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Figure 4.5 The electric potential on the thorax, produced by a step distribution of potential unit
on the surface of the heart. Simplified computational domain (left) and the electric potential on
the thorax produced by a step function epicardial voltage (Fig. 4.4a).

In mathematical and numerical modeling, a representative epicardial source is the
step function, which characterizes the abrupt variation of the potential AP front-like
shape (Fig. 4.5) and it qualifies to compare the inversion methods used in the solution
of the inverse problem that aims to find the epicardial potential. The optimal parame-
ter for Tikhonov regularization was determined with the L-curve method. For VT
regularization, α was determined by visual inspection (Mocanu, 2002).
The model ought to include all anatomical structures that could significantly influ-
ence the direct and inverse solutions (Morega et al., 2000). However, the complexity
of the internal geometry of the thorax may lead in the end to a very large number of
degrees of freedom (grid nodes). Heart geometry is much simplified here (Fig. 4.5)to
allow the smooth calculation of the gradient operator on a structured surface network.
A realistic representation of the heart shape requires an unstructured grid and compli-
cated algorithms for determining the surface gradient operator (Srinidhi, 1999). The
reconstruction of this particular type of signal (step function), out of this potential dis-
tribution on the thorax using several inversion methods, is shown in Fig. 4.6.
Tikhonov regularization, which is extensively used to stabilize the inverse solution, acts
as a linear filter. In an attempt to remove high frequency noise, this method also filters the
high frequency components in the solution, leading to the oversmooth of the AP front.
In contrast, the TV constrains the norm l 1 of the potential gradient, preserving the discon-
tinuities in the solution. In the case of smooth potential, the two regularization methods
provide similar results, but the Tikhonov method better locates the potential field extrema,
while the TV overextends them (Mocanu et al., 2002, 2005).

Image-based construction of a human heart and thorax
The models presented here are developed as a numerical instrument for the study of
normal and pathological heart conditions, supporting the experimental (in vivo)