Download Constructive and destructive addition of forward and reflected

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ej␪Zin ⫽ 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:
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0363-6135/01 $5.00 Copyright © 2001 the American Physiological Society
H1519
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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.
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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).
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Z in ⫽ 兩Zin兩ej␪Zin ⫽ 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].
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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
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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-
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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.
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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.
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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
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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
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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
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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
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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. 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 PULSE WAVE REFLECTION
30. Ting CT, Brin KP, Lin SJ, Wang SP, Chang MS, Chiang
BN, and Yin FC. Arterial hemodynamics in human hypertension. J Clin Invest 78: 1462–1471, 1986.
31. West GB, Brown JH, and Enquist BJ. A general model for the
origin of allometric scaling laws in biology. Science 276: 122–
126, 1997.
32. Westerhof N and Elzinga G. Normalized input impedance and
arterial decay time over heart period are independent of animal
size. Am J Physiol Regulatory Integrative Comp Physiol 261:
R126–R133, 1991.
33. Westerhof N and O’Rourke MF. Haemodynamic basis for the
development of left ventricular failure in systolic hypertension
and for its logical therapy. J Hypertens 13: 943–952, 1995.
34. Westerhof N, Sipkema P, van den Bos GC, and Elzinga G.
Forward and backward waves in the arterial system. Cardiovasc
Res 6: 648–656, 1972.
35. Womersley JR. An Elastic Tube Theory of Pulse Transmission and Oscillatory Flow in Mammalian Arteries. WrightPatterson Air Force Base, OH: Wright Air Development Center, 1957.
36. Yaginuma T, Avolio A, O’Rourke M, Nichols W, Morgan JJ,
Roy P, Baron D, Branson J, and Feneley M. Effect of glyceryl
trinitrate on peripheral arteries alters left ventricular hydraulic
load in man. Cardiovasc Res 20: 153–160, 1986.
37. Zhu Y. Hemodynamic Basis and Nonlinear Model Analysis of
Hypertension and Aging (MS thesis). New Brunswick, NJ: Rutgers University, 1993.
Downloaded from http://ajpheart.physiology.org/ by 10.220.33.4 on May 12, 2017
20. O’Rourke M. Arterial stiffness, systolic blood pressure, and
logical treatment of arterial hypertension. Hypertension 15:
339–347, 1990.
21. O’Rourke MF. Steady and pulsatile energy losses in the systemic circulation under normal conditions and in simulated
arterial disease. Cardiovasc Res 1: 313–326, 1967.
22. O’Rourke MF and Kelly RP. Wave reflection in the systemic
circulation and its implications in ventricular function. J Hypertens 11: 327–337, 1993.
23. O’Rourke MF and Taylor MG. Input impedance of the systemic circulation. Circ Res 20: 365–380, 1967.
24. Porjé IG. Studies of the arterial pulse wave, particularly in the
aorta. Acta Physiol Scand 13: 1–68, 1946.
25. Quick CM, Berger DS, Hettrick DA, and Noordergraaf A.
True arterial system compliance derived from apparent arterial
compliance. Ann Biomed Eng 28: 291–301, 2000.
26. Quick CM, Berger DS, and Noordergraaf A. Apparent arterial compliance. Am J Physiol Heart Circ Physiol 274: H1393–
H1403, 1998.
27. Quick CM, Berger DS, and Noordergraaf A. Direct effects of
pulse wave reflection may increase or decrease systolic blood
pressure and stroke work. 19th Annu Int Conf IEEE/EMB Soc,
1997, p. 21–24.
28. Quick CM, O’Hara DA, and Noordergraaf A. Pulse wave
reflection and arterial inefficiency. 17th Annu Int Conf IEEE/
EMB Soc, Montreal, 1995, p. 97–98.
29. Randall JE and Stacy RW. Mechanical input impedance of the dog’s
hind leg to pulsatile blood flow. Am J Physiol 187: 94–98, 1956.
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