Multipole representations of current generators in a volume conductor

As far as the potential distribution outside the current generators is concerned, any current source distribution may be replaced by a suitable collection of multipoles. If these current generators lie close to the geometrical center of the volume conductor, a central dipole is a good approximation for potentials at surface points which are at considerable distances from the center. For better accuracy and for points close to the center, additional singularities such as a central quadrupole, a central octopole, etc., should be included. Potential expressions due to such multipoles in a spherical conductor can be obtained in closed forms by means of the “interior sphere theorem”. This paper presents a method for determining successively better multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. It is shown that Einthoven's triangle and Wilson's tetrahedron in the theory of electrocardiography are first and second approximations of this method. This concept also applies to conductors of other shapes. This investigation was supported by The National Heart Institute under a research grant H-2263(c).