| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1 Dipartimento di Scienze Precliniche, Laboratorio Interdisciplinare Tecnologie Avanzate di Vialba, 3 Centro Ricerche Cardiovascolari, Consiglio Nazionale della Ricerca, Medicina Interna II, and 4 Medicina Interna I, Ospedale L. Sacco, Universitá degli Studi di Milano, 20157 Milan; and 2 Dipartimento di Bioingegneria, Politecnico di Milano, 20133 Milan, Italy
 |
ABSTRACT |
|---|
 TOP
 ABSTRACT
 INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
A double exogenous
autoregressive (XXAR) causal parametric model was used to estimate the
baroreflex gain (
XXAR) from spontaneous R-R interval and
systolic arterial pressure (SAP) variabilities in conscious dogs. This
model takes into account 1) effects of current and past SAP
variations on the R-R interval (i.e., baroreflex-mediated influences), 2) specific perturbations affecting R-R
interval independently of baroreflex circuit (e.g., rhythmic neural
inputs modulating R-R interval independently of SAP at frequencies
slower than respiration), and 3) influences of
respiration-related sources acting independently of baroreflex pathway
(e.g., rhythmic neural inputs modulating R-R interval independently of
SAP at respiratory rate, including the effect of stimulation of
low-pressure receptors). Under control conditions,
XXAR = 14.7 ± 7.2 ms/mmHg. It decreases after
nitroglycerine infusion and coronary artery occlusion, even though the
decrease is significant only after nitroglycerine, and it is completely
abolished by total arterial baroreceptor denervation. Moreover,
XXAR is comparable to or significantly smaller than
(depending on the experimental condition) the baroreflex gains derived
from sequence, power spectrum [at low frequency (LF) and high
frequency (HF)], and cross-spectrum (at LF and HF) analyses and from
less complex causal parametric models, thus demonstrating that
simpler estimates may be biased by the contemporaneous presence of
regulatory mechanisms other than baroreflex mechanisms.
cardiovascular variability; linear parametric modeling; identification techniques
| |
INTRODUCTION |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
EVALUATION OF THE BAROREFLEX GAIN is considered an important tool in clinical practice (16). The baroreflex gain is usually estimated by inducing an increase of arterial pressure by means of a vasoconstrictive drug and by evaluating the subsequent lengthening of the heart period (23). Because this procedure is based on a nonphysiological stimulus that may alter the actual value of the baroreflex gain, several techniques have been proposed to evaluate the baroreflex gain based on the spontaneous variability of systolic arterial pressure (SAP) and heart period (R-R interval). Traditional methods are based on 1) detection of sequences of SAP and R-R interval values characterized by the simultaneous increase or decrease of both variables (10), 2) calculation of the power spectrum of R-R interval and SAP variability series (19), and 3) calculation of the magnitude of the SAP-R-R transfer function (22).
The main limitation of these methods is that they are not capable of evaluating the fraction of R-R interval variability driven by SAP changes because they do not impose causality (i.e., a relationship linking the current R-R interval to past SAP values). Therefore, they cannot reveal whether the observed changes in SAP and R-R interval are the result of feedback mechanisms from SAP to R-R interval (i.e., the baroreflex pathway) or of the feedforward relationship from R-R interval to SAP (mainly related to Starling and windkessel effects). As a consequence, these methods can only produce indexes lumping together the properties of both paths, and denervation experiments or pacing are required to disentangle the feedback pathway from the feedforward pathway (2, 17). Moreover, the effects of other variables on the estimate of the baroreflex gain are not explicitly taken into account, although they are acknowledged. For example, Smyth et al. (23) suggested fixing the respiratory phase in assessing the baroreflex gain. Changes of the R-R interval synchronous with respiration can occur independently of variations in SAP and, therefore, of the arterial baroreflex circuit. For example, changes in venous return and central venous pressure related to respiration can produce activation or deactivation of the low-pressure receptors (25) and, therefore, through the Bainbridge reflex, changes in the heart period (28). Moreover, even effects of centrally mediated rhythmic variations in the autonomic drive to the sinus node (14) produce a certain amount of R-R variability not mediated by the baroreceptive circuit. The presence of a fraction of R-R interval variability independent of SAP changes, if not explicitly addressed, may lead to an overestimate of the baroreflex gain.
The aim of this study is to propose a model-based approach to the evaluation of the baroreflex gain capable of imposing causality (i.e., to evaluate the fraction of R-R variability driven by SAP changes) and directly accounting for sources capable of influencing the R-R interval independently of SAP changes (mainly the effects of slow rhythms and respiration directly driving R-R interval).
Three parametric linear model structures are considered: 1) the R-R-SAP exogenous model (X model), 2) the R-R-SAP exogenous model with autoregressive (AR) input (XAR model), and 3) the R-R-SAP exogenous model with exogenous respiration and AR input (XXAR model). In the X model the R-R interval depends on the current and several previous SAP values. In the XAR model the R-R-SAP relationship is described by an X model, but rhythmic sources affecting R-R interval independently of baroreflex pathway are also modeled. In the XXAR model, in addition to XAR features, the nonbaroreflex effects of respiration on R-R interval are described.
Comparison of the baroreflex gain estimated by the X model with those derived from traditional methods (10, 19, 22) allows us to clarify the role of causality in the estimation of baroreflex gain. Comparison of the baroreflex gains obtained by the proposed parametric models allows us to evaluate the distorting effects of sources driving R-R interval variability independently of SAP changes on the estimate of the baroreflex gain and to propose a model (i.e., the XXAR model) providing a less biased estimate of the baroreflex gain.
