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Am J Physiol Heart Circ Physiol 280: H1519–H1527, 2001. Constructive and destructive addition of forward and reflected arterial pulse waves CHRISTOPHER M. QUICK,1 DAVID S. BERGER,2 AND ABRAHAM NOORDERGRAAF3 Center for Cerebrovascular Research, University of California, San Francisco, California 94110; 2Cardiology Section, Department of Medicine, University of Chicago, Chicago, Illinois 60637; and 3Cardiovascular Studies Unit, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6392 1 Received 23 March 2000; accepted in final form 31 October 2000 hemodynamics; modeling; wave propagation 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 (Zin), peak systolic pressure (Ps), PP, and stroke work (SW). This view was based, in part, on the notion that the forward and reflected Address for reprint requests and other correspondence: C. M. Quick, Center for Cerebrovascular Research, University of California at San Francisco, 1001 Portrero Ave., Rm. 3C-38, San Francisco, CA 94110 (E-mail: [email protected]). http://www.ajpheart.org 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 Ps 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 Zin, SW, and PP. THEORY 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. Zin describes the pressure-flow relationship independent of input pressure (Pin) or flow (Qin) (16, 19, 29) Zin ⫽ Pin共兲/Qin共兲 ⫽ 兩Zin兩ejZin ⫽ Re关Zin兴 ⫹ j Im关Zin兴 (1) where is frequency and j is 公⫺1. Zin is a complex quantity and must be described by two components: The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. 0363-6135/01 $5.00 Copyright © 2001 the American Physiological Society H1519 Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 Quick, Christopher M., David S. Berger, and Abraham Noordergraaf. Constructive and destructive addition of forward and reflected arterial pulse waves. Am J Physiol Heart Circ Physiol 280: H1519–H1527, 2001.—Although the physics of arterial pulse wave propagation and reflection is well understood, there is considerable debate as to the effect of reflection on vascular input impedance (Zin), pulsatile pressure, and stroke work (SW). This may be related to how reflection is studied. Conventionally, reflection is experimentally abolished (thus radically changing unrelated parameters), or a specific model is assumed from which reflection can be removed (yielding model-dependent results). The present work proposes a simple, model-independent method to evaluate the effect of reflection directly from measured pulsatile pressure (P) and flow (Q). Because characteristic impedance (Z0) is Zin in the absence of reflection, the P with reflection theoretically removed can be calculated from Q 䡠 Z0. Applying this insight to an illustrative case indicates that reflection has the least effect on P and SW at normal pressure but a greater effect with vasodilation and vasoconstriction. Zin, P, and SW are increased or decreased depending on the relative amount of constructive and destructive addition of forward and reflected arterial pulse waves. H1520 ARTERIAL PULSE WAVE REFLECTION magnitude (兩Zin兩) and phase (Zin) or real (Re[Zin]) and imaginary (Im[Zin]) parts. Similarly, to characterize the tendency of the system to reflect antegrade waves independent of input, investigators regularly use the global reflection coefficient (⌫) ⌫ ⫽ Pr共兲/Pf共兲 ⫽ 兩⌫兩ej⌫ ⫽ Re关⌫兴 ⫹ j Im关⌫兴 (2) where Pf is the forward-traveling pressure pulse and Pr is the retrograde pressure pulse observed at the system entrance. ⌫ is also complex, having a magnitude (兩⌫兩) and a phase (⌫) or, alternatively, real (Re[⌫]) and imaginary (Im[⌫]) parts. Both parts are necessary to fully describe reflection, although ⌫ is rarely reported in the literature. Zin and ⌫ are interrelated such that (3) where Z0 is the characteristic impedance, i.e., the value of Zin in the absence of reflection. It is clear from Eq. 3 that reflection is a major determinant of Zin and, thus, the load formed by an arterial system. However, because Zin and ⌫ have complex values, it is not readily apparent whether reflection increases or decreases Zin. 