Biomedical Engineering Reference
In-Depth Information
and
r
a
r
i
σ
zz
r
d
r
r
a
d
r
r
=
(
σ
θθ
−
σ
rr
)
P
,
2
π
=
f
,
(7.13)
r
i
where
r
i
and
r
a
denote inner (intimal) and outer (adventitial) radii in
β
0
(associated
with the deformation
F
I
)and
P
and
f
are the in vivo luminal pressure and axial
force, respectively. Here, the in vivo configuration is assumed to correspond to mean
transmural pressure and basal muscular tone in the basilar artery [37]. In order to
ensure radial equilibrium, with the given in vivo luminal pressure
P
, Eq. (7.13)
1
can be solved for the adventitial radius
r
a
(or for the luminal radius in the case of a
full G&R model) while the axial force
f
is computed explicitly from Eq. (7.13)
2
.In
this way, in all the simulations, the material parameters, the intimal radius, and the
luminal pressure can be kept constant regardless of the distribution of constituent
prestretches and mass fractions, while a slight variation in the outer radius preserves
the radial equilibrium.
Equilibrium of the unloaded configuration
=
β
1
similarly requires
ρ
−
ρ
(
ρ
)=
∂
W
∂λ
ρ
λ
ρ
d
ρ
ρ
,
p
(
σ
ϑϑ
−
σ
ρρ
)
(7.14)
ρ
i
and
ρ
a
ρ
a
ρ
i
σ
ζζ
ρ
d
ρ
ρ
(
σ
ϑϑ
−
σ
ρρ
)
=
0
,
d
ρ
=
0
,
(7.15)
ρ
i
where
ρ
i
and
ρ
a
denote inner (intimal) and outer (adventitial) radii in
β
1
(associated
with the deformation
F
=
F
1
). The two global equations can be solved to determine
the inner radius
for prescribed material
properties and distribution of constituents prestretch and mass fraction.
Finally equilibrium of the excised, radially-cut configuration can be satisfied via
the following [35]
ρ
i
and the net in vivo axial prestretch
λ
∂λ
R
λ
R
R
−
R
)=
∂
W
d
R
R
,
p
(
R
(
σ
ΘΘ
−
σ
RR
)
(7.16)
R
i
and
R
a
R
i
σ
ZZ
R
d
R
R
a
R
i
σ
ΘΘ
R
d
R
R
a
d
R
R
=
(
σ
ΘΘ
−
σ
RR
)
0
,
=
0
,
=
0
,
(7.17)
R
i
where
R
i
and
R
a
denote inner (intimal) and outer (adventitial) radii in
β
2
(associated
with the deformation
F
F
2
F
1
). Note that the additional global equilibrium equa-
tion enforces zero applied moments on the radially-cut section. The three global
equations can be solved to determine the inner radius
R
i
, the net in vivo axial pre-
stretch
=
Λλ
, given
λ
from above, and the residual stress related opening angle
Φ
0
for
