On the Ill-conditioned Nature of the Intracardiac Inverse Problem

Multi-electrode catheters can be placed transvenously and positioned on the atrial endocardial surface in order to sample the chaotic electrical activity taking place during atrial fibrillation. We consider here the possibility of placing an array of electrodes over a relatively small, and hence roughly planar, region of the atrial surface in order to examine local activity patterns. This provides a spatially coarse but temporally fine sampling of electrical activity that can be expressed at each point in time as the convolution of the true electrical excitation of the tissue with a hyperbolic point spread function. We demonstrate the deconvolution of sampled signals using a polynomial approximation of the true electrical activity. When the deconvolution is unconstrained the inverse problem is poorly conditioned, showing that a high spatial sampling rate is required for accurate reconstructions of atrial activity in the vicinity of the electrode array. We discuss ways in which the conditioning of the problem might be improved through the application of constraints on the solution.