Efficient Numerical Solution of the EMI Model Representing the Extracellular Space (E), Cell Membrane (M) and Intracellular Space (I) of a Collection of Cardiac Cells

Abstract

The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512 × 256 cells, we solved linear systems with about 2.5×108 unknows. Since the computational effort scales linearly with the number of physical cells, we believe that larger computers can be used to simulate millions of excitable cells and thus allow careful analysis of physiological systems of great importance.

Operator Splitting and Finite Difference Schemes for Solving the EMI Model

“…It is therefore of importance to develop a splitting strategy for the EMI model that leads to sub-problems of the elliptic type. In (14), we showed that such a splitting can indeed be achieved. Here, we will review this convenient way of splitting the EMI model and show how to solve the system numerically using a finite difference method.…”

A computational method for identifying an optimal combination of existing drugs to repair the action potentials of SQT1 ventricular myocytes

“…This approach represents the extracellular space (E), the cell membrane (M) and the intracellular space (I), see, e.g., [ 38 – 40 ]. The EMI model equations are solved using an MFEM [ 41 , 42 ] finite element implementation of the splitting algorithm introduced in [ 43 , 44 ]. Technical specifications of the domain geometry and the EMI model solver are provided in the S1 Text .…”

Mutations change excitability and the probability of re-entry in a computational model of cardiac myocytes in the sleeve of the pulmonary vein

“…Note that, is the current density between neighbouring myocytes, specified by the passive model where G gap (in mS/cm 2 ) is the conductance of the gap junctions and R gap (in kΩcm 2 ) is the corresponding resistance of the gap junctions. The parameters used in the EMI model simulations are specified in Table 3, and the EMI model equations are solved numerically using an MFEM [54, 55] finite element implementation of the operator splitting algorithm introduced in [30, 56].…”

“…We represent essential features of the geometry of the PV sleeve by constructing a collection of coupled myocytes that form a cylinder having a diameter of 1.5 cm (similar to the diameter of a human PV [57]). Each myocyte is [30].…”

“…However, both of these models require spatial averaging over hundreds of cells and it is therefore not possible to model the significant cell-to-cell variations in the electrophysiological properties that are characteristic of the cardiac myocytes located in the ‘sleeve’ of tissue at the transition from the pulmonary vein to the left atria. In order to allow for cell-to-cell variation in ion channel densities and assess the effects of random variation in gap-junction coupling between the myocytes, we have applied a recently developed mathematical model [28, 29, 30, 31, 32] that represents each my-ocyte individually. This is referred to as the EMI model since it explicitly represents the extracellular (E) space, the membrane (M) and the intracellular (I) space, and thus follows the modeling tradition represented by, e.g., [25, 33].…”