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 Zin fully characterizes the arterial load, its use as a measure is problematic because of its complex nature (Eq. 1). Because Zin 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 Zin are presented in the literature (21, 23): measure A 兩Zin兩 (4a) Thus measure A (兩Zin兩) describes the tendency of an arterial system to produce mean pressure and PP for a given input flow. Likewise, SW has steady and oscillatory components. It is more convenient to describe average power, the average rate at which work is dissipated in the ) and arterial system. Average power has steady (W oscillatory (W̃) components ⫹ W̃兲/HR SW ⫽ 共W (7) The steady component is, again, a function of steady flow and R and is not directly affected by reflection (i.e., 2 ⫽Q in W 䡠 R). The oscillatory power is a function of flow and Re[Zin] (16). The magnitude of the nth harmonic has the form W̃n ⫽ 1⁄2兩Q̃n兩2 䡠 Re关Zin兴 Re关Zin兴 ⫽ 兩Zin兩 cos 共Zin兲 (4b) At a particular frequency, both measures have onedimensional, real values and represent two ways of viewing the same complex quantity. These two measures are also complementary, since the phase of Zin can be completely recovered when 兩Zin兩 and Re[Zin] are known: Zin ⫽ cos⫺1(Re[Zin]/兩Zin兩). To interpret the values of 兩Zin兩 and Re[Zin], their effect on pressure and SW will be considered. Pressure ) and oscillatory (P̃) components, such has steady (P that Pin ⫽ P ⫹ P̃ (8) Thus, for a given flow, measure B (Re[Zin]) describes the tendency of an arterial system to dissipate energy. Measures A and B can be viewed as input-independent transfer functions (Fig. 1). Whereas the particular PP produced and energy required to pump blood depends on Zin and properties of the heart, these transfer functions characterize the arterial system independent of the heart. This formalism provides a convenient basis from which to quantify the effects of reflection on the arterial system load, pressure, and SW. Arterial load with and without reflection. Substituting Eq. 3 into Eq. 4 expresses these two measures in terms of ⌫: measure A 冏 兩Z in兩 ⫽ Z0 and measure B 冏 1⫹⌫ 1⫺⌫ 冋 Re关Zin兴 ⫽ Re Z0 and measure B (6) (9a) 册 1⫹⌫ 1⫺⌫ (9b) In the reflectionless case, Zin ⫽ Z0; measure A degenerates into 兩Z0兩, and measure B degenerates into Re[Z0]. Generally, Re[Z0] can be approximated by 兩Z0兩 (35). The effect of reflection on arterial load can be quantified with Eq. 9. That is, the effect of reflection can be (5) ) The steady component is the product of steady flow (Q and resistance R (the value of Zin at zero frequency) and is not directly affected by reflection, since reflection is a strictly oscillatory phenomenon (2, 4, 27). Oscillatory pressure is the oscillatory flow components (Q̃) multiplied by Zin. The magnitude of the nth harmonic of pressure (P̃n) has a simple form Fig. 1. Two measures of input impedance (Zin). Top: modulus relates the magnitude of oscillatory flow at nth harmonic (Q̃n) to nth harmonic of oscillatory pressure (P̃n). Bottom: real value of Zin (Re[Zin]) relates an input flow harmonic to nth harmonic of oscillatory power (W̃n). Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 Z in ⫽ 兩Zin兩ejZin ⫽ Z0共1 ⫹ 兩⌫兩ej⌫兲/共1 ⫺ 兩⌫兩ej⌫兲 兩P̃n兩 ⫽ 兩Q̃n兩 䡠 兩Zin兩 ARTERIAL PULSE WAVE REFLECTION quantified by comparing Zin (the arterial load when reflection is present) with Z0 (the arterial load when reflection is absent, by definition). For instance, when 冏 冏 兩Z in兩 1⫹⌫ ⫽ ⬎1 兩Z0兩 1⫺⌫ waves. For instance, when ⌫ is 0°, forward and reflected waves are in phase and add constructively. This tends to make the resulting PP large and, thus, Zin large by either measure. On the other hand, when ⌫ is 180°, forward and reflected waves are out of phase and add destructively. In this case, the PP is small, and thus Zin is small by either measure. The reflectionless case and, indeed, most physiological cases lie somewhere between these extremes. In some cases, it is possible to have a mixed system (e.g., 兩⌫兩 ⫽ 0.5 and ⌫ ⫽ ⫺75°) where reflection increases one measure of arterial load but decreases the other. This time-domain approach illustrates a few potential effects of reflection. However, to illustrate the effect of all potential combinations of ⌫ and 兩⌫兩, a more general approach is necessary. 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., 兩⌫兩 ⫽ 0), and the outer boundary corresponds to the maximum value of 兩⌫兩 (i.e., 兩⌫兩 ⫽ 1). Figure 3 portrays three distinct regions. The right half of the plot (⫺90° ⬍ ⌫ ⬍ ⫹90°) corresponds to combinations of 兩⌫兩 and ⌫ that satisfy condition A (Eq. 10a). That is, reflection increases 兩Zin兩. 