modeling in physiology
Apparent arterial compliance
CHRISTOPHER M. QUICK,1 DAVID S. BERGER,2 AND ABRAHAM NOORDERGRAAF3
Research Laboratory, Department of Biomedical Engineering, Rutgers University,
Piscataway, New Jersey 08855-0909; 2Cardiology Section, Department of Medicine, University of
Chicago, Chicago, Illinois 60637; and 3Cardiovascular Studies Unit, University of Pennsylvania,
Philadelphia, Pennsylvania 19104-6392
1Cardiovascular
Quick, Christopher M., David S. Berger, and Abraham Noordergraaf. Apparent arterial compliance. Am. J.
Physiol. 274 (Heart Circ. Physiol. 43): H1393–H1403, 1998.—
Recently, there has been renewed interest in estimating total
arterial compliance. Because it cannot be measured directly, a
lumped model is usually applied to derive compliance from
aortic pressure and flow. The archetypical model, the classical
two-element windkessel, assumes 1) system linearity and 2)
infinite pulse wave velocity. To generalize this model, investigators have added more elements and have incorporated
nonlinearities. A different approach is taken here. It is
assumed that the arterial system 1) is linear and 2) has finite
pulse wave velocity. In doing so, the windkessel is generalized
by describing compliance as a complex function of frequency
that relates input pressure to volume stored. By applying
transmission theory, this relationship is shown to be a
function of heart rate, peripheral resistance, and pulse wave
reflection. Because this pressure-volume relationship is generally not equal to total arterial compliance, it is termed
‘‘apparent compliance.’’ This new concept forms the natural
counterpart to the established concept of apparent pulse
wave velocity.
windkessel; hemodynamics; input impedance; pulse wave
reflection
arose out of two distinct
competing schools of thought. In the ‘‘distributed school’’
the arterial system was viewed as an infinitely long
tube with finite pulse wave velocity. Local arterial
compliance was assumed to be a major determinant of
the pressure-flow relationship. In the ‘‘windkessel
school’’ the arterial system was viewed as a chamber of
finite length and infinite pulse wave velocity. Global
arterial compliance was assumed to determine the
pressure-flow relationship (17). These two schools have
historical and conceptual similarities that are essential
to explain.
From the mid-19th to the mid-20th century, the
distributed school was intensely interested in measuring and explaining pulse wave velocity. Described by
the Moens-Korteweg formula, it was assumed to have a
value, c0, depending only on vessel diameter, blood
density, and local arterial compliance. Thus this model’s ability to estimate local arterial compliance from
pressure measured at two locations has seemed very
promising. However, four problems arose: 1) the numerous available methods to determine c0 yielded inconsisMODERN ARTERIAL DYNAMICS
tent values (18), 2) c0 was sensitive to changes in heart
rate (20), 3) c0 was sensitive to changes in blood
pressure (5), and 4) the model underestimated measured c0 (5, 20). Investigators tried to solve these
problems by developing new models that incorporated
viscoelasticity and nonlinear mechanical properties (5,
20). Although these new models may have yielded
better phenomenological descriptions of the arterial
wall, they did not solve these problems.
The successful resolution of these problems came in
two steps. The first step was to apply Fourier analysis
to experimental data (18). It became clear that the
measured pulse wave velocity in an artery could not be
represented by a single number, as had been assumed,
but instead was a strong function of frequency. The
second step was to abandon the assumption of infinite
length and apply transmission line theory. Transmission theory predicted that the finite length of the
arterial system can give rise to pulse wave reflections.
Pulse wave reflections, sensitive to heart rate and
peripheral vasculature, can cause measured velocity to
be much different from the phase velocity (the velocity
without reflections) (18, 26). Pulse wave velocity predicted by the Moens-Korteweg formula emerged as a
special case in which frequency is very high. Because
the presence of reflected waves masks the true pulse
wave velocity, the observed pulse wave velocity was
given the name ‘‘apparent pulse wave velocity’’ (capp )
(18).
Recently, there has been much interest in determining total arterial compliance (9, 16, 24, 25, 30). Described by the windkessel model, it is generally assumed to have a value, Cw, depending only on the total
compliance of the large arteries and the peripheral
resistance. Thus this model’s ability to estimate global
arterial compliance from pressure and flow measured
at a single location has seemed very promising. However, four problems have arisen: 1) the numerous
methods to determine Cw yield inconsistent values (16,
25, 30), 2) Cw is sensitive to changes in heart rate (4, 9,
25), 3) Cw is sensitive to changes in blood pressure (4,
13, 25), and 4) the values of Cw tend to overestimate
total arterial compliance compared with a more realistic distributed model (23, 25). Investigators tried to
solve these problems by applying more complex models
(such as the 3-element model) (11, 13, 14, 23, 28) and by
incorporating nonlinear mechanical properties (4, 9,
0363-6135/98 $5.00 Copyright r 1998 the American Physiological Society
H1393
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APPARENT ARTERIAL COMPLIANCE
15, 16, 30). Although these models may yield better
descriptions of the arterial system as a whole, they may
not have solved these problems (7, 8, 25). History, it
seems, is repeating itself.
The purpose of this article is not to offer another
compliance estimation method or lumped model but
to explain these four anomalies in terms of one
coherent theory. It so happens that history offers a
solution.
arterial tree (Qin ) is equal to the flow stored (Qstored ) plus
the flow out of the arterial tree (Qout ).
Q in 5 Q stored 1 Q out
The term compliance (C) is commonly used to describe
the change in volume stored (V) per change in input
pressure (Pin ).
C5
THEORY
Generalizing the concept of compliance. Stephen Hales
qualitatively described the first lumped model of the
arterial system in 1733. As envisioned by Hales, during
systole the heart injects blood into the arterial system,
distending the large arteries. During diastole the arteries recoil, propelling the blood continuously through
the small arteries (12). As the idea evolved and was
translated into German, this description was made
analogous to early fire engines with an air chamber
or ‘‘windkessel’’ and a single outlet tube responsible
for a pressure drop (Fig. 1A). Traditionally, this
chamber has come to represent the large arteries
and the tube to represent the small arteries in parallel
(1, 9).
