## Abstract

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 (*Z*
_{in}), 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 (*Z*
_{0}) is *Z*
_{in} in the absence of reflection, the P with reflection theoretically removed can be calculated from Q · *Z*
_{0}. 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. *Z*
_{in}, P, and SW are increased or decreased depending on the relative amount of constructive and destructive addition of forward and reflected arterial pulse waves.

- 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 (*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

#### 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)
Equation 1where ω is frequency and *j* is
.*Z*
_{in} is a complex quantity and must be described by two components: magnitude (‖*Z*
_{in}‖) and phase (θ_{Z}
_{in}) or real (Re[*Z*
_{in}]) and imaginary (Im[*Z*
_{in}]) parts.

Similarly, to characterize the tendency of the system to reflect antegrade waves independent of input, investigators regularly use the global reflection coefficient (Γ)
Equation 2where P_{f} is the forward-traveling pressure pulse and P_{r} is the retrograde pressure pulse observed at the system entrance. Γ is also complex, having a magnitude (‖Γ‖) and a phase (θ_{Γ}) or, alternatively, real (Re[Γ]) and imaginary (Im[Γ]) parts. Both parts are necessary to fully describe reflection, although θ_{Γ} is rarely reported in the literature. *Z*
_{in} and Γ are interrelated such that
Equation 3where *Z*
_{0} is the characteristic impedance, i.e., the value of *Z*
_{in} in the absence of reflection. It is clear from *Eq. 3
* that reflection is a major determinant of *Z*
_{in} and, thus, the load formed by an arterial system. However, because*Z*
_{in} and Γ have complex values, it is not readily apparent whether reflection increases or decreases*Z*
_{in}.

#### 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*
Equation 4aand *measure B*
Equation 4bAt 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: θ_{Z}
_{in} = cos^{−1}(Re[*Z*
_{in}]/‖*Z*
_{in}‖).

To interpret the values of ‖*Z*
_{in}‖ and Re[*Z*
_{in}], their effect on pressure and SW will be considered. Pressure has steady (
) and oscillatory (P̃) components, such that
Equation 5The steady component is the product of steady flow (
) and resistance *R* (the value of *Z*
_{in} 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 *Z*
_{in}. The magnitude of the *n*th harmonic of pressure (P̃_{n}) has a simple form
Equation 6Thus *measure A* (‖*Z*
_{in}‖) 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 arterial system. Average power has steady (
) and oscillatory (W̃) components
Equation 7The steady component is, again, a function of steady flow and*R* and is not directly affected by reflection (i.e.,
=
· *R*). The oscillatory power is a function of flow and Re[*Z*
_{in}] (16). The magnitude of the *n*th harmonic has the form
Equation 8Thus, for a given flow, *measure B*(Re[Z_{in}]) 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 *Z*
_{in}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 Γ: *measure A*
Equation 9aand *measure B*
Equation 9bIn the reflectionless case, *Z*
_{in} =*Z*
_{0}; *measure A* degenerates into ‖*Z*
_{0}‖, and *measure B* degenerates into Re[*Z*
_{0}]. Generally, Re[*Z*
_{0}] can be approximated by ‖*Z*
_{0}‖ (35).

The effect of reflection on arterial load can be quantified with*Eq. 9*. That is, the effect of reflection can be quantified by comparing *Z*
_{in} (the arterial load when reflection is present) with *Z*
_{0} (the arterial load when reflection is absent, by definition). For instance, when
Equation 10a(*condition A*), then *Z*
_{in}described by *measure A* is increased by the presence of reflection. Similarly, when
(*condition B*), then *Z*
_{in}described by *measure B* is increased by the presence of reflection. Because the imaginary part of *Z*
_{0} is small in the aorta, *Eq. 10b* simplifies and Re[*Z*
_{in}]/Re[*Z*
_{0}] is approximately dependent on Γ only. *Equation 10* provides the means to determine whether pulse wave reflection increases or decreases the arterial load.

Because the value of Γ is complex, it is not immediately obvious how Γ affects ‖*Z*
_{in}‖ and Re[*Z*
_{in}]. 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 ‖Γ‖, but also on θ_{Γ}. This is because θ_{Γ}determines whether there is constructive or destructive addition of the forward and reflected waves. For instance, when θ_{Γ} is 0°, forward and reflected waves are in phase and add constructively. This tends to make the resulting PP large and, thus,*Z*
_{in} large by either measure. On the other hand, when θ_{Γ} is 180°, forward and reflected waves are out of phase and add destructively. In this case, the PP is small, and thus*Z*
_{in} 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., ‖Γ‖ = 0.5 and θ_{Γ} = −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 θ_{Γ} and ‖Γ‖, 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 Γ(ω) (Fig.3). To simplify,*Z*
_{0} is assumed to be real (noncomplex). The origin corresponds to the reflectionless case (i.e., ‖Γ‖ = 0), and the outer boundary corresponds to the maximum value of ‖Γ‖ (i.e., ‖Γ‖ = 1). Figure 3 portrays three distinct regions. The right half of the plot (−90° < θ_{Γ} < +90°) corresponds to combinations of ‖Γ‖ and θ_{Γ} 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 Γ that satisfy *conditions A* and *B* (*Eq. 10*). Thus reflection increases ‖*Z*
_{in}‖ and Re[*Z*
_{in}]. The left side of the graph (90° < θ_{Γ} < 270°) corresponds to combinations of ‖Γ‖ and θ_{Γ} that decrease ‖*Z*
_{in}‖ and Re[*Z*
_{in}].

