Biomedical Engineering Reference
In-Depth Information
In general, the unilateral three-compartment model is given by the following set of
equations:
q
1
¼
f
ð
t
Þþ
K
ð
K
þ
K
Þ
q
q
1
31
3
10
12
1
q
2
¼
f
ð
t
Þþ
K
ð
K
þ
K
Þ
q
q
ð
7
:
92
Þ
2
12
1
20
23
2
q
3
¼
f
ð
t
Þþ
K
23
q
2
ð
K
þ
K
31
Þ
q
3
3
30
To examine a simple unilateral three-compartment model with complex roots, consider
a closed system (i.e.,
K
10
¼
K
20
¼
K
30
¼
0). From Eq. (7.85), the roots are found from the
characteristic equation, given as
3
2
s
þ
K
12
þ
K
23
þ
K
31
ð
Þ
s
þ
K
12
K
23
þ
K
23
K
31
þ
K
31
K
12
ð
Þ
s
¼
0
ð
7
:
93
Þ
which are
s
1
¼
0, and
q
K
¼
K
12
þ
K
23
þ
K
31
ð
Þ
1
2
2
s
ð
þ
K
þ
K
Þ
4
ð
K
K
þ
K
K
þ
K
K
Þ
2
,
3
12
23
31
12
23
23
31
31
12
2
2
Complex roots occur when 4
ð
K
12
K
23
þ
K
23
K
31
þ
K
31
K
12
Þ >
K
12
þ
K
23
þ
K
31
ð
Þ
:
Repeated
2
roots occur when 4
:
Consider the case of complex roots and a zero root, which gives rise to a natural solution
of the form
ð
K
12
K
23
þ
K
23
K
31
þ
K
31
K
12
Þ ¼
K
12
þ
K
23
þ
K
31
ð
Þ
q
i
¼
B
1
þ
e
a
t
B
2
cos o
d
t
þ
B
3
sino
d
t
Þ ¼
B
1
þ
B
4
e
a
t
ð
cos o
d
t
þ
ð
f
Þ
where a and o
d
terms
are determined from initial conditions after the forced response is determined. We write
the complex roots in standardized format as
are the real and imaginary part of the complex root, and the
B
i
p
z
2
s
2
,
3
¼
zo
0
o
0
1
, which has a charac-
teristic equation of
2
o
0
¼
s
þ
2 zo
0
s
þ
0
ð
7
:
94
Þ
The system is at its most oscillatory when z
0, a pure sinusoid.
To get a better understanding of the system, we determine the extent of its oscillatory
behavior by finding the optimal values of the transfer rates to achieve maximum oscillatory
behavior (i.e., minimum z). To write an expression for z, we use the coefficients of the char-
acteristic equation (Eq. (7.93)) and set them equal to the terms in Eq. (7.94):
2zo
0
¼
K
12
þ
K
23
þ
K
31
¼
ð Þ
o
0
¼
K
12
K
23
þ
K
23
K
31
þ
K
31
K
12
ð
Þ
which gives
p
K
o
0
¼
ð
K
þ
K
K
þ
K
K
Þ
12
23
23
31
31
12
and
1
2o
0
¼
ð
K
þ
K
þ
K
Þ
12
23
31
z
¼
p
K
ð
7
:
95
Þ
2
ð
K
þ
K
K
þ
K
K
Þ
12
23
23
31
31
12
z
@
K
12
¼
@
z
@
K
23
¼
@
z
@
K
31
¼
@
To find the minimum z, we find
0,
0, and
0, which allows us to
determine the conditions that allow minimum z. First, we use the chain rule to find
