A general volume conductor can be defined as a region of volume, 
,
which has conductivity, 
, and permitivity, 
, in which
resides a source current, 
, where the 
signifies per-unit volume.
Solving a volume conductor problem means finding expressions for the
electric, 
, and potential, 
, fields everywhere within the
volume, , and/or on one of the bounded surfaces, 
.
While not unique to biophysics, a property that provides one with
significant challenges, is the complexity of the domain, , which is
subdivided into several sub-domains, each with different conductivity. In
some cases, these inhomogeneous regions are anisotropic (e.g. causing the
field to vary with direction).
The bioelectric current sources, , arise from excitable cells
undergoing an activation process. Activation of cardiac tissue, for
example, can be characterized as the process in which cells undergo rapid
depolarization (e.g. when movement of ions across the cell membrane results
in inactivation of electrical charges and a drop in potential; see
[8][7] for more details). The depolarization process
causes a propagation of excitation waves to move through the myocardium
(the muscular layer of the heart); these waves in turn produce an
extracellular potential field, . This potential field can be
characterized by the geometry and conductivity of the volume conductor, and
by the distance from, orientation to, and intensity of the source current,
.
A similar but more complicated process of depolarization occurs in the brain. While it too undergoes an activation process, the propagation occurs in a less continuous manner than in the heart. In the brain, the individual neurons determine the path that activation takes. Thus the activation process in the brain is inherently discontinuous.
Because is, in general, time-varying, the resulting field
quantities are governed by Maxwell's equations. For a discussion on
modeling the wave propagation through active myocardial tissue, the reader
is directed to [13][12][11][10][9].
For the macroscopic volume conductor problem (in which we do not consider
the individual membrane currents), we can utilize a quasi-static
approximation. Because the displacement current, (
),
is much smaller than the conduction current, (
), propagation
effects are negligible, and inductive effects are minute [7].
This leads to a reduction of Maxwell's equations to one of two forms. The
more general form is a reduction to Poisson's equation for electrical
conduction:
where is the electrical conductivity tensor, is the
electrostatic potential, is the current source per unit volume, and
is the solution domain. In this form, one includes the source
region and an understanding of the primary bioelectric sources, ,
usually in the form of a simplified mathematical model. Alternatively, one
can define a surface bounding the region which includes the sources and
recast the formulation in terms of information on that surface,
yielding Laplace's equation:
One then solves these equations subject to an appropriate set of boundary conditions. At first glance, this seems simple: we know the governing equation and we know how to solve Poisson's and Laplace's equations. However, as we shall see, there are some significant challenges when one tries to solve a large scale, inhomogeneous, anisotropic, inverse volume conductor problem that exhibits the complicated geometry of the human body.
