Biomedical Engineering Reference
In-Depth Information
direction dependent strain-energy functions
2
(n
C
+
n
D
)
2
+
λ
s
max
λ
max
λ
s
,j
−
3
λ
s
,j
−
λ
2
max
λ
s
,j
s
1
1
Ψ
s
,j
=
λ
3
+
s
2
s
,j
2
λ
max
λ
2
+
s
2
λ
2
max
λ
1
+
s
2
s
,j
s
,j
+
s
2
−
+
+
P
max
λ
s
,j
.
(5.18)
3
+
2
+
s
2
1
+
s
2
Herein,
s
1
is a stress-like material parameter and
s
2
is a dimensionless constant. Fur-
ther,
λ
s
specifies the SMC stretch and
λ
max
defines the stretch at which the generated
stress
P
max
=
κ(n
C
+
n
D
),
(5.19)
depending on the parameter
κ
, reaches its maximum. The whole contraction process
is triggered by the chemical degree of activation
(n
C
+
n
D
)
provided by Hai and
Murphy (
1988
), describing the time and calcium dependent contraction kinetics.
This model is described by the differential equation system
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
n
A
˙
˙
−
k
1
k
2
0
k
7
n
A
n
B
n
C
n
D
n
B
˙
k
1
−
(k
2
+
k
3
)
k
4
0
,
(5.20)
−
(k
4
+
n
C
˙
0
k
3
k
5
)
k
6
−
(k
6
+
n
D
0
0
k
5
k
7
)
composed of four first order differential equations in time for four chemical states
n
A
,
n
B
,
n
C
, and
n
D
. As these are fractions,
n
A
+
1 has to be hold.
The first two,
n
A
and
n
B
, represent non-force generating states whereas the last
two,
n
C
and
n
D
, are related to generated force and thus, be mechanically significant.
Further the rate constant have been used as published in Schmitz and Böl (
2011
),
in doing so
k
1
=
k
6
and
k
2
=
k
5
. SMC contraction is triggered by an increase in
intracellular calcium which is controlled by the calcium-dependent rate constants
k
1
and
k
6
. Thus, a simple relation is given by
n
B
+
n
C
+
n
D
=
Ca
2
+
]
)
2
(α
[
k
1
=
k
6
=
,
(5.21)
Ca
2
+
]
)
2
+
K
CaCaM
(α
[
Ca
2
+
]
where
α>
0 is a positive constant,
characterizes the calcium concentra-
tion, and
K
CaCaM
denotes the half-activation constant for the calcium-calmodium
complex
[
[
]
. In this approach the rate parameters
k
i
have to be identified by
experimental data (Hai and Murphy,
1988
; Yang et al.,
2003a
,
b
).
CaCaM
5.3 Numerical Examples
This section aims to study how the chemical excitation affects the mechanical be-
havior at muscle level. In doing so, we first validate the presented modeling ap-
proach with experiments by Herlihy and Murphy (
1973
) before in a second step the
