Biology Reference
In-Depth Information
S
(
t
)
S
(
t
2
)
S
(
t
1
)
S
(
t
0
)
∆
t
∆
t
∆
t
t
0
t
1
t
2
t
n
−
1
t
2
t
3
t
(
n
−
1)
t
t
0
time
n
n
n
n
FIGURE 9-21.
Approximation of the secretion rate function S(t). This approximation assumes that S(t) remains
constant over each interval
ð
k
1
Þ
kt
n
;
:
n
Using Eq. (9-14), with the specific values above, we arrive at the
following approximation for the concentration at time t:
te
k
ð
t
t
0
Þ
þ
te
k
ð
t
t
1
Þ
þ ::: þ
te
k
ð
t
t
n
1
Þ
;
C
ð
t
Þ
S
ðt
0
Þ D
S
ðt
1
Þ D
S
ðt
n
1
Þ D
i
:
e
:
X
n
1
e
k
ð
t
t
m
Þ
D
C
ð
t
Þ
S
ðt
Þ
t
:
(9-15)
m
m
¼
0
Note that Eq. (9-15) is based on two approximation assumptions. First,
we assumed that the secretion events occurred instantaneously, and,
second, we assumed that the rate of secretion S(t) remained constant
over each subinterval of length
0, both of these
approximations will become more accurate, and thus Eq. (9-15) will give
a better approximation for C(t). Because the sum in Eq. (9-15) represents
a Riemann sum for the function S
D
t. As
D
t
!
e
ð
t
r
Þ
over the interval [0,t], taking
ðtÞ
the limit as
D
t
!
0 in Eq. (9-15), gives:
Z
Z
t
t
0
X
n
1
e
k
ð
t
t
m
Þ
D
e
k
ð
t
tÞ
d
C
ð
t
Þ¼
lim
S
ðt
m
Þ
t
¼
S
ðtÞ
t ¼
S
ðtÞ
E
ð
t
tÞ
d
t;
D
t
!
m
¼
0
0
0
e
kt
. This is the integral term of Eq. (9-12) for the most
common case of exponential removal of the hormone from the
bloodstream.
¼
where E(t)
Identical derivations apply for any other form of the function E(t), such
as the sum of two exponential decays given in Eq. (9-11). For this more
general case, Eq. (9-14) changes to:
C
ð
t
Þ¼
C
0
E
ð
t
t
0
Þþ
C
1
E
ð
t
t
1
Þ þ þ
C
n
E
ð
t
t
n
Þ:
(9-16)
The first term of Eq. (9-16) represents the residual concentration at time
t remaining from the secretion event at time
t
0
. The second term is the
residual concentration at time t from the secretion event at time
t
1
, and
