Biomedical Engineering Reference
In-Depth Information
constants to be (Murtada et al.,
2012
)
q
4
k
1
=
k
6
=
η
,
(6.3)
q
1
/
2
q
4
+
where
q
is the intracellular calcium ion concentration,
η>
0 is a constant and
q
1
/
2
is the half-activation concentration.
6.2.2 Kinematics
The model presented herein is confined to homogeneous deformations and the
smooth muscle contraction is, therefore, considered to be along a well-defined direc-
tion. For a fully three-dimensional smooth muscle contraction model, the interested
reader is referred to Stålhand et al. (
2011
).
The active contraction is modeled as an additive two-step process. The first step is
a filament translation
u
ft
where the friction clutch displaces actin along the myosin,
and the second step is an elastic deformation
u
cd
of the cross-bridges. Note that the
filament translation introduces an incompatibility in the strain field, indicated by the
gap in the middle panel in Fig.
6.1
. This incompatibility arises because it is assumed
that the filament translation occurs without deforming the elastic springs. This has
little significance here and compatibility is restored by stretching the cross-bridges.
For three-dimensional contraction, however, the incompatibility becomes essential,
see Stålhand et al. (
2011
).
If the reference length is taken to be
L
, the deformed length is given by
l
=
L
+
u
ft
+
u
cd
and the total stretch can be obtained by dividing
l
by
L
giving
λ
=
1
+
ε
ft
+
ε
cd
,
(6.4)
where
ε
ft
=
u
ft
/L
and
ε
cd
=
u
cd
/L
. Note that
ε
ft
is defined to be negative in con-
traction. The time derivative of Eq. (
6.4
) gives the deformation rate
˙
λ
=
ε
ft
+
ε
cd
,
(6.5)
where the superscribed dot denotes time derivative. Because of the side-polar ar-
rangement of myosin heads (Xu et al.,
1996
), the power stroke can only generate
contraction. Consequently, the friction clutch disc always rotates counter-clockwise
in Fig.
6.1
and the velocity at the perimeter must be negative. The absolute value of
the perimeter velocity
ν
must be non-negative, however.
6.2.3 Balance Laws
The balance laws are derived using the principle of virtual power as stated by Ger-
main (
1973
). The method is based on defining virtual velocity fields and assigning