Physical Mathematical Evaluation of the Cardiac Dynamic Applying the Zipf-Mandelbrot Law


Introduction: The law of Zipf-Mandelbrot is a power law, which has been observed in natural languages. A mathematical diagnosis of fetal cardiac dynamics has been developed with this law. Objective: To develop a methodology for diagnostic aid to assess the degree of complexity of adult cardiac dynamics by Zipf-Mandelbrot law. Methodology: A mathematical induction was done for this; two groups of Holter recordings were selected: 11 with normal diagnosis and 11 with acute disease of each group, one Holter of each group was chosen for the induction, the law of Zipf-Mandelbrot was applied to evaluate the degree of complexity of each Holter, searching similarities or differences between the dynamics. A blind study was done with 20 Holters calculating sensitivity, specificity and the coefficient kappa. Results: The complexity grade of a normal cardiac dynamics varied between 0.9483 and 0.7046, and for an acute dynamic between 0.6707 and 0.4228. Conclusions: A new physical mathematical methodology for diagnostic aid was developed; it showed that the degree of complexity of normal cardiac dynamics was higher than those with acute disease, showing quantitatively how cardiac dynamics can evolve to acute state.

Share and Cite:

Rodríguez, J. , Prieto, S. , Correa, S. , Mendoza, F. , Weiz, G. , Soracipa, M. , Velásquez, N. , Pardo, J. , Martínez, M. and Barrios, F. (2015) Physical Mathematical Evaluation of the Cardiac Dynamic Applying the Zipf-Mandelbrot Law. Journal of Modern Physics, 6, 1881-1888. doi: 10.4236/jmp.2015.613193.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Zipf, G. (1949) Human Behaviour and the Principle of Least Effort: An Introduction to Human Ecology. Addison-Wesley, Cambridge.
[2] Mandelbrot, B. (1972) Scaling and Power laws without Geometry. In: The Fractal Geometry of Nature, Freeman, San Francisco, 344-348.
[3] Mandelbrot, B. (2000) Hierarchical or Classification Trees, and the Dimension. In: Fractals: Form, Chance and Dimension, Tusquets, Barcelona, 161-166.
[4] Mandelbrot, B. (1954) Structure formelle des textes et comunication. World, 10, 1-27.
[5] Adamic, L. and Huberman, B. (2002) Zipf’s Law and the Internet. Glottometrics, 3, 143-150.
[6] Larsen-Freeman, D. (1997) Chaos/Complexity Science and Second Language Acquisition. Applied Linguistics, 18, 141-165.
[7] Mandelbrot, B. and Hudson, R. (2006) Fractals and Finance. Tusquets, Barcelona.
[8] Burgos, J. and Moreno-Tovar, P. (1996) Zipf-Scaling Behavior in the Immune System. Biosystems, 39, 227-232.
[9] Burgos, J. (1996) Fractal representation of the immune B cell repertoire. Biosystems, 39, 19-24.
[10] Rodríguez, J. (2005) Fractal Behavior of T Specify Repertory against Poa p9 Alergeno. Revista de la Facultad de Medicina, 53, 72-78.
[11] Rodríguez, J., Prieto, S., Ortiz, L., Bautista, A., Bernal, P. and Avilán, N. (2006) Zipf-Mandelbrot Law and Mathematical Approach in Fetal Cardiac Monitoring Diagnosis. Revista Facultad de Medicina—Universidad Nacional de Colombia, 54, 96-107.
[12] Rodríguez, J. (2006) Dynamical Systems Theory and ZIPF—Mandelbrot Law Applied to the Development of a Fetal Monitoring Diagnostic Methodology. Proceedings of the 18th FIGO World Congress of Gynecology and Obstetrics, Kuala Lumpur, 5-10 November 2006.
[13] Robledo, R. and Escobar, F.A. (2010) Chronic Non-Communicable Diseases in Colombia. Bulletin of the Health Observatory, 3, 1-9.
