Biomedical Engineering Reference
In-Depth Information
The concentration in compartment 2 is found from Eq. (7.7) as
¼
C
10
V
1
V
1
c
1
V
c
ð
7
:
8
Þ
2
2
which when substituted into Eq. (7.6) gives
¼
K
V
Þ ¼
KC
10
V
2
Kc
1
c
2
V
ð
c
V
C
þ
V
c
2
V
ð
þ
V
Þ
1
2
1
1
10
1
1
1
2
V
V
V
1
1
or
V
þ
V
c
1
¼
KC
1
2
10
V
2
c
1
þ
K
ð
7
:
9
Þ
V
1
V
2
This is a first-order linear differential equation with forcing function
f
ðÞ¼
KC
10
V
2
ð
7
:
10
Þ
and initial condition
¼
C
10
.
Assume for simplicity that
c
1
(0)
V
1
¼
V
2
. Then Eq. (7.9) becomes
2
V
1
c
1
¼
KC
K
10
V
1
c
1
þ
ð
7
:
11
Þ
2
K
V
To solve Eq. (7.11), note that the root is
and the natural solution is
1
Kt
V
1
2
c
1
n
¼
B
e
ð
7
:
12
Þ
1
where
B
1
is a constant to be determined from the initial condition. The forced response has
the same form as the forcing function in Eq. (7.9),
c
1
f
¼
B
2
, which when substituted into
Eq. (7.11) yields
2
K
V
1
B
¼
KC
10
2
V
1
or
¼
C
10
2
B
2
Thus, the complete response is
þ
C
2
Kt
V
1
c
1
¼
c
1
n
þ
c
1
f
¼
B
1
e
10
2
To find
B
1
, the initial condition is used
2
Kt
V
1
0
þ
C
¼
B
1
þ
C
Þ¼
C
10
¼
B
1
e
10
2
10
2
c
1
ð
0
t
¼
or
¼
C
10
2
B
1
