Vol. 280, Issue 4, H1830-H1839, April 2001
Causal linear parametric model for baroreflex gain assessment
in patients with recent myocardial infarction
Giandomenico
Nollo1,
Alberto
Porta2,
Luca
Faes1,
Maurizio
Del
Greco3,
Marcello
Disertori3, and
Flavia
Ravelli1
1 Dipartimento di Fisica, Università di Trento, and
Istituto Trentino di Cultura-irst, 38050 Povo-Trento;
2 Dipartimento di Scienze Precliniche, Laboratorio
Interdisciplinare Tecnologie Avanzale di Vialba, Università
di Milano, 20157 Milano; and 3 Unità Operativa di
Cardiologia, Ospedale Santa Chiara, 38100 Trento, Italy
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ABSTRACT |
Spectral and
cross-spectral analysis of R-R interval and systolic arterial pressure
(SAP) spontaneous fluctuations have been proposed for noninvasive
evaluation of baroreflex sensitivity (BRS). However, results are not in
good agreement with clinical measurements. In this study, a bivariate
parametric autoregressive model with exogenous input (ARXAR model),
able to divide the R-R variability into SAP-related and -unrelated
parts, was used to quantify the gain (
ARXAR) of the
baroreflex regulatory mechanism. For performance assessing, two
traditional noninvasive methods based on frequency domain analysis
[spectral, baroreflex gain by autogressive model (AR);
cross-spectral, baroreflex gain by bivariate autoregressive model
(2AR)] and one based on the time domain [baroreflex
gain by sequence analysis (SEQ)] were considered and
compared with the baroreflex gain by phenylephrine test
(PHE). The BRS evaluation was performed on 30 patients
(61 ± 10 yr) with recent (10 ± 3 days) myocardial
infarction. The ARXAR model allowed dividing the R-R variability
(950 ± 1,099 ms2) into SAP-related (256 ± 418 ms2) and SAP-unrelated (694 ± 728 ms2)
parts. AR (12.2 ± 6.1 ms/mmHg) and
2AR (8.9 ± 5.6 ms/mmHg) as well as
SEQ (12.6 ± 7.1 ms/mmHg) overestimated BRS
assessed by PHE (6.4 ± 4.7 ms/mmHg), whereas the
ARXAR index gave a comparable value (ARXAR = 5.4 ± 3.3 ms/mmHg). All noninvasive methods were significantly
correlated to PHE (ARXAR and
SEQ were more correlated than the other indexes). Thus
the baroreflex gain obtained describing the causal dependence of R-R
interval on SAP showed a good agreement with PHE and may
provide additional information regarding the gain estimation in the
frequency domain.
baroreflex sensitivity; spectral analysis; phenylephrine; autoregressive models; R-R-SAP transfer function
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INTRODUCTION |
EVALUATION OF BAROREFLEX
SENSITIVITY (BRS) is considered an important clinical tool for
diagnosis and prognosis in a variety of cardiac diseases (10,
14). In humans, two techniques based on provocative tests have
been commonly used to measure the baroreflex gain. The first
method estimates the baroreflex gain by evaluating the slope of the
increase of heart period subsequent to the rise of arterial pressure
induced by injection of a vasoconstrictive drug. The second one
estimates BRS by measuring changes in heart rate and blood pressure
after the external selective manipulation of carotid baroreceptors by a
neck chamber device. Despite the encouraging results found by recent
studies (13), the need for an intravenous line and
pressure injection or neck chamber devices limits the use of this
methodology to clinical settings for risk stratification protocols.
Moreover, the induced large increase in blood pressure is a different
stimulus compared with the small amplitude pressure changes occurring
in physiological conditions. An episode of myocardial ischemia
associated to phenylephrine injection has also been recently reported
(9).
