Local Regularization and Adaptive Methods for the Inverse Problem

Abstract

The intrinsic electrical activity of the heart gives rise to an electric field within the volume conductor of the thorax. Thus, the potentials on the thorax surface are related to potentials upon the heart’s surface via the resistive properties of the intermediary tissues. The general inverse problem in electrocardiography can be stated as follows: given a set of body-surface potentials and the geometry and conductivity properties of the body, calculate the potentials on, and the source currents within, the heart. Mathematically, this can be posed as an inverse source problem in terms of the primary-current sources within the heart and described by Poisson’s equation for electrical conduction:

$$ \nabla \cdot \sigma \nabla \Phi = - {I_\upsilon }\,\,\,\,\,{\rm{in}}\,\Omega $$

(1)

with the boundary condition

$$ \sigma \nabla \Phi \cdot {\rm{n}} = 0\,\,\,\,\,{\rm{on}}\,{\Gamma _T} $$

(2)

where Φ are the potentials, σ is the conductivity tensor, I _v are the cardiac current sources per unit volume, and Γ_T and Ω represent the surface and the volume of the thorax, respectively. The goal is to recover the magnitude and location of the cardiac sources. In general, (1) does not have a unique solution. To solve the general source problem, one usually divides the volume of the heart into subregions and makes simplifying assumptions regarding the form of the source term (such as dipoles or other equivalent sources) within those subregions. One then tries to recover information regarding the magnitude and direction of the simplified model sources.