Biomedical Engineering Reference
In-Depth Information
The heart-rate time series demonstrates complex irregularity that has been
described as a fractal process because it looks similar whether it is plotted over
days, hours, or minutes. In fact, the heart rate time series has "1/ f " or power-law
scaling, in that the amplitude of oscillations ( A ) is proportional to the inverse of
oscillation frequency, according to the formula A x 1/ f . The exponent C can be
derived from the slope of the log-log transformation of the Fourier power spec-
trum. A fractal process (most complex) has a slope of 1 (i.e., amplitude is in-
versely proportional to frequency over a wide range of frequencies, indicating
the presence of long-range correlations (self-similarity) in the data). A loss of
complexity occurs as the slope approaches 0 (white noise; i.e., there is no rela-
tion between amplitude and frequency) or 2 (Brownian noise; i.e., the relation
between amplitude and frequency occurs only over a short range, then rapids
falls off, indicating a loss of long-range correlations).
Unfortunately, computation of the power spectrum using Fourier analysis
requires stationary data, which most physiologic signals are not. Another par-
ticularly useful technique that minimizes the effect of nonstationarities in the
data is "detrended fluctuation analysis" (DFA), which has been well validated in
a number of dynamic systems (10). The DFA algorithm is a two-point correla-
tion method that computes the slope of the line relating the amplitude of fluctua-
tions to the scale of measurement, after detrending the data. The root-mean-
square fluctuation of the integrated and detrended data are measured in observa-
tion windows of different size and then plotted against the size of the window on
a log-log scale. The slope of the regression line that relates log-fluctuation to
log-window size quantifies the complexity of the data (1 = fractal, 0.5 = random,
1.5 = random-walk).
Other indicators of complexity loss in physiologic systems include an in-
crease in periodicity (e.g., the tremor of Parkinson's disease), increased random-
ness (e.g., atrial fibrillation of the heart), and loss of long-range correlations
(e.g., stride-length changes during gait).
It is important to recognize that complexity and variability are not the same.
For example, a high-amplitude sine wave signal is quite variable, but not at all
complex. Alternatively, an irregular low-amplitude signal such as the heart rate
of a healthy young subject shown in Figure 2, can be quite complex but much
less variable. Similarly, irregularity and complexity are not the same. Traditional
entropy-based algorithms such as Approximate Entropy (11), used to quantify
the regularity of a time series, indicate greater irregularity (which has been in-
terpreted as greater complexity) for certain uncorrelated random signals associ-
ated with pathologic processes. However, as highlighted by the recent work
of Costa et al. (12), these algorithms fail to account for the multiple time
scales over which healthy physiologic processes operate. By calculating the en-
tropy of heart-rate dynamics over multiple time scales (multiscale entropy
analysis), these investigators were able to distinguish healthy from pathological
(e.g., atrial fibrillation and congestive heart failure) dynamics, and showed
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