Recent studies have suggested that spontaneous fluctuations of
arterial pressure and R-R intervals offer a noninvasive method for
assessing BRS in natural circumstances. They were commonly based on
spectral (21), cross-spectral (26), and
baroreflex sequence (22) analyses of simultaneous R-R
interval and systolic arterial pressure (SAP) variabilities. In most of
these studies, the noninvasive measurements of BRS have been
significantly correlated with pharmacologically derived estimates, even
though the degree of correlation decreased moving from healthy subjects
(26) to hypertensive (29) or post-myocardial
infarction (MI) (24) patients. Methodological approach and
physiological measuring conditions may be the main causes of
disagreement between the phenylephrine estimates of the baroreflex gain
and those obtained by noninvasive methods. In fact, the injection of
the vasoactive drug forces the regulatory system to work in an
open-loop condition, and the BRS slope is calculated by an open-loop
linear model. On the other side, approaches based on spontaneous
fluctuations of SAP and R-R interval are taken with all reflexes and
control mechanisms fully active (i.e., in a closed loop)
(8).
The estimation of the baroreflex gain by means of the monovariate
spectral analysis (21) and sequence analysis
(7) is performed by assuming but not testing that the
whole R-R variability is generated by SAP changes. On the other hand,
cross-spectral approaches (26) explicitly consider the
mutual interactions between R-R and SAP variabilities, but the causal
dependencies are not taken into account when the baroreflex gain is
calculated. Complex closed-loop models (3, 4) have been
proposed to describe the causal relationship from SAP to R-R interval
(the baroreflex pathway) and to separate it from the mechanical pathway (from R-R interval to SAP) in the estimation of baroreflex gain (23). In this study, a simpler approach based on an
open-loop dynamic adjustment autoregressive model (ARXAR model)
(25) was introduced. This model describes the causal
relationship between R-R interval and SAP by dividing the R-R interval
variability in SAP-related and -unrelated parts. Performance and
reliability of this method were assessed compared with classical
approaches based on frequency (i.e., spectral and cross-spectral
methods) and time domain (baroreflex sequence method) analysis of R-R
and SAP spontaneous variability and with the baroreflex slope computed by the phenylephrine method.
| |
METHODS |
Gain estimation by spectral and cross-spectral analysis.
Monovariate spectral analysis allows estimating the baroreflex gain by
a separate autoregressive (AR) description of R-R and SAP
variabilities. The application of the AR model requires reducing R-R
and SAP series to zero-mean processes (rr and sap series, respectively). According to Fig.
1A, AR analysis of the data
considers the dependence of current rr and sap values on the samples of their own past (by A1 and
A2 blocks) and on the current value of an input
noise source (i.e., wrr and
wsap). The parameter estimation of the AR model
followed the Burg identification method (12, 19), and the
model order was chosen inside the set {6, 8, 10, 12}, according to
the minimum of the Akaike figure of merit (2). For the
validation of the AR model, whiteness of the prediction error was
verified by applying the Anderson test (15) on the residuals wrr and
wsap. After the power contribution of each
oscillatory component to the overall R-R and SAP variabilities
(11) was calculated, two gain indexes were obtained as the
square root of the ratio between the power content of the low-frequency
(LF) [AR(LF)] and high-frequency (HF)
[AR(HF)] bands (21). The computation of
the gain is illustrated in Fig. 1B and described in detail
in the APPENDIX.
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Fig. 1.
Evaluation of baroreflex sensivity (BRS) by means of
spectral analysis of R-R and systolic arterial pressure (SAP) interval
series. A: autoregressive (AR) models for the separate
description of R-R and SAP variability signals. B:
evaluation of the gain indexes in the low-frequency (LF)
[AR(LF)] and high-frequency (HF)
[AR(HF)] bands as the square root of the ratio between
the power content of the R-R and SAP spectra
[Prr(f) and
Psap(f), respectively].
A1 and A2, AR block
coefficients; rr and sap, zero-mean of the R-R interval and SAP series,
respectively; wsap and
wrr, white noise input of the AR models and sap,
respectively.
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The interactions between R-R and SAP can be jointly considered for
evaluating baroreflex gain by means of a bivariate autoregressive (2AR)
model. In the diagram of Fig.
