U.S. patent number 6,144,877 [Application Number 09/132,462] was granted by the patent office on 2000-11-07 for determining the hurst exponent for time series data.
This patent grant is currently assigned to The United States of America as represented by the Department of Health and Human Services. Invention is credited to Paolo B. DePetrillo.
United States Patent |
6,144,877 |
DePetrillo |
November 7, 2000 |
Determining the hurst exponent for time series data
Abstract
Statistical information is determined for time series data of a
measurable activity. The time series data of the measurable
activity is obtained and comprises data elements representative of
the measurable activity. A Hurst exponent is determined from the
time series data and is the statistical information of the time
series data.
Inventors: |
DePetrillo; Paolo B. (Bethesda,
MD) |
Assignee: |
The United States of America as
represented by the Department of Health and Human Services
(Washington, DC)
|
Family
ID: |
22454172 |
Appl.
No.: |
09/132,462 |
Filed: |
August 11, 1998 |
Current U.S.
Class: |
600/515 |
Current CPC
Class: |
G06F
17/10 (20130101); A61B 5/318 (20210101) |
Current International
Class: |
G06F
17/10 (20060101); A61B 5/0402 (20060101); A61B
005/00 (); A61B 005/04 () |
Field of
Search: |
;600/515 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
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Experience in Diabetes," David J. Ewing et al., Diabetes Care, vol.
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.
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Lancet, p. 915, Oct. 17, 1987. .
"Sensitivity of R-R Variation and Valsalva Ratio in Assessment of
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et al., Diabetes Care, vol. 10, No. 6, pp. 735-741, Nov.-Dec. 1987.
.
"Heart rate variability: A measure of cardiac autonomic tone,"
Phyllis K. Stein et al., Am. Heart J., 127:1376-81, 1994. .
"On the fractal nature of heart rate variability in humans: effects
of data length and .beta.-adrenergic blockade," Yoshiharu Yamamoto
et al., Am. J. Physiol., 266:R40-R49, 1994. .
"Autonomic neurotoxicity of alcohol assessed by heart rate
variability," Katsuyuji Murata et al., Journal of the Autonomic
Nervous System, vol. 48, pp. 105-111, 1994. .
"On the fractal nature of heart rate variability in humans: effects
of vagal blockade," Yoshiharu Yamamoto et al., Am. J. Physiol.,
269:R830-R837, 1995. .
"Fractal Mechanisms and Heart Rate Dynamics," C-K. Peng et al.,
Journal of Electrocardiology, vol. 28 Supplement, pp. 59-65, 1995.
.
"Estrogen Control of Central Neurotransmission: Effect on Mood,
Mental State, and Memory," George Fink et al., Cellular and
Molecular Neurobiology, vol. 16, No. 3, pp. 325-336, 1996. .
"Cardiovascular Variability in Major Depressive Disorder and
Effects of Imipramine or Mirtazapine (Org 3770)," Journal of
Clinical Psychopharmacology, vol. 16, No. 2, pp. 135-145, 1996.
.
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Results from a prospective, randomized, crossover trial," Paul J.
Godin et al., Crit Care Med, vol. 24, No. 7., pp. 1117-1124, 1996.
.
"Fish Consumption, n-3 Fatty Acids in Cell Membranes, and Heart
Rate Variability in Survivors of Myocardial Infarction With Left
Ventricular Dysfunction," Jeppe Hagstrup Christensen et al., The
American Journal Of Cardiology, vol. 79, pp. 1670-1673, Jun. 15,
1997. .
"Dynamic Analysis of Heart Rate May Predict Subsequent Ventricular
Tachycardia After Myocardial Infarction," Timo H. Makikallio et
al., The American Journal Of Cardiology, vol. 80, pp. 779-783, Sep.
15, 1997. .
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Poon et al., Nature, vol. 389/2, pp. 492-495, Oct. 1997. .
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between heart rate variability, severity of illness, and outcome,"
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.
"Heart rate variability--A potential, noninvasive prognostic index
in the critically ill patient," Harold L. Kennedy, Crit Care Med,
vol. 26, No. 2, pp. 213-214, 1998..
|
Primary Examiner: Layno; Carl H.
Attorney, Agent or Firm: Venable Sartori; Michael A.
Claims
What is claimed is:
1. A method using a computer for determining statistical
information for time series data for a measurable activity, the
method comprising the steps of:
obtaining the time series data of the measurable activity, the time
of the series data comprising a plurality of data elements
representative of the measurable activity; and
using a computer to determine a Hurst exponent from the time series
data, the Hurst exponent being the statistical information of the
time series data.
2. A method according to claim 1, wherein the step of determining
the Hurst exponent comprises the step of:
determining a plurality of lengths L(r) for a plurality of sampling
resolutions r for a plurality of self-similar fractal curves for
the time series data.
3. A method according to claim 2, wherein the length L(r) and the
sampling resolution r are related by: ##EQU3## where the plurality
of data elements of the time series data are represented by S.sub.1
S.sub.2,S.sub.3, . . . , S.sub.N, r=1,2,3, . . . , M, I is the
integer part of (N-r-n+1)/r, and n.gtoreq.1 is an embedding
dimension.
4. A method according to claim 2, wherein the step of determining
the Hurst exponent further comprises the step of:
determining a linear relationship between log (r) and log
[L(r)/r].
5. A method according to claim 4, wherein the step of determining
the Hurst exponent further comprises the step of:
determining a slope of the linear relationship between log (r) and
log [L(r)/r].
6. A method according to claim 5, wherein the step of determining
the Hurst exponent further comprises the step of:
determining the Hurst exponent from the slope of the linear
relationship between log (r) and log [L(r)/r].
7. A method according to claim 1, wherein obtaining the time series
data comprises measuring the measurable activity to obtain the time
series data.
8. A method according to claim 1, wherein obtaining the time series
data comprises retrieving the time series data from a
computer-readable medium.
9. A method according to claim 1, wherein the measurable activity
is a biologically generated activity.
10. A method according to claim 1, wherein the time series data are
representative of measurements of heart rate variability.
11. A method according to claim 1, further comprising the step
of:
detecting a biological condition using the Hurst exponent.
12. A method according to claim 11, wherein the biological
condition is alcoholism.
13. A method according to claim 11, wherein the biological
condition is related to a change in serotonin activity in a
biological subject.
