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
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