Pulmonary arterial compliance in dogs and pigs:
the three-element windkessel model revisited
PATRICK SEGERS,1 SERGE BRIMIOULLE,2 NIKOS STERGIOPULOS,3 NICO WESTERHOF,4
ROBERT NAEIJE,2 MARCO MAGGIORINI,2 AND PASCAL VERDONCK1
1Hydraulics Laboratory, Institute for Biomedical Technology, University of Gent, B-9000
Ghent, Belgium; 2Laboratory for Physiology, Free University of Brussels, B-1070 Brussels, Belgium;
3Biomedical Engineering Laboratory, École Polytechnique Federale de Lausanne,
1015 Lausanne, Switzerland; and 4Laboratory for Physiology, Institute for Cardiovascular
Research, Free University of Amsterdam, 1081 BT Amsterdam, The Netherlands
pulmonary hemodynamics; pulse pressure method; threeelement windkessel; stroke volume-to-pulse pressure ratio
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).
The costs of publication of this article were defrayed in part by the
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solely to indicate this fact.
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.
MATERIALS AND METHODS
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
0363-6135/99 $5.00 Copyright r 1999 the American Physiological Society
H725
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Segers, Patrick, Serge Brimioulle, Nikos Stergiopulos, Nico Westerhof, Robert Naeije, Marco Maggiorini,
and Pascal Verdonck. Pulmonary arterial compliance in
dogs and pigs: the three-element windkessel model revisited.
Am. J. Physiol. 277 (Heart Circ. Physiol. 46): H725–H731,
1999.—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.
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PULMONARY ARTERIAL COMPLIANCE
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 (0⌫01 ) 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.
Fig. 1. A: 2-element windkessel (WK-2) model consisting of total
pulmonary artery compliance (Cpa ) and resistance (R) in parallel. B:
3-element windkessel model (WK-3) consisting of characteristic
impedance (Z0 ), Cpa, and R.
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 leastsquares 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
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).
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 · s⫺1 at baseline to 0.116 ⫾ 0.041
mmHg · ml⫺1 · s⫺1 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 · ml⫺1 · s⫺1 ) and the pigs (0.071 ⫾ 0.008 mmHg ·
ml⫺1 · s⫺1 ). At embolization level 2, Z0 in the dogs
(0.068 ⫾ 0.015 mmHg · ml⫺1 · s⫺1 ) was lower than in the
pigs (0.116 ⫾ 0.041 mmHg · ml⫺1 · s⫺1; P ⬍ 0.05). In the
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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.
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PULMONARY ARTERIAL COMPLIANCE
Fig. 2. Input impedance (Zin ) modulus, normalized to
Z0, at baseline and embolization level 1 in pig (A) and
dog (B). At baseline, dog Zin is flat, with a low R (0 Hz)
and a first harmonic Zin/Z0 close to 1. Values are
means ⫾ SD for n ⫽ 6 animals per species.
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
Table 1. Hemodynamics and compliance estimates at baseline and after pulmonary
embolization for isoflow and isobaric conditions
Isoflow Conditions
Baseline
Ppa
Pigs
Dogs
Q
Pigs
Dogs
HR
Pigs
Dogs
R
Pigs
Dogs
RP-Q
Pigs
Dogs
Z0
Pigs
