Vol. 280, Issue 1, H407-H419, January 2001
Dynamic regulation of heart rate during acute hypotension: new
insight into baroreflex function
Rong
Zhang,
Khosrow
Behbehani,
Craig G.
Crandall,
Julie H.
Zuckerman, and
Benjamin D.
Levine
Institute for Exercise and Environmental Medicine, Presbyterian
Hospital of Dallas, and University of Texas Southwestern Medical
Center Dallas, Dallas, Texas 75231
 |
ABSTRACT |
To examine the dynamic properties of
baroreflex function, we measured beat-to-beat changes in arterial blood
pressure (ABP) and heart rate (HR) during acute hypotension
induced by thigh cuff deflation in 10 healthy subjects under supine
resting conditions and during progressive lower body negative pressure
(LBNP). The quantitative, temporal relationship between ABP and HR was
fitted by a second-order autoregressive (AR) model. The frequency
response was evaluated by transfer function analysis. Results: HR
changes during acute hypotension appear to be controlled by an ABP
error signal between baseline and induced hypotension. The quantitative relationship between changes in ABP and HR is characterized by a
second-order AR model with a pure time delay of 0.75 s containing low-pass filter properties. During LBNP, the change in HR/change in ABP
during induced hypotension significantly decreased, as did the
numerator coefficients of the AR model and transfer function gain.
Conclusions: 1) Beat-to-beat HR responses to dynamic changes in ABP may be controlled by an error signal rather than directional changes in pressure, suggesting a "set point" mechanism in
short-term ABP control. 2) The quantitative relationship
between dynamic changes in ABP and HR can be described by a
second-order AR model with a pure time delay. 3) The ability
of the baroreflex to evoke a HR response to transient changes in
pressure was reduced during LBNP, which was due primarily to a
reduction of the static gain of the baroreflex.
blood pressure; mathematical modeling
| |
INTRODUCTION |
THE HEART RATE LIMB
of the baroreflex plays an important role in short-term regulation of
blood pressure (15, 31). By its nature, the underlying
control mechanisms must be dynamic. However, evaluation of dynamic
baroreflex function has been difficult in humans. This difficulty
arises not only because of the limitations in studying humans per se
but also because of the complexity of the baroreflex system. The
baroreflex functions through a complicated feedback control system with
multiple inputs and outputs and has intrinsic nonlinearity (12,
27, 43). Thus it is not surprising that analyzing such a complex
system has been a great challenge to both physiologists and medical scientists.
Traditionally, baroreflex control of heart rate has been described by a
sigmoid-shaped curve relating R-R interval or heart rate to arterial
pressure (15, 31). This relationship implies that
stimulation of baroreceptors with increases in pressure leads only to
decreases in heart rate (increases in R-R interval) and unloading of
the baroreceptors with decreases in pressure leads only to increases in
heart rate.
Although this model may correctly reflect steady-state heart rate
responses to steady-state changes in arterial pressure
(24), the fundamental assumption underlying this
description implies a static baroreflex with one pressure stimulus
associated with one heart rate response, which may not be suitable to
describe how heart rate responds to dynamic changes in pressure.
Moreover, it has been shown that the baroreflex may reset rapidly with
sustained changes in pressure (8, 10). Shifting of
multiple sigmoid baroreflex curves along the pressure and/or heart rate
axes has been observed both in human and animal studies (10,
39). These findings demonstrate that baroreflex function is not
static even when described by static techniques.
However, the limitations implicit in the static description of the
baroreflex appear to be ignored in recent studies using spontaneous
variations in arterial pressure and heart rate to quantify baroreflex
function (6, 30, 34). For example, in one proposed method,
spontaneous beat-to-beat variations in arterial pressure and cardiac
period have been classified either as "baroreflex"- or
"nonbaroreflex"-mediated sequences based on whether changes in R-R
intervals were directionally similar or different from those of
simultaneous variations in pressure (6). This
classification suggests that increases in heart rate associated with
simultaneous increases in pressure could not be mediated by the
baroreflex (6, 30, 34). However, because spontaneous variations in heart rate are most likely mediated by dynamic changes in
pressure (1, 45), this assumption may not be entirely valid. Also, recent observations in our laboratory during the recovery
from acute hypotension have raised the possibility that, under certain
conditions, increases in beat-to-beat heart rate associated with
increases in arterial pressure may still be under baroreflex control.
In this study, we present a new method for the evaluation of
baroreflex function in humans. The baroreflex was perturbed by acute hypotension induced by rapid thigh cuff deflation. The time course of changes in heart rate and arterial pressure was examined to
determine how heart rate would respond to a dynamic change in pressure.
The relationship between the transient changes in pressure and heart
rate was quantified by dynamic system analysis. With the use of this
method, baroreflex function under orthostatic stress was then evaluated
during progressive lower body negative pressure (LBNP) designed to
alter stimulation to baroreceptors and background autonomic activity.
| |
METHODS |
Subjects.
Ten healthy men and women with a mean age of 33 ± 7 yr, height of
171 ± 12 cm, and weight 70 ± 14 kg voluntarily participated in the study. All were nonsmokers and free of known cardiovascular and
pulmonary disorders. Each subject was informed of the experimental procedures and signed a written consent form approved by the
Institutional Review Boards of the University of Texas Southwestern
Medical Center and the Presbyterian Hospital of Dallas.
Instrumentation and procedures.
Heart rate was monitored continuously by electrocardiogram (ECG).
Arterial pressure was measured continuously in the finger using
photoplethysmography (Finapres, Ohmeda) and intermittently in the arm
by electrosphygmomanometry (Suntech). Respiratory frequency was
monitored by using a piezoelectric pneumograph (Protech).
All experiments were performed in the morning at least 2 h
postprandial. The subjects were constrained from using caffeinated or
alcoholic beverages at least 12 h before the tests. The
experiments were conducted in a quiet, environmentally controlled
laboratory with an ambient temperature of 25°C.
