INTRODUCTION

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IN HUMANS AND IN ANIMALS, clonidine, a centrally acting antihypertensive agent (19), decreases both blood pressure level and blood pressure variation assessed by spectral analysis (2, 5, 8, 18). In contrast, the effects of clonidine on the blood pressure variability, assessed by the standard deviation (SD), are inconsistent. Although clonidine reduced the SD of blood pressure in normotensive rats and hypertensive patients (5, 8), it did not change SD of blood pressure in hypertensive rats or normotensive volunteers (2, 9, 18). Moreover, chronic treatment with clonidine failed to decrease blood pressure level and its SD in conscious spontaneously hypertensive rats (SHRs) (2, 4). The fact that clonidine had no chronic effect on blood pressure level and on blood pressure variability was surprising. We suggest that the use of linear indexes, such as mean value, SD, and spectral method was insufficient to analyze the effect of clonidine on blood pressure dynamics.

In recent years, it has been argued that the mechanisms regulating blood pressure are most probably nonlinear (1, 20, 21). Therefore, nonlinear methods should also be used to analyze blood pressure time series. Several authors have used nonlinear techniques to analyze heart rate and blood pressure in healthy conditions and in various pathologies and have obtained useful results (see, for example, Refs. 1, 14, 15, 20, 26-28).

One marker of nonlinear dynamics that we consider as the most important is the Lyapunov exponent. The meaning of this index is simple to explain. Some dynamics have the property of "sensitivity to initial conditions." Starting from two similar values, the systems may generate two sequences that quickly (exponentially) diverge one from the other. The exponent of this divergence is called the Lyapunov exponent. In this work, we used a simple and intuitive method to estimate this exponent. Our aim was to explore whether, in chronic dosing, clonidine has modified the Lyapunov exponent of the blood pressure dynamics.

	METHODS

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Male SHRs (n = 24), aged 13 wk, and age-matched normotensive Wistar-Kyoto (WKY) rats (n = 16) (Charles River France, St. Aubin-les-Elbeuf, France) were used according to the Guide for the Care and Use of Laboratory Animals published by the US National Institute of Health (NIH Publication No. 85-23, revised 1985). The SHRs were randomized into four groups of six rats each. Groups 1 and 2 (SHRs) were treated with a vehicle (0.9% NaCl) and 0.1 mg · kg¹ · day¹ sc clonidine, respectively, for 24 h. Groups 3 and 4 (SHRs) were infused with the vehicle and with 0.1 mg · kg¹ · day¹ sc clonidine, respectively, for 4 wk. Groups 5 and 6 (WKY rats, n = 8 in each group) were infused with the vehicle for 24 h and 4 wk, respectively. Vehicle and clonidine infusions were performed via osmotic minipumps implanted subcutaneously, while animals were under ether anesthesia, in the ventral region of the abdomen (Alzet models 2ML1 and 2ML4, total volume 2 ml each, nominal rate 10 and 2.5 µl/h, respectively; Charles River France, St. Aubin-les-Elbeuf, France).

Blood Pressure Recording

Acute treatments. At 24 h before the beginning of infusion of vehicle or clonidine, the arterial catheter was implanted as described above, and baseline values of arterial pressure were recorded in conscious, freely moving rats 24 h after surgery. After blood pressure was recorded, the minipumps (Alzet model 2ML1) were implanted while animals were under anesthesia. Another recording of blood pressure was performed in the same rats 24 h later. Because of the life span of intra-arterial catheters, baseline values were recorded only in the sample of rats used in acute treatments.

Chronic treatments. In these experiments, the minipumps (Alzet model 2ML4) were implanted at day 0, and the rats were housed in individual cages for 28 days. At day 27, they were prepared for recording of blood pressure as described above. Blood pressure was recorded 24 h after implantation of intra-arterial catheters.