[14] Gallo, J., Farbiarz, J. and Alvarez, D. (1999) Spectral Analysis of Heart Rate Variability. IATREIA, 12, 61-71.
[15] Harris, P., Stein, P.K., Fung, G.L. and Drew, B.J. (2014) Heart Rate Variability Measured Early in Patients with Evolving Acute Coronary Syndrome and 1-Year Outcomes of Rehospitalization and Mortality. Journal of Vascular Health and Risk Management, 10, 451- 464.
[16] Rodríguez, J., Prieto, S., Correa, C., Bernal, P., Puerta, G., Vitery, S., et al. (2010) Theoretical Generalization of Normal and Sick Coronary Arteries with Fractal Dimensions and the Arterial Intrinsic Mathematical Harmony. BMC Medical Physics, 10, 1-6.
[17] Rodríguez, J., Prieto, S., Correa, C., Bernal, P., álvarez, L., Forero, G., et al. (2012) Fractal Diagnosis of Left Heart Ventriculograms Fractal Geometry of Ventriculogram during Cardiac Dynamics. Revista Colombiana de Cardiología, 19, 18-24.
[18] Goldberger, A., Rigney, D.R. and West, B. (1990) Science in Pictures: Chaos and Fractals in Human Physiology. Scientific American, 262, 42-49.
[19] Goldberger, A.L. and West, B.J. (1987) Applications of Nonlinear Dynamics to Clinical Cardiology. Annals of the New York Academy of Sciences, 504, 195-213.
[20] Goldberger, A.L., Rigney, D.R., Mietus, J., Antman, E.M. and Greenwald, S. (1988) Nonlinear Dynamics in Sudden Cardiac Death Syndrome: Heartrate Oscillations and Bifurcations. Experientia, 44, 983-987.
[21] Pincus, S.M. (1991) Approximate Entropy as a Measure of System Complexity. Proceedings of the National Academy of Sciences of the United States of America, 88, 2297-2301.
[22] Richman, J.S. and Moorman, J.R. (2000) Physiological Time-Series Analysis Using Approximate Entropy and Sample Entropy. American Journal of Physiology—Heart and Circulatory Physiology, 278, H2039-H2049.
[23] Rodríguez, J., Correa, C., Ortiz, L., Prieto, S., Bernal, P. and Ayala, J. (2009) Evaluación matemática de la dinámica cardiaca con la teoría de la probabilidad. Revista Mexicana de Cardiología, 20, 183-189.
[24] Rodríguez, J. (2010) Proportional Entropy of the Cardiac Dynamic Systems. Physical and Mathematical Predictions of the Cardiac Dynamic for Clinical Application..Revista Colombiana de Cardiología, 17, 115-129.
[25] Rodríguez, J. (2011) Mathematical Law of Chaotic Cardiac Dynamic: Predictions of Clinic Application. Journal of Medicine and Medical Sciences, 2, 1050-1059.
[26] Rodríguez, J. (2012) Proportional Entropy Applied to the Evolution of Cardiac Dynamics. Predictions of Clinical Application. Comunidad del Pensamiento Complejo, Argentina.
[27] Rodríguez, J., Prieto, S., Correa, C., Bernal, P., Vitery, S., álvarez, L., Aristizabal, N. and Reynolds, J. (2012) Cardiac Diagnosis Based on Probability Applied to Patients with Pacemakers. Acta Médica Colombiana, 37, 183-191.
[28] Rodríguez, J., Narváez, R., Prieto, S., Correa, C., Bernal, P., Aguirre, G., Soracipa, Y. and Mora, J. (2013) The mathematical Law of Chaotic Dynamics Applied to Cardiac Arrhythmias. Journal of Medicine and Medical Sciences, 4, 291-300.