After at least 30 min of supine rest, 6 min of data of ECG and arterial
pressure were recorded during spontaneous breathing as a baseline
steady state. Immediately after the steady-state data collection, two
thigh cuffs were inflated to a pressure of at least 30 mmHg higher than
each subject's systolic pressure (D. E. Hokanson) for 3 min to
produce temporary ischemia in the lower limbs. The thigh cuffs were
then rapidly deflated to induce a sudden decrease in arterial pressure
and a transient response in heart rate. In previous studies, it has
been conclusively shown that circulatory occlusion of resting skeletal
muscle with thigh cuff inflation does not evoke either cardiovascular
metaboreflexes or mechanoreflexes, suggesting that the cuff inflation
used in the present study caused no interference with baroreflex
function (41, 42, 58).
The above protocols were repeated during progressive LBNP. Subjects
were placed in a Plexiglas box, which was sealed at the level of the
iliac crests. Suction was provided by a vacuum pump and controlled with
a variable autotransformer. The pressure difference between the LBNP
chamber and the atmosphere was measured with a mercury manometer. LBNP
was applied with the subjects in the supine position. After the
baseline data collection, we immediately applied 
15 mmHg LBNP. After
2 min of stabilization, 6 min of data were recorded for steady state,
and 5 min of data were then recorded for the transient response (3 min
for the cuff inflation, 2 min for deflation) at this level of LBNP. The
magnitude of LBNP was then increased to 30 mmHg followed by stepwise
increments at 10 mmHg up to the subject's maximal tolerance. LBNP
was terminated if the subject developed signs and/or symptoms of
presyncope: sudden onset of nausea, sweating, light headedness,
bradycardia, or hypotension (sustained systolic blood pressure < 80 mmHg). In the present study, all subjects completed the experimental protocol through at least 40 mmHg LBNP. However, only five subjects completed the experimental protocol at 50 mmHg. Because more subjects
dropped off at high levels of LBNP, the data presented here include
only those from baseline to 50 mmHg.
Data acquisition and analysis.
Instantaneous heart rate was derived from the ECG signal
(Cardiotachometer, Quinton). Arterial pressure waveforms were sampled at 100 Hz and digitized at 12 bits to obtain beat-to-beat values of
systolic and diastolic pressure. For each cardiac cycle, defined as a
time interval between the sequential maximal upstrokes of arterial
pressure, the systolic and diastolic pressure were detected as the
maximal and minimal values within the interval. The beat-to-beat heart
rate and systolic and diastolic pressure were then linearly interpolated and resampled simultaneously at 4 Hz to construct equally
spaced time series for the modeling and data analysis.
For steady-state data analysis, arterial pressure and heart rate were
averaged over the 6-min time interval before the cuff inflation and
over the 3-min time interval during the cuff inflation. For transient
data analysis, the maximal decrease in pressure after the thigh cuff
deflation was calculated as a systolic pressure difference between an
average of 10 s of data immediately before the cuff release and a
minimal value after the cuff release. Correspondingly, the maximal
heart rate response to the acute hypotension was calculated as a heart
rate difference between an average of 10 s of data immediately
before the thigh cuff release and a maximal heart rate after the cuff
release. All the measurements were conducted under baseline conditions
and at each level of LBNP.
Mathematical modeling.
Systolic pressure and heart rate from 10 s before through 30 s after the thigh cuff release were used for identification of baroreflex function. To obtain a mathematical model relating the changes in heart rate to the changes in pressure, the autoregressive moving average model (ARMA) was used (28). The basic
assumption of the ARMA approach is that the response of heart rate at a
time point can be represented as a linear function of the present and past values of changes in pressure as well as previously observed heart
rate. A general form of an ARMA model is proposed as follows
|
(1)
|
where 
rm(k) signifies the measured
change in the heart rate at sample instant k,
pm(k) is the measured change in blood pressure at sample instant k, i and j
represent the sample instants preceding k,
ai and bj are the
weighting coefficients, and q is an integer multiple of the
sampling interval and represents the pure time delay involved in the
process. n and m represent the model order,
reflecting the number of previous sample values of heart rate and blood
pressure that are included in the model. It is noted that, without loss
of generality, a0 = 1.
To obtain an ARMA model, the values of n, m, and
the corresponding ai for i = 1, 2, ..., n and for j = 0, 1, ...,
m should be determined. Although there are some systematic
trial and error approaches to estimating n and m
(28), the step response of a system of interest often
provides a good initial estimate for these parameters. In this study,
the changes in arterial pressure were induced by rapid thigh cuff
deflation. Hence, the changes in pressure approximated a step input,
and the changes in heart rate approximated a step response. Examination
of a representative heart rate response to the changes in pressure
revealed that the response is underdamped with a slight overshoot (Fig.
1). Furthermore, it indicates that from the time that blood pressure
decreased until the heart rate began to respond, there was a latency of ~0.75 s (i.e., q = 3). Hence, it is reasonable to use
a second-order model (i.e., n = 2) as an initial
estimate. We initially selected m = 1 to simplify the
model structure, and the analysis of the experimental data showed that
this value of m was adequate. Equation 2 below
shows the resulting second-order system
|
(2)
|
For a sampling interval of 0.25 s used in the present
study, k 3 in argument of pm above
reflects the effect of 0.75 pure time delay. With the use of
system identification techniques, the values of
ai and b0 were obtained
by application of a least-square estimation technique.
Moreover, validation of model identification was conducted by
autocorrelation analysis of the model residuals. Theoretically, if the
input-output relationship of a dynamic system can be described well by
an ARMA model, time series of the model residuals is white noise, and
autocorrelation of the model residuals is a 
function
(29).
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Fig. 1.
