors of parameter estimates were calculated from the residual sum of squares and the parameter estimation variance-covariance matrix (2).
| |
RESULTS |
In Table 1 are reported the averages
(±SE) of body weight (W), heart rate (HR), left ventricular ejection
period (Ts), aortic valve to mandible distance
(AV-M), aortic valve to base of tail distance (AV-T), , ,
the ratio of descending to ascending aortic mean flow
(b/), and Rp
from ferrets 1-5 and from the five dogs used in our
previously reported study (5).
Comparison of body morphometry yielded a ratio between AV-M and AV-T,
which was 0.71 ± 0.01 in the ferret and 0.54 ± 0.02 in the dog.
In ferret 6 (1.6 kg), the analysis was focused on the
descending aortic circulation as seen from a measurement site located 2 cm below the left subclavian artery. Measured mean pressure and flow
were 55 mmHg and 1.0 cm3/s, respectively, and heart rate
was 2.7 Hz.
Data fit and parameter estimates.
Features of the ascending aortic pressure wave shape varied among
ferrets, and two extremes are displayed in Fig.
2. Three ferrets possessed wave-form
features similar to that shown in Fig. 2A, i.e., the
systolic peak of the ascending aortic pressure occurred during early
systole, and a prominent diastolic wave was seen during early diastole.
In two ferrets, the systolic peak occurred during late systole just
before the incisura giving the rather different looking waveform shown
in Fig. 2B.
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Fig. 2.
Extremes of measured pressure in the ascending aorta of
ferrets under basal states. A: systolic peak of the
ascending aortic pressure occurs during early systole and a prominent
diastolic wave is seen during early diastole (ferret
3). B: systolic peak occurs during late systole,
just before the incisura, and diastolic decay is almost exponential
(ferret 5).
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|
Measured flow wave shapes and model-generated flow waveforms are
displayed in Fig. 3, A and
B. Clearly, the difference between ascending and descending
aortic flow waveforms as well as all details of the waveforms are
reproduced with a high degree of fidelity by our T-tube model. These
good reproductions in the ferret with our T-tube model are similar to
the good reproductions obtained earlier in our previous works using the
data from the dog under normal conditions (5). Average
estimated (Zci, i, zi,
i = h and b) and calculated (all the others) parameters
of T-tube model in ferrets 1-5 are given in Table
2. Parameter estimation errors, as a
percentage of related parameter estimates, averaged 5.4 ± 0.5%
for Zch, 1.9 ± 0.2% for Zcb, 13.4 ± 4% for h,
1.5 ± 0.2% for b, 8.7 ± 1.3% for
zh, and 2.5 ± 0.1% for zb.
Estimated and calculated T-tube model parameters obtained in our
earlier study in five dogs are also given in Table 2. Note the
consistent difference between model-estimated arterial parameters of
the ferret and the dog in accord with the difference in the size of
these two animals. This is true for both the tube parameters (shorter
transmission times, larger tube inertances, smaller tube compliances,
and larger tube characteristic impedances in the ferret) and the load
parameters (lower load compliances and higher load peripheral
resistances in the ferret).
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Fig. 3.
Fits between experimental (solid line) and model
predicted (dashed line) ascending aortic flows and between experimental
(dotted line) and model predicted (dash-dot line) descending aortic
flows. Data are from ferret 3 (A) and
ferret 5 (B).
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|
Aortic impedance patterns.
Aortic input impedance frequency spectra were calculated from Fourier
analysis of ascending aortic pressure and flow waves. A representative
example of this calculation in one ferret is given for 10 harmonics
represented by the solid circles in Fig. 4. Also in Fig. 4 are the input impedance
patterns predicted by the model for the ascending aorta and for the
head-end and body-end transmission paths of the T-tube. As expected
from the close fit obtained by the model to the flow waveforms (Fig.
