Biomedical Engineering Reference
In-Depth Information
7.7 THREE-COMPARTMENT MODELING
The general form of the three-compartment model is shown in Figure 7.20. As before, we
begin with the general form of the three-compartment model and then examine special
cases. To analyze the system in Figure 7.20, conservation of mass is used to write a differ-
ential equation for each compartment describing the rate of change of the quantity of solute
in the compartment, given as accumulation
¼
input - output, where
Compartment 1
Accumulation
Compartment 2
Accumulation
Compartment 3
Accumulation
¼
q
1
¼
q
2
¼
q
3
Input
¼
f
1
ð
t
Þþ
K
21
q
2
þ
K
31
q
3
Ouput
Input
¼
f
2
ð
t
Þþ
K
12
q
1
þ
K
32
q
3
Ouput
Input
¼
f
3
ð
t
Þþ
K
13
q
1
þ
K
23
q
2
Ouput
¼ð
K
10
þ
K
12
þ
K
13
Þ
q
1
¼ð
K
20
þ
K
21
þ
K
23
Þ
q
2
¼ð
K
30
þ
K
31
þ
K
32
Þ
q
3
Therefore,
q
1
¼
f
ð
t
Þþ
K
q
þ
K
q
ð
K
10
þ
K
þ
K
Þ
q
1
q
2
¼
f
2
ð
t
Þþ
K
12
q
1
þ
K
32
q
3
ð
K
20
þ
K
21
þ
K
23
Þ
q
2
q
3
¼
f
3
ð
t
Þþ
K
13
q
1
þ
K
23
q
2
ð
K
30
þ
K
31
þ
K
32
Þ
q
3
1
21
2
31
3
12
13
ð
7
:
78
Þ
The D-Operator is used to simplify the system, where Eq. (7.78) is written in matrix form as
D
IQ
¼
AQ
þ
F
ð
7
:
79
Þ
where
2
3
2
3
2
3
q
1
q
2
q
2
ð
K
10
þ
K
12
þ
K
13
Þ
K
21
K
31
f
1
ð
t
Þ
f
2
ð
t
Þ
f
2
ð
t
Þ
4
5
,
A
4
5
,
F
4
5
Q
¼
¼
K
12
ð
K
20
þ
K
21
þ
K
23
Þ
K
32
¼
K
13
K
23
ð
K
30
þ
K
31
þ
K
32
Þ
f
1
(t)
f
2
(t)
K
12
K
10
K
20
q
1
q
2
K
21
f
3
(t)
K
23
K
13
K
31
K
32
q
3
K
30
FIGURE 7.20
A general three-compartment model. Compartment 1 has volume
V
1
, compartment 2 has volume
V
2
, and compartment 3 has volume
V
3
.