Biomedical Engineering Reference
In-Depth Information
The matrix A , given in Eq. (7.106), has nonzero elements defined as
a ii ¼ð K i 0
þ K i , i þ1
Þ
,
1
i n
1
a nn ¼ K n 0
ð
þ K n 1
Þ
ð
7
:
110
Þ
a
n ¼ K n 1
a i , i 1 ¼ K i 1 , i
1
,
2
i n
In Section 7.7.2, we investigated the three-compartment unilateral complex roots case
and determined that in the roots with the most oscillatory behavior, all the transfer rates
were equal to the same value. We will continue this investigation to explore the oscillatory
behavior for a closed system unilateral n-compartment model with equal transfer rates,
K,
and bolus input. The system matrix A is
and the determinant of (D I - A )is
As Godfrey illustrates, the roots of this system are
,
cos 2p
m
n
sin 2p
m
n
K þ K
þ j
m ¼
1, 2,
n
ð
7
:
111
Þ
and lie evenly on the unit circle of radius K, centered at (K,0) in the complex plan. For a
closed system, one of the roots is 0, and for an even
n
, there is another root at -2
K.
The
remaining roots are complex and given by Eq. (7.111). For
m ¼
1 and
m ¼
n
1, Godfrey
shows that the damping ratio is equal to
sin p
n
z
¼
ð
7
:
112
Þ
and as
increases to infinity, the damping ratio approaches zero. Since the quantity within
a compartment can never be less than zero, as
n
n
approaches infinity, the amplitude of the
sinusoid approaches 0.
Our approach to solving a unilateral n-compartment model is the same as before, letting
MATLAB do the work for us, as shown in the following example.
Search WWH ::




Custom Search