Otto Frank (9) first quantified the windkessel concept on the basis of conservation of mass. Flow into the
(1)
dV
(2)
dPin
Thus Qstored is
Q stored 5
dV
dt
5
dV dPin
dPin dt
5C
dPin
dt
(3)
The term resistance (R) is used to describe the output
load formed by the tube
Pin
Qout 5
(4)
R
This formulation implicitly assumes that venous pressure is zero. Substituting Eqs. 3 and 4 into Eq. 1
quantifies the windkessel concept in the time domain
Qin 5 C
dPin
dt
1
Pin
R
(5)
Frank (9) considered three possible cases: 1) C and R
are constants, 2) C is a function of pressure, and 3) C
and R are functions of pressure. With the assumption of
a constant value for compliance (denoted as Cw ) and
resistance (denoted as Rw ), the input impedance (Zin )
can be derived to describe the pressure-flow relationship in the frequency domain (electrical analog shown
in Fig. 1B)
Zin 5
Pin (v)
Qin (v)
5
Rw
1 1 j vCwRw
(6)
where v is the angular frequency (frequency 3 2p) and
j 5 Î21. Rw, termed the peripheral resistance, is the
input impedance at zero frequency and is conventionally calculated from the average pressure (Pin ) per
average flow (Q)
Rw 5
Pin
(7)
Q
Equation 6 has become the standard description adopted
for the classical windkessel.
Frank (9) noted that the experimental value C · Rw
increases as pressure falls in diastole. This value can be
found from Eq. 5 with Qin set to zero
CN · Rw 5 2
Fig. 1. A: conceptual representation of windkessel. C, compliance of
large arteries; R, total resistance of small arteries; Qin and Qout, flow
into and out of arterial tree. B: analog model of windkessel. Pin, input
pressure; Cw and Rw, windkessel compliance and resistance, respectively. C: analog model of 3-element windkessel. Z0, characteristic
impedance; P1, conceptual pressure.
Pin
dPin / dt
(8)
where CN represents a nonlinear compliance. Thus,
Frank concluded that, in the natural system, compliance is not constant but is pressure dependent, consistent with the observed pressure-dependent compliance
of the isolated aorta (9, 22). As noted above, Frank
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APPARENT ARTERIAL COMPLIANCE
considered that compliance described as a nonlinear
function was a natural generalization of his model. To
describe the nonlinear nature of the large arteries,
several investigators have suggested modified windkessels with nonlinear elements similar to that below
CN 5 ae2bP
(9)
where a and b are empirical positive constants (15,
16, 30).
These linear and nonlinear compliances can be viewed
as transfer functions relating stored volume to Pin (Fig.
2, A and B). Inherent in both descriptions is the
assumption of infinite pulse wave velocity (1, 17). This
is a result of assuming that the pressure is the same
throughout the large arteries and that changes in
volume immediately follow changes in pressure. This
assumption gives the windkessel compliance the useful
interpretation as the sum of all arterial compliances.
(Blood inertia is not considered in this description.)
This assumption has also made windkessel and distributed descriptions of the arterial system inconsistent (1).
The troublesome assumption of infinite pulse wave
velocity can be removed by generalizing the definitions
of compliance and resistance. In deriving the windkessel, Frank (9) used the concept of compliance to describe the ability of the arterial system to store blood.
To maintain this concept, a linear time-invariant transfer function can be defined that relates Pin and volume
stored (Fig. 2D). Here the derivative of stored volume
with respect to pressure will not be assumed constant
but will be allowed to be a linear time-invariant function of frequency. Because of conceptual similarities to
apparent pulse wave velocity (to be discussed below),
this transfer function will be termed ‘‘apparent compliance’’ (Capp )
Capp 5
dV
dPin
(v) 5
arterial system. Maintaining this concept, a linear
time-invariant transfer function can be defined that
relates Pin to Qout. Given the name apparent resistance
(Rapp ), this function will not be assumed constant but
will be allowed to be a function of frequency
V(v)
Pin(v)
(10)
From Eq. 3, Qstored can then be described in the frequency domain
Q stored 5 jvPinCapp
(11)
Likewise, Frank (9) used the concept of resistance to
describe the hindrance to blood flowing out of the
Rapp 5
(12)
Alternatively, Capp and Rapp could be derived from
classic two-port analysis (21).
Substituting Eqs. 11 and 12 into Eq. 1 and rearranging yields an expression for Capp in terms of Zin and Rapp
Capp 5
V(v)
Pin(v)
5
Rapp 2 Zin
jvRapp Zin
(13)
At this point, Capp is only defined as the pressurevolume transfer function and is assigned no physiological interpretation. The frequency dependence of Capp
allows the phase shift (time delay) between pressure
and volume stored that is caused by longitudinal
impedance (due to blood inertia and viscous effects). If
Pin changes rapidly, stored volume is no longer required
to follow instantaneously as in the traditional windkessel. Similarly, the frequency dependence of Rapp allows
there to be a time delay between Pin and Qout. Thus
these two generalizations remove the traditional necessity of assuming infinite pulse wave velocity. The
windkessel description can now be reconciled with
conventional transmission theory.