3), the model-generated impedance spectra closely approximated the
impedance magnitude and phase obtained from the Fourier analysis of
pressure and flow. In both experimentally calculated and
model-generated impedance, only one broad, indistinct minimum can be
identified in the modulus of ascending aortic impedance in the
frequency range from 0 to 10 Hz. From the T-tube model prediction, it
is seen that this minimum in the overall system impedance modulus is
correlated to the first minimum in the impedance modulus of both
head-end and body-end transmission paths. Note especially in this
example from the ferret that, although the first minimum in impedance modulus occurs at approximately the same frequency for the head-end and
body-end circulations, the zero crossings of the phase spectra for
these two paths occur at quite different frequencies. The ability of
the body-end arm of our T-tube model to approximate the experimental
input impedance data of descending aortic circulation was tested in
ferret 6 where pressure and flow data were taken from the
high descending aorta. The model-generated impedance spectrum closely
approximated impedance magnitude and phase obtained from
Fourier analysis (Fig. 5). This result
improves the reliability of our T-tube model predictions of head-end
and body-end impedance patterns. The average features of experimental
and model-predicted impedance patterns in the ferret are evident in
Fig. 6, showing the impedance calculation
from the Fourier analysis and from the T-tube predictions using average
parameters of Table 2.
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Fig. 4.
Modulus and phase angle of ascending aortic input
impedance in one ferret, calculated from Fourier analysis of ascending
aortic pressure and flow waves (solid circles), are compared with the
input impedance patterns predicted by our T-tube model for the
ascending aorta (solid line) and for the head-end (dashed line) and
body-end (dash-dot line) portions of the arterial system. Dotted
horizontal line in the phase graph is zero line.
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Fig. 5.
Modulus and phase angle of the input impedance of
descending aortic circulation in ferret 6, calculated from
Fourier analysis of descending aortic pressure and flow waves (solid
circles), are compared with the impedance patterns predicted by the
body-end arm of our T-tube model (dash-dot line). Dotted horizontal
line in the phase graph is zero line.
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Fig. 6.
Average (±SE) ascending aortic impedance data computed
over ferrets 1 to 5 (solid circles with SE bars)
are compared with predictions for the ascending aorta (solid line) and
for the head-end (dashed line) and body-end (dash-dot line) portions of
the arterial system as provided by our modified T-tube model filled
with average parameters of Table 2. Dotted horizontal line in the phase
graph is zero line.
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The average impedance calculation from Fourier analysis and prediction
from the T-tube filled with average parameters of Table 2 for the dog
are displayed in Fig. 7. These impedance
patterns are similar to those reported by O'Rourke and Taylor
(21) for moderately large (23-25 kg) dogs. In
contrast to the ferret, two distinct minima are observed in both the
Fourier calculated modulus of ascending aortic impedance and in the
model prediction in the range from 0 to 10 Hz. The first minimum
corresponds to the first minimum of body-end impedance modulus, whereas
the second minimum corresponds to the first minimum of head-end
impedance modulus. The phase angle of the overall impedance shows a low
frequency distortion that causes a zero crossing in the proximity of
the zero crossing of the head-end input impedance. The contribution of
head-end and body-end circulations to the overall impedance in the dog
is different from that in the ferret.
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Fig. 7.
Average (±SE) ascending aortic impedance data computed
over five dogs (solid circles with SE bars) are compared with
predictions for the ascending aorta (solid line) and for the head-end
(dashed line) and body-end (dash-dot line) portions of the arterial
system as provided by our modified T-tube model filled with average
parameters of Table 2. Dotted horizontal line in the phase graph is
zero line.
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Differences in the head-end and the body-end transmission path lengths
in the two differently-sized species is the cause of the differences
observed in the aortic impedance patterns. If we assume the arterial
pulse wave velocity to be almost identical in the two species
(18, 28), the ratio h/b
between the one-way transmission times estimated in the ferret
(0.51 ± 0.05) and in the dog (0.39 ± 0.04) yields the
information on the relative distances to the effective reflection sites
(effective lengths) in the two species.
Measurements of the pulse-wave velocity made in our previous study in
dogs yielded a T-tube model-based estimation of ~30.3 ± 1.4 cm
for the length db of the body-end transmission
path (effective length of the descending aortic circulation), the
average estimate of b being 68.5 ± 4.6 ms (Table
2). From the product of db by the ratio of
h/b, a head-end effective length of
11.6 ± 1.2 cm is inferred. Comparison of the average estimates of
h and b in the dog with those estimated
in the ferret yields in the latter species an estimate of 13 cm for the
body-end transmission path and an estimate of 6.5 cm for the head-end
transmission path.
