chaos in blood pressure control

Gzrdiovascular Research

Cardiovascular Research 3 1 ( 1996) 380-387

Review

Chaos in blood pressure control

CD. Wagner *, B. Nafz, P.B. Persson Physiologisches Institut der Medirinischen Fakult& der Humboldt-Unioersittit zu Berlin (Charitt!), Hessische Strasse 3-4, D-10115 Berlin, Germany

Received 9 May 1995; accepted 12 December 1995

Abstract

A number of control mechanisms are comprised within blood pressure regulation, ranging from events on the cellular level up to circulating hormones. Despite their vast number, blood pressure fluctuations occur preferably within a certain range (under physiological conditions). A specific class of dynamic systems has been extensively studied over the past several years: nonlinear coupled systems, which often reveal a characteristic form of motion termed “chaos”. The system is restricted to a certain range in phase space, but the motion is never periodic. The attractor the system moves on has a non-integer dimension. What all chaotic systems have in common is their sensitive dependence on initial conditions. The question arises as to whether blood pressure regulation can be explained by such models. Many efforts have been made to characterise heart rate variability and EEG dynamics by parameters of chaos theory (e.g., fractal dimensions and Lyapunov exponents). These method were successfully applied to dynamics observed in single organs, but very few studies have dealt with blood pressure dynamics. This mini-review first gives an overview on the history of blood pressure dynamics and the methods suitable to characterise the dynamics by means of tools derived from the field of nonlinear dynamics. Then applications to systemic blood pressure are discussed. After a short survey on heart rate variability, which is indirectly reflected in blood pressure variability, some dynamic aspects of resistance vessels are given. Intriguingly, systemic blood pressure reveals a change in fractal dimensions and Lyapunov exponents, when the major short-term control mechanism - the arterial baroreflex - is disrupted. Indeed it seems that cardiovascular time series can be described by tools from nonlinear dynamics [66]. These methods allow a novel description of some important aspects of biological systems. Both the linear and the nonlinear tools complement each other and can be useful in characterising the stability and complexity of blood pressure control.

Keywords: Baroreflex; Chaos; Blood pressure; Lyapunov exponent: Nonlinear phenomena

1. Introduction

Arterial blood pressure underlies a very complex system of controllers that are involved in its regulation. Guyton et al. compiled a complex diagram, which contained 354 blocks, “each of which represents one or more mathematical equations describing some physiological facet of circulatory function” [29]. A number of key issues of cardiovascular control, however, remain to be established: (1) Are we able to satisfactorily describe mathematically blood pressure regulation acknowledging all levels of control, and (2) if not, is our limited knowledge sufficient for an adequate description of the regulation as a black box? Unfortunately, even the attempt of an exact quantification

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on the cellular level shatters all hopes of a mathematically exact description of these mechanisms as a whole f 141. The second question may seem more promising: The mathematical field of nonlinear deterministic systems or chaos theory provides a powerful tool to describe the dynamics of systems, even in the case of only one available variable. This is often the physiologist’s situation: Biological systems are distinguished by an almost infinite number of system variables, whereas in most cases only a small number of variables are accessible to the experimentor. Chaos theory deals with systems of low dimension (i.e., a small number of variables), which may exhibit very complex behaviour. On the other hand, synergetics. a related field of research investigates systems with many degrees of freedom, often shows relatively simple behaviour. Due

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to the mutual dependence of the controllers in blood pressure regulation the time series of circulatory variables seem apt to be character&d by means of techniques from chaos theory. For reviews of application of chaos and fractals in sciences, see the introductory review or Refs. [12,20,36,41], and for a more rigorous and mathematical description, refer to Refs. [ 13,48,63].

In 1628, Harvey identified the dynamic nature of blood pressure [30]. Harvey’s contemporary, Descartes ( * 15961, was among the first who attempted to explain all processes in the organism by purely mechanistic laws. His interpreta- tion of the course of events inside the organism was deterministic: Each action causes a reaction. Isaac Newton introduced this principle into theoretical mechanics and astronomy. This principle was successfully applied to the development of statistical mechanics, where the behaviour of the whole system is character&d by the combined action of an almost infinite number of participants. Biolog- ical systems are also characterised by a vast number of degrees of freedom and, according to Descartes, his principle should allow the modelling of the behaviour of such systems and to predict its course within time. Many efforts have been made to explain and to predict the temporal development of blood pressure, but although more and more control mechanisms have been discovered, blood pressure regulation is still not entirely understood. In 1992, a review article by Cowley elucidated the complexity of systemic blood pressure regulation [ 111.

