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1 Hydraulics Laboratory, Institute for Biomedical Technology, University of Gent, B-9000 Ghent, Belgium; 2 Laboratory for Physiology, Free University of Brussels, B-1070 Brussels, Belgium; 3 Biomedical Engineering Laboratory, École Polytechnique Federale de Lausanne, 1015 Lausanne, Switzerland; and 4 Laboratory for Physiology, Institute for Cardiovascular Research, Free University of Amsterdam, 1081 BT Amsterdam, The Netherlands
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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In six dogs and
six weight-matched miniature pigs at baseline and after pulmonary
embolization, pulmonary arterial compliance was determined using the
pulse pressure method (CPPM),
the three-element windkessel model
(CWK-3), and the ratio of stroke
volume to pulse pressure (SV/PP).
CPPM was lower in pigs than in
dogs at baseline (0.72 ± 0.23 vs. 1.14 ± 0.29 ml/mmHg,
P < 0.05) and after embolism (0.37 ± 0.14 vs. 0.54 ± 0.16 ml/mmHg, P = 0.07) at
matched flow, but not at matched flow and pressure.
CPPM showed the expected inverse
relation with pressure and a direct relation with flow. CWK-3 was closely correlated with
CPPM, except for all dogs at baseline where CWK-3 was up to
100% higher than CPPM. Excluding these data, regression analysis yielded
CWK-3 = 
0.01 + 1.30 · CPPM
(r2 = 0.97).
CWK-3 was found to be unreliable
when input impedance first harmonic modulus was close to characteristic
impedance, i.e., when reflections were small. SV/PP correlated well
with CPPM (SV/PP = 0.10 + 1.76 · CPPM,
r2 = 0.89). We
conclude that 1)
CPPM is a consistent estimate of pulmonary arterial compliance in pigs and dogs,
2)
CWK-3 and SV/PP overestimate
compliance, and 3)
CWK-3 is unreliable when wave reflections are small.
pulmonary hemodynamics; pulse pressure method; three-element windkessel; stroke volume-to-pulse pressure ratio
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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RECENT STUDIES have indicated that systemic arterial pulse pressure (PP), the difference between systolic and diastolic blood pressure, is an important determinant of cardiovascular mortality and morbidity (1, 6). It has been shown in dogs that the systemic PP is determined by the pumping action of the left heart, total peripheral resistance, and total arterial compliance (17). Therefore, arterial compliance is a biomechanical property with an important diagnostic content. Also, the role of pulmonary arterial compliance as a contributor to the pulmonary input impedance has been recognized (7, 13, 14, 19). Nevertheless, to date, arterial compliance has most often been studied in the systemic circulation, for which several compliance estimation methods have been suggested (3-5, 8, 9, 11, 12, 15, 16, 18).
A well-known method is based on the three-element windkessel (WK-3) model, consisting of total peripheral resistance, characteristic impedance, and total arterial compliance (20). It has been shown that the WK-3 model generally overestimates systemic arterial compliance (5, 16) with errors between 10 and 40% (16). Nevertheless, the method is still frequently used (8, 13, 18, 19) because it generally yields excellent fits to the measured pressure waveform. The method has been used to assess compliance of the pulmonary arterial system (13, 19). Here, the WK-3 model may give compliance estimates up to 300% higher than the simple approximation of compliance as the ratio of stroke volume to pulse pressure (SV/PP) (13). This observation was never explained in detail. More recently, the pulse pressure method (PPM) has been presented as a simple and accurate method to estimate systemic arterial compliance (15, 16) from aortic pressure and flow, but the method has never been used for the pulmonary circulation.
