Biology Reference
In-Depth Information
is called a geometric series. We let S
n
denote the sum of the first n terms of
such a series, so:
ab
2
ab
3
ab
n
1
¼
þ
þ
þ
þ
...
þ
:
S
n
a
ab
(1-37)
Then:
ab
2
ab
3
ab
n
1
ab
n
andS
n
ab
n
bS
n
¼
ab
þ
þ
þ
...
þ
þ
bS
n
¼ð
1
b
Þ
S
n
¼
a
:
(1-38)
If b
6¼
1, we obtain the following compact formula for S
n
:
ab
n
a
S
n
¼
:
(1-39)
1
b
When
j
b
j
< 1, the limit below can be calculated to be:
a
lim
n
!1
S
n
¼
(1-40)
1
b
j
j
< 1, lim
n
!1
b
n
¼
:
because when
b
0
e
-Tr
, the residual
concentration after n doses from Eq. (1-34) can be written as:
Ce
-Tr
and b
Applying Eq. (1-40) with a
¼
¼
n
n
1
n
2
e
Tr
e
Tr
e
Tr
e
Tr
R
n
¼
C
ð½
þ½
þ½
þ
...
þ
Þ
n
e
Tr
Ce
Tr
1
ð
Þ
:
(1-41)
n
1
n
2
¼
Ce
Tr
ð½
e
Tr
þ½
e
Tr
þ
...
Þ¼
1
1
e
Tr
What happens to the residual amounts as the number of doses increases?
It appears from Figure 1-25 that, after several doses, the residual values
stabilize around a value slightly higher than 8
g/ml. To see if this is
m
true in general, we need to find lim R
n
as n
. Indeed, using
Eq. (1-40), the limit is now easily computed to be:
!1
n
e
Tr
Ce
Tr
Ce
Tr
1
ð
Þ
C
R
¼
lim
n
!1
R
n
¼
lim
n
!1
¼
e
Tr
¼
1
:
(1-42)
1
e
Tr
1
e
Tr
Thus, for a sufficiently large number of doses, the residual
concentrations stabilize around the value R, which depends on the dose C,
the fixed time between doses T, and the elimination rate constant r.
E
XERCISE
1-19
Is R
n
larger or smaller than R? Explain why. What is the physiological
meaning of R?
