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 ) 0
+ H(v θ 2 )/τ 3 , 0 H(v θ 2 )w 2 w 3 si ,
(14.12)
and
w 1 )/τ 1
θ 1 )w 1 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 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 +
3 , 0 , 2
+
(14.16)
τ 3 = 1
θ 2 ) τ 3 , 1 +
H(v
H(v
θ 2 3 , 2
(14.17)
τ 0 = 1
θ 0 ) τ 0 , 1 +
H(v
H(v
θ 0 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
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