In this study, we wanted to compare three methods for the estimation of pulmonary artery compliance: the WK-3 model compliance (CWK-3), SV/PP, and the more recent PPM compliance (CPPM). Therefore, we used data from a previously published study (10) in which pulmonary vascular impedance determinants were measured in dogs (n = 6) and pigs (n = 6) matched for weight and body size. Because pulmonary arterial characteristic impedance in that study was reported to be higher in the pigs (10), we expected a lower pulmonary arterial compliance in the pigs. The aim of the present study was to 1) assess the applicability of the PPM in the pulmonary circulation, 2) explain why the WK-3 model gives surprisingly high compliance values in some hemodynamic conditions, and 3) study the variation of pulmonary arterial compliance with pressure and flow.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Dog and Pig Data on Pulmonary Circulation
Six mongrel dogs (21-36 kg) and six weight-matched miniature pigs, all full-grown adults, were included in the study. In all the animals, a left lateral thoracotomy was performed, and a nonconstricting ultrasonic flow probe (Transonic Systems, Ithaca, NY) was positioned around the main pulmonary artery to measure instantaneous pulmonary flow. Instantaneous pulmonary arterial pressure (Ppa) was measured with a 5-F high-fidelity manometer-tipped catheter (model SPC 350, Millar Instruments, Houston, TX) with its tip positioned immediately distal to the flow probe. Disposable pressure transducers (Gould-Spectramed, Binchoven, The Netherlands) were connected to polyethylene catheters inserted in a branch of the main pulmonary artery and the aorta for arterial and venous blood sampling and for additional pressure measurements. To reduce venous return and control cardiac output (Q), a balloon catheter (Percor Stat-DL 10.5-F, Datascope, Paramus, NJ) was advanced into the inferior vena cava through a right femoral venotomy. After instrumentation, the chest was closed, but no attempt was made to restore a negative pleural pressure. A more complete description of the animal preparation and anesthesia has been published previously (10).After baseline assessment, Q was altered by stepwise inflation of the inferior vena cava balloon. For each level of Q, complete hemodynamic data were obtained. The same procedure was repeated after injection of autologous blood clots to cause pulmonary hypertension. Embolization was carried out progressively until two levels of Ppa were obtained (~40 and 50 mmHg in dogs and ~45 and 55 mmHg in pigs). At each level of Ppa, the animals were allowed to stabilize for 30 min. All measurements were performed according to the "Guiding Principles in the Care and Use of Animals" approved by the American Physiological Society.
For each species, data sets were selected at baseline and embolization (levels 1 and 2) to obtain a series of 1) isoflow conditions in which Q is matched while Ppa varies and 2) isobaric conditions in which Ppa is matched while Q varies. Fourier analysis was performed on the instantaneous pressure and flow measurements for a series of three to six consecutive end-expiratory heart beats. From these pressure and flow harmonics, average pressure and flow cycles were reconstructed in the time domain by summation of individual harmonics.
Data Analysis
Stroke volume (SV) was obtained by numerical integration of the instantaneous flow. Total vascular resistance was calculated as either the ratio of Ppa to Q (R) or the slope of Ppa-Q relations (RP-Q) assessed by linear regression using at least three points. For each species and at each condition, the input impedance (Zin) was derived as the ratio of corresponding pressure and flow harmonics. The characteristic impedance (Z0) was calculated as the average of the modulus of Zin between 3 and 10 Hz. Zin was normalized to Z0 (Zin/Z0) and expressed as a function of harmonic number. The reflection coefficient (
) was derived as
(Zin Z0)/(Zin + Z0), and the
modulus of the first harmonic
(||1) was used to characterize wave-reflection intensity.
Estimation of Pulmonary Arterial Compliance
PPM.
The measured instantaneous pulmonary arterial flow was applied to a
two-element windkessel (WK-2) model (Fig.
1A)
with an initial value of total pulmonary arterial compliance
(Cpa) and known resistance. The
resulting pulse pressure
(PPWK-2) was computed. Cpa was adjusted by minimization
of the difference between PPWK-2 and measured PP (15). The WK-2 response was first calculated in the
frequency domain by computing individual harmonics (multiplying flow
harmonics with the WK-2 impedance) and then transformed into the time
domain. More details about the method are found in Ref. 15. PPM was
applied using either R or
RP-Q as estimates
for pulmonary vascular resistance, and results are indicated as
CPPM or
CPPM(P-Q), respectively. The
windkessel time constant was computed as
R · CPPM
and
RP-Q · CPPM,
respectively.
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WK-3. The pressure response of the WK-3 model (PWK-3) (Fig. 1B) was computed with the flow wave as input. PWK-3 was fitted to the measured pressure wave using a least-squares fitting algorithm (18). Resulting parameters were R, Z0, and Cpa.
