Biomedical Engineering Reference
In-Depth Information
and
K
fi
¼
k
fi
i ¼ 1; 2; 3
(E10-2.52)
k
fi
Equations
(E10-2.38)
,
(E10-2.40)
, together with Eqns
(E10-2.41)
and
(E10-2.43)
, give
k
1c
½
M
1
½
E
2
T
k
3c
½
M
3
½
E
4
T
k
5c
½
M
3
½
E
5
T
1
¼
3
þ
(E10-2.53)
K
1
þ K
1
K
1
K
3
þ K
3
K
1
K
5
þ K
5
K
1
f1
½
M
3
þ½
M
f2
½
P
1
þ½
M
f3
½
P
2
þ½
M
3
Therefore, the feedback control has rendered the system nonlinear to the degree that
multiple steady states are possible. The intermediate concentration [M
3
] can be deter-
mined by solving Eqn
(E10-2.53)
.
The rate of formation for P
1
can be obtained by
r
P
1
¼
N
R
i¼1
n
P
1
i
r
i
¼ k
4c
½
M
4
k
f2
½
P
1
½
E
4
þk
f2
½
P
1
E
4
4
¼k
3c
K
3
½
¼ k
4c
½
M
4
¼k
3c
½
M
3
E
M
3
½
E
4
(E10-2.54)
k
3c
½
M
3
½
E
4
T
¼
K
3
þ K
3
K
1
f2
½
P
1
þ½
M
3
and the rate of disappearance of M
1
is given by
r
M
1
¼
N
R
i¼1
n
M
1
i
r
i
¼ k
1
½
M
1
½
E
2
k
1
½
M
1
E
2
(E10-2.55)
k
1c
½
M
1
½
E
2
T
2
¼k
1c
K
1
½
¼ k
1c
½
M
1
E
M
1
½
E
2
¼
K
1
þ K
1
K
1
f1
½
M
3
þ½
M
1
Also, substituting Eqn
(E10-2.53)
into
(E10-2.55)
yields
k
3c
½
M
3
½
E
4
T
k
5c
½
M
3
½
E
5
T
r
M
1
¼
3
þ
(E10-2.56)
K
3
þ K
3
K
1
K
5
þ K
5
K
1
f2
½
P
1
þ½
M
f3
½
P
2
þ½
M
3
2. From Eqn
(E10-2.54)
, one can conclude that if P
1
is accumulated to high levels the rate of
formation of P
1
is lowered. If both P
1
and P
2
are accumulated to high levels, the rate for M
1
consumption is to be very low according to Eqn
(E10-2.56)
. Therefore, the sequential
feedback control is effective.
3. We have derived two rate expressions as shown in Eqns
(E10-2.54) and (10.2-56)
. However,
these two expressions cannot be utilized directly to analyze the measured concentration
profile data. In Chapter 8, we learned that rate can be estimated by the rate at any step by
adjusting the rate coefficients. Also, the intermediate concentrations can be obtained via
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