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Symmetry: Culture and Science
Volume 35, Number 2, pages 141-154 (2024)
https://doi.org/10.26830/symmetry_2024_2_141

FRACTAL ANALYSIS OF CARDIAC SPECTRA

Onder Pekcan¹, Taner Arsan^2*

¹ Department of Molecular Biology and Genetics, Kadir Has University, Istanbul 34083, Turkey.
Email: pekcan@khas.edu.tr
Web: https://fens.khas.edu.tr/en/academic/1307
ORCID: 0000-0002-0082-8209

² Department of Computer Engineering, Kadir Has University, Istanbul 34083, Turkey.
Email: arsan@khas.edu.tr
Web: https://fens.khas.edu.tr/en/academic/10
ORCID: 0000-0002-4453-3218

^* corresponding author

Abstract: Cardiac diseases are one of the main reasons for mortality in modern, industrialized societies, and they cause high expenses in public health systems. Therefore, it is important to develop analytical methods to improve cardiac diagnostics. The heart's electric activity was first modeled using a set of nonlinear differential equations. Variations of cardiac spectra originating from deterministic dynamics are investigated. Analyzing the power spectra of a normal human heart presents the His-Purkinje network, which possesses a fractal-like structure. Phase space trajectories are extracted from the time series graph of ECG. Lower values of fractal dimension, , indicate dynamics that are more coherent. If has non-integer values greater than two when the system becomes chaotic or strange attractor. Recently, the development of a fast and robust method, that can be applied to multichannel physiologic signals, was reported. The convolutional Neural Networks (CNNs) method was also applied to patient-specific ECG classification for real-time heart monitoring. This manuscript investigates two different ECG systems produced from normal and abnormal human hearts to introduce an auxiliary phase space method in conjunction with ECG signals for diagnosing heart diseases. Here, the data for each person includes two signals based on and modified lead III (MLIII), respectively.

The fractal analysis method is employed on the trajectories constructed in phase space, from which the fractal dimension is obtained using the box-counting method. It is observed that, the second signals (i.e., MLIII) have larger values than the first signals (i.e., ), predicting more randomness yet more information. The lowest value of (i.e., ) indicates the perfect oscillation of the normal heart, and the highest value of (i.e., ) presents the abnormal heart's randomness. Our significant finding is that the phase space picture presents the distribution of the peak heights from the ECG spectra, giving valuable information about heart activities in conjunction with ECG.

Keywords: Fractals, Chaotic Systems, Cardiac Dynamics, Time Series Analysis, Random Processes. PACS 2010: 05.45.Df, 05.45.Pq, 87.19.Hh, 05.45.Tp, 05.40.-a

References:
Babloyantz, A., Salazar, J. M., and Nicolis, C. (1985) Evidence of chaotic dynamics of brain activity during the sleep cycle, Physics Letters A, 111(3), 152-156. https://doi.org/10.1016/0375-9601(85)90444-X

Babloyantz, A., and Destexhe, A. (1986) Low-dimensional chaos in an instance of epilepsy, Proceedings of the National Academy of Sciences, 83(10), 3513-3517. https://doi.org/10.1073/pnas.83.10.3513

Babloyantz, A., and Destexhe, A. (1988) Is the normal heart a periodic oscillator? Biological Cybernetics, 58(3), 203-211. https://doi.org/10.1007/BF00364139

Bergé, P., Pomeau, Y., and Vidal, C. (1984) L’ordre dans le chaos: Vers une approche déterministe de la turbulence, Hermann. https://books.google.com.tr/books?id=-vixAAAAIAAJ

Burden, R. L., and Faires, J. D. (1995) Numerical analysis (5. ed., 4. print), PWS Publ. Comp.

