Biomedical Engineering Reference
In-Depth Information
•
the Mitchell-Schaeffer model [43]:
w
τ
in
V
m
(
V
m
τ
out
,
I
ion
(
V
m
,
w
)=
V
m
−
1
)
−
⎧
⎨
−
1
w
τ
open
if
V
m
≤
v
gate
,
(4.13)
(
,
)=
g
V
m
w
⎩
w
τ
close
−
if
V
m
>
v
gate
.
Here, 0
<
a
<
1,
k
,
ε
,
γ
,
τ
in
,
τ
out
,
τ
open
,
τ
close
and 0
<
v
gate
<
1 are given positive
constants.
The next theorem asserts the existence of solutions for the coupled heart-torso
model. It also provides the uniqueness of solutions if the ionic current is described
by FitzHugh-Nagumo model.
L
2
Ω
)
,
L
∞
(
Ω
σ
i
,
σ
Theorem 1.
Let T
>
0
,I
app
∈
×
(
0
,
T
∈
)
symmetric and
H
e
H
L
2
and V
m
∈
H
1
uniformly definite positive, w
0
∈
be given data. Assume
that I
ion
and g are given by
(4.10)
,
(4.11)
,
(4.12)
or a regularized version of
(4.13)
.
Then, the heart-torso system
(4.8)
has a weak solution V
m
(
Ω
H
)
(
Ω
H
)
L
∞
(
T
;
H
1
∈
0
,
(
Ω
))
∩
H
H
1
T
;
L
2
H
1
T
;
L
2
L
∞
(
T
;
H
1
(
,
(
Ω
))
∈
(
,
(
Ω
))
∈
,
(
Ω
))
0
,w
0
and u
0
with
H
H
u
e
in
Ω
H
,
u
def
=
u
T
in
Ω
T
,
def
=
Ω
∪
Ω
and
Ω
H
. Moreover, for the FitzHugh-Nagumo model
(4.10)
, the solution
T
is unique.
The proof of Theorem 1 is reported in [8] and [72, Part II] and generalizes some of the
arguments used for the analysis of the bidomain problem in [5, 9] to the case of the
heart-torso coupling. The main idea consists in reformulating the bidomain system as
a couple of degenerate reaction-diffusion equations and approximating the resulting
heart-torso system by a suitably regularized problem in finite dimension; the latter
is then analyzed through a Faedo-Galerkin/compactness procedure and a specific
treatment of the non-linear terms. The heart-torso coupling is handled through an
adequate definition of the Galerkin basis. Compared to models (4.10)-(4.12), the
Mitchell-Schaeffer ionic model has a different structure that makes the existence
proof slightly more involved. As shown in the next section, realistic ECG signals
can be simulated with this ionic model.
4.4 ECG simulations
The aim of this section is twofold: first, provide realistic simulations of the 12-lead
ECG based on the model given by system (4.8); secondly, discuss through numerical
simulations the impact of various modelling options (e.g., uncoupling, monodomain,
cell homogeneity, tissue anisotropy). More details can be found in [7].
