Biomedical Engineering Reference
In-Depth Information
5.2 Mathematical models of the cardiac bioelectric activity
The bioelectric activity of the cellular membrane of a myocyte is described by the
time course of the transmembrane potential, i. e. the potential jump
v
across the cel-
lular membrane surface separating the intra-
(i)
and extracellular
(e)
media, usually
called
action potential
. The whole process of action-potential generation and prop-
agation is due to ionic membrane currents and to the electrotonic diffusion in the
(i)
and
(e)
conducting media.
Starting from the sino-atrial node, which acts as a pacemaker, a front-like varia-
tion of the transmembrane potential
v
spreads first in the atria and then reaches the
ventricles through the Purkinje network, with a very fast transition from the rest-
ing value
v
r
to the plateau value
v
p
. This phase constitutes the excitation or depol-
arization phase, and it is followed by fast and slow repolarization phases with a
subsequent return to the initial state. The time profile of the transmembrane poten-
tial
v
may depend in general on the position
x
and on the local state of the heart;
the whole bioelectric cycle lasts about 300 msec in the human heart. Moreover, the
fibre structure strongly affects both the excitation and repolarization processes and
in particular is the main cause of the anisotropic conductivity in the cardiac tissue,
see [71, 130].
5.2.1 Ionic current membrane models
The electrical behavior of the membrane is represented by a circuit consisting of a
capacitor connected in parallel with a resistor, modelling the various ionic channels
regulating the selective and independent ionic fluxes through the membrane. The
total transmembrane current
I
m
is given by
C
m
dv
I
m
=
dt
+
I
ion
=
I
app
,
(5.1)
with
I
ion
the ionic membrane current,
C
m
the membrane capacity and
I
app
the applied
current per unit area of the membrane surface.
Most mathematical models of the ionic currents are based on appropriate exten-
sions of the formalism introduced by Hodgkin and Huxley in [59]. Current progress
in molecular biology continues to produce more detailed data and knowledge on the
dynamics of the ionic fluxes through the cellular membrane, see e. g. the review
papers [17, 81, 113]. The general form of the ionic current is given by
P
k
=
1
g
k
(
c
)
M
j
=
1
w
p
j
k
(
,
,
)=
(
−
v
k
(
)) +
(
,
)
,
I
ion
v
w
c
v
c
I
0
v
c
(5.2)
j
log
c
c
i
k
and
c
i
,
k
are the maximal conductance of the ion
channel, the Nernst potential and the intra- and extracellular ion concentrations for
the
k
-th ionic species, respectively, and
p
j
k
are integers. In (5.2), we have split the
ionic current as the sum of a term related to ionic fluxes modulated by the gating dy-
where
g
k
(
c
)
,
v
k
(
c
)=(
RT
/
zF
)
