Biomedical Engineering Reference
In-Depth Information
MATLAB is used to reconstruct the differential equations, as before, in terms of a single
variable and the inputs. The characteristic equation, det(
...
q
n
,
as well as the form of the natural response. The roots of the characteristic equation are
determined using the MATLAB command
“
eig(A)
”
and may be underdamped, over-
damped, or critically damped, depending on the transfer rates. The expression for the roots
is far too complex to be usable and will not be written here. Most models will have many
elements in
A
as zero, which makes the solution much more tractable.
In the remainder of this section, we consider special cases of the multicompartment
model: mammillary, catenary, and unilateral. Each model may be closed and may have sink
and source compartments.
A
), is identical for
q
1
,
q
2
,
D
I
7.8.1 Mammillary Multicompartment Model
A mammillary n-compartment model is shown in Figure 7.26, which is characterized by a
central compartment connected to
1 peripheral compartments. All exchange of solute is
through the central compartment, and there is no direct exchange of solute among the other
compartments. Each compartment can have an input and an output to the environment.
The matrix
A
, given in Eq. (7.106), has nonzero elements defined as
n
a
¼ð
K
þ
K
þ
K
þþ
K
n
Þ
11
10
12
13
1
a
ii
¼
K
i
1
ð
þ
K
i
0
Þ
,
2
i
n
ð
7
:
108
Þ
a
i
¼
K
i
1
,
2
i
n
1
a
i
1
¼
K
,
2
i
n
1
i
This system only has real roots.
f
2
(t)
q
2
K
20
K
12
K
21
f
1
(t)
f
i
(t)
K
i1
q
1
q
i
K
10
K
1i
K
i0
K
1n
K
n1
f
n
(t
)
q
n
K
n0
FIGURE 7.26
A mammillary n-compartment model.