Beat-to-Beat Blood Pressure

Lyapunov Exponent: The l_max Index of the Recurrence Plot

To illustrate the method, Fig. 1A shows a series of 20 systolic blood pressure values, x_i (in mmHg), i = 1, 2,..., 20. The recurrence plot looks for repeated sequences in the data. We consider that two blood pressure values are the same if their difference is less than a small number (r), say 2 mmHg. Starting from x₁, we are interested to see whether the same value (±2 mmHg) occurs later on in the series (i.e., for some j with j > 1). In Fig. 1A, the same values are found at j = 4, 7, 10, 14, and 18. To mark these recurrences, we plot on a 20 × 20 square the points (1, 4), (1, 7), (1, 10), (1, 14), and (1, 18) (Fig. 1B). Then we start from x₂ and plot the recurrent points (2, 8), (2, 11), and (2, 16). Figure 1B shows the recurrent points of the total series. Of particular interest are the diagonals in the figure. One example is the line (1, 7)-(2, 8)-(3, 9). This line means that when a recurrence is found, the two sequences starting from these recurrent points remained close together for several subsequent beats: the trajectory x₁-x₂-x₃ was parallel to the trajectory x₇-x₈-x₉. We recall that the Lyapunov exponent of a dynamic is a measure of the divergence of two trajectories that start from two points close to each other. Therefore, the Lyapunov exponent is inversely related to the l_max index (3). A high Lyapunov exponent, i.e., a short l_max, expresses a chaotic dynamic.

View larger version (19K): [in this window] [in a new window]   Fig. 1.   Method of recurrence plot. A: example of blood pressure time series, ×1, ×2, ...., ×20. B: recurrence plot: a 20 × 20 square is drawn. If blood pressure xi differs from xj of less than a given value (here, 2 mmHg), then we plot the point (i,j). Diagonal lines, for example (1,7)-(2,8)-(3,9), represent parallel trajectories. Lengths of diagonal lines account for dependence of dynamics on initial values. C: histogram of lengths of diagonal lines. Here, longest length is lmax = 4. L(k) and N(k), length of and no. of occurrences of diagonal lines, respectively (k = 1-4).

To analyze blood pressure data, we embedded the one-dimensional blood pressure data in a p-dimensional Euclidean space by using the time-delay reconstruction of Takens (16; see also Ref. 22). The delay of reconstrution was one beat. With use of the false neighbors method of Kennel et al. (10), the embedding dimension was taken as P = 10. The recurrence plot method was applied in dimension 10 by using the Euclidean norm. The cutoff value in the recurrence plot method in dimension 1, r₁, was generally taken as r₁ = f · SD where f is a fraction of one unit or less. In dimension p, we used r = r₁ ; this formula gives the diagonal length of a hypercube of side r₁. The SD of blood pressure was ~4-5 mmHg. Here we used f = 1. Because systolic blood pressure was somewhat more dispersed than diastolic blood pressure, we selected finally, r =  mmHg for systolic blood pressure and r =  mmHg for diastolic blood pressure. Each blood pressure series included ~10,000 values. We calculated the l_max for nonoverlapping segments of 1,000 consecutive readings and took the mean values of the l_max obtained in the different segments. As suggested by Eckmann et al. (3), we also plotted the histograms of the density of the recurrences in the diagonal lines to see if some drift might exist in these segments.

Statistical Analysis

	RESULTS

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The effects of clonidine were similar on systolic and diastolic blood pressures. In the following, the term blood pressure means either of the two pressures.

Effects on Blood Pressure Level and Blood Pressure Variability

After 24 h of treatment, mean value and SD of blood pressure in the clonidine-treated SHRs were significantly decreased compared with the vehicle-treated SHRs (P < 0.05 and P < 0.01, respectively). However, the level of blood pressure in the clonidine-treated SHRs was still significantly higher (P < 0.01) than in the vehicle-treated WKY rats, whereas the SD of blood pressure in the clonidine-treated SHRs was reduced to the level observed in the vehicle-treated WKY rats (Fig. 2).

View larger version (30K): [in this window] [in a new window]   Fig. 2.   Systolic blood pressure (SBP; A), diastolic blood pressure (DBP; B) (mean ± SD taken over groups of rats), and their SD (C and D, respectively) taken over the time series: at baseline, after 24-h infusion (acute), and after 28 days of infusion (chronic) with vehicle or clonidine. SHR, spontaneously hypertensive rats; WKY, Wistar-Kyoto rats; NS, not significant. Significantly different, SHRs treated by clonidine vs. SHRs treated by vehicle, * P < 0.05, ** P < 0.01. Significantly different, SHRs treated by clonidine vs. WKY rats treated by vehicle, £ P < 0.05, ££ P < 0.01.