14. A method according to claim 11, wherein the biological
condition is related to the heart of a biological subject.
15. A method according to claim 11, wherein the biological
condition is a psychiatric disorder.
16. An apparatus for determining statistical information for time
series data for a measurable activity, comprising:
means for obtaining the time series data of the measurable
activity, the time series data comprising a plurality of data
elements representative of the measurable activity; and
means for determining a Hurst exponent from the time series data,
the Hurst exponent being the statistical information of the time
series data.
17. An apparatus according to claim 16, wherein said means for
determining the Hurst exponent comprises:
means for determining a plurality of lengths L(r) for a plurality
of sampling resolutions r for a plurality of self-similar fractal
curves for the time series data.
18. An apparatus according to claim 17, wherein the length L(r) and
the sampling resolution r are related by: ##EQU4## where the
plurality of data elements of the time series data are represented
by S.sub.1,S.sub.2,S.sub.3, . . . , S.sub.N, r= 1,2,3, . . . , M, I
is the integer part of (N-r-n+1)/r, and n.gtoreq.1 is an embedding
dimension.
19. An apparatus according to claim 17, wherein said means for
determining the Hurst exponent further comprises:
means for determining a linear relationship between log (r) and log
[L(r)/r].
20. An apparatus according to claim 19, wherein said means for
determining the Hurst exponent further comprises:
means for determining a slope of the linear relationship between
log (r) and log [L(r)/r].
21. An apparatus according to claim 20, wherein said means for
determining the Hurst exponent further comprises:
means for determining the Hurst exponent from the slope of the
linear relationship between log (r) and log [L(r)/r].
22. An apparatus according to claim 16, wherein said means for
obtaining the time series data comprises means for measuring the
measurable activity to obtain the time series data.
23. An apparatus according to claim 16, wherein said means for
obtaining the time series data comprises means for retrieving the
time series data from a computer-readable medium.
24. An apparatus according to claim 16, wherein the measurable
activity is a biologically generated activity.
25. An apparatus according to claim 16, wherein the time series
data are representative of measurements of heart rate
variability.
26. An apparatus according to claim 16, further comprising:
means for detecting a biological condition using the Hurst
exponent.
27. An apparatus according to claim 26, wherein the biological
condition is alcoholism.
28. An apparatus according to claim 26, wherein the biological
condition is related to a change in serotonin activity in a
biological subject.
29. An apparatus according to claim 26, wherein the biological
condition is related to the heart of a biological subject.
30. An apparatus according to claim 26, wherein the biological
condition is a psychiatric disorder.
31. A computer-readable medium embodying code segments for
determining statistical information for time series data for a
measurable activity, comprising:
code segments for obtaining the time series data of the measurable
activity, the time series data comprising a plurality of data
elements representative of the measurable activity; and
code segments for determining a Hurst exponent from the time series
data, the Hurst exponent being the statistical information of the
time series data.
32. A computer-readable medium according to claim 31, wherein said
code segments for determining the Hurst exponent comprise:
code segments for determining a plurality of lengths L(r) for a
plurality of sampling resolutions r for a plurality of self-similar
fractal curves for the time series data.
33. A computer-readable medium according to claim 32, wherein the
length L(r) and the sampling resolution r are related by: ##EQU5##
where the plurality of data elements of the time series data are
represented by S.sub.1,S.sub.2,S.sub.3, . . . , S.sub.N, r=1,2,3, .
. . , M, I is the integer part of (N-r-n+1)/r, and n.gtoreq.1 is an
embedding dimension.
34. A computer-readable medium according to claim 32, wherein said
code segments for determining the Hurst exponent further
comprise:
code segments for determining a linear relationship between log (r)
and log [L(r)/r].
35. A computer-readable medium according to claim 34, wherein said
code segments for determining the Hurst exponent further
comprise:
code segments for determining a slope of the linear relationship
between log (r) and log [L(r)/r].
36. A computer-readable medium according to claim 35, wherein said
code segments for determining the Hurst exponent further
comprise:
code segments for determining the Hurst exponent from the slope of
the linear relationship between log (r) and log [L(r)/r].
37. A computer-readable medium according to claim 31, wherein said
code segments for obtaining the time series data comprise code
segments for measuring the measurable activity to obtain the time
series data.
38. A computer-readable medium according to claim 31, wherein said
code segments for obtaining the time series data comprise code
segments for retrieving the time series data from a
computer-readable medium.
39. A computer-readable medium according to claim 31, wherein the
measurable activity is a biologically generated activity.
40. A computer-readable medium according to claim 31, wherein the
time series data are representative of measurements of heart rate
variability.
41. A computer-readable medium according to claim 31, further
comprising:
code segments for detecting a biological condition using the Hurst
exponent.
42. A computer-readable medium according to claim 41, wherein the
biological condition is alcoholism.
43. A computer-readable medium according to claim 41, wherein the
biological condition is related to a change in serotonin activity
in a biological subject.
44. A computer-readable medium according to claim 41, wherein the
biological condition is related to the heart of a biological
subject.
45. A computer-readable medium according to claim 41, wherein the
biological condition is a psychiatric disorder.
Description
FIELD OF THE INVENTION
The invention relates to a method, an apparatus, and a
computer-readable medium for determining statistical information
for time series data and for detecting a biological condition of a
biological system from the statistical information.
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For the convenience of the reader, the publications referred to in
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BACKGROUND OF THE INVENTION
Measures of heart rate variability (HRV) have been shown to be a
powerful means of assessing the influences of autonomic tone on
cardiac function (8). However, because some methodological problems
exist in obtaining reliable estimates of this measure, the
practical application of HRV is limited, and the interpretation of
such results is confounded. For example, in time domain analysis,
measures of dispersion, such as the standard deviation, increase
with increasing data length, making cross-study comparisons
difficult. Further, dispersion measures also do not take into
account the degree of temporal autocorrelation.
Conventionally, problems due to autocorrelations can be avoided by
conducting the analysis in the frequency domain using a fast
Fourier transform. However, this method assumes that for the epoch
investigated, the time series remains stationary. This assumption
is less likely to hold true as longer time intervals are sampled.
On the other hand, short data lengths are also problematic because
the contributions of low frequencies to the overall power spectrum
cannot properly be estimated. Consequently, the outcomes of
applying time domain analysis or frequency domain analysis are not
easily interpreted when the recording time of the HRV is relatively
short, such as less than five minutes, or when the length of the
recorded time series varies significantly between individuals.