2A, A11
and A22 blocks contain the AR parameters of rr
and sap signals, whereas A12 and
A21 blocks pertain to the effects of SAP on R-R
interval and vice versa. The model order P was chosen, in
the set {6, 8, 10, 12}, minimizing the Akaike figure of merit
(2) for the bivariate joint process |rr(n) or
sap(n)|, where n is the current value of the rr
or sap series. The same model order P was assigned to all
the model blocks, thus avoiding advantaging one regulation mechanism
with respect to the other. The model identification is based on the
generalization of the Burg maximum entropy spectral estimation to the
multichannel case (18). With the use of the 2AR
model, we estimated a power spectral density (PSD) matrix whose
elements were used to compute the gain function
2AR(f) and the coherence function
K2(f) as outlined in the
APPENDIX. The coherence was used to estimate the strength
of the coupling between R-R and SAP at each frequency. Thus the gain
indexes at LF and HF [2AR(LF) and
2AR(HF), respectively] were obtained by sampling
2AR(f) on the maximum of the coherence function inside the specific band (Fig. 2B).
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Fig. 2.
Evaluation of BRS by means of cross-spectral analysis.
A: block diagram of the bivariate AR (2AR) model for the
joint description of R-R and SAP series. B: gain indexes for
LF and HF bands [2AR(LF) and 2AR(HF),
respectively] are obtained by sampling the modulus of the R-R-SAP
transfer function [2AR(f)] in
correspondence with the maximum of coherence function
K2(f). Only
coherence values >0.5 were considered for a reliable estimation
of the gain function.
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Causal open-loop model for baroreflex gain estimation.
The model considered in Fig. 3 belongs to
the class of single-output ARXAR models (5, 25) and is
defined by the equation
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(1)
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The R-R interval is affected by both P values of the
sap sequence [by a12(k)
coefficients] and the current value of the noise source
urr. Moreover, Eq. 1 takes the
possible dependence of the R-R interval on P samples of its
own past into account [by a11(k) coefficients]. As outlined in Fig. 3, sap and
urr signals are described as AR processes with
wsap and wrr zero-mean
input white noises. The blocks A22 and
D1 are formed by the AR parameters of sap and
urr, respectively. In the open-loop ARXAR model,
the variability of SAP around its mean value (i.e., the sap signal) is
considered as an exogenous input, i.e., it may affect the R-R interval
variability without being affected. The effects of other sources
independent from SAP on R-R variability, considered as noise in this
context, are accounted for in the model by means of the
urr signal.
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Fig. 3.
Bivariate parametric AR model with exogenous input (ARXAR model)
for the description of the causal effects of SAP on R-R. In the
open-loop scheme, the sap signal constitutes an exogenous input. Thus
R-R changes are separately driven by SAP variations and independent
inputs different from SAP intervals. urr,
colored noise.
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The coefficient estimation follows an iterative identification task
based on the generalized least-squares method (28). The
model order P was chosen, in the set {6, 8, 10, 12},
minimizing the Akaike figure of merit (2) for the
bivariate joint process |rr(n), sap(n)|. The
model validation required us to check the whiteness of the model inputs
and the uncorrelation, even at zero lag, from
wsap to wrr.
The ARXAR model allows computing the PSD of R-R interval as a sum of
two partial spectra (5), which represent the variability of R-R dependent and independent of SAP. In this way, the total R-R
power, as well as its amount in the LF and HF bands, was decomposed in
two parts, quantifying the SAP-related and -unrelated contribution to
the R-R interval variability.
The gain of the R-R-SAP transfer function
[ARXAR(f)] was estimated in the
frequency domain directly from the coefficients of
A12 and A11 blocks (see
APPENDIX for details). The function
ARXAR(f) was sampled in connection
with the main oscillations of the driving signal sap inside the two
major bands LF and HF, thus providing the corresponding gain indexes
ARXAR(LF) and ARXAR(HF) (Fig. 4).
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Fig. 4.
Estimation of gain index by the ARXAR model. The gain
function [ARXAR(f);
bottom] is estimated from the coefficients of the blocks
A11 and A12 of Fig. 3.
The index is calculated by sampling
ARXAR(f) in correspondence with the
central frequencies of LF and HF spectral components (11)
(dotted lines) of SAP power spectra. Top:
Psap(f) by the ARXAR model.
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Sequence analysis.
