Biomedical Engineering Reference
In-Depth Information
Space discretization.
Many different approaches have been developed for the space
discretization of the Bidomain and Monodomain equations. Finite differences were
studied in [56, 84, 94, 99, 115, 143, 156, 157, 161]. Finite elements have been widely
used, see e. g. [20, 29, 51, 89, 119, 148]. The finite-volume method, which has the
advantage of conserving local flux, was developed in [11, 39, 55, 142]. Finally, some
researchers recently investigated the spectral element method, see [13]. In order
to reduce the computational load of the Bidomain system, also adaptive remesh-
ing techniques have been developed, see e. g. [6, 7, 16, 20, 40, 141, 152]; these
techniques have proved successful for problems of moderate size and are currently
under investigation. Interested readers can find numerical approximation schemes
and numerical simulations based on eikonal approaches in [64, 66, 67, 83] and in
[24, 25, 28, 139], using mesh-size of the order of 1 mm.
Semi-implicit scheme with FEM.
We briefly describe here the approximation for
the anisotropic Bidomain system (5.4), obtained applying the finite element method
in space and a semi-implicit method in time. In the following, we will denote vectors
representing finite element functions by boldface symbols, for both product space
functions and their components.
Let
and
V
h
the associated space
of trilinear finite elements. We obtain a semidiscrete problem by applying a standard
Galerkin procedure. Choosing a finite element basis
h
be a uniform hexahedral triangulation of
T
Ω
for
V
h
,wedenoteby
{
g
i
}
a
i
,
e
rs
T
D
i
,
e
∇
M
=
{
m
rs
=
g
r
g
s
dx
},
A
i
,
e
=
{
=
Ω
(
∇
g
r
)
g
s
dx
},
Ω
the symmetric mass matrix and stiffness matrices, respectively, and by I
ion
,
I
app
the
finite element interpolants of
I
ion
and
I
app
, respectively.
We sketch now a semi-implicit time discretization of the Bidomain system, using
for the diffusion term the implicit Euler method, while the nonlinear reaction term
I
ion
is treated explicitly. The implicit treatment of the diffusion terms appearing in
the Monodomain or Bidomain models is essential in order to adapt the time step
according to the stiffness of the various phases of the heart-beat. The ODE system for
the gating variables is discretized by the semi-implicit Euler method and the explicit
Euler method is applied for solving the ODE system for the ion concentrations. As
a consequence, the full evolution system is decoupled by first solving the gating and
ion concentrations system (given the potential
v
n
M
at the previous time-step)
w
n
+
1
v
n
M
,
w
n
+
1
w
n
c
n
+
1
c
n
v
n
M
,
w
n
+
1
c
n
−
Δ
tR
(
)=
,
=
+
Δ
tS
(
,
)
,
and then solving for
u
n
+
1
u
n
+
1
i
u
n
+
1
e
=(
,
)
the system
u
n
+
1
A
=
F
(5.27)
where
M
A
i
0
0A
e
−
M
c
m
Δ
A
:
=
γ
+
,
γ
:
=
t
,
(5.28)
−
MM
