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Research Article | DOI: https://doi.org/10.31579/2641-0419/008
1 EC-leasing.Tikhonov Moscow Institute of Electronics and Mathematics, Higher School of Economics, Russia.
*Corresponding Author: Novopashin MA, EC-leasing.Tikhonov Moscow Institute of Electronics and Mathematics, Higher School of Economics, Russia.
Citation: Shmid AV, Novopashin MA, Zimina EY , Berezin AA, Big Data Clustering in Cardiology Based on Modeling of Electrical Dynamics of the Heart in the form of Fermi-Pasta-Ulam Auto-Recurrence as a New Tool for the Study of Cardiac Activity. JClinical Cardiology and Cardiovascular Interventions. 1(2); Doi: 10.31579/2641-0419/008
Copyright: © 2018. Berezin AA, This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Received: 20 July 2021 | Accepted: 27 July 2018 | Published: 13 August 2018
Keywords: Electrocardiogram, Veloergometer, Pacemaker, Dechaotization, Cardiobarometer. General Clustering.
The mass application of mobile cardiographs already leads to both explosive quantitative growth of the number of patients available for ECG study, registered daily outside the hospital (Big DATA in cardiology), and to the emergence of new qualitative opportunities for the study of long-term oscillatory processes (weeks, months, years) of the dynamics of the individual state of the Cardiovascular system of any patient
The article demonstrates that new opportunities of long-term continuous monitoring of the Cardiovascular system state of patients ' mass allow to reveal the regularities (DATA MINING) of Cardiovascular system dynamics, leading to the hypothesis of the existence of an adequate Cardiovascular system model as a distributed nonlinear self-oscillating system of the FPU recurrence model class [1]. The presence of a meaningful mathematical model of Cardiovascular system within the framework of the FPU auto–recurrence [2], as a refinement of the traditional model of studying black box, further allows us to offer new computational methods for ECG analysis and prediction of Cardiovascular system dynamics for a refined diagnosis and evaluation of the effectiveness of the treatment.
In our model of the heart based on the of the FPU recurrence phenomenon, the basic concepts are traditionally the "states" of the heart as a registered oscillatory process. However, since the heart is a self-oscillating system (in contrast to the previously studied theoretical and experimental models of the FPU recurrence based on passive nonlinear systems (plasma, nonlinear electrical lines, deep and shallow water waves dynamics [3,4,5,6] the perturbation energy is not required to start the oscillations, and in the study of the heart electrical dynamics, we can introduce the concept of the auto FPU recurrence.
The ideal ECG in the hypothetical case of the absence of any external influences and represents a periodically reappearing FPU auto-recurrence, characterizing the patient's condition.
In real life, various physical and emotional loads, therapy, etc. are external to the heart model "disturbances" and will be periodically displayed in the picture of the FPU auto-recurrence. Their influence on auto-recurrence picture can theoretically be revealed, allowing to establish biologically significant parameters of the equations of the FPU recurrence model, diagnosing the actual biological condition of the heart (for example, the area of the myocardium) predicting the future picture of the dynamics of cardiac activity.
Theoretically, the model of the FPU recurrence may contain an infinite number of states of the heart activity. However, in case of the ECG, for reasons of common sense for a practical cardiologist all the infinite number of pictures of auto-recurrences of heart conditions should fit into a finite number of possible diagnoses. Therefore, the implementation of the clustering process is constructive: the integration of similar self-oscillating processes of the heart into subsets (clusters) based on belonging to the diagnosis.
In this case, the set of clusters becomes finite which seems obvious.
However, from the FPU auto-recurrence model [2] it also follows that the real picture of the heart's behavior can be much more complicated: in the ECG patterns may be such variants of heart behavior that "do not fit" any of the known diagnoses, and reflect the processes of the heart transition from state to state, including in completely healthy patients.
Theoretical (before the experiment) prediction of the possibility of such "strange" ECGs and their subsequent experimental detection can serve as a proof of the adequacy of the FPU model with automatic recovery of the physical picture of the heart.
The FPU model with the introduction of the concept of "auto-recurrence" [2] can adequately describe many processes taking place in the heart, at a level practically acceptable for solving the problem of clustering of ECG diagnosis.
The solution of the problem of content-based ECG clustering, in turn, opens up opportunities for a better solution to the problem of diagnosis and prediction of scenarios of heart disease development.
The phenomenon of the canonical FPU recurrence in passive systems was first described as a result of numerical study of solutions of differential – difference equations describing a chain of nonlinear coupled vibrators [1]. In these model chains, the phenomenon of dissipation has not been included.
To describe real dynamic processes, such as cardiac activity, a more adequate model was required, capable of describing the recurrences in autonomously functioning or self-oscillating systems.
A significant contribution in this direction was the work of American researchers Zabusky and Kruskal, who proposed to describe the FPU recurrence within the framework of the Korteweg de Vries (KDV) equation with periodically changing boundary and initial conditions [6].
Using the results of this work, it is possible to simplify the solution of the KDV equation in the form of knoidal waves, replacing them with the solutions of the Van der Pol equation, close to harmonic, periodic and using the low-frequency changes in boundary and initial conditions by the relaxation solutions of the Van der Pol equation.
At the same time, if we apply the theorem on the possibility of replacing the wave links by the delayed ones [6] , then we can present a mathematical model of the electrical activity of the heart (ECG) in the form of the FPU auto-recurrence, described in the framework of coupled equations of Van der Pol with a time lag [7].
The computer study of the FPU auto-recurrence model [2] shows that in case of intact myocardium and under the absence of external influence at main heart 1 Hz frequency, the model reproduces the oscillations similar to a regular ECG (Fig. 1) with its Fourier spectrum given in Fig.2.
The spectra shown in Figs. 3 and 4 correspond qualitatively to the model spectrum of the FPU auto-recurrence of the normal ECG obtained from the study of the FPU auto-recurrence model [2].
The obtained data result in a prediction of consequences of resonant external influence at 1 Hz frequency following from the hypothesis. In particular, an enough intense external resonant influence can discompose the regular model oscillations. The computer study shows that increasing the amplitude of the external perturbation by 50% brings the breakage of the rhythm of model oscillations corresponding to the state of the heart fibrillation (Fig.5.) This can be called a model infarct. At the same time, the hypothesis predicted the threshold of the external perturbation amplitude that would stop the discomposing of the model.
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