The bioelectric fields produced in the human thorax can be mathematically
described by Maxwell's equations. For electrostatic problems in dielectric
volume conductors, an electric field, , can be described in
terms of the gradient of a scalar potential field [18]

In the frequency range
of the electrocardiographic signals (0-100 Hz [19]), the volume
conductor can be considered purely resistive so that a quasi-static
approximation is justified [20].

Thus the current density associated with the electric field is given by
Ohm's law,

where is the current density and is the electrical
conductivity of the tissue, represented mathematically as a 3 x 3
symmetric tensor. If active current sources are also present, this
equation becomes

Under quasi-static conditions, the net current flow is solenoidal such that
the divergence of equation (3) is zero. Therefore,

Substituting equation (1) into equation (5),
we obtain

or

where

which is a form of Poisson's equation.

For this study, *I*_{sv} = 0, because we bound the volume of interest by
the epicardial and body surfaces so that all sources are outside the
bounded volume domain. As such, equation (7) simplifies to a
general form of Laplace's equation. By applying the Dirichlet and Neumann
boundary conditions, we can formulate an electrocardiographic forward
problem in terms of epicardial sources as

with

where

Scientific Computing and Imaging