It has emerged that almost all laws of nature are nonlinear. This means that within the mathematical formu- lation of a law of nature (by differential equations) some system variables, as well as their derivatives with respect to time, are coupled nonlinearly. As a result, remarkable dynamics may appear: besides the known stochastic systems, fixed point solutions, as well as periodic motion, aperiodic behaviour may occur, a form of motion which is distinct from all of the above-mentioned forms. The motion is not periodic, but the system is restricted to a limited volume in phase-space.

2. New tools for quantifying nonlinear properties of blood pressure control

As mentioned above, in real life one has only a few variables available, at worst only one recording is attained. Nonetheless, it is possible (at least theoretically) to extract the system’s dynamics from this variable only, which is due to the fact that each variable is coupled with one or more other variables [49,61]. Fluctuation in one variable impinges on all other variables after a certain time. A graphic tool to visual& the dynamics of a system is producing a phase portrait.

A phase portrait relates the given state of the system to the state at some later time (Fig. 1). One plots a point of a given time series against a point later in time, then this

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Fig. 1. Original arterial blood pressure recording in a conscious foxhound at rest (upper panel). In this phase portrait, the value of blood pressure at a given time is plotted against a value at a fixed time later. The state space (reduced to two dimensions) is not completely filled. Consequently the trajectory is restricted to a closed region in state space, the attractor. It is unclear if this structure represents a torus (periodic or multiperiodic motion) or a strange attractor (chaos). ‘Ihe attractor exhibits forbidden zones, which is typical of strange attractors. 1000 points were plotted, the sampling frequency was 50 Hz and the time lag d t = 100 ms. In the lower panel a phase portrait is depicted from blood pressure mean values after low-pass filtering of the original time series @OOO data points, sampling frequency I Hz, dt = 40 s).

procedure is repeated. In stochastic systems, the points are uncorrelated and densely fill the state-space. If the signal is correlated with itself, then some structure would appear in the phase portrait. Especially if the underlying dynamics form a strange attractor, then the phase portrait will also result in points lying on a strange attractor. This attractor may appear quite different from the original one but has the same characteristic measures as fractal dimension and Lyapunov exponents. Examples of phase portraits of periodic systems, stochastic systems, as well as of chaotic systems can be found in Ref. [12].

Precautions have to be taken regarding the nonstationar- ity of time series as well as noise contaminating the measured signal. Stationarity is the invariance of all statistical properties of the signal to location or time index. This rarely holds for biological systems. In consequence, most of the time, short-term sequences (in which shift invariance is fulfilled) should be extracted from the long-term signal. Noise is a more crucial point in time series analysis. First, noisy fluctuations originating from the system itself may overlap the measured signal. Second, measures such as Lyapunov exponents are very sensitive to fluctuations in the system parameters. Finally, the analog-to-dig- ital conversion (ADC) of a signal may be regarded as a source of noise. Commercially available ADC equipment

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maps the signal on to a set of 12 digits. Thus, the recording of blood pressure within a range of 200 mmHg, for example, results in a maximum resolution of ca. 0.05 mmHg. The trailing zeroes in these values can be understood as noise corrupting the exactness of the signal. Noise handling is not a trivial issue, thus, a large number of noise reduction algorithms do exist (for a review, see Ref. [39]).

3. Blood pressure regulation

Systemic blood pressure is not constant but shows a considerable amount of variability (Fig. 2). In the range from seconds to minutes, some fluctuations in blood pressure correspond to heart rate variability (see the mini-review on heart rate variability in this issue).

The baroreflex is of particular importance in short-term blood pressure control. Other factors also affect cardiovascular control in different frequency ranges, such as the renin-angiotensin system [441, vasopressin [lo,371 and as vasoactive substances, which are produced in the resistance vessels [34,40,46,57,58].

4. Nonlinear properties of short-term blood pressure regulation

The major short-term mechanism of arterial blood pressure control is the arterial and cardiopulmonary baroreceptors, which impinge on the autonomic nervous tone. If

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Fig. 2. Arterial blood pressure recording of a conscious resting foxhound (upper panel). The major oscillations are due to cardiac activity, which manifests itself in the peak at approximately 1 Hz (lower panel).

arterial blood pressure rises, the integral activity of the baroreceptors will also increase. In consequence, periph- eral resistance and heart rate will diminish via a lower sympathetic tone. This negative feedback has nonlinear response characteristics: the sum activity of the baroreceptors is a sigmoid function of arterial pressure. (In addition, the heart rate response to arterial blood pressure is also a sigmoid function, but with opposite orientation.)