SV/PP. PP was calculated as the difference between the maximal and minimal values of instantaneous Ppa. Compliance was obtained as SV/PP (4, 12, 13).
Statistical Analysis
The isoflow and isobaric data series were analyzed separately. Measurements were averaged per species and per embolization level (baseline, embolization levels 1 and 2). Data were analyzed by a two-way repeated-measures ANOVA. When the significance level reached P < 0.05, further testing was performed with repeated-measures ANOVA and Bonferroni t-tests to evaluate differences between baseline and embolization and with the Mann-Whitney U test for comparison of data from pigs and dogs (SigmaStat 2.0, Jandel Scientific). CPPM, CWK-3, and SV/PP were compared by correlation analysis and by Bland-Altman agreement analysis (2).| |
RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Pulmonary Hemodynamics in Pigs and Dogs
The normalized input impedance at baseline and embolization level 1 is given in Fig. 2. In the pig, the pattern of Zin/Z0 was similar for both conditions (Fig. 2A). In the dog, there was a large discrepancy between baseline and embolization (Fig. 2B). At baseline, resistance was only ~5 times higher than Z0, and the first harmonic impedance modulus was only 1.05 ± 0.43 times Z0 (compared with 1.92 ± 0.56 at embolization level 1).
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Isoflow conditions.
An overview of hemodynamic data and estimates of compliance,
resistance, and characteristic impedance are given in Table
1. For a flow rate matched around 2.3 l/min, mean Ppa was significantly lower in the dogs than in the pigs at baseline and at embolization (P < 0.01). Consequently,
R was lower in the dogs for these
conditions as well. In the pigs,
Z0 increased with
Ppa from 0.071 ± 0.008 mmHg · ml
1 · s1
at baseline to 0.116 ± 0.041 mmHg · ml1 · s1
at embolization level 2 (P < 0.05). In the dogs,
Z0 was
independent of Ppa. At baseline,
there was no difference in
Z0 between the dogs (0.069 ± 0.023 mmHg · ml1 · s1)
and the pigs (0.071 ± 0.008 mmHg · ml1 · s1).
At embolization level 2,
Z0 in the dogs
(0.068 ± 0.015 mmHg · ml1 · s1)
was lower than in the pigs (0.116 ± 0.041 mmHg · ml1 · s1;
P < 0.05). In the pigs,
||1 was unchanged
from baseline to embolization (range 0.54-0.59). In the dogs,
||1 was lower at baseline than at embolization level 2 (0.29 ± 0.10 vs. 0.55 ± 0.07;
P < 0.001).
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Isobaric conditions.
In the dogs at baseline, the data points at the highest
Ppa were selected. Nevertheless,
Ppa reached only 17.8 ± 2.3 mmHg, which was lower than the pressure range of 30-32 mmHg that
was obtained in the dogs after embolization and in the pigs in all conditions (Table 1). With pressure matched around 32 mmHg, flow was
higher in the dogs at all conditions. In the pigs,
Z0 was higher at
embolization (P < 0.01),
whereas Z0 was
independent of flow in the dogs. In the pigs,
||1 increased from
baseline (0.54 ± 0.06) to embolization (level
2: 0.69 ± 0.13; P < 0.05). In the dogs,
||1 was lower at
baseline than after embolization (P < 0.001).
Pulmonary Arterial Compliance
Isoflow conditions.
PPM.
In both pigs and dogs, compliance was inversely related to
Ppa (Fig.
3, Table 1). In the pigs,
CPPM changed from 0.72 ± 0.23 ml/mmHg at baseline to 0.37 ± 0.14 ml/mmHg at
embolization level 2. In the dogs,
CPPM varied from 1.14 ± 0.29 ml/mmHg at baseline to 0.54 ± 0.16 ml/mmHg at
embolization level 2. The difference between pigs and dogs was significant at baseline and
embolization level 1 (P < 0.05) but not at
embolization level 2.
