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An assessment of numerical methods for cardiac simulation

Solving the mathematical models of the electrical activity in the heart is difficult mainly due to the complexity of the models required to capture the electrochemical details of the organ. A variety of mathematical models has been developed to describe the electrical activity of individual heart cells. Cardiac cells respond to an electrical stimulus, causing ions to flow across the cell membrane, changing the electrical potential difference between the interior and the exterior of the cell. Cardiac cell models describe the potential difference across the cell membrane, and depending on the complexity of the model, the ion concentrations and the movement of ions through the cell membrane. This thesis studies 37 cardiac cell models and compares the efficiency of the Forward Euler and Rush–Larsen methods, as well as two generalized Rush–Larsen methods (of order one and order two). The Backward Euler method is compared for six of the 37 models and type-insensitive methods are compared for four of the 37 models. From the results, it is determined that the Rush–Larsen method is the most efficient for moderately stiff models and that the generalized Rush–Larsen methods perform well on the stiff models. The type-insensitive methods are more efficient than the single methods for each of the four models considered. The bidomain model combines a cardiac cell model with two partial differential equations that model the propagation of the electrical activity throughout the entire heart. Simulations of the bidomain model are computationally expensive, necessitating improvements to the numerical methods used to solve the model. In order to determine whether the cell model results can be applied to the bidomain model, a one-dimensional simulation of the bidomain model is considered and solved using the Chaste software package developed by the Computational Biology Group at Oxford University Computing Laboratory, together with additions written for this thesis. The cellular electrical activity within the bidomain model is modelled by eight of the 37 cell models previously studied, chosen to represent a range of the available models. From these results, it is shown that the cell model results are directly applicable to the one-dimensional bidomain problem. The firstorder generalized Rush–Larsen method drastically reduces computation time for the two stiffest models, and the Rush–Larsen method performs optimally for the moderately stiff models. One of the de facto standards, the Forward Euler method, is shown to perform poorly for seven of the eight one-dimensional bidomain simulations.