A dual coordinate system finite difference method for forward and inverse solutions of the volume conductor problem in neurology.

Finite difference methods for the volume conductor problem have used a single coordinate system for the mesh and made approximations of Laplace's equation. This method is simple but has two major problems. Firstly, to deal with boundary conditions properly, the normal potential gradient at the boundary must be known. However it is complicated to compute at a curved surface point. Secondly, for an inverse solution the equation on a curved boundary is difficult to reverse since more than one inner mesh node appears in the approximation equation for each surface point. The new method developed in this paper is a dual coordinate system. One system serves as a frame mesh, the other is a sub-coordinate system in which surface points become mesh points (regular nodes). The equation at each surface point is then directly reversible since only one inner point appears in the equation. The forward solution is applied to both centric and eccentric bone models and uses the conventional successive over-relaxation (SOR) method. Noise is added to this solution for input to the inverse procedure which is a direct step-in non-iterative method. Low pass filtering was effective in reducing the effects of noise. In the examples given, only one coordinate subsystem is used but, for complex shape boundaries, multiple subsystems would be necessary.