Biomedical Engineering Reference
In-Depth Information
Sect.
13.3
. Akin to the momentum balance, the equation of excitation is also fur-
nished by the corresponding essential and natural boundary conditions, Fig.
13.2
(right),
=
Φ
on
∂
Φ
S
φ
and
q
=¯
q
on
∂
S
q
,
(13.7)
respectively. Evidently, the surface subdomains
∂
S
φ
and
∂
S
q
are disjoint,
∂
S
φ
∩
∂
S
q
=∅
, and complementary,
∂
S
=
∂
S
φ
∪
∂
S
t
. The electrical surface flux term
q
¯
in (
13.7
)
2
is related to the spatial flux vector through the Cauchy-type formula
q
¯
:=
J
−
1
ˆ
·
n
in terms of the spatial surface normal
n
. Owing to the transient term in the
excitation Eq. (
13.6
), its solution necessitates an initial condition for the potential
field at
t
q
=
t
0
Φ
0
(
X
)
=
Φ(
X
,t
0
)
in
B
.
(13.8)
I
φ
indicates that these
Note that the 'hat' sign used along with the terms
τ
,
ˆ
q
, and
ˆ
variables are dependent on the primary fields.
13.2.3 Constitutive Equations
The solution of the field equations requires the knowledge of constitutive equations
describing the Kirchhoff stress tensor
q
, and the current source
I
φ
. In contrast to the constitutive approaches suggested in the literature (Cherubini
et al.,
2008
; Ambrosi et al.,
2011a
), we additively decompose the free-energy func-
tion into the passive part
ψ
p
τ
, the potential flux
and the active part
ψ
a
, Ask et al. (
2012a
,
2012b
),
+
ψ
a
g
F
e
,
=
ψ
p
(
g
ψ
;
F
)
;
(13.9)
where the former depends solely on the total deformation gradient, while the latter
depends on the elastic part of the deformation gradient, thus both on the deformation
and on the potential. This additive form results in the decoupled stress response
τ
=
τ
p
(
g
;
F
)
+
τ
a
g
;
F
e
,
(13.10)
where the Kirchhoff stress tensor is obtained by the Doyle-Ericksen formula
τ
:=
FF
a
−
1
from Eq. (
13.1
). Since the formulation is laid out in the Eulerian setting, the current
metric
g
is explicitly included in the arguments of the constitutive functions.
The potential flux
2
∂
g
ψ
and the elastic part of the deformation gradient is defined as
F
e
=
ˆ
q
is assumed to depend linearly on the spatial potential gradi-
ent
∇
x
Φ
q
=
D
(
g
;
F
)
·∇
x
Φ,
(13.11)
through the deformation-dependent anisotropic spatial conduction tensor
D
(
g
F
)
that governs the conduction speed of the non-planar depolarization front in three-
dimensional anisotropic cardiac tissue.
;
