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Nonlinear dynamics of an artificial baroreflex system.

Nose et al, concluded from many experiments that no man-made computer servo system should be incorporated for TAH control. To evaluate the TAH automatic control algorithm, the total system must be analyzed as an entity, not as single parts like the autonomic nervous system, hormonal factors, and so on. Recent progress in nonlinear mathematics enables us to estimate the whole system.

During the past two decades, many researchers have investigated nonlinear dynamics in the cardiovascular system. Guevara et al. described the cellular and subcellular mechanisms that produce the cardiac action potential known to be characterized by a chaotic attractor. Other investigators reported that non-oscillatory cardiac tissues also manifest nonlinear dynamics. Of course, cardiovascular function is characterized by a complex interaction of many control mechanisms which permit adaptation to the changing environment. These complexities from the field of nonlinear dynamics made it difficult to analyze the circulatory regulatory system quantitatively.

Nonlinear dynamics coinciding with deterministic chaos may generate random-like time series, which on closer analysis result highly ordered and critically dependent on the initial conditions. Mathematically, all nonlinear dynamic systems with more than two degrees of freedom can generate deterministic chaos, becoming unpredictable [1-3]. Using nonlinear mathematical methodology, we can obtain attractors of nonlinear dynamic behavior in the phase space. Attractors displaying chaotic behavior are termed "strange attractors", which have a sensitive dependence on initial conditions.

In linear systems, attractors are, of course, not strange. Attractors of ordinary differential equations with a degree of freedom below 2 are limited to either a fixed point or a limited cycle, and have proved not to be strange. In a strange attractor, the stretching and folding operations smear out the initial volume, thereby destroying the initial information as the system evolves and the dynamics create new information. In this study, to evaluate the circulation with the TAH with experimentally constructed baroreflex system from the viewpoint of the whole system, the hemodynamic parameters with TAH were analyzed by nonlinear mathematical analyzing techniques useful in the study of deterministic chaos. Hemodynamic parameters were recorded and analyzed in chronic animal experiments using healthy adult goats.

MATERIALS AND METHODS

After resection of the natural heart under extracorporeal circulation, pneumatically driven sac type blood pumps were connected and set outside the body on the chest wall in chronic animal experiments using two adult female goats weighing 48 kg and 60 kg. The only medication used in these experiments was antibiotics to avoid infection. Pump output was measured at the outlet port of the left pump with an electromagnetic flowmeter. Aortic pressure, and left and right atrial pressure were measured with pressure transducers through the side tube of the cannulae. These data were fed through an A/D converter into a personal computer in which the control logic was run.

In this experiments, TAH automatic control algorithm were based upon these concept. Firstly, optimal drive point of the inner sac were maintained by the automatic control algorithm for the pneumatic driver. It was needed to prevent the thrombus formation in the inner sac. Secondary, by the alteration of the stroke volume, left and right flow balances were maintained. Stroke volume changes were maintained within an optimal operating point. And at last, maintain the hemodynamics within normal range and an automatic TAH control algorithm based upon artificial baroreflex concept were added to these basic control algorithm. The TAH drive parameters used in the experiments were: pulse rate, drive air pressure, and systolic-diastolic rate. The L-R balance was controlled by dynamic balancing of blood volumes between the pulmonary and systolic circulations.

In this study, we had used the embedding technique proposed by Takens et al to evaluate the hemodynamics of 1/R control [24]. For a variable x(t) denoting a time series datum of the hemodynamics, we tabled new variables y(t)=x(t+T), z(t)=x(t+2T), w(t)=x(t+3T)..., where T was the order of the correlation time. Using this embedding technique, we reconstructed the attractor in the phase space. It is well known that time series data of the chaotic system are similar to the time series data of the random system, but the phase portrait of the chaotic system shows a completely different pattern from the random system. The reconstructed attractor of the chaotic dynamics shows itself to be a strange attractor.

