INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY EXPERIMENTAL ANALYSIS DISCUSSION REFERENCES

AS THE HEART BEATS, pressure and flow pulse waves travel away from the heart and are reflected back toward the heart from various locations in the arterial system. Within a particular beat, a reflected wave, reaching the heart, is rereflected. The observed pulsatile pressure (PP) and flow are thus conventionally viewed as the sum of multiple forward and reflected pulse waves (2). Although the physics of pulse wave propagation and reflection is well understood, it is not clear how reflections contribute to arterial load or how they affect blood pressure and flow in the dynamically coupled heart-arterial system.

Traditionally, reflection is believed to significantly increase input impedance (Z_in), peak systolic pressure (P_s), PP, and stroke work (SW). This view was based, in part, on the notion that the forward and reflected pressure waves can only add constructively and, thus, always increase pressure. This view seems to be corroborated by experimental and clinical evidence (22, 36). As a corollary, investigators have suggested that reducing reflections should be a clinical goal for those with isolated systolic hypertension (20, 33). This viewpoint also suggests that the mammalian arterial system has evolved to minimize reflection (31).

Recently, however, the traditional view has been challenged. Propagating waves were recognized to be strictly oscillatory phenomena and, thus, can raise and lower pressure (2). With the use of single and T-tube models to represent the arterial system, model and experimental studies were performed to determine the effects of reflection while other arterial and ventricular parameters were controlled (4, 5). Results suggested that, in actuality, reflection can decrease SW and has a minor effect on peak P_s and mean arterial pressure (3-5). On retrospection, the experimental and clinical evidence that supports the traditional view is considered flawed because of confounding changes in other factors that strongly affect pressure and flow, primarily peripheral resistance, preload, and heart rate (HR) (4, 5).

The goal of the present work is to provide a simple model-independent method to determine, from experimental data, the effect of constructive and destructive addition of forward and reflected pulse waves on measured Z_in, SW, and PP.

	THEORY

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Relationship of arterial load to pulse wave reflection. There is little disagreement about how to describe the dynamic load formed by an arterial system when the system is predominantly linear. Z_in describes the pressure-flow relationship independent of input pressure (P_in) or flow (Q_in) (16, 19, 29)

(1)

where  is frequency and j is . Z_in is a complex quantity and must be described by two components: magnitude (|Z_in|) and phase (_Zin) or real (Re[Z_in]) and imaginary (Im[Z_in]) parts.

(2)

(3)

Interpreting arterial system load. To determine whether reflection increases or decreases arterial load, a measure or index of arterial load must be defined. This is more challenging than it first may appear. Although Z_in fully characterizes the arterial load, its use as a measure is problematic because of its complex nature (Eq. 1). Because Z_in is two-dimensional, there are several measures one can use to determine whether one complex load is larger than another. Two practical measures derived from Z_in are presented in the literature (21, 23): measure A

(4a)

and measure B

(4b)

At a particular frequency, both measures have one-dimensional, real values and represent two ways of viewing the same complex quantity. These two measures are also complementary, since the phase of Z_in can be completely recovered when |Z_in| and Re[Z_in] are known: _Zin = cos¹(Re[Z_in]/|Z_in|).

(5)

(6)

(7)

(8)

View larger version (10K): [in this window] [in a new window]   Fig. 1.   Two measures of input impedance (Zin). Top: modulus relates the magnitude of oscillatory flow at nth harmonic (n) to nth harmonic of oscillatory pressure (n). Bottom: real value of Zin (Re[Zin]) relates an input flow harmonic to nth harmonic of oscillatory power (n).

Arterial load with and without reflection. Substituting Eq. 3 into Eq. 4 expresses these two measures in terms of : measure A

(9a)

and measure B

(9b)

In the reflectionless case, Z_in = Z₀; measure A degenerates into |Z₀|, and measure B degenerates into Re[Z₀]. Generally, Re[Z₀] can be approximated by |Z₀| (35).

(10a)

View larger version (18K): [in this window] [in a new window]   Fig. 2.   Importance of phase of reflection coefficient (). Circles represent no effect. Orientation of triangles indicates how reflection influences a measure of arterial load. Reflection increases or decreases Zin depending on whether the forward and reflected waves (Pf and Pr, respectively) are adding predominantly constructively or destructively. Z0, characteristic impedance; Re[Z0], real part of Z0; , global reflection coefficient.

General graphical approach to relate arterial load to reflection. This can be provided by a single polar plot of () (Fig. 3). To simplify, Z₀ is assumed to be real (noncomplex). The origin corresponds to the reflectionless case (i.e., || = 0), and the outer boundary corresponds to the maximum value of || (i.e., || = 1). Figure 3 portrays three distinct regions. The right half of the plot (90° <  < +90°) corresponds to combinations of || and that satisfy condition A (Eq. 10a). That is, reflection increases |Z_in|. The inner circle on the right half of the plot corresponds to all magnitudes and phases of  that satisfy conditions A and B (Eq. 10). Thus reflection increases |Z_in| and Re[Z_in]. The left side of the graph (90° <  < 270°) corresponds to combinations of || and that decrease |Z_in| and Re[Z_in].

