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Am J Physiol Regul Integr Comp Physiol 275: R1661-R1666, 1998;
0363-6119/98 $5.00
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Vol. 275, Issue 5, R1661-R1666, November 1998

Correlation integral of blood pressure as a marker for exercise intensities

C. D. Wagner1, H. M. Stauss1, P. B. Persson1, and K. C. Kregel2

1 Department of Physiology, Humboldt University of Berlin, D-10117 Berlin, Germany; and 2 Department of Exercise Science, University of Iowa, Iowa City, Iowa 52242

    ABSTRACT
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References

The purpose of this study was to test the hypothesis that the correlation integral technique detects altered regulation of cardiovascular function during graded treadmill exercise. Arterial blood pressure (BP) was measured via telemetry before and during graded treadmill exercise in Sprague-Dawley rats. During treadmill running at mild, moderate, and heavy exercise intensities, the slope of the correlation integrals (SCI) continuously increased from 5.45 ± 0.17 to 7.12 ± 0.18, 7.92 ± 0.23, and 8.40 ± 0.23, respectively. However, corresponding changes in pulse interval, blood pressure, and systolic blood pressure with increasing workload were not consistently observed. Low-frequency, midfrequency, and high-frequency powers of BP were not different between adjacent exercise grades; only the low-frequency component of pulse interval was different between resting state and mild exercise, and BP variance was significantly different between mild and moderate grades. Comparison of the SCI values with those obtained from surrogate data sets suggests that these differences originate mainly from nonlinear components in the cardiovascular control system. These findings support the hypothesis that SCI detects alterations in cardiovascular regulation associated with graded exercise. Furthermore, SCI may be superior to linear techniques in detecting altered regulation with changing exercise intensities.

blood pressure power spectrum; blood pressure variability; dynamic exercise; nonlinear dynamics; rats

    INTRODUCTION
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References

REGULATION OF SYSTOLIC BLOOD PRESSURE (SBP) during exercise is primarily a function of cardiac output and sympathetic nerve activity. Moreover, steady-state responses of heart rate (HR) and arterial blood pressure (BP) are under the influence of both the sympathetic and parasympathetic nervous system. Thus, with the assumption that a steady-state condition is achieved, investigation of BP and HR variability can provide a means to evaluate the balance of neural outflow in a number of physiological settings, including exercise (8, 9).

Conventional techniques used to assess cardiovascular control during exercise involve the measurement of BP, HR, and power spectra (i.e., linear techniques). However, mean values and power spectra of BP and HR provide only incomplete information because the maintenance of cardiovascular homeostasis during exercise will involve spontaneous fluctuations in these hemodynamic parameters, which include nonlinear components. Therefore, the analysis of these fluctuations with the use of nonlinear techniques should provide greater insight into the dynamics of the various control systems that are critical for proper cardiovascular function during exercise.

In the present study, we tested for nonlinear components in the BP signal by applying the correlation integral technique (CI) to the continuously recorded BP signal (3) from rats that had undergone graded treadmill exercise. The CI of a signal is the probability of two randomly chosen data points from the pool of available data having a distance closer than a fixed value epsilon . CI is not just estimated from points of the signal, but from vectors whose components are time-shifted data points. Therefore, the CI is able to provide a quantity which is a sensitive index of the temporal dynamics of the signal and its generating system. Moreover, the CI is independent of translations of the time series and multiplication with a constant value; i.e., the outcome of CI is independent of the mean value and variance of the signal.

The purpose of the present study was to clarify the possible usefulness of the CI in assessing the altered regulation of cardiovascular function associated with gradations in exercise intensity. We were particularly interested in clarifying whether the CI is suitable for the discrimination between mild and moderate exercise intensities and in investigating whether this measure is superior to conventional linear techniques. It was hypothesized that the CI provides a better discriminating statistic of cardiovascular changes during graded exercise than power spectra of BP, pulse interval (PI), and mean values of BP, SBP, or PI.

    METHODS
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Abstract
Introduction
Methods
Results
Discussion
Appendix
References

Surgical Procedure and Experimental Setup

Animals. The experiments were performed in 3-mo-old male Sprague-Dawley rats (n = 6; Harlan, Indianapolis, IN). After implantation of telemetric transmitters, rats were housed individually in clear plastic cages. Temperature (24 ± 2°C), humidity (60 ± 10%), and light periods (12:12-h light-dark cycle; lighting 6:00 AM to 6:00 PM) were controlled. Rats had free access to a standard rat chow and were provided with tap water ad libitum. All experiments were performed in accordance with institutional guidelines for health and care of experimental animals.

