AN ANALYSIS OF THE SHOCK STRENGTH NEEDED TO ACHIEVE DEFIBRILLATION IN A SIMPLIFIED MATHEMATICAL MODEL OF CARDIAC TISSUE

The pumping function of the heart is driven by an electrical wave traversing the cardiac muscle in a well-organized manner. Perturbations to this wave are referred to as arrhythmias. Such arrhythmias can, under unfortunate circumstances, turn into fibrillation, which is often lethal. The only known therapy for fibrillation is a strong electrical shock. This process, referred to as defibrillation, is routinely used in clinical practice. Despite the importance of this procedure and the fact that it is used frequently, the reasons for defibrillation's effectiveness are not fully understood. For instance, theoretical estimates of the shock strength needed to defibrillate are much higher than what is actually used in practice. Several authors have pointed out that, in theoretical models, the strength of the shock can be decreased if the cardiac tissue is modeled as a heterogeneous substrate. In this paper, we address this issue using the bidomain model and the Courtemanche ionic model; we also consider a linear approximation of the Courtemanche model here. We present analytical considerations showing that for the linear model, the necessary shock strength needed to achieve defibrillation (defined in terms of a sufficiently strong change of the resting state) decreases as a function of an increasing perturbation of the intracellular conductivities. Qualitatively, these theoretical results compare well with computations based on the Courtemanche model. The analysis is based on an energy estimate of the difference between the linear solution of the bidomain system and the equilibrium solution. The estimate states that the difference between the linear solution and the equilibrium solution is bounded in terms of the shock strength. Since defibrillation can be defined in terms of a certain deviation from equilibrium, we use the energy estimate to derive a necessary condition for the shock strength.