The study was carried out on conscious dogs (21) undergoing the following experimental conditions: 1) control, 2) nitroglycerine infusion (NT) producing a sympathetic activation by decreasing SAP, 3) coronary artery occlusion (CAO) increasing sympathetic activity without an important change in mean arterial pressure, and 4) total arterial baroreceptive denervation (TABD) preventing SAP changes to produce baroreflex-mediated R-R interval variations.
| |
CAUSAL PARAMETRIC MODELING APPROACH TO ESTIMATE OF BAROREFLEX GAIN |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
This section describes the three parametric models (X, XAR, and XXAR) capable of extracting the baroreflex gain from spontaneous variability of R-R interval and SAP.
The main characteristics of these models are that 1) they describe the linear relationship among changes of the variables around a set point (the mean value); 2) they are dynamic because several past samples are considered; 3) they are causal because they fix a temporal direction of the influences (R-R interval changes depend on current and past variations of SAP); and 4) they are defined in the beat-to-beat domain, thus allowing us to reduce the complexity of the model structure (we do not have to reconstruct all the features of the waveform) while maintaining information about the short-term regulatory mechanisms (6, 11).
In the discussion of the three models rri, sapi, and respi represent the difference between the actual sample RRi, SAPi, and Respi and their mean values, where RR is R-R interval, Resp is the respiratory signal, and i is the progressive cardiac beat number.
R-R-SAP X model.
The simplest parametric model describing the rr series and the causal
influences of sap on rr can be defined as
|
(1) |
wrr2. Therefore, the
rri value depends on the sap (the exogenous signal) at the same cardiac beat (i.e., sapi)
and on p past values of sap. In this way, causal effects of
SAP on R-R interval (i.e., the baroreflex-mediated influences) are
explicitly addressed. Fig. 1A
represents this relationship in terms of a block diagram. All the rr
variability independent of sap is lumped into
wrr. The sap series is modeled as an AR process
of order p
|
(2) |
wsap2.
|
R-R-SAP XAR model.
To take into account both direct effects of sap and possible influences
(indicated as urr), including rhythmic
influences, affecting rr independently of sap, the model described by
Eq. 1 can be modified as
|
(3) |
|
(4) |
R-R-SAP XXAR model.
The most complex model is introduced to separate the influences of
respiration on rr acting independently of sap. It is defined as
|
(5) |
|
(6) |
wresp2.
Therefore, three inputs are considered to be capable of driving rr
changes (Fig. 1C): 1) current and past sap
variations producing R-R interval variability via the baroreflex
pathway, 2) current and past respiratory changes not
mediated by sap variations (e.g., oscillations mediated by respiratory
centers and by the activation of low-pressure receptors), and
3) unmeasurable inputs, particularly slow oscillations,
independent of sap and resp (e.g., slow rhythms accompanying myocardial ischemia).
Evaluation of baroreflex gain. After the coefficients of the models are identified (see Identification and validation of X, XAR, and XXAR models in the APPENDIX), the block Hrr-sap (Fig. 1) is fed by an artificial signal simulating a unitary ramplike rise of SAP and the slope of the relevant increase of R-R interval is taken as an index of the baroreflex gain. The slope of the R-R interval response is calculated with least-squares linear fitting evaluated on the first 15 samples.
Hypothesis testing and evaluation of performance of model. The adequacy of the model in describing the dynamics of the R-R signal and its interactions with SAP and Resp signals is tested by evaluating whether the residual signals (wrr and wsap for X and XAR models and wrr, wsap, and wresp for the XXAR model) are white and not correlated with each other (see Identification and validation of X, XAR and XXAR models in the APPENDIX).
The performance of the model is evaluated by calculating the goodness of fit (
) of the R-R interval series. It is defined as
|
(7) |
rr2 and
wrr2 represent the variance
of R-R interval and of its residue, respectively, thus measuring the
percent variance of the R-R interval series explained by the proposed model. rr = 1 means that the R-R interval power is
completely described by the model; an insufficient model structure
produces rr closer to 0.
| |
TRADITIONAL APPROACHES TO ESTIMATE OF BAROREFLEX GAIN |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
This section summarizes the most utilized techniques for estimating the baroreflex gain from spontaneous variability series of R-R interval and SAP. These techniques are based on baroreflex sequence (10), power spectrum (19), and transfer function (22) analysis.
Baroreflex sequence analysis.
This method is based on the search for sequences characterized by the
contemporaneous increase (positive sequence) or decrease (negative
sequence) of R-R interval and SAP. The lag (
) between R-R interval
and SAP samples is chosen as the lag producing the maximum in the
normalized R-R-SAP cross-correlation function. Both positive and
negative sequences are referred to as baroreflex sequences if they
match the following prerequisites: 1) the total of R-R
variations is >5 ms; 2) the total of SAP variations is >1
mmHg; and 3) the length of the sequences is four beats (3 variations). For each sequence the slope of the regression line
in the plane (SAPi-
,RRi)
is calculated. All the slopes with correlation coefficient >0.85 are
averaged, and this average, represented by BS, has been
taken as a measure of the baroreflex gain (10). The
reliability of BS depends on the number of baroreflex sequences detected in the series.
Power spectral analysis.
The method relies on the calculation of the power spectrum of the rr
and sap series (see Power Spectrum Estimate in the
APPENDIX) and on the evaluation of the power inside two
bands, the low frequency (LF, from 0.04 to 0.14 Hz) and high frequency
(HF, ±0.04 Hz around the respiratory rate) bands
(26). Two indexes estimating the baroreflex gain are
calculated (19) as
|
(8) |
Transfer function analysis.