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 兩Zin兩 and Re[Zin]. The left side of the graph (90° ⬍ ⌫ ⬍ 270°) corresponds to combinations of 兩⌫兩 and ⌫ that decrease 兩Zin兩 and Re[Zin]. It is possible for the different harmonics to be splayed across the different regions shown here. With this representation, the potential is clearly illustrated for reflections to increase or decrease the arterial load (28). To determine whether reflection increases or de- (10a) (condition A), then Zin described by measure A is increased by the presence of reflection. Similarly, when Re关Zin兴 ⫽ Re关Z0兴 冋 册 1⫹⌫ 1⫺⌫ 1⫹⌫ ⬎1 ⬇ Re Re关Z0兴 1⫺⌫ Re Z0 冋 册 共10b) (condition B), then Zin described by measure B is increased by the presence of reflection. Because the imaginary part of Z0 is small in the aorta, Eq. 10b simplifies and Re[Zin]/Re[Z0] is approximately dependent on ⌫ only. Equation 10 provides the means to determine whether pulse wave reflection increases or decreases the arterial load. Because the value of ⌫ is complex, it is not immediately obvious how ⌫ affects 兩Zin兩 and Re[Zin]. For illustrative purposes, the interaction of forward and reflected waves is presented in Fig. 2. For clarity, only one harmonic is shown. From consideration of Fig. 2, it becomes clear that the direct effect of reflection on the pressure depends not only on 兩⌫兩, but also on ⌫. This is because ⌫ determines whether there is constructive or destructive addition of the forward and reflected Fig. 3. Polar plot of regions of 兩⌫兩ej⌫. Arrows indicate whether reflection increases (1) or decreases (2) 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 兩Zin兩 (i.e., 兩Zin兩 ⬍ 兩Z0兩) 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 兩Zin兩 and Re[Zin]. The region outside the inner circle on the right side corresponds to values of ⌫ that increase 兩Zin兩 yet decrease Re[Zin]. Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 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. H1521 H1522 ARTERIAL PULSE WAVE REFLECTION creases arterial load in an actual arterial system, values of Zin and Z0 must be determined experimentally. 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? This question can be answered by relying on a simple mathematical trick. Because Z0 is Zin in the absence of reflection, the oscillatory pressure that would result from the same oscillatory flow entering a reflectionless system is simply Qin 䡠 Z0. [As emphasized by Westerhof et al. (34), Qin 䡠 Z0 is not equivalent to the antegrade wave.] Comparing measured pressure with Qin 䡠 Z0 reveals the direct effect of reflection in a particular system. That is, the simple technique, illustrated in Fig. 4, mathematically removes reflection without disturbing the system experimentally. Comparison of the two pressure curves in Fig. 4 reveals that the effect of reflection is to redistribute pressure. Reflection does not affect steady pressure Fig. 5. Example of 兩Zin兩 (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.) EXPERIMENTAL ANALYSIS 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 兩Zin兩 and Re[Zin] 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 Zin). The details of the calculation can be found in Westerhof et al. (34). Although 兩Zin兩 ⬎ 兩Zo兩 for most frequencies, Re[Zin] ⬍ Re[Z0] for frequencies between 1.3 and 3.0 Hz. There is a minimum in Re[Zin] at 2.4 Hz (corre- Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 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. and flow, and thus the instantaneous change in pressure due to reflection must average zero throughout a cardiac cycle. Therefore, if reflection increases pressure in one part of the cardiac cycle, it must decrease pressure in another part of the cardiac cycle by a commensurate amount. That is, the reflected wave must swing positive and negative to average zero. This point was first made by Berger et al. (4), who quantified the effect of reflection in a particular arterial model. This analysis has the unique ability to remove reflection in a particular case without altering other important properties. For instance, the pressure with reflection theoretically removed has the same mean value, ejection period, and HR as the measured pressure. This contrasts with the traditional approach to studying reflection, where the arterial system is perturbed (e.g., with a vasodilator) to alter reflection. Such perturbations may diminish or augment reflection but generally affect other cardiovascular properties. Vasodilators, for