Multiscale multifractal analysis of heart rate variability recordings with a large number of occurrences of arrhythmia

Human heart rate variability, in the form of time series of intervals between heart beats, shows complex, fractal properties. Recently, it was demonstrated many times that the fractal properties vary from point to point along the series, leading to multifractality. In this paper, we concentrate not only on the fact that the human heart rate has multifractal properties but also that these properties depend on the time scale in which the multifractality is measured. This time scale is related to the frequency band of the signal. We find that human heart rate variability appears to be far more complex than hitherto reported in the studies using a fixed time scale. We introduce a method called multiscale multifractal analysis (MMA), which allows us to extend the description of heart rate variability to include the dependence on the magnitude of the variability and time scale (or frequency band). MMA is relatively immune to additive noise and nonstationarity, including the nonstationarity due to inclusions into the time series of events of a different dynamics (e.g., arrhythmic events in sinus rhythm). The MMA method may provide new ways of measuring the nonlinearity of a signal, and it may help to develop new methods of medical diagnostics.

Figure 1

Nighttime RR interval series of a healthy, 25-year-old man (subject code: CHM).

Figure 2

(a) Plot of the fluctuation functions $F_q\left(s\right)$ (black points) calculated for the series presented in Fig. 1. The curves correspond to $q$ between $-5$ and 5 in steps of 1. Vertical (blue) lines mark two examples of the fitting windows for the small $s\in \langle 10,50\rangle$ range and for the large $s\in \langle 120,600\rangle$ range. Fits made within these two windows are shown by solid lines (red or dark gray in the left window and green or light gray in the right window). (b), (c) Two examples of the $h\left(q\right)$ curves, obtained in the fitting windows of (a), for the small and the large scales $s$, respectively.

Figure 3

(a) Family of curves $F_q\left(s\right)$ calculated for the first test signal (multifractality visible for the small scales and monofractality for the larger scales). Vertical (blue) lines mark two examples of the fitting windows for the small $s\in \langle 10,50\rangle$ range and for the large $s\in \langle 120,600\rangle$ range. Fits made within these two windows are shown by solid lines (red or dark gray in the left window and green or light gray in the right window). (b) The $h\left(q\right)$ dependence calculated for the small (red or dark gray points, upper curve) and for the large (green or light gray points, lower curve) scales $s$.

Figure 4

(a) Family of curves $F_q\left(s\right)$ calculated for the second test signal (monofractality visible in the small scales and multifractality in the large scales). (b) The $h\left(q\right)$ dependence calculated for the small (red or dark gray points, lower curve) and for the large (green or light gray points, upper curve) scales $s$.

Figure 5

$h(q,s)$ dependence (Hurst surface) calculated for a multifractal nondeterministic binomial cascade of length 32 768. The two red (dark gray) points correspond to ${\alpha }_1$ and ${\alpha }_2$ calculated using the standard, monofractal DFA method. The green (light gray) line at the back of the plot corresponds to $h\left(q\right)$ calculated with the standard MF-DFA method.

Figure 6

(a)–(c) The $f(\alpha ,s)$ dependence for the RR interval series of the healthy subject CHM seen from three different viewing angles. Numerous artifacts make the plot difficult to read and interpret. Light gray points mark the artifacts recognized by the artifact removing procedure. (d) An example cross section of $f(\alpha ,s)$ in (a)–(c) obtained for $s=360$.

Figure 7

(a) The standard $h\left(q\right)$ dependence calculated using MF-DFA for a monofractal series of random numbers (uniform distribution, range $\langle 0,2000\rangle$, length of series 16 384). (b) The $h(q,s)$ dependence calculated for the monofractal series in (a). (c) $h\left(q\right)$ for a multifractal series of random numbers (Cauchy distribution, location 0, scale 1, length of series 65 536). (d) The $h(q,s)$ dependence calculated for the series in (c).

Figure 8

(a) The $h(q,s)$ surface calculated for the nighttime heart rate variability of a healthy 25-year-old man (CHM, length of series 29 700, cf. Fig. 1). (b) $h(q,s)$ calculated for the shuffled version of the same series.

Figure 9

$h(q,s)$ for the RR interval series of the healthy subject CHM, shortened to different lengths: (a) 20 000, (b) 15 000, (c) 10 000, (d) 7500, (e) 5000, and (f) 2000. Presented are results averaged over 1000 realizations of the test series.

Figure 10

$h(q,s)$ calculated for the multifractal series of random numbers [compare with Fig. 7d, Cauchy distribution, location 0, scale 1] of different lengths: (a) 20 000, (b) 15 000, (c) 10 000, (d) 7500, (e) 5000, (f) 2000. The presented results have been averaged over 1000 realizations of the test series.

Figure 11

$h(q,s)$ for the RR interval series of the healthy subject CHM, with different levels of noise added: (a) $10\%$, (b) $20\%$, (c) $30\%$, (d) $40\%$, (e) $50\%$, and (f) $60\%$. The presented results have been averaged over 1000 realizations of the test series.

Figure 12

$h(q,s)$ calculated for the multifractal series of random numbers [compare with Fig. 7d, Cauchy distribution, location 0, scale 1] with noise added: (a) $0\%$ and (b) $100\%$. The results presented have been averaged over 1000 realizations of the test series.

Figure 13

(a) Nighttime RR interval series of the subject DWD, with many ectopic beats (arrhythmia). (b) The RR interval series of the subject CHM with $10\%$ of the short arrhythmic segments incorporated. (c) Enlargement with three short, arrhythmic segments included randomly.

Figure 14

$h(q,s)$ for the RR interval series of the healthy subject CHM, with five-interval-long arrhythmic segments included: (a) $10\%$ of arrhythmic intervals, (b) $20\%,$ (c) $30\%,$ (d) $40\%,$ (e) $50\%$, and (f) $60\%$. The presented results have been averaged over 1000 realizations of the test series.

Figure 15

(a) Example of a 300-interval-long arrhythmic segment extracted from the RR interval time series of the patient DWD, ending approximately 1 h before cardiac arrest. The arrhythmic segments were rescaled (see text) and incorporated into the RR interval series of a healthy man (CHM) at random locations. (b) RR interval series recorded for CHM with $10\%$ of 300-interval-long arrhythmic segments incorporated (arrows).

Figure 16

The $h(q,s)$ surface for the RR interval series of the healthy subject CHM, with 300-intervals-long arrhythmic segments added at random locations: (a) $10\%$ of arrhythmic intervals added, (b) $20\%,$ (c) $30\%,$ (d) $40\%,$ (e) $50\%,$ and (f) $60\%$. The presented results have been averaged over 1000 realizations of the test series.

Figure 17

(a) “Reversed” multifractality: $h(q,s)$ calculated for the nighttime RR interval series shown below. (b) Nighttime RR interval series of a healthy 14-year-old girl (MKL).

Figure 18

(a) Cross section of the $h(q,s)$ dependence for healthy subject CHM, taken for scale $s=30$. (b) The cross section for $s=75$ and (c) for $s=360$.

Figure 19

(a) $h(q,s)$ dependence for a 25-year-old man at risk of cardiac arrest (BHD) but without an organic heart disease [cf. Fig. 8a]. (b) RR interval series for BHD.

Figure 20

(a) $h(q,s)$ for a subject at risk of cardiac arrest (DWD); cardiac arrest occurred about 1 h after the time series analyzed. (b) $h(q,s)$ for the same subject but after a very successful, nine-month pharmacological treatment. (c) The nighttime RR interval series of the subject DWD after successful pharmacological treatment, the fragment analyzed in (b).