Biomedical Engineering Reference
In-Depth Information
The assumption of a linear viscosity element appears to be in error, since the analysis in
Example Problem 13.5 predicts a linear relationship between load and maximum velocity
(according to the assumptions of the solution), and the data from the isotonic experiment
shown in Figure 13.22 is clearly nonlinear. Thus, a reasonable assumption is that the
viscosity element is nonlinear.
Traditionally, muscle viscosity is characterized by the nonlinear Hill hyperbola, given by
V
max
P
þ
a
ð
Þ¼
bP
0
P
ð
Þ
ð
13
:
28
Þ
where
V
max
is the maximum velocity,
P
is the external force,
P
o
the isometric tension, and
a
are the empirical constants representing the asymptotes of the hyperbola. As
described previously,
and
b
represents the isometric tension, which is the largest weight that
the muscle can move, and
P
o
P
is the weight
Mg
. Hill's data suggests that
a
¼
P
b
¼
V
0
4
max
4
and
Therefore, with these values for
a
and
b
, the Hill equation is rewritten from Eq. (13.28) as
P
¼
P
0
V
max
P
0
þ ð Þ
b
þ
V
max
¼
P
0
BV
max
ð
13
:
29
Þ
where
P
0
þ
a
b
þ
V
B
¼
ð
13
:
30
Þ
max
The term
represents the viscosity of the element. Clearly, the force due to viscosity is
nonlinear due to the velocity term,
B
V
max
, in the denominator of Eq. (13.29).
x
In oculomotor models,
V
max
is usually replaced by
2
,
P
is replaced by muscle tension,
T,
and
P
o
is replaced by the active-state tension,
F,
as defined from Figure 13.24. Therefore,
Eqs. (13.29) and (13.30) are rewritten as
T
¼
F
BV
ð
13
:
31
Þ
where
B
¼
F
þ
a
b
þ
x
ð
13
:
32
Þ
2
in the Hill equation that
depend on whether the muscle is being stretched or contracted. There is some evidence to
suggest that stretch dynamics are different from contraction dynamics. However, the form
of the viscosity expression for muscle shortening or lengthening is given by Eq. (13.32), with
values for
Some oculomotor investigators have reported values for
a
and
b
parameterized appropriately. For instance, Hsu and coworkers [33]
described the viscosity for shortening and lengthening for oculomotor muscles as
a
and
b
B
ag
¼
F
ag
þ
AG
a
x
ð
13
:
33
Þ
þ
AG
b
2