example, cause a reduction in peripheral resistance and, through venous pooling, an increase in ventricular preload; these and other secondary changes can overwhelm any affect of reflections (3, 4). Thus, instead of comparing one vasoactive state with another and ascribing the difference in Zin, PP, and SW to reflection, the effect of reflection for each vasoactive state can be determined independently. The simple approach proposed here to determine the effect of reflection on pressure, SW, and Zin does not require the assumption of a particular model. Moreover, it can be applied directly to experimental data to evaluate the effect of pulse wave reflection. H1523 ARTERIAL PULSE WAVE REFLECTION 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 兩Zin兩 for harmonics 4–10. control case (for a total PP of 26 mmHg) but causes a larger increase in PP during vasoconstriction and vasodilation. Interestingly, the direct effects of reflection during vasodilation are qualitatively quite similar to those during vasoconstriction, although they are numerically smaller. Figure 7 illustrates the relative change in all indexes due to reflection, with the particular interaction of the heart and the vasculature neglected. The values are expressed relative to the theoretical reflectionless case. In other words, the change in the index due to reflection (⌬index) can be expressed as the difference in the indexes derived from the two pressures in Fig. 6 ⌬index ⫽ index共Pin兲 ⫺ index共Qin 䡠 Z0兲 (11) In this methodology, the measured control pressure (thick line, Fig. 6B) is compared with the control pressure with reflection theoretically removed (thin line, Fig. 6B). In a separate analysis, the measured vasodilated pressure (thick line, Fig. 6A) is compared with the vasodilated pressure with reflection theoretically removed (thin line, Fig. 6B). The measured vasodilated pressure is not compared with the control pressure. This approach is fundamentally different from that 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 䡠 Z0 (Eq. 11), where Q is flow. Ps, systolic pressure; P s, mean d, mean Pd; PP, pulse pressure; SW, Ps; Pd, diastolic pressure; P stroke work. Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 sponding to 145 beats/min), which is similar to the dog’s resting HR (2 Hz). 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 Zin, 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 Ps and end-diastolic pressure (Pd) are most often used clinically. However, some have championed indexes such as mean Ps (P s), critical to ventricular afterload during ejection, and mean Pd (P 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. Figure 6 shows pressure measured from a single anesthetized dog in various vasoactive states. The experimental details are reported by Berger and Li (1). Briefly, pressure was measured with a catheter-tipped pressure transducer, and flow was measured with a cuff-type electromagnetic flow probe. Both were digitized at a sampling rate of 100 s⫺1. After baseline data were recorded, vasoconstriction was induced with a bolus of methoxamine (5 mg/ml). After steady-state conditions were reestablished, vasodilation was induced with a bolus of nitroprusside (10 mg/ml) (1). Zin was calculated from pressure-flow pairs by standard methods (16). Also shown in Fig. 6 are the theoretical reflectionless pressures calculated from Qin 䡠 Z0 (as in Fig. 4B). The difference in the curves in Fig. 6 illustrates the effect of reflection in each particular case, with the assumption that total input flow (and thus ventricular preload, cardiac contractility, HR, and ejection period) is unaffected. For instance, in the control case, reflection lowers late Pd and early Ps and raises late Ps and early Pd. In this case, reflection has little effect on Ps and SW, whereas it reduces Pd. In contrast, the effects of reflection are quite large during vasoconstriction, causing a large increase in Ps and a large decrease in Pd. Similarly, reflection increases PP ⬍10 mmHg in the H1524 ARTERIAL PULSE WAVE REFLECTION usually taken (10, 37), where the effect of reflection is inferred by comparing control pressure with measured pressure after administration of a potent vasodilator that abolishes reflection. DISCUSSION Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 The present work illustrates that reflection can potentially increase and decrease vascular Zin, 