Reconciliation of the windkessel with transmission
line theory. The measured pressure and flow at a
particular point in the arterial system consist of a sum
of forward and reflected waves. In a linear system, they
can be decomposed into separate frequencies, with Pa
representing the sum of forward-traveling pressure
waves at a particular frequency and Pr representing the
sum of reflected pressure waves (17, 27, 29). In a
uniform vessel, pressure and flow can be described by
the following equations
P(z,t) 5 (Pae2jvz/cph 1 Pr e jvz/cph)e jvt
Q(z,t) 5
Fig. 2. A: interpretation of windkessel compliance as a constant
(noncomplex), time-invariant transfer function relating volume stored
(V) and Pin. B: Frank’s generalization of compliance as a nonlinear,
time-varying function of pressure. Defined in time domain [V(t)], this
is not a true transfer function. C: interpretation of 3-element
windkessel compliance (Cw3 ) as a time-invariant transfer function
relating volume stored to P1. D: apparent compliance (Capp ) is a linear
time-invariant transfer function relating stored volume to Pin. Capp is
allowed to be a complex function of frequency (v).
Pin(v)
Qout(v)
1
Z0
(Pae2jvz/c ph 2 Pr e jvz/cph)e jv t
(14)
(15)
where cph is the complex phase velocity and Z0 is the
characteristic impedance. Whereas the windkessel is
formulated as a function of time only, this description,
derived from the linearized Navier-Stokes equations, is
formulated as a function of time as well as position
(z) (17).
With Eqs. 14 and 15, Zin for a distributed linear
system can be calculated. First, it is convenient to
define the global reflection coefficient (G) as the ratio of
Pr to the incident pressure wave (Pa ) at the entrance of
an arterial tree
G5
Pr
Pa
(16)
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APPARENT ARTERIAL COMPLIANCE
Zin can then be expressed in terms of G and Z0
Zin 5
P(0,t)
Q(0,t)
5 Z0
11G
12G
(17)
Any mismatch in Z0 values from one point to another in
an arterial tree will cause pulse wave reflection and a
nonzero G (3, 17, 26, 27, 29). From Eq. 17, it is evident
that Z0 is the Zin in the absence of pulse wave reflection.
Z0, in turn, can be related to local compliance per unit
length (Cl ) and the longitudinal impedance per unit
length (Zl ) or cph (17)
Z0 5
Î
Zl
jvCl
5
1
cphCl
(18)
From Eqs. 13, 16, and 17, Capp can now be interpreted
as a function of pulse wave reflection
Capp 5
Rapp(1 2 G) 2 Z0(1 1 G)
jv Rapp Z0(1 1 G)
5
Rapp(Pa 2 Pr ) 2 Z0(Pa 1 Pr )
(19)
jvRapp Z0(Pa 1 Pr )
From this general equation, it is clear that the value of
Capp depends on Rapp. This should not be surprising,
since Rapp defines how volume leaves the system. The
control volume of interest, therefore, depends on the
functional form of Rapp. The particular control volume
of interest to Frank (9) was the systemic arterial
system excluding the arterioles.
The complicated expression in Eq. 19 can be clarified
by analyzing the three special cases for a given v
illustrated in Fig. 3. For instance, when G 5 1, incident
and reflected pressure waves have the same magnitude
and phase. This causes them to add constructively,
making the pulse pressure large. Thus Capp would be
relatively small. When G 5 21, the incident and
reflected pressure waves have the same magnitude but
are 180° out of phase. This causes them to add destructively, making the pulse pressure zero. Thus Capp would
be infinite. A reflection coefficient with a value of zero
lies somewhere between these extremes. In this special
case, Capp is related only to the local compliance embedded in Z0 (Eq. 18). It has been shown that reflection
depends on how compliance is distributed (3, 27). Thus,
when reflection is nonzero, Capp depends on the dis-
Fig. 4. Distributed model of arterial system consisting of a uniform
linear transmission line terminated with a 3-element windkessel (see
Fig. 1C). Model was proposed originally by Berger et al. (3) to relate
systemic arterial pressure to pulse wave reflection. Ct and Cp, total
tube and terminal compliance; L, length; cph, complex phase velocity;
ZL, complex load.
tribution of arterial compliance, not just the total
arterial compliance.
Relating apparent compliance to total arterial compliance. To illustrate how Capp is related to total vessel
compliance in a specific case, a simple distributed
model can be applied (3). As shown in Fig. 4, the first
part of this model consists of a transmission line. It is
described by its Z0, cph, length (L), and total tube
compliance (Ct ). The second part of this model consists
of a complex load (ZL ), which also contains a compliance
(Cp ).
This model is completely described by its Zin. To
determine it, the first step is to set ZL to P(L,t)/Q(L,t)
and solve for G
G5
ZL 2 Z0
ZL 1 Z0
e22jvL/c ph
(20)
The second step is to specify the load. Here the load will
be described by the three-element windkessel (Fig. 1C)
ZL 5 Z0 1
Rp
(21)
1 1 jvCp Rp
with Rp and Cp representing the terminal resistance
and compliance (27, 28). This choice of load allows pulse
wave reflection to disappear at high frequencies, much
like an actual system (3, 17, 18, 27, 29). Equations 17,
20, and 21 completely specify the Zin of this model.
To calculate the Capp of this model, it is necessary to
specify the control volume and, thus, Rapp. For this
illustration, the control volume will be taken to be the
entire model including the load. Thus Qout can be
calculated from the pressure drop across Rp (Fig. 4)
Rapp 5
Pin
Qout
5
Pin
P1 / Rp
5
P(0,t)
(22)
[ P(L,t) 2 Z0Q(L,t)]/Rp
where P1 is a conceptual pressure and P(0,t), P(L,t),
and Q(L,t) are specified by Eqs. 14 and 15.
Substituting Eqs. 18 and 22 into Eq. 19 yields Capp
Capp 5
Fig. 3. Forward and reflected pressure waves resulting in Capp for 3
cases of reflection coefficient (G). Pr, sum of reflected pressure waves;
Pa, sum of forward-traveling pressure waves; j 5 Î21.