The sequences (7) in which R-R and SAP values concurrently
increased or decreased progressively over three variations (four beats)
were extracted, and a linear regression analysis was performed on them.
The sequences in which total R-R or SAP changes were smaller than 5 ms
and 1 mmHg, respectively, and/or the correlation coefficients were
smaller than 0.85 were excluded. The absolute values of the slopes of
the regression lines were then averaged to accomplish a time
domain-based estimation of BRS (SEQ).
Experimental protocol and data analysis.
Thirty consecutive patients (25 men and 5 women; mean age 61 ± 10 yr) were studied 10 ± 3 days after acute MI; the diagnosis of
which was based on currently accepted criteria. The signal recordings
and the BRS evaluation were executed in the electrophysiology laboratory between 9 AM and 12 AM, in comparably comfortable and quiet
ambience conditions with patients in sinus rhythm and breathing spontaneously. After a period of 15 min (allowed for patient
stabilization), the electrocardiogram, respiratory, and blood pressure
signals were recorded in supine position for 10 min. Electrocardiograms were continuously traced by Siemens Mingograph 7 system. The
respiratory activity was recorded in the nostril by using a
differential pressure transducer. Arterial blood pressure was recorded
at finger level (20) by a photoplethysmographic Finapres
device (Ohmeda 2300, Finapres; Englewood, CO). All signals were
digitized at the sampling frequency of 1 kHz by a 12-bit precision
analog-to-digital converter.
Successively, patients underwent the phenylephrine test. A bolus of
phenylephrine (2 µg/kg) was injected via peripheral vein to raise
blood pressure from 15 to 40 mmHg. The test was repeated to obtain at
least three recordings with sufficient pressure rise. Phenylephrine-induced beat-to-beat SAP increases were plotted against
the corresponding R-R interval increases (Fig.
5). Linear least-squares fit was used to
calculate the slope of the regression line. PHE was
obtained by averaging the slopes of the successive recordings.
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Fig. 5.
BRS assessment by the phenylephrine test. A:
phenylephrine open-loop model. After injection of phenylephrine, R-R
changes are linearly correlated to SAP changes. B: example
of BRS estimate. According to the model, R-R-SAP gain
(PHE) is measured as the slope of the regression line
between R-R interval changes ( RR) and SAP changes (SAP).
Equation for the solid line is RR = 4.13 × SAP + 21.9 (r = 0.72, P < 0.001).
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R-R intervals and SAP values were automatically measured on recorded
electrocardiograms and arterial blood pressure signals. Variability
series were then built up with the nth SAP value inside the
nth R-R interval. From each series, mean values were
subtracted to obtain zero-mean processes. Sequences of 300 samples that
fulfilled the stationarity criterion were then analyzed by means of
spectral methods. The PSD of respiratory activity was considered to
locate the HF power content of SAP and R-R spectra in the AR model
(Fig. 1B), sample the coherence function at HF in the 2AR
model (Fig. 2B), and detect the respiratory-driven
oscillation of SAP in the ARXAR model (Fig. 4). For each frequency
domain approach, the mean of baroreflex gain was computed as the
average of baroreflex gain estimated in the LF and HF bands
{AR = [AR(LF) + AR(HF)]/2, 2AR = [2AR(LF) + 2AR(HF)]/2, and
ARXAR = [ARXAR(LF) + ARXAR(HF)]/2} (16).
Statistical analysis.
All results are expressed as means ± SD. The ANOVA test was used
for comparison of BRS measures. Regression analysis was used to assess
the BRS slope and compare different measures of baroreflex gain.
The agreement between the invasive and noninvasive tests was further
assessed by sensitivity and specificity analysis. Data were divided
into true positive and true negative on the basis of a threshold, set
at 4 ms/mmHg for PHE. For each noninvasive test, the
corresponding cutoff was defined by the equation found by linear
regression analysis between PHE and the noninvasive gain index.
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RESULTS |
Model validation.
To verify the whiteness of the inputs of the parametric models, the
Anderson test was performed over 40 lags of the normalized autocorrelation functions of wrr and
wsap. The autocorrelation functions of
wsap and wrr were zero
for each lag > 0 with 5% confidence (
2 points out of the confidence
intervals) in all 30 patients. In addition, the causal structure of the
ARXAR model required verification of the uncorrelation from
wsap to wrr even at zero lag. The normalized cross-correlation was zero for each lag 
0 with
5% confidence in all patients.