To make an independent test of this inference, we made an estimate of
head-end and body-end effective lengths in our ferret 5,
making use of the measurement of pressure taken from the femoral artery
24 cm below the location of the ascending aortic pressure measurement
and in proximity to trifurcation. From dividing this distance between
transducers by the pressure foot-to-foot time delay, we estimated a
pulse-wave velocity of 480 cm/s. The product of this velocity times the
estimate of b (0.028 s) yielded a body-end effective
length of 13.5 cm. This approximated the measured length of ~15 cm
between the proximal pressure measurement site and the renal artery
branching site. Assuming the pulse-wave velocity of 480 cm/s to also be
applicable to the head-end circulation, with the estimate of
h being 0.013 s, we obtained a head-end effective length
of 6.3 cm.
Forward and reflected waves.
With confidence that our T-tube model accurately represented the
overall pressure-flow relationships in the aorta, we proceeded to use
the model to evaluate forward and backward propagated components of the
arterial pressure wave. This evaluation for the two kinds of aortic
pressure wave shapes observed in these ferrets (Fig. 2) is displayed in
Figs. 8 and
9. Note the timing of the foot of the
reflected wave (solid line) in each panel relative to the foot of the
composite (dotted line) and forward (dashed line) waves. Also note the
relative role of the amplitude of the reflected waves from the head-end
and body-end circulations as contributors to the reflected wave and the
composite wave in the ascending aorta; this body-end reflection makes a
major contribution to the composite wave at the time of the incisura
and immediately after. Thus the presence of a prominent diastolic
oscillation in the ascending aortic pressure of Fig. 2A
appears correlated to the peak of the body-end reflected wave
(Fig. 8). On the other hand, the absence of a prominent diastolic
oscillation in the ascending aortic pressure of Fig. 2B (and
its almost exponential decay) is explained by the fact that the peak of
body-end reflected wave boosts the late systolic pressure and the
incisura (Fig. 9).
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Fig. 8.
Measured ascending aortic pressure minus mean pressure in
ferret 3 (dotted line) is compared with forward (dashed
line) and backward (solid line) pressure wave components predicted by
or T-tube model in the ascending aorta just upstream of the junction
with head-end and body-end directed transmission paths (A)
and at the input (just downstream the junction) of the body-end
(B) and the head-end (C) directed transmission
paths.
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Fig. 9.
Measured ascending aortic pressure minus mean pressure in
ferret 5 (dotted line) is compared with forward (dashed
line) and backward (solid line) pressure wave components predicted by
our T-tube model in the ascending aorta just upstream of the junction
with head-end and body-end directed transmission paths (A),
and at the input (just downstream the junction) of the body-end
(B) and the head-end (C) directed transmission
paths.
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Timing and amplitude of forward, reflected, and composite waves at the
distal end of the tubes (i.e., the effective reflection sites) are
shown in Fig. 10 (to be correlated with
Fig. 2A and Fig. 8) and Fig.
11 (to be correlated with Fig.
2B and Fig. 9). Note the apparent change in the shape of the
composite wave at the termination of the body-end tube due to
transmission and reflection effects. Note also, that, in contrast to
proximal locations: 1) the foot of the forward and backward
waves occurs at approximately the same point in time at the reflection
site, and 2) the peak of the reflected wave occurs earlier
during systole (Figs. 10 and 11) with the result that the reflected
wave contributes to an increase in the pressure pulse over that at the
proximal location (Figs. 8 and 9). Our measurement of arterial pressure
in the femoral artery just beyond the aortic trifurcation has been
added to Fig. 11A. This measured pressure (dash-dot line) is
to be compared with the model predicted, composite pressure wave (dot)
at the termination of the body-end tube. It is seen that factors
responsible for the transformation in wave shape that distinguishes
pressure waves measured in ascending aorta and femoral artery are well
represented at the termination of the body-end transmission tube.
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Fig. 10.
Pressure pulse (dotted line) predicted by our T-tube
model at the termination of the body-end (A) and head-end
(B) transmission paths in ferret 3 is displayed
with forward (dashed line) and backward (solid line) pressure wave
components.
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Fig. 11.
Pressure pulse (dotted line) predicted by our T-tube
model at the distal end of the body-end (A) and head-end
(B) transmission paths in ferret 5 is displayed
with forward (dashed line) and backward (solid line) pressure wave
components. Femoral artery pressure pulse (dash-dot line) measured in
proximity of trifurcation (about 24 cm below the ascending aortic
pressure measurement site) is displayed in A.
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The wave reflection and transmission phenomena observed in the ferret
may be compared with the same phenomena observed in the dog (Figs.
