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 rr_i, sap_i, and resp_i represent the difference between the actual sample RR_i, SAP_i, and Resp_i 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)

where h_rr-sap,k is the kth coefficient setting the influences of sap on rr and w_rr is a zero mean white noise with variance _wrr². Therefore, the rr_i value depends on the sap (the exogenous signal) at the same cardiac beat (i.e., sap_i) 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 w_rr. The sap series is modeled as an AR process of order p

(2)

where h_sap-sap,k is the kth coefficient of the AR model and w_sap is a zero mean white noise with variance _wsap².

View larger version (8K): [in this window] [in a new window]   Fig. 1.   The 3 causal parametric models proposed to describe R-R interval dynamics are shown: exogenous (X) model (A), X with autoregressive (AR) input (XAR) model (B), and XAR with exogenous input (XXAR) model (C). The models differ in describing the effects of inputs affecting R-R interval (rr) independently of systolic arterial pressure (sap). These influences are noise (wrr) without any rhythmic pattern (A), rhythmic patterns (urr) (B), and rhythmic patterns driven by respiration (uresp) and independent of respiration (urr) (C). H, transfer function.

R-R-SAP XAR model. To take into account both direct effects of sap and possible influences (indicated as u_rr), including rhythmic influences, affecting rr independently of sap, the model described by Eq. 1 can be modified as

(3)

where sap is described as in Eq. 2 and u_rr is the AR process

(4)

where h_urr-urr,k is the kth coefficient of the AR model. Therefore, rr_i depends not only on current and past sap values but also on a source (u_rr) describing all the influences unrelated to sap but capable of producing rhythmic changes in rr (Fig. 1B). In this way, oscillations generated by mechanisms involving nonbaroreflex circuits (e.g., cardiogenic reflexes mediated by myocardial ischemia) are explicitly addressed.

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)

where h_rr-resp,k is the kth coefficient setting the influences of resp on rr, and u_rr is described by Eq. 4. The respiratory signal is modeled as an AR process

(6)

where w_resp is zero mean white noise with variance _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 H_rr-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 (w_rr and w_sap for X and XAR models and w_rr, w_sap, and w_resp 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).

(7)

	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 (SAP_i-,RR_i) 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)

where P_rr(LF), P_sap(LF), P_rr(HF), and P_sap(HF) represent the power in LF and HF bands detected on R-R interval and SAP series, respectively. These indexes are reliable if the coherence function (K²) sampled at LF and HF is >0.5 (11), thus meaning that the rr and sap signals are significantly correlated at that specific frequency and, therefore, the power calculated in LF and HF bands is not only the effect of independent noises impinging on both signals.

Transfer function analysis. The R-R-SAP transfer function modulus can be calculated from the zero mean sap and rr series as

(9)

where C_rr-sap is the cross spectrum between rr and sap series and P_sap is the spectrum of sap series (see Cross-Spectrum Estimate in the APPENDIX). The transfer function modulus can be sampled at LF and HF detected on sap series (22), thus obtaining two indexes of baroreflex gain that are referred as to _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).

1

1

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.

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).

	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.

View larger version (37K): [in this window] [in a new window]   Fig. 2.   Beat-to-beat series of R-R interval (A) and SAP (B) in a conscious dog at control. The relevant AR power spectral densities exhibit a dominant high-frequency (HF) component at 0.23 Hz (C and D, respectively).

View larger version (9K): [in this window] [in a new window]   Fig. 3.   Squared coherence function (K2) between the R-R and SAP series of Fig. 2. K2 is >0.5 at HF and <0.5 in the range from 0 to 0.1 Hz.

View larger version (19K): [in this window] [in a new window]   Fig. 4.   Response of the Hrr-sap block to the ramplike increase of sap for the X (), XAR (), and XXAR () models calculated on the R-R and SAP series of Fig. 2. The slope of the linear fitting (solid lines) represents the estimate of the baroreflex gain (X, XAR, and XXAR, respectively).

View larger version (17K): [in this window] [in a new window]   Fig. 5.   Straight segments representing in the plane (SAPi-1,RRi) the linear regression on the baroreflex sequences (both positive and negative) detected in the R-R and SAP series of Fig. 2. The mean slope is BS.

View larger version (15K): [in this window] [in a new window]   Fig. 6.   Spectral components at HF (filled area) relevant to the R-R (A) and SAP (B) power spectra calculated on the R-R and SAP series depicted in Fig. 2. The area (i.e., the power) of these components is used to estimate PS(HF). No component is detectable in the low-frequency (LF) band in the R-R power spectrum (A).

View larger version (12K): [in this window] [in a new window]   Fig. 7.   Magnitude of the Hrr-sap transfer function between R-R and SAP series of Fig. 2. This function, sampled at HF, provides CS(HF). It is not sampled in the LF band because the coherence function K2, depicted in Fig. 3, is <0.5 at LF.

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, w_sap was always white, whereas w_rr 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 w_rr and w_sap 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).

View larger version (23K): [in this window] [in a new window]   Fig. 8.   Example of R-R interval (A) and SAP (B) series in which no baroreflex sequence is found. Indeed, the magnification of 2 cycles of the dominant respiratory rhythm (C) present on the R-R (solid line) and SAP (dotted line) series demonstrates that 3 contemporaneous increases or decreases of R-R interval and SAP cannot be found.

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.

View this table: [in this window] [in a new window]   Table 1.   Summary of baroreflex gains estimated by traditional noninvasive methods [BS, PS(LF), PS(HF), CS(LF), and CS(HF)] and by causal parametric models (X, XAR, and XXAR) under different experimental conditions

	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) (K_LF² 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).