Top: representative changes in arterial blood pressure
(ABP) induced by the thigh cuff deflation. Point A indicates
the time of cuff deflation; point B indicates recovery of
pressure to the level before the cuff deflation; and point C
indicates the ending of pressure overshoot. Middle: changes
in heart rate (HR) during the acute hypotension. Bottom:
measured and simulated HR of the second-order autoregressive (AR) model
in response to the changes in systolic blood pressure (SBP). bpm, Beats
per minute.
|
|
In analyzing the model in Eq. 2, it is helpful to obtain the
transfer function between the pressure and heart rate by taking the
z-transform of Eq. 2. A transfer function can be
obtained as
|
(3)
|
where Rm(z) and
Pm(z) represent the z-transform of
changes in heart rate [rm(k)] and pressure
[pm(k)], respectively.
Equation 3 provides a means to examine the frequency
response of the system by letting z = ej2
fT,
where j is the complex operator, f is the
frequency, and T is the sampling interval. In the present
study, the transfer function gain (G) between the changes in pressure
and heart rate was calculated as a magnitude of G(z), and
the phase spectrum was estimated from real and imaginary parts of
G(z) (25).
An advantage of transfer function analysis is that it allows the
examination of both the transient and steady-state response of a system
to a given input. In APPENDIX A, we show that for a step
change in the blood pressure, the steady-state value of the heart rate
is predicted to be
|
(4)
|
where p0 is the magnitude of the step input and
rss signifies the steady-state value of the heart rate.
Thus it is important to note that the term
b0/(1 + a1 + a2) reflects the static gain (i.e., zero
frequency) of the system. It represents how the magnitude of the input
affects the magnitude of the output after the transient response has
diminished. From Eq. 4, it is clear that the static transfer
function gain is determined by the coefficients of both the numerator
and denominator polynomials of Eq. 3. However, the shape of
the frequency distribution of the transfer function and, therefore, the
transient response of the system is determined only by the coefficients
of the denominator polynomials of Eq. 3 (see APPENDIX
A).
Statistical analysis.
The parameters of the autoregressive (AR) model were identified for
each individual subject at baseline and at each level of LBNP and then
averaged to obtain the group mean values. In addition, the time course
of pressure and heart rate for each individual subject during the acute
hypotension were aligned at the time point of thigh cuff deflation and
then averaged to obtain group-averaged changes. The group-averaged
pressure and heart rate were also fitted by the second AR model at the
baseline and at each level of LBNP. Transfer function gain [i.e.,
|G(z)| for z = e j2fT]
was averaged in low (0.0156~0.15 Hz)-, high (0.15~0.50 Hz)-, and
very high-frequency ranges (0.50~2.00 Hz), first for each individual
subject and then group averaged for statistical analysis. The selection
of frequency ranges was arbitrary in this study to facilitate the data
analysis. However, the low- and high-frequency ranges were selected
based on the consideration that a consistency between the present study
and those in the study of spontaneous changes in arterial pressure and
heart rate may facilitate further comparisons regarding the frequency
domain properties of baroreflex function (9, 45). The
steady-state and transient changes in pressure and heart rate, and the
AR model parameters as well as the transfer function gain at baseline
and at each level of LBNP, were compared by using one-way ANOVA with
Duncan's post hoc tests for multiple comparisons. The effects of thigh
cuff inflation on blood pressure and heart rate at baseline and during LBNP were evaluated by two-way ANOVA with cuff inflation and magnitude of LBNP as intervention factors. A P value <0.05 was
considered statistically significant, and all data were represented as
means ± SE. Statistics were performed by using a PC-based
software program (ABstat, AndersonBell, Arvada, CO).
| |
RESULTS |
Heart rate response to acute hypotension.
The thigh cuff inflation alone had no effects on blood pressure or
heart rate at baseline (Table 1). A
representative thigh cuff deflation, with its consequent induced acute
hypotension, and the heart rate response are shown in Fig.
1. Three phases with different
characteristics can be identified. In phase 1, after the
cuff deflation (Fig. 1, point A), arterial pressure decreased quickly to a nadir with a magnitude about 16 ± 2 mmHg under supine rest (Figs. 1 and 2 and
Table 1). In response to this reduction in pressure, heart rate
increased after a slight time delay, suggesting a typical baroreflex
mediated cardioacceleration (Figs. 1 and 2). In phase 2, in
association with the restoration of pressure from the nadir to the
level before the cuff release (Fig. 1, point B), heart rate
continuously increased with simultaneous increases in pressure, which
is in contrast to what might be expected from a static model of
baroreflex function. In phase 3, with the overshoot of
pressure (Fig. 1, from point B to C), heart rate decreased from the maximal response and recovered to the level before
the cuff release in association with the recovery of pressure. These
data suggest that beat-to-beat heart rate response to transient changes
in pressure are likely controlled by an error signal in the pressure
generated by the difference between the pressure level before and after
the thigh cuff release. Specifically, it appears that heart rate
increased when the pressure was below the level attained before the
cuff release and decreased when pressure was above the level.
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Fig. 2.
Left: group-averaged SBP and HR as well as
simulated HR during acute hypotension induced by the thigh cuff
deflation at baseline and at each level of lower body negative pressure
(LBNP). Right: estimates of autocorrelation function of the
AR model residuals at the baseline and at each level of LBNP. The 99%
confidence intervals for the estimated autocorrelation function were
calculated and displayed as dotted lines. Note that the confidence
intervals were calculated assuming that the model residuals are white
noise.
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Mathematical modeling.
Figures 1 and 2 show that changes in heart rate and pressure were
fitted well by the second-order AR model with a pure time delay of
0.75 s. This result was obtained not only at baseline but also
during LBNP and is further supported by the model residual analysis
(Fig. 2).