Previous work suggests a strong correlation between alcohol
dependence and altered heart rate dynamics. Heart rate dynamics are
usually estimated using parameters obtained from time-domain or
frequency-domain analyses (11, 12). Results obtained with these
methods are confounded by the changing statistical properties of
heartbeat interval time series over time, called non-stationary
signals. These signals are difficult to interpret with dispersional
measures in the time-domain, such as the standard deviation,
because the results are not stable with increasing data length.
These measures also provide no information regarding the internal
dynamics of the time series. Frequency-domain measures rely on
assumptions of stationarity that are not met with interbeat
interval (IBI) time series data, especially with longer recording
times.
SUMMARY OF THE INVENTION
It is an object of the invention to determine the Hurst exponent
from time series data.
It is another object of the invention to detect a biological
condition of a biological subject by determining the Hurst exponent
from time series data measured from the biological subject.
It is a further object of the invention to determine the Hurst
exponent for an IBI time series from a human subject.
It is still another object of the invention to determine the Hurst
exponent from a small number of data points in a time series.
The above objects and advantages of the present invention are
achieved by a method, an apparatus, and an article of manufacture
for determining the Hurst exponent for time series data.
The method of the invention includes a method for determining
statistical information for time series data for a measurable
activity, the method comprising the steps of: obtaining the time
series data of the measurable activity, the time series data
comprising a plurality of data elements representative of the
measurable activity; and determining a Hurst exponent from the time
series data, the Hurst exponent being the statistical information
of the time series data.
The apparatus of the invention includes an apparatus for
determining statistical information for time series data for a
measurable activity, comprising: means for obtaining the time
series data of the measurable activity, the time series data
comprising a plurality of data elements representative of the
measurable activity; and means for determining a Hurst exponent
from the time series data, the Hurst exponent being the statistical
information of the time series data.
The article of manufacture of the invention includes a
computer-readable medium embodying code segments for determining
statistical information for time series data for a measurable
activity, comprising: code segments for obtaining the time series
data of the measurable activity, the time series data comprising a
plurality of data elements representative of the measurable
activity; and code segments for determining a Hurst exponent from
the time series data, the Hurst exponent being the statistical
information of the time series data.
The apparatus of the invention includes a computer programmed with
software to operate the computer in accordance with the invention.
Examples of "computer" include: a general purpose computer; an
interactive television; a hybrid combination of a general purpose
computer and an interactive television; and any apparatus
comprising a processor, memory, the capability to receive input,
and the capability to generate output.
The article of manufacture of the invention comprises a
computer-readable medium embodying code segments to control a
computer to perform the invention. Examples of a "computer-readable
medium" include: a magnetic hard disk; a floppy disk; an optical
disk, such as a CD-ROM or one using the DVD standard; a magnetic
tape; a memory chip; a carrier wave used to carry computer-readable
electronic data, such as those used in transmitting and receiving
electronic mail or in accessing a network, such as the Internet or
a local area network ("LAN"); and any storage device used for
storing data accessible by a computer. Further, examples of "code
segments" include: software; instructions; computer programs; or
any means for controlling a computer.
Moreover, the above objects and advantages of the invention are
illustrative, and not exhaustive, of those which can be achieved by
the invention. Thus, these and other objects and advantages of the
invention will be apparent from the description herein or can be
learned from practicing the invention, both as embodied here and as
modified in view of any variations which may be apparent to those
skilled in the art.
BRIEF DESCRIPTION OF THE DRAWINGS
Embodiments of the invention are explained in greater detail by way
of the drawings.
FIG. 1 illustrates a flow diagram for the invention.
FIG. 2 illustrates five time series data for various values of the
Hurst experiment.
FIG. 3 illustrates a phase-space plot for an IBI of a human
subject.
FIG. 4 illustrates a log (r) v. log [L(r)/r] plot for the IBI time
series taken from subject 6 in Table 1.
FIG. 5 illustrates a log (r) v. log [L(r)/r] plot for a
computer-generated IBI times series.
FIG. 6 illustrates the results from varying the data length using
the invention.
FIG. 7 illustrates the experimental results obtained from
determining the Hurst exponent for human subjects with their eyes
closed.
FIG. 8 illustrates the experimental results obtained from
determining the Hurst exponent for human subjects with their eyes
open.
FIG. 9 illustrates an apparatus and an article of manufacture for
implementing the invention.
DETAILED DESCRIPTION OF THE INVENTION
The Hurst exponent H is a measure of the dynamics of a time series,
as conceived by Hurst (3) and formalized by Mandelbrot (4). In
particular, H is a measure of the persistence, or anti-persistence,
of trends within serial measurements. The value of H varies between
0 and 1, namely 0.ltoreq.H.ltoreq.1. As H approaches 1, increasing
or decreasing trends in consecutive data points in a time series
are reinforced, and the time series is said to be persistent.
Consequently, a time series having an H value close to 1 shows
relatively little variation between consecutive data samples.
Conversely, as H approaches 0, positive and negative trends
alternate rapidly, and the time series is said to be
anti-persistent. In this extreme, the absolute differences in
values between consecutive time points of the time series tend to
be large. A time series of the increments between adjacent values
tends to show a linear relationship between the power spectrum and
frequency as H approaches 0. As H approaches the value of 0.5, the
correlation between consecutive elements of the time series
vanishes, displaying the characteristics of (one-dimensional)
Brownian motion. Mandelbrot (5) formally related H to fractional
Brownian motion, which made the Hurst exponent available as a
descriptor of the scaling behavior of fractal curves, called
self-similarity.
FIG. 1 illustrates a flow diagram for the invention. In block 1,
time series data is obtained. For example, the time series data can
be obtained from measuring a measurable activity, such as measuring
the heartbeat of a biological subject. The biological subject can
be, for example, a human subject. The time series can also be
obtained by retrieving the time series data from a
computer-readable medium. Another example of a time series is an
IBI time series, which can be extracted from a heartbeat time
series.
