Thesis (M. Eng.)--Carleton University, 1996. / Also available in electronic format on the Internet.

Modelling in cardiac electrophysiology results in a complex system of partial differential equations (PDE) describing the propagation of the electrical wave in the heart muscle coupled with a highly nonlinear system of ordinary differential equations (ODE) describing the ionic activity in the cardiac cells. This system forms the widely accepted bidomain model or its slightly simpler version, the monodomain model. To a large extent, the stiffness of the whole model depends on the choice of the ionic model, which varies in terms of complexity and realism. These simulations require accurate and, depending on the ionic model used, possibly very stable numerical methods. At this time, solving these models numerically requires CPU time of around one day per heartbeat. Therefore, it is necessary to use the most efficient method for these simulations. This research focuses on the comparison and analysis of several time-stepping methods: explicit or semi-implicit, operator splitting, deferred correction and Rush-Larsen methods. The goal is to find the optimal method for the ionic model used. For our analysis, we used the monodomain model but our results apply to the bidomain model as well. We compare the methods for three ionic models of varying complexity and stiffness: the Mitchell-Schaeffer models with only 2 variables, the more realistic Beeler-Reuter model with 8 variables, and the stiff and very complex ten Tuscher-Noble-Noble-Panfilov (TNNP) models with 17 variables. For each method, we derived absolute stability criteria of the spatially discretized monodomain model and verified that the theoretical critical time steps obtained closely match the ones in numerical experiments. Convergence tests were also conducted to verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. We looked at the efficiency of the different methods by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured respectively by the most negative eigenvalue of the model's Jacobian and the number of variables, and based on strict stability and accuracy criteria.

Design and analysis of conceptually different cooling systems for the human heart preservation are numerically investigated. A heart cooling container with required connections was designed for a normal size human heart. A three-dimensional, high resolution human heart geometric model obtained from CT-angio data was used for simulations. Nine different cooling designs are introduced in this research. The first cooling design (Case 1) used a cooling gelatin only outside of the heart. In the second cooling design (Case 2), the internal parts of the heart were cooled via pumping a cooling liquid inside both the heart’s pulmonary and systemic circulation systems. An unsteady conjugate heat transfer analysis is performed to simulate the temperature field variations within the heart during the cooling process. Case 3 simulated the currently used cooling method in which the coolant is stagnant. Case 4 was a combination of Case 1 and Case 2. A linear thermoelasticity analysis was performed to assess the stresses applied on the heart during the cooling process. In Cases 5 through 9, the coolant solution was used for both internal and external cooling. For external circulation in Case 5 and Case 6, two inlets and two outlets were designed on the walls of the cooling container. Case 5 used laminar flows for coolant circulations inside and outside of the heart. Effects of turbulent flow on cooling of the heart were studied in Case 6. In Case 7, an additional inlet was designed on the cooling container wall to create a jet impinging the hot region of the heart’s wall. Unsteady periodic inlet velocities were applied in Case 8 and Case 9. The average temperature of the heart in Case 5 was +5.0oC after 1500 s of cooling. Multi-objective constrained optimization was performed for Case 5. Inlet velocities for two internal and one external coolant circulations were the three design variables for optimization. Minimizing the average temperature of the heart, wall shear stress and total volumetric flow rates were the three objectives. The only constraint was to keep von Mises stress below the ultimate tensile stress of the heart’s tissue.