Biomedical Engineering Reference
In-Depth Information
P
int
in Eq. (
6.10
)isthe
internal power obtained by replacing the virtual velocities in Eq. (
6.6
) for their real
counterparts. Following Stålhand et al. (
2008
) we take the free energy to be given
by
for all admissible evolutions of the state variables. The term
D
ψ
p
(λ)
f(ε
ft
)ψ
cb
(ε
cd
,
n
)
ψ
q
(q)
ψ
n
(n
i
),
ψ
=
+
+
+
(6.11)
i
=
A
where
ψ
p
is the strain energy stored in the parallel spring,
ψ
cb
is the free energy
for the cross-bridges,
ψ
q
and
ψ
n
are the free energies associated with the calcium
ion concentration and myosin states, respectively, and
f
∈[
describes the strain
dependence of the force generation due to filament overlap. Replacing the virtual ve-
locities in Eq. (
6.6
) for their true counterparts and substituting the result into (
6.10
)
together with Eqs. (
6.5
) and (
6.11
), gives, after some rearrangement of the terms,
T
p
+
0
,
1
]
˙
λ
−
∂ψ
p
∂λ
−
f
∂ψ
cd
∂ε
cd
f
∂ψ
cd
∂ε
cd
∂f
∂ε
ft
ψ
cb
T
cd
−
+
T
cd
−
+
ε
ft
˙
T
q
−
f
∂ψ
cb
∂
n
∂ψ
q
∂q
∂ψ
n
∂
n
+
T
ft
(
ε
ft
+
˙
ν)
+
q
˙
−
+
· ˙
n
≥
0
,
(6.12)
where the dot denotes a scalar product. Assuming the first and fourth terms in
Eq. (
6.12
) to be non-dissipative gives
∂ψ
p
∂λ
−
f
∂ψ
cd
t
=
∂ε
cd
,
(6.13)
and
∂ψ
q
∂q
,
t
q
=
(6.14)
where Eqs. (
6.8
) have been substituted. Back-substitution of Eqs. (
6.13
) and (
6.14
)
into (
6.12
) gives the reduced dissipation inequality
f
∂ψ
cb
f
∂ψ
cd
∂ε
cd
∂ψ
n
∂
n
∂f
∂ε
ft
ψ
cb
−
T
cd
−
+
ε
ft
+
˙
T
ft
(
ε
ft
+
˙
ν)
+
∂
n
+
·
n
≥
0
.
(6.15)
To guarantee that energy dissipates during contraction, first assume the stronger
condition that each term in Eq. (
6.15
) satisfies the inequality, i.e., is greater than
zero. Second, take the friction clutch stress to be
n
C
μf (
˙
ε
ft
+
ν),
if
˙
ε
ft
≤
0
,
T
ft
=
(6.16)
n
C
μf (
ε
ft
+
˙
ν)
+
n
D
μf
ε
ft
,
if
˙
˙
ε
ft
>
0
,
where
μ>
0, and the factors
n
C
μf
and
n
D
μf
may be thought of as friction coeffi-
cients. Equation (
6.16
) is inspired by the contractile force in Murtada et al. (
2010
)
and it is assumed that only cycling cross-bridges contribute to the force generation,