In block 2, using the time series data, the lengths L(r) for a
number of self-similar fractal curves are determined for various
values of r up to a maximum value of M and for an embedding
dimension n. The length of a self-similar fractal curve is given by
the following relationship (9):
where the length L(r) is measured as the Euclidean distance between
sampled values of the time series, r is the sampling resolution or
"yardstick size" and is an integer greater than 0 (i.e., r>0),
and D is the similarity dimension and 0.ltoreq.D.ltoreq.2. When
r=1, L is computed using every sample value of the time series and
reaches a maximum length L.sub.max. For r=2, every second sample is
used in the length computation, and L<L.sub.max because "peaks"
and "valleys" of width <2 do not contribute to the total length.
With increasing values of r, progressively wider local maxima and
local minima are disregarded and, if the curve is self-similar, the
total length decreases according to equation (1).
Before the time series can be scaled, the time series must be
embedded in n-dimensional space as an n-dimensional curve. For an
embedding dimension of n, the coordinates of each point within the
so-called phase-space are determined by the values of the n
consecutive samples of the time series. When n=3, each sample of
the time series is represented in three-dimensional space as a
point P=(x,y,z)=(S.sub.i,S.sub.i+1,S.sub.i+2). The x-axis
represents the sample S.sub.i of the time series
T(S.sub.1,S.sub.2,S.sub.3, . . . ,S.sub.i, . . . ,S.sub.N), and the
y-axis and z-axis represent the magnitude of the phase advanced
elements of the time series, namely, time+1 and time+2,
respectively.
FIG. 3 illustrates a phase-space plot for n=2 of an IBI time series
of 1000 data points taken from a healthy human subject during mild
exertion, namely, walking. The coordinates on the x-axis represent
the original time series in milliseconds. The coordinates on the
y-axis represent the phase-advanced value of the time series, also
in milliseconds. Consecutive points are connected by a straight
line.
Scaling of the time series is accomplished by computing the sum of
the lengths of all vectors in n-space that connect each succeeding
point on the n-dimensional curve. For any embedding dimension, the
geometric length L of the curve can be determined for any sampling
resolution r as L(r). When r=1, L(r) is calculated by the summation
of the geometric length of the curve joining every point in
phase-space. When r=2, L(r) is calculated by summing the geometric
length of the curve joining every other point in phase space. This
process is continued for increasing values of r to yield a
relationship between L(r) and r. For succeeding computations with
different r, a new curve is generated, connecting points along the
original curve, but sampled at a lower resolution.
In general, the relationship between the length L(r) and the
sampling resolution r is given by: ##EQU1## where the time series
data is represented by S.sub.1,S.sub.2,S.sub.3, . . . ,SN, r=1,2,3,
. . . ,M, I is the integer part of (N-r-n+1)/r, n.gtoreq.1 is an
embedding dimension, N is the number of data points of the time
series vector T(S.sub.1,S.sub.2,S.sub.3, . . . ,S.sub.i, . . .
,S.sub.N) and M is the maximum number of scales. Thus, in block 2,
for an embedding dimension of n, equation (2) can be used to
determine L(r) for various r for a times series of N data
points.
For the embedding dimension of n=3, the value of L(r) for the
three-dimensional phase-space is given by: ##EQU2## where the time
series data is represented by S.sub.1,S.sub.2,S.sub.3, . . .
,S.sub.N, r=1,2,3, . . . ,M, and I is the integer part of
(N-r-n+1)/r.
For equations (2) and (3), the choice of M should be dictated by
the length of the time series. The inventor has discovered that the
relationship between log (r) and log [L(r)/r] remains relatively
linear when M<(N).sup.1/2. If the M is set too high, some
deviation from linearity will occur at higher values of r in the
relationship, which will bias the results.
Since each point in the phase-space contains information with
respect to adjacent elements of the time series, the choice of the
embedding dimension depends on the dynamics of the time series. In
general, the best choice of the embedding dimension corresponds to
the largest number of elements which encompass the "memory" of the
time series. For a true fractional Brownian process, this is known
to be infinite (5). However, for a particular physiological system,
the memory of the process appears to degrade rapidly, so that the
appropriate embedding dimension n can be experimentally determined
by testing for convergence of the slope of the log--log plot of r
vs. L(r)/r. Choosing higher embedding dimensions than is required
by the dynamics of the time series exponentially increases
computing time, while not significantly altering the final value of
D, and therefore H.
In block 3, a linear relationship between log (r) and log [L(r)/r]
is determined. The log transformation of equation (1) yields:
where r>0. Equation (4) has the form of a linear equation with
y-axis intercept log (L.sub.max) and a slope of -D. Hence, if the
time series exhibits self-similar scaling behavior, a plot of log
(r) vs. log [L(r)/r] will result in a straight line with a y-axis
intercept of log (L.sub.max) and a slope of -D. To determine the
linear relationship in equation (4), a linear regression with a
least mean squares fit of the log (r) vs. log [L(r)/r] plot can be
performed.
In block 4, the slope -D of the linear relationship from block 3 is
determined.
In block 5, the Hurst exponent H is determined from the slope -D of
block 4. The similarity dimension D of the curve is related to H by
the following equation:
FIG. 9 illustrates an apparatus and an article of manufacture for
implementing the invention. As discussed above, the computer in
accordance with the invention includes a computer 11 programmed
with software to operate the computer in accordance with the
invention, an embodiment of which is depicted in the flow diagram
of FIG. 1. Further, the article of manufacture of the invention
includes a computer-readable medium 12 embodying code segments to
control the computer 11 to perform the invention, an embodiment of
which is depicted in the flow diagram of FIG. 1. In addition, FIG.
9 illustrates the computer 11 for obtaining time series data from a
human subject 13 and for obtaining time series data by retrieving
the time series data from the computer-readable medium 12.
TESTING THE INVENTION
FIG. 2 illustrates five cardiac IBI time series for various Hurst
exponents. The y-axes represent the IBI time in milliseconds (ms),
and the x-axis represents the beat number. The time series A to D
were synthesized and correspond to the H value shown on the right
hand side of the figure. The time series E was taken from data from
a human subject, and the H value was derived using the
invention.
The synthetic IBI time series A to D shown in FIG. 1 were computed
using a spectral synthesis method (19) from the Time-Series
Statistical Analysis System TSAS 3.01.01b, written by Yoshiharu
Yamamoto, Ph.D., Lab. For Exercise Physiol. & Biomechanics,
Grad. School of Education, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113, Japan, and were compiled and run on a DEC
3000/600S AXP (Digital Equipment Corp., Maynard, Mass.) This module
conveniently allows varying the length, mean, standard deviation,
and spectral characteristics of the resulting time series using the
spectral synthesis method.
