Skip to main content
SpringerLink
Log in

Fractal character of the electrocardiogram: Distinguishing heart-failure and normal patients

  • Published:
Annals of Biomedical Engineering Aims and scope Submit manuscript

Abstract

Statistical analysis of the sequence of heartbeats can provide information about the state of health of the heart. We used a variety of statistical measures to identify the form of the point process that describes the human heartbeat. These measures are based on both intervent intervals and counts, and include the intervent-interval histogram, interval-based periodogram, rescaled range analysis, the event-number histogram, Fano-factor, Allan Factor, and generalized-rate-based periodogram. All of these measures have been applied to data from both normal and heart-failure patients, and various surrogate versions thereof. The results show that almost all of the interevent-interval and the long-term counting statistics differ in statistically significant ways for the two classes of data. Several measures reveal 1/f-type fluctuations (long-duration power-law correlation). The analysis that we have conducted suggests the use of a conveniently calculated, quantitative index, based on the Allan factor, that indicates whether a particular patient does or does not suffer from heart failure. The Allan factor turns out to be particularly useful because it is easily calculated and is jointly responsive to both short-term and long-term characteristics of the heartbeat time series. A phase-space reconstruction based on the generalized heart rate is used to obtain a putative attractor's capacity dimension. Though the dependence of this dimension on the embedding dimension is consistent with that of a low-dimensional dynamical system (with a larger apparent dimension for normal subjects), surrogate-data analysis shows that identical behavior emerges from temporal correlation in a stochastic process. We present simulated results for a purely stochastic integrate-and-fire model, comprising a fractal-Gaussian-noise kernel, in which the sequence of heartbeats is determined by level crossings of fractional Brownian motion. This model characterizes the statistical behavior of the human electrocardiogram remarkably well, properly accounting for the behavior of all of the measures studied, over all time scales.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

AF:

Allan factor

CD:

Capacity dimension

DFT:

Discrete Fourier transform

DTMP:

Dead-time-modified Poisson point process

ECG:

Electrocardiogram

ENH:

Event-number histogram

FBM:

Fractional Brownian motion

FF:

Fano factor

FGN:

Fractal Gaussian noise

FSPP:

Fractal stochastic point process

HPP:

Homogeneous Poisson point process

IAF:

Integrate-and-fire

IBP:

Interval-based periodogram

HH:

Interevent-interval histogram

NCR:

Normalized coincidence rate

PSD:

Power spectral density

PSR:

Phase-space reconstruction

RBP:

Generalized rate-based periodogram

R/S:

Rescaled range analysis

WAF:

Wavelet Allan factor

WFF:

Wavelet Fano factor

References

  1. Babloyantz, A., and A. Destexhe. Is the normal heart a periodic oscillator?Biol. Cybern. 58:203–211, 1988.

    Article  PubMed  CAS  Google Scholar 

  2. Barnes, J. A. and D. W. Allan. A statistical model of flicker noise.Proc. IEEE 54:176–178, 1966.

    Google Scholar 

  3. Bassingthwaighte, J. B., L. S. Liebovitch, and B. J. West.Fractal Physiology, New York; American Physiological Society/Oxford, 1994, 364 pp.

    Google Scholar 

  4. Bassingthwaighte, J. B. and G. M. Raymond. Evaluating rescaled range analysis for time series.Ann. Biomed. Eng. 22:432–444, 1994.

    Article  PubMed  CAS  Google Scholar 

  5. Berger, R. D., S. Akselrod, D. Gordon, and R. J. Cohen. An efficient algorithm for spectral analysis of heart rate variability.IEEE Trans. Biomed. Eng. BME-33:900–904, 1986.

    Article  PubMed  CAS  Google Scholar 

  6. Bigger, J. T., Jr., J. L. Fleiss, R. C. Steinman, L. M. Rolnitzky, R. E. Kleiger, and J. N. Rottman. Frequency domain measures of heart period variability and mortality after myocardial infarction.Circulation 85:164–171, 1992.

    PubMed  Google Scholar 

  7. Cox, D. R. and P. A. W. Lewis.The Statistical Analysis of Series of Events. London: Methuen, 1966, 285 pp.

    Google Scholar 

  8. DeBoer, R. W., J. M. Karemaker, and J. Strackee. Comparing spectra of a series of point events particularly for heart rate variability data.IEEE Trans. Biomed. Eng. BME-31: 384–387, 1984.

    Article  PubMed  CAS  Google Scholar 

  9. Ding, M., C. Grebogi, E. Ott, T. Sauer, and J. A. Yorke. Plateau onset for correlation dimension: when does it occur?Phys. Rev. Lett. 70:3872–3875, 1993.

