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112 Computational Modeling in Biomedical Engineering and Medical Physics

The AP is provided, here, through the FitzHugh Nagumo model that renders the
normal electrical activity of the heart (Kogan et al., 1991)
@u
5 Δu 1 α 2 uÞ u 2 1Þu 2 w;
@t ð ð ð4:18Þ
@w
5 εβu 2 γw 2 δÞ; ð4:19Þ
ð
@t
where u is the “fast” variable which models the AP propagation, w is the “slow” vari-
able which models the inhibitor that reflects the probability of a transmembrane ionic
channel to conduct an ionic current, influencing the rest state comeback (Malmivuo
and Plonsey, 1995). The (α u)(u 1)u term is specific to the membrane depolarization
of the myocardial nodal cells, α is the excitation threshold, ε is an empirical parameter
which triggers the cells rest state comeback while β, γ, δ are simulating the electrical
activity dynamics of the heart, when properly adjusted.
The AP propagation is happening very fast due to the short time intervals needed
for the sodium ionic channel opening and conducting a transmembrane depolarization
ionic current. On the other side, there are models for simulating the pathological
regimes for the heart’s electrical activity (e.g., the arrhythmias), described by self-
sustained reentrant spiral waves. Landau Ginzburg is one of the many oscillator mod-
els that describes these abnormal states (Aranson and Kramer, 2002)

@u 2 2
5 Δ u 2 χ w 1 u 2 u 2 χ w u 1 w ;
1
2
@t ð4:20Þ
@w 2 2
5 Δχ u 1 w 1 w 2 χ u 1 w u 1 w ;
@t 1 2 ð4:21Þ
where χ 1 and χ 2 are material property values that influence the solution existence and
stability.
In general, Landau Ginzburg equations are used to model chaotic phenomena and
were introduced to describe the superconductivity theory and chaotic phenomena (Du
et al., 1992). In other studies, the same equations model the LASER pulse generators
functioning (Akhmediev et al., 2001) or the dynamics of Belousov Zhabotinsky chemical
reactions (Petrov et al., 1993).
In the numerical simulations of the AP propagation presented next homogeneous
flux (Neumann) boundary conditions are set on the endocardium and epicardium
(Fig. 4.13) assuming that no ionic current flows between the myocardium and the
inside and outside neighboring domains of the heart. The computational domain is
generated using the 3D model of the myocardium after applying voxel and volumetric
marching cube (VoMaC)-based meshing algorithms (Keyak et al., 1990; Mueller and
Ruegsegger, 1994; Simpleware, 2010).