Biomedical Engineering Reference
In-Depth Information
6.3 A Numerical Example
In this section we describe a model tailored to simulate a type of smooth muscle
experiment described in Arner ( 1982 ). Briefly, strips of skinned smooth muscles
are stimulated in an open organ bath and contract isometrically at the optimal force
generating length. The contraction is elicited by adding chemicals to the bath and
the specimen is allowed to contract for three minutes after which it relaxes for seven
minutes in a normalizing bath. This process is repeated several times. For this ex-
periment, the model inputs are the isometric stretch and calcium ion concentration
while the output is the total stress. The isometric stretch is taken to be λ
=
1 . 5 and
the calcium ion concentrations are taken to be 100 nM and 350 nM for the nor-
malizing and contracting baths, respectively, see Murtada et al. ( 2012 ). Further, we
assume λ HM =
1inEq.( 6.20 ) to be consistent with the model in the
same reference. All computations are summarized in Algorithm 6.1 .
1 . 5 and a m =
Algorithm 6.1 Given a stretch λ(t) and the calcium ion concentration q(t)
Step 1. Compute consistent initial conditions at isometric conditions
Step 1a. Set
1
Step 1b. Compute n ( 0 ) from the null space of the matrix in Eq. ( 6.19 )
Step 1c. Compute ε ft ( 0 ) from Eq. ( 6.17 )
Step 1d. Set ε ft , 0 =
n
˙
=
0 ,
ε ft =
˙
0 and f
=
ε ft ( 0 )
Step 2. For each time step k , compute
˙
Step 2a.
n (k) from Eq. ( 6.18 ), and
Step 2b.
ε ft (k) by using Eqs. ( 6.16 ) and ( 6.17 )
Step 2c. Compute n (k
˙
+
1 ) and ε ft (k
+
1 ) using a time-stepping
scheme
Step 2d. Compute the total stress t(k) from Eq. ( 6.13 )
The passive free-energy function is taken from Murtada et al. ( 2012 ), i.e.
2 λ 2
2 λ 1 +
2 c 2 exp c 2 λ 2
1 2
1 ,
c 0
c 1
ψ p
=
+
(6.21)
where c 0 ,c 1 ,c 2 > 0 are constants. Equation ( 6.21 ) is derived by specializing the
three-dimensional strain energy proposed in Holzapfel et al. ( 2000 ) to a uniaxial
extension case. For cross-bridges, assume the free energy to be
n C +
n D
ψ cb
cd ,
=
(6.22)
2
where E is the stiffness of the cross-bridges. The free energy in ( 6.22 ) gives a linear
stress response in ε cd which has been observed for cross-bridges in striated muscles
using extremely small steps in quick-release experiments (McMahon, 1984 ).
Search WWH ::




Custom Search