12 and
13). In general, most phenomena are
represented equally in the dog and the ferret with the following
important exceptions: 1) the reflected wave from the
body-end circulation is delayed to a greater amount in the dog due to
the significantly longer transmission path and, as a consequence, the
peak of this reflected wave contributes more to proximal aortic
pressure during diastole, and 2) the reflected wave from the
head-end circulation is of relative greater magnitude and its peak
contributes more substantially to ascending aortic pressure during
systole.
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Fig. 12.
Measured ascending aortic pressure minus mean pressure
in one dog (dotted line) is displayed with forward (dashed line) and
backward (solid line) pressure wave components predicted by our T-tube
model in the ascending aorta just upstream of the junction with
head-end and body-end directed transmission paths (A) and at
the input (just downstream the junction) of the body-end (B)
and the head-end (C) directed transmission paths.
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Fig. 13.
Pressure pulse (dotted line) predicted by our T-tube
model at the termination of the body-end (A) and head-end
(B) transmission paths in the dog is displayed with forward
(dashed line) and backward (solid line) pressure wave components.
Pressure pulse measured in the terminal aorta (dash dot line), just
below the renal arteries origin, is displayed in A.
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DISCUSSION |
It is generally agreed that differences among animals in the
contour of the aortic pressure wave must be due to differences among
animals in vascular properties and so must be explicable in terms of
modulus and phase of the vascular input impedance (18,
22). This is the reason why Fourier analysis has been extensively applied to determine impedance patterns in a wide variety
of mammals and in various physiological conditions. Inferring physiological meaning from vascular impedance in terms of wave reflection, however, requires the formulation of distributed, (or
partially distributed) parameter models that incorporate hypotheses to
be experimentally tested in a trial-and-error approach (3, 15-18, 24-29). According to the literature, two
important questions need to be addressed in the process of a reliable
interpretation of arterial wave reflection as seen from the heart. The
first question is whether the reflections arise from one or two
functionally discrete reflection sites. The second question is whether
the reflections are seen to arise from 1) a purely resistive
load constituted by the peripheral resistance vessels, or 2)
a terminal load of complex nature incorporating the reactive and the
resistive properties of the lumped, downstream vascular beds.
It appears from the literature that the most acceptable answer to the
first question is that impedance patterns and wave reflections are due
to two functionally discrete, effective reflecting sites, although it
has been shown that in some circumstances these two sites may appear as
one to the heart (1, 3-6, 9, 11, 15, 16-21, 25).
The second question finds an answer in that, just as the random
branching of the arterial system together with arterial compliance
serves to uncouple the heart from the peripheral resistances
(26), these same factors also act to uncouple the terminal
end of the T-tube transmission paths from the most distant resistant
vessels. Such uncoupling gives the terminal load of the transmission
paths a complex, low-pass filter appearance. Indeed, others and we
(3-6, 9, 11, 25) have shown in previous works that
replacing the terminal resistance loads of the original T-tube model
with the complex and frequency-dependent loads described by Eq. 1 yields a decisive improvement that allows accurate description of pressure-flow data and identification of parameters of physiological interest.
The present study was designed to test whether the model was able to
discriminate between differences in body shape. These differences in
body size with concomitant differences in arterial system size appear
reflected clearly by the differences in model-estimated arterial
parameters given in Table 2. The ferret heart is located more centrally
in the body than is the heart of the dog. For instance, when we
measured the ratio of (AV-M)/(AV-T), we found 0.71 ± 0.01 in the
ferret, whereas it was 0.54 ± 0.02 in the dog. When we computed
the ratio of h/b as an equivalent
parameter of T-tube arterial morphometry, we found values of ~0.51
and 0.39 for the ferret and the dog, respectively. Thus
model-predicted, head-end and body-end transmission paths supplied by
the ascending aorta are consistent with externally measured long-axis
body morphometry parameters with respect to the heart location.
According to the results of our present and previous studies (3,
5, 6), the effective lengths of the head-end
(dh) and body-end (db)
transmission paths are ~12 and 30 cm, respectively, in the dog and
6.5 and 13 cm, respectively, in the ferret. Comparison of
db estimates with measurements of the distance
between the ascending aorta and the abdominal aortic site where major
branching occurs in dogs located the body-end effective reflection site at level of the renal arteries origin. Measurements of the distance between the ascending aorta and the renal arteries branching site in
the ferret (14-15 cm) was close enough to the
db estimate of ~13 cm to conclude that also in
the ferret, under normal conditions, the body-end effective reflection
site is seen at the level of the abdominal aortic region where major
branching occurs. This conclusion also fits with the experimental
finding of a major reflection site at level of renal arteries origin in
humans reported by Latham et al. (14).