The Lyapunov exponent calculated from the reconstructed attractor in the phase space was widely used the application of non-linear mathematics for a pragmatic measure of chaos as we shown before.

Results

In this study, all the hemodynamic parameters were embedded into the phase space and projected into the three dimensional phase space. Time series data of the pressure pattern recorded through the side branch of cannulae, and continuous monitoring of the hemodynamics were easily obtained by this side branch system, while the mean pressures were useful for the automatic control of TAH. However, the water hammer effect of the prosthetic valves in the TAH cannot be avoided. Thus, the most evident phenomenon in the pressure pattern attractor is thought to be the effect of the water hammer, so we cannot detect the hemodynamic behavior exactly using this method.

So, the phase portrait of the pump output flow pattern during the TAH fixed driving condition embedded into the four-dimensional phase space and projected into the three-dimensional phase space is shown in Fig.22. This attractor was projected from a higher dimensional phase space, so we cannot describe the dimensions of the axis. A simple attractor suggesting a limited cycle attractor coinciding with a periodic system is shown. Fig.23 shows the reconstructed attractor of the pump output with an automatic TAH control. To evaluate the patterns of the phase portrait, we calculated the Lyapunov exponents from the reconstructed attractor of the time series data. At least one positive large Lyapunov exponent suggests the strange attractor, which is a feature of nonlinear dynamics with deterministic chaos. Fig.24 shows the calculation of the largest Lyapunov exponents during TAH automatic control. Positive Lyapunov exponents suggest a sensitive dependence on the reconstructed attractor.

Discussion

One of the main findings in this study is that the reconstructed attractor of the artificial heart pump output suggests the formation of a strange attractor in the phase space during automatic TAH control algorithm. The strangeness of the reconstructed attractor was measured by reconstructing it and using Lyapunov exponents. A wider band in the reconstructed attractor, and positive Lyapunov exponents during the segmented time series were seen. Our results suggest that strangeness of the reconstructed attractor appears to have the largest value, indicating larger dimensional strong chaos with TAH automatic control algorithm.

In this study, the chaos was determined by reconstruction of the attractor in the four-dimensional phase space and calculating the largest Lyapunov exponent. A simple reconstructed attractor in the phase space during fixed TAH driving, and the positive largest Lyapunov exponents suggest a lower dimensional chaotic system. Compared with the fixed driving condition, TAH automatic control algorithm showed more complicated nonlinear dynamics suggesting higher dimensional deterministic chaos. Though several investigators have been studying nonlinear dynamics including deterministic chaos, there is no universally accepted definition. The widely accepted characteristics of deterministic chaos are:

1) a deterministic dynamic system

2) a sensitive dependence on initial conditions

3) an attractor.

Many investigators have noted the functional advantages of the chaotic system [2-6]. In some clinical cases, pathologies exhibit increasing periodic behavior and a loss of variability [2-6]. In nonlinear dynamics, a chaotic system operates under a wide range of conditions and is therefore adaptable and flexible compared with the periodic system [6]. When unexpected stimuli are fed from outside, this plasticity allows whole systems to cope with the requirements of an unpredictable and changing environment [2-6]. For example, Tsuda et al found deterministic chaos in a pressure waveform of the finger capillary vessels in normal and psychiatric patients and proposed the notion of a "homeochaotic" concepts have been proposed state [6]. Similar by several other investigators. From this standpoint a cardiovascular regulatory system with TAH fixed driving shows a lower dimensional limit cycle attractor, suggesting a lower dimensional dynamic system. So fixed TAH driving may be in the lower dimensional homeochaotic state. Thus, hemodynamic parameters can be thought of as irritated by external turbulence compared to automatic TAH driving. These results suggest that TAH automatic control algorithm may be suitable when unexpected stimuli may be fed from outside.

Last modified:2006/04/10 22:42:01
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References:[Welcome Dr. Yambe's Home Page!! (eng)] [Origin of Chaos]

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