Biomedical Engineering Reference
In-Depth Information
Substituting these values from MATLAB into Eq. (7.49) gives
Q
2
D
þ
DK
ð
þ
K
þ
K
þ
K
Þ þ
K
ð
þ
K
þ
K
Þ
K
K
K
10
12
20
21
10
20
10
21
12
20
F
D
þð
K
þ
K
Þ
K
ð
7
:
50
Þ
20
21
21
¼
K
12
D
þð
K
10
þ
K
12
Þ
Returning to the time domain, we have the following independent differential equations:
q
1
þ
K
10
þ
K
12
þ
K
20
þ
K
21
ð
Þ
q
1
þ
K
10
K
20
þ
K
10
K
21
þ
K
12
K
20
ð
Þ
q
1
¼
df
ð
t
Þ
dt
þð
K
20
1
þ
K
Þ
f
ð
t
Þþ
K
f
ð
t
Þ
21
1
21
2
ð
7
:
51
Þ
q
2
þ
K
Þ
q
2
þ
K
ð
þ
K
þ
K
þ
K
ð
K
þ
K
K
þ
K
K
Þ
q
10
12
20
21
10
20
10
21
12
20
2
f
1
ð
t
Þþ
df
2
ð
t
Þ
dt
¼
K
12
þð
K
10
þ
K
12
Þ
f
2
ð
t
Þ
Note that the characteristic equation, det(D
I
q
2
, and the
form of the natural response is the same for either variable. Also note that the coefficients
in the natural response are not identical for
A
), is identical for both
q
1
and
q
1
and
q
2
, and depend on the input to the com-
partment and the initial conditions.
The roots of the characteristic equation are determined using MATLAB as
>>
eig(A)
ans
¼
-1/2*K10-1/2*K12-1/2*K20-1/2*K21
þ
1/2*(K10
^
2
þ
2*K10*K12-
2*K10*K20-2*K10*K21
þ
K12
^
2-2*K12*K20
þ
2*K21*K12
þ
K20
^
2
þ
2*K20*K21
þ
K21
^
2)
^
(1/2)
-1/2*K10-1/2*K12-1/2*K20-1/2*K21-1/2*(K10
^
2
þ
2*K10*K12-
þ
^
þ
þ
^
þ
2*K10*K20-2*K10*K21
K12
2-2*K12*K20
2*K21*K12
K20
2
þ
^
^
(1/2)
This expression simplifies to
2*K20*K21
K21
2)
q
K
20
þ
K
21
K
10
K
12
ð
þ
K
þ
K
þ
K
Þ
1
2
s
1
,
2
¼
K
10
12
20
21
2
ð
Þ
þ
4
K
21
K
12
ð
7
:
52
Þ
2
From Eq. (7.52), we note that there can be no positive real roots and no imaginary roots if all
K
ij
K
20
þ
K
21
K
10
K
12
)
2
0. If (
þ
4
K
21
K
12
¼
0, then the roots are repeated and equal to
s
1
,
2
¼
K
ð
þ
K
þ
K
20
þ
K
Þ
10
12
21
. For repeated roots to happen, (
K
20
þ
K
21
) must equal (
K
10
þ
2
K
12
), and either
K
12
are both equal to zero, then there is
no movement of solute between the compartments.
In the following example, we revisit Fick's Law of diffusion using compartmental analy-
sis and compute the concentration. The difference in analysis involves the transfer rates as
K
21
or
K
12
must be zero. If
K
21
and