Glass, L., and Mackey, M.C. (1979) Pathological conditions resulting from instabilities in physiological control systems, Annals of the New York Academy of Sciences, 316(1), 214-235. https://doi.org/10.1111/j.1749-6632.1979.tb29471.x

Goldberger, A. L., Bhargava, V., West, B. J., & Mandell, A. J. (1985) On a mechanism of cardiac electrical stability. The fractal hypothesis, Biophysical Journal, 48(3), 525-528. https://doi.org/10.1016/S0006-3495(85)83808-X

Goldberger, A. L., Amaral, L. A. N., Glass, L., Hausdorff, J. M., Ivanov, P. Ch., Mark, R. G., Mietus, J. E., Moody, G. B., Peng, C.-K., & Stanley, H. E. (2000) PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals, Circulation, 101(23). https://doi.org/10.1161/01.CIR.101.23.e215

Grassberger, P., and Procaccia, I. (1983a) Measuring the strangeness of strange attractors, Physica D: Nonlinear Phenomena, 9(1-2), 189-208. https://doi.org/10.1016/0167-2789(83)90298-1

Grassberger, P., and Procaccia, I. (1983b) Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, 28(4), 2591-2593. https://doi.org/10.1103/PhysRevA.28.2591

Katholi, C. R., Urthaler, F., Macy, J., and James, T. N. (1977) A mathematical model of automaticity in the sinus node and AV junction based on weakly coupled relaxation oscillators, Computers and Biomedical Research, 10(6), 529-543. https://doi.org/10.1016/0010-4809(77)90011-8

Khan I.R., and Ohba, R. (1999) Closed form expressions for the finite difference approximations of first and higher derivatives based on taylor series. J. Comp. Appl. Math., 107:179–193. https://doi.org/10.1016/S0377-0427(99)00088-6

Kiranyaz, S., Ince, T., and Gabbouj, M. (2016) Real-Time Patient-Specific ECG Classification by 1-D Convolutional Neural Networks, IEEE Transactions on Biomedical Engineering, 63(3), 664-675. https://doi.org/10.1109/TBME.2015.2468589

La Rovere, M.T., Pinna, G.D., Maestri, R., Mortara, A., Capomolla, S., Febo, O., Ferrari, R., Franchini, M., Gnemmi, M., Opasich, C., Riccardi, P.G., Traversi, E., and Cobelli, F. (2003) Short-term heart rate variability strongly predicts sudden cardiac death in chronic heart failure patients, Circulation, 107(4), 565-570. https://doi.org/10.1161/01.CIR.0000047275.25795.17

Liebovitch L.S. and Toth, T. (1989) A fast algorithm to determine fractal dimensions by box counting. Physics Letters A, 141(8):386–390, 1989. https://doi.org/10.1016/0375-9601(89)90854-2

Liebovitch, L.S. (1998) Fractals and chaos simplified for the life sciences.New York:Oxford University Press.

Mackey, M.C., and Glass, L. (1977) Oscillation and Chaos in Physiological Control Systems. Science, 197(4300), 287-289. https://doi.org/10.1126/science.267326

Pekcan, O., and Arsan, T. (2023) Chaotic – deterministic or random nature of earthquakes – a phase space analysis. Symmetry: Culture and Science, 34(1), 45-59. https://doi.org/10.26830/symmetry_2023_ 1_045

van der Pol, B. and van der Mark, J. (1928) The Heartbeat Considered as a Relaxation Oscillation, and an Electrical Model of the Heart, Philosophical Magazine, 6, 763-775. https://doi.org/10.1080/14786441108564652

Wessel, N., Riedl, M., & Kurths, J. (2009) Is the normal heart rate “chaotic” due to respiration? Chaos: An Interdisciplinary Journal of Nonlinear Science, 19(2), 028508. https://doi.org/10.1063/1.3133128

West, B. J., Goldberger, A. L., Rovner, G., & Bhargava, V. (1985) Nonlinear dynamics of the heartbeat, Physica D: Nonlinear Phenomena, 17(2), 198-206. https://doi.org/10.1016/0167-2789(85)90004-1

Wilson J.D. and Haueisen, J. (2017) Separation of physiological signals using minimum norm projection operators, IEEE Transactions on Biomedical Engineering, 64(4), 904-916. https://doi.org/10.1109/TBME.2016.2582643