Mathematical Modeling of Spiral Wave

The mechanisms of cardiac de brillation, the most e ective clinical procedure for termination of lethal cardiac arrhythmias, are not well understood. Shocks establish an electric eld in the heart, which, in turn, induces changes in the transmembrane potential leading to de brillation. The complexity of cardiac structure makes it di cult to explain, based on experiments alone, how exactly the electric eld affects transmembrane potential. In an attempt to provide additional insight into the issue, in this computational study we explore the e ects of strong electric shocks on myocardium in brillation. Cardiac tissue is represented by the two-dimensional anisotropic bidomain model with active membrane kinetics and various ber geometries. A single reentrant circuit of excitation serves as a simple model of the arrhythmic behavior. Bidomain model allows us to study currents introduced by the shock into the extracellular space of the myocardium. We illustrate various mechanisms by which cardiac tissue structure, as well as the electrode shape, assists the changes in transmembrane potential throughout the myocardium that result in de brillation. We observe the shock-induced regions of the reverse polarity of transmembrane potential, so-called virtual electrodes, that form at a distance from the physical electrodes. These regions a ect the electrical state of the tissue globally by perturbing reentrant wavefronts and generating new excitations. Our results support the hypothesis that virtual electrodes play an important role in de brillation. The variety of simulation results described here o ers a new level of understanding of the virtual electrode concept and provides a theoretical ground for future experimental studies. Mathematical Modeling of Spiral Wave Reentry and De brillation Shocks in Ventricular Myocardium A Dissertation Submitted on the Twenty-Seventh Day of May, 1998, to the Department of Mathematics of the Graduate School of Tulane University in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy by Kirill B. Skouibine Approved by: Peter Moore, Ph.D., Chair Natalia Trayanova, Ph.D. Lisa Fauci, Ph.D. Steven Rosencrans, Ph.D. Acknowledgments I thank my advisors, Peter Moore in mathematics and Natalia Trayanova in biomedical engineering for their understanding and patience that kept the project going through the times of doubt and frustration. Their timely advice was invaluable. Close contact and even friendship with my advisors was not something I quite expected when coming to the U.S., but it happened and made a world of di erence. I would like to say many thanks to other professors, graduate students and sta in our department: Dr. Fauci and Dr. Rosencrans, members of my disseration committee, Dr. Moll and Dr. Kalka, Dave, Matt, Angel and Earl, Les and Susie (in mathematics department), Felipe, Greg and Jason (in biomedical engineering department). It would be simply impossible to imagine life here without all my New Orleans friends: Ken, John, Jeremy, Mike Cusac, Kristina, Cindy, Alex, Lana, Gonzalo, Mike Choo, Noraer, Alim and Olya. I am truly honored by your friendship, people, and will drink to it any time. Separate and very special thanks to my closest friend, Heather, whose hard work in Medical School made my graduate student life look like a vacation, and left no excuses for procrastination or wining (I did wine a bit anyway). ii I thank my family and all my friends back home for their understanding and putting up with my long-term absense from Moscow. Short summer visits don't count, even though they were the happiest days of the ve years spent away from home. I once again thank my parents, Boris Georgievich and Irina Petrovna, and my sisters, Zhenya and Irina, and dedicate this work to them, my Family.