Monodomain reduction

Equation (11.26) can be reduced to a monodomain equation for the membrane potential in one special case. Notice that

so that the balance of transmembrane currents becomes

¿(Cm'dF + lion) = V ■ (a,(ai + ae)—1aeVV) — V ■ ai(ai + ae)—1ii. (11.37)

Here we see that there is possibly a contribution to the transmembrane current from the divergence of the total current. We know that V-it = 0, so this source term is zero if the matrix ai(ai + ae)— 1 is proportional to a constant multiple of the identity matrix. In other words, if the two conductivity matrices ai and ae are proportional, ai = aae, with a a constant, then the source term disappears, and the bidomain model reduces to the monodomain model.

where a = ai(ai + ae)—1ae. When ai = aae, the tissue is said to have equal anisotropy ratios. A one-dimensional model with constant conductivities can always be reduced to a monodomain problem.

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