Biomedical Engineering Reference
In-Depth Information
Figure 10.2
Flow of water through a tank fi lled with drug-containing water.
of a drug is remaining in the tank after certain time, one can use the conservation
of mass principle. The general form is
Input [I]
−
Output [O]
+
Generation [G]
=
Accumulation [A]
(10.1)
Generation or disappearance is due to a reaction occurring in the system where
a product is generated or a reactant is consumed. There is no reaction occurring
in the tank. However, concentration of the drug decreases due to that carried by
water leaving the tank. Let
C
(
t
) be the concentration of the drug at any time, and
Δ
C
is the concentration of the drug carried by the water leaving at time
Δ
t
. Then,
(
)
Q
*0
−
QCt
()
Δ
t
=
V
⎡
Ct
()
+ Δ
Ct
()
−
Ct
()
⎤
⎣
⎦
o
o
D
t has to be very small or approaches zero.
From the definition of differential calculus, the above equation can be simplified
and written as
To obtain a continuous function,
Δ
dC t
()
Q
=−
o
Ct
()
(10.2)
dt
V
D
An assumption is that the volume of water does not change during the time of
study. The ratio of
Q
0
to
V
D
is termed as a rate constant and represented by
k
with
units of time
−1
.
dC t
()
(10.3)
=−
kC t
()
dt
Rearranging and integrating with the initial condition (i.e., at time =zero),
C
(
t
)
=
C
0
to obtain
Ce
−
kt
ln
Ct
( )
−
ln
C
= −
kt
or
Ct
( )
=
(10.4)
0
0
Thus, knowing the rate constant, the concentration of the drug within the tank
at any time can be determined. This forms the basis for a single compartmental
