Approximate analytical solutions of the Bidomain equations for electrical stimulation of cardiac tissue with curving fibers

The mechanism by which an applied electric field stimulates cardiac tissue far from the stimulating electrodes is not wholly understood. One possible mechanism relates the curving cardiac fibers to the induced membrane currents and transmembrane potentials. However, we lack a qualitative understanding of where these areas of polarization will occur when an electric field is applied to a sheet of cardiac tissue with curving fibers. In our study, we derive an analytical model for the transmembrane potential, dependent on the gradient of the fiber angle theta, for a two-dimensional passive sheet of cardiac tissue exhibiting various fiber geometries. Unequal anisotropy ratios are crucial for our results. We compare the results from our analytical solution to a numerical calculation using the full bidomain model. The results of our comparison are qualitatively consistent, albeit numerically different. We believe that our analytical approximation provides a reliable prediction of the polarization associated with an electric field applied to cardiac tissue with any fiber geometry and a qualitative understanding of the mechanisms behind the virtual electrode polarization.