INTRODUCTION

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THE ASSESSMENT of short-term cardiovascular control by means of nonlinear analysis has become a subject of interest. The complex mechanisms involved in cardiovascular regulation very likely interact with each other in a nonlinear way, and therefore nonlinear techniques would be better suited for the study of this regulation than traditional spectral methods. However, the question as to whether cardiovascular signals are appropriate for the applications of such modern techniques is still under discussion. In fact, fractal analysis techniques (especially those based on the power spectral density of the signal) have been proposed as a valid alternative for the study of the complex temporal evolution of such kind of signals (30). Recently, although some authors (18, 20, 23) have claimed that R-R interval data are inadequate for nonlinear analysis, others (2) have demonstrated, by using advanced statistical tests, the existence of nonlinear components in the same signal. These contradictory results could be explained, bearing in mind that cardiovascular time series (especially R-R interval data) comprise an almost periodic component, which is slightly modulated for the action of autonomic nervous system and respiratory sinus arrhythmia, among others. Although it would be impossible to regard the time evolution of the periodic component alone as nonlinear, the resulting variability signal could certainly possess some nonlinear structure due to the interaction of these modulating variables. In fact, the indexes derived from nonlinear time series analysis have shown to be sensitive markers of changes in the autonomic function in both healthy and pathological conditions (19, 28). Furthermore, advanced nonlinear methods have been used to quantify the degree of coupling between respiration and blood pressure variability, improving the results obtained from the coherence analysis (13).

The influence on nonlinear indexes of some of the variables regulating R-R interval variability (RRV) and systolic blood pressure (SP) variability (SPV) has been assessed either by locking these variables to a fixed value (8, 16) or by altering them with proper pharmacological blockers (5, 10, 14-15). However, what is not yet clear is the degree of involvement of the different cardiovascular control systems in the appearance of the nonlinear components of SPV and RRV. Therefore, the aim of this study is to clarify this question using the joint analysis of these signals in conscious rats in both baseline and after the administration of drugs affecting cardiovascular functions. The dependence of these nonlinear characteristics on the high-frequency respiratory component of the spectra is also investigated. Additionally, the usefulness of this analysis in assessing the degree of coupling between these two cardiovascular variables is studied.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Animals, drugs, and recording methods. The data used in this study are the same as reported in earlier studies (3-4, 9), in which only the results from spectral analysis were presented. Male Sprague-Dawley rats, weighing between 450 and 550 g, were housed in controlled conditions (temperature 22 ± 2°C, humidity between 60 and 70%, and 12:12-h light-dark cycle) and received food and water ad libitum. All procedures conformed to Directive 86/609 of the European Community and were approved by the Ethical Committee of La Laguna University. With the rats under light ether anesthesia, catheters filled with heparinized saline (250 IU/ml) were inserted into the lower abdominal aorta for continuous blood pressure (BP) recording and into the jugular vein for drug injection. Two metallic coil electrodes were positioned subcutaneously in a lead II configuration to continuously record electrocardiography (ECG) waveforms. Both catheters and electrodes were tunnelled subcutaneously to exit at the neck. After surgery, we administered 5 mg/kg of gentamicin intramuscularly to prevent postsurgical infection.

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Signal preprocessing. ECG and BP signals from the polygraph were fed into an analog-to-digital converter card inserted in a personal computer (PC) for on-line processing. The beat-to-beat data of each recording were obtained by means of a PC assembler program developed in our laboratory that simultaneously sampled the ECG and the BP signals at a 1 kHz rate. The series of consecutive R-R intervals and the series of SP values within each R-R interval were computed and stored on disc for further analysis. From each recording, the best segment of 2,048 consecutive R-R intervals (and its corresponding values of SP) was selected after automatic checking for stationarity and lack of artifacts in the following way: a 2,048-points window was moved along the data recording; each 2,048-data-point segment was divided into eight sequences of 256 points; and the mean of each sequence was calculated. The segment with the lowest coefficient of variation in its mean was selected. Segments greater than 2,048 points normally exhibited nonstationarities difficult to resolve. Subsequently, the epochs selected were linearly detrended through a least square fit and then transformed to reach zero mean and unit variance.

Nonlinear techniques. To calculate the correlation integral (CI) of the signal, we used Takens' theorem (24). Although initially derived for evenly sampled continuous time series, recent theoretical results (11) have demonstrated that it could be extended to deal with series of time intervals between relevant events. The procedure involves the construction of time delay vectors whose components are the consecutive values of RRV or the SPV. A delay time ensuring lack of temporal correlation between successive components was obtained from the mutual information functions of the RRV and SPV (1) as the greater of the first minima of both signals (5 R-R intervals in our data). We used an improved version of the false nearest neighbor method (12) to estimate the minimal embedding dimension necessary for a proper reconstruction of the dynamics of our time series. In most baseline signals checked, the fraction of false nearest neighbors always converged to zero for embedding dimensions below 10. So, calculations were performed for values of the embedding dimension (m) greater or equal to this lower bound. The calculation of CI was carried out using the Grassberguer-Procaccia algorithm modified by Theiler (25).

