Fractal Analysis of Blood Flow Based on Heart Valve Dynamics
DOI:
https://doi.org/10.51699/k724s832Keywords:
HRV, mitral valve, multifractal analysis, MFDFA, heart signals, visualization.Abstract
Introduction. This article develops a mathematical model and algorithms of blood flow taking into account the dynamics of heart valves. The main goal of the research is to determine the nonlinear and complex dynamics of heart activity.
Methods. The model considers the heart chambers as elastic reservoirs that can deform to a certain volume. A point dynamic model is used to represent the dynamics of blood flow. The movement of the valves follows a specific dynamic equation, and valve function is expressed as a smoothly transitioning monotonic function. The complex dynamic characteristics of heart signals are determined using the MFDFA (Multifractal Detrended Fluctuation Analysis) method..
Results. The proposed model allows for the differential analysis of healthy and pathological signals. The main advantages of the model are: low demand for computational resources, a relatively small number of parameters, and the fact that most of them are practically measurable quantities. The results were implemented in the Python environment and the solutions were presented in a visual format.
Conclusion. The mathematical model has made it possible to gain a deeper understanding of the normal and pathological states of the heart. This, in turn, provides practical assistance in developing new diagnostic and treatment methods in medicine. The parameters of the model can be assessed in medical practice through indirectly measurable quantities (dynamics of chamber volume, ejection fraction, and changes in pressures).
References
1. Li, Sh., “Multifractal Detrended Fluctuation Analysis of Congestive Heart Failure Disease Based on Constructed Heartbeat Sequence,” IEEE Access, vol. 8, pp. 205244–205249, 2020.
2. Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., and Stanley, H. E., “Multifractal detrended fluctuation analysis of the German stock index,” Physica A: Statistical Mechanics and its Applications, vol. 316, no. 1–4, pp. 87–114, 2002.
3. Ivanov, P. Ch., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. G., Struzik, Z. R., and Stanley, H. E., “Multifractality in human heartbeat dynamics,” Nature, vol. 399, no. 6735, pp. 461–465, Jun. 1999.
4. Wang, J., Ning, X., and Chen, Y., “Multifractal analysis of electronic cardiograms taken from healthy and unhealthy adult subjects,” Physica A: Statistical Mechanics and its Applications, vol. 323, pp. 561–568, 2003.
5. Bucelli, A. A. A. B., et al., “A mathematical model that integrates cardiac electrophysiology, mechanics, and fluid dynamics: Application to the human left heart,” International Journal for Numerical Methods in Biomedical Engineering, vol. 39, no. 3, p. e3678, 2023.
6. Al-Mekhlafi, S. M. D. L., and Alkhaled, Y. E., “A Lumped Parameter Heart Model with Valve Dynamics,” in Proceedings of the 2019 International Conference on Electrical and Computing Technologies and Applications (ICECTA), Ras Al Khaimah, UAE, pp. 1–4, 2019.
7. Goldberger, A. L., Amaral, L. A. N., Hausdorff, J. M., Ivanov, P. Ch., Peng, C. K., and Stanley, H. E., “Fractal dynamics in physiology: Alterations with disease and aging,” Proceedings of the National Academy of Sciences of the USA, vol. 99, suppl. 1, pp. 2466–2472, 2002.
8. Peng, C. K., Havlin, S., Stanley, H. E., and Goldberger, A. L., “Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series,” Chaos, vol. 5, no. 1, pp. 82–87, 1995.
9. Costa, M., Goldberger, A. L., and Peng, C. K., “Multiscale entropy analysis of complex physiologic time series,” Physical Review Letters, vol. 89, no. 6, p. 068102, 2002.
10. Malik, M., et al., “Heart rate variability: Standards of measurement, physiological interpretation and clinical use,” European Heart Journal, vol. 17, no. 3, pp. 354–381, 1996.
11. Acharya, U. R., Joseph, P. K., Kannathal, N., Lim, C. M., and Suri, J. S., “Heart rate variability: A review,” Medical and Biological Engineering and Computing, vol. 44, no. 12, pp. 1031–1051, 2006.
12. Glass, L., “Synchronization and rhythmic processes in physiology,” Nature, vol. 410, pp. 277–284, 2001.
13. Suga, H., “Cardiac energetics: From Emax to pressure–volume area,” Clinical and Experimental Pharmacology and Physiology, vol. 30, no. 8, pp. 580–585, 2003.
14. Quarteroni, A., Veneziani, A., and Vergara, C., “Geometric multiscale modeling of the cardiovascular system,” Computing and Visualization in Science, vol. 12, pp. 93–104, 2009.
15. Shaffer, F., and Ginsberg, J. P., “An overview of heart rate variability metrics and norms,” Frontiers in Public Health, vol. 5, p. 258, 2017.
