Page 121 - Computational Modeling in Biomedical Engineering and Medical Physics

110 Computational Modeling in Biomedical Engineering and Medical Physics

Figure 4.12 Spiral crumbling due to block (squares) regions of unexcitable pieces of tissue (e.g.,
infarct), which are simulated by setting the diffusion coefficient in x and y directions in Eq. (4.14)
— before (A) and (B) after breakup (Mocanu et al., 2007).

fibrillation (Fig. 4.12). In this state the electrical activity turns chaotic, leading to high
frequency and low amplitude contractions of the cardiac muscle.
Hodgkin Huxley model simulates the variation of the conductance of the neural
membrane during the AP, using four differential equations that describe the dynam-
ics of ion channels. The model includes a nonlinear model of the membrane in the
“cable” equation. The waves travelling through excitable media (autowaves) are self-
sustained, powered by the medium in which they propagate. Unlike classical waves,
which attenuate and distort after a certain distance, they do not dissipate, have no
reflection and interference properties (when meeting they do not annihilate).
Autowaves are characteristic not only of neural axons or cardiac tissues, but also
occur in other nonlinear systems, chemical or physical. A famous example is the
Belousov Zabotinski chemical reaction (Bergé and Pomeau, 1984), which is an
oscillating chemical system with spatial and temporal self-organizing properties that
seem to question the second law of thermodynamics: an isolated system evolves irre-
versibly from order to disorder, and finally to thermal equilibrium (thermodynamic
death)—a consequence of the fundamental law of thermodynamics. However, for
biological systems, perpetuity is supported by other energy consuming physiological
phenomena that develop in parallel, which are intended to regenerate the initial
conditions.
Different abstractions are known to render the nonlinear dynamic of the
electrophysiological model for the heart under various conditions using the two
variables (excitation and inhibition) whose time evolution is described through
two to four coupled PDEs for continuous media (Wiener and Rosenblueth, 1946;
Fitzhugh, 1961; Zaikin and Zhabotinsky, 1970; Bergé and Pomeau, 1984; Keener,
1988; Bär et al., 1994; Boulakia et al., 2015; Chen et al., 2018), solved using
numerical methods, (Bürger et al., 2010; Filippi and Cherubini, 2006, 2009;