V
Pin
5
Ct cph
jvL
3
3 122
Ct Rp 1 (L/cph)e jvL/cph
4
Ct Rp 1 (Ct Rp 1 2 L/cph
1 2 jvCp Rp L/cph)e2jvL/cph
(23)
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APPARENT ARTERIAL COMPLIANCE
Expanding Eq. 23 in a Maclaurin series yields a
simpler expression for Capp in terms of phase velocity
Capp 5 (Ct 1 Cp) 2
Cp
C t Rp
(1 1 jvCp Rp)
Capp <
L
1c 2
ph
1O
(24)
L
2
1c 2
ph
As the phase velocity approaches infinity, higher-order
terms drop out, and Capp approaches the sum of all
compliances (Ct 1 Cp ). This is the central tenet of the
reasoning that led to the classical windkessel. When
pulse wave velocity remains finite, as it does in reality,
Capp depends on the relative amounts of Ct and Cp as
well as the total compliance.
Viewed in another way, as Zl decreases, this distributed system becomes more like a windkessel. This can
be shown by substituting Eq. 18 into Eq. 23 and taking
the limit as Zl approaches zero (corresponding to eliminating inertial and viscous effects)
Lim Capp 5 Ct 1 Cp
(25)
Zl=0
Although Eqs. 13 and 19 were applied to the specific
model of Fig. 4, both are model independent. They can
be applied to any linear system constructed from a
branching assembly of tubes, including an actual arterial system.
To illustrate the relationships of Rapp and Capp to
arterial resistances and compliances, Rapp and Capp of
the model above are plotted for a specific case (Figs. 5
and 6). The parameter values Ct 5 0.126 ml/mmHg,
Cp 5 0.3 ml/mmHg, cph 5 420 cm/s, L 5 14 cm, and Rp 5
3.5 mmHg · s · cm23 were chosen to simulate Zin of a dog
(3). By choosing a real (noncomplex) value for cph, the
transmission line is implicitly assumed to be lossless.
Approximating apparent resistance. Capp of the model
of Fig. 4 could be found exactly from Eq. 23, since Zin
and Rapp are known. However, in an actual arterial
system, Rapp is unknown. Qout cannot be measured
directly, since blood flows out of the arterial system
through millions of arterioles. This difficulty can be
overcome by intelligently assuming a value for Rapp.
At one extreme, Rapp degenerates into the peripheral
resistance at low frequencies (Fig. 5). This can be
illustrated in the model described above by substituting Eqs. 14 and 15 into Eq. 22 and taking the limit
Lim Rapp 5 Lim Pin /Qout 5 Z0 1 Rp 5 R w
v=0
v=0
ing Rw for Rapp in Eq. 13
Rw 2 Zin
(27)
jvRw Zin
The other approximation can be calculated by taking
the limit of Capp in Eq. 13 as Rapp approaches infinity
Capp < Lim
Rapp=`
Rapp 2 Zin
jvRapp Zin
5
1
jvZin
(28)
This second approximation is exact when oscillatory
outflow from the capillaries is zero. These two approximations are plotted for the model in Fig. 6 and can be
compared with the model’s Capp without approximation.
Because Rapp is large in comparison to Zin, these two
approximations yield very similar results. Thus, for
practical purposes, Capp is insensitive to the particular
value of Rapp assumed.
Apparent viscoelasticity. Viscoelasticity is a basic
property of an arterial wall that causes an artery’s
compliance to be frequency dependent. Specifically, it
makes the phase of compliance negative and the magnitude of compliance decrease with frequency (2, 17).
This is qualitatively similar to the model’s Capp shown
in Fig. 6. If Fig. 6 were calculated from experimentally
measured data, it would be reasonable to assume
that this frequency dependence is the result of viscoelasticity of the large arteries. However, Capp in Fig. 6
originates from a model with constant, nonviscoelastic
(26)
At the other extreme, Rapp approaches infinity at high
frequencies. This can be illustrated by evaluating the
limit described above, but with v approaching infinity.
This follows from an oscillatory flow out of the system
that decreases to zero at high frequencies. It is expected
that Rapp in an actual arterial system is bounded
between these two extremes.
These extremes yield two different approximations of
Capp. One approximation can be calculated by substitut-
Fig. 5. Magnitude (0Rapp0) and phase (uRapp ) of apparent resistance
(Rapp ) of model shown in Fig. 4 as a function of frequency (Eq. 22).
Total resistance of model (Z0 1 Rp, where Rp is terminal resistance) is
3.765 mmHg · s · ml21.
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APPARENT ARTERIAL COMPLIANCE
Fig. 6. Magnitude (0Capp0) and phase (uCapp ) of Capp of model shown in
Fig. 4 as a function of frequency (Eq. 23). Total compliance of model
(Ct 1 Cp ) is 0.426 ml/mmHg, illustrated by straight line. Parameter
values are given in THEORY, Relating Capp to total arterial compliance.
Three cases are Capp, Capp approximated assuming Rapp 5 Rw (Eq. 27),
and Capp approximated assuming Rapp 5 ` (Eq. 28). Capp approaches
total compliance at low frequencies. Low-frequency Capp equals total
compliance only when cph approaches infinity, and thus Z0 approaches zero (see Eqs. 18 and 24).
elements. This ‘‘apparent viscoelasticity’’ is due only to
pulse wave propagation and reflection (Eq. 20).
Apparent nonlinear compliance. Thus far, compliance
has been analyzed in the frequency domain. This
approach, when applied to the model in Fig. 4, allowed
the deviation of Capp from the total arterial compliance
to be correctly attributed to finite pulse wave velocity
and wave reflection (Eq. 23). However, most compliance
estimation methods assume a windkessel model and
analyze diastolic data in the time domain. The question
now arises whether this approach can correctly characterize total arterial compliance.
Fig. 7. A: measured aortic flow (same as dog 2
control in Fig. 9D) to be input into model described in Fig. 4 and by Eqs. 17, 20, and 21. B:
predicted theoretical pressure.
