Application of An Exponential Mathematical Law of Cardiac Dynamics In 16 Hours

Introduction and objectives: nonlinear dynamics and fractal geometry have allowed the advent of an exponential mathematical law applicable to diagnose cardiac dynamics in 21 hours, however, it would be benecial to reduce the time required to diagnose cardiac dynamics with this method in critical scenarios, in order to detect earlier complications that may require medical attention. The objective of this research is to conrm the clinical applicability of the mathematical law in 16 hours, with a comparative study against the Gold Standard. Methods: There were taken 450 electrocardiographic records of healthy patients and with cardiac diseases. A physical-mathematical diagnosis was applied to study cardiac dynamics, which consists of generating cardiac chaotic attractors based on the sequence of heart rate values during 16 hours, which were then measured with two overlapping grids according to the Box-Counting method to quantify the spatial occupation and the fractal dimension of each cardiac dynamic, with its respective statistical validation. Results: The occupation spaces of normal dynamics calculated in 16 hours were compatible with previous parameters established, evidencing the precision of the methodology to differentiate normality from abnormality. Sensitivity and specicity values of 100% were found, as well as a Kappa coecient of 1. Conclusions: it was possible to establish differences between cardiac dynamics for 16 hours, suggesting that this method could be clinically applicable to analyze and diagnose cardiac dynamics in real time.


Introduction
The theory of dynamic systems allows analyzing the state and evolution of systems, which can be predictable or unpredictable. Its analysis has been possible through mathematical procedures that evaluate the changes of its dynamic variables over time [1]. The evolution of the systems can be represented graphically in the phase space from attractors. According to the trajectories of the dynamics [2], the different attractors originate; if the trajectories exhibit a predictable character, punctual or cyclic attractors are obtained and if the trajectories have an unpredictable character, the chaotic attractors, characterized by their irregularity, originate. This characteristic means that its quanti cation is done using fractal geometry, developed by Benoit Mandelbrot [3,4].
Cardiovascular diseases are consolidated as one of the most important health problems of the century.
According to estimates by the World Health Organization, the number of deaths from this cause is projected to increase to 25,000.00 in the year 2030 [5], with low-and middle-income countries being the most affected [6]. Considering the impact of these diseases on public health, it is necessary to develop and improve methods that allow the timely detection of cardiovascular diseases. The Holter is a diagnostic tool, with which it is possible to detect alterations of the cardiac rhythm [7]; in Intensive Care Units, continuous electrocardiographic recordings are made through monitors.
The analysis of the RR interval guides the clinical experts in the differentiation of normality and disease [5,8,9]; however, there are still limitations in its clinical application, given that its evaluation is made from the homeostatic notions, according to which, normality is related to regular patterns, while in aging or abnormality there is a decrease in capacity to maintain the heart rate [10]. Some studies show that this perspective presents limitations and has established new forms of analysis based on theories and methods of physics and mathematics, such as dynamic systems, the law of chaos, fractal geometry, among others [11][12][13][14].
Goldberger et al., Have conducted studies based on the theory of dynamic systems; these investigations allowed the establishment of a new conception of normality and illness, according to which the disease is associated with very irregular or very periodic dynamics and normality is found between these two behaviors [15]. From this line of thought, Huikuri et al, established new predictive mortality rates in patients with acute coronary syndrome such as acute myocardial infarction and severely decreased ejection fraction [16].
From this same perspective, Rodríguez and Cols [17] developed an exponential mathematical law based on dynamic systems and fractal geometry, with which it was possible to make precise diagnoses, determine the severity of cardiac dynamics, differentiate normality, disease and evolution towards exacerbation in 21 hours; Likewise, it was possible to establish the totality of possible cardiac dynamics.
Its clinical usefulness was corroborated in different investigations, including a blind study with 115 Holters [18], evidencing sensitivity values, 100% speci city and a Kappa coe cient of 1. This methodology was applied in subjects enrolling with arrhythmias [19][20], a study was also developed where the ability of this exponential mathematical law to perform diagnoses in 18 hours was demonstrated [21]. The purpose of the present investigation is to con rm the clinical applicability of the aforementioned law in 16 hours.

Procedure
The clinical diagnoses established by the expert were masked. The maximum and minimum values of the heart rate were taken every hour and the beats per hour, in 21 and 16 hours. With these data, a quasirandom sequence of heart frequencies was generated by an equiprobable algorithm. Then, the cardiac attractors were constructed, plotting one frequency against the next in time in the phase space. The fractal dimension was calculated with the Box-Counting method (Equation 2) through the superposition of two grids, with which the spatial occupancy in Kp and Kg of the attractors was determined.
From Equation 3, the mathematical diagnosis of the records was determined, according to the previously developed law17-20, according to which, the dynamics of acute cardiac pathologies exhibit occupation spaces in the Kp grid below 73, normality it presents occupation spaces in this grid over 200, and the evolution towards the disease presents occupation spaces between 73 and 200. The mathematical diagnosis was compared in 16 and 21 hours, with which the agreement between both was established.