Implantation of telemetric transmitters. Implantation of telemetric transmitters (TL11M2-C50-PXT; Data Sciences, St. Paul, MN) was accomplished using a method described by Casati et al. (2). Rats were anesthetized with pentobarbital sodium (Nembutal, 60 mg/kg ip), the abdominal area was shaved and washed with 70% alcohol, and the skin and abdominal wall were opened via a midline incision. All surgeries were performed under sterile conditions. The intestine and abdominal wall were mechanically retracted to allow access to the abdominal aorta and vena cava. The abdominal aorta and vena cava were separated and dissected free of connective tissue at an area 5 mm in length caudal from the branch of the left renal artery. Blood flow in the abdominal aorta was interrupted by a temporary ligature at the area where the aorta and vena cava were previously separated. The tip of the catheter was inserted into the abdominal aorta via a small hole punched into the vessel 3-5 mm cranial to the iliac bifurcation with the use of an injection needle. After sliding the catheter forward up to the temporary ligation, we closed the hole in the aorta with a small amount of polyacrylic tissue glue (Enbucrilat; Braun, Melsungen, Germany). The body of the sensor was sutured to the abdominal wall. The abdominal wound was closed by separate continuous sutures of the muscular and skin layers of the abdominal wall.

Experimental protocol. Two weeks before implantation of telemetric transmitters, rats were familiarized to exercise on a motorized treadmill by running every other day for either 5 min at speeds of 15, 25, and 35 m/min each or for 30 min at a speed of 11 m/min. Five days after surgery, rats were subjected to another familiarization run on the treadmill (5 min at speeds of 15, 25, and 35 m/min each). Experimental protocols were performed on days 7 and 8 after implantation of the transmitters. On the first experimental day, rats were placed onto the treadmill belt without running and resting BP was recorded for a time period of 30 min. Then the treadmill was switched on, and the rats ran on the treadmill for another 30 min at a speed of 11 m/min while measurement of arterial BP was continued. On the second experimental day, arterial BP was recorded during resting conditions and during runs on the treadmill at stepwise increasing speeds of 15, 25, and 35 m/min. The duration of the resting and running periods were 5 min each.

Data recording. A telemetric receiver (RLA1020; Data Sciences, St. Paul, MN) was placed inside the Plexiglas chamber of the treadmill. The digital signals obtained from the receiver were transmitted to a digital-to-analog converter (UA10, Data Sciences) via a consolidation matrix (BCM100, Data Sciences). The analog outputs of the UA10 unit were connected to a computerized recording and analyzing system (MacLab/8s, AD Instruments). The arterial BP signal was recorded on the MacLab system with sampling rates of 400 Hz for each signal.

Data Analysis

Arterial BP was sampled over 300 s (5 min), with 400 Hz for each exercise grade. SBP sequences were identified offline. Before the computation of power spectra, PI series were obtained by calculating the time differences between two successive systolic pressure values. Mean BP was obtained by integrating the BP signal between two systolic pressure values. These data were then transformed into time-equidistant BP and PI time series (step functions) with a sampling frequency of 200 Hz, and the power spectra were extracted from the time-equidistant data. The spectra were calculated as absolute spectra, relative spectra being obtained after dividing by the variance of the signal (i.e., total power). Thereafter, the spectra were divided into three frequency bands: the low-frequency band (LF) was defined as the range from 0.02 to 0.2 Hz, the midfrequency band (MF) was defined as the range from 0.2 to 0.8 Hz, and the high-frequency band (HF) was defined as the range from 1 to 6 Hz. The technique of embedding and the CI are explained in the APPENDIX.

Statistics. For the global evaluation of group differences of the mean values in PI, BP, SBP, slope of the CI (SCI), and the powers of BP and PI in the LF, MF, and HF band, we used the Friedman two-way ANOVA. To assess pair differences in these quantities, we applied the Wilcoxon signed-rank test for paired differences. Rejection regions were defined for the two-sided significance level alpha  = 0.05.