To answer this question, an experimentally measured canine aortic flow shown in Fig. 7A will be taken
as the input to the linear system described in Fig 4. The
parameter values are the same as those described
above, except Rp is given a new value to be consistent
with the data in Fig. 9D. The pressure shown in Fig. 7B
is the resulting theoretical pressure.
Following Frank’s example, the windkessel compliance of this model can be expressed in terms of its
diastolic pressure (1, 9). With the use of Eq. 8, windkessel compliance is plotted as a function of diastolic
pressure, just as Frank did a century ago (Fig. 8A). The
system’s compliance, for a significant portion of the
curve, is apparently increasing as pressure is decreasing. This is consistent with Frank’s argument that the
system has pressure-dependent arterial compliance (9,
15, 16, 22). However, in this case the system is completely
linear, and the compliance is known to be a constant. This
‘‘apparent nonlinear (pressure-dependent) compliance’’ is
the result of wave reflection and finite pulse wave velocity.
To further make this point, the nonlinear model
introduced in Eq . 9 is fit to the diastolic portion of these
data (Fig. 8B). The resulting root mean squared error
(RMSE) is less for the nonlinear windkessel (with
RMSE 5 0.58) than for the linear two- or three-element
windkessels (with RMSE 5 0.88). In this special case,
two unknown constants (a 5 1.25 ml/mmHg and b 5
0.019 mmHg21 ) better describe the data than one
unknown constant (Cw 5 0.33 ml/mmHg). Although
this nonlinear model describes the data well, it fundamentally mischaracterizes the model compliance.
From these two analyses, it becomes clear that when
data from this system are analyzed in the time domain,
a system with constant compliance can appear to have
nonlinear compliance. In this special case, this apparent nonlinear compliance is due to the frequencydependent phenomena that arise in a distributed system with finite pulse wave velocity.
EXPERIMENTAL ANALYSIS
It is desirable to know the Capp of an actual arterial
system. However, from the discussion above, it is clear
that Rapp cannot be measured directly. Therefore, the
two approximations to Capp (Eqs. 27 and 28) will be
utilized along with measured Zin. Of course, inherent in
this treatment is the assumption that the system is
approximately linear during the sampling period.
The pressure and flow data shown in Fig. 9 were
collected from the root of the aorta of two dogs. The
APPARENT ARTERIAL COMPLIANCE
Fig. 8. A: nonparametric plot of windkessel compliance as a function
of diastolic pressure in Fig. 7B predicted from simple linear model in
Fig. 4. CN, nonlinear compliance. (Only 1st 15 harmonics are utilized.) Compliance appears to be a function of pressure. However,
data were generated with a model with constant compliance. B: plot
of linear and nonlinear windkessel model fits to simulated diastolic
pressure (after 0.24 s) shown in Fig. 7B. Linear model is solution to
Eq. 5 given zero inflow and constant compliance (P 5 P0et/RwCw).
Nonlinear model is solution to Eq. 5 given zero inflow and nonlinear
compliance described by Eq. 9. Nonlinear model has a better fit,
although data were generated with a linear model.
experimental details are described elsewhere (23). In
dog 1, after a baseline was recorded, vasodilation was
induced with nitroprusside (Fig. 9, A and B). In dog 2,
vasoconstriction was induced with phenylephrine (Fig.
9, C and D). From these data, Zin was calculated and
inserted into Eqs. 27 and 28 (Fig. 10).
DISCUSSION
Interpreting conventional difficulties in estimating
windkessel compliance. The present theory, assuming a
strictly linear system, is able to explain the four general
problems in estimating total arterial compliance presented in the introduction. Similar to the historical
solution to problems estimating pulse wave velocity,
this explanation has two parts. First, Capp, like capp, is a
function of frequency, not a constant, as had been
assumed. Second, Capp, also like capp, depends on pulse
wave reflection. In light of these two parts, these four
problems will now be discussed in detail.
H1399
1) Different estimation methods yield inconsistent
estimates. This can result from fitting a constant
compliance (Cw ) to a system with complex compliance
(0Capp0ejuCapp). This problem will arise if any n-element
model is assumed (when n , `). One reason that
different estimation methods yield inconsistent estimates (while using the same model and data) is that
they weigh the contributions of the various frequency
components differently. For instance, methods that
integrate pressure with respect to time (area methods)
tend to minimize the effects of the high-frequency
components in the data. The classic time-decay method,
on the other hand, can be sensitive to them (30).
2) Compliance estimates depend on heart rate. This
can result from a Capp that is a strong function of
frequency. If the heart rate were to double, for instance,
the first harmonic at the original rate would disappear.
As a result, the lowest harmonic of Capp would be
significantly different (Fig. 10). A fit of the windkessel
to data would thus yield different values of Cw for
different heart rates (Fig. 6) (25). This would be expected, even if the actual total arterial compliance were
unaffected.
3) Compliance estimates depend on blood pressure.
This can result from a Capp that is a function of Rw and G
(Eq. 19). Changes in the vasoactive state of the peripheral vessels will alter not only Rw, and thus mean
pressure, but also G and thus pulsatile pressure (3, 6,
14, 17, 27, 29). A fit of the windkessel to data would thus
yield different values of Cw for different amounts of
reflection or levels of peripheral resistance. This would
be expected even if the actual total arterial compliance
were unaffected.
4) Compliance estimates diverge from actual total
arterial compliance. This is at the heart of the present
theory. Capp is not necessarily equal to the total arterial
compliance but depends heavily on the magnitude and
phase of the reflected pulse waves (Eq. 19). Thus Capp
depends not only on the system’s total compliance but
also on how compliance is distributed. Only under very
limited conditions do Capp, Cw, and total arterial compliance converge (Eqs. 24 and 25). When there is divergence, Cw can only approximate Capp.