In contrast to the descending aortic circulation, with the presence of
the aorta as the dominant vessel, several equivalent-sized vessels
directed to the forelimbs and the head characterize the head-end
portion of the circulation so that no discrete anatomic landmarks can
be identified at present among this group of vessels, which can be
associated with the effective lengths dh = 12 cm in the dog and dh = 6.5 cm in the ferret.
Average estimates for dh and
db for the dog, taken from the literature and
reported by Nichols and O'Rourke (see Ref. 18, their Fig. 11.27) are 20 and 38 cm, respectively. These
distances, calculated by using the quarter wavelength formula, appear
significantly overestimated with respect to those obtained from our
modified T-tube model. The reason of this overestimation and the
shortcomings of the quarter wavelengths formula and of the model from
which it is derived were extensively discussed in our previous works (3, 5-7).
One of the major goals of arterial models is to represent compliance
with respect to its role in arterial function. Most attempts to
quantify arterial compliance have been based on the windkessel model
(3, 10, 13, 27, 30). This approach assumes that the
windkessel compliance represents the sum of all compliances throughout
the arterial system, so that evaluation of the distributed and
heterogeneous changes that may occur in compliance is not possible. In
contrast to these models, our modified T-tube model allows
discrimination between proximal and distal compliant properties of the
arterial system. This aspect was investigated in previous studies in
the dog (5, 25). Here the question arises as to how the
estimates of proximal and distal compliance in the ferret correlate
with those from the dog to understand the impact of the difference in
body size.
Total T-tube model compliance (Ct) averaged 15.0 ± 1.4 10
6
g1 · cm4 · s2 in
our five ferrets and 415 ± 43 106
g1 · cm4 · s2 in
our five dogs (Table 2). To test the consistency between these
compliance estimates with respect to the animal size, we assumed
compliance equal to C = CoW1.23,
with Co being a constant and W the body weight, in
accordance with the finding by Westerhof and Elzinga (28)
from a comparative analysis of normalized input impedance in mammals of
different size. With this approach, the ratio between total compliance
in the dog and total compliance in the ferret equals the ratio between respective body weights to the power 1.23. Application of this ratio to
our preparations averaged 32.8 ± 3.7 and resulted consistent with
the ratio of 28.5 ± 3.5 between T-tube model-based estimates of
total compliance from our dogs and ferrets.
A further test of the reliability of T-tube model estimates of
compliance was to compare head-end and body-end compliance estimates in
the ferret and in the dog as they relate to long-axis body morphometry.
The ratio Ch/Cb between total compliance of the
head-end circulation (Ch = chdh + CLh) and total
compliance of the body-end circulation (Cb = cbdb + CLb) was 0.70 ± 0.10 in the ferret and 0.51 ± 0.06 in the dog. Thus the
model-predicted Ch/Cb ratio is consistent with
externally measured long-axis body morphometry ratio of AV-M/AV-T
discussed above.
Differences in body morphometry in the ferret and in the dog yield
differences in the features of ascending aortic impedance patterns that
are clearly visible in the modulus (Figs. 6 and 7). In the ferret, only
one broad minimum is seen in the modulus of ascending aortic impedance
in the frequency range from 0 to 10 Hz, whereas, in this frequency
range, two minima are distinguishable in the ascending aortic impedance
modulus of the dog. Coincidentally, impedance data reported for humans
show a broad minimum in the low-frequency range, rather than two
distinct minima as seen in large dogs (18). Thus the
ascending aortic impedance modulus in the ferret resembles the
ascending aortic impedance of adult humans and differs from the
corresponding impedance pattern in the dog in much the same manner as
does the human. The similarity of ascending aortic impedance patterns
in humans and ferrets is likely due to the fact that these two species
have a heart that is more centrally located along the long axis of the
body than is the case with the dog. Indeed, using estimates from the
quarter wavelength formula, Nichols and O'Rourke (18)
reported head-end and body-end effective lengths to reflection sites in
the human as 29 and 41 cm, respectively, for a ratio of 0.71, and in
the dog as 20 and 38 cm, respectively, for a ratio of 0.53. These numbers indicate that the heart in humans is more centrally located than in dogs just as we found in the present study that in ferrets the
heart is more centrally located than in dogs. It is likely that in
humans the low-frequency minimum in the ascending aortic impedance
modulus is a result of nearly coincident first minima in the
input impedances of the head-end and body-end sections of modified
T-tube model as shown for the ferret in Figs. 4 and 6.