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Isobaric conditions. PPM. In the pigs, CPPM changed with Q and dropped from 0.80 ± 0.40 ml/mmHg at baseline (high flow) to 0.31 ± 0.08 ml/mmHg at embolization level 2 (low flow; Fig. 3). In the dogs, only the measurements during embolization were at the same pressure; CPPM was reduced from 0.38 ± 0.14 (level 1) to 0.31 ± 0.08 ml/mmHg (level 2). CPPM was higher in dogs than in pigs at both embolization levels.
WK-3 AND SV/PP. The compliance estimates obtained with WK-3 and SV/PP were higher than those obtained with PPM (Table 1) but showed similar differences between species and similar changes after embolization.Relation of PPM to WK-3 and SV/PP
PPM versus WK-3.
The relation between CPPM and
CWK-3 is shown in Fig. 4. With all
data, except dog data at baseline, regression analysis gave CWK-3 = 0.01 + 1.30 · CPPM
(r2 = 0.967). It can be seen that the correlation is lost for
the dog data at baseline.
PPM versus SV/PP.
Regression analysis on all data yielded SV/PP = 0.10 + 1.76 · CPPM
(r2 = 0.887). The
correlation improved after the data set was separated into pigs and
dogs. For pigs SV/PP = 0.21 + 2.15 · CPPM
(r2 = 0.942), and
for dogs SV/PP = 0.11 + 1.63 · CPPM
(r2 = 0.869). The
latter correlation did not improve after exclusion of the dog data at baseline.
Sensitivity of PPM to Computation of Total Vascular Resistance
The values of RP-Q, obtained from a linear regression analysis, are given in Table 1. A single resistance value was obtained from multipoint linear regression analysis at baseline and at each embolization level. The same value was used for isoflow and isobaric conditions. There was an excellent correlation between CPPM and CPPM(P-Q): CPPM(P-Q) =0.02 + 0.976 · CPPM
(r2 = 0.988). The
mean difference between both methods was small (0.03 ± 0.04 ml/mmHg) but significant, as indicated by a paired t-test
(P < 0.001) and the Bland-Altman (2)
plot (Fig. 5). The windkessel time constant
was calculated as
R · CPPM
and
RP-Q · CPPM,
respectively. For the isoflow measurements,
R · CPPM
was not different between pigs and dogs, but there was an increase from
baseline (~0.4 s) to embolization (~0.6 s). For the isobaric measurements, the increase from baseline (~0.45 s) to embolization (~1 s) was more pronounced (Table 1).
RP-Q · CPPM
was lower than R · CPPM
but showed no difference between pigs and dogs or between baseline and
embolization.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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We estimated pulmonary artery compliance in dogs and weight-matched miniature pigs using PPM, WK-3, and SV/PP. The WK-3 method has been used earlier by other investigators. In dogs, van den Bos et al. (19) found a mean Cpa of 2.36 ml/mmHg at control (Ppa = 20.1 mmHg) and 0.74 ml/mmHg after administration of serotonin (Ppa = 28 mmHg). These values are comparable to our findings using the WK-3 model (see Table 1). We found a good correlation between PPM and WK-3, except for the dogs at baseline conditions. Excluding these data, r2 = 0.96 and the coefficient of the linear regression line is 1.30 with a zero intercept. This suggests a systematic overestimation of WK-3 by ~30%. Others have reported this rate of overestimation in studies on the systemic arterial compliance (5, 16).
For the dog at baseline, however, WK-3 yields compliance estimates 128% higher than those of PPM. For one dog, WK-3 even gives a value of 13.3 ml/mmHg. Such high estimates of the WK-3 model have been reported previously (13, 19) without obvious explanation. Because PPM estimates are not available in the literature, we reviewed studies that allowed us to compare CWK-3 with SV/PP. From the dog data of van den Bos et al. (19), total systemic arterial compliance estimated with the WK-3 method (0.45 ± 0.29 ml/mmHg) corresponds well with the SV/PP approximation of 0.45 ± 0.32 ml/mmHg for control conditions. After administration of a vasodilator, WK-3 gives 1.23 ± 0.60 ml/mmHg, versus 0.42 ± 0.13 ml/mmHg for SV/PP. In the pulmonary circulation, WK-3 yields 2.36 ± 1.24 versus only 1.00 ± 0.58 ml/mmHg for SV/PP at control conditions and 0.74 ± 0.51 versus 1.02 ± 0.76 ml/mmHg, respectively, after administration of a vasoconstrictor. Slife et al. (13) calculated compliance in the human pulmonary circulation. Using WK-3, they found 20 ± 15 ml/mmHg compared with only 8 ± 2 ml/mmHg for SV/PP at baseline. Thus WK-3 sometimes yields compliance estimates that are markedly higher than PPM or the SV/PP approximation. In our study, this was the case in the pulmonary circulation of the dog at baseline. This condition is characterized by a flat input impedance pattern and low wave-reflection intensity (Fig. 2). The data of Slife et al. (13) suggest that analog conditions are met in the human pulmonary circulation. Van den Bos et al. (19) have shown that overestimation of compliance can be obtained in the systemic circulation using vasodilating drugs.