Testing for nonlinearity. The presence of nonlinear components in the series was tested by means of the surrogate data method (26). It works by creating a modified version of the original signal (called the surrogate) that shares all its linear properties but is devoid of coherent phase relationship. The thorough analysis of all the baseline signals was achieved by obtaining a set of 39 surrogate versions from each signal. The difference between the original Takens' best estimator of dimension (Q) and the average Q from the ensemble of surrogates was quantified as a function of the SD of the ensemble (21). As a further check, the nonlinear prediction error for both the original and the surrogate data was calculated as a function of the prediction time by using a simple nonlinear prediction algorithm (17). In the rest of the signals, nonlinearity was tested through the paired comparison of the original Q with that from one of its surrogates for all the segments corresponding to the same pharmacological treatment. When the differences were significant, it was possible to define an index of nonlinearity (NLI), which was computed as the ratio of the difference between the original and the surrogate Q to the original Q and expressed as a percentage of the difference. The more negative the NLI, the greater the difference and the more important the effect on the correlation integral of the phase randomization. Thus the NLI can be regarded as an index of the relative importance of the nonlinear structure in the original signal. In all the cases, surrogates were obtained by means of the amplitude-adjusted Fourier transform method, which precludes the possibility of obtaining spurious identification of nonrandom structure due to the long-term linear correlation of the data (22).

Complexity and predictability measures. The complexity of the signals was estimated from the CI by calculating their correlation dimension (D₂), which was obtained as the mean slope of the linear region of the plots log CI versus the logarithm of the intervector distance r in the three highest embedding dimensions (here, from m = 10 to m = 12). In addition, a measure of the predictability of the time series (PM) was obtained as the ratio of the CI in two consecutive embedding dimension (m = 10 and m = 11) for r = 1 SD (13). It must be stressed that the assessment of nonlinear parameters from time series using time-delay embedding presents some restrictions. In particular, there is a fundamental limitation on the highest value of D₂ that can be reliably estimated from a time series, which depends on the data length (6-7). This upper boundary, according to (6-7), is D₂  2log₁₀ N, where N is the number of data points. In our series, with N = 2,048, the result is D₂  6.67. As we show in the RESULTS (Tables 1 and 2), the values of D₂ for RRV and SPV fall below this limit, indicating that such a segment size is large enough for a proper estimation of the nonlinear parameters of the time series.

Stationarity and regularity measures. Recurrence quantification analysis (RQA) allowed the quantitative evaluation of other important features of the signals, such as their recurrence percentage (RC%), determinism percentage (DT%), and the length of the longest diagonal line in the plot (L_max) (29). Both RC% and DT% are a measure of the stationarity and regularity of the signal over time (31), whereas L_max is inversely related to the Lyapunov exponent, which is a measure of the sensibility of the system to the initial conditions (5, 31). The recurrence plot was constructed using the Euclidean norm for intervector distances lower than .

Estimating RRV-SPV coupling. The coupling between RRV and SPV was assessed using two nonlinear parameters: the independence of complexity (I_CM) and the independence of predictability (I_PM), both obtained from the comparison of the individual CI with those obtained from the joint embedding of both time series (13). For both parameters, a result equal to 0 indicates completely coupled systems, and a result equal to 1 indicates completely independent systems; values within these limits indicate partially coupled systems. The three RQA indexes, as well as the PM, Q, mean prediction error, I_CM, and I_PM, were calculated for m = 10.

Statistical comparisons. In all the cases, differences between parameters were tested by a t-test for dependent samples or a Wilcoxon matched paired test when appropriate and were considered significant when P < 0.05.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Analysis of basal signals. The results from the surrogate data test for baseline RRV and SPV are presented in Fig. 1. In both signals, the requirement of nonlinearity was fulfilled for embedding dimensions lower than 10. Below this limit, the average Q for the ensemble of surrogates was significantly greater than that of the original signal. This result was confirmed by the paired comparison of Q from each original signal with that from one of its surrogates (Table 1); this parameter being significantly greater for the latter. Moreover, this effect was also clear when comparing the mean nonlinear prediction error of the ensemble of original signals with those of their surrogates (Fig. 2). Although the error increased with increased prediction times for both the original and the surrogate data, it was always significantly greater (P < 0.05) for the latter.

