Biomedical Engineering Reference
In-Depth Information
uniform. The time rate of change of concentration
C
i
of the
i
th ionic species in a
given compartment is given by
dC
()
t
I
()
t
i
=
i
,
[12]
dt
z FV
i
where
C
i
(
t
) is the concentration (typically mM) of species
i
;
t
is time (typically
in msec),
I
i
(
t
) is net current into the compartment carried by species
i
(typically
in pA);
z
i
is the valence of the
i
th species,
F
is Faraday's constant; and
V
is the
compartment volume (typically in units of pL). One such equation may be de-
fined for the concentration of each ionic species in each model compartment.
Ion flux between compartments, related to the term
I
i
(
t
) in Eq. [12], is produced
either by: (a) diffusion due to differences in ion species concentration between
adjacent compartments (as is the case for the flux term
J
xfer
in Figure 1B repre-
senting Ca
2+
diffusion from the subspace to the cytosol); (b) gating of ion chan-
nels in the sarcolemmal or JSR membrane (as is the case for Ca
2+
flux
J
rel
in
Figure 1B from the JSR into the subspace through RyR channels); or (c) the
action of membrane transporters and exchangers (for example, Ca
2+
flux through
the SR Ca
2+
-ATPase, labeled
J
up
in Figure 1B). The form of the algebraic equa-
tions describing the function of membrane transporters and exchangers, includ-
ing their concentration, voltage, and in some instances ATP dependence, may be
found in the published equations for a number of myocyte models. In addition,
buffering of Ca
2+
by negatively charged phospholipid head groups in the sar-
colemmal and JSR subspace membrane, by cytosolic myofilaments (troponin)
and by calsequestrin in the JSR is modeled. Buffering due to mechanisms other
than myofilaments is described using the rapid buffer approximation of Wagner
and Keizer (34).
2.5. Composite Equations for Common Pool Models
Common pool models of the cardiac myocyte consist of systems of nonlin-
ear ordinary differential-algebraic equations describing the time evolution of
model state variables. These state variables are: (a) probability of occupancy of
ion channel states (Eq. [2]) and current flux through open channels (Eq. [4]); (b)
concentrations of ion species in model compartments (Eq. [12]); and (c) time
evolution of membrane potential. Currently, all biophysically detailed models of
the myocyte assume that since these cells are spatially compact they are isopo-
tential, with time-rate-of-change of membrane potential given by
£
²
dv t
()
¦
¦
¦
¦
,
=
I
ion
[()]
vt
+
I
pump
[(),()]
vt ct
[13]
¤
»
¦
i
i
¦
dt
¦
¥
¦
¼
i
i