Evaluating conventional explanations for problems
using the windkessel to estimate compliance. These four
phenomena can also be explained in terms of the
nonideal elastic properties of the large arteries. Investigators have been aware that finite wave speed and
nonlinear arterial compliance can cause deviation from
the linear windkessel (9, 13, 14, 16, 23, 25, 30). However, of the two, most investigators have focused on the
effect of nonlinear compliance. Two major reasons for
this can be identified. First, the nonlinear mechanical
properties of the aorta have been quantified for a long
time (22) and were originally identified by Frank (9) as
the culprit. Second, linear system analysis was applied
more recently, and there was no way to quantify the
impact of finite pulse wave velocity on compliance
estimation for an actual system.
Furthermore, a linear system with finite pulse wave
velocity and constant compliance can mimic a system
with infinite pulse wave velocity and nonlinear compli-
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APPARENT ARTERIAL COMPLIANCE
Fig. 9. A and B: measured aortic pressure and flow in dog 1 before and after
vasodilation with nitroprusside. C and
D: measured aortic pressure and flow in
dog 2 before and after vasoconstriction
with phenylephrine.
ance (described above as apparent nonlinear compliance). In an actual arterial system, if heart rate or
peripheral resistance were to change, mean pressure
and pulse wave reflection would be altered. A change in
Cw would thus reflect changes in Capp (from altered
propagation and reflection) and actual total arterial
compliance (from altered pressure). It is unknown how
much of the four anomalies described above should be
attributed to nonlinear arterial compliance or to reflection effects. Although not specifically stated above, the
present mathematical treatment can be extended by
treating a nonlinear system as piecewise linear.
Three-element windkessel. The three-element model
was introduced, because the traditional windkessel
fails to describe Zin at high frequencies (27, 28). By
incorporating Z0, the model includes the effect of inertia
and can describe Zin for very high and low frequencies.
For this reason, it is often used to estimate total
arterial compliance, with the implicit interpretation of
the model’s compliance as the total arterial compliance.
However, because the three-element windkessel incorporates the classic windkessel, it shares many of the
same inherent weaknesses.
For instance, the value of the three-element windkessel’s compliance (Cw3 ) is governed by pulse wave reflection. This can be shown by setting the Zin of the
three-element windkessel equal to Zin described by Eq.
17 and solving for Cw3 (Fig. 2C)
Cw3 5
dV
dP1
5
Rw(1 2 G) 2 2GZ0
jv2GZ0 Rw
(29)
Similar to Eq. 19, it can be shown that the term dV/dP1
equals the sum of all arterial compliances only if pulse
wave velocity is infinite (and thus Z0 disappears).
Clearly, interpreting Cw3 as the total arterial compliance can be hazardous.
Apparent compliance calculated from data. Analysis
of Capp of an actual arterial system has three notable
features (Fig. 10). First, the phase of the Capp in all
cases approaches zero at lower frequencies, similar to
that of the model in Fig. 6. However, the heart rates
were not low enough for the phases to reach zero. Also,
the plateau of Capp magnitudes evident in the simple
model is not evident in the data. Thus there may be
information about the actual compliances in frequencies between zero and heart rate that cannot be recovered from the analysis of a single beat.
Second, the phase of Capp became more negative in
dog 1 during vasodilation (Fig. 10B) and less negative
in dog 2 during vasoconstriction (Fig. 10D). This is to be
expected, since phase velocity increases with pressure.
Thus Pin and volume stored will be more in phase at
high than at low pressures. This can be predicted
theoretically from Eq. 24.
Third, the two approximations of Capp yielded similar
results (Fig. 10). Thus, similar to the model estimates
(Fig. 6), the value of Rapp assumed makes little practical
difference. This is because Rapp is so much larger than
the oscillatory components of Zin that the particular
value of Rapp becomes unimportant.
Role of arterial viscoelasticity. Isolated arteries have
been shown to be viscoelastic, meaning that their
APPARENT ARTERIAL COMPLIANCE
H1401
Fig. 10. Magnitude (0Capp0) and phase
(uCapp ) of Capp of a dog calculated by
applying Eq. 13 to data in Fig. 9. Open
symbols, approximation assuming
Rapp 5 Rw (Eq. 27); solid symbols, approximation assuming Rapp 5 ` (Eq.
28).
compliance is a function of frequency. This implies that
the total arterial compliance must also be frequency
dependent. If arterial viscoelasticity can impart a frequency dependence on the global pressure-volume relationship (Capp ), why is it necessary to invoke pulse wave
propagation and reflection? To answer this, the dynamic compliance of an isolated artery can be considered. Bergel (2) found that the magnitude of thoracic
aortic compliance (reported as aortic elastance) decreases by 6.8% as frequency is increased from 0 to 2 Hz
and by 20% as frequency is further increased to 18 Hz.
Volume lags pressure by ,5 degrees at 2 Hz and ,10
degrees at 18 Hz. In contrast, the magnitude of Capp
reported in Fig. 10 decreases an order of magnitude for
dogs A and B as frequencies increased from heart rate
(,2 Hz) to ,18 Hz. Furthermore, the phase of Capp in
Fig. 10 is several times larger than the phase shift
caused by viscoelasticity. Thus viscoelasticity alone
could only impart a small frequency dependence on
Capp. On the other hand, the simple distributed model
introduced above (Fig. 6) illustrates that pulse wave
propagation and reflection are capable of producing the
necessary frequency dependence.
Because arterial viscoelasticity and pulse wave propagation can have similar effects on the system’s global
pressure-volume relationship, it is impossible to separate actual viscoelastic compliance from ‘‘apparent
viscoelastic compliance’’ given only Pin and flow. However, the presence of viscoelasticity does not present a
limitation for the present theory. Although not explic-
itly stated, the equations derived above are valid for
viscoelastic, as well as distributed, systems (17).