The reason why the WK-3 model yields these high compliance values is
found in the input impedance pattern of the WK-3 model and the
time-domain fitting algorithm: 1)
the WK-3 input impedance can only decay monotonically from the value of
total peripheral resistance to the characteristic impedance, and
2) to obtain a good fit in the time
domain, the WK-3 model impedance has to be close to the actual input
impedance at all frequencies. For most hemodynamic conditions, actual
input impedance decays monotonically for the first three to five
harmonics (Fig.
6B) and
WK-3 can easily fit the input impedance at all frequencies. However, in
the pulmonary circulation of the dog at baseline (Fig.
6A), the input impedance pattern is
flat and the impedance modulus for the first harmonic is close to the
characteristic impedance. Using the WK-3 fitting algorithm, the model
will first fit to the characteristic impedance at high frequencies; to
catch the lowest frequencies (and match to PP), it can only increase
its compliance to get a good time-domain fit. In Fig. 6, the actual
input impedance is compared with the input impedance of a three-element
windkessel model with 1) fitted WK-3
parameters and 2) calculated
R and
Z0 and the PPM
estimate for compliance (CPPM).
In Fig. 6B (dog at embolization), the
WK-3 fit is close to the calculated impedance using
R,
Z0, and
CPPM. At baseline (fig.
6A), the fitted values give an
impedance pattern going through all data points, including the first
harmonic. To get this pattern, the fitting algorithm can only increase
compliance.
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The resulting large WK-3 compliance estimation does not represent a
true physical increase of the storage capacity of the arterial tree. It
is caused by the flat input impedance pattern, which is the result of
low wave-reflection intensity. The input impedance of a straight
elastic tube with characteristic impedance Z0 is given as
Zin = Z0(1 +)/(1 ). The input impedance pattern can be modified via without altering the compliance
(Z0) of the
tube (11). This ambiguous behavior of the WK-3 model has important
consequences for hemodynamic studies with large alterations in pressure
caused by vasoactive drugs. Vasodilators not only lower pressure but
also reduce wave-reflection intensity. The converse is true for
vasoconstrictors (19, 21). Because of the nonlinear pressure-compliance
relation of arteries, compliance is effectively increased at low
pressures. However, the overestimation of compliance given by the WK-3
model will be higher at the low pressures than at the high pressures.
This may create the impression of an exaggerated increase in arterial
compliance at the lower pressures. The applicability of the WK-3 model
for the evaluation of the hemodynamic action of drugs can therefore be questioned.
Recently, Quick et al. (11) commented on the three-element windkessel
model in an analytical study of arterial system compliance, using a
classic windkessel model, that addressed the coupling between
compliance estimation and wave-propagation phenomena. Two important
observations, supporting our experimental study results, were made.
1) From their mathematical
expression it can be calculated that for approaching zero, WK-3
compliance tends toward infinity. 2)
Quick et al. stress the fact that total compliance is a low-frequency
property and that compliance estimation methods should utilize the
lowest frequencies. Because PPM basically uses the lowest three to four
harmonics, this method is preferable over the three-element windkessel
model fit, which takes into account all harmonics. Quick et al. have
also shown that heart rate may be a disturbing factor in the
interpretation of compliance estimation results. The frequency
dependency of compliance is also addressed by Burattini et al. (3).
