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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.

Since the first half of twentieth century, linguistics has an empirical law which quantifies the relationship between the range and frequency of words in a text, called Zipf law [

This law has been the foundation of new methodologies to interpret the information in different areas [

Cardio Vascular Diseases (CVD), according to reports from the World Health Organization, are the leading cause of death worldwide. In Colombia, DANE (Departamento Administrativo Nacional de Estadística) reported for 2009 that the CVD were one of the five principal causes of mortality, among them was ischemic heart disease with 28,650 cases and cerebrovascular disease with 14,555 cases [

Classically, in cardiac physiology, periodic normal behaviors that characterize a system were searched, so the analysis of heart rate variability (HRV) has been widely studied [

Pioneering work done from this perspective, like that done by Goldberger et al, allowed observing spectral abrupt changes and oscillations sustained of low frequency in patients with high risk of sudden cardiac death; the loss of complexity was associated with reduced dynamic response before sudden cardiac death and aging [

It has been developed new methodologies based on theoretical physics and mathematics, which have allowed assessing cardiac dynamic system. These diagnostic methods characterize normal and diseased cardiac dynamics, achieving to foresee states that evolve to the intensification. These new mathematical and physic diagnoses are based on theories and concepts like the probability theory [

The purpose of this work is to develop a methodology for clinical diagnostic aid application to evaluate the complexity degree of adult cardiac dynamics from Zipf’s law―Mandelbrot, differentiating the degree of complexity of normal dynamics regarding dynamics with acute myocardial infarction.

Definitions:

Ranges of 15 lat/min: heart rate (HR) ranges, which includes all HR within a range of 15 beats/min.

Zipf-Mandelbrot statistical fractal dimension: It is obtained through of logarithmic linearization, in this case, of occurrence frequencies distribution of ranges of 15 beats/min.

where D: is the statistical fractal dimension,

22 Holter registers were selected which correspond to patients over 21 years old. Those come from the Insight database, whose Holter registers were assessed by an experienced electrophysiologist. For the mathematical induction a Holter register was taken with a normal cardiac dynamic and one with acute myocardial infarction (AMI), the values of minimum and maximum frequency were taken during each hour of each Holder test. Subsequently, for the application of Zipf-Mandelbrot law, the distribution number heart rates are in ranges of 15 beats/min was found, as well as the frequency of occurrence for each range. These values were organized in order from highest to lowest, associating to each frequency appearance a hierarchical range from one to the highest value, and increasing as the frequency of occurrence values were decreasing.

These values, the frequency of appearance associated to hierarchical ranks, are brought to a graph, seeking a hyperbolic behavior (

For statistical analysis, a blind study was conducted with 20 Holters, which included 10 normal Holter registers and 10 with acute heart disease. The clinical findings of Holter registers and the diagnosis made by clinical

parameters were assumed as Gold standard. To do this the specificity and sensivity was calculated through a binary classification, where true positives (TP) corresponding to the number of abnormal patients according to the Gold-Standard and are within the mathematical values corresponding to abnormality, false positives (FP) are those which mathematically have been diagnosed as normal but whose clinical values correspond to abnormal patients, and finally true negative (TN) defined as the number of heart registers clinically diagnosed as normal and whose mathematical values also correspond to the normal.

The correlation between the physical mathematical values and the conventional clinical diagnosis was calculated through Kappa coefficient using the following formula [

where: Co: number of concordances observed, that is, the number of patients with the same diagnosis according to the new methodology proposed and the Gold- Standard. To: All the observations, that is, all normal and disease cases. Ca: random concordances, which are calculated according to the following formula:

where f_{1} is the number of patients with mathematical values within normal limits, C_{1} is the number of patients clinically diagnosed within normality; f_{2} is the number of patients showing abnormalities associated to mathematical values, C_{2} is the number of patients clinically diagnosed with any kind of pathology and To is the total number of normal and abnormal cases.

Once the fractal dimensions of the two Holter statistics records used in the induction had been realized, it was found that for normal Holter, this value was 0.9225 (

infarction, which evinced the optimal quantification of the differentiation between this type of dynamics.

Hyperbolic behavior between the frequencies of appearance of the ranges 15 beats/min and ranges partners in the implementation of Zipf’s law was evident-Mandelbrot, and likewise succeeded in establishing the linearization for all the dynamics, which can be seen is the R2 that were found, which ranged between 0.5924 and 0.9016 (see

P | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Normal | 0.9225 | 0.8343 | 0.7406 | 0.7046 | 0.9483 | 0.7505 | 0.8741 | 0.7145 | 0.9002 | 0.9179 | 0.7948 |

Acute dynamic | 0.5267 | 0.6707 | 0.4716 | 0.4659 | 0.6093 | 0.6057 | 0.5640 | 0.4228 | 0.5488 | 0.5823 | 0.5660 |

P | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Normal | 0.8251 | 0.835 | 0.8808 | 0.8571 | 0.8356 | 0.808 | 0.7164 | 0.8005 | 0.7815 | 0.6422 | 0.7664 |

Acute dynamic | 0.8327 | 0.8896 | 0.5924 | 0.8183 | 0.6981 | 0.5961 | 0.7826 | 0.7058 | 0.8264 | 0.9016 | 0.7433 |

This is the first work in which a diagnostic aid methodology was developed from application of the Zipf-Man- delbrot law into cardiac dynamics, by analyzing hierarchical ordering of the maximum and minimum heart rates in each hour, in which was found a hyperbolic behavior for heart rates that were in ranges of 15 beats/min, showing a self-organization statistic fractal of the dynamic. It was evidenced that the complexity degree of normal cardiac dynamics was higher than the dynamic with acute pathologies for all Holters, showing quantitatively how the cardiac dynamics can be sharpen. This new methodology constitutes a new way to assess the complexity degree of cardiac dynamics, which simplify other methods of assessment [

This work is based on the theoretical physics method [

For example, the first work developed in the context of the theory of dynamical systems and fractal geometry contradicted the principle of homeostasis. Goldberger et al. [

Other works based on this investigation line have resulted in a new clinical diagnosis and practices predictions in cardiology, as already described; likewise predictions have been established in other fields like immunology [

We thank the Universidad el Bosque, especially the Investigations Division of the Universidad el Bosque, for their support. This work is part of the products made in the PIC-2013-369 project approved by the Research Division of the Universidad el Bosque. We thank the Research Center Clinic Country, especially Dr. Tito Tulio Roa, Director of Medical Education; Dr. Jorge Ospina, Medical Director; Dr. Alfonso Correa, Director of Research Center; also Dr. Adriana Lizbeth Ortiz, epidemiologist, and Silvia Ortiz, head nurse; nurse Sandra Rodríguez and Juan Camilo Benítez, bacteriologist of clinical studies, for the continued support of our research group.

Javier OswaldoRodríguez,Signed EsperanzaPrieto,Sandra CatalinaCorrea,FernánMendoza,GioraWeiz,María YolandaSoracipa,NellyVelásquez,Juan MauricioPardo,MiguelMartínez,FreddyBarrios, (2015) Physical Mathematical Evaluation of the Cardiac Dynamic Applying the Zipf-Mandelbrot Law. Journal of Modern Physics,06,1881-1888. doi: 10.4236/jmp.2015.613193