Biomedical Engineering Reference
In-Depth Information
EXAMPLE PROBLEM 7.12
Consider a two-compartment system for the distribution of creatinine in the body illustrated in
Figure 7.19. Compartment 1 represents the plasma and compartment 2 the muscle. Creatinine is a
waste product of metabolism in the muscle that's cleared from the body through the urine (trans-
fer rate
K
10
). Assume creatinine production in the muscle is
f
2
(
t
) and is given by a step input. Find
the concentration of creatinine in the plasma compartment.
Solution
The differential equations describing the rate of change of creatinine in the compartments 1
and 2 are written by using the conservation of mass equation as
q
1
¼
K
21
q
2
K
10
þ
K
12
ð
Þ
q
1
ð
7
:
75
Þ
q
2
¼
K
q
K
q
þ
f
¼
K
q
K
21
q
þ
1
ð
7
:
76
Þ
12
1
21
2
2
12
1
2
The D-Operator is used to remove
q
2
, giving
q
1
þ
K
10
þ
K
12
þ
K
21
Þ
q
1
þ
K
10
K
21
q
1
¼
K
21
ð
ð
7
:
77
Þ
The roots of the characteristic equation are
q
K
21
K
10
K
12
s
1
,
2
¼
K
10
þ
K
12
þ
K
21
ð
Þ
1
2
2
ð
Þ
4
K
21
K
10
2
The natural response is an overdamped response:
q
1
n
¼
B
1
e
s
1
t
þ
B
2
e
s
2
t
The forced response is a constant (
B
3
) and when substituted into the differential equation
1
K
10
:
yields
B
3
¼
The complete response is
1
K
10
q
1
¼
B
1
e
s
1
t
þ
B
2
e
s
2
t
þ
and
1
V
1
B
1
e
s
1
t
þ
B
2
e
s
2
t
þ
1
K
10
c
1
¼
for
t
0
:
The constants
B
1
and
B
2
are determined using the initial conditions.
f
2
(t)=u(t)
K
12
q
1
q
2
K
21
K
10
FIGURE 7.19
Illustration for Example Problem 7.12.
