Abstract
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 twoelement windkessel, assumes1) system linearity and2) 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 pressurevolume 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
modern arterial dynamics 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 pressureflow 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 pressureflow relationship (17). These two schools have historical and conceptual similarities that are essential to explain.
From the mid19th to the mid20th century, the distributed school was intensely interested in measuring and explaining pulse wave velocity. Described by the MoensKorteweg formula, it was assumed to have a value,c _{0}, 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 determinec _{0}yielded inconsistent values (18), 2)c _{0} was sensitive to changes in heart rate (20), 3)c _{0} was sensitive to changes in blood pressure (5), and4) the model underestimated measuredc _{0} (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 MoensKorteweg 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” (c _{app}) (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, C_{w}, 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 C_{w} yield inconsistent values (16,25, 30), 2) C_{w} is sensitive to changes in heart rate (4, 9, 25), 3) C_{w} is sensitive to changes in blood pressure (4, 13, 25), and 4) the values of C_{w} 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 3element model) (11, 13,14, 23, 28) and by incorporating nonlinear mechanical properties (4, 9,15, 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.
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.1 A). 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 arterial tree (Q_{in}) is equal to the flow stored (Q_{stored}) plus the flow out of the arterial tree (Q_{out}).
Frank (9) noted that the experimental value C ⋅ R
_{w}increases as pressure falls in diastole. This value can be found fromEq. 5
with Q_{in} set to zero
These linear and nonlinear compliances can be viewed as transfer functions relating stored volume to P_{in} (Fig.2, A andB). 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 timeinvariant transfer function can be defined that relates P_{in} and volume stored (Fig. 2
D). Here the derivative of stored volume with respect to pressure will not be assumed constant but will be allowed to be a linear timeinvariant function of frequency. Because of conceptual similarities to apparent pulse wave velocity (to be discussed below), this transfer function will be termed “apparent compliance” (C_{app})
Substituting Eqs. 11
and
12
into Eq.1
and rearranging yields an expression for C_{app} in terms ofZ
_{in} andR
_{app}
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 P_{a} representing the sum of forwardtraveling pressure waves at a particular frequency and P_{r} representing the sum of reflected pressure waves (17, 27, 29). In a uniform vessel, pressure and flow can be described by the following equations
With Eqs. 14
and
15
,Z
_{in} for a distributed linear system can be calculated. First, it is convenient to define the global reflection coefficient (Γ) as the ratio of P_{r} to the incident pressure wave (P_{a}) at the entrance of an arterial tree
The complicated expression in Eq. 19 can be clarified by analyzing the three special cases for a given ω illustrated in Fig. 3. For instance, when Γ = 1, incident and reflected pressure waves have the same magnitude and phase. This causes them to add constructively, making the pulse pressure large. Thus C_{app} would be relatively small. When Γ = −1, 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 C_{app} would be infinite. A reflection coefficient with a value of zero lies somewhere between these extremes. In this special case, C_{app} is related only to the local compliance embedded inZ _{0}(Eq. 18 ). It has been shown that reflection depends on how compliance is distributed (3, 27). Thus, when reflection is nonzero, C_{app}depends on the distribution of arterial compliance, not just the total arterial compliance.
Relating apparent compliance to total arterial compliance.
To illustrate how C_{app} 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 itsZ _{0},c _{ph}, length (L), and total tube compliance (C_{t}). The second part of this model consists of a complex load (Z _{L}), which also contains a compliance (C_{p}).
This model is completely described by itsZ
_{in}. To determine it, the first step is to setZ
_{L}to P(L,t)/Q(L,t) and solve for Γ
To calculate the C_{app} of this model, it is necessary to specify the control volume and, thus,R
_{app}. For this illustration, the control volume will be taken to be the entire model including the load. Thus Q_{out} can be calculated from the pressure drop acrossR
_{p}(Fig. 4)
Substituting Eqs. 18
and
22
into Eq.19
yields C_{app}
Viewed in another way, asZ_{l}
decreases, this distributed system becomes more like a windkessel. This can be shown by substituting Eq. 18
intoEq. 23
and taking the limit asZ_{l}
approaches zero (corresponding to eliminating inertial and viscous effects)
To illustrate the relationships ofR _{app}and C_{app} to arterial resistances and compliances,R _{app}and C_{app} of the model above are plotted for a specific case (Figs. 5 and6). The parameter values C_{t} = 0.126 ml/mmHg, C_{p} = 0.3 ml/mmHg,c _{ph}= 420 cm/s, L = 14 cm, andR _{p}= 3.5 mmHg ⋅ s ⋅ cm^{−3}were chosen to simulateZ _{in} of a dog (3). By choosing a real (noncomplex) value forc _{ph}, the transmission line is implicitly assumed to be lossless.
Approximating apparent resistance.
C_{app} of the model of Fig. 4 could be found exactly from Eq. 23 , sinceZ _{in}andR _{app}are known. However, in an actual arterial system,R _{app}is unknown. Q_{out} 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 forR _{app}.
At one extreme,R
_{app} degenerates into the peripheral resistance at low frequencies (Fig. 5). This can be illustrated in the model described above by substitutingEqs. 14
and
15
into Eq.22
and taking the limit
These extremes yield two different approximations of C_{app}. One approximation can be calculated by substitutingR
_{w}forR
_{app}in Eq. 13
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 C_{app} 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, C_{app} in Fig. 6 originates from a model with constant, nonviscoelastic 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 C_{app} 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.