View larger version (47K): [in this window] [in a new window]   Fig. 3.   Polar plot of regions of ||ej. Arrows indicate whether reflection increases () or decreases () the 2 measures of arterial load. The center corresponds to reflectionless case. The left half of the plane (90° <  < +270°) corresponds to values of  that decrease |Zin| (i.e., |Zin| < |Z0|) and Re[Zin] (i.e., Re[Zin] < Re[Z0]). The right half of the plane (90° <  < +90°) has 2 distinct regions. The region inside the inner circle corresponds to values of  that increase |Zin| and Re[Zin]. The region outside the inner circle on the right side corresponds to values of  that increase |Zin| yet decrease Re[Zin].

Aortic pressure and SW in a system with and without reflection. Aortic pressure and SW depend on pulse wave reflection and input aortic flow. This presents a singular obstacle to determining the effect of reflection independent of flow. Experimental changes in reflection usually evoke confounding changes in flow. Instead of investigating whether changing reflection raises or lowers pressure and SW, a fundamentally different question can be posed: How does reflection transform the input flow into aortic pressure and SW?

View larger version (13K): [in this window] [in a new window]   Fig. 4.   Schema for determining the effect of reflection in a particular experiment. Dashed lines represent mean pressures, which are assumed to be constant. A: experimentally measured pressure and flow are related by Zin. B: pressure predicted if the system were reflectionless. Pressure and flow are related by Z0. Z0 is assumed to be constant, as is Zin at zero frequency. Comparison of A and B reveals effect of reflection on pressure for a given flow.

	EXPERIMENTAL ANALYSIS

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Input impedance. The proposed methodologies do not require any particular arterial system model and can be easily used to derive information from measured data. Figure 5 illustrates |Z_in| and Re[Z_in] calculated from pressure and flow measured at the aortic root of an open-chest anesthetized dog with stable sinus arrhythmia (11). The experimental details are reported by Hettrick et al. (11). For reference, Z₀ is also plotted (calculated from high-frequency components of Z_in). The details of the calculation can be found in Westerhof et al. (34). Although |Z_in| > |Z_o| for most frequencies, Re[Z_in] < Re[Z₀] for frequencies between 1.3 and 3.0 Hz. There is a minimum in Re[Z_in] at 2.4 Hz (corresponding to 145 beats/min), which is similar to the dog's resting HR (2 Hz).

View larger version (11K): [in this window] [in a new window]   Fig. 5.   Example of |Zin| (A) and Re[Zin] (B) measured at the aortic root of an open-chest anesthetized dog. Arrows indicate resting heart rate. All values with coherence <0.95 were eliminated (thus eliminating data significantly influenced by noise and nonlinear effects). Z0 was calculated from frequencies >4 Hz. (For experimental details of study see Ref. 11.)

Pressure and power in a reflectionless system. Because aortic pressure and arterial system power dissipation are functions of time, it is inherently difficult to compare values of these measures under different conditions. This situation recalls the difficulty in comparing two values of Z_in, and it is likewise necessary to define relevant indexes of pressure and SW (or power). Certain indexes have already emerged in the literature. For instance, peak P_s and end-diastolic pressure (P_d) are most often used clinically. However, some have championed indexes such as mean P_s (_s), critical to ventricular afterload during ejection, and mean P_d (_d), critical to coronary perfusion (22). PP may also be an important index, because it has been associated with coronary heart disease (9). To describe power dissipation, SW has become a standard index. These indexes of pressure and power are by no means exhaustive and merely serve as a convenient means to compare two time-varying pressure and power curves.

1

View larger version (11K): [in this window] [in a new window]   Fig. 6.   Measured pressure (thick lines) and theoretical reflectionless pressure (thin lines) for a dog in the control case (B), vasoconstricted by methoxamine (C), and vasodilated by nitroprusside (A). Z0 was calculated by averaging |Zin| for harmonics 4-10.

(11)

View larger version (22K): [in this window] [in a new window]   Fig. 7.   Effect of reflection on various indexes for a dog in the control case, vasodilated by nitroprusside, and vasoconstricted by methoxamine. Each index value is calculated from the difference between the index derived from measured pressure and the index derived from Q · Z0 (Eq. 11), where Q is flow. Ps, systolic pressure; s, mean Ps; Pd, diastolic pressure; d, mean Pd; PP, pulse pressure; SW, stroke work.

	DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY EXPERIMENTAL ANALYSIS DISCUSSION REFERENCES

The present work illustrates that reflection can potentially increase and decrease vascular Z_in, PP, and SW. The proposed approach allows measured data to be analyzed, while the inevitable changes in other variables that occur with most experimental approaches, such as peripheral resistance, ventricular preload, cardiac contractility, mean pressure, HR, and ejection period, are avoided. It also clarifies the role of reflection in a model-independent manner. It thus has a generality similar to methods to estimate phase velocity from apparent phase velocity (24) and total arterial compliance from apparent arterial compliance (25, 26). By applying this model-independent analysis to an illustrative example, reflection was shown to have a relatively small effect on PP and SW under normal blood pressure conditions and to increase PP and SW in pharmacologically induced vasodilation and vasoconstriction.

Effect of reflection on arterial load. Two measures of arterial load were presented. These measures are independent of heart properties, much like an ideal index of myocardial contractility is independent of vascular load. These measures, both derived from Z_in, emphasize two aspects of the load formed by the arterial tree. It is possible for reflection to decrease one measure of load and increase another. This is indeed the case for an experimental condition illustrated in Fig. 5. The fact that |Z_in| is reported more often than Re[Z_in] is consistent with the bias, prevalent in the literature, toward a view that reflection only increases arterial load. This does not mean that the dominant view is wrong; it is only myopic. Because Z_in is two-dimensional, other indexes of load, besides |Z_in|, may be no less important when the effects of reflection on arterial load are evaluated.

Theoretically removing reflection. This work has a narrowly defined goal: to characterize the effect of reflection in a particular system in a particular state. That is, the question addressed is how reflection impacts the transformation of an input aortic flow into aortic pressure and power. This work does not explicitly address the effect of an incremental change in reflection on pressure and SW. This fundamentally different question requires a fundamentally different approach. This is because a change in the reflection coefficient has a direct effect on pressure and SW (i.e., constructive and destructive wave interference) and an indirect effect via changes in flow (Eqs. 6-8). This indirect effect depends on properties of the heart as well as properties of the arterial system. The approach used here can only elucidate the direct effect of reflection (27).

Effect of reflection on pressure and power. Analysis of three specific experimental conditions (Figs. 6 and 7) clarifies how constructive and destructive interference of forward and reflected waves affects pressure and power in a particular system. In the control case, reflection alters pressure morphology but has a relatively small effect on most of the indexes of pressure and SW analyzed here. In contrast, reflection may have a larger influence in vasoconstriction or vasodilation. In both cases, reflection increased PP and SW more than in control. It seems, then, that reflection is certainly tolerable and perhaps optimized for the system under normal conditions. Although the results apply only to the illustrative case analyzed here, this approach can be used to evaluate the role of reflection in humans and other animals under various experimental conditions.

Identifying prevalent misconceptions. In the light of these findings, a number of common misconceptions in the literature become apparent. First, |Z_in| is often mistakenly believed to determine the SW given a particular input flow (32). |Z_in| is only a piece of the story; oscillatory power (and thus SW) depends on Re[Z_in] (=|Z_in|cos[_Zin]). The critical role of _Zin, like that of , is often overlooked. This oversight can lead to confusion, for instance, when determining the frequency at which arterial load is minimized. According to the dog data illustrated in Fig. 5, the minimum in Re[Z_in] occurs at a frequency close to resting HR.

(12a)

(12b)

View larger version (16K): [in this window] [in a new window]   Fig. 8.   A: incorrect analysis of pressure into forward (Pf) and reflected (i.e., backward, Pb) waves. Mistakenly substituting steady plus oscillator pressure components ( + ) into Eq. 12 yields a reflected pressure pulse with a value that appears to be positive throughout the cardiac cycle. B: correct analysis that yields a reflected pressure that is positive and negative. Figure was adapted from Campbell et al. (7) and Berger et al. (4).

Limitations of estimating Z₀. This work provides a new method to determine the effect of reflection on a measured pressure-flow pair. The particular results, presented in Figs. 5-7, depend on an accurate estimate of Z₀. The literature provides several competing methods to estimate Z₀ from measured pressure and flow (8, 13, 15); the most widely used method, averaging higher-frequency components of |Z_in|, was applied here. There may be some cases where a more accurate measure of Z₀ might impact the determination of the effects of reflection. As better methods to estimate Z₀ from pressure and flow are developed, the stronger the interpretive value of the present methodology will become.

Implications for the optimum design of the mammalian arterial system. It has been shown that the ratio Z_in/Z₀ is similar in different mammals of widely varying body size (32). This implies that  is similar in different mammals, since Z_in/Z₀ is equal to (1 + )/(1  ) (14). Noordergraaf et al. (18) originally conjectured that resting HR of a particular mammal is set at a frequency that minimizes SW. However, this position has been challenged because minimum |Z_in| occurs at a frequency well above that corresponding to normal resting HR (32). This stance is based on the belief that |Z_in| solely determines SW. Because SW is determined by Re[Z_in] and not |Z_in| (Eqs. 7 and 8), it can now be established that the original conjecture may be essentially correct. However, the design of the mammalian arterial system may be subtler and less constrained than previously believed.