In order to character& differences in blood pressure regulation after disruption of the baroreflex arc, experiments in two groups of conscious dogs were performed 1671: a control group and a group subjected to total sinoaortic and cardiopulmonary baroreceptor denervation. As a measure of variability, standard deviation was deter- mined and power spectra were calculated (fast Fourier transform, FFI). A Fourier spectrum describes a signal in the frequency domain and divides the signal into its har- monic components. The FFI is a special type of Fourier transform, where symmetries between the sine and cosine functions in the Fourier transform algorithm are exploit, reducing calculation time. In the lower frequency range (f< 0.1 Hz), power density was inversely related to frequency in both groups, indicating l/f noise [42,65]. In many physical systems, as well as in living systems, this specific pattern is observed: in a low frequency range, log(power density) depends linearly on log(frequency). In spite of its ubiquity, this behavious is not well understood. It is even observed in the dynamics of interplanetary magnetic fields [43] and in music and speech [33,64]. A possible explanation for these noise spectra may be the complexity of the whole system itself [4]: the vast amount of variables act in concert to regulate blood pressure in such a way that the interaction of two or more of them is dependent on the behaviour of the other variables. The overall system behaviour is the product of many individual influences that are linked together, resulting in a log-normal distribution [56]. The Fourier transform of the log-normal distribution’s tail bears a strong resemblance to an inverse power law [4,47,56]. This model can be applied to blood pressure regulation: a great number of control ele- ments are involved in the neurohumoral control, which yields the l/f spectra. Within the frequency range where l/f noise is observed, no specific frequency is preferred by the overall system (i.e., it has no resonance frequency or characteristic time constant).

Surprisingly, there are two l/f ranges in the blood pressure spectra (Fig. 3, [65]). The first l/f region is located within a low-frequency range (f< lo-‘.’ Hz; slope -0.9). The second l/f range is sited at 1O-‘.4 <f< lo- ‘.’ Hz with slope - 1.2. After baroreceptor denervation, the steepness of both slopes increased significantly (slopes in the lower and higher range - 1.2 and -3.1, respectively). Neither a-receptor nor p-receptor blockade considerably changed the slopes after denervation. However, autonomic blockade with the ganglionic blocking agent hexametho- nium, restored the slope in the lower frequency range.

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The Grassberger-Procaccia algorithm [24,25] was applied to estimate the correlation dimension of the time series; 1024 data points of the original time series were used [67]. The fractal dimension is a measure that describes how points lying along an attractor fill up phase- space [ 13,15,23]. In many cases, the dimension is termed “fractal”, due to its non-integer value. The correlation dimension D, is a lower estimation of the fractal dimension and is commonly computed by the algorithm described by Grassberger and Procaccia [24,25]. A fixed time lag of 100 ms was chosen, which is the time after which the autocorrelation functions of the time series has the first zero. Estimating the correlation dimensions of the blood pressure time series as a quantification of complexity revealed a decrease after baroreceptor denervation (1.74 vs. 3.05 control; Figs. 4 and 6). The fractal dimension is a measure of a system’s complexity, and it is the lowest estimate of the minimal number of relevant degrees of freedom or variables which constitute the dynamics of the system [26,32]. Fractal means non-integer, which refers to the geometry of the strange attractor, along which the system moves in its phase-space.

Another quantity to characterise the dynamics of a system are Lyapunov exponents (hi). In general, an n-dimensional system has n Lyapunov exponents. The set of

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Fig. 3. Log-log plot of power spectrum for blood pressme. (A): l/f relationship for mean values obtained from control group, (B) mean data after denervation. After denervation, the difference between both l/f ranges becomes apparent.

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Fig. 4. Sequences of original blood pressure time series of a representa- tive of the control group (A) and after baroreceptor denervation (B); the sampling frequency was 10 Hz. Panels C and D show the estimates of the correlation dimension by the algoithm of Grassberger and Procaccia. The slope of the linear segment in the plot log (correlation integral) vs. log (radius) is taken as the correlation dimension. Each curve refers to a different embedding dimension with the uppermost for d = 2 and the rest for d increasing in steps of 1, The curves contain a linear segment whose slope converges to a constant value as the embedding dimension is increased. For each estimate we used 1024 data points.

all Lyapunov exponents (often referred to as the “Lyapunov spectrum”) gives the average rate of volume growth under the flow in the system’s phase-space. We are particularly interested in the largest Lyapunov exponent A,: this is a direct measure of the predictability of the time course of a system. It stands for the average rate of divergence in phase-space of two adjacent trajectories. Periodic systems have a h, of zero. The case h, > 0 can be used for the identification of chaotic systems [13,48] (i.e., the sensitivity on initial conditions). Since stochastic signals may reveal positive Lyapunov exponents as well, it is necessary to further distinguish both cases (e.g., by checking for nonlinear predictability [60]).

