Biomedical Engineering Reference
In-Depth Information
¼
K
V
¼
K
V
functions of the volume of each compartment—that is
K
and
K
, where
12
21
1
2
K
¼
DA
D
x
. The system in Example 7.7 is called a
closed compartment
because there is no output
to the environment in a closed system.
EXAMPLE PROBLEM 7.7
Consider the two-compartment model in Figure 7.16, with
q
1
(0)
¼ z and q
2
(0)
¼
0. Solve for the
concentration in each compartment.
K
12
q
1
q
2
K
21
FIGURE 7.16
Illustration for Example Problem 7.7.
Solution
Conservation of mass for each compartment is
q
1
¼
K
21
q
2
K
12
q
1
q
2
¼
K
12
q
1
K
21
q
2
Using the D-Operator method gives
q
1
þ
K
ð
þ
K
Þ
q
1
¼
0
12
21
q
1
þ
K
12
þ
K
21
Þ
q
2
¼
ð
0
The roots are
s
1,2
¼
0,
(
K
12
þ
K
21
), which gives
q
1
ð
t
Þ¼
B
1
þ
B
2
e
K
12
þ
K
21
ð
Þ
t
q
2
ð
t
Þ¼
B
3
þ
B
4
e
K
12
þ
K
21
ð
Þ
t
We use the initial conditions to solve for
B
i
as follows
Þ¼z ¼
B
1
þ
B
2
e
ð
K
12
þ
K
21
Þ
t
q
1
ð
0
j
¼
B
1
þ
B
2
t
¼
0
q
1
ð
q
1
at time zero
To find
0
Þ
, we use the conservation of mass equation for
q
1
ð
0
Þ¼
K
21
q
2
ð
0
Þ
K
12
q
1
ð
0
Þ¼
K
12
z
and from the solution,
q
1
¼
dB
1
þ
B
2
e
K
12
þ
K
21
ð
Þ
t
Þ
B
2
e
K
12
þ
K
21
ð
Þ
t
¼
K
12
þ
K
21
ð
dt
q
1
ð
0
Þ¼
K
12
z ¼
K
12
þ
K
21
ð
Þ
B
2
