Prof. Chay with others, has successfully demonstrated normal and abnormalchains of voltage pulses in a ring of cardiac cells. Generally each cell response is modelled in terms of standard ionic currents in membranes. Bifurcation analysis has been used to determine conditions giving standing waves, persistent traveling waves, wave destruction, etc. This work was confirmed by computational methods working with rings in the range of 6 to 600 cells, and time periods and pulse shapes very similar to cardiac observations. A very simplified statement of the numerical method used for waves in a ring is that the Euler method was used to solve a diffusion partial differential equation (PDE) (variables time and ring position) of considerable complexity. Small time-steps were required (about 1/50 millisecond) and about 2000 steps for test display, for example, of a persistent wave train. If the model is reduced to only 2 ions, and a little attention is given to vectorization, a test of this kind can be reduced to about 10 seconds of cp time (single node) on the Cray 90. Certain concerns of particular dependence of the ion currents on (t,x), as well as concerns about numerical convergence, can quickly lead to computations taking 2 or 3 times longer that this. Obviously a computation long enough to simulate a typical EKG could take an hour of running on a single cpu of the C90. I have learned some of these methods from Prof. Chay and have been collaborating with her in several ways. One concern has been to apply the """"""""implicit"""""""" Crank-Nicolson (CN) PDE method to be sure that the """"""""explicit"""""""" Euler computations were truly convergent, etc. Quite good agreement has been found for short tests between the CN and Euler methods. This gives considerable encouragement regarding the studies already done with the Euler method. The CN computation, per step, takes longer than the Euler method, but this is offset by the fact that we can safely use time steps 5 times larger with the CN method. A deeper concern is that a few hundred cells in a ring configuration only approximates heart anatomy. For example, do the conclusions of Prof. Chay's computations still hold for a cells arranged in a stack of rings (i.e.,a cylindrical arrangement)? This could be computed for test purposes as a 600x600 array of cells, for example, with """"""""wrap-around"""""""" continuous boundary conditions at opposite edges. The PDE formulation would be very similar to that already used, except that there would be (t,x,y) independent variables. Even this configuration is too small, as it could be thought to represent less than 1 square millimeter of tissue.
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