    Article  PubMed  Google Scholar 

  10. Fano, U. Ionization yield of radiations. II. The fluctuations of the number of ions.Phys. Rev. 72:26–29, 1947.

    Article  CAS  Google Scholar 

  11. Feller, W. The asymptotic distribution of the range of sums of independent random variables.Ann. Math. Stat. 22:427–432, 1951.

    Google Scholar 

  12. Freeman, R., J. P. Saul, M. S. Roberts, R. D. Berger, C. Broadbridge, and R. J. Cohen. Spectral analysis of heart rate in diabetic autonomic neuropathy.Arch. Neurol. 48: 185–190, 1991.

    PubMed  CAS  Google Scholar 

  13. Garfinkel, A., M. L. Spano, W. L. Ditto, and J. N. Weiss. Controlling cardiac chaos.Science 257:1230–1235, 1992.

    Article  PubMed  CAS  Google Scholar 

  14. Glass, L., A. Shrier, and J. Bélair. Chaotic cardiac rhythms. InChaos, edited by A. V. Holden. Princeton: Princeton University Press, 1986, pp. 237–256.

    Google Scholar 

  15. Goldberger, A. L., D. R. Rigney, J. Mietus, E. M., Antman, and S. Greenwald. Nonlinear dynamics in sudden cardiac death syndrome: heartrate oscillations and bifurcations.Experientia 44:983–987, 1988.

    Article  PubMed  CAS  Google Scholar 

  16. Goldberger, A. L., Nonlinear dynamics, fractals and chaos: applications to cardiac electrophysiology.Ann. Biomed. Eng. 18:195–198, 1990.

    Article  PubMed  CAS  Google Scholar 

  17. Goldberger, A. L., D. R. Rigney, and B. J. West. Chaos and fractals in human physiology.Sci. Am. 262(2):42–49, 1990.

    PubMed  CAS  Google Scholar 

  18. Grassberger, P. and I. Procaccia. Measuring the strangeness of strange attractors.Physica D 9:189–208, 1983.

    Article  Google Scholar 

  19. Grüneis, F., M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, and T. Musha. Further study of 1/f fluctuations observed in central single neurons during REM sleep.Biol. Cybern. 68:193–198, 1993.

    Article  PubMed  Google Scholar 

  20. Guevara, M. R. and A. Shrier. Rhythms produced by high-amplitude periodic stimulation of spontaneously beating aggregates of embryonic chick ventricular myocytes.Ann. N.Y. Acad. Sci. 591:11–22, 1990.

    Article  PubMed  CAS  Google Scholar 

  21. Guzzetti, S., E. Piccaluga, R. Casati, S. Cerutti, F. Lombardi, M. Pagani, and A. Malliani. Sympathetic predominance in essential hypertension: a study employing spectral analysis of heart rate variability.J. Hypertens. 6:711–717, 1988.

    Article  PubMed  CAS  Google Scholar 

  22. Haight, F. A.Handbook of the Poisson Distribution, New York: Wiley, 1967, 168 pp.

    Google Scholar 

  23. Hoop, B., H. Kazemi, and L. Liebovitch. Rescaled range analysis of resting respiration.Chaos 3:27–29, 1993.

    Article  PubMed  Google Scholar 

  24. Hurst, H. E.. Long-term storage capacity of reservoirs.Trans. Amer. Soc. Civil Eng. 116:770–808, 1951.

    Google Scholar 

  25. Hyndman, B. W. and R. K. Mohn. A pulse modulator model for pacemaker activity.Dig. 10th Int. Conf. Med. Biol. Eng. (Dresden, Germany), p. 223, 1973.

  26. Hyndman, B. W. and R. K. Mohn. Model of the cardiac pacemaker and its use in decoding the information content of cardiac intervals.Automedica 1:239–252, 1975.

    Google Scholar 

  27. Kanters, J. K., N.-H. Holstein-Rathlou, and E. Agner. Lack of evidence for low-dimensional chaos in heart rate variability.J Cardiovasc. Electrophysiol. 5:591–601, 1994.

    Article  PubMed  CAS  Google Scholar 

  28. Kaplan, D. T. and R. J. Cohen. Searching for chaos in fibrillation.Ann. N.Y. Acad. Sci. 591:367–374, 1990.

    Article  PubMed  CAS  Google Scholar 

  29. Kluge, K. A., R. M. Harper, V. L. Schechtman, A. J. Wilson, H. J. Hoffman, and D. P. Southall. Spectral analysis assessment of respiratory sinus arrhythmia in normal infants and infants who subsequently died of sudden infant death syndrome.Pediatr. Res. 24:677–682, 1988.

    Article  PubMed  CAS  Google Scholar 

  30. Kobayashi, M. and T. Musha, 1/f fluctuation of heartbeat period.IEEE Trans. Biomed. Eng. BME-29:456–457, 1982.

    Article  PubMed  CAS  Google Scholar 

  31. Liebovitch, L. S. Introduction to the properties and analysis of fractal objects, processes, and data. In:Advanced Methods of Physiological System Modeling, Vol. 2, edited by V. Z. Marmarelis. New York: Plenum Press, 1989, pp. 225–239.

