The Effect of Fiber Curvature on Stimulation of Cardiac Tissue

Trayanova et al.120 were the first to realize that fiber curvature can induce polarization in cardiac tissue, and Trayanova's group has examined this effect in detail.121'150'151 Roth and Langrill Beaudoin152 found approximate analytical solutions to the bidomain equations for electrical stimulation of cardiac tissue with curving fibers and illustrated two mechanisms of polarization, both of which require unequal anisotropy ratios. In the first mechanism, the fiber orientation changes in the direction parallel to the electric field (Fig. 19a). On the left, the current is distributed evenly between the intracellular and extracellular spaces because they have similar conductivities in the direction parallel to the fibers.153 On the right, most of the current is in the extracellular space because of the relatively small conductivity of the intracellular space in the direction perpendicular to the fibers. In the middle, the current must be moving from the intracellular to the extracellular space, thereby depolarizing the membrane. The key insight is that the current distributes according to the ratio of the conductivities (g) in the intracellular (i) and extracellular (e) spaces, and that this ratio is different in the longitudinal (L) and transverse (T) directions (gih/geh = giT/ger). This inequality is equivalent to the condition of unequal anisotropy ratios (gih/giT = geL/ger).

In the second mechanism, the fiber orientation changes in the direction perpendicular to the electric field (Fig. 19b). In this case, the current density J on the left and right

Figure 17: The transmembrane potential induced by an insulating heterogeneity in a uniform electric field (30V/cm), with fibers (a) horizontal and (b) vertical. The right panels are numerical simulations. The middle and left panels are experimental data, for two polarities of the electric field. (Woods et al.)146

Figure 17: The transmembrane potential induced by an insulating heterogeneity in a uniform electric field (30V/cm), with fibers (a) horizontal and (b) vertical. The right panels are numerical simulations. The middle and left panels are experimental data, for two polarities of the electric field. (Woods et al.)146

-120 mV

Figure 18: The calculated transmembrane potential under an epicardial electrode during an applied electric field (Patel and Roth, with kind permission of Springer Science and Business Media)148

-120 mV

120 mV

Figure 18: The calculated transmembrane potential under an epicardial electrode during an applied electric field (Patel and Roth, with kind permission of Springer Science and Business Media)148

sides is in the same direction as the electric field E. However, J in the middle is not aligned with E because of the anisotropy. The higher anisotropy in the intracellular space causes the intracellular current to rotate toward the fiber direction more than the extracellular current. This induces a horizontal component of the current density that is larger inside the cells than outside. The net result is current entering the cells on the left (thereby hyperpolarizing the tissue) and exiting the cells on the right (thereby depolarizing the tissue).

Figure 20 shows the transmembrane potential induced by a curving fiber geometry. The inset shows the individual contributions of the two mechanisms. Although the approximate analytical model used to calculate the results in Fig. 20 has significant limitations, it does provide useful insight into the mechanisms underlying polarization by fiber curvature.

The impact of fiber curvature during stimulation of the heart has been studied using a combination of whole-heart modeling and optical mapping. Efimov et al.105 observed defibrillation shock-induced virtual electrodes that correlate well with simulations, albeit for epicardial polarizations.119 Similarly, Knisley et al.,122 using stimulation parallel to the surface of the heart, and Tung and Kleber,154 using cultured, two-dimensional strands of cells, have found excellent agreement between theory and experiment. As discussed above, there remains a need for quantitative comparisons between model and experiment for intramyocardial fibers.

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