Group-averaged transfer functions derived from the AR model are shown
in Figs. 3 and
4. Examination of the gain plots reveals low-pass filter properties of the baroreflex with a corner frequency at
0.09 ± 0.01 Hz under supine rest. The phase spectra show a degree of
~180° at the very low frequencies, suggesting that at steady state,
a fall in pressure results in an increase in the heart rate and vice
versa, which is consistent with the predications of static model of
baroreflex. With the increase in frequency, phase fell gradually and
crossed 0° at ~0.20 Hz (Fig. 3). At higher frequencies, the slope
of the fall in phase was steeper, reflecting the pure time delay of the
baroreflex, which was incorporated in the model as 0.75 s.
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Fig. 3.
Group-averaged transfer function gain (top) and phase
(bottom) at baseline and each level of LBNP. Solid lines,
averaged estimates; dotted lines, SE.
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Fig. 4.
Overlap of averaged transfer function gain at baseline (BL) with
those obtained at each level of LBNP in zoomed scales. Left:
low-frequency (LF) range (0.0156~0.15 Hz); middle:
high-frequency (HF) range (0.15~0.50 Hz); right:
very high-frequency (VHF) range (0.50~2.00 Hz).
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|
Lower body negative pressure.
During LBNP, the thigh cuff inflation caused a slight increase in
diastolic pressure at 40 mmHg LBNP (Table 1). In association with the
typical changes in the steady-state hemodynamics, the maximal decreases
in systolic pressure after the thigh cuff deflation were significantly
augmented at 30, 40, and 50 mmHg LBNP. In contrast, the maximal
responses of heart rate remained unchanged (Table 1). Hence, the ratio
of maximal heart rate response to the maximal pressure decrease showed
a trend toward a reduction (Table 1). Moreover, the numerator
coefficients of the AR model significantly decreased during LBNP (Table
2). However, the denominator coefficients
of the AR model and the corner frequency of transfer function remained
unchanged (Figs. 3 and 4, Table 2). Consequently, the transfer function
gain in the low-, high-, and very high-frequency ranges all were
significantly reduced during LBNP (Fig. 4, Table 2). Taken together,
these data suggest that the ability of the baroreflex to evoke a heart
rate response to transient changes in pressure was reduced during LBNP,
which was primarily due to a reduction of the static gain of the
baroreflex.
| |
DISCUSSION |
The primary new findings of the present study are threefold.
1) With novel application of thigh cuff deflation as a
perturbation to the baroreflex, we found that beat-to-beat heart rate
responses to acute hypotension are likely controlled by an error signal rather than by directional changes in pressure. These data suggest a
"set-point" mechanism in short-term beat-to-beat blood pressure control. 2) The quantitative relationship between changes in
pressure and heart rate can be described well by a second-order AR
model with a pure time delay of 0.75s, confirming dynamic properties of
baroreflex function. 3) Dynamic baroreflex function, as
reflected by the shape of the transfer function gain, was preserved
during LBNP. However, the ability of the baroreflex to evoke a heart rate response to transient changes in pressure was reduced during LBNP,
which was primarily due to a reduction of the static gain of baroreflex function.
Baroreflex regulation of heart rate: methodological considerations.
The regulatory mechanisms and the properties of the baroreflex have
been extensively studied over the last several decades (15,
31). Two basic methods have been developed for evaluation of the
heart rate limb of the baroreflex in humans (14, 24, 38,
48). First, by using vasoactive drugs to alter arterial pressure, baroreflex function has been described either as a linear or
sigmoid curve relating changes in heart rate to changes in pressure (24, 38, 48). Second, by using a specially
designed neck chamber, the carotid baroreceptor can be unloaded or
stretched by a sequence of stepwise pressure changes in the neck
chamber (14). Heart rate responses to the pressure
stimuli have been quantified by a sigmoid curve (15). The
slope of the linear or the maximal slope of the sigmoid curve, obtained
with either infusion of vasoactive drugs or application of the neck
chamber, has been calculated to reflect the baroreflex gain for heart
rate control (15, 38).
Although modeling of the baroreflex with these methods has provided
valuable insights into cardiovascular control mechanisms, a fundamental
limitation with these methods is that these descriptions imply a static
baroreflex. That is, either the linear or the sigmoid curve is
constructed from "one-to-one" pressure-heart rate mapping, suggesting that one stimulus induces only one response. Thus these curves are static in nature, containing relatively little information regarding the dynamic properties of baroreflex function.
The concepts of dynamic system analysis deserve a brief comment here.
First, on the basis of engineering principles and common sense, it is
not difficult to appreciate that the output of a dynamic system at any
given moment is determined not only by the current input to the system
but also by the preceding inputs and outputs of the system
(25). This fact emphasizes the importance of history and
memory properties of a dynamic system in determining its behavior.
Mathematically, contributions of the preceding inputs and outputs to
the current output of the system are characterized by the impulse
response function of the system in the time domain, which is equivalent
to the transfer function of the system in the frequency domain
(25). Second, as shown in APPENDIX A, responses of a dynamic system to a step function input include two
components. One is the transient response; the other is the steady-state response. Specifically, the transient response contains multiple frequencies, whereas the steady-state response is determined only by the frequency response of the system at the zero frequency. Hence, a proper system analysis must reflect both the steady state and
transient characteristics of the system.
Dynamic properties of baroreflex function in humans have been
recognized in a previous study (4), which indicated that the heart rate response to a brief baroreceptor stimulation could last
for several seconds. More recently, dynamic properties of baroreflex function have been studied by using spontaneous variations in arterial pressure and heart rate (45). Estimation of
the transfer function between these two variables confirms the
frequency-dependent properties of the baroreflex (9, 30, 34,
45). However, because the underlying mechanisms for generating
the rhythmic variations in pressure and heart rate are not always
clear, the physiological significance of the transfer function
estimated under these conditions has yet to be elucidated (1, 45,
51). For example, an estimated phase between the spontaneous
changes in pressure and heart rate at most frequencies below 0.5 Hz has been found neither near 0° nor near 180° (45, 54).
