ABSTRACT
One of the most important problems in bioelectric theory is the calculation of the electrical potential, (V), throughout a volume conductor. The calculation of is important in impedance imaging, cardiac pacing and defibrillation, electrocardiogram and electroencephalogram analysis, and functional electrical stimulation. In bioelectric problems, often changes slowly enough that we can assume it is quasistatic [Plonsey, 1969]; we ignore capacitive and inductive effects and the finite speed of propagation of electromagnetic radiation. (Usually for bioelectric phenomena, this assumption is valid for frequencies below about 100 kHz.) Under the quasistatic approximation, the continuity equation states that the divergence,∇·, of the current density, J (A/m2), is equal to the applied or endogenous source of electrical current, S (A/m3):
∇ · J = S. (21.1)
In regions of tissue where there are no sources, S is zero. In these cases, the divergenceless of J is equivalent to the law of conservation of current that is often invoked when analyzing electrical circuits. Another property of a volume conductor is that the current density and the electric field, E (V/m), are related linearly by Ohm’s Law,
J = g E, (21.2)
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where g is the electrical conductivity (S/m). Finally, the relationship between the electric field and the gradient,∇, of the potential is
E = −∇. (21.3) The purpose of this chapter is to characterize the electrical conductivity. This task is not easy, because g
is generally a macroscopic parameter (an “effective conductivity”) that represents the electrical properties of the tissue averaged over many cells. The effective conductivity can vary with direction, can be complex (contain real and imaginary parts), and can depend on the temporal and spatial frequencies.