Implications for appropriate windkessel use. Two
basic uses of the various windkessels have been cited in
the literature. First, windkessels have been used as
empirical models that describe the load formed by an
arterial tree (14, 27, 28). As such, any identifiable
model that reproduces this load is appropriate. The use
of the three-element model is particularly attractive as
an artificial termination device in a distributed model
(3, 6, 23, 25) (as in Fig. 4) or as a physical load to
study an ex vivo heart (28). Likewise, nonlinear windkessels may be useful to empirically describe the
load the heart sees over a large range of pressures (4,
15, 16, 30).
Second, windkessels have been used as interpretive
models to relate the load the heart sees to properties of
the arterial system. However, windkessel compliance is
not the same as vascular compliance, except for the
special conditions described above. Pharmacological
treatments that purportedly increase or decrease arterial compliance may only be changing pulse wave reflection
or pulse wave velocity, a conclusion reached experimentally (14). The central role of wave reflections in determining windkessel compliance had been appreciated by several investigators but has not been quantified (6, 14, 23).
If one wishes to recover total arterial compliance by
fitting a windkessel to pressure and flow data, it is
necessary to know whether the central tenet of the
windkessel is valid. That is, it is necessary to know
H1402
APPARENT ARTERIAL COMPLIANCE
whether the pulse wave velocity is fast enough for Pin to
be in phase with the volume stored. The phase of Capp
provides this information. If the phase shift is large,
then clearly the central tenet of the windkessel is
violated, and conventional estimation methods will fail
to quantify total arterial compliance. Thus it is more
appropriate to apply the windkessel to the control case
in dog 1 (Fig. 10B) than to the control case in dog 2 (Fig.
10D). In addition, the windkessel is more applicable to
conditions that increase pulse wave velocity (e.g., aging
and hypertension). By the same reasoning, it is more
appropriate to apply the windkessel to data with high
frequencies filtered out of the data.
All methods to estimate arterial compliance from
pressure and flow data are limited by the amount of
information contained in the data. A pulse wave must
travel away from the heart and then back for remote
compliance to have an effect on the heart. Because
reflected waves tend to add destructively at high frequencies, little information about the peripheral compliance is transmitted back to the heart at higher frequencies. At very high frequencies, the reflected wave
disappears (3, 17, 27, 29), and Zin approaches Z0 (Eq.
17). Z0 contains only information about the compliance
per unit length at the entrance of the aorta (Eq. 18).
Although pulse wave reflection confounds estimation of
true pulse wave velocity, it is essential to determine
total arterial compliance. Compliance estimation methods should therefore utilize the lowest frequency components experimentally measurable. In his experimental
procedure, Frank (9) slowed heart rate via vagal stimulation to collect data suitable for analysis. Even though
the three-element windkessel describes high frequencies better than the classic windkessel, little is gained,
since the higher frequencies contain little useful information about total arterial compliance.
Reconciliation of distributed and windkessel descriptions of the arterial system. To derive cardiac output
from measured aortic pressure, Frank (9, 10) attempted to combine distributed and lumped descriptions of the arterial system. Frank’s approach was
criticized by Apéria (1), who took issue with his inconsistent assumptions of finite and infinite pulse wave
velocity. These two competing descriptions were once
again linked within the three-element model presented
by Westerhof and co-workers (27, 28). This model
degenerates into the distributed school’s description at
high frequencies and the windkessel school’s description at low frequencies (17, 27). The three-element
windkessel thus represents a useful compromise between the two schools. However, neither school fully
embraced pulse wave propagation and reflection. Pulse
wave propagation and reflection are the phenomena
that determine the load formed by the arterial system
for all frequencies between zero and infinity. This
frequency range is covered only by a full-fledged transmission theory.
To apply transmission theory to the windkessel concept, the assumption of infinite pulse wave velocity had
to be abandoned. This invalidated the classic interpretation of the observed volume-pressure relationship as
equivalent to the sum of arterial compliances. However,
transmission line theory provides a new interpretation
(19). It may not have occurred to previous investigators
to reconcile these two schools that seemingly had
mutually exclusive bases. The key was to understand
that the windkessel’s basis is conservation of mass. The
assumption of infinite pulse wave velocity was shown to
be unnecessary. Frank (9) understood the limitations of
his model but failed to generalize compliance and
resistance as linear time-invariant transfer functions,
since the description of Zin did not become popular until
after his death. Out of this reconciliation, the classic
windkessel was shown to be a first-order approximation of a distributed system.
In conclusion, the concept of apparent compliance is
similar to that of apparent pulse wave velocity, because
both are functions of pulse wave reflection described by
transmission theory. Apparent pulse wave velocity diverges from true pulse wave velocity, because the
arterial system has a finite length. Apparent compliance diverges from the actual total arterial compliance,
because the arterial system has a finite pulse wave
velocity. With these two complementary concepts, the
windkessel and distributed schools are unified within
the domain of full-fledged transmission theory.
The authors are grateful to Dr. Sanjeev G. Shroff for generously
providing the dog data.
This material is based on work supported by an American Heart
Association Predoctoral Fellowship (C. M. Quick) and American
Heart Association Grant-in-Aid 96009940 (D. S. Berger).
Address for reprint requests: C. M. Quick, Cardiovascular Research Laboratory, Dept. of Biomedical Engineering, Rutgers University, Piscataway, NJ 08855-0909.
Received 25 June 1997; accepted in final form 30 December 1997.
REFERENCES
1. Apéria, A. Hemodynamical studies. Skand. Arch. Physiol. 83,
Suppl. 16: 1–230, 1940.
2. Bergel, D. H. The dynamic elastic properties of the arterial wall.
J. Physiol. (Lond.) 156: 458–469, 1961.
3. Berger, D. S., J. K.-J. Li, and A. Noordergraaf. Differential
effects of wave reflections and peripheral resistance on aortic
blood pressure: a model-based study. Am. J. Physiol. 266 (Heart
Circ. Physiol. 35): H1626–H1642, 1994.