Our modified T-tube model allows interpretation of the morphology of
measured aortic pulse waves, as seen at the heart, in terms of timing
and shape of reflected wave components arising from two functionally
discrete reflecting sites, respectively, located in the body-end and
head-end portions of the circulation supplied by the ascending aorta.
According to the model, it may happen that the body-end reflected wave
peaks during early diastole and results in a prominent diastolic
oscillation in measured ascending aortic pressure as seen in Figs. 8
and 12. It is also possible that the body-end reflected wave peaks at
the level of incisura and boosts early diastolic pressure in such a way
that its decay appears smooth and almost exponential (Fig. 9). The
relatively longer body-end transmission path in the dog determines a
more evident contribution of the body-end reflected wave in diastole (Fig. 12). In contrast to body-end reflection, the head-end reflected wave travels across a relatively short pathway so that it gets to the
ascending aorta earlier than the body-end reflected wave, thus
affecting systolic more than diastolic pressure (Figs. 8, 9, and 12).
Our comparative analysis, however, showed a lower impact of head-end
wave reflection on ascending aortic pressure in the ferret than in the
dog (compare Figs. 8 and 9 with Fig. 12). This suggests a better
matching between terminal load and transmission path in the ferret. In
the ascending aorta, the head-end and body-end reflected waves compose
such that the positive contribution of the head-end reflected wave to
systolic pressure is attenuated by the negative contribution of the
body-end reflected wave. The result is that systolic pressure is not
elevated to the extent that would occur if the timing of the arrival of
these reflected waves was more coincident (Figs. 8, 9 and 12).
Furthermore, the positive peak of the reflected wave from the body-end
tube boosts aortic pressure during diastole. Diastolic pressure boost
is desirable because it aids coronary perfusion, whereas the
concomitant reduction of the pulse pressure helps the pumping function
of the heart (18-20).
This favorable correspondence between cardiac performance and timing of
body-end reflected wave is preserved in different mammals, irrespective
of body size, because the duration of LV ejection changes in an
appropriate fashion with body size and related timing of wave
reflection as seen at the heart. The relationship between average
body-end transmission time (b), and average LV ejection
period (Ts) in our ferrets and dogs, having
similar mean aortic pressure level (Table 1), is displayed in Fig.
14.
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Fig. 14.
Plot of average (±SE) body-end transmission time
(b), over five ferrets and five dogs vs. average (±SE)
duration (Ts) of LV ejection.
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Variability of LV ejection period among individual animals of a species
(SE bars in Fig. 14) may explain differences in the morphology of
measured aortic pressure waves as seen in Fig. 2. The pressure wave
displayed in Fig. 2A is characterized by an ejection period
that is almost 30 ms shorter than that characterizing the pressure wave
of Fig. 2B. As a consequence, in the case of Fig.
2A the body-end reflected wave returns by the end of systole and its peak, occurring beyond the incisura, contributes to diastolic pressure bump (Fig. 8). In the case of Fig. 2B, the peak of
the body-end reflected wave is reached in correspondence of the
incisura and causes a late augmented aortic pressure peak in systole
(Fig. 9).
In conclusion, whereas our T-tube arterial model (Fig. 1) is a reduced
representation of aortic circulation, it captures essential features of
hydraulic input impedance. This enables our model to be used to
describe physiologically important features of the arterial system, to
predict pressure and flow events at the aortic entrance and at various
locations along the transmission path, and to interpret the composition
of aortic pressure and flow waves in terms of forward and reflected
components. Furthermore, we demonstrate here that the asymmetric T-tube
model is sufficiently versatile and general in its representation of
mammalian arterial systems that it is able to discriminate between
substantial differences in body size and location of the heart along
the body's axis of mammals of different size and shape.
| |
ACKNOWLEDGEMENTS |
This work was supported in part by the Italian Ministero
dell'Università e della Ricerca Scientifica e Tecnologica.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: R. Burattini, Dept. of Electronics and Automatics, Univ. of Ancona, Via
Brecce Bianche, 60131 Ancona, Italy (E-mail:
r.burattini{at}popcsi.unian.it).
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. Section 1734 solely to indicate this fact.
Received 11 May 2001; accepted in final form 17 September 2001.
| |
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ABSTRACT
INTRODUCTION
