Biomedical Engineering Reference
In-Depth Information
The specifications for (
14.1
) are in accordance with the minimal model of Bueno-
Orovio et al. (
2008
), a four-equations phenomenological model for human ventri-
cles. The auxiliary variables are
w
(w
1
,w
2
,w
3
)
which are phenomenological
quantities (no direct physical interpretation), however
w
3
behaves like a re-scaled
intracellular calcium concentration. The reaction terms are defined as
I
ion
(v,
w
)
=−
w
1
H(v
−
θ
1
)(v
−
θ
1
)(v
v
−
v)/τ
fi
+
(v
−
v
0
)
1
=
−
H(v
−
θ
2
)
/τ
0
+
H(v
−
θ
2
)/τ
3
,
0
−
H(v
−
θ
2
)w
2
w
3
/τ
si
,
(14.12)
and
⎛
⎝
⎞
⎠
w
1
)/τ
1
θ
1
)w
1
/τ
1
((
1
−
H(v
−
θ
1
))(w
1
,
inf
−
−
H(v
−
−
H(v
−
θ
2
))(w
2
,
inf
−
w
2
)/τ
2
−
H(v
−
θ
2
)w
2
/τ
2
m
(v,
w
)
=
(
1
,
(14.13)
w
3
)/τ
3
)
T
+
−
−
((
1
tanh
(k
3
(v
v
3
)))/
2
where
H
stands for the usual Heaviside function. The switches and infinite values
are defined as follows:
=
1
H
v
θ
1
τ
1
,
1
+
H
v
θ
1
τ
1
,
2
τ
1
−
−
−
(14.14)
τ
2
,
1
+
τ
2
,
2
−
τ
2
,
1
1
tanh
k
2
v
v
2
/
2
τ
2
=
+
−
(14.15)
τ
3
,
0
,
1
)
1
tanh
k
3
,
0
(v
v
3
,
0
)
/
2
τ
3
,
0
=
τ
3
,
0
,
1
+
(τ
3
,
0
,
2
−
+
−
(14.16)
τ
3
=
1
θ
2
)
τ
3
,
1
+
−
H(v
−
H(v
−
θ
2
)τ
3
,
2
(14.17)
τ
0
=
1
θ
0
)
τ
0
,
1
+
−
H(v
−
H(v
−
θ
0
)τ
0
,
2
(14.18)
1
,v<θ
1
0
,u
w
1
,
inf
=
(14.19)
θ
1
≥
w
2
,
inf
=
1
−
H(v
−
θ
0
)
(
1
−
v/τ
2
,
∞
)
+
H(v
−
θ
0
)w
2
,
∞
.
(14.20)
The model reproduces the correct shape of the voltage wave. For the sake of sim-
plicity, we use the epicardial parameters for the whole cardiac muscle:
θ
0
=
0
.
005,
0
.
13,
θ
1
θ
1
=
0
.
3,
θ
2
=
=
0
.
1,
τ
3
,
0
,
1
=
91,
τ
3
,
0
,
2
=
0
.
8,
τ
3
,
1
=
2
.
7342,
τ
3
,
2
=
4,
1
.
61,
τ
1
,
1
=
80,
τ
1
,
2
=
τ
1
7,
w
2
,
∞
=
τ
0
,
1
=
410,
τ
0
,
2
=
0
.
5,
v
v
=
=
1
.
4506,
τ
2
,
1
=
70,
τ
2
,
2
=
8,
τ
2
280,
k
2
200,
v
2
=
=
=
0
.
016,
τ
fi
=
0
.
078,
k
3
,
0
=
2
.
1,
v
3
,
0
=
0
.
6,
k
3
=
2
.
0994,
v
3
=
0
.
9087,
τ
si
=
3
.
3849,
τ
2
,
∞
=
0
.
01. The initial data
corresponds to
w
1
=
w
2
=
0.
The governing ODE for the activation corresponds to (
14.9
), as introduced in
Rossi et al. (
2011
) and Nobile et al. (
2012
), with the specification
G(γ
f
,w
3
)
1,
w
3
=
=
−
0
.
04
γ
f
.
The time sequence of transmembrane potential, activation
γ
f
and other ionic
concentrations are illustrated in Fig.
14.3
for a point on the epicardial surface. The
highest activation value is attained with a delay of about 120 ms with respect to that
of the action potential.
0
.
02
w
3
−