INTRODUCTION

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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 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 inconsistent 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" (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 3-element 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. 1A). Traditionally, this chamber has come to represent the large arteries and the tube to represent the small arteries in parallel (1, 9).

View larger version (8K): [in this window] [in a new window]   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.

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View larger version (15K): [in this window] [in a new window]   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 ().

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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 forward-traveling 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

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where c_ph 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).

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View larger version (9K): [in this window] [in a new window]   Fig. 3.   Forward and reflected pressure waves resulting in Capp for 3 cases of reflection coefficient (). Pr, sum of reflected pressure waves; Pa, sum of forward-traveling pressure waves; j = .

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 its Z₀, 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).

View larger version (7K): [in this window] [in a new window]   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.

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View larger version (11K): [in this window] [in a new window]   Fig. 5.   Magnitude (|Rapp|) and phase (Rapp) of apparent resistance (Rapp) of model shown in Fig. 4 as a function of frequency (Eq. 22). Total resistance of model (Z0 + Rp, where Rp is terminal resistance) is 3.765 mmHg · s · ml1.

View larger version (14K): [in this window] [in a new window]   Fig. 6.   Magnitude (|Capp|) and phase (Capp) of Capp of model shown in Fig. 4 as a function of frequency (Eq. 23). Total compliance of model (Ct + 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 = Rw (Eq. 27), and Capp approximated assuming Rapp =  (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).

Approximating apparent resistance. C_app of the model of Fig. 4 could be found exactly from Eq. 23, since Z_in and R_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 for R_app.

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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.

View larger version (10K): [in this window] [in a new window]   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.

View larger version (13K): [in this window] [in a new window]   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 = 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.

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	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 measured Z_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. 9 were 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, A and B). In dog 2, vasoconstriction was induced with phenylephrine (Fig. 9, C and D). From these data, Z_in was calculated and inserted into Eqs. 27 and 28 (Fig. 10).

View larger version (23K): [in this window] [in a new window]   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.

View larger version (28K): [in this window] [in a new window]   Fig. 10.   Magnitude (|Capp|) and phase (Capp) of Capp of a dog calculated by applying Eq. 13 to data in Fig. 9. Open symbols, approximation assuming Rapp = Rw (Eq. 27); solid symbols, approximation assuming Rapp =  (Eq. 28).

	DISCUSSION

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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, like c_app, is a function of frequency, not a constant, as had been assumed. Second, C_app, also like c_app, depends on pulse wave reflection. In light of these two parts, these four problems will now be discussed in detail.

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.

Three-element windkessel. The three-element model was introduced, because the traditional windkessel fails to describe Z_in at high frequencies (27, 28). By incorporating Z₀, the model includes the effect of inertia and can describe Z_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 three-element windkessel incorporates the classic windkessel, it shares many of the same inherent weaknesses.

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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.

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 pressure-volume 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 and B 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.

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).

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.