Biomedical Engineering Reference
In-Depth Information
4.3.3 Length-Tension and Force-Velocity Relationships
The ability for smooth muscle to produce active tension over a broad range of mus-
cle lengths, with a maximal active tension development at an optimal muscle length,
is an important characteristic to capture when simulating active smooth muscle con-
traction under large deformation. We have worked out the modeling of the length-
tension behavior through two different approaches, briefly reviewed here. The model
of Murtada et al. (
2010a
) served as a basis.
In the first approach, the effect of the intracellular calcium concentration
Ca
2
+
]
i
and the dispersion of contractile fibers in smooth muscles was investigated (Murtada
et al.,
2010b
). In the second approach, the effect of filament overlap and filament
sliding behavior in the smooth muscle contractile unit was analyzed (Murtada et al.,
2012
).
[
4.3.3.1 Agonist Sensitivity and Dispersion of Contractile Fibers
In the first approach of Murtada et al. (
2010b
), two experimental studies of smooth
muscle were used to analyze stretch-dependent agonist sensitivity and the dispersion
effects of contractile fibers in smooth muscles, see Fig.
4.5
. This was then used as a
basis for studying the smooth muscle length-tension behavior.
The intracellular calcium measurements at different muscle lengths was coupled
with the Hai and Murphy reaction rates
k
1
through Eqs. (
4.4
) and (
4.5
). The smooth
muscle contractile units were modeled as in Murtada et al. (
2010a
) with an equiva-
lent evolution law for the filament sliding
u
fs
and a constant filament overlap. The
passive components in the surrounding matrix was modeled by elastin and one fam-
ily of collagen fibers aligned along the main direction of the contractile units. The
neo-Hookean material was used to model elastin and an anisotropic exponential
function was used to model the anisotropic response (Holzapfel et al.,
2000
). The
passive stress
P
p
of the surrounding matrix was derived as
¯
μ
p
λ
2
c
1
λ
exp
c
2
λ
2
1
2
λ
2
1
,
1
λ
2
P
p
=
−
+
−
−
(4.12)
where
λ
denotes the stretch in the loading direction and
μ
p
,
c
1
and
c
1
are material
parameters. The passive material parameters
(μ
p
,c
1
,c
2
)
were estimated by com-
paring the simulated stress-stretch behavior
P
p
through Eq. (
4.12
) with the passive
length-tension experimental behavior of a carotid media (Kamm et al.,
1989
), with
the results of
μ
p
=
0
.
20 Pa.
The contractile unit orientation dispersion was modeled by introducing an orien-
tation density function
ρ(θ,γ)
with rotational symmetry as a function of the angle
θ
and the parameter
γ
which describes the shape of the density function. Hence, the
active stress
P
a
was expressed as
1680 Pa,
c
1
=
5040 Pa and
c
2
=
χ
λ
3
χ)λ
,
(4.13)
n
AM
)
λ
+¯
u
fs
−
1
1
λ
2
μ
a
L
o
(n
AMp
+
P
a
=
−
+
(
1
−
λ
