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Similitude methods in the model of the rabbit intact sinoatrial node

Galanin V.V.

Medical university Reaviz, Russia, 443001, Samara, Chapayevskaya st 227, galanin_v@hotmail.com

This work demonstrates how the methods of the similarity theory can be used to reduction of the mathematical model of the intact sinoatrial node of the rabbit heart. The study of similarity allows decreasing the order of the equation by reducing the number of quantities that define the object. Such simplifications are useful for simulation of the complex biological systems.

In this study, the intact sinoatrial node and the associated atrial myocardium are simulated as a chain of pacemaker cells of the sinus node connected to three chains of atrial contractile cells. The presented model is based on the reduction of the detailed model by H. Zhang et al. [1], in which the oscillations of the membrane potential of pacemaker cells is described using a system of nonlinear differential equations taking into account the gradual change in their electrophysiological properties in the direction from the center of the sinus node to its border [2]. After the similitude transformation of the system of differential equations, dimensionless power coefficients appear in it, which are the similarity parameters (invariants of the similarity transformation). The obtained similarity parameters are numerically the same for all similar cell chains. Similarity parameters describe the investigated physical object as fully as the primary dimensional quantities. In this case, the number of dimensionless similarity parameters is less than the number of quantities that initially define the model. By the numerical value of the similarity parameters, it can be concluded about the mode of electrical activity of the pacemaker cells and the atrial contractile cells interacting with them. Also, the problem of lowering the order of the system of differential equations based on reducing the number of interacting elements in the model while maintaining the most important information about the cellular system as a whole has been solved.

References.

1. Zhang H.G., Holden A.V., Kodama I., et al. // Am. J. Physiol. Heart Circ. Physiol. 279, 2000, P.397.

2. Inada S., Zhang H., Tellez J.O., Shibata N., Nakazawa K., et al. // PLoS One 9 (4), e94565. doi: 10.1371/journal.pone.0094565, 2014.

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