Three measures of nonlinear chaos (fractal dimension, Approximate Entropy (ApEn), and Lyapunov exponents) were studied as potential measures of cardiovascular condition. It is suggested that these measures have potential in the assessment of cardiovascular condition in environments of normal cardiovascular stress (normal gravity on the Earth surface), cardiovascular deconditioning (microgravity of space), and increased cardiovascular stress (lower body negative pressure (LBNP) treatments)

Hooker, John C.

English

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THE APPLICABILITY OF 
NONLINEAR SYSTEMS DYNAMICS 
CHAOS MEASURES TO 
CARDIOVASCULAR PHYSIOLOGY VARIABLES 
Final Report 
NASA/ASEE Summer Faculty Fellowship Program 1991 
Johnson Space Center 
Prepared By: John C. Hooker, Ph.D. 
Academic Rank: Professor 
University & Department: Houston Baptist University 
Department of Biology 
Houston, Texas 77074 
NASA/JSC 
Directorate: 
Division: 
Branch: 
JSC Colleague: 
Date Submitted: 
Contract Number: 
Space and Lite Sciences 
Medical Sciences 
Space Biomedical Research 
Institute 
John B. Charles, Ph.D. 
August 16, 1991 
NGT-44-001-800 
10-1 
https://ntrs.nasa.gov/search.jsp?R=19920012031 2020-03-24T07:12:59+00:00Z
ABSTRACT 
Three measures of nonlinear chaos--fractal dimension, 
Approximate Entropy (ApEn), and Lyapunov exponents--were 
investigated as potential measures of cardiovascular 
condition. 
It is suggested that these measures have potential in 
the assessment of cardiovascular condition in environments 
of normal cardiovascular stress (normal gravity on the earth 
surface), cardiovascular deconditioning (microgravity of 
space), and increased cardiovascular stress (lower body 
negative pressure (LBNP) treatments). 
10-2 
INTRODUCTION 
Many cardiovascular variables, such as heart rate and 
blood pressure, when recorded over long periods of time, 
appear to be random variables. Using statistics, such as 
the mean and the standard deviation, and measures of 
confidence in these statistics, statistical analyses of 
stochastic systems producing true random variables, are well 
understood and extensively used. The cardiovascular system, 
however, is not a stochastic system producing true random 
variables, but, instead, is a highly developed, very 
complicated. nonlinear. dynamic system producing tightly 
regulated output variables. 
Recently, several techniques have been developed for 
the analysis of the behavior of deterministic nonlinear 
dynamic systems. These systems produce chaotic output 
variables that appear to be random functions of time, but 
which are actually completely determined by the 
deterministiC, though nonlinear, differential equations 
describing the system. Even though a deterministic 
description of the cardiovascular system is not known, it 
seems appropriate to apply these techniques to its analysis. 
This report presents some of the findings of a 
preliminary study of the applicability of three measures of 
nonlinear chaos to the assessment of cardiovascular 
condition based on recordings of cardiovascular physiology 
variables. These measures are fractal dimension, Lyapunov 
exponents, and Approximate Entropy (ApEn). 
NONLINEAR CHAOS MEASURES 
The three measures considered here are attempts at the 
quantification of the variability, or, inversely, the 
"patternness", of chaotic behavior. Two of these, fractal 
dimension and Lyapunov exponents. are used extensively in 
the characterization of deterministic nonlinear dynamic 
systems and have been used to a lesser extent in the attempt 
to describe the behavior of chaotic systems with unknown 
descriptions. The third, ApEn, has been developed for the 
analysis of heart rate data and, specifically, used in the 
analysis of fetal heart rate. A short discussion of each of 
these measures follows. 
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Fractal Dimension 
. Fractal dimension is a means ot classifying structures 
that exhibit self-similarity. A self-similar structure can 
be described as follows. If a section of a self-similar 
structure is viewed in magnification, the smaller-scale 
section is similar in form to its larger-scale parent. As 
an example, consider a straight line segment which is broken 
into smaller segments with gaps between each segment. Then 
each segment is again broken into smaller segments with gaps 
between each of these segments, etc.. If the ratio of 
segment length to gap length is maintained, a self-similar 
structure is generated, and as each segment is viewed in 
magnification the same form is apparent. 
For self-similar structures the fractal dimension can 
be thought of as a measure of how the structure fills the 
space of the original structure. Generally, the 
self-similar structure will have a fractal dimension less 
than the dimension of the original structure. A continuous 
line segment has a dimension of one, while a self-similar 
line structure as described above will have a fractal 
dimension less than one. 
Consider the following larger dimensioned structures. 
A filled square has a dimension of two, since it completely 
fills a two-dimensional space. A self-similar square only 
partially filled with smaller self-similar squares will have 
a fractal dimension less than two, since the blank spaces 
between the squares are unfilled. Similarly, a solid cube 
has a dimension of three, since it completely fills a three 
dimensional space. A self-similar cube only partially 
filled with smaller self-similar cubes will have a fractal 
dimension less than three, 
are unfilled. 