After 4 wk of infusion, blood pressure in the clonidine-treated SHRs unexpectedly rose and recovered to the level observed in the vehicle-treated SHRs. Similarly, the SD of blood pressure were the same in the clonidine-treated SHRs and in the vehicle-treated SHRs. Thus, in contrast to its acute effects, clonidine in chronic treatments did not change the level and the SD of blood pressure (Fig. 2).

Effects on the l_max Index of the Blood Pressure Dynamics

After 24 h of treatment, the l_max of blood pressure was significantly increased (P < 0.01) in the clonidine-treated SHRs compared with the vehicle-treated SHRs. The index in the clonidine-treated SHRs had joined the level observed in the vehicle-treated WKY rats (Fig. 3).

View larger version (15K): [in this window] [in a new window]   Fig. 3.   lmax index of SBP [lmax(SBP); A] and of DBP [lmax(DBP); B] at baseline, after 24-h infusion (acute), and after 28 days of infusion (chronic) in SHRs and WKY rats treated with vehicle or clonidine. Symbols same as in Fig. 2.

After 4 wk, the l_max index of blood pressure in the clonidine-treated SHRs was still significantly increased (P < 0.01) compared with the vehicle-treated SHRs. This index remained at the same level as in the vehicle-treated WKY rats (Fig. 3).

	DISCUSSION

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After 24 h of treatment with clonidine in conscious SHRs, blood pressure level and its SD significantly decreased. These effects are consistent with previously published results (5, 8, 9, 18) and may be interpreted as the consequence of the centrally mediated inhibition of sympathetic tone induced by the drug (19).

In contrast to acute treatments, after 4 wk of infusion, clonidine no longer changed blood pressure level and its SD. The lack of effectiveness of clonidine on blood pressure level may be ascribed to the development of tolerance rather than to a malfunctioning of the micropumps. Indeed, after 4 wk of infusion, flesinoxan, another centrally acting antihypertensive agent (13, 24), significantly reduced blood pressure level in SHRs (2). On the other hand, tolerance also develops to the antihypertensive effects of rilmenidine when it is infused in SHRs (4). Observations of results on the SD of blood pressure were also inconsistent. Whereas clonidine reduced the SD of blood pressure in normotensive rats and hypertensive patients (5, 8), it did not change the SD in hypertensive rats and in normotensive volunteers (2, 9, 18).

We suggest that the absence of chronic effect of, and/or inconsistent results with, clonidine might be caused by the method of analysis. The use of blood pressure level and SD or spectral method might not be instructive enough, and the method was unable to explore the deep impact of clonidine on the blood pressure dynamics. It is clear that if we randomly perturb the initial order of the blood pressure series, then we obtain a new series with the same mean value and the same SD as the initial series. Therefore, mean value and SD cannot account for the temporal structure of the initial series. On the other hand, in the Fourier method, one decomposes the blood pressure signal into a sum of many cosine functions with different amplitudes and phases. However, the spectrum of the series uses only the amplitudes of the cosines and cannot account for a change in the phases of the different cosines. More subtle is the reason why the Fourier method cannot account for a possible sequential nonlinear relationship in the initial series. Theiler et al. (17) have shown that one can random modify a time series without modifying its linear sequential relationship (i.e., its linear autocorrelation coefficients). In the new series, the Fourier spectra are the same as in the initial series. For these reasons, nonlinear methods are needed to explore a possible nonlinear time-dependent relationship in the original data.

In this work, we have used a nonlinear l_max index, obtained by the recurrence plot method, to characterize the dynamic of blood pressure variation before and after treatment with clonidine. The meaning of l_max is simple (see details in the METHODS section). This index is approximately proportional to the inverse of the well-known Lyapunov exponent, which characterizes the dependency of the dynamic on initial conditions.

