Biomedical Engineering Reference
In-Depth Information
contraction, substitute Eq. (
6.16
)
1
in (
6.17
) and rearrange the terms to have
(g
+
n
C
μf )
˙
ε
ft
=
f
∂ψ
cb
∂ε
ft
−
n
C
μf ν.
(6.27)
The first term on the right-hand side can be interpreted as the stress measured in
isotonic experiments and is, hence, associated with the constant afterload. The sec-
ond term is the isometric force generated by the muscle which is obtained by setting
˙
0inEq.(
6.16
)
1
. Second, introduce
T
0
=
n
C
μf ν
and
T
=
f∂(ψ
cb
)/∂ε
cb
for
ε
ft
=
T
/b
.
the isometric stress and afterload, respectively, and take
a
=
n
C
μf/b
and
g
=
Finally, if the deformation rate in the cross-bridges
ε
cd
is assumed to be negligible
after the elastic recoil, an initial step-like contraction following the quick release,
Eq. (
6.5
) gives that
˙
ε
ft
can be replaced by
˙
λ
(McMahon,
1984
). Equation (
6.27
),
therefore, has the same functional form as Hill's equation.
˙
6.4 Discussion
In this paper we have exemplified how a thermodynamically consistent model can be
derived using a very general continuum thermodynamic framework. Despite the in-
creased complexity, the model has several advantages compared to the models men-
tioned in Sect.
6.1
. First, it has a clear kinematic description which is a cornerstone
in any mechanical model, and particularly so for active materials. By considering an
additive decomposition of the deformation, the model distinguishes between defor-
mations associated with filament translation and force generation, i.e., cross-bridge
deformation. This higher resolution allows for a model which is closer to the real
physiological situation inside the muscle cell. Second, it has derived couplings be-
tween the electrochemical and the mechanical scales as can be seen in Eq. (
6.16
)or
(
6.20
), for example. These coupling are still introduced intuitively to some extent in
the free energies, but the kinematic analysis and the dissipation inequality restricts
the number of appropriate choices for the free energies considerably.
The model presented in the previous sections is homogeneous and only accounts
for muscle contraction along one direction. This is not as restrictive as may ap-
pear at first glance, however. Even though contractile units are less organized in a
smooth muscle cell than in other muscle cells, they still have a preferred direction
and a certain dispersion around this direction, see Walmsley and Murphy (
1987
).
It is, therefore, possible to extend the smooth muscle model to three dimensions
by considering
λ
to be the stretch along the preferred direction and include a fiber
dispersion. The interested reader is referred to Murtada et al. (
2010
) for more infor-
mation.
To show the parabolic behavior of the evolution law for
ε
ft
, it was assumed that
the stress in the parallel spring was negligible. For skeletal muscle this is an accept-
able assumption (McMahon,
1984
) since passive structures are, generally, recruited
further down on the descending portion of the bell-shaped function
f
. For smooth
