Chaos in blood pressure control
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Cardiovascular Research 3 1 ( 1996) 380-387
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
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 . 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
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 mathemati- cal equations describing some physiological facet of circu- latory function . A number of key issues of cardio- vascular 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 mathe- matical 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 physiologists situation: Biological sys- tems 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 com- plex 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 . Harveys 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 princi- ple 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 sys- tems, 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 mo- tion 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 systems 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 peri- odic systems, stochastic systems, as well as of chaotic systems can be found in Ref. .
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 statis- tical 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 invari- ance is fulfilled) should be extracted from the long-term signal. Noise is a more crucial point in time series analy- sis. First, noisy fluctuations originating from the system itself may overlap the measured signal. Second, measures such as Lyapunov exponents are very sensitive to fluctua- tions 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 under- stood 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. ).
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 pres- sure correspond to heart rate variability (see the mini-re- view on heart rate variability in this issue).
The baroreflex is of particular importance in short-term blood pressure control. Other factors also affect cardiovas- cular control in different frequency ranges, such as the renin-angiotensin system [441, vasopressin [lo,371 and as vasoactive substances, which are produced in the resis- tance vessels [34,40,46,57,58].
4. Nonlinear properties of short-term blood pressure regulation
The major short-term mechanism of arterial blood pres- sure control is the arterial and cardiopulmonary barorecep- tors, 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 barorecep- tors 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, experi- ments 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 fre- quency 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  and in music and speech [33,64]. A possible explanation for these noise spectra may be the complexity of the whole system itself : 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-nor- mal distribution . The Fourier transform of the log-nor- mal distributions 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, ). 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
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The Grassberger-Procaccia algorithm [24,25] was ap- plied to estimate the correlation dimension of the time series; 1024 data points of the original time series were used . The fractal dimension is a measure that de- scribes 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 dimen- sion and is commonly computed by the algorithm de- scribed 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 quantifica...