Numerical technique for bioelectromagnetic modeling
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 written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. 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.
Historical development.
Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. 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. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications.
In 2009, Greengard et al. 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. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics.
Physical background - surface charges in biological media.
The charge-based BEM is based on the concept of an "impressed" (or primary) electric field formula_1 and a "secondary" electric field formula_2. The impressed field is usually known "a priori" or is trivial to find. 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 formula_11 appears on the conductivity interfaces. This charge density induces a secondary conservative electric field formula_2 following Coulomb's law.
One example is a human under a direct current powerline with the known field formula_1 directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions.
Another example is a human head with electrodes attached. At any conductivity interface with a normal vector formula_14 pointing from an "inside" (-) compartment of conductivity formula_15to an "outside" (+) compartment of conductivity formula_16, Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form
Its derivation does not involve Green's identities (integrations by parts) and is applicable to non-nested geometries. 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 formula_17 and formula_18 of triangles formula_19 and formula_20, respectively, by
formula_21
where formula_22 is the surface area of the triangle formula_23. This approximation is only valid when formula_24 is much larger than a typical facet size i.e., in the "far field". Otherwise, semi-analytical formulae and Gaussian quadratures for triangles should be used. Typically, 4 to 32 such neighbor integrals per facet should be precomputed, stored, and then used at every iteration. 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.
Applications and limitations.
Applications of the charge-based BEM-FMM include modeling brain stimulation with near real-time accurate TMS computations as well as neurophysiological recordings. They also include modeling challenging mesoscale head topologies such as thin brain membranes (dura mater, arachnoid mater, and pia mater). This is particularly important for accurate transcranial direct-current stimulation and electroconvulsive therapy dosage predictions. The BEM-FMM allows for straightforward adaptive mesh refinement including multiple extracerebral brain compartments. Another application is modeling electric field perturbations within densely packed neuronal/axonal arbor. Such perturbations change the biophysical activating function. A charge-based BEM formulation is being developed for promising bi-domain biophysical modeling of axonal processes.
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 formula_25 for typical academic computer hardware resources used as of 2023.