Dogs
0 ⌫ 01
Pigs
Dogs
CPPM
Pigs
Dogs
CWK-3
Pigs
Dogs
SV/PP
Pigs
Dogs
R · CPPM
Pigs
Dogs
RP-Q · CPPM
Pigs
Dogs
Embolization 1
Isobaric Conditions
Embolization 2
P
Baseline
Embolization 1
Embolization 2
32.7 ⫾ 4.3
32.4 ⫾ 3.4
P
24.3 ⫾ 1.2
15.3 ⫾ 2.3†
40.4 ⫾ 3.4
32.8 ⫾ 4.0†
52.1 ⫾ 6.6
39.1 ⫾ 6.9†
⬍0.001
⬍0.001
29.5 ⫾ 2.7
17.8 ⫾ 2.3†
32.3 ⫾ 4.6
31.3 ⫾ 2.0
2.4 ⫾ 0.4
3.0 ⫾ 0.7
2.3 ⫾ 0.6
2.3 ⫾ 0.5
2.0 ⫾ 0.7
2.3 ⫾ 0.5
NS
NS
3.0 ⫾ 0.7
3.5 ⫾ 1.0
1.5 ⫾ 0.8
2.6 ⫾ 0.7
0.6 ⫾ 0.6
1.6 ⫾ 0.7*
⬍0.001
⬍0.001
132 ⫾ 28
132 ⫾ 32
138 ⫾ 24
127 ⫾ 26
139 ⫾ 26
140 ⫾ 28
NS
NS
123 ⫾ 27
116 ⫾ 29
150 ⫾ 24
127 ⫾ 28
130 ⫾ 40
148 ⫾ 24
NS
⬍0.001
0.64 ⫾ 0.15
0.32 ⫾ 0.09†
1.11 ⫾ 0.33
0.90 ⫾ 0.16
1.67 ⫾ 0.45
1.05 ⫾ 0.20†
⬍0.001
⬍0.001
0.62 ⫾ 0.16
0.32 ⫾ 0.08†
1.54 ⫾ 0.55
0.78 ⫾ 0.20*
4.42 ⫾ 2.27
1.35 ⫾ 0.53†
⬍0.001
⬍0.001
0.44 ⫾ 0.16
0.28 ⫾ 0.17
0.68 ⫾ 0.35
0.43 ⫾ 0.07
0.90 ⫾ 0.40
0.58 ⫾ 0.20
⬍0.05
⬍0.001
0.44 ⫾ 0.16
0.28 ⫾ 0.17
0.68 ⫾ 0.35
0.43 ⫾ 0.07
0.90 ⫾ 0.40
0.58 ⫾ 0.20
⬍0.05
⬍0.001
0.071 ⫾ 0.008
0.069 ⫾ 0.023
0.106 ⫾ 0.030
0.066 ⫾ 0.022*
0.116 ⫾ 0.041
0.068 ⫾ 0.015*
⬍0.05
NS
0.064 ⫾ 0.012
0.069 ⫾ 0.022
0.138 ⫾ 0.029
0.064 ⫾ 0.027†
0.120 ⫾ 0.031
0.061 ⫾ 0.017†
⬍0.01
NS
0.54 ⫾ 0.04
0.29 ⫾ 0.10†
0.59 ⫾ 0.10
0.58 ⫾ 0.09
0.58 ⫾ 0.08
0.55 ⫾ 0.07
NS
⬍0.001
0.54 ⫾ 0.06
0.33 ⫾ 0.08†
0.56 ⫾ 0.11
0.55 ⫾ 0.10
0.69 ⫾ 0.13
0.62 ⫾ 0.07
⬍0.05
⬍0.001
0.72 ⫾ 0.23
1.14 ⫾ 0.29*
0.44 ⫾ 0.11
0.62 ⫾ 0.18*
0.37 ⫾ 0.14
0.54 ⫾ 0.16
⬍0.001
⬍0.001
0.80 ⫾ 0.40
1.33 ⫾ 0.33*
0.38 ⫾ 0.14
0.72 ⫾ 0.23*
0.31 ⫾ 0.08
0.60 ⫾ 0.19†
⬍0.001
⬍0.001
0.95 ⫾ 0.34
2.61 ⫾ 0.89†
0.56 ⫾ 0.20
0.80 ⫾ 0.19*
0.45 ⫾ 0.18
0.76 ⫾ 0.17*
⬍0.001
⬍0.001
1.04 ⫾ 0.54
2.54 ⫾ 0.60†
0.51 ⫾ 0.22
1.00 ⫾ 0.25*
0.39 ⫾ 0.11
0.79 ⫾ 0.20†
⬍0.01
⬍0.001
1.36 ⫾ 0.46
2.20 ⫾ 0.58*
0.80 ⫾ 0.21
0.90 ⫾ 0.22
0.58 ⫾ 0.17
0.79 ⫾ 0.26
⬍0.001
⬍0.001
1.61 ⫾ 0.84
2.36 ⫾ 0.50
0.57 ⫾ 0.28
1.07 ⫾ 0.30*
0.32 ⫾ 0.10
0.86 ⫾ 0.33†
⬍0.001
⬍0.001
0.44 ⫾ 0.09
0.35 ⫾ 0.08
0.48 ⫾ 0.16
0.56 ⫾ 0.21
0.62 ⫾ 0.31
0.55 ⫾ 0.08
NS
⬍0.05
0.46 ⫾ 0.13
0.42 ⫾ 0.12
0.54 ⫾ 0.14
0.54 ⫾ 0.15
1.37 ⫾ 0.72
0.74 ⫾ 0.19
⬍0.01
⬍0.001
0.31 ⫾ 0.07
0.30 ⫾ 0.08
0.27 ⫾ 0.13
0.26 ⫾ 0.06
0.32 ⫾ 0.08
0.30 ⫾ 0.07
NS
NS
0.35 ⫾ 0.08
0.36 ⫾ 0.22
0.22 ⫾ 0.07
0.27 ⫾ 0.08
0.37 ⫾ 0.21
0.34 ⫾ 0.16
NS
NS
NS
⬍0.001
Values are means ⫾ SD (n ⫽ 6 animals in each group, except CWK-3 in dogs at baseline, where n ⫽ 5) at conditions of matched flow (isoflow)
and matched pulmonary arterial pressure (isobaric). Ppa , mean pulmonary arterial pressure (in mmHg); Q, cardiac output (in l/min); HR,
heart rate (in beats/min); R, resistance as ratio of mean pressure to flow (in mmHg · ml⫺1 · s⫺1 ); RP-Q , resistance as slope of pressure-flow
relation (in mmHg · ml⫺1 · s⫺1 ); Z0 , characteristic impedance (in mmHg · ml⫺1 · s⫺1 ); 0 ⌫ 01 , first harmonic modulus reflection coefficient; CPPM ,
pulse pressure method compliance (in ml/mmHg); CWK-3 , windkessel compliance (in ml/mmHg); SV/PP, stroke volume-to-pulse pressure ratio
(in ml/mmHg); R · CPPM and RP-Q · CPPM , windkessel time constant (in s). * P ⬍ 0.05, † P ⬍ 0.01, dogs vs. pigs; NS, not significant.
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pigs, 0⌫01 was unchanged from baseline to embolization
(range 0.54–0.59). In the dogs, 0⌫01 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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PULMONARY ARTERIAL COMPLIANCE