They express compliance as a complex, frequency-dependent property
using a viscoelastic windkessel model formulation, with a modulus and a
phase angle. They have shown that, for the systemic circulation in
dogs, the modulus of the dynamic compliance at the heart rate frequency corresponds well with CPPM. In our
experiments, heart rate is not different for pigs and dogs and
therefore is not responsible for observed differences in pulmonary
artery compliance.
SV/PP has been reported as a simple and useful index of systemic arterial compliance (4, 12) and has been used to estimate pulmonary arterial compliance in humans (13). However, as argued by others (9), we believe that SV/PP inherently overestimates true arterial compliance. SV/PP can be seen as the total arterial compliance if the complete stroke volume is buffered in the large elastic arteries in systole, without any peripheral outflow. SV is then the volume increase in this (closed) system, and PP is the associated pressure increase. However, there is a continuous flow toward the periphery, and the volume increase during ejection is only a fraction of SV. SV/PP therefore always overestimates true compliance. Note that the explanation is more complex because of wave-reflection and viscoelastic effects. In our data, SV/PP overestimated CPPM by 81% in the pigs and 60% in the dogs. Apart from this overestimation, the correlation between CPPM and SV/PP is remarkable. Because SV/PP is the upper limit of true compliance, CWK-3 values higher than SV/PP are necessarily incorrect. The high values of CWK-3 at baseline in dogs are therefore undoubtedly overestimations of true compliance.
Before compliance can be estimated with PPM, total vascular resistance has to be calculated (15). When resistance is calculated as the ratio of mean pressure and mean flow (R) in the pulmonary circulation, a hyperbolic function of R with flow rate is obtained (10). Alternatively, one may use the slope of the pressure-flow relationship as a flow-independent resistance (RP-Q). We have applied PPM using both R and RP-Q. As shown in Fig. 5, the difference between CPPM and CPPM(P-Q) is small, indicating that the flow dependency of R is not transferred into the compliance estimate. This is mainly because PPM does not fit to the mean pressure level, only to the pulse pressure, thus decoupling the steady and pulsatile flow problem. The covariation of CPPM with flow (Fig. 3B) therefore cannot be attributed to a flow dependency of the compliance estimation method itself. The windkessel time constant, calculated as R · CPPM, is not different between pigs and dogs but increases after embolization. The variation is more pronounced for the isobaric measurements, indicating a (strong) dependency of R · CPPM of flow. This is the result of the hyperbolic relation of R with flow rate. Using RP-Q as a flow-independent resistance, we obtained the time constant RP-Q · CPPM (~0.3 s), which is independent of species and of hemodynamic condition.
On the basis of these arguments, we consider CPPM to be the best approximation of true pulmonary arterial compliance (Cpa). CPPM is a function of both pressure and flow rate (Fig. 3 and Table 1). At baseline, CPPM is higher in the dog than in the weight-matched pig, but this is mainly the effect of the lower Ppa. At matched pressure and flow (embolization level 2 for the dog and level 1 for the pig), there is still a tendency for a somewhat higher compliance in the dog, but the difference is not significant.
In conclusion, using PPM for the estimation of total pulmonary arterial compliance yields consistent results in the pig and dog at baseline and after embolization. Using PPM, we could demonstrate the expected inverse relation between compliance and pressure as well as a covariation of compliance with flow. Pulmonary arterial compliance is higher in dogs than in weight-matched pigs at baseline, but it is not different at matched pressure and flow.
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ACKNOWLEDGEMENTS |
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This research is funded by postdoctoral Grant IWT 960250 of the Flemish Institute for the Promotion of the Scientific-Technological Research in Industry, a concerted action program of the University of Gent, supported by the Flemish government (GOA 95003) and Fondation pour la Recherche Scientifique Médicale Grants 9.4513.94 and 3.4517.95.
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: P. Segers, Hydraulics Laboratory, Inst. of Biomedical Technology, Univ. of Gent, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium (E-mail: patrick.segers{at}rug.ac.be).
Received 23 July 1998; accepted in final form 31 March 1999.
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
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) compared with WK-3
Zin modulus based
on parameter values obtained via WK-3 model fit (solid line) and
modulus calculated from
Z0,
R, and
CPPM (dashed line).
A: dog data at baseline.
B: data for same dog at
embolization.