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Electrical activity of the heart 99

where S j are the interfaces between the homogeneous regions. The sources, conve-
niently placed inside the heart volume (e.g., in the myocardium), may be utilized to
generate the electric potential on the epicardium, which is actually mapped on the
thorax surface, accessible for voltage measurements, provided there is no epicardial
electrical current outflow to the thorax.

4.2 Coupled direct and inverse ECG problems for electrical imaging

The inverse ECG problem, I-ECG, (or electrocardiographic imaging) aims to find the
sources on the epicardium (electric potential) from voltage measurements on the chest
surface. Clinically, I-ECG is important because it helps identify cardiac arrhythmias. The
measurement data on the chest used as input are actually filtered images of the epicardial
potential, due to the smoothing and attenuating properties of the thorax volume
between the source, on epicardium, and the observation surface, the thorax outline.
As with all ill-posed problems (Chapter 1: Physical, Mathematical and Numerical
Modeling), the difficulty of recovering the source here has to be alleviated by using
some supplementary, consistent information on the source. Therefore a companion D-
ECG problem has to be formulated and solved in the first place. The
D-ECG here consists in finding the electric field inside the volume conductor of the
thorax—the volume conductor between the epicardium (inner boundary) and the tho-
rax surface (outer boundary), with known geometry and material properties—when the
field sources, on the epicardium are known. The boundary, initial, and interface condi-

σ rVÞUn 2 σvrVÞUnv 5 0,
0 0 0
tions are V epicardium 5 0; J n thorax 5 0, V 5 Vv, ð ð
j
respectively, and there are no internal sources, J i 5 0. Its solution is found usually using
numerical methods such as the finite element method or the boundary element method
(Mocanu, 2002). The analytic solution of D-ECG problem is then a Fredholm integral
of first type (Yamashita, 1982)
ð
VPðÞ 5 KðP; QÞVQðÞ dS H ; PAS T ; QAS H ; ð4:3Þ

S H
where S H is the epicardium, S T is the thorax surface, and V is the electric potential.
The cardiac sources are given through the epicardial electric potential. The kernel K
2
(P,Q) is a compact operator of L class. Given K(P,Q) (e.g., by the D-ECG) and the
function V(P), the problem is to find the function V(Q).
The kernel may be seen as the normal component of the current density at a point
QAS H , produced by a unit current source at a point PAS T , when the epicardium is of

zero potential. Another interpretation is that K(P,Q) is the potential at PAS T
produced by a unit potential at the QAS H point, while the rest of the epicardium is of
zero potential.