since spaces between the cubes 
For self-similar structures the fractal dimension is a 
function of the logarithms of the number of repeated 
structures and the dimension of the original structure. 
Self-similar structures can be much more complicated 
than the examples given above. Line segments can branch 
into the two-dimensional plane creating a fractal dimension 
between one and two. Other three dimensional structures 
such as spheres may be nested to created fractal dimensions 
between two and three. 
Others (1) have illustrated the appearance of 
self-similarity in heart rate recordings of several minutes 
duration. Normant and Tricot (2) have suggested an approach 
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to the evaluation of fractal dimension of experimentally 
derived curves. Their method is presently under 
investigation in the Cardiovascular Laboratory. It may be 
used to determine the fractal dimension of heart rate data 
and other physiological variables of interest to this lab. 
Lyapunov Exponents 
Lyapunov exponents are used to obtain a measure of the 
sensitivity of a nonlinear dynamic system to its initial 
conditions. Nonlinear dynamic systems are often studied in 
the phase space in which trajectories appear. In the case of 
chaotic systems these trajectories nearly repeat themselves. 
Lyapunov exponents are also a measure of the average rate of 
divergence of these trajectories. As the order of a system 
increases the number of Lyapunov exponents necessary to 
characterize the system increases. 
For deterministic systems Lyapunov exponents are 
determined by taking the limit of the sum of the logarithms 
of the derivatives of the iterations of the trajectories 
through the phase space. This is a complicated procedure 
for deterministic systems, and much more difficult for 
systems \~ith unknown dynamics such as the cardiovascular 
system. In addition, since the order of the cardiovascular 
system is also unknown, the number of Lyapunov exponents 
necessary to describe it is also unknown. However, Wolf, 
et.al. (3) have suggested a method of determining Lyapunov 
exponents from experimental time series. Their method is 
also under study in the Cardiovascular Laboratory. 
Approximate Entropy (ApEn) 
Pincus (4) has proposed another measure in the ApEn. 
ApEn is a measure of the deviation of a time series away 
from a pattern or somewhat regularity. It is a measure of 
the increased information available in the time series with 
increased variability. The.algorithmic computation of ApEn 
is similar to the computation of Lyapunov exponents but much 
simpler. It involves the summation of the logarithms of the 
number of deviations of different magnitudes. Pincus has 
evaluated the level of fetal distress by determining the 
ApEn of fetal heart rate (4). This approach is also under 
consideration in the Cardiovascular Laboratory. 
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COMMENTS AND RECOMMENDATIONS 
This preliminary study indicates that a more detailed 
analysis of nonlinear chaos measures applied to 
cardiovascular physiology variables should be vigorously 
pursued. There is the potential for the development of a 
very useful tool in the evaluation of cardiovascular 
condition in these measures. The analysis of chaotic 
systems must make use of measures other than the traditional 
statistics. 
Present analyses indicate that these nonlinear chaos 
measures require time series on the order of 1000 data 
points in length for good resolution. For baseline data 
gathering situations time series of this length are easily 
obtained, but, for active tests such as LBNP ramp studies 
conducted both in the lab and in space, such time periods 
may not be practical. Techniques using shorter time 
intervals must be developed. 
The following recommendations are offered: 
1) The continued development of an analysis capability 
to be used in the evaluation of the three measures 
using Cardiovascular Laboratory data. 
2) Using relatively long term time series of heart rate 
data evaluated the three measures and compare with 
spectral analyses of the same data. To do this and 
associa.ted correlation studies an improved FFT 
should be incorporated. 
3) Investigate the usefulness of incorporating other 
cardiovascular variables, such as blood pressure, 
into the analyses. This could have the effect of 
changing trom a single function ot time to a vector 
function of time, and possibly reduce the long time 
series requirement. 
4) Consider variations of the nonlinear chaos measures 
to facilitate the use of shorter time series. This 
is of significant importance in the analysis of 
active LBNP ramp tests. 
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REFERENCES 
1. Goldberger, A.L., Rigney, D.R., and West, B.J., Chaos and 
Fractals in Human Physiology, Scientific American, 
February, 1990, vol. 262:42-48. 
2. Normant, F. and Tricot, C., Method for Evaluating the 
Fractal Dimension of Curves using Convex Hulls, Physical 
Review A, June, 1991, vol. 43:6518-6525. 
3. Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A., 
Determining Lyapunov Exponents from a Time Series, 
Physica D, 1985, vol. 16:285-317. 
4. Pincus, S.M., Hobbins, J.C., and Viscarello, R.P., ApEn: 
A Regularity Measure for Fetal Heart Beat Analysis, 
presented to the Society for Gynecologic Investigation, 
San Antonio, TX, March 20-23, 1991. 
10-7 


The Applicability of Nonlinear Systems Dynamics Chaos Measures to Cardiovascular Physiology Variables

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