A solution of the forward problem is an important component of any method for computing the spatio-temporal activity of the neural sources of magnetoencephalography (MEG) and electroencephalography (EEG) data. The forward problem involves computing the scalp potentials or external magnetic field at a finite set of sensor locations for a putative source configuration. We present a unified treatment of analytical and numerical solutions of the forward problem in a form suitable for use in inverse methods. This formulation is achieved through factorization of the lead field into the product of the moment of the elemental current dipole source with a "kernel matrix" that depends on the head geometry and source and sensor locations, and a "sensor matrix" that models sensor orientation and gradiometer effects in MEG and differential measurements in EEG. Using this formulation and a recently developed approximation formula for EEG, based on the "Berg parameters", we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different boundary element methods (BEMs) and present evidence that improvements over currently published BEM methods can be realized using alternative error-weighting methods. Explicit expressions for the matrix kernels for MEG and EEG for spherical and realistic head geometries are included.

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EEG and MEG: forward solutions for inverse methods

A novel analytical model of inductively coupled wireless power transfer is presented. For the first time, the effects of coil misalignment and geometry are addressed in a single mathematical expression. In the applications envisaged, such as radio frequency identification (RFID) and biomedical implants, the receiving coil is normally significantly smaller than the transmitting coil. Formulas are derived for the magnetic field at the receiving coil when it is laterally and angularly misaligned from the transmitting coil. Incorporating this magnetic field solution with an equivalent circuit for the inductive link allows us to introduce a power transfer formula that combines coil characteristics and misalignment factors. The coil geometries considered are spiral and short solenoid structures which are currently popular in the RFID and biomedical domains. The novel analytical power transfer efficiency expressions introduced in this study allow the optimization of coil geometry for maximum power transfer and misalignment tolerance. The experimental results show close correlation with the theoretical predictions. This analytic technique can be widely applied to inductive wireless power transfer links without the limitations imposed by numerical methods.

Wireless Power Transfer in Loosely Coupled Links: Coil Misalignment Model

An improved boundary element method (BEM) with a virtual triangle refinement using the vertex normals, an optimized auto solid angle approximation, and a weighted isolated problem approach is presented. The performance of this new approach is compared to analytically solvable spherical shell models and highly refined reference BEM models for tangentially and radially oriented dipoles at different eccentricities. The lead fields of several electroencephalography (EEG) and magnetoencephalography (MEG) setups are analyzed by singular-value decompositions for realistically shaped volume-conductor models. Dipole mislocalizations due to simplified volume-conductor models are investigated for EEG and MEG examinations for points on a three dimensional (3-D) grid with 10-mm spacing inside the conductor and all principal dipole orientations. The applicability of the BEM in view of the computational effort is tested with a standard workstation. Finally, an application of the new method to epileptic spike data is studied and the results are compared to the spherical-shells approximation.

An improved boundary element method for realistic volume-conductor modeling

A 3D magnetostatics computer code optimized for Undulators and Wigglers is described. The code uses a boundary integral method and makes extensive use of analytical expressions for the field and field integrals along a straight line. The code outperforms currently available finite element packages in the area of simple data input, CPU time of the solver and accuracy reached for the estimation of field integrals. It is written in C++ and takes advantage of object-oriented programming. The code is interfaced to Mathematica. Pre- and post-processing of the field data is done in the Mathematica Language. It has been extensively benchmarked with respect to a commercial finite element code. All ESRF Insertion Devices built during the last 4 years have been designed using this code or an older version.

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Computing 3D magnetic fields from insertion devices

The purpose of this paper is to determine the mutual inductance between two noncoaxial circular coils. In many cases, such as coil guns or tubular linear motors, one of them is fixed while the other one is moving, and if not supported, its axis may not coincide with that of the fixed coil. This paper presents a method for the calculation of the mutual inductance in the case of noncoaxial coupled coils, the characteristics of this inductance, and experimental results. The computation is based on complete elliptic integrals and on the mesh-matrix technique. The method enables one to obtain accurate results from a relatively simple procedure and calculation program.

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Mutual inductance of noncoaxial circular coils with constant current density

Analytical expressions for the components of the vector potential and magnetic field of a circular arc segment of a current-carrying conductor of rectangular cross section and arbitrary azimuthal length are derived. Components of the field gradient are also calculated. All the expressions developed consist of known functions such as Jacobian elliptic functions, complete and incomplete elliptic integrals of the first, second, and third kind, and thus permit a new timesaving efficient computation algorithm.

Vector potential and magnetic field of current-carrying finite arc segment in analytical form, Part III: Exact computation for rectangular cross section

Analytical expressions for the components of the vector potential and magnetic field of a thin circular conic cylinder segment of arbitrary azimuthal and axial length, and carrying a constant peripheral current are derived. All the expressions developed consist of known functions such as Jacobian elliptic functions, and complete and incomplete elliptic integrals of the first, second, and third kind and thus permit a new time-saving efficient computation algorithm.

Vector potential and magnetic field of current-carying finite arc segment in analytical form, part II: Thin sheet approximation

Analytical expressions for the components of the vector potential and magnetic field derived in Part II [1] and in Part III [2] for azimuthal constant current density are now extended to general three-dimensional constant vector current density.

Vector potential and magnetic field of current-carrying finite arc segment in analytical form, part IV: General three-dimensional current density

For pt.IV see ibid., vol.MAG-20, p.2145-50 (1984). Analytical expressions for the components of the vector potential and magnetic field of a circular-arc segment of a current-carrying conductor of polygonal cross section and arbitrary azimuthal length are derived. The expressions developed consists of known analytical functions such as Jacobian elliptic functions and complete and incomplete elliptic integrals of the first, second, and third kind, which permit the formulation of a time-saving efficient computation algorithm. An example of application is given. The case of a beam with polygonal cross-section carrying a longitudinal current is also examined. >

Vector potential and magnetic field of current-carrying circular finite arc segment in analytical form. V. Polygon cross section

A boundary element formulation for biomagnetic problems is described. The use of an analytical element integration and tuning of the boundary element method (BEM) forward solution for implementation in the inverse problem algorithm results in a far better relationship between accuracy and computational effort. The computational accuracy of the realistic heart and torso model compared with the simplified models commonly used in the literature is shown. >

L. Urankar

Papers

Vector potential and magnetic field of current-carrying finite arc segment in analytical form, Part III: Exact computation for rectangular cross section

Vector potential and magnetic field of current-carying finite arc segment in analytical form, part II: Thin sheet approximation

Vector potential and magnetic field of current-carrying finite arc segment in analytical form, part IV: General three-dimensional current density

Vector potential and magnetic field of current-carrying circular finite arc segment in analytical form. V. Polygon cross section

Boundary Element Solution of Biomagnefic Problems