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.