Biomedical Engineering Reference
In-Depth Information
To solve for
B 1 and
B 2 , we evaluate
1
¼
1
B 1
B 2
z
K 12 z
0
K 12 þ K 21
ð
Þ
which gives
2
4
3
5
z K 21
¼
ð
K
þ K
Þ
12
21
B 1
B 2
z
K 12
K 12 þ K 21
ð
Þ
and
z
K 12 þ K 21
K 21 þ K 12 e K 12 þ K 21
ð
Þ t
q 1 ð t Þ¼
u ð t Þ
ð
Þ
The concentration is
z
V 1 K 12 þ K 21
K 21 þ K 12 e K 12 þ K 21
ð
Þ t
c 1 ð t Þ¼
u ð t Þ
ð
Þ
Repeating the same steps for
q 2 as before gives
K 12 z
K 12 þ K 21
e K 12 þ K 21
ð
Þ t
q 2 ð t Þ¼
1
u ð t Þ
ð
Þ
and
K 12 z
V 2 K 12 þ K 21
e K 12 þ K 21
ð
Þ t
c 2 ð t Þ¼
1
u ð t Þ
ð
Þ
or using
q 2 ¼ z q 1 gives the same result.
A more straightforward solution involves substituting
q 1 ¼ K 21 q 2 K 12 q 1 ,
q 2 ¼ z q 1 into
and solving q 1 ¼ K 21 z q 1
ð
Þ K 12 q 1 ¼ K 21 z K 21 þ K 12
ð
Þ q 1 :
K 12 ¼ K
K 21 ¼ K
If
V 1 and
V 2 , and
V 1 ¼ V 2 ¼ V
, then these results are the same as those computed
using Fick's Law of diffusion in Section 7.2.2:
u ð t Þ
z
2
Kt
V þ
c 1 ¼
e
1
and
u ð t Þ
z
2
Kt
V
c 2 ¼
1
e
In a two-compartment model, the half-life is defined using two terms based on the roots
of the characteristic equation. The half-life associated with the smaller root is called the
elimination half-life, and the distribution half-life is used for the larger root.
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