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 Zin, 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 兩Zin兩 is reported more often than Re[Zin] 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 Zin is two-dimensional, other indexes of load, besides 兩Zin兩, may be no less important when the effects of reflection on arterial load are evaluated. The phase of ⌫ determines whether there is constructive or destructive addition of pulse waves. Furthermore, 兩⌫兩 and ⌫ (or Re[⌫] and Im[⌫]) are necessary to fully characterize reflection. Thus reporting the magnitude of ⌫ without its phase (5, 6, 12, 34) may be misleading, inasmuch as hemodynamic variables thought to be influenced by 兩⌫兩 may also be affected by ⌫ (Fig. 2). The constructive and destructive addition of waves determines whether wave reflection increases or decreases arterial load. 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). In addition to the problem of direct vs. indirect effects mentioned above, experimentally modifying reflection in the intact animal inevitably modifies other properties that influence aortic pressure and flow. For this reason, interpreting the effects of reflection becomes difficult. For example, in the reflectionless system, aortic pressure and flow have the same shape (Fig. 4B). This phenomenon is predicted from fundamental theory and has been observed experimentally after extreme vasodilation (10). (Consider, for instance, the measured pressure for the vasodilated case in Fig. 6.) However, experimentally abolishing the reflected wave results in, among other things, a large change in peripheral resistance, which itself yields concomitant changes in mean pressure, arterial compliance, and pulse wave velocity. Vasodilation also can induce large changes in ventricular preload, HR, and ejection period (Fig. 6A). Thus the system after vasodilation is scarcely similar to the arterial system before vasodilation. Because there are changes in critical parameters other than reflection, the extent to which changes in pressure and power should be attributed to wave reflection alone is unclear. This difficulty in interpreting experimental data was discussed in more detail by Berger et al. (3–5). The novel approach presented above eliminates these interpretive problems by avoiding direct comparison among different vasoactive states. Instead, the measured pressure of a particular vasoactive state is compared with the same state with reflection theoretically removed. For instance, the measured pressure in the control case in Fig. 6B (thick line) is compared with the control case when reflections are theoretically removed (thin line). Thus this theoretical approach keeps critical parameters, such as cardiac contractility, ventricular preload, mean pressure, and cardiac period, theoretically constant. Therefore, this approach is particularly useful to compare the effect of reflection in separate populations where many critical parameters differ. This is important because reflection is known to be different in various physiological conditions, such as hypertension, exercise, and arteriosclerosis. For example, the present approach can be used to determine how the effect of reflection changes throughout the aging process. Although arterial compliance decreases, pulse wave velocity increases, and resistance increases (16), the effect of reflection within any age group can be determined separately. 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 H1525 ARTERIAL PULSE WAVE REFLECTION phase of ⌫ changes when the magnitude of ⌫ is altered. The effect of changing the magnitude of reflection depends primarily on whether there is predominantly constructive or destructive addition of forward and reflected waves. Finally, a lingering misconception arises from the incorrect analyses of pressure and flow into forward and reflected components. If Z0 is treated as a constant, the forward and reflected waves can be calculated in the time domain (13, 17) via P̃f ⫽ 共P̃ ⫹ Z0Q̃兲/2 (12a) P̃r ⫽ 共P̃ ⫺ Z0Q̃兲/2 (12b) Many investigators, not realizing that propagation and reflection of a traveling wave are, by definition, strictly ⫹ P̃ oscillatory phenomena, mistakenly substitute P and Q ⫹ Q̃ (i.e., entire measured waveforms) into Eq. 12 (6, 7, 13, 17) instead of only the oscillatory components. This yields a calculated reflected wave in Fig. 8A that is always positive. This misconception understandably leads to the conviction that reflection must increase PP and SW. By removing the steady component from analysis first, forward and reflected waves are rightly shown oscillating about 0, having positive and negative values (Fig. 8B). This misconception was first clarified by Berger et al. (3, 4). Limitations of estimating Z0. This work provides a new method to determine the effect of reflection on a Fig. 8. A: incorrect analysis of pressure into forward (Pf ) and reflected (i.e., backward, Pb) waves. Mistakenly substituting steady plus oscillator pressure components (P ⫹ P̃) 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). Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 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. To compare pressures with and without reflection, several indexes of pressure and power were used: Ps, Pd, P s, P d, PP, and SW. This is not an exhaustive list, and there may be many other ways to compare pressure curves, each emphasizing different aspects. For instance, in the data displayed in Fig. 6, reflection caused the peak pressure to be delayed 39 ms in the control case, 14 ms with vasodilation, and 49 ms with vasoconstriction. The shift in time to peak Ps induced by reflection may have an effect on ventricular contraction. Although the six indexes of pressure and power illustrated here tell a relatively consistent story, this may change when different indexes of pressure are considered. The present technique indicates how reflection changes the time course of pressure. The interpretation of these changes is not a closed issue. Identifying prevalent misconceptions. In the light of these findings, a number of common misconceptions in the literature become apparent. First, 兩Zin兩 is often mistakenly believed to determine the SW given a particular input flow (32). 兩Zin兩 is only a piece of the story; oscillatory power (and thus SW) depends on Re[Zin] (⫽兩Zin兩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[Zin] occurs at a frequency close to resting HR. Second, there is a common misconception that a reflectionless system theoretically requires the least energy to pump blood and yields the smallest PP in distributed systems (31). Actually, Zin is minimized when ⌫ equals 1ej (i.e., 兩⌫兩 ⫽ 1 and ⌫ ⫽ 180°). This value would theoretically lead to a negligible PP and minimal SW. In fact, it can be shown that 兩⌫兩 ⫽ 1 makes Re[Zin] ⫽ 0 for all phases except at ⌫ ⫽ 0° (a singularity). A reflectionless system does not minimize PP and SW but, instead, maximizes the power transfer from the heart to the periphery. A third and related misconception is that decreasing the magnitude of reflection must decrease Zin and PP (6, 30). As indicated in the polar plot (Fig. 3), the effect of decreasing the magnitude of reflection depends on the initial phase of the reflection coefficient (and the measure of Zin considered). For instance, if the phase of ⌫ is 0°, then reducing the magnitude of reflection indeed decreases Zin. However, if the phase is 180°, then reducing the magnitude of reflection actually increases Zin. For intermediate values of ⌫, it is critical how the H1526 ARTERIAL PULSE WAVE REFLECTION arterial properties would most likely introduce reflections, which might be detrimental in a system that evolved to minimize them. Thus the mammalian arterial system may not be constructed to minimize reflection per se but, instead, to minimize the effect of reflection. The authors are grateful to Douglas A. Hettrick and Sanjeev G. Shroff for generously providing the dog data. This material is based on work supported by an American Heart Association Predoctoral Fellowship (to C. M. Quick) and American Heart Association Grant-in-Aid 96009940 (to D. S. Berger). REFERENCES 1. Berger DS and Li JK. Concurrent compliance reduction and increased peripheral resistance in the manifestation of isolated systolic hypertension. Am J Cardiol 65: 67–71, 1990. 2. Berger DS, Li JK, Laskey WK, and Noordergraaf A. Repeated reflection of waves in the systemic arterial system. Am J Physiol Heart Circ Physiol 264: H269–H281, 1993. 3. Berger DS, Li JK, and Noordergraaf A. Arterial wave propagation phenomena, ventricular work, and power dissipation. Ann Biomed Eng 23: 804–811, 1995. 4. Berger DS, Li JK, and Noordergraaf A. Differential effects of wave reflections and peripheral resistance on aortic blood pressure: a model-based study. Am J Physiol Heart Circ Physiol 266: H1626–H1642, 1994. 