In the sequence analysis, regression slopes were carried out on the
2.5% of the total number of sequences on average. Moreover, it could
not be performed on 3 of 30 subjects due to the absence of valid
sequences in the variability series.
Spectral decomposition.
According to the causal ARXAR open-loop model, R-R spectrum was the sum
of the R-R interval variability driven by SAP changes (due to
baroreflex mechanisms) and that independent of SAP changes. An example
of R-R interval spectrum decomposition accomplished by the ARXAR model
is plotted in Fig. 6 along with the PSDs
of SAP and the respiratory series. Both the SAP-driven R-R variability (Fig. 6, top; dotted line) and that owing to different
inputs (Fig. 6, top; dashed line) showed two main components
in LF and HF bands well synchronized with those of SAP and respiratory
spectra. On the whole population, the mean variance of R-R series
(950 ± 1,099 ms2) was divided by the model in 256 ms2 as induced by SAP changes and in 694 ms2 as
owing to unpredictable inputs. As shown in Table
1, the LF rhythms were present both in
SAP-related and -unrelated R-R variabilities and were larger than the
HF rhythms.
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Fig. 6.
Example of spectral decomposition given by the ARXAR
model. Top: R-R interval spectrum (solid line) and its
decomposition in the SAP-related (dotted line) and SAP-unrelated
(dashed line) spectra. Middle and bottom: SAP
spectrum and respiratory spectrum, respectively. n.u., Normalized
units.
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Gain assessment.
Mean values of BRS estimates are shown in Table
2. The central tendency and variability
of the different methods used for baroreflex gain assessment are
reported in Fig. 7. The box and whisker
plots display the mean of the baroreflex gains (PHE, SEQ, AR, 2AR, and
ARXAR) and the dispersion by means ± SE (box) and
means ± SD (whisker). Only the gain index obtained by the ARXAR
model resulted comparable with PHE, whereas
SEQ, AR, and 2AR
overestimated the invasive BRS gain.
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Fig. 7.
Box and whisker diagram of the central tendency and
variability of the different methods used for BRS assessment. The index
obtained by the sequence analysis (SEQ) as well as AR
and 2AR gain indexes (AR and 2AR,
respectively) overestimated values assessed by PHE,
whereas the ARXAR index ARXAR gave comparable values.
Indexes provided by parametric models are the average of the gain in
the two major bands (LF and HF). *P < 0.02 and
**P < 0.01 vs. PHE.
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The reliability of noninvasive methods for BRS estimation was assessed
by performing linear regression analysis between each index and
PHE. Although the correlation coefficients were
relatively low, all the noninvasive indexes were significantly
correlated (P < 0.05) with the reference. The low
correlation may be explained by considering that the relationship
between PHE and noninvasive indexes may change for
different levels of BRS. In fact, PHE distribution
showed a skew for high values. Therefore, to limit the spread of
PHE values, the distribution was cut at the 90th percentile. The results of linear regression analysis after exclusion of the upper tail from the PHE distribution (3 patients)
are reported in Table 3. The baroreflex
gain provided by the sequence analysis showed the best linear
correlation coefficient (r = 0.80). The gain obtained
by the ARXAR model was better correlated with PHE
(r = 0.76; Fig. 8) than
those obtained by the AR and 2AR models. Furthermore, the slope of the
regression line for the ARXAR model was closer to one than the slope
for all other approaches.
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Table 3.
Summary of linear regression analysis versus PHE and of
specificity and sensitivity analysis for each noninvasive gain
index
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Fig. 8.
Correlation between estimates of baroreflex gain by the
ARXAR model {ARXAR = [ARXAR(LF) + ARXAR(HF)]/2} and
the phenylephrine method (PHE) in postmyocardial
infarction patients (n = 27). Equation for solid line
is ARXAR = 0.86 × PHE + 0.86 (r = 0.76, P < 0.001).
