A bidomain reaction-diffusion model of the human heart was developed, and potentials resulting from normal depolarization and repolarization were compared with results from a compatible monodomain model. Comparisons were made for an empty isolated heart and for a heart with fluid-filled ventricles. Both sinus rhythm and ectopic activation were simulated. The bidomain model took 2 days on 32 processors to simulate a complete cardiac cycle. Differences between monodomain and bidomain results were extremely small, even for the extracellular potentials, which in case of the monodomain model were computed with a high-resolution forward model. Propagation of activation was 2% faster in the bidomain model than in the monodomain model. Electrograms computed with monodomain and bidomain models were visually indistinguishable. We conclude that, in the absence of applied currents, propagating action potentials on the scale of a human heart can be studied with a monodomain model.
Computing the potentials on the heart's epicardial surface from the body surface potentials constitutes one form of the inverse problem of electrocardiography. An often-used approach to overcoming the ill-posed nature of the inverse problem and stabilizing the solution is via zero-order Tikhonov regularization, where the squared norms of both the surface potential residual and the solution are minimized, with a relative weight determined by a so-called regularization parameter. This paper looks at the composite residual and smoothing operator (CRESO) and L-curve methods currently used to determine a suitable value for this regularization parameter, t, and proposes a third method that works just as well and is much simpler to compute. This new zero-crossing method selects t such that the squared norm of the surface potential residual is equal to t times the squared norm of the solution. Its performance was compared with those of the other two methods, using three stimulation protocols of increasing complexity. The first of these protocols involved a concentric spheres model for the heart and torso and three current dipoles placed inside the inner sphere as the source distribution. The second replaced the spheres with realistic epicardial and torso geometries, while keeping the three-dipole source configuration. The final simulation kept the realistic epicardial and torso geometries, but used epicardial potential distributions corresponding to both normal and ectopic activation of the heart as the source model. Inverse solutions were computed in the presence of both geometry noise, involving assumed erroneous shifts in the heart position, and of Gaussian measurement noise added to the torso surface potentials. It was verified that in an idealistic situation, in which correlated geometry noise dominated the uncorrelated Gaussian measurement noise, only the CRESO approach arrived at a value for t. Both L-curve and zero-crossing approaches did not work. Once measurement noise dominated geometry noise, all three approaches resulted in comparable t values. It was also shown, however, that often under low measurement noise conditions none of the three resulted in an optimum solution.
The performance of two methods for selecting the corner in the L-curve approach to Tikhonov regularization is evaluated via computer simulation. These methods are selecting the corner as the point of maximum curvature in the L-curve, and selecting it as the point where the product of abcissa and ordinate is a minimum. It is shown that both these methods resulted in significantly better regularization parameters than that obtained with an often-used empirical Composite REsidual and Smoothing Operator approach, particularly in conditions where correlated geometry noise exceeds Gaussian measurement noise. It is also shown that the regularization parameter that results with the minimum-product method is identical to that selected with another empirical zero-crossing approach proposed earlier.
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