This result has been interpreted to reflect both a feedforward effect
of heart rate on arterial pressure and a feedback effect of pressure on heart rate (45, 51, 53, 54). However, this interpretation is derived from the static model of the baroreflex, which assumes that
baroreflex mediated changes in heart rate occur only as a consequence
of those directionally opposite from changes in arterial pressure
(45). On the basis of similar assumptions, a "sequence method" has been proposed to classify the spontaneous variations in
pressure and heart rate into either baroreflex or nonbaroreflex mediated sequences (6, 30, 34). However, if spontaneous variations in arterial pressure and heart rate truly reflect a dynamic
process, then why should these changes be dictated by the rules derived
from the static model of baroreflex function?
In the present study, by using thigh cuff deflation, a moderate acute
hypotensive stimulus was generated to perturb the baroreflex without
"clamping" or restricting the physiological responses by injection
of vasoactive drugs (16). Heart rate responses to the
changes in pressure are consistent with a baroreflex-mediated stimulus-response relationship. Identification of this relationship with an AR process confirmed the dynamic properties of the baroreflex. Most importantly, we have observed that the beat-to-beat heart rate
response to transient changes in pressure is not always controlled by
directional changes in pressure, as would be expected from a static
model of baroreflex function (15, 38). Rather, the heart
rate response to changes in pressure appears to be controlled by an
error signal in the pressure generated by the differences between the
pressure values before and after the thigh cuff deflation. These data
suggest a "set-point" mechanism for beat-to-beat blood pressure
control in a time scale of <1 min. On the basis of previous studies
(23, 46), we speculate the following chain of events (Fig.
5): 1) a disturbance to the
systemic pressure would cause the baroreceptors to signal the central
nervous system; 2) a deviation between the set point (which
may reflect the steady-state value of arterial pressure) and the actual
pressure occurs; 3) the error signal generated will then
control the heart rate response as well as other cardiovascular
variables (e.g., vascular resistance and stroke volume) to restore the
pressure back to the set-point level before the perturbations. On the
basis of this hypothesis, a simplified mathematical model has been
developed in APPENDIX B for deriving the transfer function
relationship between the changes in the pressure and the heart rate.
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Fig. 5.
Schematic representation of baroreflex control system. CNS, central
nervous system; BP, blood pressure; SV, stroke volume; TPR, total
peripheral resistance.
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The concept of a set point for blood pressure control is not new. A
previous study (23) indicated that autonomic neural activity, which controls blood pressure and heart rate, is reflexively regulated by an error signal in pressure in reference to its set point.
Furthermore, it has been proposed that the set point for blood pressure
control may reset centrally and/or peripherally to accommodate
steady-state changes in pressure and heart rate (23).
However, recent studies (8, 10) suggest that, instead of a
single set point, the whole sigmoid curve of the baroreflex may reset
rapidly within 20 to 30 s under sustained changes in arterial
pressure. These findings have raised doubts about whether an absolute
set point exists in the blood pressure control system (10,
41).
Moreover, for long-term time scales over hours and days, it has been
well known that blood pressure is controlled by the balance of body
fluid volume regulated by the kidney (17, 18). A pressure higher than the set point will cause loss of body fluid, whereas a
pressure lower than the set point will cause retention of body fluid.
Thus an error signal in pressure plays an essential role in the control
of body fluid volume, which in turn buffers the changes in the
steady-state pressure.
The present study extends these previous studies by demonstrating that
even during short-term intervals of seconds to minutes, a set-point
regulatory mechanism may still play an essential role in beat-to-beat
blood pressure and heart rate control. We cannot exclude the
possibility that a rapid reseting of the baroreflex curves may explain
the simultaneous increases in pressure and heart rate observed in the
present study. However, recent carefully performed animal studies
(8, 32) have shown that rapid baroreflex reseting does not
occur or is attenuated when changes in pressure are transient and
pulsatile. We hypothesize that a set-point regulatory scheme across
different time scales may contribute to an overall homeostasis in blood
pressure control even though the regulatory mechanisms for short-
versus long-term blood pressure control may be distinctly different.
Interestingly, although the static model of the baroreflex may not be
completely accurate when heart rate or R-R interval is used as the
dependent variable during acute changes in pressure, the behavior of
the baroreflex does appear to be modeled accurately by a sigmoid curve
if baroreceptor afferent nerve activity is used as the dependent
variable and the frequency of changes in pressure is low (<1 Hz)
(36, 44). Thus whenever baroreceptor afferent nerve
activity deviates from its own operating point by a perturbation in
pressure, continuous compensatory responses of pressure and heart rate
will be elicited until the stimulus to the baroreceptors is diminished.
Thus increases or decreases in arterial pressure always lead to
directionally similar increases or decreases in baroreceptor afferent
nerve activity regardless of the absolute value of heart rate or
pressure (36). This speculation is consistent with the
characteristics of dynamic changes in pressure and heart rate observed
in the present study.
It is important to note, however, that the set point in short-term
blood pressure control may reset to a new level if the steady-state
arterial pressure has been altered (10). Consequently, heart rate responses to changes in pressure would be controlled by the
new set point. The transient heart rate response to acute hypotension
during progressive LBNP suggests this possibility. During LBNP, a
similar pattern of heart rate responses was observed as that at
baseline (Fig. 2). These data suggest a "resetting" of the set
point associated with gradual decreases of cardiac filling, stroke
volume, and steady-state systolic pressure (26). However,
we have also observed that the transient changes in pressure and heart
rate did not always follow the set-point predictions precisely at the
highest levels of LBNP, suggesting either an interference from the
enhanced spontaneous variations in arterial pressure or a degradation
of blood pressure control during high degrees of orthostatic stress
(59).
Limitations.
Two limitations of the present study should be emphasized. First, the
specific regulatory mechanisms underlying the transient changes in
pressure and heart rate cannot be determined from the data collected.