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Data of sensitivity and specificity carried out for the four
noninvasive tests are also shown in Table 3. Better agreement was found
between PHE and 2AR or
ARXAR then between PHE and AR. The sequence analysis demonstrated a higher
specificity but a lower sensibility than the ARXAR model.
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DISCUSSION |
Regression analysis demonstrates that all noninvasive gain indexes
were significantly correlated with PHE. The correlations found by this study were smaller than those previously reported in
healthy subjects (26), thus reflecting the characteristics of the considered sample of post-MI patients showing a huge range of
PHE values (1.7 
22.2 ms/mmHg). This result is
not surprising because baroreceptor response caused by phenylephrine
injection can be affected in a different way in post-MI patients than
in healthy subjects. The degree of correlation found in our study was
indeed comparable to the one reported in other works analyzing patients
with hypertension (29) or coronary artery disease (1, 24). Furthermore, the correlation turned out to be higher by
cutting the distribution of PHE to the 90th percentile.
Indeed, it is unlikely that the linear relation between invasive and
noninvasive methods is held throughout the whole range of BRS values.
To explain the low degree of correlation found between the
pharmacological test and noninvasive methods, the methodological
differences in BRS measurement also have to be considered. Indeed, the
phenylephrine test and noninvasive methods for BRS estimation explore
different physiological conditions. Because of phenylephrine infusion,
the reflex changes in peripheral vasoconstriction and the heart
rate-SAP mechanism are basically overridden, thus approximating an
open-loop condition. On the other hand, measurements based on the
analysis of spontaneous fluctuations of SAP and heart rate consider all reflexes and control mechanisms fully active; consequently, they are
carried out in closed-loop condition. Thus, according to other authors
(1, 23, 24), invasive and noninvasive approaches seem to
provide reliable but not identical information.
Spectral approaches.
The computation of the baroreflex gain based on separate spectral
analysis of R-R and SAP variabilities (21) assumes that all R-R changes are caused by SAP variations. This approach does not
explicitly consider the closed-loop interactions between R-R and SAP.
In the bivariate analysis, the causal dependencies between R-R and SAP
are not taken into account even if the strength of the link is assured
by the coherence function (17, 26). This means that the
effect of SAP on the R-R interval cannot be disentangled from the
effect of the R-R interval on SAP. On the contrary, in the present
study, the causality relationships are accounted for by utilizing an
ARXAR model designed to separate the contribution of a driving signal
to the variance of a driven process from the effects of independent
unpredictable inputs (25). Thus the proposed open-loop
model makes it possible to estimate the R-R-SAP transfer function on a
specific path. In this way, only the amount of R-R variability that can
be ascribed to SAP variations is exploited for gain computation. For
these reasons, the spectral estimation of BRS seemed to be improved by
introducing causality. Indeed, better agreement with the phenylephrine
method was shown by the ARXAR model with respect to the AR and 2AR
models. Moreover, because in our post-MI population the
SAP-unrelated R-R power is about three times greater than the
SAP-related one, the spectral decomposition of R-R interval variability
is mandatory to reliably evaluate the baroreflex gain based on the
analysis of SAP and R-R spontaneous fluctuations. Otherwise, the
baroreflex gain is overestimated as a result of considering the overall
R-R variability as completely driven by SAP changes.
Although disagreement exists concerning the nature of the LF and HF
rhythms of R-R variability (6, 27), in humans, a contribution of the baroreflex mechanism to the genesis of both of
these oscillations cannot be excluded. Therefore, to provide a global
measure of the baroreflex-mediated adjustments of the heart rate, the
average of LF and HF gain indexes was introduced. Independently of the
kind of model, a closer correlation and an improved agreement with
PHE were found using this average index. However,
averaging LF and HF gain indexes has to be done very cautiously,
because one needs to consider the possibly different autonomic
contribution of LF and HF coupling between R-R and SAP. Indeed,
experimental evidence (17) has recently suggested the baroreflex nature of LF gain, whereas in the HF frequency band the
coupling between R-R and SAP does not seem to exclusively depend on the
baroreflex mechanism. Our results confirm these findings because a
greater power content in the SAP-related R-R spectrum was observed in
the LF (124 ms2) than HF (40 ms2) band. Again,
introduction of causality seems to make more reliable the spectral
estimates of BRS in the HF band.
