Integral curves and the variational principle of intermittent normal conduction in the tachycardia-dependent right bundle branch block.

Abstract

Electrocardiographic (ECG) tracings of atrial fibrillation with the tachycardia-dependent right bundle branch block (RBBB) in a patient under digitalis therapy are presented, in which supernormal phase of intraventricular conduction was well-documented. The development of long intervals terminated by a normally conducted beat was attributed to the occurrence of concealed atrio-ventricular (A-V) conduction of atrial fibrillation impulse during the supernormal phase. The ventricular interval caused by a normally conducted beat (x seconds) in the interval of 1.01-1.50 seconds (s) was transformable into from of In(x + a) + l, where parameter a = 0, 0.04, 8.08 and 0.16. The vector differential equation dy/dx = ln (x + a) + 1l which is piecewise continuous and integrable, was assumed to describe the process of optimal control. This is equivalent to the problem of finding Green's function of the region, i.e. to solving, the Dirichlet problem for the region. The solution curves given by y = (x + a) ln(x + a) + l can be interpreted as distributions. This is the first study to apply integral curves and the variational principle to mathematical modelling of normal A-V conduction in tachycardia-dependent RBBB with atrial fibrillation. Furthermore, this is the first topological study to derive some definite properties of states of the phase space of a dynamical system. There are structurally stable vector field points so-called attractors on a differential manifold or surface. The optimal control is characterized by the fixed point property. In general, mathematical modelling may be applicable to the tachycardia-dependent RBBB when the cycle lengths caused by normally conducted beats range from 1.00 to 1.50 s as a feasible region. A solution of Dirichlet problem in the region was found. It was derived from this study that the normally conducted beats appeared as period doubling of 0.38-0.59 s, i.e. the interval of supernormal phase.