Biomedical Engineering Reference
In-Depth Information
Our reference mathematical model of ECG relies on (4.8) and the following ad-
ditional modelling ingredients:
•
the heart geometry only includes the ventricles (see Fig. 4.3). Note that this sim-
plification prevents from computing the P-wave of the ECG;
•
the “torso” geometry contains three regions: the lungs, the bones and the remain-
ing extracardiac tissues (see Fig. 4.3). These regions are modelled with three dif-
ferent values of
σ
T
;
•
the fast conduction system (His bundle and Purkinje fibres) is modelled by ini-
tializing the activation with a time-dependent external volume current
I
app
, acting
on a thin sub-endocardial layer of left and right ventricles. The propagation speed
of this external stimulus is a parameter of the model (see [7] for details);
•
the dynamics of the cardiac cell's membrane are based on the Mitchell-Schaeffer
ionic model (4.13);
•
cells are heterogeneous in terms of
Action Potential Duration
(APD), which
varies transmurally within the left ventricle. In practice, this amounts to consider
a parameter
τ
close
in (4.13) which takes three different values in the left ventricle
epi
close
in
endo
intra
(
τ
close
in the subendocardial region,
τ
close
in the intracardial region and
τ
rv
the subepicardial region) and another value
close
in the right ventricle. This is an
important factor to obtain the T-wave with a correct polarity;
τ
•
the heart conductivities are anisotropic:
def
=
σ
t
i
t
i
,
e
σ
i
,
e
(
x
)
i
,
e
I
+(
σ
−
σ
)
a
(
x
)
⊗
a
(
x
)
,
,
e
l
t
where
a
i
,
e
are,
respectively, the conductivity coefficients in the intra- and extra-cellular media,
measured along the fibre and transverse directions.
(
x
)
is a unit vector parallel to the local fibre direction and
σ
i
,
e
and
σ
4.4.1 Numerical approximation
Problem (4.8) can be cast into weak form as follows: for
t
∈
(
0
,
T
)
,find
w
(
·,
t
)
∈
L
∞
(
Ω
H
)
H
1
H
1
L
0
(
Ω
H
)
H
1
,
V
m
(
·,
t
)
∈
(
Ω
H
)
,
u
e
(
·,
t
)
∈
(
Ω
H
)
∩
and
u
T
(
·,
t
)
∈
(
Ω
T
)
with
u
e
(
·,
t
)=
u
T
(
·,
t
)
on
Σ
, such that
∂
t
w
)
ξ
=
+
g
(
V
m
,
w
0
,
Ω
H
χ
m
∂
t
V
m
+
)
φ
+
I
ion
(
V
m
,
w
Ω
H
σ
i
∇
(
V
m
+
u
e
)
·
∇φ
=
I
app
φ
,
(4.14)
Ω
H
Ω
H
Ω
H
(
σ
i
+
σ
)
∇
·
∇ψ
+
Ω
H
σ
i
∇
·
∇ψ
+
Ω
T
σ
∇
·
∇ζ
=
u
e
V
m
u
T
0
e
T
H
1
L
2
H
1
L
0
(
Ω
H
1
for all
(
ξ
,
φ
,
ψ
,
ζ
)
∈
(
Ω
)
×
(
Ω
)
×
(
Ω
)
∩
)
×
(
Ω
)
with
ψ
=
H
H
H
H
T
ζ
. The weak formulation (4.14) is discretized in space using finite elements
and in time using a semi-implicit scheme based on a backward difference formula
(BDF).
on
Σ
