Biomedical Engineering Reference
In-Depth Information
¯
filament sliding
u
fs
, which was supported by Guilford and Warshaw (
1998
), a simi-
lar hyperbolic function was used to redefine the evolution law of the relative filament
sliding
u
fs
, thus
¯
(P
a
+
α)(
−
u
fs
+
β)
=
(P
c
+
α)β,
(4.21)
which can be rewritten as
β
P
a
−
P
c
P
a
+
¯
u
fs
=
α
.
(4.22)
¯
When comparing the evolution law of the filament sliding
u
fs
, as presented in Mur-
tada et al. (
2010a
), with the updated evolution law (Murtada et al.,
2012
) it can be
seen that the two evolution laws do not differ that much from each other.
The evolution law for
u
fs
was further extended to also allow the simulation of
isotonic muscle extension such as
¯
P
a
−
P
c
P
a
+
α
−
P
c
P
a
−
P
LBC
P
a
−
¯
u
fs
=
β
1
β
2
,
(4.23)
where
P
LBC
is the maximal load-bearing capacity of the contractile units (yield
stress) (Dillon et al.,
1981
), and
β
1
and
β
2
are fitting parameters. The internal stress
P
c
, which is governed by the number of attached cycling and non-cycling cross-
bridges (depending on the mechanical state of the smooth muscle) is dependent on
the varying filament overlap
L
o
as well. Hence, based on Eq. (
4.11
), the internal
stress
P
c
during contraction was quantified as
κ
AMp
L
o
(
P
c
=
u
fs
)n
AMp
,
¯
(4.24)
and during muscle relaxation (extension) as
κ
AMp
L
o
(
κ
AM
L
o
(
P
c
=
u
fs
)n
AMp
+
¯
¯
u
fs
)n
AM
,
(4.25)
where
κ
AMp
is a parameter related to the force due to a power-stroke of a single
cross-bridge and
κ
AM
is related to the force-bearing capacity of a dephosphorylated
attached (latch) cross-bridge during muscle extension.
The material parameters in the mechanical model were fitted to isometric tension
development and to isotonic shortening velocities from swine carotid media (Dillon
et al.,
1981
; Murtada et al.,
2012
), resulting in
μ
a
=
5
.
3MPa,
α
=
26
.
7kPa,
β
=
0
.
0083 s
−
1
β
1
=
and
κ
AMp
=
204 kPa. The material parameters
β
2
and
κ
AM
were
fitted to sudden extension experiments resulting to
β
2
=
0
.
0021 s
−
1
and
κ
AM
=
u
opt
fs
L
o
were fitted to
61
.
1 kPa. The parameters in the filament overlap function
¯
=
0
.
48 and
0
.
8544 by means of the conditions in Eqs. (
4.17
) and (
4.18
) together
with length-tension experimental data from swine carotid media (Murtada et al.,
2012
). By using Eq. (
4.19
) together with a filament sliding evolution law and the
kinetic model by Hai and Murphy (
1988
), the active stress
P
a
was simulated for
different stretches
λ
, see Fig.
4.8
. The simulated results show very good correlations
with experimental data obtained from swine carotid media.
x
0
=
¯
