Biomedical Engineering Reference
In-Depth Information
a steady state can be maintained due to the recycling of drug molecules by the
circulatory system. If the environment is heterogeneous, the system is described by
the equations:
d
x
d
t
=
C
X
k
1
(
e
0
−
x
)
−
(
k
−
1
+
k
2
)
x
,
(39)
d
p
d
t
=
k
2
x
.
(40)
Therefore, the Michaelis-Menten equation becomes
v
max
C
X
K
M
+
v
=
C
X
.
(41)
It can be noted that Eq. (
41
) has the same form as the Hill equation that describes
the response of the system as a function of the drug concentration. Incorporating
this formula into a one-compartment model with an intravenous infusion yields
v
max
C
X
k
M
+
d
C
d
t
=
−
i
)
V
d
,
(
t
C
X
+
(42)
where
i
is the infusion rate in units of mass/time and
V
d
is the volume of
distribution in units of volume. To investigate the asymptotics of Eq. (
42
), we
consider the model post-infusion. For high concentrations (those occurring at or
above
K
M
):
(
t
)
d
C
d
t
=
−
v
max
.
(43)
For low concentrations (those occurring far below
K
M
):
d
C
d
t
=
−
v
max
K
M
C
X
.
(44)
Integrating Eq. (
44
) leads to the asymptotic power law behavior
t
1
/
(
1
−
x
)
.
C
(
t
)
∼
(45)
Comparing to Eq. (
29
) yields the relationship
1
γ
=
X
,
(46)
−
1
or
1
γ
.
X
=
1
−
(47)
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