[29] Rodríguez, J., Prieto, S., Flórez, M., Alarcón, C., López, R., Aguirre, G., Morales, L., Lima, L. and Méndez, L. (2014) Physical-Mathematical Diagnosis of Cardiac Dynamic on Neonatal Sepsis: Predictions of Clinical Application.. Journal of Medicine and Medical Sciences, 5, 102-108.
[30] Rodríguez, J. (2012) New Physical and Mathematical Diagnosis of Fetal Monitoring: Clinical Application Prediction. Momento Revista de Física, 44, 49-65.
[31] Borgatta, L., Shrout, P.E. and Divon, M.Y. (1988) Reliability and Reproducibility of Nonstress Test Readings. American Journal of Obstetrics & Gynecology, 159, 554-558.
[32] Cohen, J. (1960) A Coefficient of Agreement for Nominal Scales. Educational and Psychological Measurement, 20, 37-46.
[33] Ksela, J., Avbelj, V. and Kalisnik, J.M. (2015) Multifractality in Heartbeat Dynamics in Patients Undergoing Beating-Heart Myocardial Revascularization. Computers in Biology and Medicine, 60, 66-73.
[34] Chang, M.C., Peng, C.K. and Stanley, H.E. (2014) Emergence of Dynamical Complexity Related to Human Heart Rate Variability. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 90, Article ID: 062806.
[35] Einstein, A. (1934) On the Method of Theoretical Physics. Philosophy of Science, 1, 163-169.
[36] Huikuri, H.V., Mökikallio, T., Peng, C.K., Goldberger, A.L., Hintze, U., Mogens Møller, M., et al. (2000) Fractal Correlation Properties of R-R Interval Dynamics and Mortality in Patients with Depressed Left Ventricular Function after an Acute Myocardial Infarction. Circulation, 101, 47-53.
[37] Voss, A., Schulz, S., Schroeder, R., Baumert, M. and Caminal, P. (2009) Methods Derived from Nonlinear Dynamics for Analysing Heart Rate Variability. Philosophical Transactions of the Royal Society A, 367, 277-296.
[38] Rodríguez, J. (2008) Binding to Class II HLA Theory: Probability, Combinatory and Entropy Applied to Peptide Sequences. Inmunología, 27, 151-166.
[39] Rodríguez, J. (2010) A Method for Forecasting the Seasonal Dynamic of Malaria in the Municipalities of Colombia.. Revista Panamericana de Salud Pública, 27, 211-218.
[40] Rodríguez, J., Prieto, S., Correa, C., Forero, M., Pérez, C., Soracipa, Y., Mora, J., Rojas, N., Pineda, D. and López, F. (2013) Set Theory Applied to White Cell and Lymphocyte Counts: Prediction of CD4 T Lymphocytes in Patients with Human Immunodeficiency Virus/Aids. Inmunología, 32, 50-56.
[41] Rodríguez, J., Prieto, S., Correa, C., Pérez, C., Mora, J., Bravo, J., Soracipa, Y. and álvarez, L. (2013) Predictions of CD4 Lymphocytes’ Count in HIV Patients from Complete Blood Count. BMC Medical Physics, 13, 3.
[42] Rodríguez, J. (2011) New Diagnosis Aid Method with Fractal Geometry for Pre-Neoplasic Cervical Epithelial Cells.. Revista U.D.C.A Actualidad & Divulgación Científica, 14, 15-22.
[43] Prieto, S., Rodríguez, J., Correa, C. and Soracipa, Y. (2014) Diagnosis of Cervical Cells Based on Fractal and Euclidian Geometrical Measurements: Intrinsic Geometric Cellular Organization. BMC Medical Physics, 14, 1-9.
[44] Rodríguez, J., Prieto, S., Catalina, C., Dominguez, D., Cardona, D.M. and Melo, M. (2015) Geometrical Nuclear Diagnosis and Total Paths of Cervical Cell Evolution from Normality to Cancer. Journal of Cancer Research and Therapeutics, 11, 98-104.

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.