    RESULTS
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Abstract
Introduction
Methods
Results
Discussion
Appendix
References

For each rat, PI, SBP, BP, SCI (Figs. 1 and 2), and spectra of PI (Fig. 3) and BP were computed. Comparison of these quantities under exercise conditions using the Friedman ANOVA test suggests that there are group differences between the workloads.


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  		Fig. 1.   Mean arterial blood pressure (BP), mean systolic blood pressure (SBP), mean pulse interval (PI), and slope of the correlation integral (SCI) during 3 exercise grades compared with resting state. Values are means ± SE. Rest, resting state; Ex 1, 2, and 3, mild, moderate, and heavy exercise grades, respectively.


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  		Fig. 2.   A: trace of an original blood pressure (AP) recording. B: trace of an amplitude-adjusted surrogate of A. C: CIs of a recorded data set and 20 surrogate data sets; 1.2 × 108 mutual distances using 120,000 data points of BP were used. Maximum slopes of curves were evaluated in region between arrows. In all calculations, an embedding dimension of d = 19 was used. Slope of each surrogate data set was different from slope of corresponding recorded data set. In this figure, slope for original data was 4.90 and was 6.30 ± 0.16 for set of surrogates (experiment 1, resting condition; see Table 2).


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  		Fig. 3.   Relative (rel.) PI power spectra from a single animal. A: resting state. B: mild exercise. C: moderate exercise. D: heavy exercise.

PI was different between the resting state (152.8 ± 5.5 ms) and mild exercise (131.3 ± 5.1 ms; Fig. 1) and between moderate and heavy exercise (131.5 ± 5.7 and 125.0 ± 5.7 ms; Table 1). No differences were seen in PI between mild and moderate grades.

                              
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  		Table 1.   BP, SBP, PI, relative LF, MF, and HF of BP and PI, and variances of BP and PI during 3 exercise grades compared with resting state

BP was not significantly different between resting state and the value during mild exercise grade (128 ± 3 vs. 136 ± 2 mmHg), between mild and moderate exercise, or between moderate and heavy exercise (140 ± 2 and 141 ± 2 mmHg, respectively; Table 1). Systolic pressure was elevated from the resting state (151 ± 3 mmHg) to the three exercise levels (161 ± 2, 165 ± 2, and 167 ± 3 mmHg, respectively). In BP and SBP, the differences between successive groups were not significant (Table 1).

Relative BP Spectra

The LF component of BP decreased from resting state to mild exercise and further decreased during moderate and heavy exercise, but the group differences to the preceding groups were not significant. MF power increased to a similar extent at the three exercise grades. Relative HF power increased steadily from rest during the three exercise grades. There were no significant differences between two successive groups in all three experimental groups. Total power (variance) of BP decreased from resting conditions throughout all exercise grades; the only significant difference was observed between the mild and moderate grades (Table 1).

Relative PI Spectra

The relative LF component continuously decreased from rest during the three exercise grades. MF power was also highest during rest, dropped during mild exercise, and remained on this level during moderate and heavy exercise. In contrast to LF, the HF component steadily increased from rest during graded exercise (differences not significant, Table 1). PI variance decreased from resting state to moderate exercise and then increased during heavy exercise, but the differences were not significant.

The SCI continuously rose from 5.45 ± 0.17 to 7.12 ± 0.18 during the first exercise grade and was elevated to 7.92 ± 0.23 and 8.40 ± 0.23 during the moderate and heavy exercise intensities, respectively (Table 2). Compared with PI, BP, SBP, and BP and PI spectra, SCI progressively increased for each animal during the exercise grades.

                              
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  		Table 2.   SCI of BP during 3 exercise grades compared with resting state

    DISCUSSION
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References

The purpose of the present study was to test the hypothesis that the altered regulation of cardiovascular function during graded treadmill exercise is mirrored by changes in the CI. It was of particular interest to determine whether this method is able to distinguish between mild and moderate exercise intensities, which are not readily detected by HR, mean BP, and SBP responses.

Because differences in the SCI of pulsatile BP recordings were observed between mild exercise and resting conditions, moderate and mild exercise levels, and moderate and heavy workloads, we conclude that SCI is indeed able to detect the alterations in cardiovascular regulation associated with graded exercise. Moreover, because the differences of BP and SBP were not significant among the four conditions (i.e., rest and mild, moderate, and heavy intensities of exercise), and PI and LF power of PI only showed differences between rest and mild exercise and moderate and heavy exercise, the findings suggest that SCI may be superior to linear techniques in detecting slight workload differences.