View larger version (18K): [in this window] [in a new window]   Fig. 1.   Differences between the original Takens' best estimator of dimension (Q) and the average Q for the ensemble of surrogates as a function of the SD of the ensemble (sigmas) for the R-R interval (RRV) (A) and systolic blood pressure variability (SPV) (B) in baseline (n = 53). Error bars represent 95% confidence limit for the mean value of sigmas, whereas the dashed line indicates the limit for statistical significance.

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Mean normalized prediction error as a function of the prediction time for the baseline (n = 53) RRV (A) and SPV (B). Squares represent original signals (solid line), and triangles represent the surrogates.

Effect of drugs. The results concerning the effects of the different drugs on the nonlinear structure of RRV and SPV showed that only atropine decreased the NLI in RRV (5.7 ± 1.9 vs. 2.9 ± 1.5%; P < 0.01). The NLI of SPV was increased by prazosin (3.4 ± 0.7 vs. 13.9 ± 3.2%, P < 0.001) and L-NMMA (3.6 ± 0.6 vs. 8.9 ± 1.4%, P < 0.01). As for the other indexes (Tables 2 and 3), whereas atropine decreased both D₂ and PM (both signals) and increased RC% and DT% (RRV), propanolol decreased PM (in SPV) and only decreased the indexes derived from the RQA in RRV. Prazosin decreased D₂ and PM (both signals) and increased the RC% of the SPV. Finally, atropine was able to change the indexes of nonlinear coupling, increasing both I_PM (0.21 ± 0.04 vs. 0.38 ± 0.04, P < 0.05) and I_CM (0.23 ± 0.03 vs. 0.42 ± 0.03, P < 0.01), whereas propranolol decreased only I_PM (0.28 ± 0.05 vs. 0.12 ± 0.03; P < 0.01). As for L-NMMA and phenylephrine, they modified neither the differences between original and surrogate data nor the NLI. L-NMMA induced a decrease of D₂ and PM in the SPV, whereas phenylephrine decreased the PM but also increased the DT% and RC% of this signal.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

In this study, we have carried out an analysis of the nonlinear properties of RRV and SPV in baseline conditions as well as after different pharmacological treatments in conscious rats. The effect of cancelling the phase relationship in baseline RRV and SPV on both the CI and the mean prediction error has proven the existence of nonlinear structure in these signals. However, most of their characteristics changed very little after the elimination of these phase relationships, indicating a lesser importance of the nonlinear structure of the signals compared with the linear long-term correlation. In particular, the overall profile of the mean prediction error, which increases with the prediction time for the original signals, remains unchanged for the surrogate data and therefore cannot be ascribed to the presence of nonlinearity in the RRV and SPV. Indeed, such a chaoslike profile of the prediction error is also present in fractal signals (27). The results obtained after removing the respiratory component by LPF in baseline signals suggest that these nonlinear features were partly due to this component of the spectra for the RRV but more conclusive for the SPV. A clearer evidence of nonlinearity in the heart rate variability with increased respiratory component has been reported elsewhere (8, 16), although not quantitatively assessed. The filtering operation also produced a decrease of D₂ and PM along with an increase of DT%, RC%, and L_max. In fact, this seems a predictable result because of the reduction of the effective degrees of freedom of the signals due to the filtering, which makes the signals more regular and predictable.

Regarding the effects of pharmacological treatments, only atropine decreased the nonlinearity of the RRV, suggesting that RRV nonlinear structure is mainly mediated by the parasympathetic system. In addition, atropine produces an overall depletion of the RRV spectrum, which is more pronounced for the respiratory component (9). This supports the influence of the respiratory component on the RRV nonlinearity already mentioned. On the other hand, whereas both L-NMMA and phenylephrine have been reported to increase the respiratory component of RRV (3-4), they did not produce alterations in the NLI of RRV. Therefore, the nonlinearity of RRV seems to be affected by other factors besides the parasympathetic activity. Atropine induced a decrease in the RRV irregularity (lowering D₂ and PM and enhancing RC% and DT%). This effect seems to be a consequence of NLI decrease and could also be attributed to the dependence of CI on long-term linear correlations. Indeed, this dependence has been described in the heart rate variability signal as a clear correlation between D₂ and the mean heart rate (15). This effect, which seems to work for atropine, does not work for propranolol, atenolol, or L-NMMA, which produce an increase in the R-R intervals without altering D₂, thus questioning the R-R interval-D₂ correlation. The decrease of D₂ and the increase of the RQA parameters after atropine (in RRV) agree with previous results (5, 14).

Prazosin increased SPV nonlinearity; this effect could be considered a consequence of the increase reported in the respiratory component of SPV after this blocker (9). However, L-NMMA, which is not known to produce an increase of this component (4), also increased the NLI of SPV. This result and the one found after low-pass filtering support the idea that SPV nonlinearity comes from nonneural pathways (i.e., effects of thoracic respiratory movements on BP) and that ₁-sympathetic activity and nitric oxide production lowered the nonlinearity of SPV. In addition, the lower complexity and predictability of SPV after prazosin, L-NMMA, and phenylephrine can be interpreted due to its buffering effects on its respiratory component.