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 decreases arterial load in an actual arterial system, values of *Z*
_{in} and *Z*
_{0} 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 *Z*
_{0} is *Z*
_{in}in the absence of reflection, the oscillatory pressure that would result from the same oscillatory flow entering a reflectionless system is simply Q_{in} · *Z*
_{0}. [As emphasized by Westerhof et al. (34), Q_{in} · *Z*
_{0} is not equivalent to the antegrade wave.] Comparing measured pressure with Q_{in} · *Z*
_{0} 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 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 *Z*
_{in}, 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 *Z*
_{in} 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.

## 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 ‖*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, *Z*
_{0} 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[*Z*
_{0}] 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).

#### 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.

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). *Z*
_{in} was calculated from pressure-flow pairs by standard methods (16). Also shown in Fig. 6 are the theoretical reflectionless pressures calculated from Q_{in} · *Z*
_{0} (as in Fig.4
*B*).

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 P_{d} and early P_{s} and raises late P_{s} and early P_{d}. In this case, reflection has little effect on P_{s} and SW, whereas it reduces P_{d}. In contrast, the effects of reflection are quite large during vasoconstriction, causing a large increase in P_{s} and a large decrease in P_{d}. Similarly, reflection increases PP <10 mmHg in the 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. Figure7 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 (Δindex) can be expressed as the difference in the indexes derived from the two pressures in Fig. 6
Equation 11In this methodology, the measured control pressure (thick line, Fig. 6
*B*) is compared with the control pressure with reflection theoretically removed (thin line, Fig. 6
*B*). In a separate analysis, the measured vasodilated pressure (thick line, Fig.6
*A*) is compared with the vasodilated pressure with reflection theoretically removed (thin line, Fig. 6
*B*). The measured vasodilated pressure is not compared with the control pressure. This approach is fundamentally different from that 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

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.

The phase of Γ determines whether there is constructive or destructive addition of pulse waves. Furthermore, ‖Γ‖ and θ_{Γ} (or Re[Γ] and Im[Γ]) are necessary to fully characterize reflection. Thus reporting the magnitude of Γ without its phase (5, 6, 12, 34) may be misleading, inasmuch as hemodynamic variables thought to be influenced by ‖Γ‖ may also be affected by θ_{Γ} (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. 4
*B*). 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. 6
*A*). 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.6
*B* (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 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: P_{s}, P_{d},
_{s},
_{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 P_{s} 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, ‖*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[θ_{Z}
_{in}]). The critical role of θ_{Z}
_{in}, like that of θ_{Γ}, 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.

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,*Z*
_{in} is minimized when Γ equals 1*e ^{j}
*

^{π}(i.e., ‖Γ‖ = 1 and θ

_{Γ}= 180°). This value would theoretically lead to a negligible PP and minimal SW. In fact, it can be shown that ‖Γ‖ = 1 makes Re[

*Z*

_{in}] = 0 for all phases except at θ

_{Γ}= 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 *Z*
_{in} 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*Z*
_{in} considered). For instance, if the phase of Γ is 0°, then reducing the magnitude of reflection indeed decreases*Z*
_{in}. However, if the phase is 180°, then reducing the magnitude of reflection actually increases*Z*
_{in}. For intermediate values of θ_{Γ}, it is critical how the phase of Γ changes when the magnitude of Γ 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*Z*
_{0} is treated as a constant, the forward and reflected waves can be calculated in the time domain (13,17) via
Equation 12a
Equation 12bMany investigators, not realizing that propagation and reflection of a traveling wave are, by definition, strictly oscillatory phenomena, mistakenly substitute
+ P̃ and
+ 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.8
*A* 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.8
*B*). This misconception was first clarified by Berger et al. (3, 4).

#### Limitations of estimating Z_{0}.

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*Z*
_{0}. The literature provides several competing methods to estimate *Z*
_{0} 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 *Z*
_{0} might impact the determination of the effects of reflection. As better methods to estimate *Z*
_{0} 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}/*Z*
_{0} is similar in different mammals of widely varying body size (32). This implies that Γ is similar in different mammals, since*Z*
_{in}/*Z*
_{0} is equal to (1 + Γ)/(1 − Γ) (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.

The present work challenges the traditional conception of the optimal design of the mammalian arterial system. Conventionally, it is assumed that reflection increases *Z*
_{in}, 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[*Z*
_{in}] has been illustrated with experimental data (Fig. 5). Furthermore, in the control case, reflection had a negligible effect on SW, P_{s},
_{s}, and
_{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 ‖Γ‖) and the appropriate pulse wave velocities and arterial lengths (determining θ_{Γ}). 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. There would be little latitude for arterial system adaptation to acute or chronic environmental changes, such as those arising from exercise, fight or flight, temperature, elevation, pregnancy, aging, or disease. In addition, any change in 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.

## Acknowledgments

The authors are grateful to Douglas A. Hettrick and Sanjeev G. Shroff for generously providing the dog data.

## Footnotes

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).

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: quickc{at}anesthesia.ucsf.edu).

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.

- Copyright © 2001 the American Physiological Society