Charge based boundary element fast multipole method
The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns.
The charge-based BEM solves an integral equation of the potential theory[1] written in terms of the induced surface charge density.
The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method.
For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart.
[7] A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn.
[8] The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, [9] and used in different applications.
[10] In 2009, Greengard et al.[11] successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics.
A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al.[12] in 2018.
Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies[13] as well as in acoustics.
[14][15] The charge-based BEM is based on the concept of an impressed (or primary) electric field
For the human brain, the impressed electric field can be classified as one of the following: When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density
This leads to the interfacial boundary condition in the form for every facet at a triangulated interface.
The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field
If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values.
just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign.
, we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface.
For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for
The more rigorous generalized minimum residual method (GMRES) yields a much faster convergence of the BEM-FMM.
[2][3][16][17][18] In either case, the major work is in computing the underbraced sum in the system of equations above for every
It is therefore unnecessary to form or store the dense system matrix typical for the standard BEM.
When the Galerkin method is applied and the same zeroth-order basis functions (with a constant charge density for each facet) are still used on triangulated interfaces, we obtain exactly the same discretization as before if we replace the double integrals over surfaces
[12][2][17][18][27] This is an important correction to the plain fast multipole method in the "near field" which should also be used in the simple discrete formulation derived above.
Such a correction makes it possible to obtain an unconstrained numerical (but not anatomical) resolution in the brain.
[17] Applications of the charge-based BEM-FMM include modeling brain stimulation[3][17][18][21] with near real-time accurate TMS computations[28][4] as well as neurophysiological recordings.
This is particularly important for accurate transcranial direct-current stimulation and electroconvulsive therapy dosage predictions.
[30] The BEM-FMM allows for straightforward adaptive mesh refinement including multiple extracerebral brain compartments.
[27][29] Another application is modeling electric field perturbations within densely packed neuronal/axonal arbor.
A charge-based BEM formulation is being developed for promising bi-domain biophysical modeling of axonal processes.
[31] In its present form, the charge-based BEM-FMM is applicable to multi-compartment piecewise homogeneous media only; it cannot handle macroscopically anisotropic tissues.
Additionally, the maximum number of facets (degrees of freedom) is limited to approximately