The R-R-SAP transfer function modulus can be calculated from the zero
mean sap and rr series as
|
(9) |
CS(LF) and
CS(HF). This approach also requires us to test whether
K2 is >0.5 at LF and HF (7, 11).
| |
METHODS |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
Surgical preparation, experimental protocol, and recorded variables. We used a subset of experiments carried out on chronically instrumented, conscious dogs to evaluate the effects of several experimental maneuvers on the cardiovascular variability series of R-R interval and SAP (21). The surgical procedure and the experimental protocol were described previously by Rimoldi et al. (21).
Briefly, all the dogs were chronically instrumented to measure electrocardiogram (ECG) and arterial pressure. Under anesthesia, the ECG electrodes were subcutaneously fixed to intercostal muscles (lead II). A catheter with strain-gauge transducers (Statham Instruments, Oxnard, CA) was inserted in the femoral artery and advanced to the abdominal aorta. In addition to ECG and arterial pressure, the respiratory movements were monitored by means of a thoracic belt connected to strain gauges. The respiratory signal was used to extract the breathing rate. In six dogs, a hydraulic occluder was positioned around the left circumflex coronary artery to occlude the artery when inflated with a volume of saline. In four dogs, TABD was performed in two steps. First, after both carotid artery bifurcations were identified, the sinus nerves were located and severed. Second, at the time of the thoracotomy, the adventitia surrounding the aortic arc and its branches was carefully dissected. Completeness of denervation was assessed by the loss of the usual heart rate response to pressure increases mechanically produced by occluding the distal thoracic aorta. Attention was paid to avoid visible damages to the vagi or to the sympathetic nerve branches. Recordings were performed after a recovery period of 1-2 wk. All the dogs were acquainted with the laboratory and were trained to lie unrestrained on the recording table. Recordings were carried out: 1) under control conditions (n = 16), 2) 5-10 min after intravenous NT (32 µg · kg
1 · min1)
(n = 8), 3) during brief (2 min) CAO
(n = 6), and 4) after TABD (n = 4). Experimental protocols were approved by the
Committee on Animal Use and Care of the University of Milan.
At control conditions, mean R-R and SAP were 739 ± 128 ms and
126 ± 22 mmHg, respectively. After NT, a moderate hypotension (104 ± 13 mmHg) and a reflex tachycardia (539 ± 139 ms)
were observed. During CAO, the decrease of R-R interval was even more
marked (499 ± 44 ms), without marked changes in mean arterial
pressure. After TABD, the R-R was 621 ± 179 ms and SAP strongly
increased (167 ± 9 mmHg).
Beat-to-beat variability series extraction and baroreflex gain evaluation. ECG, arterial pressure, and respiratory signals were analog/digital converted at 300 Hz. The QRS peak was detected by using a threshold on the first derivative of ECG. Jitters in the location of the R peak were minimized by parabolic interpolation. The ith R-R measure was obtained as the temporal distance between two consecutive R peaks. The maximum of the arterial pressure inside the ith R-R interval was the ith SAP. The respiratory signal was sampled in correspondence with the first QRS defining the ith R-R interval.
Indexes of the baroreflex gain () were derived directly from
sequences of ~300 consecutive R-R intervals and SAP values by means
of the proposed models (X, XAR, and
XXAR from X, XAR, and XXAR models, respectively). We
also calculated BS (10), PS(LF) and PS(HF) (19), and
CS(LF) and CS(HF) (22).
Statistical analysis. Comparisons between control and other experimental conditions were performed by means of one-way analysis of variance (Tukey test). If the normality test was not passed, Kruskal-Wallis one-way analysis of variance on ranks was used (Dunn test).
The indexX was compared with BS,
PS, and CS in the same experimental
condition by using one-way repeated-measures analysis of variance
(Tukey test). If the hypothesis of normality was not fulfilled,
Friedman one-way repeated-measures analysis of variance on ranks was
used (Tukey test). This comparison is performed to evaluate the extent
to which the estimate of the baroreflex gain changed when causality was
imposed and variability sources different from SAP were disregarded.
The same statistical analysis was used to compare XXAR
to X and XAR in the same experimental
condition. This comparison was performed to evaluate the extent to
which the estimate of the baroreflex gain varied when sources
independent of SAP variations but capable of driving R-R interval
changes were explicitly disentangled. A P < 0.05 was
considered significant.
| |
RESULTS |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
Estimating baroreflex gain from spontaneous beat-to-beat
variability of R-R interval and SAP: an example.
An example of beat-to-beat variability series of R-R interval and SAP
under control conditions is depicted in Fig.
2, A and B. The
power spectra (Fig. 2, C and D) point out that
the R-R interval and SAP variabilities were dominated by a clear HF
rhythm (at 0.23 Hz in this example) synchronous with respiration (not shown). The relevant K2 (Fig.