Inspecting time series D with H=1.0 in FIG. 1, an overall relative
smoothness can be seen. As H approaches 0, trends are more rapidly
reversed, as shown in time series A, where large variations occur
between adjacent values, which give the time series A an irregular
look. When H=0.5, as shown in time series C, the magnitudes of the
sequential points of the time series are independent and
uncorrelated, and time series C can be considered a random walk.
Thus, H values approaching 0.5 from either extreme are symptomatic
of a breakdown in the long-range correlations of the IBI time
series.
The four time series A to D illustrated in FIG. 2 were generated to
have the same mean and standard deviation as the IBI time series
obtained from the healthy human subject, illustrated in time series
E. In particular, the mean and standard devication are
1286.97.+-.63.19 ms. As shown in time series E, the time series
from the healthy human subject had an associated H value of
0.16.+-.-0.05. By visual inspection, the "roughness" of time series
E best matches the synthesized IBI series from time series B with
H=0.16.
The invention was tested using electrocardiogram (EKG) data from
nine healthy human subjects. Results for the calculation of H
derived from equation (5) are shown in Table 1. In Table 1, the
first column lists the subject identification number, the second
column lists the determined H and its standard error (H.+-.SEM),
the third column lists the maximum value of the scale (M), the
fourth column lists the square of the correlation coefficient
(R.sup.2), and the fifth column lists the mean of the IBI and its
standard deviation in milliseconds (mean.+-.SD). In Table 1,
subject 4 was engaged in moderately heavy exercise, while subject 5
was examined during very heavy exercise (entry 5a), and at rest
(entry 5b). The values of H obtained for resting subjects 1, 2, 3,
5a, 7, and 8 are consistent with values of H obtained by
coarse-graining spectral analysis (CGSA) (9). The higher value of H
seen in subject 5b is consistent with increased H values obtained
during exercise (10).
TABLE 1 ______________________________________ Results of Analysis
of Human Subject EKGs Subject ID H .+-. SEM M R.sup.2 mean .+-. SD
______________________________________ 1 0.07 .+-. 0.02 31 0.998
1034.33 .+-. 91.32 2 0.14 .+-. 0.02 31 0.997 984.27 .+-. 94.54 3
0.08 .+-. 0.02 31 0.998 1092.88 .+-. 107.40 4 0.16 .+-. 0.02 41
0.995 578.36 .+-. 44.56 (moderate exercise) 5a (rest) 0.09 .+-.
0.02 31 0.996 1219.29 .+-. 73.98 5b 0.47 .+-. 0.03 20 0.995 289.60
.+-. 106.60 (heavy exercise) 6 0.164 .+-. 0.005 100 0.999 858.87
.+-. 191.94 7 0.148 .+-. 0.005 100 0.999 741.47 .+-. 87.55 8 0.156
.+-. 0.003 100 0.999 913.69 .+-. 129.84
______________________________________
FIG. 4 illustrates a plot of log (r) vs. log [L(r)/r] calculated
from a set of 98,355 IBI data points obtained from subject 6 with
embedding dimension n=3 and scales M=100. The linear slope of the
plot suggests that a power-law relationship is present, supporting
the assumption of self-similarity of the IBI time series for the
range of scales investigated. Similar analyses of data sets from
the other subjects resulted in qualitatively similar straight line
plots of log (r) vs. log [L(r)/r].
The invention was also tested using synthesized IBI time series
data. The synthetic IBI time series were computed using a spectral
synthesis method (1) from the TSAS 3.01.1b, and were compiled and
run on a DEC 3000/600S AXP.
The invention was tested using a computer-generated IBI time series
with mean IBI=1000 ms, standard deviation SD=200, and data length
N=1000. The spectral slope in the synthesis was set to two values:
0 and 2. L(r) was determined using equation (3) with an embedding
dimension of n=3 and a scale of M=31, and D was determine using
equation (4). FIG. 5 illustrates the log (r) vs. log [L(r)/r] plot
for the two synthesized IBI time series with the same means and
standard deviations but differing spectral slopes. The circles and
squares represent points determined from time series generated
using the spectral synthesis method with .beta.=0 and .beta.=-2,
respectively, where .beta. refers to the spectral slope in the
generating function. Respective values for D and its standard error
for regression are 2.01.+-.0.02 and 1.53.+-.0.01. FIG. 5
illustrates that the approximate expected results are obtained for
D=2 and H=0 for the time series with a spectral slope .beta.=0, and
for D=1.5 and H=0.5 for the time series with a spectral slope
.beta.=-2.
The determination of H by the invention was tested by varying the
statistical moments of the time series while holding the spectral
noise characteristics steady. Several time series of data length
N=1000 were generated, and the statistical moments were varied
while maintaining the spectral slope at 0 or 2. Further, D was
calculated using a scale of M=31, and an embedding dimension of
n=3. In Table 2, the first and second columns list the mean and
standard deviation, respectively, in milliseconds for IBI, the
third column lists the coefficient of variation (CV), the fourth
column lists computed values of H when H=0, and the fifth column
lists computed values of H when H=0.5.
TABLE 2 ______________________________________ Effect of signal
characteristics on computed H-values Mean IBI (ms) SD (ms) CV H = 0
H = 0.5 ______________________________________ 1034.33 91.32 0.088
0.00 0.47 834.33 73.66 0.088 0.00 0.47 634.33 56.01 0.088 0.00 0.45
1034.33 182.65 0.177 0.00 0.42 1034.33 365.30 0.354 0.01 0.45
______________________________________
In Table 2, the results show good agreement with theoretical values
of H=0 for a spectral slope .beta.=0, and H=0.5 for a spectral
slope .beta.=-2. These results suggest that the invention is
insensitive to mean and standard deviation differences of the time
series. For H=0.5, a bias existed in that an underestimation of the
true value occurred with an embedding dimension of n=3.
To determine the effect of the embedding dimension n on the
invention, the embedding dimension n was varied, and H was
calculated. The results are shown in Table 3. In Table 3, the first
column lists the six embedding dimensions used, the second column
lists H and its standard error (H.+-.SEM) for a synthesized time
series with a spectral slope of .beta.=0 and an expected H=0, the
third column lists H and its standard error (H.+-.SEM) for a
synthesized time series with a spectral slope of .beta.=2 and an
expected H=0.5, and the fourth column lists H and its standard
error (H.+-.SEM) for a human subject. The second, third, and fourth
columns in Table 3 were calculated for N=1000 IBI data points,
M=31, and a mean IBI and its standard deviation of 864.29.+-.76.39
ms.
