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. Re-entrant spirals have been optically mapped in thin slices of epicardial tissue (Jalife & Davidenko 1993) and on the epicardial surface of the rabbit ventricle (Gray et al. 1995). 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 re-entrant propagation in a two-dimensional, homogeneous model of ventricular tissue, and use these to account for the linear regions of unidirectional conduction blocks seen in mapping studies (Davidenko et al. 1993), to explain why it is difficult to establish re-entrant propagation in the healthy ventricle, and to quantitatively assess low amplitude, repetitive stimulation by a spatially uniform electric field as a means of eliminating re-entry from ventricular tissue. A spiral wave in a two-dimensional, homogeneous, isotropic excitable medium provides a model for re-entry. Spiral waves rotate around a central core, and may be characterized by their period of rotation, size of core, and movement of the tip of the spiral. At any specified instant in time a rotating spiral wave has a location, given by the position of its tip, and a spatial orientation of rotation phase.
A spiral wave can be forced to move by a spatially uniform, time periodic perturbation of appropriate frequency (Davydov et al. 1988). Resonant drift in the location of a spiral occurs when the frequency of perturbation is the same as the frequency of rotation of the spiral. In principle, resonant drift under feedback control could provide a means of eliminating re-entrant activity in cardiac tissue (Biktashev & Holden 1994, 1995). This will be practical only if any re-entry is eliminated within a reasonable time, say less than 30 s, and so estimation of the velocities that can be achieved by resonant drift is important in assessing the feasibility of resonant drift as a means of controlling re-entrant arrhythmias.
We construct an excitable medium model for mammalian ventricular tissue by incorporating ordinary differential equations for ventricular cell excitability into a partial differential equation model. There are a number of models available for ventricular excitation that summarize the results of voltage clamp experiments -- these include the Beeler-Reuter (1989) model, the Oxsoft guinea-pig ventricular cell model specified in (Noble 1991) and the phase 2 Luo-Rudy (1994) model. None of these models are definitive, they all represent steps in an on-going process of modelling the behaviour of ventricular cells by a description of membrane currents and pumps, and intracellular ion binding and concentration changes (Noble, 1995). Spiral waves solutions for rabbit atrial tissue models (Biktashev & Holden 1995, and Winslow et al. 1995), show a reasonable agreement with experimental data. The Beeler-Reuter simplified ventricular model (Beeler & Reuter 1977) has been incorporated into two-dimensional tissue models, and re-entrant solutions characterised by Courtemanche & Winfree (1991) and Efimov et al. (1995). Study of current ventricular models has been inhibited by the computational costs of this problem, which is a reflection of the stiffness of the kinetics of ventricular excitability.
In this paper we use equations of the Oxsoft guinea pig ventricle model. These equations provide a convenient starting point, as they have been extensively used to reproduce experimental results e.g. see Noble et al., 1991, Kiyosue, et al., 1993, LeGuennec & Noble, 1994, Rice et al. 1995. There are alternative models of mammalian ventricular cells, the most recent is that of Luo & Rudy (1994). The Oxsoft equations for cell excitability are incorporated into a reaction-diffusion equation with the voltage diffusion coefficient (hence, spatial scaling) selected to give an appropriate conduction velocity. Re-entrant spiral waves are initiated from a cut wavefront, or a twin-pulse protocol, and the evolution of their meander pattern followed. The velocity of resonant drift, produced by spatially uniform perturbations is obtained.