3) is close to 1 at HF and even at
frequencies higher than HF. In contrast,
K2 is <0.5 at frequencies lower than HF
(from 0 to 0.1 in this example). Both the proposed causal parametric
approach estimating X, XAR, and
XXAR and traditional methods producing
BS, PS, and CS were
applied to this sequence of data.
|
|
X, XAR, and
XXAR, respectively), were superposed. The X model provided the largest estimate of the baroreflex gain
(X = 48.9 ms/mmHg), whereas the smallest estimate
was furnished by the XXAR model (XXAR = 31.6 ms/mmHg). XAR, based on the XAR model, was 41.2 ms/mmHg.
|
BS is based on the detection of the baroreflex sequences
present in the two series. In this set of data they are only 4%. All
these sequences, when plotted in the plane (SAP, RR), appeared as
straight segments (Fig. 5). These
segments were covered from the lower left corner to the upper right
corner for the positive sequences and in the opposite direction for the
negative sequences. The index BS = 54.2 ms/mmHg was
obtained as the average slope of these segments.
|
PS is based on power spectral analysis of R-R interval
and SAP series and on the calculation of the power associated with spectral components. The HF components are represented in Fig. 6. The power associated with the HF
components (Fig. 6, A and B) was used to evaluate
an PS(HF) = 39.2 ms/mmHg, whereas a consistent estimate of PS(LF) was prevented by the absence of LF
components of R-R interval variability. The reliability of the estimate
of PS(HF) was confirmed by the high degree of squared
coherence (Fig. 3) at HF (KHF2 = 0.99).
|
CS is based on the estimation of the SAP-R-R transfer
function magnitude (Fig. 7). It was
sampled at HF detected on the SAP series, thus producing
CS(HF) = 41.9 ms/mmHg. CS(LF) could
not be calculated consistently because
KLF2 < 0.5 (Fig. 3).
KHF2 > 0.5 confirmed the
reliability of CS(HF).
|
Testing reliability of baroreflex gain estimates.
The reliability of the baroreflex gain estimates based on the causal
parametric approach depends on the ability of the model to produce
white and noncorrelated residual signals. In the X model,
wsap was always white, whereas
wrr passed the whiteness test in 14 of 16 animals under control conditions, in 1 of 8 after NT, in 4 of 6 during
CAO, and in 2 of 4 in TABD. These results pointed out that the
structure of this model is too simple to describe the R-R interval
dynamics. However, the two residual signals wrr
and wsap in the X model were not correlated,
even at zero lag, in any condition. The inability of the X model to describe the R-R interval dynamics is confirmed by the relatively low
value of the goodness of fit rr (0.59 ± 0.16 at
control, 0.59 ± 0.16 after NT, 0.45 ± 0.14 during CAO, and
0.57 ± 0.19 after TABD). When the structure of the model was
rendered more complex (XAR and XXAR models), the tests on the whiteness
of the residual signals and on their noncorrelation were fulfilled in all conditions. rr was increased accordingly in the XAR
model (0.69 ± 0.13 at control, 0.71 ± 0.17 after NT,
0.75 ± 0.06 during CAO, and 0.76 ± 0.17 after TABD) and
revealed the largest values in the XXAR model (0.78 ± 0.12 at
control, 0.73 ± 0.16 after NT, 0.79 ± 0.06 during CAO and
0.81 ± 0.12 after TABD).
BS depends of the number of
baroreflex sequences detected in the variability series. Under control
conditions the mean percentage of baroreflex sequences (over 300 samples) was 5%. In 8 of 16 dogs the percentage was <3%, and in 4 animals no slope was present. An example of R-R interval and SAP
beat-to-beat variability in which at least three contemporaneous
increases or decreases of R-R interval and SAP were not present (no
baroreflex sequence could be detected) is depicted in Fig.
8, A and B. The magnification of two periods of dominant HF rhythm (Fig. 8C)
shows that, whereas three consecutive increases can be found in SAP series, R-R interval can increase or decrease only for two beats, thus
producing the absence of baroreflex sequences. The percentage of
baroreflex sequences was larger after NT (18%; only 1 dog exhibited <3%), during CAO, and after TABD (12% and 24%; no animal exhibited <3%).
|
PS(LF), CS(LF),
PS(HF), and CS(HF) was tested by
evaluating the value of the squared coherence at LF and HF
(KLF2 and
KHF2). PS(LF) and
CS(LF) could not be calculated in all animals because LF
oscillations were not always found. Moreover, even when LF oscillations
were observed, they exhibited a value of KLF2 > 0.5 only in a few cases
(7 of 16 at control, 2 of 8 after NT, 5 of 6 during CAO, and 0 of 4 after TABD). In contrast, KHF2 was
>0.5 in all animals in all the conditions (0.98 ± 0.03 at control, 0.95 ± 0.05 after NT, 0.91 ± 0.16 after CAO, and
0.98 ± 0.01 after TABD).
Causal modeling approach vs. traditional methods.
The results are summarized in Table 1. At
control, BS, PS(HF), and
CS(HF) were very high (>40 ms/mmHg).
PS(LF) and CS(LF) were smaller and
similar to X and XAR. XXAR
showed the smallest value, significantly smaller than X
and XAR.
|
PS(LF),
CS(LF), X, XAR, and
XXAR comparable and smaller than BS,
PS(HF), and CS(HF). During CAO, the
indexes BS, PS(HF),
CS(HF), and X decreased significantly
whereas PS(LF), CS(LF), and indexes based
on more complex models (XAR and XXAR)
exhibited a nonsignificant fall. It is worth noting that, during CAO,
XAR and XXAR were comparable but
significantly smaller than PS(LF) and
CS(LF). After TABD, all the indexes showed a drastic
reduction. However, only the indexes based on a causal parametric
approach (X, XAR, and
XXAR) were close to zero. After TABD,
PS(LF) and CS(LF) could not be calculated
because of the lack of coherence in all animals.
| |
DISCUSSION |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
All the methods considered here for estimating the baroreflex gain
from spontaneous R-R interval and SAP variabilities are able to detect
the decrease of the baroreflex sensitivity during baroreceptive
unloading provoked by NT, during sympathetic activation induced by CAO,
and after TABD, thus confirming the capability of these parameters to
measure the baroreflex gain (Table 1). However, Table 1 points out that
the values of the estimated baroreflex gains can be very different.