The most widespread method used to compute the Lyapunov exponent was provided by Wolf and colleagues [68]. From the delayed time series two adjacent trajectories are observed over time. From the average divergence rate A, ( = h,,) is extracted. The Lyapunov exponents were estimated by Wolf’s algorithm with the fixed evolution time program for A, (1024 data points). Determination of the largest Lyapunov exponents, which indicate the sensitive dependence on initial conditions - the hallmark of chaos - also yielded a diminution after denervation (0.74 vs. 1.8 bits/s control; Figs. 5 and 6). The positive values of the Lyapunov exponents of both groups mean that the underlying control mechanisms are nonlinearly coupled and reveal chaotic behaviour. However, the average Lya- punov exponent of the denervated group is only about 40% of the control value. This means that the temporal develop-

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Fig. 5. Relative frequency of mean arterial blood pressure of one dog of the control group and after baroreceptor denervation. The increase in variability after denervation is visible (A). Panel B shows the relation of standard deviation of the sample time series as a function of the largest Lyapunov exponent. The error bars indicate standard deviations.

ment of systemic blood pressure of denervated dogs can be more accurately predicted (less chaotic): after baroreceptor denervation, blood pressure control is less complex and less sensitive on initial conditions. This is indirectly supported by the observation of slow periodic components in the power spectra after denervation [52]. In contrast, variability (standard deviation) is increased after baroreceptor denervation [9] (Fig. 5).

For cross-checking of the results, we also computed surrogate data sets from the original time series. We obtained surrogate data based on randomising the phases of the Fourier transforms of the time series [62]. From the surrogates, we calculated correlation dimensions and Lya- punov exponents. Theiler et al. suggest a measure of significance between the original data and the surrogates [62]. The null hypothesis being tested is that the data arise

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Fig. 6. Correlation dimensions and greatest Lyapunov exponent of 7 resting foxhounds studied after baroreceptor denervation compared with 7 control dogs. Statistical significance is indicated by asterisks (* l P < 0.01).

from a stochastic linear process. In the surrogate data sets, several linear properties of the original data are preserved (mean, variance, and power density), but there are no further nonlinear structures in the time series. We applied this method and found the correlation dimensions and Lyapunov exponents for the original data and the surrogates to be significantly different [67].

It is emphasised that the results above refer to blood pressure variations in a time range of about 100 s (1024 data points in the time series with a sampling frequency of 10 Hz). Hence, this study can only show that the baroreflex contributes largely to the dynamic properties of blood pressure regulation within the time range of approximately 2 min. Longer time series may yield different Lyapunov exponents due to the different controllers responsible for producing the variability seen in this range. The baroreceptors efficiently buffer blood pressure fluctuations in a range > approx. 25 s [34], but also other regulating mechanisms may act more strongly after denervation in blood pressure regulation. After blocking of the major short-term pressure-controlling system, the contribution of the remain- ing system (e.g., the renin-angiotensin system, vasopressin secretion, etc.) increases [63, and therefore the dynamics of the regulating system is perturbed in a way that is simpler and less complex in nature, but with greater variability.

Hence, under physiological conditions, arterial and cardiopulmonary baroreceptors reduce the variability of blood pressure [9], yet at the cost of blood pressure being less predictable. Thus the regulation has a higher degree of chaos under healthy circumstances. Elimination of the baroreceptor feedback on blood pressure control also de- creases the complexity of arterial blood pressure control. A positive effect of chaos in blood pressure regulation may be seen in the ability to react to altered external influences. According to several authors, a transition from physiological conditions of blood pressure regulation to pathological ones (also manifesting in other physiological variables) is accompanied by a reduction in the largest Lyapunov exponents and, therefore, by a loss of amount of chaos. This has been shown in a study of multiple sclerosis [16] and heart rate dynamics prior to sudden death [22]. Finally, Skinner et al. reported a reduction of the point estimates of the heartbeat correlation dimension that precedes lethal arrhythmias by hours [59]. This observation also fits the hypothesis that disturbed or handicapped systems loose complexity and stability.