TABLE 3 ______________________________________ Dependence of H on
Embedding Dimension H .+-. SEM Embedding Expected Expected Human
Dimension n H = 0.00 H = 0.50 Subject
______________________________________ 6 0.00 .+-. 0.01 0.49 .+-.
0.01 0.29 .+-. 0.03 5 0.00 .+-. 0.01 0.48 .+-. 0.01 0.29 .+-. 0.03
4 0.01 .+-. 0.01 0.46 .+-. 0.01 0.29 .+-. 0.03 3 0.00 .+-. 0.01
0.46 .+-. 0.01 0.29 .+-. 0.03 2 0.01 .+-. 0.02 0.45 .+-. 0.01 0.30
.+-. 0.03 1 0.02 .+-. 0.03 0.43 .+-. 0.02 0.31 .+-. 0.03
______________________________________
As can be seen in Table 3 for the synthetic data in the second and
third columns, a higher embedding dimension is required to achieve
less bias in the estimation of H. This is expected because the
synthetic data has a longer "memory" and is thus closer to an ideal
fractal. However, for the human subject data in the fourth column,
an embedding dimension of n=3 was sufficient.
When the embedding dimension is n>1, the scaling behavior of the
time series with respect to phase, as well as time, can be
examined. Given the results in Table 3, it appears that the
influence of the magnitude of previous IBIs on a particular IBI is
exerted within 2 to 3 heartbeats, at least within the data lengths
examined in the present study of up to 1197 data points. Since no
human EKG data that was analyzed resulted in H>0.5, and since
EKG data had values of 0.01<H<0.5, an embedding dimension of
n=3 is preferably used for calculating H in human subjects.
The effects of varying data length N on the determination of H was
tested by synthesizing a set of time series by varying H from 0 to
1 in increments of 0.25 with a mean of 1000 and a standard
deviation of SD=200. For each value of H and N, five time series
and the mean and standard deviation of the estimated value of H
were determined. The embedding dimension was n=6. To avoid problems
with bias inherent in the spectral synthesis method, only the
initial N data points of the generated time series were used for
the analysis, and the total length of the original time series was
10N. The results are shown in FIG. 6. The values represented on the
y-axis show the calculated values of H, and the values of H shown
on the x-axis are those used to generate the time series. The bar
plots show the mean values of H for each of the time series, and
the error bars represent their standard deviations.
For short series with N.gtoreq.1000 points, the estimate of H
obtained using the invention is more accurate than estimates
obtained with other regression-based methods, including the power
spectral density, discrete wavelet transform, and dispersive
analysis methodologies (4), (2). With N.gtoreq.500 points, H
converges reasonably to the expected values determined by the
generating algorithm.
Neither differences in the means and standard deviations of the
signal nor variations significantly alter the estimated value of H.
These strengths of the invention allow the estimation of H from
relatively short data sets, such as might be obtained from 3 to 5
minutes of EKG recordings. More importantly, data sets from
different subjects or different physiological states of a subject
can be compared and contrasted because, by using the invention,
these data sets will be relatively independent of differences in
data collection time.
An important implication of the invention relates to the problem of
measuring the continually changing underlying dynamics of many
biologically generated time series. Since any estimation of H must,
of necessity, be a time averaged value, the ability of the
invention to discriminate variations in the dynamics of time series
within resolutions of 100-500 data points is an advantage in the
study of biological systems.
The characterization of biological signals, such as heart rate
variability, can be improved by assessing the internal underlying
dynamics of such time series at relatively high resolutions.
Studying changes of the dynamic properties using the invention can
provide insight into the generation of deterministic chaos by the
interplay of oscillatory homeostatic feedback loops, as they appear
to operate in the generation of IBI time series (6).
With the invention, a proper embedding dimension n must be chosen
for the time series. In part, the choice of the embedding dimension
is dependent on the complexity of the underlying dynamics, which
may change with time. The choice is straightforward for a
relatively short time series, since the embedding constant can be
incrementally changed and H calculated after each increment. At
some point, further increases in the embedding dimension do not
appreciably change the value of H, as shown in Table 3. For a
longer time series, it is possible that epochs with very complex
internal dynamics might co-exist and be contiguous to epochs with
low complexity. H could thus be overestimated or underestimated if
an "average" embedding dimension were chosen based on the results
of the whole time series. As discussed above, an embedding
dimension of n=3 is preferably used for human subjects. However, as
noted, minimal experimentation with a particular time series data
set may result in a more optimal embedding dimension.
In the case of a synthetic time series with an "infinite" memory,
the choice of a low value for the embedding dimension may result in
a bias in the estimation of H. As shown in Table 3, H tended to be
underestimated for time series with H close to 0.50, unless the
embedding dimension was set to at least 6.
Further, with the invention, H can be calculated in real-time from
a stream of input time series data, including biologically
generated time series data, such as IBI time series data. This
permits the immediate examination of the internal dynamics of any
input time series.
Detecting Biological Conditions Using the Invention
The invention can be used to detect a variety of biological
conditions. First, alcohol dependence and gender are discussed, and
then other types of biological conditions are described for
detection using the invention.
The inventor has discovered that determining H from IBI time series
data can be used to detect biological conditions by providing a
measure of cardiac signal complexity in relationship to the
biological condition, such as alcohol dependence or gender. With
the invention, the internal dynamics of an IBI time series are
understood by determining H. Further, with the invention, a better
representation of the internal dynamics of a system can be obtained
than from simple dispersional measures, such as mean and standard
deviation determinations. Moreover, with the invention, a special
transform of the IBI data is not required prior to analysis, and
short lengths of IBI data with differing statistical
characteristics can be compared.
The invention can be used to determine H using IBI time series data
obtained from human subjects stratified based on gender and the
presence or absence of alcohol dependence. Decreases in HRV,
defined as beat-to-beat variations of human heartbeat intervals,
measured both in the frequency domain and the time domain, are
associated with higher values of H (15). The inventor has
discovered that a higher H value characterizes EKG data from
alcoholic subjects as compared to controls. This discovery is
supported by previous studies in this population group that have
shown decreases in HRV measures in alcoholic subjects compared to
non-alcoholic subjects (16).