These differences are the result of the different capabilities of the
various methods to separate the baroreflex pathway from mechanisms
capable of producing R-R interval and SAP variability independently of
baroreflex circuit (Table 2). Three main
mechanisms are responsible for R-R interval and SAP variabilities
independent of baroreflex pathway: 1) Starling and
windkessel effects producing SAP variations driven by R-R interval
changes (i.e., the feedforward mechanisms), 2) neural
modulations at HF capable of driving the R-R interval independently of
SAP changes [e.g., neural influences projecting the activity of
central respiratory oscillators (14) and neural reflexes involving low-pressure receptors activated by respiration-related changes in blood volume and central venous pressure (13,
28)], and 3) neural modulations affecting the sinus
node at LF independently of SAP variations [e.g., neural influences
projecting the activity of central slow oscillators (20)
and cardiogenic reflexes initiated by ischemia]. Also, mechanical
influences of respiration on sinus node produce R-R interval changes
independent of SAP variations, but they are usually irrelevant [they
become apparent in the denervated heart (9)].
|
Traditional methods for estimation of baroreflex gain.
The indexes PS and CS (19,
22) are calculated without separating all these mechanisms from
the baroreflex pathway (Table 2). Indeed, they are estimated without
imposing any specific causal direction in the interactions between R-R
interval and SAP variabilities (i.e., the feedforward mechanical path
from R-R interval to SAP is merged with the feedback baroreflex pathway from SAP to R-R interval). In contrast, the index BS
(10) is actually calculated by separating the baroreflex
sequences from the nonbaroreflex sequences, but no memory is allowed in
feedback and feedforward relationships, thus preventing the full
separation of feedback from feedforward influences (Table 2). As a
consequence, the traditional indexes BS,
PS, and CS mix the gains of both paths
and are closer to the baroreflex gain only if the R-R interval is
mainly driven by SAP variations. This condition is more likely to be
fulfilled at LF (2, 11). In contrast, R-R interval variability leads SAP oscillations at HF (2, 27), thus
rendering PS(HF) and CS(HF) sensible to
the mechanical pathway (i.e., the Starling and windkessel effects) and
introducing a bias when these indexes are used to evaluate the
baroreflex gain. According to this consideration, PS(HF)
and CS(HF) can be different from PS(LF)
and CS(LF), respectively, and can be significantly
larger than 0 even after TABD (Table 1). Because the index
BS is not conceptually different from PS
and CS, it could be more similar to
PS(LF) and CS(LF) or to
PS(HF) and CS(HF), depending on the
predominant dynamics in the R-R interval and SAP variabilities. In
dogs, as a result of a dominant HF dynamic in every experimental condition, BS is similar to PS(HF) and
CS(HF) (Table 1). These observations suggest the use of
PS(LF) and CS(LF) instead of BS, PS(HF), and CS(HF) in
dogs and in other experimental preparations in which respiratory
influences might directly drive the sinus node. Unfortunately, the
presence of a small amount of R-R power in the LF band in dogs
(21) makes it difficult to obtain a robust estimation of
PS(LF) and CS(LF)
(KLF2 is usually <0.5), thus
rendering the estimate of baroreflex gain unreliable even in the LF
band and necessitating new tools based on a causal modeling approach.
Role of causality in estimate of baroreflex gain.
The proposed modeling approach fixes the temporal direction of the
influences of SAP on R-R interval (i.e., causality), thus exploring the
baroreflex pathway directly. Indeed, the R-R interval depends on SAP
inside the same cardiac beat (fast effect) and on past SAP values
(dynamic effect). The fast effect has been ascribed to the action of
vagal circuits and the dynamic, slow effect to the sympathetic branch
(12). This feature gives the model the possibility of
separating the baroreflex pathway (the feedback path) from the
mechanical relationship (mainly windkessel and Starling effects, the
feedforward path) (Table 2). In other words, only that part of the R-R
variability causally related to SAP changes is exploited to evaluate
the baroreflex gain. As a result, X, XAR,
and XXAR are smaller than BS,
PS(HF), and CS(HF) in all experimental
conditions, comparable to PS(LF) and
CS(LF) under control conditions and after NT but
significantly smaller during CAO and close to 0 after TABD.
Role of rhythmic influences independent of sap changes in estimate
of baroreflex gain
The simplest model (i.e., the X model) is able to evaluate the amount
of R-R interval variability driven by SAP changes (i.e., the
baroreflex-mediated R-R variability) but cannot model rhythmic inputs
capable of driving R-R interval independently of SAP variations (Table
2). Therefore, X is calculated without separating the baroreflex pathway from mechanisms [e.g., central slow oscillators (20), central respiratory oscillators (14),
and cardiopulmonary reflexes (13, 28)] producing LF and
HF oscillations on R-R interval not mediated by the baroreflex circuit
(Table 2). In contrast, the XAR and XXAR models can disentangle the
baroreflex pathway from mechanisms driving R-R interval independently
of SAP (Table 2). The XXAR model can be considered a refinement of the
XAR model (Table 2). Indeed, among the rhythmic sources driving R-R
interval independently of SAP the XXAR model distinguishes those driven
by respiration and, therefore, in the HF band, from those independent
of respiration (i.e., in the LF band).