However, on the basis of reported differences in the latency of vagal
and sympathetic responses to a sudden change in pressure (11, 12,
57), it is likely that the initial cardioacceleration associated
with the rapid fall in pressure after the thigh cuff deflation was
mediated primarily by vagal withdrawal. However, the continuous
augmentation of heart rate that follows, associated with the
simultaneous increases in pressure, was more likely mediated by both
vagal withdrawal and/or concurrent activation of cardiac sympathetic
nerve activity (56). Direct recording of sympathetic nerve
activity or pharmacological blockade might provide additional insight
into these specific mechanisms.
Second, because the magnitude, direction, and rate of pressure changes
induced by the thigh cuff deflation could not be controlled in the
present study, baroreflex function was not evaluated over a large range
of pressure changes in both directions, as might be obtained with other
conventional methods (14, 48). Considering the well-known
nonlinear properties of baroreceptors in response to transient changes
in pressure (12, 27), we are cautious about extending the
results obtained in the present study to other situations where changes
in blood pressure may have different magnitudes, rates, and directions.
However, it should be noted that the thigh cuff deflation-induced
changes in pressure were biphasic rather than unidirectional (Fig. 1).
Moreover, the transient changes in pressure and heart rate were fitted
very well by the linear AR model not only at baseline but also during
LBNP and associated with significantly augmented changes in
pressure. These data suggest that the method used in the
present study for baroreflex identification may be efficacious even
when changes in pressure are of different magnitudes, as induced by the
thigh cuff deflation.
Mathematical modeling of baroreflex function.
In the present study, we demonstrated that the dynamic changes in
pressure and heart rate during thigh cuff deflation can be fitted well
by a second-order AR model with a pure time delay of 0.75 s.
Moreover, the frequency response of the baroreflex showed properties of
a low-pass filter with a gradual phase decline from ~180° at the
very low frequencies. The low-pass filter properties of the baroreflex
with a gradual phase decline is intuitive in that as the frequency of
changes in pressure rises, the heart rate response to the pressure
stimuli will lag more and diminish in amplitude.
However, two limitations should be addressed regarding the modeling
strategies in the present study. First, it has been shown that heart
rate responses to brief stimuli to baroreceptors are asymmetrical, and,
second, adaptation occurs with sustained changes in pressure (12,
27). In addition, the baroreflex saturates with large variations
in pressure and has thresholds with small variations in pressure
(15, 31). These data suggest intrinsic nonlinearity of the
baroreflex in the regulation of heart rate. An important question then
arises, Is the linear model assumption of the baroreflex suitable in
the present study?
Theoretically, the linearity of a system can be tested according to the
principle of superposition. That is, if two or more stimuli are acting
on a system, the responses of a linear system are the sum of the
effects of each individual stimuli on the system (25).
Direct testing of linearity of the baroreflex is difficult to conduct
under the current experimental conditions. However, the efficacy of
using linear system analysis methods in the evaluation of baroreflex
function has been shown by other investigators (3, 33,
44). For example, in animal studies (22, 44), with open-loop control of the baroreflex, linear properties have been demonstrated in both the afferent traffic of carotid and aortic baroreceptors and in the efferent sympathetic nerve activity in response to dynamic changes in pressure. Additionally, linearity has
been shown in the responses of the sinus node to vagal and sympathetic
nerve stimuli (5). In humans, under closed-loop conditions, evaluation of baroreflex function using spontaneous variations in arterial pressure and heart rate also has revealed linear
properties of baroreflex function (3, 33).
In the present study, we found that heart rate responses to acute
hypotension can be fitted well by a second-order AR model, as evidenced
by the randomness of the model residuals. These findings support the
efficacy of using linear system methods in the present study. Although
we caution that a nonlinear relationship between steady-state changes
in pressure and heart rate may modulate temporal responses of heart
rate to dynamic changes in pressure (2, 15), it is
possible that, for a given steady state, when transient changes in
pressure are in the physiological range, contribution of nonlinearities
may not be significant for the baroreflex control of heart rate. In the
present study, the maximal pressure fall induced by the thigh cuff
deflation was ~16 mmHg at baseline and ~34 mmHg at 50 mmHg LBNP.
Hence, as a first approximation, linear system analysis should be
applicable under the current experimental conditions.
Second, in the present study, for model identification, the transient
changes in pressure and heart rate were considered as the input and
output of the baroreflex system respectively. This strategy was
implemented as if the system performed under open-loop conditions
(49). However, we caution that the baroreflex by nature is
a proprioceptive reflex (43). Thus, in normal situations, the system must operate under closed-loop conditions.
Identifiability of a system under closed-loop conditions is determined
by the specific features of the feedback loop of the system and the
properties of external inputs to the system (49). It has
been shown that identifiability of a closed-loop system based on
input-output data cannot be guaranteed if the input is determined
through a linear low-order noise-free feedback from the output
(49). Because effects of baroreflex-mediated changes in
heart rate (output) on arterial pressure (input) are likely determined
through the "triple product" of heart rate, stroke volume, and
total peripheral resistance (Fig. 5), it appears unlikely that the
effects of heart rate on arterial pressure could be modeled by a linear
low-order noise-free system (26). Thus the aforementioned unidentifiable condition may not be applicable in the present study.
Furthermore, in the present study, an external perturbation to the
arterial pressure was generated by the thigh cuff deflation. A typical
baroreflex mediated response in heart rate was observed. We speculate
that the identifiability of baroreflex function may be enhanced by the
external perturbations used in the present study (49). In
the simplified model of Fig. 5, under linear assumptions, with the
presence of external perturbations, it is possible to show that the
transfer function between the dynamic changes in pressure and heart
rate will reflect the fundamental operating properties of
baroreflex function even under closed-loop conditions (see
APPENDIX B).
Attempts have been made to quantify baroreflex function under
closed-loop conditions from spontaneous changes in arterial pressure
and heart rate (1, 3, 33). While the interpretation of
these results has been controversial (3, 45), the
usefulness of this technique for the evaluation of baroreflex function
has been demonstrated in many recent studies (9, 30, 34).
