Biomedical Engineering Reference
In-Depth Information
To determine
q
S
at steady state for this one-compartment system, we set
q
S
¼
0in
Eq. (8.54), which gives
z
K
M
q
S
ðÞ¼
ð
8
:
58
Þ
V
z
max
Since
q
S
needs to be positive, this requires
V
>
z. Thus, the maximum removal rate must
max
be larger than the input for a bound solution for
q
S
.
To examine the case when
V
¼
z, we substitute these values into Eq. (8.54), giving
max
K
M
q
S
þ
K
M
z
q
S
¼
ð
8
:
59
Þ
ð
Þ
Separating
q
S
and
t
in Eq. (8.59) gives
1
dt
¼
K
M
q
S
þ
K
M
ð
Þ
dq
S
z
and after integrating
þ
1
1
z
2
S
q
2
S
ð
t
¼
q
0
Þ
q
S
q
S
ð
0
Þ
ð
8
:
60
Þ
2z
K
M
which increases without bound,
q
S
!1:
If z
V
max
, then the substrate is not eliminated
quickly enough, continuously increasing, and
q
S
!1:
Exponential Input
Next, consider an exponential input to a one-compartment model of Eq. (8.51), a type of
input previously observed in Example Problem 7.8, when a substrate is digested and moves
into the plasma. For
t
0, the model is
q
S
¼
V
max
q
S
þ
K
M
Þ
K
21
e
K
21
t
Þ
q
S
þ
q
2
ð
0
ð
8
:
61
Þ
ð
where
K
21
is the transfer
rate from the digestive system to the compartment. Simulating Eq. (8.61) is the only way to
solve this problem.
Before simulation became an easy solution method, the quasi-steady-state approximation
was linearized with a lower and upper bound, with the actual solution falling in between
these two bounds. For the lower bound, the quasi-steady-state approximation is
V
max
K
M
q
2
ð
0
Þ
is the initial amount of substrate in the digestive system and
, where
the quasi-steady-state approximation is a constant. As we will see, the lower bound is a
good approximation for small values of
q
S
. For the upper bound, the quasi-steady-state
V
max
q
S
max
þ
K
M
approximation is
, where
q
S
max
is the maximum
q
S
. As before with the lower
bound, the quasi-steady-state approximation is a constant for the upper bound. The upper
bound is a good approximation for large values of
q
S
:
This approach works for all inputs.
