Cardiac Dynamics

The temporal and spectral properties of the time series of RR intervals can provide information about the ability of the heart to adapt to changes in physical activity. Loss of adaptability has been shown to be associated with disease and the effect of ageing [1]. This adaptability can be quantified by studying how the power-spectral density scales as a function of the frequency. Measures of the self-similarity of biomedical signals provides a means of distinguishing between health and disease and monitoring the effect of ageing [2-3].

DFA exponent Using the power-spectral density, S(f), it is possible to estimate the powerlaw scaling exponent, $&beta$ , that satisfies $S(f) ~ f$ ^-&beta. This scaling exponent, $&beta$ , provides a measure of the roughness of the time series, with smoother self-similar time series having larger values of $&beta$ . Pink noise, also known as 1/f-like noise ( $&beta$ = 1), is characteristic of cardiac inter-beat interval time series [4-5] and presents a balance between the randomness of white noise ( $&beta$ = 0) and the much smoother Brownian motion ( $&beta$ = 2). Detrended fluctuation analysis (DFA) may also be used to quantify the self-similarity of nonstationary time series and is widely used for the analysis of biomedical signals [6]. DFA provides a scaling exponent, $&alpha$ , which is related to $&beta$ through $&beta = 2&alpha -1$ . The figures above and to the left show the time series and resulting DFA analysis for cardiac interbeat intervals from patient nsr001 of the Physionet database NSR2DB alongside realisations of white noise and Brownian motion. The routines powspecscale.m and dfa.m provide estimates of exponents $&alpha$ and $&beta$ respectively. These and other techniques for quantifying self-similarity are described and compared using synthetic signals in [7].

Probability Density Functions An interesting feature of the probability distribution function (PDF) of increments of RR intervals is a stretch-exponential-like PDF over short time scales and a near Gaussian PDF over long time scales [8]. The routine pdfdxdt.m produces plots of the increment PDF of a time series using kernel density estimation. The figure to the right shows how the increment PDFs change over different time scales for patient nsr001 of the Physionet database NSR2DB.

Recent observations of sleep-wake transitions observed during sleep have been incorporated into a new model of beat-to-beat heart rate variations over the course of 24 hours [9]. This new model faithfully represents some of the key empirical characteristics that have been observed in actual recordings of healthy human subjects, for example the existence of scale-free power law distributions during wakefulness and the exponential distribution of sleep-wake activity during prolonged sleep. This model can be used to assess biomedical signal processing methods applied to the RR interval time series.

A simple physiological model of the cardiovascular system, that produces realistic heart rate and blood pressure signals is also available.

Matlab files are available for download:

powspecscale.m	A matlab file for estimating the power-law scaling exponent $&beta$
dfa.m	A matlab file for estimating the DFA scaling exponent $&alpha$ . Also see FastDFA for very long time series.
pdfdxdt.m	A matlab file for visualising the probability distribution function (PDF) of increments of a time series over different temporal scales

Further information is available from Patrick McSharry. Suggestions and comments are welcome.

[1] P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley (1999), Multifractality in human heartbeat dynamics. Nature, 399:461–465.

[2] L. A. N. Amaral, A. L. Goldberger, P. C. Ivanov, and H. E. Stanley (1998), Scale-independent measures and pathologic cardiac dynamics. Phys. Rev. Lett., 81(11):2388–2391.

[3] C.-K. Peng, J. M. Hausdorff, and A. L. Goldberger (2000), Fractal mechanisms in neural control: Human heartbeat and gait dynamics in health and disease. In Nonlinear Dynamics, Self-Organization, and Biomedicine, pages 66–96, Cambridge, UK. Cambridge University Press.

[4] M. Kobayashi and T. Musha (1982), 1/f fluctuation of heartbeat period. IEEE Trans. Biomed. Eng., 29:456.

[5] A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C. K. Peng, and H. E. Stanley (2002), Fractal dynamics in physiology: alterations with disease and ageing. Proc. Natl. Acad. Sci., 99:2466–2472.

[6] C-K. Peng, S. Havlin, H. E. Stanley, and A. L. Goldberger (1995), Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos, 5:82–87.

[7] P. E. McSharry and B. D. Malamud (2005), Quantifying self-similarity in cardiac inter-beat interval time series. Computers in Cardiology, 32, September.

[8] D. C. Lin and R. L. Hughson (2001), Modeling heart rate variability in healthy humans: a turbulence analogy. Physical Review Letters, 86(8):1650–1653.

[9] P. E. McSharry and G. D. Clifford (2005), A statistical model of the sleep-wake dynamics of the cardiac rhythm, in Computers in Cardiology, Vol 32, IEEE Publishers: New York. pp. 591-594.