Furthermore, with the application of external perturbations to the
carotid transmural pressure, it has been shown that the transfer
function between the changes in pressure and efferent sympathetic nerve activity identified under closed-loop conditions did not significantly differ from those identified under open-loop conditions
(22).
Taken together, although perfect applicability of the modeling strategy
for the system identification cannot be demonstrated under the
conditions of the present study, considering the remarkable fit of the
stimulus-response data with the second-order AR model, the proposed
methods appear able to evaluate well the dynamic properties of
baroreflex function in humans.
Baroreflex function during LBNP.
LBNP has been used extensively to simulate the effects of orthostatic
stress on the cardiovascular system (26). Typical cardiovascular responses, including augmented sympathetic nerve activity with LBNP, have been consistently reported in the literature (21, 26, 55). However, findings regarding baroreflex
regulation of heart rate during othostatic stress are much more
controversial (13, 35, 37, 50). Earlier work showed that
prolongation of R-R intervals provoked with neck suction or infusion of
vasoactive drugs either did not change or was diminished with changes
in either posture or application of LBNP (13, 37, 50).
However, these reports have been challenged by data using both neck
pressure and neck suction during LBNP (35). The results of
the latter study showed that maximal baroreflex gain calculated from
the steep portion of the sigmoidal baroreflex curve significantly increased during LBNP, suggesting a reduction of inhibitory effects from "cardiopulmonary receptors" on the arterial baroreflex
regulation of heart rate (35). However, recent studies
(9, 19, 20) evaluating baroreflex function using
spontaneous changes in arterial pressure and heart rate showed that
baroreflex gain significantly decreased during LBNP and with head-up tilt.
One explanation for this discrepancy may be that the stimulation of
different baroreceptor populations (i.e., carotid vs. aortic) and
different methods used for estimating the baroreflex gain may result in
different conclusions, particularly if the operating point has shifted
closer to the threshold for baroreceptor stimulation associated with
the significant increases in heart rate during orthostatic stress.
In the present study, we found that the heart response to acute
hypotension induced by thigh cuff deflation was attenuated at high
levels of LBNP. We also found, consistently, that the numerator
coefficients of the AR model significantly decreased during LBNP and
associated with an overall reduction of transfer function gain. These
data suggest that the ability of the baroreflex to regulate heart rate
was reduced during LBNP.
However, further analysis showed that the denominator coefficients of
the AR model and the curve shape of the transfer function gain remained
unchanged during LBNP. These data suggest that the reduced ability of
the baroreflex to buffer changes in pressure was due primarily to a
reduction of the static gain rather than changes in the dynamic
properties of baroreflex function. In other words, these data suggest
that the heart rate response to dynamic changes in pressure are
preserved during LBNP, whereas the magnitude of steady-state heart rate
responses to the steady-state changes in pressure is diminished with
the orthostatic stress.
We speculate that, during LBNP, a reduction in pulsatile pressure
and/or flow associated with the fall in stroke volume may modulate
transduction properties of baroreceptors and therefore attenuate its
responsiveness to the maximal changes in pressure (7, 26).
However, it is also possible that an augmented sympathetic nerve
activity and/or a diminished vagal reserve (i.e., nearly complete vagal
withdrawal and therefore limited capacity to shorten R-R interval with
further unloading of the baroreceptors) in control of heart rate may
contribute to the static gain reduction (37, 40, 47).
Interestingly, the significant reduction in the transfer function gain
also occurred at 15 mmHg LBNP and are associated with no significant
changes in steady-state arterial pressure and heart rate. Whether the
arterial baroreflex is provoked at this low level of LBNP is
controversial, even though recent evidence clearly shows changes in
aortic dimension making unloading of arterial baroreceptors very likely
(52). These data support our findings that the arterial
baroreflex may be altered even at low levels of LBNP.
In summary, we have presented a new noninvasive method for the
evaluation of baroreflex function in humans. The primary findings are
that beat-to-beat heart rate responses to dynamic changes in arterial
pressure appear to be controlled by an error signal rather than by
directional changes in pressure, suggesting a set-point mechanism in
short-term beat-to-beat blood pressure control. Furthermore, we found
that dynamic changes in pressure and heart rate during acute
hypotension induced by thigh cuff deflation can be fitted well by a
second-order AR model, confirming dynamic properties of baroreflex
function. Moreover, the frequency response of the baroreflex
demonstrated low-pass filter properties. Finally, in applying this
method for the evaluation of baroreflex function during LBNP, we found
that the ability of the baroreflex to evoke a heart rate response to
transient changes in arterial pressure was reduced during orthosatic
stress and this reduction was due primarily to a reduction of the
static gain of the baroreflex.
| |
APPENDIX A |
Assuming that changes in arterial pressure (i.e., the input) is
a step function, the change in heart rate can be obtained from
|
(1a)
|
where Rm(z) is the
z-transform of changes in heart rate,
Pm(z) is the z-transform of
changes in pressure, and G(z) represents the transfer
function of the system. Equation 1a provides the total response of the baroreflex function with respect to heart rate. That
is, it contains both the transient and steady-state response. To
explore this, assume that G(z) is a second-order system with a pure time delay of 0.75 s, as we have shown previously in
Eq. 3. Allowing the input, Pm(z),
to be a step function with a magnitude of p0, the
expression for Pm (z) becomes
|
(2a)
|
Hence, the total change in heart rate,
Rm(z), is given by
|
(3a)
|
Taking the inverse z-transform of
Rm(z), we get
![&Dgr;r<SUB>m</SUB>(<IT>k</IT>)<IT>=</IT>[(<IT>&bgr;e<SUP>−j&thgr;</SUP></IT>)<SUP>(<IT>k−3</IT>)</SUP><IT>+</IT>(<IT>&ggr;e<SUP>j&thgr;</SUP></IT>)<SUP>(<IT>k−3</IT>)</SUP><IT>+&agr;</IT>]<IT>u</IT>(<IT>k−3</IT>)](/content/vol280/issue1/fulltext/H407/img008.gif)
|
(4a)
|
where rm(k) is the total change in
heart rate in time domain; 
, 
, 
, and 
are functions of
b0, p0, a1,
and a2, respectively; and
[u(k 3)] is the discrete unit step
function delayed by three sampling intervals (0.75 s). For a stable
system, | | < 1 and | | < 1.
