Objective. To formulate, validate, and apply an alternative to the finite element method (FEM) high-resolution modeling technique for electrical brain stimulation—the boundary element fast multipole method (BEM-FMM). To include practical electrode models for both surface and embedded electrodes. Approach. Integral equations of the boundary element method in terms of surface charge density are combined with a general-purpose fast multipole method and are expanded for voltage, shunt, current, and floating electrodes. The solution of coupled and properly weighted/preconditioned integral equations is accompanied by enforcing global conservation laws: charge conservation law and Kirchhoff’s current law. Main results. A sub-percent accuracy is reported as compared to the analytical solutions and simple validation geometries. Comparison to FEM considering realistic head models resulted in relative differences of the electric field magnitude in the range of 3%–6% or less. Quantities that contain higher order spatial derivatives, such as the activating function, are determined with a higher accuracy and a faster speed as compared to the FEM. The method can be easily combined with existing head modeling pipelines such as headreco or mri2mesh. Significance. The BEM-FMM does not rely on a volumetric mesh and is therefore particularly suitable for modeling some mesoscale problems with submillimeter (and possibly finer) resolution with high accuracy at moderate computational cost. Utilizing Helmholtz reciprocity principle makes it possible to expand the method to a solution of EEG forward problems with a very large number of cortical dipoles.

客观。制定、验证和应用用于脑电刺激的有限元法 (FEM) 高分辨率建模技术的替代方案 - 边界元快速多极法 (BEM-FMM)。包括表面电极和嵌入式电极的实用电极模型。方法。边界元法关于表面电荷密度的积分方程与通用快速多极方法相结合，并针对电压、分流、电流和浮动电极进行了扩展。耦合和适当加权/预处理积分方程的解伴随着执行全局守恒定律：电荷守恒定律和基尔霍夫电流定律。主要结果。与分析解决方案和简单的验证几何形状相比，报告的准确度低于百分之几。与考虑真实头部模型的 FEM 进行比较，导致电场幅度的相对差异在 3%–6% 或更小的范围内。与 FEM 相比，包含更高阶空间导数的量（例如激活函数）以更高的精度和更快的速度确定。该方法可以很容易地与现有的头部建模管道（例如 headreco 或 mri2mesh）相结合。意义。BEM-FMM 不依赖于体积网格，因此特别适用于对一些具有亚毫米（可能更精细）分辨率的中尺度问题进行建模，并且具有较高的精度和适中的计算成本。利用亥姆霍兹互易原理可以将该方法扩展到解决具有大量皮质偶极子的 EEG 前向问题。