The method of the CI, as well as other algorithms from nonlinear dynamics theory, has rarely been applied to direct pulsatile BP signals. Moreover, to our knowledge, this is the first study to use the CI technique to gain insight into mechanisms of cardiovascular regulation in a physiologically relevant situation, such as dynamic exercise. Application of the CI to BP signals of conscious dogs led to the assumption that BP control in resting dogs reflects some aspects of nonlinear regulation (14). SCI changed after baroreceptor denervation in conscious dogs, a model of cardiovascular control that is characterized by a high sympathetic tone (13). Almog et al. (1) used the CI to detect differences in the cardiovascular control mechanisms of spontaneously hypertensive rats and a Wistar-Kyoto (WKY) control strain. In all cases, application of SCI detected nonlinear components in the time series, originating from nonlinear control mechanisms that create the signals.

The interpretation of results from SCI must be cross- checked with the calculations from surrogate data (SD). Surrogates are "alternative signals" derived from the original time series preserving autocorrelation function and power spectrum. Because mean values and standard deviation of the surrogates are unchanged, linear techniques fail in discriminating between the measured time series and the surrogates. Differences in the SCI statistic may therefore result from nonlinearities in the genuine signal that are destroyed in the SD (12).

The results of resting BP SCI values obtained in this study are in line with the previously presented values from WKY rats [5.48 ± 0.30, (1)]. As was emphasized by these authors, differences in SCI are due to alterations in the nonlinear components of cardiovascular control mechanisms. A possible source of nonlinearity in cardiovascular variables may be changing dynamics in sympathetic and vagal tone to the heart and resistance vessels (1), regional heterogeneity in redistribution of vascular resistance, and hormonal secretion dynamics.

The power spectra of BP and PI contain relevant information about the status of the autonomic nervous system. The LF component accounts for the long-term regulatory mechanism, perhaps related to endocrine factors and thermoregulation. The MF component seems predominantly due to sympathetic discharge, whereas the HF component of BP, at the respiratory frequency, is the result of mechanical effects of respiration on stroke volume (4, 5). The accompanying HF component in PI is related to vagal input to the heart, which is also related to respiration. Stauss et al. (10) found the LF and MF spectral components of BP not to be suitable markers for sympathetic nerve activity, which is in line with the present results, whereas we only found a significant difference of LF component of PI between rest and mild exercise.

In the present study, CI analysis of BP variability demonstrated that SCI increases with increasing workload. The differences among all exercise grades are significant, whereas PI, BP, SBP, MF, and HF of BP and PI failed to identify significant differences between mild and moderate exercise levels. PI power spectrum only distinguishes workload differences in the LF range, and BP variance was only significantly different between mild and moderate exercise grades. Comparison of the results of SCI with those obtained from the SD sets suggests that these differences originate mainly from nonlinear features of the cardiovascular control system.

Perspectives

The demand for noninvasive measures for cardiovascular control parameters has prompted several studies using linear analysis of BP and HR. A large proportion of hemodynamic variability, however, consists of nonrhythmic variations that can only be adequately analyzed by nonlinear techniques, such as the CI. In this study, we tested the feasibility of the CI to detect moderate-to-severe alterations in cardiovascular control as elicited by treadmill exercise. There were consistent and significant changes of CI in response to three different degrees of this cardiovascular challenge. This was not observed for either systolic or mean BP or PI or for the spectral estimates for the MF and HF range. Future studies are required to test whether the CI can provide a tool for clinically assessing derangement in the control of BP and HR.

    APPENDIX
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References

Embedding and CI

In physiology, one is often faced with the problem that only a single variable has been measured, although the entire system has an almost infinite number of degrees of freedom. The dynamics of this single variable, however, result from the entity of the control network. The one-dimensional time series can be transformed into a multidimensional artificial phase space. This reconstructed space is defined by vectors, whose components consist of data values from the original time series, which are separated by a suitable time lag (3, 7).

