Mathematical and numerical methods for reaction-diffusion models in electrocardiology - Modeling of Physiological Flows - page 129

Table 5.2. Conductivity coefficients of the intra and extracellular media in Siemenscm − 1 and

i , e i , e i , e e

tn

i , i , e i , e

anisotropy ratios ρ = σ / σ , ρ = σ / σ n

t t

lt l

Unequal anisotropy

l t n lt tn

medium σ σ σ ρ ρ

intra 2.31724e-3 2.43504e-4 5.69083e-5 9.51622 4.27889

l t n lt tn

medium σ σ σ ρ ρ

extra 1.54483e-3 1.04385e-3 3.7221e-4 1.47993 2.80447

Equal anisotropy

i
l

i
t

i
n

i
lt

i
tn

medium σ σ σ ρ ρ

intra 1.34511e-3 3.36278e-4 8.40695e-5 4 4

e
l

t n e

lt

tn

medium σ σ σ ρ ρ

extra 5.35282e-3 1.3382e-3 3.34551e-4 4 4

with the unequal anisotropy ratios of Table 5.2, while the panels on the left col-
umn refer to a tissue slab with equal anisotropy ratio. In the left panel of Fig. 5.2,
the elliptical intra- and extracellular potential patterns have the same major to mi-
nor semi-axis ratio, yielding a transmembrane potential pattern (bottom left panel)
with more rounded elliptical lines around the anodal electrode. Conversely, in a slab
with unequal anisotropy ratio, the intra- and extracellular equipotential contour lines,
displayed in the top and middle right panels of Fig. 5.2, have elliptical shapes with
different ratio of the two semi-axes. The difference between these two epicardial
patterns yields a transmembrane potential distribution with the typical virtual elec-
trode response (Fig. 5.2, bottom right panel). In fact, the tissue within and around
the anodal electrode is negatively polarized (hyperpolarized) and it exhibits an epi-
cardial dog-bone shape, developing perpendicularly to the epicardial fibre direction
of 45 ◦ . Two regions of positive polarization (depolarization), i.e. two virtual cath-

odes, develop along the fibre direction adjacent to the concave parts of the epicardial
virtual anode boundary.
The same anodal stimulation was applied to a tissue slab with transmural fibre
rotation. In case of unequal anisotropy ratio, see the right panels of Fig. 5.3, the epi-
cardial transmembrane potential pattern exhibits a virtual electrode response with
a twisted hyperpolarized dog-bone shaped region, which is not symmetric with re-
spect to the epicardial fibre direction because of the counterclockwise fibre rotation
from epicardium to endocardium. In fact, the equipotential surfaces inside the virtual
cathode volumes, shown in Fig. 5.4, are shaped as two horns pointing counterclock-
wise when proceeding from the upper (epicardial) face to the lower (endocardial)
face. On intramural sections parallel to the epicardial face, the equipotential lines
inside the virtual anode preserve the same dog-bone shape as on the epicardium but
with a counterclockwise rotation, thus reducing their area and yielding a twisted tote
bag; see Fig. 5.4. On the contrary, the same anodal stimulus applied to a slab with
equal anisotropy ratios and transmural fibre rotation does not yield virtual cathodal

−