5. Berger DS, Robinson KA, and Shroff SG. Wave propagation in coupled left ventricle-arterial system. Implications for aortic pressure. Hypertension 27: 1079–1089, 1996. 6. Brin KP and Yin FC. Effect of nitroprusside on wave reflections in patients with heart failure. Ann Biomed Eng 12: 135–150, 1984. 7. Campbell KB, Lee LC, Frasch HF, and Noordergraaf A. Pulse reflection sites and effective length of the arterial system. Am J Physiol Heart Circ Physiol 256: H1684–H1689, 1989. 8. Dujardin JP and Stone DN. Characteristic impedance of the proximal aorta determined in the time and frequency domain: a comparison. Med Biol Eng Comput 19: 565–568, 1981. 9. Franklin SS, Khan SA, Wong ND, Larson MG, and Levy D. Is pulse pressure useful in predicting risk for coronary heart disease? The Framingham Heart Study. Circulation 100: 354– 360, 1999. 10. Geipel PS. Wave Reflections During Altered Ventricular States and Afterload (MS thesis). New Brunswick, NJ: Rutgers University, 1989. 11. Hettrick DA, Pagel PS, and Warltier DC. Differential effects of isoflurane and halothane on aortic input impedance quantified using a three-element windkessel model. Anesthesiology 83: 361–373, 1995. 12. Latson TW, Hunter WC, Katoh N, and Sagawa K. Effect of nitroglycerin on aortic impedance, diameter, and pulse-wave velocity. Circ Res 62: 884–890, 1988. 13. Li JK. Time domain resolution of forward and reflected waves in the aorta. IEEE Trans Biomed Eng 33: 783–785, 1986. 14. Li JK and Noordergraaf A. Similar pressure pulse propagation and reflection characteristics in aortas of mammals. Am J Physiol Regulatory Integrative Comp Physiol 261: R519–R521, 1991. 15. Lucas CL, Wilcox BR, Ha B, and Henry GW. Comparison of time domain algorithms for estimating aortic characteristic impedance in humans. IEEE Trans Biomed Eng 35: 62–68, 1988. 16. Milnor WR. Hemodynamics. Baltimore, MD: Williams & Wilkins, 1989. 17. Nichols WW and O’Rourke MF. McDonald’s Blood Flow in Arteries: Theoretic, Experimental, and Clinical Principles. Philadelphia, PA: Arnold, 1990. 18. Noordergraaf A, Li JK, and Campbell KB. Mammalian hemodynamics: a new similarity principle. J Theor Biol 79: 485– 489, 1979. 19. Noordergraaf A and Melbin J. Ventricular afterload: a succinct yet comprehensive definition. Am Heart J 95: 545–547, 1978. Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017 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 兩Zin兩, 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 Zin/Z0 is similar in different mammals of widely varying body size (32). This implies that ⌫ is similar in different mammals, since Zin/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 兩Zin兩 occurs at a frequency well above that corresponding to normal resting HR (32). This stance is based on the belief that 兩Zin兩 solely determines SW. Because SW is determined by Re[Zin] and not 兩Zin兩 (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. The present work challenges the traditional conception of the optimal design of the mammalian arterial system. Conventionally, it is assumed that reflection increases Zin, and thus a reflectionless system is optimal in terms of minimal PP and SW (31). However, because it is now understood that reflection can potentially decrease the arterial load, this view must be reexamined. The ability of reflection to actively lower Re[Zin] has been illustrated with experimental data (Fig. 5). Furthermore, in the control case, reflection had a negligible effect on SW, Ps, P s, and P d and a ⬍10 mmHg augmentation of PP (for a total PP of 26 mmHg; Fig. 7). Instead of postulating that the mammalian arterial system is optimized for minimal reflection, a new principle is proposed. Apparently, there can be a large amount of reflection without a large effect on several of the important hemodynamic parameters. To achieve this, the arterial system must have an architecture with appropriate impedance mismatches (determining 兩⌫兩) and the appropriate pulse wave velocities and arterial lengths (determining ⌫). Through a balance of constructive and destructive addition of forward and reflected waves, the effect of reflection is minimized. Although a system with minimized reflections is conceivable, there is little to gain and perhaps much to lose. Consider a mammalian arterial system in which reflections are minimized. 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