Some cardiac arrhythmias are due to re-entrant propagation, in which the same wavefront repeatedly re-invades the same piece of tissue after propagating around an anatomical or functional block. Monomorphic ventricular tachycardia is probably produced by simple re-entry [1, 2] and the order remaining in ventricular fibrillation [3, 4] may be due to re-entrant waves. The relatively thick wall of the ventricle means that propagation in ventricular muscle could be a predominantly three-dimensional phenomenon that occurs in an anisotropic and heterogeneous tissue, or could be explained by two-dimensional phenomena that result from the excitation properties rather than heterogeneity and anisotropy. In this paper we obtain the characteristics of propagation in one- and two-dimensional, homogeneous model of ventricular tissue, and use these to account for the linear regions of temporal conduction blocks seen in mapping studies [5, 6], to interpret the changes in conduction velocity of a re-entrant wave around an extended obstacle [7], to explain why it is difficult to establish re-entrant propagation in the healthy ventricle, and to quantitatively assess single shock and resonant drift methods of eliminating re-entry from ventricular tissue.
There are a number of published and available models for ventricular excitation that summarise the results of voltage clamp experiments on ventricular tissue and cells -- these include the Beeler-Reuter model [8], the Oxsoft guinea pig ventricular cell model specified in Noble et al. [9, 10], the Nordin [11] model and the phase 2 Luo-Rudy [12] model and its recent modification [13]. None of these models are definitive, they all represent steps in an on-going process of modelling the behaviour of different types of ventricular cells by a description of membrane currents and pumps, and intracellular ion binding and concentration changes [14].
In this paper we use equations of the Oxsoft guinea pig ventricle model [9, 10], later referred to as OGPV. These equations provide a convenient starting point, and can be modified to simulate e.g. the effects of epicardial to endocardial changes in ventricular action potential described by Antzelevitch [15, 16], long QT syndrome [17], and ischemia - see Boyett et al. [18]. We construct an excitable medium model for mammalian ventricular tissue by incorporating ordinary differential equations for ventricular cell excitability into a reaction-diffusion system of differential equations with the voltage diffusion coefficient selected to give an appropriate conduction velocity. For a homogeneous two dimensional medium, the effects of homogeneous anisotropy can be included by a simple rescaling of coordinates.