Biomedical Engineering Reference
In-Depth Information
__
q
2
a
11
þ
a
22
þ
a
33
ð
Þ
q
2
þ
a
11
a
22
þ
a
22
a
33
þ
a
33
a
11
a
12
a
21
a
13
a
31
a
23
a
32
ð
Þ
q
2
þ
a
11
a
23
a
32
a
11
a
22
a
33
þ
a
22
a
13
a
31
þ
a
33
a
12
a
21
a
12
a
23
a
31
a
13
a
32
a
21
ð
Þ
q
2
ð
7
:
83
Þ
¼
a
21
f
1
a
21
a
33
a
23
a
31
Þ
f
1
þ
f
2
a
33
þ
a
11
Þ
f
2
þ
a
11
a
33
a
13
a
31
ð
ð
ð
Þ
f
2
þ
a
23
f
3
a
23
a
11
a
13
a
21
ð
Þ
f
3
Þ
q
3
þ
a
11
a
22
þ
a
22
a
33
þ
a
33
a
11
a
12
a
21
a
13
a
31
a
23
a
32
Þ
q
3
__
q
3
a
11
þ
a
22
þ
a
33
ð
ð
þ
a
11
a
23
a
32
a
11
a
22
a
33
þ
a
22
a
13
a
31
þ
a
33
a
12
a
21
a
12
a
23
a
31
a
13
a
32
a
21
ð
Þ
q
3
ð
7
:
84
Þ
¼
a
31
f
1
þ
a
21
a
32
a
31
a
22
Þ
f
1
þ
a
32
f
2
a
32
a
11
a
12
a
31
ð
ð
Þ
f
2
þ
f
Þ
f
3
þ
a
a
ð
þ
a
ð
a
a
a
Þ
f
3
22
11
11
22
12
21
3
The characteristic equation, det(
q
3
, as well as the form of
the natural response. Note that the coefficients in the natural response are not identical and
depend on the input to the compartments and the initial conditions. The characteristic
equation is
A
), is identical for
q
1
,
q
2
, and
D
I
3
2
s
a
11
þ
a
22
þ
a
33
ð
Þ
s
þ
a
11
a
22
þ
a
22
a
33
þ
a
33
a
11
a
12
a
21
a
13
a
31
a
23
a
32
ð
Þ
s
ð
7
:
85
Þ
a
11
a
22
a
33
þ
a
11
a
23
a
32
þ
a
22
a
13
a
31
þ
a
33
a
12
a
21
a
12
a
23
a
31
a
13
a
32
a
21
¼
0
The roots of the characteristic equation are determined using the MATLAB command
“
eig
”
and may be underdamped, overdamped, or critically damped, depending on the
transfer rates. The expressions for the roots are far too complex to be usable and will not
be written here. With a two-compartment model, there was only one way for repeat roots
to occur. With a three-compartment model, there are many more configurations for repeat
roots to occur. Complex roots can occur under certain conditions, which are discussed in
Section 7.7.2.
In the remainder of this section and the next, we consider special cases of the three-
compartment model: mammillary, catenary, and unilateral. Each model may be closed
and may have sink and source compartments.
7.7.1 Mammillary Three-Compartment Model
A mammillary three-compartment model is shown in Figure 7.21, which is characterized
by a central compartment connected to two peripheral compartments. All exchanges of the
solute are through the central compartment, and there is no direct exchange of solute
f
1
(t)
f
2
(t)
f
3
(t)
K
32
K
12
q
1
q
2
q
3
K
21
K
23
K
10
K
20
K
30
FIGURE 7.21
A mammillary three-compartment model.
