Biomedical Engineering Reference
In-Depth Information
Fig. 13.2
Depiction of the
mechanical (
left
)and
electrophysiological (
right
)
natural and essential
boundary conditions
13.2.2 Governing Differential Equations
The balance of linear momentum with its following well-known local spatial form
J
div
J
−
1
τ
+
ˆ
B
=
0
in
B
(13.4)
describes the quasi-static stress equilibrium in terms of the Eulerian Kirchhoff stress
tensor
[•]
denotes the divergence with respect to the spatial coordinates
x
. Note that the mo-
mentum balance depends on the primary field variables through the Kirchhoff stress
tensor
τ
and a given body force
B
per unit reference volume. The operator div
ˆ
τ
, whose particular form is elaborated in Sect.
13.2.3
. The essential (Dirich-
let) and natural (Neumann) boundary conditions, see Fig.
13.2
(left),
ˆ
=
t
ϕ
=¯
ϕ
on
∂
S
ϕ
and
t
on
∂
S
t
,
(13.5)
complete the description of the mechanical problem. Clearly, the surface subdo-
mains
∂
S
ϕ
and
∂
S
t
fulfill the conditions
∂
S
=
∂
S
ϕ
∪
∂
S
t
and
∂
S
ϕ
∩
∂
S
t
=∅
.
The surface stress traction vector
t
, defined on
∂
S
t
, is related to the Cauchy stress
t
J
−
1
τ
tensor through the Cauchy stress theorem
:=
·
n
where
n
is the outward
surface normal on
∂
.
The second field equation of the coupled problem, the excitation equation of the
following form
S
J
div
J
−
1
q
−
I
φ
Φ
−
ˆ
=
0in
B
(13.6)
describes the spatio-temporal evolution of the action potential field
Φ(
X
,t)
in terms
of the diffusion term div
I
φ
. The notation
J
−
1
[
ˆ
]
q
and the nonlinear current term
˙
[•] :=
[•]
/
D
t
is utilized to express the material time derivative. Within the frame-
work of Fitzhugh-Nagumo-type models of electrophysiology (Fitzhugh,
1961
), the
current source
I
φ
controls characteristics of the action potential regarding its shape,
duration, restitution, and hyperpolarization along with another variable, the so-
called recovery variable
r
whose evolution is governed by an additional ordinary
differential equation. Since the recovery variable
r
chiefly controls the local repo-
larization behavior of the action potential, we treat it as a local internal variable.
This will be more transparent as we introduce the explicit functional form of
D
I
φ
in
