Biomedical Engineering Reference
In-Depth Information
E
Cl
dz
dt
=−
ʳ
x
x
+
x
Cl
(
z
−
z
Cl
)
Cl
−
channels
z
−
z
Ca
1
+
2
E
Ca
e
−
(
z
−
z
Ca
2
)/
R
Ca
VOCC influx
1
+
x
Na
/
Ca
z
−
z
Na
/
Ca
NCx
x
x
x
+
ʲ
e
−
(
z
−
z
Ca
3
)/
R
K
K
+
efflux
(1c)
+
E
Na
/
Ca
+
E
K
(
z
−
z
K
)
x
+
where the electric reversal potentials with respect to
Ca
2
+
and
Na
+
are determined
from the Nernst equation according to:
ln
[
ln
[
Ca
2
+
]
i
,
0
x
Na
+
]
i
,
0
[
RT
FZ
Ca
RT
FZ
Na
z
Ca
1
=
z
Na
=
(2)
Na
+
]
i
ln
[
Na
+
]
i
=
2,
RT F
−
1
=0.024,
Na
+
]
i
,
0
/
[
andwhere
Z
Ca
=1,
Z
Na
=
(
RT
/
FZ
Na
)
M,
R
is the gas constant in J K
−
1
mol
−
1
,
F
is the
Faraday constant in A s mol
−
1
and
T
Ca
2
+
]
i
,
0
0
.
069V,
[
=
0
.
0025
µ
=
310K. The reversal potential of the NCX
is calculated as
z
NCx
2
z
Ca
1
. All values of the coefficients are found in
Table
1
. A complete description of the mathematical model can be found in [
19
,
51
].
The subscript
r
refers to RyR-mediated CICR, as opposed to IICR (not included
in the present formulation). The same general conclusions would not significantly
differ whether the CICR positive feedback mechanism is modelled as an InsP
3
-or
RyR-dependent process, as was also pointed out in [
19
]. Admittedly, the addition
of InsP
3
as another independent variable could allow to asses the importance of
crosstalks between RyR and Insp
3
R, and extend the present formulation to include
coupling between the endothelium-fluid domain and the SM layer.
=
3
z
Na
−
2.2 Cell-Cell Coupling
The model has been extensively validated, with analysis and results presented for the
case of a single SMC [
19
,
51
]. The generalization of these results to a population of
SMCs, using image-based geometries, allows to characterize the effects of a number
of parameters, while enabling a more 'intuitive representation' of the phenomenon.
The nature of coupling between adjacent cells remains controversial. In the current
formulation we explore the consequences of either
Ca
2
+
or electrical coupling, by
assuming a simple gradient driven flux between neighbouring cells coupled via gap
junctions. Two terms:
g
Ca
j
x
j
−
x
i
z
j
−
z
i
J
Ca
,
i
=
V
m
,
i
=
g
z
(3)
∈
N
i
j
∈
N
i
are added to (
1a
) and (
1c
), respectively, and for each SMC
i
, to model
Ca
2
+
and
electrical coupling with all nearest neighbouring SMCs
j
∈
N
i
. Depending on