If the BP time series is denoted by pi. then a vector-valued time series is obtained using the method of time-delay coordinates by
<B>x</B><SUB><IT>i</IT></SUB> = [<IT> p</IT><SUB><IT>i</IT></SUB>, <IT>p</IT><SUB><IT>i</IT>−&tgr;</SUB>, <IT>p</IT><SUB><IT>i</IT>−2&tgr;</SUB>, …, <IT>p</IT><SUB><IT>i</IT>−(<IT>d</IT>−1)&tgr;</SUB>]
In this notation, the embedding dimension d denotes the number of components of the constructed vector, whereas the time lag tau  stands for the temporal displacement of each two succeeding vector components.

The CI of BP (3) is now defined as
C<SUP><IT>d</IT></SUP>(<IT>l</IT>) = <FR><NU>2</NU><DE><IT>N</IT>(<IT>N</IT> − 1)</DE></FR><LIM><OP>∑</OP><LL><IT>i</IT><<IT>j</IT></LL></LIM> &THgr;(<IT>l</IT> − ‖<B>x</B><SUB><IT>i</IT></SUB> − <B>x</B><SUB><IT>j</IT></SUB>‖)
where | xi - xj| is the distance between two points i and j. Theta  is the Heavyside step function and is equal to 1 if its argument is positive or zero and equal to 0 otherwise. N is the number of points in the vector series under consideration. Therefore, Cd(l) is the probability that the mutual distance of two randomly chosen vectors is <= l. In a statistical sense, Cd(l) means a frequency distribution of the mutual distances between each two vectors.

For this study, CI was evaluated by analyzing a subset (11) of all possible pair differences xi - xj
C(<IT>l</IT>) = 1/(<IT>NL</IT>) × [number of pairs (<IT>i</IT>, <IT>j</IT>) 
whose distance ‖<B>x</B><SUB><IT>i</IT></SUB> − <B>x</B><SUB><IT>j</IT></SUB>‖ is < <IT>l</IT>]
For the estimation of the CI, we used L = 1,024. Figures of CIs are displayed as double-logarithmic plots. The embedding dimension was fixed to 19, which is two times the largest value obtained for the SCIs. The tau  between successive components of the embedding vectors was set to 7.

The common approach to the analysis of CIs is the determination of the slope (SCI) of log C(l) versus log l, provided the slope is almost constant over some range (l1, l2). In chaos theory, the SCI is an estimate for the correlation dimension (or fractal dimension) of a strange attractor. In the case of the BP recordings, the range of constant slope of CI is very narrow, and often it declined to a single point. Only in this point is the slope maximal, not over a finite range of l. Because there was no extended range (l1, l2) with constant SCI, we determined the maximum value of SCI.

SD Sets

Because our hypothesis was that the dynamics generating BP fluctuations are, at least in part, nonlinear, we used a statistical procedure capable of discriminating between linear dynamics and an alternative conjecture. This was done in three steps. First, a null hypothesis was defined (linear dynamics hypothesis). Then a number of data sets (surrogates) were compiled from the experimental signal, compatible with the null hypothesis. Finally, a discriminating statistic was applied to both the original series and the set of surrogates. This statistic must be able to discriminate between the null and the alternative hypothesis.

The null hypothesis was that the observed signal is a monotonic nonlinear transformation (i.e., monotonically increasing measurement function) of a linear Gaussian process (6, 12). To this end, a Gaussian white noise series was created, and amplitude was transformed to the original data (i.e., so that the ranks of both signals agree). From the adjusted Gaussian series, the Fourier transformation was obtained and the phases were shuffled in a random manner. Subsequently, the spectrum was transformed back to the time domain. Finally, the original time series was reordered so that the ranks of both the original and the phase-shuffled series agreed. These surrogates shared the same distribution with the original data and the power spectra (and also the autocorrelation) and were quite similar to the original ones.

As a measure of significance with the SD sets, we used the following quantity (12)
<IT>S</IT> = <FR><NU>‖SCI<SUB>orig</SUB> − &mgr;<SUB><IT>S</IT></SUB>‖</NU><DE>&sfgr;<SUB><IT>S</IT></SUB></DE></FR>
where SCIorig is the SCI of the original series, µS the mean value of the surrogates' SCI, and sigma S is the standard deviation of the surrogates' SCI. Because the surrogates are compatible with the null hypothesis of a linear Gaussian process, the rejection range lies outside the defined region. If SCIorig lies outside µS ± sigma S, the null hypothesis is rejected. For each BP time series, 20 SD sets were evaluated.