It is noted that after the input, pm(k), is
applied to the system, the total response of the system is determined
by all the terms in Eq. 4a. However, as time elapses (i.e.,
k 
), the first two terms inside the bracket in
Eq. 4a will diminish for a stable system (because
| | < 1 and | | < 1) and only remains, which
reflects the steady-state response. It can be shown that is given
by
|
(5a)
|
which is the same as shown in Eq. 4.
The first two terms inside the parentheses in Eq. 4a [i.e.,
(e
j
)(k 3) and
(e j)(k 3)] reflect the transient response of the system because
they diminish over an extended time horizon. These two terms are
functions of the roots of the denominator polynomial of the transfer
function G(z) in Eq. 3 [i.e., poles of
G(z)]. Here, the poles are
ej and
e j. It is well established in
the linear control system theory that the poles determine the dynamic
(i.e., transient) response of a linear system. That is, the position of
the poles in the complex z-plane determines whether the
system is stable, underdamped (overshoots present in the response), or
overdamped (no overshoot present). Furthermore, the poles determine the
frequency response characteristics of the system. Specifically, the
transfer function poles determine whether the system acts as a low- or
high-pass filter.
| |
APPENDIX B |
To gain insight into the experimental results obtained in this
study, a simplified model of changes in the heart rate and arterial
pressure after the thigh cuff deflation is proposed. Figure
6 shows the block diagram of the proposed
model.
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|
Fig. 6.
Simplified model of baroreflex control system with
perturbations in arterial pressure induced by thigh cuff deflation.
Ps(z), physiological set point (or desired
value) of change in ABP; A(z), change in the afferent signal
of baroreceptors, reflecting changes in ABP sensed by the
baroreceptros; E(z), difference between
Ps(z) and A(z); B(z),
G1(z), and G2(z), linear
transform functions of the baroreceptor, HR regulator, and ABP
regulator, respectively; Rm(z), change in HR
in response to E(z); Pa(z), change
in ABP relating to changes in HR; Pd(z), change
in ABP induced by thigh cuff deflation;
Pm(z), change in measured ABP determined by
the combined effects of Pd(z) and
Pa(z).
|
|
It is important to note that the model in Fig. 6 is intended to reflect
the control process for changes in the arterial pressure and heart rate
with respect to their respective levels before the cuff deflation. It
does not represent the absolute value of the pressure and the heart
rate. Although it is likely that the system governing the changes in
pressure and heart rate is more complicated than what is shown in Fig.
6, however, the model in this figure can provide useful insight.
For this model, it is assumed that components are linear. Specifically,
it is assumed that B(z), G1(z), and
G2(z) are linear transfer functions of the
baroreceptor, heart rate regulator, and blood pressure regulator,
respectively. In this regard, the following parameters are defined:
Ps(z), physiological set point (or desired
value) of change in arterial pressure; A(z), change in the
afferent signal of baroreceptors, reflecting changes in arterial
pressure sensed by the baroreceptors; E(z), difference between Ps(z) and A(z);
Rm(z), change in heart rate in response to
E(z); and Pa(z), change in
arterial pressure relating to the changes in the heart rate. It is
important to note that Pa(z) depends not only
on Rm(z) but also on the change in the stroke volume (SV) and the total peripheral resistance (TPR), as we have shown
in the Fig. 5. However, for the sake of simplicity, it is assumed that
the effects of changes in SV and TPR on Pa(z)
could be integrated into the transfer function of pressure regulator. Pd(z) is the change in the blood pressure
induced by the thigh cuff deflation, and
Pm(z) is the change in the measured blood pressure determined by the combined effects of
Pd(z) and Pa(z).
With the use of the model in Fig. 6, it can be shown that the changes
in the heart rate, Rm(z), is related to the
changes in the internal set point, Ps(z), and
pressure perturbation induced by the thigh cuff deflation,
Pd(z), in the following manner
|
(1b)
|
Likewise, the measured changes in blood pressure,
Pm(z), can be expressed in terms of
Ps(z) and Pd(z) as
|
(2b)
|
Hence, the relationship between the measured changes in the
heart rate, Rm(z), and the measured changes
in the pressure, Pm(z), can be obtained from
the following equation
|
(3b)
|
Equations 1b and 2b provide that when
there is no perturbation [i.e., Pd(z) = 0] and Ps(z) is nonzero, the
Rm(z) and Pm(z) are
then affected by Ps(z) as follows
|
(4b)
|
|
(5b)
|
Alternatively, when Ps(z) = 0, that is, assuming that the physiological set point of changes in
pressure remains constant for the short-term blood pressure control and
Pd(z) 
0, the changes in the heart rate
and blood pressure are related as (from Eq. 3b)
|
(6b)
|
Equation 6b is of particular interest because it
shows that the transfer function between the dynamic changes in
pressure and heart rate reflect properties of baroreflex function even under a closed-loop condition. Moreover, it suggests that at the steady
state (i.e., f 0 in z = e j2fT;
hence, z 1), a rise in the measured blood pressure
(i.e., p0 > 0) may yield a drop in the heart rate.
Particularly, let
|
(7b)
|
where p0 > 0, reflecting a positive step
change in the measured pressure. Note that the resulting change in the
heart rate, Rm(z), can be obtained from the
following equation
|
(8b)
|
At the steady state, we have
|
(9b)
|
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ABSTRACT
INTRODUCTION