4. Burattini, R., G. Gnudi, N. Westerhof, and S. Fioretti. Total
systemic arterial compliance and aortic characteristic impedance
in the dog as a function of pressure: a model based study.
Comput. Biomed. Res. 20: 154–165, 1987.
5. Bramwell, J. C., and A. V. Hill. The velocity of the pulse wave
in man. Proc. R. Soc. Lond. B Biol. Sci. 93: 298–306, 1922.
6. Campbell, K. B., R. Burattini, D. L. Bell, R. D. Kirkpatrick,
and G. G. Knowlen. Time-domain formulation of asymmetric
T-tube model of arterial system. Am. J. Physiol. 258 (Heart Circ.
Physiol. 27): H1761–H1774, 1990.
7. Fogliardi, R., R. Burattini, S. G. Shroff, and K. B. Campbell. Fit to diastolic arterial pressure by third-order lumped
model yields unreliable estimates of arterial compliance. Med.
Eng. Phys. 18: 225–233, 1996.
8. Fogliardi, R., M. Di Donfrancesco, and R. Burattini. Comparison of linear and nonlinear formulations of the threeelement windkessel model. Am. J. Physiol. 271 (Heart Circ.
Physiol. 40): H2661–H2668, 1996.
9. Frank, O. Die Grundform des arteriellen Pulses. Z. Biol. 37:
483–526, 1899. [Translated by K. Sagawa, R. K. Lie, and J.
Schaefer. J. Mol. Cell Cardiol. 22: 253–277, 1990.]
10. Frank, O. Die Elastizität der Blutgefässe. Z. Biol. 71: 255–272,
1920.
11. Goldwin, R. M., and T. B. Watt. Arterial pressure pulse contour
analysis via a mathematical model for the clinical quantification
of human vascular properties. IEEE Trans. Biomed. Eng. 14:
11–17, 1967.
APPARENT ARTERIAL COMPLIANCE
12. Hales, S. Statical Essays: Containing Haemostaticks. London,
UK: Innys and Manby, 1733, vol. II.
13. Laskey, W. K., H. G. Parker, V. A. Ferrari, W. G. Kussmaul,
and A. Noordergraaf. Estimation of total systemic arterial
compliance in humans. J. Appl. Physiol. 69: 112–119, 1990.
14. Latson, T. W., W. C. Hunter, N. Katoh, and K. Sagawa. Effect
of nitroglycerin on aortic impedance, diameter, and pulse-wave
velocity. Circ. Res. 62: 884–890, 1988.
15. Li, J. K.-J., T. Cui, and G. M. Drzewiecki. A nonlinear model
of the arterial system incorporating a pressure-dependent compliance. IEEE Trans. Biomed. Eng. 37: 673–678, 1990.
16. Liu, Z., K. P. Brin, and F. C. P. Yin. Estimation of total arterial
compliance: an improved method and evaluation of current
methods. Am. J. Physiol. 251 (Heart Circ. Physiol. 20): H588–
H600, 1986.
17. Noordergraaf, A. Circulatory System Dynamics. New York:
Academic, 1978.
18. Porjé, I. G. Studies of the arterial pulse wave, particularly in the
aorta. Acta Physiol. Scand. Suppl. 13: 1–68, 1946.
19. Quick, C. M., J. K.-J. Li, D. A. O’Hara, and A. Noordergraaf.
Reconciliation of windkessel and distributed descriptions of
linear arterial systems (Abstract). Proc. Bioeng. Conf. ASME
BED 29: 469–470, 1995.
20. Remington, J. W., W. F. Hamilton, and P. Dow. Some difficulties involved in the prediction of the stroke volume from the pulse
wave velocity. Am. J. Physiol. 144: 536–545, 1945.
21. Rose, W. C., and A. A. Shoukas. Two-port analysis of systemic
venous and arterial impedances. Am. J. Physiol. 265 (Heart Circ.
Physiol. 34): H1577–H1587, 1993.
H1403
22. Roy, C. S. The elastic properties of the arterial wall. J. Physiol.
(Lond.) 3: 125–159, 1880–1882.
23. Shroff, S. G., D. S. Berger, C. Korcarz, R. M. Lang,
R. H. Marcus, and D. E. Miller. Physiological relevance of
T-tube model parameters with emphasis on arterial compliances. Am. J. Physiol. 269 (Heart Circ. Physiol. 38): H365–H374,
1995.
24. Stergiopulos, N., J.-J. Meister, and N. Westerhof. Simple
and accurate way for estimating total and segmental arterial
compliance: the pulse pressure method. Ann. Biomed. Eng. 22:
392–397, 1994.
25. Stergiopulos, N., J.-J. Meister, and N. Westerhof. Evaluation of methods for estimation of total arterial compliance. Am. J.
Physiol. 268 (Heart Circ. Physiol. 37): H1540–H1548, 1995.
26. Taylor, M. G. An approach to an analysis of the arterial pulse
wave. I. Oscillations in an attenuating line. Phys. Med. Biol. 1:
258–269, 1957.
27. Westerhof, N. Analog Studies of Human Systemic Arterial
Hemodynamics (Ph.D. thesis). Philadelphia, PA: University of
Pennsylvania, 1968.
28. Westerhof, N., G. Elzinga, and P. Sipkema. An artificial arterial
system for pumping hearts. J. Appl. Physiol. 31: 776–781, 1971.
29. Westerhof, N., P. Sipkema, G. C. van den Bos, and G.
Elzinga. Forward and backward waves in the arterial system.
Cardiovasc. Res. 6: 648–656, 1972.
30. Yin, F. C. P., and Z. Liu. Arterial compliance physiological viewpoint. In: Vascular Dynamics. Physiological Perspectives, edited by N. Westerhof and D. R. Gross. New York: Plenum,
1989.