Methodological features of the analysis of the fractal dimension of the heart rate

Objectives. The aim of the present work is to determine the fractal dimension parameter calculated for a sequence of R–R intervals in order to identify the boundaries of its change for healthy and sick patients, as well as the possibility of its use as an additional factor in the detection of cardiac pathology.

Methods. In order to determine the fractal dimension parameter, the Hurst-, Barrow-, minimum coverage area-, and Higuchi methods are used. For assessing the stationarity of a number of electrocardiography (ECG) intervals, a standard method is used to compare arithmetic averages and variances from samples of the total data array of ECG intervals. To identify differences in fractal dimensions of healthy and sick patients, this parameter was ranked. Using the Kolmogorov–Smirnov two-sample criterion, the difference between the distribution laws in the samples for healthy and sick patients is shown.

Results. Among the considered methods for calculating the fractal dimension, the Higuchi method demonstrates the smallest data spread between healthy patients. By ranking the calculated fractional dimension values, it was possible to identify the difference between this parameter for healthy and sick patients. The difference in the distribution of fractal dimension of healthy and sick patients is shown to be statistically significant for the coverage and Higuchi methods. At the same time, when using the traditional Hurst method, there is no reason to reject the null hypothesis that two groups of patients belong to the same general population.

Conclusions. Based on the obtained data, the difference between the fractal dimension indicators of the duration of R–R intervals of healthy and sick patients is shown to be statistically significant when using the Higuchi method. The fractal dimensions of healthy and sick patients can be effectively distinguished by ranking samples. The results of the research substantiate prospects for further studies aimed at using fractal characteristics of the heart rhythm to identify abnormalities of the latter, which can serve as an additional factor in determining heart pathologies.

1. Baevskii P.M. Cybernetic analysis of heart rate control processes. In: Aktual’nye problemy fiziologii i patologii krovoobrashcheniya (Actual Problems of Physiology and Pathology of Blood Circulation). Moscow: Meditsina; 1976. P. 161−175 (in Russ.).

2. Baevskii P.M., Berseneva A.P. Otsenka adaptatsionnykh vozmozhnostei organizma i risk razvitiya zabolevanii (Assessment of the Adaptive Capabilities of the Body and the Risk of Developing Diseases). Moscow: Meditsina; 1997. P. 265. (in Russ.).

3. Baevskii P.M. Prognozirovanie sostoyanii na grani normy i patologii (Prediction of Conditions on the Verge of Norm and Pathology). Moscow: Meditsina; 1979. 205 p. (in Russ.)

4. Baevskii P.M., Ivanov G.G. Cardiac rhythm variability: the theoretical aspects and the opportunities of clinical application (lecture). Ul’trazvukovaya i funktsional’naya diagnostika = Ultrasound and Functional Diagnostics. 2001;3:106−127 (in Russ.).

5. Antonov V.I., Zagainov A.I., Vu van Quang. Dynamic fractal analysis of heart rate variability. Nauchnotekhnicheskie vedomosti Sankt-Peterburgskogo gosudarstvennogo politekhnicheskogo universiteta. Informatika. Telekommunikatsii. Upravlenie = St. Petersburg Polytechnical University Journal. Computer Science. Telecommunication and Control Systems. 2012;1(140):88−94 (in Russ.).

6. Fedorov V.A., MizirinA.V., Khramtsov P.I.,Agafonova N.A. Fractal analysis of heart rhythm. Voprosy sovremennoi pediatrii = Current Pediatrics. 2006;5(1):596 (in Russ.).

7. Yatsyk V.Z., Paramzin V.B., Bolotin A.E., Vorotova M.S. Fractal analysis of variability of heart rhythm among female biathletes with a diff training level. Fizicheskaya kul’tura, sport - nauka i praktika = Physical Education, Sports - Science and Practice. 2018;4:95−102 (in Russ.).