XXAR is significantly smaller
than X and XAR, as a result of the effect
of taking into account the respiration-related changes of R-R interval
independent of SAP variations, which are particularly prominent in dogs
(2, 4). During NT, X, XAR,
and XXAR are similar, as a result of a more important
role of the baroreflex circuit in regulating R-R interval than that of
central respiratory oscillators and low-pressure areas. During CAO,
X is significantly smaller than XAR and
XXAR because of the technical inability of the X model to fit the data. The use of XAR and XXAR models allows us to solve the
identification problem correctly and to find out that
XAR and XXAR are similar. During CAO, the
use of an XAR or XXAR model is necessary to take into account LF
oscillations directly impinging on the sinus node as previously
reported (4). In this experimental condition, XAR and XXAR
models perform similarly because of the reduced importance of HF direct
effects on the sinus node. It is worth noting that
PS(LF) and CS(LF) remain significantly larger than XAR and XXAR as a result of
the influence of LF fluctuations on R-R interval independent of SAP,
thus pointing out that even in the LF band traditional indexes may be
biased and making obvious the improvement of the causal model-based approach.
In conclusion, our data suggest that XXAR provides a
less biased value of baroreflex gain than any other indexes derived from sequence, power spectrum, and transfer function analyses or from
simpler causal models. XXAR is calculated by exploiting the R-R variability driven by SAP changes after disentangling different
variability sources capable of producing changes in the R-R interval
variability independent of the baroreflex circuit. If R-R variability
was completely driven by SAP changes, traditional indexes would be
equal to XXAR. However, the former indexes may be larger
than the latter because of the bias of direct effects of respiration on
R-R variability, of slow fluctuations of R-R variability independent of
the baroreflex circuit, and of the mechanical feedforward pathway. The
estimation of XXAR requires the additional recording of
a respiratory signal but, in contrast to the sequence analysis, it uses
all the information content of R-R interval series and, unlike spectral
and transfer function analyses, it does not require the setting of
specific frequency bands.
| |
APPENDIX |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
In this appendix we summarize the methods used to estimate the coefficients of the X, XAR, and XXAR models, the power spectrum and the cross spectrum.
Identification and validation of X, XAR, and XXAR models. The coefficients of the X model (i.e., Eq. 1) and of the separate AR models describing the dynamics of sap and resp signals (i.e., Eqs. 2 and 6) are identified via a least-squares procedure (8, 15). The coefficients of the XAR model (i.e., Eqs. 3 and 4) and of the XXAR model (i.e., Eqs. 5 and 4) are identified using a generalized least-squares approach (8, 24). This iterative procedure is stopped when the current iteration does not produce a significant percent decrease in the variance of the residue wrr with respect to the previous one (the threshold is 0.001). The solution of both the traditional and generalized least-squares problems is performed by means of the Cholesky decomposition method (15).
The model adequately describes the dynamics of the signal if the residual signals (wrr and wsap for X and XAR models and wrr, wsap, and wresp for the XXAR model) are white (all the information is captured by the model parameters). The whiteness of the residual signals is verified by the Anderson test (15). This test checks that the normalized autocorrelation functionswrr(), wsap(), and
wresp() (the autocorrelation functions divided by
wrr2,
wsap2, and
wresp2,
respectively) are equal to 0 for 
0 and 
40. If this
is true for 95% of the values of , the test is fulfilled with 5%
confidence, [e.g., with max = 40, () 0 is allowed for <3 values of ]. The structure of the model is
adequate to describe interactions among the signals if the residues are
not correlated with each other (all relationships among the series are
explained by the model structure). The same test used to check the
whiteness of the residual signals is carried out to verify their
noncorrelation; it checks that the normalized cross-correlation
functions
wrr-wsap() and
wrr-wresp()
(the cross-correlation functions divided by the product of
the standard deviation
wrrwsap and
wrrwresp,
respectively) are 0 for 40 (even at = 0).
Obviously, in the case of X and XAR models, it is sufficient to test
the whiteness of wrr and
wsap and their noncorrelation. The choice of a
high model order can help to fulfill these hypotheses. However, this
solution should be avoided because the fitting is performed even on the
noise superposed on the data. To favor small model orders, the model
order is selected according to the minimum of the Akaike figure of
merit for multivariate processes (1). The best model order
is searched in the range from 6 to 16. After choosing the best model
order, whiteness and noncorrelation on the residual signals are tested
and the goodness of fit rr (i.e., Eq. 7) is calculated.
Power spectrum estimate. The power spectrum is calculated by using a parametric approach (18) instead of a nonparametric approach (3). The rr and sap series are described as AR processes considering the current value of the series as a linear combination of its p past values (e.g., see Eq. 2 for the AR model of sap). The coefficients of this linear combination are estimated via Levinson-Durbin recursion (15), and the number of the coefficients is chosen according to the Akaike criterion (1). The Anderson test is used to test whether the residue is white (15). After the power spectral densities of these two series are calculated, the power spectral decomposition procedure (29) is carried out to evaluate the contribution of each oscillation (described by a real pole or a pair of complex and conjugated poles) to the total power (i.e., the variance) of the process, thus allowing us to estimate the power in LF and HF bands.
Cross-spectrum estimate. The calculation of the cross spectrum between rr and sap series requires a bivariate approach instead of the monovariate approach required by spectral analysis. We choose a parametric approach based on a bivariate AR model (5) to estimate the autospectrum of sap series and the cross spectrum between sap and rr instead of a nonparametric method (11). The model order is fixed at 10, and the coefficients of the bivariate AR model are identified via least-squares methods (8, 15).
| |
FOOTNOTES |
|---|
Address for reprint requests and other correspondence: A. Porta, Universitá degli Studi di Milano, Dipartimento di Scienze Precliniche, LITA di Vialba, Via G.B. Grassi 74 20157 Milano, Italy (E-mail: alberto.porta{at}unimi.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 13 December 1999; accepted in final form 20 May 2000.
| |
REFERENCES |
|---|
|
TOP
ABSTRACT
INTRODUCTION
CAUSAL PARAMETRIC MODELING...
