1. health and fitness
2. disease
3. cholesterol

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
Keywords:
Baroreflex; Chaos; Blood pressure; Lyapunov exponent: Nonlinear phenomena
1. Introduction
Arterial blood pressureunderlies 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 representsone or more mathematical equations describing somephysiological facet of circulatory function” [29]. A number of key issuesof cardiovascular control, however, remain to be established: (1)
Are we able to satisfactorily describemathematically 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
l
Corresponding
author. Tel.: ( + 49-30)28468-511;
Fax: ( + 49-
30)28468-608.
0008-6363/%/$15.00
SSDI OOOS-6363(96)00007-7
on the cellular level shattersall hopes of a mathematically
exact description of thesemechanismsas 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 caseof 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 systemsof 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
Tie
6 1996 Elsevier Science B.V. All rights reserved
for primary
review
21 days.
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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.
C.D. Wagner
et al./Cardiovascular
2. New tools for quantifying
blood pressure control
nonlinear
properties
of
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
31 (1996)
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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 @OOOdata 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 nonstationarity 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-digital conversion (ADC) of a signal may be regarded as a
source of noise. Commercially available ADC equipment
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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 interpretation 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. Biological 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 formulation 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.
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C.D. Wagner
382
et al. / Cardiovascular
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].
properties
of short-term
blood pressure
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).
31 (1996)
380-387
arterial blood pressure rises, the integral activity of the
baroreceptors will also increase. In consequence, peripheral 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 determined 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 harmonic 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 elements 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 hexamethonium, restored the slope in the lower frequency range.
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4. Nonlinear
regulation
Research
C.D. Wagner
et al./Cardiovascular
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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.
31 (1996)
383
380-387
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Fig. 4. Sequences of original blood pressure time series of a representative 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 Lyapunov exponent of the denervated group is only about 40%
of the control value. This means that the temporal develop-
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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 phasespace [ 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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384
C.D. Wagner
et al./ Cardiovascular
50
100
150
200
mean arterialpressure (mmiig)
0
0.5
1 1.5 2 2.5
Lyapunov exponents
3
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 Lyapunov 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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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 remaining 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 decreases 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.
2.0
1.0
5. Cardiac chaos
0.5
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 (* P <
l
0.01).
31 (1996) 380-387
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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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.
Research
C.D. Wagner
et al./Cardiovascular
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 dissected 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-
31 (1996) 380-387
385
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 concentration of intracellular Ca*+. This model of coupled
oscillators producing chaos was supported by calculations
of correlation dimensions: after administration of verapamil 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 characteristical 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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