Biomedical Engineering Reference
In-Depth Information
4.3.2.1 Mechanical Framework
In the recent works by Murtada et al. (
2010a
,
2010b
,
2012
) a description of a smooth
muscle contractile unit based on structural observations (Herrera et al.,
2005
) and
filament sliding theory was presented. The model of the smooth muscle contractile
unit consists of two thin actin filaments, each with a certain length, and one thick
myosin filament, with a certain length, which are overlapped. The actin filaments
are organized on each side of the myosin filament from which the filament overlap
L
o
can be distinguished. Based on the Hill's muscle model, the length-change in a
smooth muscle contractile unit is described by the relative actin and myosin filament
sliding
u
fs
caused by the phosphorylated cycling cross-bridges or by an external
load/deformation, and the elastic elongation
u
e
of the attached cross-bridges. Hence,
the stretch of a contractile unit with reference length
L
CU
can be expressed as
l
CU
L
CU
=
L
CU
+
u
fs
+
u
e
λ
=
,
(4.7)
L
CU
where
l
CU
is the deformed length of a contractile unit. Note that
u
fs
is denoted
positive in extension. When looking at a half contractile unit, the average elastic
elongation of the attached cross-bridges can be described by the average force acting
on the contractile unit and the total elastic stiffness from all the attached cross-
bridges. Thus,
P
a
N
CU
u
e
2
=
,
(4.8)
n
AM
)
L
δ
E
cb
(n
AMp
+
where
P
a
is the measurable active (averaged) first Piola-Kirchhoff stress (engineer-
ing stress),
N
CU
is the number of contractile units per unit area in the reference con-
figuration,
δ
is the average distance between the cross-bridges,
(n
AMp
+
n
AM
)L
o
/δ
is the total number of the attached cross-bridges and
E
cb
is the elastic stiffness of a
single phosphorylated/dephosphorylated cross-bridge with the unit force per length.
Together with Eq. (
4.7
) the active stress
P
a
can be derived as a function of the fila-
ment sliding
u
fs
and the stretch
λ
,i.e.
μ
a
L
o
(n
AMp
+
P
a
=
n
AM
)(λ
−¯
u
fs
−
1
),
(4.9)
L
CU
E
cb
N
CU
/(
2
δ)
is a stiffness constant,
where
μ
a
=
u
fs
=
¯
u
fs
/L
CU
is the normal-
L
o
=
ized filament sliding and
L
o
/L
CU
is the normalized filament overlap.
4.3.2.2 Evolution Law of Filament Sliding
The normalized filament sliding
u
fs
depends on the mechanical state (contraction
and relaxation) of the smooth muscle contractile unit. During muscle contraction,
¯
¯
u
fs
is driven by the difference of the internal force of the cycling phosphorylated
cross-bridges (AMp) and any external force acting on the contractile unit. During
