Biomedical Engineering Reference
In-Depth Information
which amounts to working with an isolated bidomain model. The reliability of this
approximation will be discussed later on. Moreover, isolating the heart might not
be applicable to several studies. For instance this approximation is not physically
acceptable while studying cardiac defibrillation mechanisms [66].
As will be shown in Sect. 4.4, this model has to be furnished by important
modelling ingredients like cell heterogeneity, tissue anisotropy, modelling of fast-
conducting networks (His bundle, Purkinje fibres) [6].
4.3 Mathematical analysis
Results on the existence and uniqueness of the solution for the isolated bidomain
system (4.4) have been reported in a number of works. A first result for the bido-
main model coupled with the FitzHugh-Nagumo ionic model, described below by
Eq. (4.10), has been reported in [18]. The proof is based on a reformulation of the
system of equations in terms of an abstract evolutionary variational inequality. The
analysis for a simplified ionic model, namely
I
ion
(
, has been ad-
dressed in [5]. In paper [9], existence, uniqueness and regularity of a local solution
in time are proved for the bidomain model with a general ionic model by using a
semi-group approach. Existence of a global in time solution of the bidomain prob-
lem is also proved in [9] for a wide class of ionic models - including FitzHugh-
Nagumo (4.10), Aliev-Panfilov (4.11), and Roger-McCulloch (4.12) - through a
compactness argument. As in the result presented below, uniqueness is only achieved
for the FitzHugh-Nagumo ionic model. Finally, in [70], existence, uniqueness and
some regularity results are proved with the Luo-Rudy ionic model [41]. All these
papers deal with a model for the heart alone, without any coupling with the external
medium.
Let us now present the result contained in [8], which addresses the mathematical
analysis of the coupled heart-torso system (4.8). The well-posedness analysis of this
system is obtained for an abstract class of two-variable ionic models including:
•
V
m
,
w
)=
I
ion
(
V
m
)
the FitzHugh-Nagumo model [26, 45]:
I
ion
(
V
m
,
w
)=
kV
m
(
V
m
−
a
)(
V
m
−
1
)+
w
,
g
(
V
m
,
w
)=
−
ε
(
γ
V
m
−
w
)
;
(4.10)
•
the Aliev-Panfilov model [2]:
I
ion
(
V
m
,
w
)=
kV
m
(
V
m
−
a
)(
V
m
−
1
)+
V
m
w
,
g
(
V
m
,
w
)=
ε
(
γ
V
m
(
V
m
−
1
−
a
)+
w
)
;
(4.11)
•
the Roger-McCulloch model [57]:
I
ion
(
V
m
,
w
)=
kV
m
(
V
m
−
a
)(
V
m
−
1
)+
V
m
w
,
g
(
V
m
,
w
)=
−
ε
(
γ
V
m
−
w
)
; (4.12)