    Google Scholar 

  32. Liebovitch, L. S. and T. Tóth. A fast algorithm to determine fractal dimensions by box counting.Phys. Lett. A 141:386–390, 1989.

    Article  Google Scholar 

  33. Lishner, M., S. Akselrod, V. Mor Avi, O. Oz, M. Divon, and M. Ravid. Spectral analysis of heart rate fluctuations. A non-invasive, sensitive method for the early diagnosis of autonomic neuropathy in diabetes mellitus.J. Auton. Nerv. Sys. 19:119–125, 1987.

    Article  CAS  Google Scholar 

  34. Lowen, S. B. and M. C. Teich. Estimation and simulation of fractal stochastic point processes.Fractals 3:183–210. 1995.

    Article  Google Scholar 

  35. Lowen, S. B. and M. C. Teich. The periodogram and Allan variance reveal fractal exponents greater than unity in auditory-nerve spike trains.J. Acoust. Soc. Am. 99:in press, 1996.

  36. Mandelbrot, B. B. and J. W. van Ness. Fractional brownian motions, fractional noises and applications.SIAM Rev. 10:422–437, 1968.

    Article  Google Scholar 

  37. Meesmann, M., F. Grüneis, P. Flachenecker, and K.-D. Kniffki. A new method for analysis of heart rate variability: counting statistics of 1/f fluctuations.Biol. Cybern. 68:299–306, 1993.

    Article  PubMed  CAS  Google Scholar 

  38. Moody, G. B. and R. G. Mark. Development and evaluation of a two-lead ECG analysis program.Proc. IEEE Comp. in Cardiol. Conf. 9:39–44, 1982.

    Google Scholar 

  39. Osborne, A. R. and A. Provenzale. Finite correlation dimension for stochastic systems with power-law spectra.Physica D 35:357–381, 1989.

    Article  Google Scholar 

  40. Papoulis, A.Probability Random Variables, and Stochastic Processes. 2nd Ed. New York: McGraw-Hill, 1984, 576 pp.

    Google Scholar 

  41. Peng, C.-K., J. Mietus, J. M. Hausdorff, S. Havlin, H. E., Stanley, and A. L. Goldberger. Long-range anticorrelations and non-Gaussian behavior of the heartbeat.Phys. Rev. Lett. 70:1343–1346, 1993.

    Article  Google Scholar 

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

    Article  PubMed  CAS  Google Scholar 

  43. Powers, N. L., R. J. Salvi, and S. S. Saunders. Discharge rate fluctuations in the auditory nerve of the chinchilla. Abstracts of the XIV Midwinter Research Meeting, Assoc. for Res. in Otolaryngology, Des Moines, IA; D. J. Lim, Ed., Abstract No. 411, p. 129, 1991.

  44. Powers, N. L. and R. J. Salvi. Comparison of discharge rate fluctuations in the auditory nerve of chickens and chinchillas. Abstracts of the XV Midwinter Research Meeting, Assoc. for Res. in Otolaryngology, Des Moines, IA; D. J. Lim, Ed., Abstract No. 292, p. 101, 1992.

  45. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery.Numerical Recipes in C, 2nd Ed. Cambridge, U.K.: Cambridge University Press 1992, 994 pp.

    Google Scholar 

  46. Ravelli, F. and R. Antolini. Complex dynamics underlying the human electrocardiogram.Biol. Cybern. 67:57–65. 1992.

    Article  PubMed  CAS  Google Scholar 

  47. Ricciardi, L. M. and F. Esposito. On some distribution functions for non-linear switching elements with finite dead time.Kybernetik (Biol. Cybern.) 3:148–152, 1966.

    CAS  Google Scholar 

  48. Rompelman, O., J. B. I. M. Snijders, and C. J. van Spronsen. The measurement of heart rate variability spectra with the help of a personal computer.IEEE Trans. Biomed. Eng. BME-29:503–510, 1982.

    Article  PubMed  CAS  Google Scholar 

  49. Saul, J. P., P. Albrecht, R. D. Berger, and R. J. Cohen. Analysis of long term heart rate variability: methods, 1/f scaling and implications.Proc. IEEE Comput. in Cardiol. Conf. 14:419–422, 1988.

    CAS  Google Scholar 

  50. Scharf, R., M. Meesmann, J. Boese, D. R. Chialvo, and K.-D. Kniffki. General relation between variance-time curve and power spectral density for point processes exhibiting 1/f β-fluctuations, with special reference to heart rate variability.Biol. Cybern. 73:255–263, 1995.

