Page 233 - Computational Modeling in Biomedical Engineering and Medical Physics

222 Computational Modeling in Biomedical Engineering and Medical Physics

Combined Ohm’s law and Kirchhoff’s theorems applied to the equivalent circuit
with distributed elements presented here lead to the spatial and temporal distribution
of the transmembrane voltage, which is illustrated by the second-order partial differen-
tial equation:

2
@ U m
5 r e i S 1 r i 1 r e Þi m ; ð7:1Þ
ð
@x 2
and where the expression of the transmembrane current, i m , shows the difference
between the resting and active states of the membrane, as in the following equations:

@U m
i m U m ; tð Þ 5 U m 2 U m0 Þg m 1 c m ; ð7:2aÞ
ð
@t
for the resting (polarized) membrane, and

@U m
i m U m ; tð Þ 5 U m 2 U Na Þg Na 1 U m 2 U K Þg K 1 U m 2 U L Þg L 1 c m ; ð7:2bÞ
ð
ð
ð
@t
for the active (depolarized) membrane; Na and K channels conductances are not con-
stant, they depend on time and transmembrane voltage, as the Hodgkin Huxley
model states (Chapter 4: Electrical Activity of The Heart).
When the stimulus is applied, the transmembrane voltage, U m , rises from the resting
value, U m0 (specific to the polarized state), toward a threshold value, U p , which marks the
limit between polarized and depolarized states; this process is characterized by Eq. (7.1)
with i m given by Eq. (7.2a). For a sufficiently strong stimulus, the threshold is exceeded
and an action potential is generated, a process described by Eqs. (7.1) and (7.2b),which is a
mathematical model much more complicated than the previous one, solvable only by
numerical methods. Practically if only the effectiveness of the stimulus is evaluated, no
complexity is needed, and the first path allows the study of the membrane electrical
behavior from the rest state to reach the threshold. Equations (7.1) and (7.2a) yield the
expression of the polarized membrane behavior under the action of an electric stimulus
2
@ U m @U m
ð
ð
5 r e i s 1 r i 1 r e Þ U m 2 U m0 Þg m 1 c m ; ð7:3aÞ
@x 2 @t
where, after a few processing, the canonical form of a second-order PDE (Chapter 1:
Physical, Mathematical, and Numerical Modeling) is obtained
2 @
λ 2 @ u m 2 τ u m 2 u m 5 r e λ i s : ð7:3bÞ
2
@x 2 @t
The transmembrane true voltage U m was replaced considering its variation relative
to the resting state, so that the new voltage variable is u m 5 U m 2 U m0 , while space