Biomedical Engineering Reference
In-Depth Information
release in these directions (as done in Usyk et al.,
2000
) and then translating this
information into active strains, or assuming transverse isotropy of the mechanical
response at the cell level (as in Rossi et al.,
2012
). Moreover, if the components of
the activated deformation
F
o
satisfy the condition
(
1
+
γ
f
)(
1
+
γ
s
)(
1
+
γ
n
)
=
1
,
(14.7)
then det
F
o
≡
1, a very convenient choice from a numerical point of view for this
nonlinear problem, since convergence has to be incrementally reached.
The thermodynamic assumption of the multiplicative decomposition (
14.5
)is
that the active deformation det
F
o
stores no energy, so that the strain-energy function
is
W
=
W
(
F
e
)
and
det
F
o
W
FF
−
o
.
det
F
o
W
=
W
strain
=
(14.8)
The activation
γ
f
depends on the concentration of ionic species as can be deduced
from ordinary differential equation models (Rice et al.,
2008
; Murtada et al.,
2010
;
Nobile et al.,
2012
), which can be summarized in the symbolic equation
∂
t
γ
f
−
G(
w
,γ
f
)
=
0
,
(14.9)
where
G
defines the activation dynamics depending on ionic concentrations denoted
by the vector
w
.
In the reference configuration
Ω
o
, the equations governing the electromechanical
interaction under active strain read
⎨
det
F
o
p
F
−
T
(
FF
−
o
)
∂
F
∂
W
−∇·
−
=
0
,
in
Ω
o
;
det
F
=
1
,
−∇·
F
−
1
D
e
F
−
T
u
e
+
I
app
,
∇
χ(I
ion
+
=
χc
m
∂
t
v
I
sac
)
⎩
+∇·
F
−
1
D
i
F
−
T
u
i
+
I
app
,
χc
m
∂
t
v
∇
χ(I
ion
+
I
sac
)
=
in
Ω
o
×
(
0
,T).
∂
t
w
−
m
(v,
w
)
=
0
,
∂
t
γ
f
−
G(
w
,γ
f
)
=
0
,
(14.10)
System (
14.10
) is to be completed with suitable initial data for
v
,
w
as well as with
boundary conditions for all fields. The usual prescription of voltage at the initial
time is that a large enough perturbation is located at the apex, so that an electric
wave starts traveling up to the base, producing the due ionic currents and mechanical
contraction. This initial condition corresponds to immaterial assumption that the
electric signal, actually produced at the sinoatrial node, has been traveling very fast
along the Purkinje fibers down to the apex, where they finely branch producing a
volumetric diffusion at
t
0. Since the Purkinje fibers also branch up from the apex
towards the base at the subepicardial level, different protocols are often used to
=