We have calculated the l_max in dimension 10. This dimension was suggested by the application of the false-neighbors method of Kennel et al. (10) to our observations. This result is in agreement with previous reports analyzing physiological data. Webber and Zbilut (22) used the same dimension to study electromyogram recording. Layne et al. (12), estimating fractal dimension of electroencephalogram data, showed that the fractal dimension was stabilized for an embedding dimension larger than ~10.

Nonlinear methods, in particular the recurrence plot method, have been used in previous reports. It was shown that the new methods could discover some important properties in the time series that were unrevealed by linear methods (14, 22).

It is interesting to notice that, before treatment, the l_max index is shorter in the SHRs than in the WKY rats. This means that the blood pressure dynamic is more sensitive to initial conditions in the hypertensive rats than in the normotensive rats. It has been suggested that physiological dynamics show a large variability in the healthy condition whereas a loss of variability is a sign of some pathology (7). Wagner et al. (20) showed that blood pressure regulation in the intact dog has a positive Lyapunov exponent and a decreased exponent was seen after baroreceptor denervation. Similarly, when analyzing electroencephalogram data of patients suffering from multiple sclerosis, Ganz et al. (6) also observed a decrease of Lyapunov exponent in pathological conditions. We also observed a longer l_max of heart rate variation in patients with diabetes mellitus and dysautonomy compared with control patients without dysautonomy (14). However, in contrast to the above results, Yip et al. (25) observed almost periodic oscillations of tubular pressure in the kidneys of normal rats. The oscillations had more aperiodic character after arterial clipping of one kidney and in SHRs. Results of the present study were in the same direction as those of Yip et al. (25). Therefore, it seems that the question of whether physiological dynamics are more or less "chaotic" in healthy status compared with some disease status cannot be stated in a general manner. The answer depends on the parameter analyzed, the type of pathology, the species, or both.

We have shown that 24 h after dosing, clonidine significantly increased the l_max index of systolic and diastolic blood pressures in the direction of normality, and this effect was maintained until the end of the period of treatment (28 days). One interesting point is the fact that the index, at the 24th h after treatment, and on day 28, reached the level observed in the normotensive rats.

In contrast to l_max, blood pressure level and its SD returned to the pretreatment level 28 days after clonidine infusion. It might be surprising that blood pressure dynamics in the SHRs before and after 4 wk of clonidine may have different regulations, as witnessed by two different l_max coefficients, and yet have the same mean level and the same SD. An explanation may be the fact that many factors might act on mean level, SD, and the sensitivity of blood pressure to initial conditions. We give an example to illustrate this point. Fig. 4 shows two series generated by the recurrence equation x_n+1 = a x_n  b x²_n, with two coefficients of regulation, a and b. In this model, coefficient a acts on the sensitivity to initial conditions and mean values of the time series, while coefficient b acts solely on the mean level. We chose a = 3.95, b = 4.45 in series A and a = 3.86, b = 4.36 in series B. The two series have almost the same mean values (0.51 and 0.52, respectively) and SD (0.28 and 0.27, respectively), but they have quite different l_max indexes: l_max is equal to 18 in series A, and l_max is equal to 37 in series B.

View larger version (60K): [in this window] [in a new window]   Fig. 4.   Left, A and B: 2 time series generated by recurrence equation xn+1 = a xn  b x2n, with a = 3.95, b = 4.45 in series A and a = 3.86, b = 4.36 in series B. The 2 series have almost the same mean value (0.51 and 0.52, respectively) and SD (0.28 and 0.27, respectively). Right, A and B: histograms of lengths of diagonal lines (see Lyapunov Exponent: The lmax Index of the Recurrence Plot). Series A and B have quite different lmax indexes: lmax = 18 in series A; lmax = 37 in series B. L, length of diagonal lines.

In conclusion, the present study demonstrates that clonidine has a significant chronic effect on blood pressure dynamics, but this effect can be evidenced only by nonlinear methods. Our study also suggests that the mechanisms governing blood pressure variations are nonlinear, and nonlinear methods should be used in complement to linear methods (mean values, SD, and spectral power). One important point is the relationship between the l_max index and the known physiological mechanisms that govern blood pressure and heart rate variations. More data are needed to document this issue.