TRADITIONAL APPROACHES TO...
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
1.
Akaike, H.
A new look at the statistical model identification.
IEEE Trans Autom Control
19:
716-723,
1974.
2.
Akselrod, S,
Gordon D,
Madwed JB,
Snidman NC,
Shannon DC,
and
Cohen RJ.
Hemodynamic regulation: investigation by spectral analysis.
Am J Physiol Heart Circ Physiol
249:
H867-H875,
1985
3.
Akselrod, S,
Gordon D,
Ubel FA,
Shannon DC,
Berger RD,
and
Cohen RJ.
Power spectrum analysis of heart rate fluctuations: a quantitative probe of beat-to-beat cardiovascular control.
Science
213:
220-223,
1981
4.
Baselli, G,
Cerutti S,
Badilini F,
Biancardi L,
Porta A,
Pagani M,
Lombardi F,
Rimoldi O,
Furlan R,
and
Malliani A.
Model for the assessment of heart period and arterial pressure variability interactions and respiratory influences.
Med Biol Eng Comput
32:
143-152,
1994[Web of Science][Medline].
5.
Baselli, G,
Cerutti S,
Civardi S,
Liberati D,
Lombardi F,
Malliani A,
and
Pagani M.
Spectral and cross-spectral analysis of heart rate and arterial blood pressure variability signals.
Comput Biomed Res
19:
520-534,
1986[Web of Science][Medline].
6.
Baselli, G,
Cerutti S,
Civardi S,
Malliani A,
and
Pagani M.
Cardiovascular variability signals: towards the identification of a closed-loop model of the neural control mechanisms.
IEEE Trans Biomed Eng
35:
1033-1046,
1988[Web of Science][Medline].
7.
Baselli, G,
Porta A,
and
Ferrari G.
Models for the analysis of cardiovascular variability signals.
In: Heart Rate Variability, edited by Malik M,
and Camm AJ.. Armonk, NY: Futura, 1995, p. 135-145.
8.
Baselli, G,
Porta A,
Rimoldi O,
Pagani M,
and
Cerutti S.
Spectral decomposition in multichannel recordings based on multivariate parametric identification.
IEEE Trans Biomed Eng
44:
1092-1101,
1997[Web of Science][Medline].
9.
Bernardi, L,
Keller F,
Sanders M,
Reddy PS,
Griffith B,
Meno F,
and
Pinsky MR.
Respiratory sinus arrhythmia in the denervated human heart.
J Appl Physiol
67:
1447-1455,
1989
10.
Bertinieri, G,
di Rienzo M,
Cavallazzi A,
Ferrari AU,
Pedotti A,
and
Mancia G.
A new approach to analysis of the arterial baroreflex.
J Hypertens
3:
S79-S81,
1985[Web of Science].
11.
De Boer, RW,
Karemaker JM,
and
Strackee J.
Relationships between short-term blood pressure fluctuations and heart rate variability in resting subjects. I. A spectral analysis approach.
Med Biol Eng Comput
23:
352-358,
1985[Web of Science][Medline].
12.
De Boer, RW,
Karemaker JM,
and
Strackee J.
Hemodynamic fluctuations and baroreflex sensitivity in humans: a beat-to-beat model.
Am J Physiol Heart Circ Physiol
253:
H680-H689,
1987
13.
Desai, TH,
Collins JC,
Snell M,
and
Moscheda-Garcia R.
Modeling of arterial and cardiopulmonary baroreflex control of heart rate.
Am J Physiol Heart Circ Physiol
272:
H2343-H2352,
1997
14.
Hirsch, JA,
and
Bishop B.
Respiratory sinus arrhythmia in humans: how breathing pattern modulates heart rate.
Am J Physiol Heart Circ Physiol
241:
H620-H629,
1981
15.
Kay, SM.
Modern Spectral Analysis: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1989.
16.
La Rovere, MT,
Bigger JT,
Marcus FI,
Mortara A,
and
Schwartz PJ.
Baroreflex sensitivity and heart-rate variability in prediction of total cardiac mortality after myocardial infarction.
Lancet
351:
478-484,
1999.
17.
Legramante, JM,
Raimondi G,
Massaro M,
Cassarino S,
Peruzzi G,
and
Iellamo F.
Investigating feed-forward neural regulation of circulation from analysis of spontaneous arterial pressure and heart rate fluctuations.
Circulation
99:
1760-1766,
1999
18.
Pagani, M,
Lombardi F,
Guzzetti S,
Rimoldi O,
Furlan R,
Pizzinelli P,
Sandrone G,
Malfatto G,
Dell'Orto S,
Piccaluga E,
Turiel M,
Baselli G,
Cerutti S,
and
Malliani A.
Power spectral analysis of heart rate and arterial pressure variabilities as a marker of sympatho-vagal interaction in man and conscious dog.
Circ Res
59:
178-193,
1986
19.
Pagani, M,
Somers VK,
Furlan R,
Dell'Orto S,
Conway J,
Baselli G,
Cerutti S,
Sleight P,
and
Malliani A.
Changes in autonomic regulation induced by physical training in mild hypertension.
