Biomedical Engineering Reference
In-Depth Information
Fig. 6.2
Results from the
numerical example. Only two
pulses are shown in the
interest of space.
Top panel
:
calcium ion concentration
(
solid line
) and isometric
stretch (
dashed line
) inputs.
Second panel form top
:
myosin fractions for states
n
C
(
solid line
)and
n
D
(
dashed
curve
).
Third panel form top
:
active filament translation
strain
ε
ft
.
Bottom panel
:
resulting stress
t
where
b
=
5. This choice allows us to recover the same parabolic form for the evo-
lution of
ε
ft
as for the shortening velocity in Hill's equation, see Remark
6.1
.
All constants used in this example are listed in Table
6.1
and the results are shown
in Fig.
6.2
. The most notable result is the rapid transformation of myosin from state
C to state D when the muscle relaxes (second panel, 4-11 minutes). From an energy
point of view this is an efficient behavior. The majority of energy consumption is
associated with cycling cross-bridges, i.e., states B and C, and the transformation
allows the muscle to maintain a basal tone at low energy cost. Finally,
ε
ft
in the
third panel undergoes a rapid shortening when the muscle is activated (at 1 and 11
minutes). The shortening takes about 10 s and has an exponential decay although
it is difficult to see in the graph. This is in qualitative agreement with experimental
studies, see, e.g., Arner (
1982
). If a better agreement is sought, a nonlinear parame-
ter identification can be used to tune the model to experiments, but it is outside the
scope of this paper. The interested reader is referred to Stålhand et al. (
2008
,
2011
).
Remark 6.1
In quick-release experiments, smooth muscle show the same parabolic
relation between shortening velocity and afterload as skeletal muscle, see Arner
(
1982
). This parabolic relation is also known as Hill's equation and is given by
b
T
−
T
0
V
=
a
,
(6.26)
T
+
where
V
is the shortening velocity,
T
and
T
0
are the isometric stress and after-
load, respectively, and
a
and
b
are constants. Note that the order of the terms in the
nominator is shifted relative to the classical definition since contraction is defined
as negative herein. Since the model presented herein uses stress and strain rather
than force and velocity,
V
is replaced by
˙
λ
in Eq. (
6.26
). Following Murtada et al.
(
2012
), we choose to specialize the evolution law in Eq. (
6.17
) such that it has the
same functional form as Eq. (
6.26
) under the assumption of negligible stress con-
tribution from the parallel spring in Fig.
6.1
. First, since Hill's equation applies to