Time domain approaches.
The sequence analysis (7) has been previously applied in a
variety (22, 24) of clinical conditions with promising
results. Also in our study, the estimates of BRS accomplished with this technique gave good results in terms of correlation with the
phenylephrine test. This good correlation can be explained by
considering that sequence analysis, calculating the slope of regression
line between changes of R-R and SAP values, attempted to spontaneously
reproduce the procedure of the drug test.
Nevertheless, some limitations are implicitly present in this approach.
First, in case of low amplitude and/or fast changes of R-R and SAP
values, as could happen in elderly and post-MI subjects, the sequence
technique could fail due to the low number of sequences (<3% in our
study) useful for the analysis. Second, the comparison of gain values
and slopes of the regression line documented an overestimate of the BRS
values. This result can be considered as due to independent inputs (as
respiration or enhanced sympathetic tone) acting on the cardiac rhythm,
but not on the systolic pressure, and erroneously ascribed by the
sequence technique to the baroreflexes. Indeed, even though the
baroreflex nature of this technique has been demonstrated on an
experimental animal preparation (7), it does not provide
information on causality.
Closed-loop parametric models should be introduced to overcome these
limitations. In these models, the whole dynamics of the investigated
series is exploited, and the causal feedback effects of SAP on R-R are
separated from the feedforward influences of R-R on SAP (3,
4). In a recent study (23), these approaches were
followed to accomplish a time domain estimation of the baroreflex gain.
In the study, causality was accounted for by measuring the open-loop
baroreflex gain under closed-loop global conditions. Differently, the
ARXAR model utilized in this work is simpler and specifically addressed
to describe the baroreflex pathway; thus the causal dependence of R-R
from SAP is imposed by its open-loop structure. In the proposed model,
the influences of respiration impinging directly on R-R interval are
not directly taken into account and are treated as an unmeasurable
input uncorrelated with SAP (described by urr in
Fig. 3). On the contrary, the effects of respiration on R-R interval
mediated by SAP are accounted for both in LF and HF bands by the
R-R-SAP block (A12 block). The lack of
evaluation of respiration independently of SAP is a limitation for the
model, but these influences are not accounted for by any traditional
noninvasive method based on spectral, cross-spectral, and sequence
analyses. In Ref. 23, the respiratory
influences were explicitly modeled and utilized to explore the
baroreflex modulation during controlled random interval breathing, thus
making possible the investigation of the broadband dynamic effect of SAP on R-R. However, in our study, patients were allowed to
spontaneously breathe. This choice was considered useful to disclose
the performance of the ARXAR model for evaluating BRS in post-MI
patients and also for future applications in a clinical setting.
Clinical implications.
With traditional noncausal approaches, noninvasive evaluation of BRS is
difficult due to the presence of the feedforward effects of R-R
interval on arterial pressure and more specifically of other inputs
directly affecting the sinus node. Thus the introduction of
dynamic causal models should be suggested to avoid spurious effects in
the calculation of baroreflex gain. This becomes mandatory in post-MI
patients, characterized by low amplitude of R-R and SAP variability and
enhanced sympathetic tone.
At present, the phenylephrine test still remains the only accepted
technique for risk stratification by BRS assessment in postinfarction
(13). Specificity and sensitivity values obtained by
comparing invasive and noninvasive tests for BRS evaluation were
generally high. Particularly, a high correspondence with the clinical
classification was found when the gain index was computed after
assessing the strength of the link between R-R and SAP variabilities,
as was done by checking the correlation coefficient in the sequence
analysis, the coherence function in the 2AR model, and the tests of the
modeling hypotheses in the ARXAR model. The agreement found between the
phenylephrine test and noninvasive tests supports the feasibility of
the noninvasive measures of baroreflex gain. However, because of the
different aspects of the baroreflex regulation investigated by invasive and noninvasive approaches, the clinical information provided by the
phenylephrine test cannot be fully extrapolated by the "spontaneous" methods. Therefore, further studies are needed to facilitate the introduction of approaches describing the causal interactions between R-R and SAP for baroreflex gain quantification into clinical practice, for instance, addressing the correlation analysis of the results directly to the prognosis instead of to the
invasive method. Finally, the results provided by the proposed causal
model should be validated in conditions of normal and impaired baroreflex modulation (e.g., in an experimental animal preparation before and after sinoaortic denervation).
