Modeling of Smooth Muscle Activation - Computer Models in Biomechanics - page 80

in contrast to extension, where both cycling and latched cross-bridges contribute to
the force generation. Third, substituting Eqs. ( 6.8 ) 2 into ( 6.15 ) and taking the first
term to be linear in ˙ ε ft ,gives

f ∂ψ cd

∂f
∂ε ft ψ cb

−

T ft −

+

∂ε ft =

˙

g ε ft , (6.17)

where g 0 is a function. Equation ( 6.17 ) constitutes an evolution law from which

ε ft can be computed. Finally, for the third term in Eq. ( 6.16 ), define

≥

∂ψ cb
∂ n

∂ψ n
∂ n −

+ r 1 = n , ˙ (6.18)

K

where 1 ( 1 , 1 , 1 , 1 ) , r is an arbitrary Lagrangian multiplier introduced to satisfy

the constraint in Eq. ( 6.2 ), and K is a four-by-four positive definite matrix. Note that
the term

=

0by
Eq. ( 6.2 ). On closer inspection, the structure of Eq. ( 6.18 ) is reminicent of the kinetic
evolution law by Hai and Murphy in Eq. ( 6.1 ), although the former is nonlinear. It
is possible to recover Eq. ( 6.1 ) by linearizing the left-hand side of Eq. ( 6.18 ) around
the experimental stretch in Hai and Murphy ( 1988 ) and choosing ψ cb to be linear
in n , see Stålhand et al. ( 2008 ). After some straightforward but non-trivial steps, the
result reads

− r 1 disappears when Eq. ( 6.18 ) is substituted in ( 6.15 ) because 1 · ˙ n =

⎡
⎣

⎤
⎦ =[

⎡
⎣

⎤
⎦

n A
n B
n C
n D

n A
n B
n C
n D

d
d t

k 1 k 2 k 3 k 4 ] , (6.19)

where k m (m = 1 , 2 , 3 , 4 ) are the column vectors

η 1 − k 1 , 1 , 0 , 0 T , η 2 k 2 , k 3 , 3 , 0 T ,

k 1 = k 2 = − k 2 −

η 3 0 , 4 , k 5 , 5 T , η 4 k 7 , 0 , 6 , k 7 T ,

(6.20)

k 3 = − k 4 − k 4 = − k 6 −

where a superscribed T denotes the transpose and η m = 1, where

λ HM is the stretch at which the experiments in Hai and Murphy ( 1988 ) are per-
formed, see Stålhand et al. ( 2008 ). Note that the column vectors in Eq. ( 6.20 )are
equal to the column vectors in Eq. ( 6.1 ) when λ

a m (λ − λ HM ) +

λ HM .
In summary, the model presented is governed by Eq. ( 6.13 ) for the external stress,
Eq. ( 6.14 ) for the calcium ion concentration, Eqs. ( 6.16 ) and ( 6.17 ) for the evolu-
tion of ε ft , and ( 6.19 ) for the myosin transformation. The unknowns variables in the
model are λ , ε ft , n , t , and t q . In order to proceed, we must specify the model struc-
ture, i.e., what is considered as input and output. Further, we must also particularize
the free-energy functions, μ , ν , and g . This will be exemplified in the next section.

=