An experimental study using the invention was approved by the
Institutional Review Board at the National Institutes of Health,
and was carried out with written consent from the subjects.
Potential subjects were evaluated using the SADS (Schedule for
Affective Disorders and Schizophrenia) and the SCID (Structured
Clinical Interview for DSM-III-r) (17). Subjects were instructed
not to consume any prescription or non-prescription drug for 24
hours prior to the start of testing. Urine toxicology testing was
performed, and blood alcohol concentrations were estimated using a
breathalyzer. Subjects who were positive for drugs of abuse or
alcohol were excluded.
From the surrounding community, 120 volunteer subjects were
obtained and evaluated. Two groups of subjects were abstracted from
this study population. Subjects with a primary diagnosis of alcohol
dependence, and no other Axis I or Axis II disorders were
designated for the alcoholic group. Healthy comparison subjects
without evidence of Axis I or Axis II disorders were placed in the
control group. The final study group consisted of 61 subjects, 13
who met criteria for alcohol dependence (6 females and 7 males,
mean age.+-.SEM of 40.0.+-.3.8 years), and 48 healthy comparison
subjects (33 females and 15 males, mean age.+-.SEM of 40.3.+-.2.7
years). Racial composition of the study subjects included 48
Caucasians and 13 African-Americans, proportionally distributed
across gender and alcohol groups.
Subjects were seated in a comfortable, quiet room monitored through
a closed circuit video camera. On each subject, two silver-silver
chloride electrodes were attached laterally, along the mid-axillary
line, beneath the last rib. The electrodes were referenced to each
other. Pre-testing impedance was measured at 5K ohms or below. The
EKG signals were amplified via 12 A amplifiers (Grass, Quincy,
Mass.) with standard amplification filters set between 0.01 and 100
Hz. Data was calibrated against an average 100 uV 5 Hz sine wave
and digitized with the use of a 12-bit A-to-D converter (Data
Translation, Marlboro, Mass.).
Subjects were asked to minimize movement while EKG data was
recorded for approximately 7 minutes. For the first approximately
3.5 minutes of data collection, the subject was asked to keep his
or her eyes open. For the second approximately 3.5 minutes of data
collection, the subject was asked to keep his or her eyes
closed.
Data was analyzed in blind fashion, both during the EKG filtering
process, as well as during the determination of H. The code
relating the subjects and the results was broken after all data was
processed and ready for statistical analysis.
The digitized EKG data was mathematically filtered with a band pass
filter to reduce noise-to-signal ratio. A computer program, written
in C (Borland, Scotts Valley, Calif.) was used to extract the IBIs
from the data set. The IBIs were then manually checked for
anomalies. The criteria for the admissibility for anomalies in the
EKG data was extremely stringent. Corrections were made only if the
irregular IBI differed from the mean IBI by a factor of 2. Abnormal
beat-to-beat intervals were corrected by removing beats, depending
on the situation. Out of the 61 subjects, each with approximately
200 data points per condition, only four anomalous readings were
removed.
Using the invention, the IBI time series data was processed, as
previously discussed with FIG. 1, with an embedding dimension of
n=3 and a maximum value of r as M=10. Linear regression, using a
least mean square fit of the log (r) vs. log [L(r)/r]plot was
performed, and the slope of -D was used to derive H according to
the relationship in equation (3). For all linear regressions,
R.sup.2 .gtoreq.0.98.
The value of H was compared and contrasted for the alcoholic and
control groups using a two-factor (alcohol status and gender)
parametric analysis of variance (ANOVA) for each condition (eyes
open or eyes closed). Using the parameters derived from the ANOVA,
group means were explored for significant differences using a
Bonferroni/Dunn correction. The significance level was set to
p>0.05.
Upon examining the experimental results, there were no significant
differences in mean IBI between the two groups under each
experimental condition, as shown in Table 4 below. In Table 4, the
first column lists the two groups, the second column lists the two
experimental conditions, the third column lists the count for the
number of subjects, and the fourth and fifth columns list the mean
IBI and the standard deviation (SD) in milliseconds,
respectively.
TABLE 4 ______________________________________ Mean IBI .+-.
Standard Deviation by Group and Condition Group Condition Count
Mean IBI (ms) SD ______________________________________ Alcoholic
Open Eyes 13 922.60 153.52 Alcoholic Closed Eyes 13 946.57 143.52
Control Open Eyes 48 939.62 133.39 Control Closed Eyes 48 949.83
132.00 ______________________________________
As shown in FIGS. 7 and 8, both alcohol use and male gender are
correlated with higher values of H. In FIG. 7, the values of H
obtained in the closed eyes condition are plotted by group,
alcoholic vs. control, as indicated in the legend and segregated by
gender: women (F) and men (M). The error bars above the columns
represent the standard error (SEM). In FIG. 8, the values of H
obtained in the open eyes condition are plotted by group, alcoholic
vs. control, as indicated in the legend and segregated by gender:
women (F) and men (M). The error bars above the columns represent
SEM.
Table 5 lists the values of H and its standard error (H.+-.SEM)
obtained from each sub-group under the two experimental conditions
in the experimental study.
TABLE 5
__________________________________________________________________________
H .+-. SEM Alcoholic Control Female Male All Female Male All
Condition (n = 6) (n = 7) (n = 13) (n = 33) (n = 15) (n = 48)
__________________________________________________________________________
Closed Eyes 0.09 .+-. 0.07 0.25 .+-. 0.04 0.18 .+-. 0.05 0.07 .+-.
0.02 0.13 .+-. 0.03 0.09 .+-. 0.02 Open Eyes 0.12 .+-. 0.09 0.21
.+-. 0.04 0.17 .+-. 0.05 0.05 .+-. 0.02 0.11 .+-. 0.02 0.07 .+-.
0.02
__________________________________________________________________________
Table 6 lists the parameters for the experimental study derived
from the ANOVA.
TABLE 6 ______________________________________ ANOVA Results
Conditions Effects Closed Eyes Open Eyes
______________________________________ Alcohol vs. Control p
.ltoreq. 0.014 p .ltoreq. 0.011 Male vs. Female p .ltoreq. 0.003 p
.ltoreq. 0.018 ______________________________________
Sub-group analysis shows that for males, alcohol use is associated
with an increase in H (p.ltoreq.0.05) under both experimental
conditions. Further, H values from females in the alcoholic group
do not differ significantly from H values from females in the
control group. However, given the low number of subjects in the
alcoholic group and the high dispersion of H, the probability of a
Type II error is high.