8. Sen J., McGill D. Fractal analysis of heart rate variability as a predictor of mortality: A systematic review and meta-analysis. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2018;28(7):072101. https://doi. org/10.1063/1.5038818

9. Rakhimov N.G., Olimzoda N.Kh., Muradov A.M., Muradov A.A., Khusainova M.B. Some indicators of fractal analysis heart rate variability as predictors of the development of severe preeclampsia and eclampsia. Vestnik poslediplomnogo obrazovaniya v sfere zdravookhraneniya = Herald of Postgraduate Education Health Sphere. 2018;1:70−75 (in Russ.).

10. Goldberger E. L. Goldberger A.L., Rigney D.R., West B.J. Chaos and fractals in human physiology. V mire nauki (Russian version) = Sci. American. 1990;4:25−32 (in Russ.).

11. Shuster G. Determinirovannyi khaos: Vvedenie (Deterministic chaos: An Introduction). Moscow: Mir; 1988. 248 p. (in Russ.).

12. Lorenz E. Deterministic non-periodic motion. In: Strange attractors: collection of articles. Moscow: Fizmatlit; 1981. P. 88−116 (in Russ.).

13. Klimontovich Yu.L. Turbulentnoe dvizhenie i struktura khaosa: Novyi podkhod k statisticheskoi teorii otkrytykh system (Turbulent motion and the structure of chaos: A new approach to the statistical theory of open systems). Moscow: URSS; 2021. 326 p (in Russ.).

14. Mandelbrot B. Fractal geometry of nature. Moscow: Institut komp’yuternykh issledovanii; 2002. 656 p. (in Russ.).

15. Cervantes-De la Torre F., González-Trejo J.I., Real-Ramírez C.A., Hoyos-Reyes L.F. Fractal dimension algorithms and their application to time series associated with natural phenomena. J. Phys.: Conf. Ser. 2013;475(1):012002. https://doi.org/10.1088/1742-6596/475/1/012002

16. Antipov O.I., Nagornaya M.Yu. Bioenergetic signal Hearst index. Infokommunikatsionnye tekhnologii = Infocommunication Technologies. 2011;9(1):75−77 (in Russ.).

17. Aptukov V.N., Mitin V.Yu. Fractal analysis of meteorological series based on the minimum covering method. Geograficheskii vestnik = Geographical Bulletin. 2019;2(49):67-79 (in Russ.). https://doi.org/10.17072/2079-7877-2019-2-67-79

18. Gladun K.V. Higuchi fractal dimension as a method for assessing response to sound stimuli in patients with diffuse axonal brain injury. Sovremennye tekhnologii v meditsine = Modern Technologies in Medicine. 2020;12(4):63-71 (in Russ.). https://doi.org/10.17691/stm2020.12.4.08

19. Koichubekov B.K., Sorokina M.A., Mkhitaryan K.E. Sample size determination in planning of scientific research. Mezhdunarodnyi zhurnal prikladnykh i fundamental’nykh issledovanii = International Journal of Applied and Fundamental Research. 2014;4:71−74 (in Russ.).

20. Anishchenko V.S. Deterministic chaos. Sorosovskii obrazovatel’nyi zhurnal = Soros Educational Journal. 1997;6:70-76 (in Russ.).

21. Meganur R., Zadidul K., Maksudul H., Jarin S. Successive RR interval analysis of PVC with sinus rhythm using fractal dimension, Poincaré plot and sample entropy method. Int. J. Image, Graphics and Signal Processing (IJIGSP). 2013;5(2):17−24. https://doi.org/10.5815/ijigsp.2013.02.03

22. Orlov A.I. System of models and methods of testing the homogeneity of two independent samples. Politematicheskii setevoi elektronnyi nauchnyi zhurnal Kubanskogo gosudarstvennogo agrarnogo universiteta = Polythematic online Electronic Scientific Journal of Kuban State Agrarian University. 2020;157:145−169 (in Russ.). https://doi.org/10.21515/1990-4665-157-012