Biomedical Engineering Reference
In-Depth Information
membrane current per unit volume, respectively. Then the
anisotropic Bidomain
model
in term of the potential unknowns
u
i
(
x
,
t
)
,
u
e
(
x
,
t
)
,
v
(
x
,
t
)=
u
i
(
x
,
t
)
−
u
e
(
x
,
t
)
,
the gating variables
w
(
x
,
t
)
and ion concentrations
c
(
x
,
t
)
can be written as
⎧
⎨
c
m
∂
v
I
i
app
t
−
div
(
D
i
∇
u
i
)+
i
ion
(
v
,
w
,
c
)=
in
Ω
H
×
(
0
,
T
)
∂
c
m
∂
v
I
app
in
−
t
−
div
(
D
e
∇
u
e
)
−
i
ion
(
v
,
w
,
c
)=
Ω
H
×
(
0
,
T
)
(5.4)
∂
⎩
∂
w
∂
∂
c
t
−
R
(
v
,
w
)=
0
,
t
−
S
(
v
,
w
,
c
)=
0 n
Ω
H
×
(
0
,
T
)
.
∂
The Reaction-Diffusion (R-D) system (5.4) uniquely determines
v
, while the po-
tentials
u
i
and
u
e
are defined only up to the same time-dependent additive constant
relating to the reference potential. Until now the Bidomain model has been formu-
lated in terms of the potential fields
u
i
and
u
e
, but it can be equivalently expressed in
terms of the transmembrane and extracellular potentials
v
(
x
,
t
)
and
u
e
(
x
,
t
)
; in fact,
=
+
adding the two evolution equations of system (5.4) and substituting
u
i
v
u
e
,we
(
,
)
obtain an elliptic equation in the unknown
which, coupled with one of the
evolution equations, gives the following equivalent formulation of the anisotropic
Bidomain model
⎧
⎨
v
u
e
I
i
app
+
I
app
−
div
((
D
i
+
D
e
)
∇
u
e
)
−
div
(
D
i
∇
v
)=
in
Ω
H
×
(
0
,
T
)
(5.5)
c
m
∂
v
I
i
app
⎩
t
+
(
,
,
)
−
(
)
−
(
)=
×
(
,
)
.
i
ion
v
w
c
div
D
i
∇
v
div
D
i
∇
u
e
in
Ω
0
T
H
∂
The system must be supplemented by the initial conditions
v
(
x
,
0
)=
v
0
(
x
)
,
w
(
x
,
0
)=
w
0
(
x
)
,
c
(
x
,
0
)=
c
0
(
x
)
in
Ω
H
and by boundary conditions. In the case of an insulated heart surface
Γ
H
, both the the
intra- and extracellular current vector are tangent to the interface, i. e.
n
T
j
i
=
n
T
j
e
=
0, where
n
denotes the outward normal with respect to
Ω
H
.
In the non-insulated case, we must couple the macroscopic Bidomain model of
the cardiac tissue with the description of the current conduction in the extracar-
diac medium in order to relate the noninvasive potential measurements on the body
surface to the bioelectric cardiac source currents. Let us denote by
Ω
0
,
D
0
,
j
0
=
−
u
0
, the extracardiac volume, the conductivity tensor, the current density
and the extracardiac potential respectively, and by
D
0
∇
u
0
,
\
Γ
H
the body sur-
face. Disregarding, for instance, the presence of extracardiac applied currents, no
current sources lie outside the heart, thus div
j
0
Γ
=
∂Ω
0
0
=
0
. Moreover, the body
is embedded in the air, which is an insulated medium, hence
n
T
j
0
0in
Ω
=
0 n
Γ
0
.
H
requires that
n
T
n
T
j
0
,
Current conservation on the heart interface
Γ
(
j
i
+
j
e
)) =
where
n
denotes the outward normal to
Ω
H
, and the zero intracellular flux condi-
tion
n
T
j
i
=
0. In terms of potentials, the system must be coupled with the following