-Blockade with propranolol or atenolol altered neither the NLI nor the CI measures of RRV, in agreement with previous results in rabbits (32) and humans (10); in this respect, we have reported that propranolol, contrariwise to atropine, does not modify the respiratory component of RRV spectrum (9). In addition, propranolol decreased the regularity of RRV (decreasing DT% and RC%), an effect that was opposite to that produced by atropine. The different behavior of these RQA indexes after unspecific -sympathetic and parasympathetic blockades highlights the advantages of these measures compared with those derived from spectral analysis to discriminate the origin of the autonomic alterations. In fact, the source of the decreased slow frequencies (that takes place under atropine and propranolol, see Ref. 9) could be discriminated from the opposite effects on RQA indexes of both the blockades. In addition, the ₁-sympathetic blockade (prazosin) also produces an overall depletion of the spectral components of RRV (9) without changing the RQA indexes. Therefore, these measures could serve as a tool that, together with RRV spectral analysis, allows one to elucidate the origin of an autonomic dysfunction. The results of RQA after -blockade did not agree with previous results (5) in which an increment of RC% of the RRV after atenolol is reported; it was claimed that this effect was due to the selective blockade of ₁-receptors. On the contrary, present data showed not only a lack of effect of atenolol on RRV but also a decrease in all the RQA parameters after unspecific -blockade. Furthermore, the trend of the changes produced in RQA by atenolol is the same as that observed for propranolol but not so marked as to yield statistical significance.

Another interesting result concerns the nonlinear coupling between RRV and SPV as assessed by the I_CM and I_PM indexes. The usefulness of these indexes has been tested before (13), but this is the first study dealing with the effects of drugs on the RRV-SPV coupling. The values of I_CM and I_PM, which were always close to zero, indicated a high degree of nonlinear coordination influenced by the respiratory component, because that coupling was significantly reduced after the low-pass filtering of both signals. This was further supported by the parasympathetic blockade, which always modified the respiratory component of the RRV (9), and also produced a decoupling. The decrease in the RRV-SPV nonlinear coupling after atropine and its increase after propranolol stress the relevance of the parasympathetic system in the heart rate baroreflex operation. In addition, the D₂ and PM decrease in the SPV after atropine could be considered a consequence of the reduction of baroreflex sensibility due to RRV-SPV decoupling. Following the same line of argument, the increase of SPV predictability after propranolol could be attributed to a better performance of the heart rate baroreflex under -sympathetic blockade, as suggested by the improvement of the RRV-SPV coupling under this blockade. Previous results (9) on coherence spectral analysis showed that atropine, prazosin, and propanolol decreased the RRV-SPV coherence in the low-frequency band and atropine decreased the RRV-SPV coherence in the high-frequency respiratory band. These results agree with those of the nonlinear indexes only for atropine, whereas the results disagree with the increased coupling after propanolol and the lack of effect of prazosin on I_PM and I_CM. We should bear in mind that the coherence function and the indexes of nonlinear coupling are measuring different things, because coherence is unable to detect any nonlinearity in the interdependence between two signals.

Finally, the effects of atropine and propranolol on SPV were only reflected in the indexes derived from the CI, whereas those from the RQA were only affected by prazosin, as previously reported (5). All the above results stress the importance of the joint use of both the RQA and CI to analyze the complex behavior of cardiovascular controls, because the information they provide is not redundant. Indeed, changes in the CI are not always detected by D₂ but rather from the PM, indicating that both indexes should be simultaneously calculated to correctly assess variations in the CI. The same can be asserted about the three RQA indexes. The results concerning the CI indexes were not so conclusive as those derived from RQA.

In short, we have enlightened some aspects related to the nonlinear components of RRV and SPV. The present study supports the idea that nonlinear measures represent a robust approach to the cardiovascular research that complements the traditional spectral and time-domain analysis and are able to provide further insight about the underlying physiopathological processes associated with normal or altered RRV and SPV patterns. Nevertheless, it is clear that the reliability of both the CI and RQA in assessing relevant characteristics of these signals, as well as their mutual interaction, depends on our ability to find out what properties of the RRV and SPV are stressed by the indexes. Because it is very difficult to define a range for the baseline values of nonlinear indexes (due to the evident interanimal variation), these tools can be useful only when differences between original and modified signals (either mathematically or after pharmacological treatments) are previously known. Therefore, it is possible to understand what physiological changes are responsible for the modification of either the indexes or the nonlinear characteristics.