Biomedical Engineering Reference
In-Depth Information
Note that when there are a molecules of
A
reacting, the
q
A
term in Eq. (8.26) is raised to the
a power, and similarly, when b molecules of
q
B
term in Eq. (8.26) is raised
to the b power. In problems like this, we resort to using SIMULINK for the solution.
Now consider the sequential reaction in Eq. (8.27)
react, then the
B
K
1
K
2
A
!
B
!
P
ð
8
:
27
Þ
in which reactant
A
produces reactant
B,
which then produces reactant
P.
The law of mass
action for Eq. (8.27) is
q
A
¼
K
1
q
A
q
B
¼
K
q
A
K
q
B
ð
8
:
28
Þ
1
2
q
P
¼
K
q
B
2
Equation (8.28) is easily solved, assuming nonzero initial condition for reactant
A, q
A
ð
0
Þ
,
and zero for the other two reactants, as
q
A
¼
q
A
ð
Þ
e
K
1
t
u
ð
t
Þ
0
u
ð
t
Þ
q
B
¼
K
1
q
A
ð
0
Þ
1
e
K
1
t
e
K
2
t
K
K
2
ð
8
:
29
Þ
0
@
1
A
u
ð
t
Þ
e
K
1
t
K
e
K
2
t
K
q
P
¼
K
K
q
A
ð
0
Þ
1
1
1
2
K
K
2
1
1
2
Sequential reactions are typical of those occurring in the body and can be quite complex,
as described in Sections 8.4-8.6. As we will see, enzymes play a role in sequential reactions,
and reactants can be of higher orders. Some sequential reactions involve many sequences in
which some branch backward to previous reactions.
When there are profound differences in the transfer rates in a sequential reaction, simpli-
fications occur that make analysis much easier. For instance, suppose
1
. Here, the
second reaction is much faster than the first reaction, and it appears that reactant
K
K
2
A
imme-
diately produces reactant
is often used to describe this behavior—
that is, the first reaction slows the creation of the product.
For a rate limiting reaction, an approximation to Eq. (8.29) for
P.
The term
rate limiting
q
B
and
q
P
is to eliminate the
e
K
2
t
e
K
1
t
term, since it goes to zero almost immediately, as compared to the
term, giving
ð
8
:
30
Þ
8.1.5 Quasi-Steady-State
Another way to look at a rate limiting reaction is to assume that reactant
B
is in a quasi-
steady-state mode—that is
q
B
¼
0. From
q
B
¼
0andEq. (8.28),wehavethat0
¼
K
1
q
A
K
2
q
B
,