    ACKNOWLEDGEMENTS

H. M. Stauss was supported by a grant from the German Humboldt Foundation (Feodor-Lynen program). This study was supported by National Institute on Aging Grant AG-12350.

    FOOTNOTES

Address for reprint requests: C. D. Wagner, Humboldt University of Berlin-Charité, Dept. of Physiology, Tucholskystrasse 2, D-10117 Berlin, Germany.

Received 19 November 1997; accepted in final form 28 July 1998.

    REFERENCES
Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References

1.   Almog, Y., S. Eliash, O. Oz, and S. Akselrod. Nonlinear analysis of BP signal. Can it detect malfunctions in BP control? Am. J. Physiol. 271 (Heart Circ. Physiol. 40): H396-H403, 1996[Abstract/Free Full Text].

2.   Casati, C., A. Monopoli, A. Forlani, E. Bonizzoni, and E. Ongini. Telemetry monitoring of hemodynamic effects induced over time by adenosine agonists in spontaneously hypertensive rats. J. Pharmacol. Exp. Ther. 275: 914-919, 1995[Abstract/Free Full Text].

3.   Grassberger, P., and I. Procaccia. Characterization of strange attractors. Phys. Rev. Let. 50: 346-349, 1983.

4.   Janssen, B. J. A., J. Oosting, D. W. Slaaf, P. B. Persson, and H. A. J. Struijker-Boudier. Hemodynamic basis of oscillations in systemic arterial pressure in conscious rats. Am. J. Physiol. 269 (Heart Circ. Physiol. 38): H62-H71, 1995[Abstract/Free Full Text].

5.   Julien, C., Z.-Q. Zhang, C. Cerutti, and C. Barres. Hemodynamic analysis of arterial pressure oscillations in conscious rats. J. Auton. Nerv. Syst. 50: 239-252, 1995[Medline].

6.   Kaplan, D., and L. Glass. Understanding Nonlinear Dynamics. Texts in Applied Mathematics. New York: Springer-Verlag, 1995, vol. 19.

7.   Packard, N. H., J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. Geometry from a time series. Phys. Rev. Lett. 45: 712-716, 1980.

8.   Pagani, M., F. Lombardi, S. Guzzetti, O. Rimoldi, R. Furlan, P. Pizzinelli, G. Sandrone, G. Malfatto, S. Dell'Orto, E. Piccaluga, M. Turiel, G. Baselli, S. Cerutti, and A. Malliani. Power spectral analysis of heart rate and arterial pressure variabilities as a marker of sympathovagal interaction in man and conscious dog. Circ. Res. 59: 178-193, 1986[Abstract/Free Full Text].

9.   Parati, G., G. Pomidossi, R. Casadei, A. Groppelli, S. Trazzi, M. di Rienzo, and G. Mancia. Role of heart rate variability in the production of blood pressure variability in man. J. Hypertens. 5: 557-560, 1987[Medline].

10.   Stauss, H. M., R. Mrowka, B. Nafz, A. Patzak, T. Unger, and P. B. Persson. Does low frequency power of arterial blood pressure reflect sympathetic tone? J. Auton. Nerv. Syst. 54: 145-154, 1995[Medline].

11.   Takens, F. Detecting nonlinearities in stationary time series. Int. J. Bifurc. Chaos 3: 241-256, 1993.

12.   Theiler, J., S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer. Testing for nonlinearity in time series: the method of surrogate data. Physica D. 58: 77-94, 1992.

13.   Wagner, C. D., R. Mrowka, B. Nafz, and P. B. Persson. Complexity and "chaos" in blood pressure after baroreceptor denervation of conscious dogs. Am. J. Physiol. 269 (Heart Circ. Physiol. 38): H1760-H1766, 1995[Abstract/Free Full Text].

14.   Wagner, C. D., and P. B. Persson. Nonlinear chaotic dynamics of arterial blood pressure and renal blood flow. Am. J. Physiol. 268 (Heart Circ. Physiol. 37): H621-H627, 1995[Abstract/Free Full Text].

Am J Physiol Regul Integr Compar Physiol 275(5):R1661-R1666
0002-9513/98 $5.00 Copyright © 1998 the American Physiological Society


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