5. Cardiac chaos

Related rhythms to those observed in cardiac activity are reflected in blood pressure variability. Blood pressure and heart rate reveal both circadian and ultradian rhythms, as well as respiration activity [5,551. The autonomous balance can possibly be character&d by means of spectral

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analysis, as suggested by several authors [1,2,3,50,53], as well as by linear modelling (e.g., autoregressive methods [51]). Besides pure oscillatory behaviour, a less specific variability exists which extends over a wide frequency range. In a double logarithmic plot, power density and frequency are in inverse proportion, the l/f [21,38,54]. This is typically found in circulatory variables [42,65]. Experiments utilising ventricular muscle from isolated dog hearts [8] reveal period-doubling bifurcations and aperiodic activity (chaos) dependent on the relationship between pacing basic cycle length and beat-to-beat action potential amplitudes, whereas experiments with periodically stimulated excitable cells and cell assemblies also show phase- locking, period-doubling bifurcations, and aperiodic behaviour [7,14,28]. Mathematical models in which a normal sinus rhythm interacts with an ectopic pacemaker also reveal these characteristic patterns [ 18,311. Finally, the existence of cardiac chaos is best demonstrated by the controlling of interbeat intervals in isolated hearts 1171. But it remains an open question, to date, whether the normal healthy in vivo heartbeat reveals aperiodic behaviour or just some very complicated form of noise. Glass and Kaplan suggested that heart rate signals did not show evidence of determinism, but were indistinguishable from a stochastically forced linear system with the same power spectrum [19]. A study by Kanters and colleagues seems to confirm that R-R interval variability is a process with an element of nonlinear determinism, but without evidence of low-dimensional chaos in those time series [35]. In contrast, the work of Michaels and co-workers showed clear evidence of chaotic behaviour in a mathematical model of the vagally driven sinoatrial node, also with additional respect to the case of the baroreceptor reflex loop ([45] and references therein).

6. Resistance vessels

Recently, Griffith and Edwards demonstrated in an impressive way how vasomotion may exhibit nonlinear chaotic behaviour [26]. First branch generation arteries, arising from the central ear artery in rabbits, were dis- sected and perfused in situ. Perfusion pressure dynamics were observed via a sidearm connected to the perfusion circuit.

In a large number of preparations, they found Period 2 as well as Period 4 behaviour prior to the onset of chaos, which is strong evidence for a period-doubling route to chaos. For the regular pressure oscillations, the mean amplitude and frequency remained inversely related to each other, which is evidence for a nonlinear coupling of the oscillators. In some preparations, sequences of nearly periodic behaviour alternated with aperiodic motion, a phenomenon called “intermittency”. The frequency of the periodic sequences as a function of the length of the sequences fits very well with a model of Pommeau-Man-

neville [48]. The major finding is the occurrence of quasiperiodicity: two principal frequencies (a fast and a slow component with periods of 5-20 s and l-5 min, respectively) were found in the power spectra. It seems reasonable to assume that two oscillators act in concert to produce this behaviour.

Edwards and Griffith also calculated correlation dimensions of the pressure time series [26]. They found the correlation dimensions D,, in almost all cases, to be less than 3 (although in 16% of cases 3 < D, < 4). The fast oscillator involved ion movements at the cell membrane, whereas the slow oscillator was intracellular [27]. The coupling between both oscillators seemed to be the con- centration of intracellular Ca*+. This model of coupled oscillators producing chaos was supported by calculations of correlation dimensions: after administration of vera- pamil or ryanodine, D, decreased significantly to values < 2. Thus, after removal of one oscillator, the system became less complex.

7. Summary and conclusions

Chaos is located in EEG data, R-R intervals from electrocardiograms, and at the cellular level. Only a few studies deal with chaos in blood pressure control.

Positive Lyapunov exponents and non-integer fractal dimensions can be found in arterial blood pressure time series. After disruption of a crucial feedback control system - the baroreflex - the regulation becomes less complex and less sensitive under initial conditions. Also the l/f increase in the power spectrum alters in a charac- teristical manner.

The results indicate that analytical techniques derived from the field of chaos theory can be useful in characterising the stability and complexity of blood pressure control, which may provide important measures for the prediction of cardiovascular risk.

Acknowledgements

This study was supported by the German Research Foundation (Gerhard-Hess Program Pe 388/2-2).

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