Page 122 - Advances in Biomechanics and Tissue Regeneration

118 7. MULTISCALE NUMERICAL SIMULATION OF HEART ELECTROPHYSIOLOGY

The charge of the cell membrane directly depends on the difference of the membrane potential, V ¼ V i V e and the
capacitance of the membrane
q
,
V ¼ (7.8)
χC m
where C m is the membrane capacitance and
q i q e
: (7.9)
2
q ¼
Combining Eqs. (7.8), (7.9) and deriving with respect to time, we get
∂V 1∂ðq i q e Þ
χC m ¼ :
∂t 2 ∂t
Using Eq. (7.3), we get the relation

∂q i ∂q e ∂V
¼ χC m :
∂t ∂t ∂t
¼
Replacing this latter expression in Eq. (7.4) and using Eq. (7.1), we obtain
∂V
r D i rV i Þ ¼ C m + J ion , (7.10)
∂t
ð
where D i ¼M i /χ.
Eqs. (7.7), (7.10) depend on three potentials V i , V e , and V. Eliminating V i from Eqs. (7.7), (7.10), the equations for the
bidomain model are obtained
∂V
Þ + r D i rV e Þ ¼ C m + J ion , (7.11)
∂t
r D i rVð ð
Þ + r ðD i + D e ÞrV e Þ ¼ 0: (7.12)
r D i rVð ð
Assuming that the heart is surrounded by a nonconductive medium, the normal components of both currents (intra-
cellular and extracellular) are zero at the boundary, by which we have
n J i ¼ 0,
(7.13)
n J e ¼ 0,
where n is the outer normal. Using the expression for both currents and eliminating V i , the boundary conditions of the
model are obtained
n D i rV + D i rV e Þ ¼ 0, (7.14)
ð
n r D e rV e Þ ¼ 0: (7.15)
ð
7.2.1.2 Monodomain Model
As can be observed, the bidomain model represents the electric currents in both the intracellular and extracellular
medium. It is represented by a nonlinear parabolic equation coupled with an elliptical equation. Under particular con-
ditions, the bidomain model can be decoupled, allowing us to calculate the transmembrane potential independently
of the extracellular potential. Assuming that the conductivity tensors have the same variation in the anisotropy, that is,
D e ¼ λD i , where λ is a scalar, so D e can be eliminated from Eqs. (7.11), (7.12), obtaining
∂V
Þ + r D i rV e Þ ¼ C m + J ion , (7.16)
∂t
r D i rVð ð
Þ + ð1+ λÞr D i rV e Þ ¼ 0, (7.17)
r D i rVð ð
from Eq. (7.17), we have
1
Þ,
1+ λ
ð
r D i rV e Þ ¼ r D i rVð
replacing in Eq. (7.16) and operating, we get the standard formulation for the monodomain model:
∂V
r D rVÞ ¼ C m + I ion , (7.18)
ð
∂t

I. BIOMECHANICS