JMP  Vol.6 No.13 , October 2015
Physical Mathematical Evaluation of the Cardiac Dynamic Applying the Zipf-Mandelbrot Law
Abstract: 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.
Cite this paper: 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.

[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.