Biomedical Engineering Reference
In-Depth Information
Using the initial conditions, we solve for
B
1
,
B
2
, and
B
3
from
10
3
q
1
ð
0
Þ¼
0
:
1
¼
B
1
þ
B
2
þ
B
3
10
3
q
1
ð
0
Þ¼
0
:
06
¼
1
:
26
B
2
0
:
14
B
3
10
5
q
1
ð
0
Þ¼
6
:
6
¼
1
:
4
B
2
þ
0
:
02
B
3
giving
10
4
,
10
4
, and
10
4
, and
B
1
¼
0
:
48
B
2
¼
0
:
47
B
3
¼
0
:
05
e
1:26
t
þ
e
0:14
t
10
4
q
1
¼
0
:
48
þ
0
:
47
0
:
05
u
ð
t
Þ
The model used in Example Problem 7.16 is too simple to capture the real transport
dynamics of thyroid hormone. Some investigators have included multiple compartments
for the hepatic duct and many other compartments. Some have included chemical reactions
in the model. We will investigate these models in a later chapter.
7.8 MULTICOMPARTMENT MODELING
Realistic models of the body typically involve more than three compartments. The
concepts described in the previous sections can be applied to a compartment model of
any size. Each compartment is characterized by a conservation of mass differential equation
describing the rate of change of the solute. Thus, for the case of
n
compartments, there are
n
equations of the form
dq
i
dt
¼
input
output
where
q
i
is the quantity of solute in compartment
i
, which can be generalized for the system to
D
IQ
¼
AQ
þ
F
ð
7
:
106
Þ
where
and for the first row in
A
, we have
and so on for the other rows in
A
. Equation (7.106) is solved as before from
1
Þ
1
F
Q
¼
D
ð
I
A
¼
adj
ð
D
I
A
Þ
F
det
ð
I
A
Þ
D
ð
7
:
107
Þ
or
det
ð
D
I
A
Þ
Q
¼
adj
ð
D
I
A
Þ
F