Statistical analysis
The diagnoses established by the clinical expert were taken as Gold Standard, according to conventional clinical parameters and were compared with the mathematical diagnosis in 16 hours, once the agreement of the diagnoses established by the mathematical law was con rmed at 16 and 21 hours. For the purposes of the statistical analysis, cases with acute and normal disease were taken. False negatives, false positives, true positives and true negatives were calculated. From a 2x2 contingency table, sensitivity and speci city were determined. Also, the Kappa coe cient was calculated.
Ethical aspects

Results
The diagnosis established by clinical parameters of the different continuous and ambulatory electrocardiographic records is shown in Table 1. For the cardiac dynamics evaluated in 21 hours, the fractal dimension of the normal attractors was found between 0,9170 and 1,9068, and for the Abnormal dynamics were found between 0,8113 and 1,9349. The fractal dimensions of the normal cardiac chaotic attractors in 16 hours showed values between 0,9251 and 1,8744, while in the case of abnormal dynamics they presented values between 0,8212 and 1,9926. This is in line with previous ndings, which show that the fractal dimension does not establish objective distinctions between the different cardiac dynamics, neither for 16 nor for 21 hours (see Table 1). There was agreement between the mathematical diagnosis in 16 and 21 hours in all the cases of the study.  Table 2). Regarding the grid Kg, the occupation spaces in 21 hours for the dynamics without alterations were between 60 and 188, while for the abnormal dynamics were between 8 and 102. The occupation spaces for the dynamics in the grid Kg in 16 hours they presented values for normality between 60 and 188, and for abnormality between 6 and 103 (see Table 2). Figure 1 shows three attractors, corresponding to normality, acute disease and evolution towards exacerbation. This gure shows the decrease in the spatial occupation of the chaotic attractor when the dynamics approaches the acute disease, which con rms the previously mentioned ndings.

Discussion
This is the rst work in which the clinical applicability and the diagnostic capacity of the exponential chaotic mathematical law are corroborated, in the context of a reduction of the evaluation time to 16 hours in a blind study with 450 cases, evidenced that the methodology allows establish precise quantitative differences between normality and disease, with values of sensitivity and speci city of 100% and a Kappa coe cient of 1, after a comparison between the mathematical and conventional diagnosis. This mathematical method objectively quanti es the severity of the pathologies, establishing itself as a predictive and diagnostic methodology capable of differentiating normal and pathological states, by decreasing the occupation spaces of the attractors in the fractal space of Box Counting.
The mentioned mathematical exponential law, developed from an inductive reasoning coming from the theoretical physics, allowed to establish the totality of possible cardiac dynamics. This method could be automated for its wide use in the clinic, given its objectivity and reproducibility. He has shown his ability to make mathematical distinctions between cardiac dynamics [18], even in patients taking arrhythmias [19,20,22]. In the present work, the evaluation time of cardiac dynamics was reduced, which would be important at the clinical level due to the greater opportunity in the establishment of the diagnoses.
Classical physiology has been governed by homeostatic conceptions, according to which normality is evidenced by regular or periodic behaviors [22]. However, this approach has been controverted by studies carried out from the theory of dynamic systems [15]. These investigations have proposed new routes in the analysis of heart rate variability, which is usually evaluated based on pre-established homeostatic notions. With the work done on the basis of dynamic systems, partial results have been obtained, associating the decrease in the variability of the heart rate with disease [23]. On the other hand, the mathematical law applied in this research allows the realization of precise diagnoses and the detection of evolution towards disease, based on the self-organization of the cardiac system, establishing clear numerical limits.
The theory of chaos [1,13,24], quantum mechanics [25] and statistical mechanics [26,27] have been the starting point for predictions in science, regardless of causal relationships. This acausal perspective, typical of modern theoretical physics, is the substratum of the exponential mathematical law object of this work, for which, it makes independent diagnoses of statistical, population or other considerations.
Physical and mathematical theories have led to diagnoses and predictions of cardiac systems, based on the Zipf / Mandelbrot law [28,29], the probability theory and entropy [30]. Likewise, diagnoses have been established at the cellular level [31,32,33] and arterial [34]; Predictions of malaria have been made in Colombia [35] and the population of CD4 T lymphocytes has been determined in patients seropositive for HIV [36]. Predictions have also been developed in immunology [37] and mortality in the Intensive Care Unit [38].

Conclusions
In this work, the clinical and diagnostic utility of the exponential mathematical law was con rmed. It was evidenced that he could adequately differentiate the normal cardiac pathological dynamics, depending on the occupation spaces of the chaotic attractors, showing their predictive character.
Declarations Figure 1 Cardiac chaotic attractors exhibiting evolution from normality towards chronic and acute disease.