A systematic approach for potentials on closely packed cells using the null-field boundary integral equation in conjunction with the degenerate kernel and eigenfunction expansion

Abstract

A systematic approach based on the null-field integral formula is used to determine the electric potential of a tissue with many cells stimulated by remote electric fields. When the cells are very close to each other, the problem becomes nearly singular and the accuracy of the solution deteriorates. However, in the proposed approach, the highly accurate results are obtained because the separable kernel (degenerate kernel) and eigenfunction expansion are introduced to capture the geometry property in the integral formulation. Only boundary nodes are required instead of boundary elements to satisfy the boundary conditions or interface conditions. The proposed approach could be seen as one kind of meshless and semi-analytical methods. In addition, the error just stems from the number of truncation terms of the eigenfuntion expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. For the problem of closely packed cells, the boundary density of sharp variation could be accurately simulated or captured by increasing the number of terms of eigenfunctions. Finally, the acceptable results are shown to see the efficiency and accuracy of the proposed approach by the given numerical examples including one, three and twenty cells.