Dimensionality in cardiac modelling

The development of mathematical models of the heart has been an ongoing concern for many decades. The initial focus of this work was on single cell models that incorporate varyingly detailed descriptions of the mechanisms that give rise to experimentally observed action potential shapes. Clinically relevant heart rhythm disturbances, however, are multicellular phenomena, and there have been many initiatives to develop multidimensional representations of cardiac electromechanical activity. Here, we discuss the merits of dimensionality, from 0D single cell models, to 1D cell strands, 2D planes and 3D volumes, for the simulation of normal and disturbed rhythmicity. We specifically look at models of: (i) the origin and spread of cardiac excitation from the sino-atrial node into atrial tissue, and (ii) stretch-activated channel effects on ventricular cell and tissue activity. Simulation of the spread of normal and disturbed cardiac excitation requires multicellular models. 1D architectures suffer from limitations in neighbouring tissue effects on individual cells, but they can (with some modification) be applied to the simulation of normal spread of excitation or, in ring-like structures, re-entry simulation (colliding wave fronts, tachycardia). 2D models overcome many of the limitations imposed by models of lower dimensionality, and can be applied to the study of complex co-existing re-entry patterns or even fibrillation. 3D implementations are closest to reality, as they allow investigation of scroll waves. Our results suggest that 2D models offer a good compromise between computational resources, complexity of electrophysiological models, and applicability to basic research, and that they should be considered as an important stepping-stone towards anatomically detailed simulations. This highlights the need to identify and use the most appropriate model for any given task. The notion of a single and ultimate model is as useful as the idea of a universal mechanical tool for all possible repairs and servicing requirements in daily life. The ideal model will be as simple as possible and as complex as necessary for the particular question raised.