Hypertension
12:
600-610,
1988
20.
Preiss, G,
and
Polosa C.
Patterns of sympathetic neuron activity associated with Mayer waves.
Am J Physiol
226:
724-730,
1974.
21.
Rimoldi, O,
Pierini S,
Ferrari A,
Cerutti S,
Pagani M,
and
Malliani A.
Analysis of short-term oscillations of R-R and arterial pressure in conscious dogs.
Am J Physiol Heart Circ Physiol
258:
H967-H976,
1990
22.
Robbe, HWJ,
Mulder LJM,
Ruddel H,
Langewitz WA,
Veldman JB,
and
Mulder G.
Assessment of baroreceptor reflex sensitivity by means of spectral analysis.
Hypertension
10:
538-543,
1987
23.
Smyth, HS,
Sleight P,
and
Pickering GW.
Reflex regulation of the arterial pressure during sleep in man. A quantitative method of assessing baroreflex sensitivity.
Circ Res
24:
109-121,
1969
24.
Soderstrom, T.
On the convergence properties of the generalized least squares identification method.
Automatica
13:
59-75,
1974.
25.
Stevens, PM,
and
Lamb LE.
Effects of lower body negative pressure on the cardiovascular system.
Am J Cardiol
16:
506-515,
1965[Web of Science][Medline].
26.
Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology.
Standard of measurement, physiological interpretation and clinical use.
Circulation
93:
1043-1065,
1996
27.
Taylor, JA,
and
Eckberg DL.
Fundamental relations between short-term RR interval and arterial pressure oscillations in humans.
Circulation
93:
1527-1532,
1996
28.
Vatner, SF,
and
Zimpfer M.
Bainbridge reflex in conscious, unrestrained, and tranquilized baboons.
Am J Physiol Heart Circ Physiol
240:
H164-H167,
1981.
29.
Zetterberg, LH.
Estimation of parameters for a linear difference equation with application to ECG analysis.
Math Biosci
5:
227-275,
1969.
This article has been cited by other articles:
|
|
 |
|
  C. Julien, M. J. Parkes, S. Y. C. Tzeng, P. Y. W. Sin, P. N. Ainslie, P. van de Borne, J.-O. Fortrat, M.-A. Custaud, C. Gharib, A. Porta, et al. Comments on Point:Counterpoint: Respiratory sinus arrhythmia is due to a central mechanism vs. respiratory sinus arrhythmia is due to the baroreflex mechanism J Appl Physiol, May 1, 2009; 106(5): 1745 - 1749. [Full Text] [PDF]
|
|
|
|
 |
|
 M Di Rienzo, G Parati, A Radaelli, and P Castiglioni Baroreflex contribution to blood pressure and heart rate oscillations: time scales, time-variant characteristics and nonlinearities Phil Trans R Soc A, April 13, 2009; 367(1892): 1301 - 1318. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Porta, F. Aletti, F. Vallais, and G. Baselli Multimodal signal processing for the analysis of cardiovascular variability Phil Trans R Soc A, January 28, 2009; 367(1887): 391 - 409. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 F. Cottin, C. Medigue, and Y. Papelier Effect of heavy exercise on spectral baroreflex sensitivity, heart rate, and blood pressure variability in well-trained humans Am J Physiol Heart Circ Physiol, September 1, 2008; 295(3): H1150 - H1155. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
X. Chen and R. Mukkamala Selective quantification of the cardiac sympathetic and parasympathetic nervous systems by multisignal analysis of cardiorespiratory variability Am J Physiol Heart Circ Physiol, January 1, 2008; 294(1): H362 - H371. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 J.-P. Fauvel, C. Cerutti, I. Mpio, and M. Ducher Aging Process on Spectrally Determined Spontaneous Baroreflex Sensitivity: A 5-Year Prospective Study Hypertension, September 1, 2007; 50(3): 543 - 546. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Pagani To Dip or Not to Dip: Of Arterial Pressure Physiology, System's Analysis, and Information Hypertension, May 1, 2007; 49(5): 979 - 980. [Full Text] [PDF]
|
|
|
|
|
|
G. Nollo, L. Faes, A. Porta, R. Antolini, and F. Ravelli Exploring directionality in spontaneous heart period and systolic pressure variability interactions in humans: implications in the evaluation of baroreflex gain Am J Physiol Heart Circ Physiol, April 1, 2005; 288(4): H1777 - H1785. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 D. Laude, J.-L. Elghozi, A. Girard, E. Bellard, M. Bouhaddi, P. Castiglioni, C. Cerutti, A. Cividjian, M. Di Rienzo, J.-O. Fortrat, et al. Comparison of various techniques used to estimate spontaneous baroreflex sensitivity (the EuroBaVar study) Am J Physiol Regulatory Integrative Comp Physiol, January 1, 2004; 286(1): R226 - R231. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
P. A. Lanfranchi and V. K Somers Arterial baroreflex function and cardiovascular variability: interactions and implications Am J Physiol Regulatory Integrative Comp Physiol, October 1, 2002; 283(4): R815 - R826. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 R. Furlan, G. Jacob, L. Palazzolo, A. Rimoldi, A. Diedrich, P. A. Harris, A. Porta, A. Malliani, R. Mosqueda-Garcia, and D. Robertson Sequential Modulation of Cardiac Autonomic Control Induced by Cardiopulmonary and Arterial Baroreflex Mechanisms Circulation, December 11, 2001; 104(24): 2932 - 2937. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
), XAR
(
), and XXAR (
) models calculated on
the R-R and SAP series of Fig. 