Fig. 3. Estimated pulmonary arterial
compliance (CPPM ) in pigs and dogs as a
function of pulmonary arterial pressure (Ppa ) (A) and flow (B) using pulse
pressure method (PPM). Compliance
decreases with increasing pressure and
increases with increasing flow. Values
are means ⫾ SD for n ⫽ 6 animals per
species, except for dogs at baseline
where n ⫽ 5.
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.
WK-3. The WK-3 model yielded higher compliance
estimates than PPM (P ⬍ 0.001) (Fig. 4, Table 1). In one
dog, at baseline, the WK-3 model yielded the unrealistically high value of 13.3 ml/mmHg; this single value was
excluded for computation of the mean ⫾ SD of CWK-3 at
baseline. In the pigs, CWK-3 ⫽ 0.95 ⫾ 0.34 ml/mmHg at
baseline and 0.45 ⫾ 0.18 ml/mmHg at embolization
level 2; in the dogs, CWK-3 ⫽ 2.61 ⫾ 0.88 and 0.76 ⫾ 0.17
ml/mmHg, respectively. The differences between pigs
and dogs were significant at baseline (P ⬍ 0.01) and at
both embolization levels (P ⬍ 0.05).
SV/PP. The changes of SV/PP with embolization level
were similar to what was found with the other two
methods (Table 1). At baseline, SV/PP was higher in the
dogs (2.20 ⫾ 0.58 vs. 1.36 ⫾ 0.46 ml/mmHg; P ⬍ 0.05).
At embolization, compliance was higher in the dogs,
but the difference was no longer statistically significant.
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
Fig. 4. Relation between compliance estimates using PPM (CPPM )
and WK-3 model (CWK-3 ). Solid line, regression line for all data except
dogs at baseline (dogs-BL); dotted line, line of identity.
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
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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, 0⌫01 increased from baseline (0.54 ⫾
0.06) to embolization (level 2: 0.69 ⫾ 0.13; P ⬍ 0.05). In
the dogs, 0⌫01 was lower at baseline than after embolization (P ⬍ 0.001).
PULMONARY ARTERIAL COMPLIANCE
DISCUSSION
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
Fig. 5. Bland-Altman plot showing difference between CPPM, using
flow-dependent R (ratio of mean pressure to flow), and CPPM(P-Q),
using flow-independent resistance (RP-Q; slope of pressure-flow relation). Solid line, mean difference; dashed lines, mean value ⫾ 2SD.
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.
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
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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.
H729
H730
PULMONARY ARTERIAL COMPLIANCE
Fig. 6. Measured Zin modulus (r) 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.
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 pressureflow 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
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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 threeelement 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.
PULMONARY ARTERIAL COMPLIANCE
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.
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:
[email protected]).
Received 23 July 1998; accepted in final form 31 March 1999.
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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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