In conclusion, this study provides evidence that the ARXAR model,
specifically designed to describe the causal influences of SAP on R-R
interval, is able to quantify baroreflex gain in humans without
altering blood pressure through pharmacological interference. Thanks to
the model structure and to the estimation in the frequency domain,
reliable measures of baroreflex gain in both the LF and HF bands can be
obtained by the ARXAR method. Moreover, the model allows us to quantify
the amount of R-R variability imputable to arterial pressure changes.
Our results confirm the correlation between the baroreflex gain
estimated by noninvasive measurements and by the phenylephrine method,
but this agreement was dependent on the structure of the model and the
methodology used. The findings of this study suggest that the
introduction of dynamic causal models could provide additional information on the estimation of baroreflex gain by noninvasive approaches. In any case, further investigation will be necessary to
delineate the stratification of patients at increased risk of mortality
associated with cardiovascular disease.
| |
APPENDIX |
Computation of baroreflex gain by linear parametric models.
The autoregressive (AR) description of the zero-mean series of the R-R
interval and systolic arterial pressure (SAP) (rr and sap series,
respectively) is given by the equations
|
(A1)
|
|
(A2)
|
where the coefficients a1(k)
and a2(k) represent the regression of
rr and sap (respectively) on P and Q samples of
their own past (respectively), k is the delay of the rr or
sap sample series, n is the current value of the rr or sap
sample series, and the wrr and
wsap are set to be zero-mean white noise inputs with variance 
and
. The power spectral
density (PSD) of R-R interval and systolic arterial pressure
variabilities are computed as follows
|
(A3)
|
|
(A4)
|
where z is the complex frequency,
f is frequency, and T is the sampling period. The
AR spectral decomposition method (11) can be used to
calculate the power contribution of the poles of the
Z-transform of the rr and sap series
[Prr(z) and Psap(z)] and, consequently, the percentage of rr and sap variance inside a
specific frequency band.
In the bivariate AR model, the interactions between rr and sap series
are considered by accounting for the dependence of a series on the
samples of the other by a12 and
a21 coefficients
|
(A5)
|
|
(A6)
|
The uncorrelation between the white noises
wrr and wsap allows
to evaluate the cross-spectrum function Prr 
sap(f) and the autospectra
Psap(f) and
Prr(f) from the model coefficients and the variances of the noises (12). The coherence
function
|
(A7)
|
is the best estimate (in a least-square sense) of the proportion
of nonrandom variance common to a couple of variables at a given
frequency and ranges from 0 to 1. The coherence can be used to evaluate
the reliability of the gain of the transfer function from sap to rr
|
(A8)
|
where 2AR(f) is the baroreflex
gain function of the bivariate AR model. It is worth pointing out that
2AR(f) can be expressed in terms of
K(f)
|
(A9)
|
showing how, at a given frequency, the gain function computed
via the bivariate AR model is modulated from the coherence between R-R
and SAP.
The bivariate parametric AR model with exogenous input (ARXAR)
evaluates the effects of sap on rr separately from those deriving from
immeasurable inputs uncorrelated with sap and excludes the backward
influences of sap on rr
|
(A10)
|
The colored noise (urr) sequence, being
described by the coefficients of D1 block in
Fig. 3, represents the fraction of R-R variability that cannot be
explained by SAP changes. The ARXAR model is used to estimate the gain
of the R-R-SAP transfer function directly from the coefficients
a11(k) and
a12(k)
|
(A11)
|
| |
FOOTNOTES |
Address for reprint requests and other correspondence: G. Nollo, Biofisica Medica, ITC-irst, via Sommarive 18, 38050 Povo-Trento, Italy (E-mail: nollo{at}itc.it).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 2 February 2000; accepted in final form 2 November 2000.
| |
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ABSTRACT
INTRODUCTION