The experimental results extend previous observations that heart
rhythms of healthy subjects generate IBI time series with H close
to 0. The value of H for all healthy control subjects in the open
eye condition was 0.07.+-.0.11 (mean.+-.SD). This value compares
with H=0.083.+-.0.139 calculated using CGSA with a sample of 5
healthy subjects and 512 data points (19). Although significant
differences in H between the closed eyes condition and open eyes
condition were not observed, a trend appears as a slightly lower H
in the closed eyes condition as compared to the open-eyes condition
in the control group. This would be expected if oculomotor and
vestibular activities are associated with inhibition of the vagally
mediated baroreflex control of heart rate (19).
Further, the experimental results using the invention to determine
H confirm previous observations that women appear to have more
complex heart-rate dynamics than men. Instead of using H, the
previous studies used an approximate entropy measure (20).
The experimental results strongly suggest that chronic excessive
alcohol use results in simplification of cardiac heart rate
dynamics, as determined by a measure of auto-correlation, namely H.
In particular, alcoholic subjects show increased values of H
compared to non-alcoholic subjects. This decreased level of
auto-correlation is associated with decreased signal complexity.
Particular inherited traits, which might be associated with
pre-morbid alterations in HRV, might increase the risk of
developing alcohol dependence. It is also possible that alcohol use
partially or completely determined the observed changes in heart
rate dynamics as a consequence of effects on the autonomic nervous
system.
The presence of alcohol withdrawal syndrome (AWS) in the alcoholic
subjects, and its consequent disruption of autonomic equilibrium,
might partially account for the experimental results. Alcoholic
subjects were studied more than 24 hours from their last alcohol
consumption, by self-report, and were determined to be alcohol free
at the time of the study. The lack of significant differences in
heart rate between the alcoholic and control groups suggests that
if AWS was occurring at the time of the study, its effects on the
sympathetic nervous system were not significant enough to be
reflected by relative tachycardia. Nevertheless, sub-clinical AWS
might have altered the reflex regulation of heart rate, perhaps by
increasing sympathetic nervous system activity. This might result
in a decrease in auto-correlation and contribute to an elevation of
H.
One explanation for the decreased HRV in alcoholics is decreased
parasympathetic nervous system (PNS) modulation of the heart rate.
However, the finding of lower HRV would not necessarily imply a
decrease in PNS activity because, although PNS blockade decreases
measures of HRV, decreases in PNS activity are not associated with
changes of HRV in the absence of central nervous system (CNS) input
and modulation (21). These data, taken together, suggest that an
abnormality in central and peripheral feedback regulation of heart
rate may be present in alcoholic subjects. The dysregulation may be
sufficient, even in the absence of changes in PNS activity, to
result in measurable changes in HRV parameters. This dysregulation
could be due to either a pre-existing trait, as a result of chronic
excessive alcohol consumption, or an interaction between these two
factors.
In addition to detecting alcoholism in human subjects, the
invention can also be used to detect other biological conditions in
biological subjects that are manifested as a change in H.
For example, the invention can be used to detect changes in
serotonin in a biological subject, such as a human. Serotonin is a
key neurochemical effector of both central and peripheral
modulation of PNS activity. Acting via 5-HT3 receptors present on
vagal fibers, peripherally administered 5-HT3 receptor agonists
stimulate acetylcholine release inducing bradycardia (22).
Centrally administered, these same agents decrease peripheral
acetylcholine release via inhibitory 5HT3 receptors present in the
nucleus tractus solitarius (23). Evidence of decreased central 5HT3
receptor function has been found in rats consuming alcohol on a
chronic basis (24). Low serotonin turnover is positively correlated
with Type II alcoholism (25), while chronic excessive alcohol
consumption appears to decrease serotonergic function (26). Taken
together, these data suggest that decreases in both 5-HT3-receptor
sensitivity and in serotonergic turnover might contribute to
decreases in HRV seen in alcoholics. Additionally, HRV studies in
women show elevations in the high-frequency spectral power
associated with PNS activation compared to males, while estrogens
are known to strongly increase serotonin turnover (20, 27, 28).
Furthermore, treatment of psychiatric disorders with selective
serotonin reuptake inhibitors, which also increase serotonin
turnover, results in similar increases in HRV (29).
Both autonomic neuropathy and cardiomyopathy have been shown to
result in decreases in HRV and simplification of cardiac dynamics.
These conditions are relatively common in alcoholics (30, 31, 32).
Decreased cardiac complexity has been found to be predictive of
ventricular tachycardia after myocardial infarction and in patients
with coronary artery disease (33, 34). Since increased
cardiovascular mortality in alcoholics is known to be associated
with the degree of autonomic neuropathy, measures of HRV can be
used in predicting end-organ damage and future cardiovascular
mortality in an alcoholic population.
Further, the invention can be used to detect other biological
conditions, such as a psychiatric disorder. Dysregulation of the
serotonergic axis is associated with certain psychiatric disorders
which have also been shown to result in alterations in HRV
measures. Examples include major depression (35), generalized
anxiety disorder (36), post-traumatic stress disorder (37), and
panic disorder (38). Because the invention is sensitive to
perturbations of HRV, changes in H determined from EKG data can be
associated with these and other psychiatric disorders which involve
disruption of the serotonergic axis.
Moreover, the invention can be used to detect other biological
conditions associated with the heart, such as cardiac arrhythmia,
coronary disease, autonomic neuropathy, and cardiomyopathy.
Alteration of HRV has been shown to be predictive of cardiac
arrhythmias (39, 40, 41, 42) and coronary artery disease (43).
Further, alterations in HRV, reflected by changes in the H
determined from EKG data, can be used to assess the risk factors
for cardiac arrhythmias, both as a result of cardiovascular disease
processes, as well as adverse effects of drugs which alter HRV
(44).
The invention has been described in detail with respect to
preferred embodiments, and it will now be apparent from the
foregoing to those skilled in the art that changes and
modifications may be made without departing from the invention in
its broader aspects, and the invention, therefore, as defined in
the appended claims is intended to cover all such changes and
modifications as fall within the true spirit of the invention.
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