Biomedical Engineering Reference
In-Depth Information
exp
.
k
S
2
W
k
WS
2
=
G
W
(x)
−
G
S
2
(x)
−
(7.14)
RT
To describe the activation of the cross-bridges we use a phenomenological model,
which is able to reproduce the main properties of heart muscle to generate the
stress during the heart beat, which depends on time and the length of the sarcomere
(Jewell,
1977
). To activate the contraction, the concentration of Ca
2
+
changes. For
that we use in the activation model the intermediate state B for reactions between
tropomyosin C and Ca
2
+
d
A
d
t
=
A
−
−
c
1
B(
1
A)
c
2
(l
s
)
A
,
(7.15)
+
Q
where
c
2
is a function of sarcomere length
l
s
,i.e.
c
2
F
l
max
−
l
s
c
2
=
c
2
MX
+
l
min
,
(7.16)
l
s
−
and normalized concentration
B
is a function of time, i.e.
⎧
⎨
(
t
−
T
p
T
a
)
2
t
≤
T
p
,
exp
[−
]
,
=
B
(7.17)
⎩
[−
(
t
−
T
p
T
D
)
2
otherwise
,
exp
]
.
The symbols
c
2
,
c
2
MX
and
c
2
F
are rate constants for reactions between tropomyosin
and Ca
2
+
,
Q
describes the cooperativity for Ca
2
+
to bind with tropomyosin C
(Tobacman and Sawyer,
1990
) and
l
max
and
l
min
are the maximum and minimum
lengths of the half sarcomere, respectively. The constant
T
p
is the time that is needed
to develop maximal contraction and
T
a
is a time constant. The characteristic time
T
D
is dependent on the half sarcomere length and is defined as
T
D
=
T
d
0
1
,
+
T
d
1
l
s
−
l
min
l
max
−
(7.18)
l
min
where
T
d
0
is a time constant and
T
d
1
is a relative change of the time constant. The
ATP consumption rate during one beat is according to Eq. (
7.6
)
T
c
d
2
k
S
2
W
n
S
2
(x,t)
k
WS
2
n
W
(x,t)
d
x
d
t.
1
d
V
beat
AT P
=
−
(7.19)
2
0
−
7.3.2 Optimization Strategy
The goal of the optimization process is to find the set of model parameters for a
cross-bridge model which allows us to replicate the experimentally measured linear
dependence of oxygen consumption on SSA. For that we divided the model pa-
rameters into two sets: (a) parameters describing free energy profiles (Fig.
7.2
) and