    PubMed  CAS  Google Scholar 

  51. Schepers, H. E., J. H. G. M. van Beek, and J. B. Bassingthwaighte. Four methods to estimate the fractal dimension from self-affine signals.IEEE Eng. Med. Biol. Mag. 11:57–71, 1992.

    Article  Google Scholar 

  52. Schiff, S. J. and T. Chang. Differentiation of linearly correlated noise from chaos in a biologic system using surrogate data.Biol. Cybern. 67:387–393, 1992.

    Article  PubMed  CAS  Google Scholar 

  53. Shono, H., M. Yamasaki, M. Muro, M. Oga, Y. Ito, K. Shimomura, and H. Sugimori Chaos and fractals with 1/f spectrum below 10−2 Hz demonstrates full-term fetal heart rate changes during active phase.Early Human Dev. 27: 111–117, 1991.

    Article  CAS  Google Scholar 

  54. Teich, M. C. and S. M. Khanna Pulse-number distribution for the neural spike train in the cat's auditory nerve.J. Acoust. Soc. Am. 77:1110–1128, 1985.

    Article  PubMed  CAS  Google Scholar 

  55. Teich, M. C. Fractal character of the auditory neural spike train.IEEE Trans. Biomed. Eng 36:150–160, 1989.

    Article  PubMed  CAS  Google Scholar 

  56. Teich, M. C. Fractal neuronal firing patterns. In:Single Neuron Computation, edited by T. McKenna, J. Davis, and S. Zornetzer. Boston: Academic Press, 1992, pp. 589–625.

    Google Scholar 

  57. Teich, M. C. and S. B. Lowen. Fractal patterns in auditory nerve-spike trains.IEEE Eng. Med. Biol. Mag. 12:197–202, 1994.

    Article  Google Scholar 

  58. Teich, M. C., C. Heneghan, S. B. Lowen, and R. G. Turcott. Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods. In:Wavelets in Medicine and Biology edited by A. Aldroubi and M. Unser. Boca Raton, FL: CRC Press. 1996.

    Google Scholar 

  59. Teich, M. C., R. G. Turcott, and R. M. Siegel, Variability and long-duration correlation in the sequence of action potentials in cat striate-cortex neurons.IEEE Eng. Biol. Med. Mag. 14: in press 1996.

  60. Theiler, J. Estimating fractal dimensionJ. Opt. Soc. Am. A 7:1055–1073, 1990.

    Google Scholar 

  61. Theiler, J., S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer. Testing for nonlinearity in time series: the method of surrogate data.Physica D 58:77–94, 1992.

    Article  Google Scholar 

  62. Tuckwell H. C.:Stochastic Processes in the Neurosciences. Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia. 1989, 129 pp.

    Google Scholar 

  63. Turcott, R. G. and M. C. Teich. Long-duration correlation and attractor topology of the heartbeat rate differ for normal patients and those with heart failure.Proc. SPIE (Chaos in Biology and Medicine) 2036:22–39, 1993.

    Google Scholar 

  64. Turcott, R. G., P. D. R. Barker, and M. C. Teich. Long-duration correlation in the sequence of action potentials in an insect visual interneuron.J. Statist. Comput. Simul. 52: 253–271, 1995.

    Article  Google Scholar 

  65. West, B. J. and W. Deering. Fractal physiology for physicists: Lévy statistics.Physics Reports 246:1–100, 1994.

    Article  Google Scholar 

  66. Witkowski, F. X., K. M. Kavanagh, P. A. Penkoske, R. Plonsey, M. L. Spano, W. L. Ditto, and D. T. Kaplan. Evidence for determinism in ventricular fibrillation.Phys. Rev. Lett. 75:1230–1233, 1995.

    Article  PubMed  CAS  Google Scholar 

  67. Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining Lyapunov exponents from a time series.Physica D 16:285–317, 1985.

    Article  Google Scholar 

  68. Zbilut, J. P., G. Mayer-Kress, P. A. Sobotka, M. O'Toole, and J. X. Thomas, Jr. Bifurcations and intrinsic chaotic and 1/f dynamics in an isolated perfused rat heart.Biol. Cybern. 61:371–378, 1989.

    Article  PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Columbia University, New York, NY

    Robert G. Turcott & Malvin C. Teich

  2. School of Medicine, Stanford University, Stanford, CA

    Robert G. Turcott

  3. Department of Electrical, Computer & Systems Engineering and Department of Biomedical Engineering, Boston University, 44 Cummington Street, 02215, Boston, MA

    Malvin C. Teich

Authors
  1. Robert G. Turcott
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Malvin C. Teich
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Turcott, R.G., Teich, M.C. Fractal character of the electrocardiogram: Distinguishing heart-failure and normal patients. Ann Biomed Eng 24, 269–293 (1996). https://doi.org/10.1007/BF02667355

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02667355

Keywords

Search

Navigation