To answer this question, an experimentally measured canine aortic flow shown in Fig.7 A will be taken as the input to the linear system described in Fig 4. The parameter values are the same as those described above, exceptR _{p}is given a new value to be consistent with the data in Fig.9 D. The pressure shown in Fig.7 B 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 ofEq. 8 , windkessel compliance is plotted as a function of diastolic pressure, just as Frank did a century ago (Fig.8 A). 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 pressuredependent 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 (pressuredependent) compliance” is the result of wave reflection and finite pulse wave velocity.
To further make this point, the nonlinear model introduced inEq . 9 is fit to the diastolic portion of these data (Fig. 8 B). The resulting root mean squared error (RMSE) is less for the nonlinear windkessel (with RMSE = 0.58) than for the linear two or threeelement windkessels (with RMSE = 0.88). In this special case, two unknown constants (a = 1.25 ml/mmHg and b = 0.019 mmHg^{−1}) better describe the data than one unknown constant (C_{w} = 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 C_{app}of an actual arterial system. However, from the discussion above, it is clear that R _{app}cannot be measured directly. Therefore, the two approximations to C_{app} (Eqs.27 and 28 ) will be utilized along with measuredZ _{in}. 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. 9were collected from the root of the aorta of two dogs. The experimental details are described elsewhere (23). In dog 1, after a baseline was recorded, vasodilation was induced with nitroprusside (Fig. 9, Aand B). In dog 2, vasoconstriction was induced with phenylephrine (Fig. 9, C andD). From these data,Z _{in} 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, C_{app}, likec _{app}, is a function of frequency, not a constant, as had been assumed. Second, C_{app}, also likec _{app}, depends on pulse wave reflection. In light of these two parts, these four problems will now be discussed in detail.
1) Different estimation methods yield inconsistent estimates. This can result from fitting a constant compliance (C_{w}) to a system with complex compliance (‖C_{app}‖
2) Compliance estimates depend on heart rate. This can result from a C_{app} 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 C_{app} would be significantly different (Fig. 10). A fit of the windkessel to data would thus yield different values of C_{w} 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 C_{app} that is a function ofR _{w}and Γ (Eq. 19 ). Changes in the vasoactive state of the peripheral vessels will alter not onlyR _{w}, and thus mean pressure, but also Γ and thus pulsatile pressure (3, 6,14, 17, 27, 29). A fit of the windkessel to data would thus yield different values of C_{w} 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. C_{app} 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 C_{app}depends not only on the system’s total compliance but also on how compliance is distributed. Only under very limited conditions do C_{app}, C_{w}, and total arterial compliance converge (Eqs. 24 and 25 ). When there is divergence, C_{w} can only approximate C_{app}.
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 compliance (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 C_{w} would thus reflect changes in C_{app} (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.
Threeelement windkessel.
The threeelement model was introduced, because the traditional windkessel fails to describeZ _{in} at high frequencies (27, 28). By incorporatingZ _{0}, the model includes the effect of inertia and can describeZ _{in} 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 threeelement windkessel incorporates the classic windkessel, it shares many of the same inherent weaknesses.
For instance, the value of the threeelement windkessel’s compliance (C_{w3}) is governed by pulse wave reflection. This can be shown by setting theZ
_{in} of the threeelement windkessel equal toZ
_{in}described by Eq. 17
and solving for C_{w3} (Fig.2
C)
Apparent compliance calculated from data.
Analysis of C_{app} of an actual arterial system has three notable features (Fig. 10). First, the phase of the C_{app} 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 C_{app} 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 C_{app} became more negative in dog 1 during vasodilation (Fig. 10 B) and less negative in dog 2 during vasoconstriction (Fig. 10 D). This is to be expected, since phase velocity increases with pressure. Thus P_{in} 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 C_{app} yielded similar results (Fig.10). Thus, similar to the model estimates (Fig. 6), the value ofR _{app} assumed makes little practical difference. This is becauseR _{app}is so much larger than the oscillatory components ofZ _{in}that the particular value ofR _{app}becomes unimportant.
Role of arterial viscoelasticity.
Isolated arteries have been shown to be viscoelastic, meaning that their 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 pressurevolume relationship (C_{app}), 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 C_{app} reported in Fig. 10 decreases an order of magnitude for dogs A andB as frequencies increased from heart rate (∼2 Hz) to ∼18 Hz. Furthermore, the phase of C_{app} 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 C_{app}. 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 pressurevolume relationship, it is impossible to separate actual viscoelastic compliance from “apparent viscoelastic compliance” given only P_{in} and flow. However, the presence of viscoelasticity does not present a limitation for the present theory. Although not explicitly 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 threeelement 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 whether the pulse wave velocity is fast enough for P_{in} to be in phase with the volume stored. The phase of C_{app} 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 indog 1 (Fig.10 B) than to the control case indog 2 (Fig.10 D). 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), andZ _{in} approachesZ _{0}(Eq. 17 ).Z _{0} 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 threeelement 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 threeelement model presented by Westerhof and coworkers (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 threeelement 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 fullfledged 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 volumepressure 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 timeinvariant transfer functions, since the description ofZ _{in} did not become popular until after his death. Out of this reconciliation, the classic windkessel was shown to be a firstorder 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 fullfledged transmission theory.
Acknowledgments
The authors are grateful to Dr. Sanjeev G. Shroff for generously providing the dog data.
Footnotes

Address for reprint requests: C. M. Quick, Cardiovascular Research Laboratory, Dept. of Biomedical Engineering, Rutgers University, Piscataway, NJ 088550909.

This material is based on work supported by an American Heart Association Predoctoral Fellowship (C. M. Quick) and American Heart Association GrantinAid 96009940 (D. S. Berger).
 Copyright © 1998 the American Physiological Society