Biology Reference
In-Depth Information
a fixed moment
t, then we are in a situation like that in
Chapter 1; namely, it is only a small inaccuracy to consider the amount
of hormone S(
t
as being S(
t
)
D
t
)
D
t to be totally and instantaneously delivered at time
t
.
Now, as in Chapter 1, if t is some time later than
, the concentration of
the hormone caused by this one secretion event is S
t
ðtÞD
te
k
ð
t
tÞ
:
The result will become exact after passing to a limit
D
t
!
0, where
D
t is
the width of the rectangles. Eqs. (9-12) and (9-13) give the resulting
mathematical form of the combination of secretion and elimination in the
form of a convolution integral for the concentration at time t.
Z
t
C
ð
t
Þ¼
S
ðtÞ
E
ð
t
tÞ
d
t þ
C
ð
0
Þ
E
ð
t
Þ
(9-12)
0
Z
t
C
ð
t
Þ¼
S
ðtÞ
E
ð
t
tÞ
d
t ¼
S
ð
t
Þ
E
ð
t
Þ:
(9-13)
1
We now formalize this idea and derive Eq. (9-12). Divide the interval
1
n
and assume
n instantaneous secretory events of magnitude C
0
,C
1
,
[0,t] into n equal subintervals of length
D
t
¼
C
n
1
have taken
...
place at time instances
t
0
<
t
1
<
<
t
1
<t
;
where
t
0
is in the
n
; t
, and, in general,
t
n
t
n
;
2
t
n
interval 0
;
1
is in the interval
t
m
is in
the interval m
t
t
n
n
; ð
m
þ
1
Þ
;
m
¼
0
;
1
; ;
n
1 (see the horizontal axis
in Figure 9-21). Assume also that the elimination function E(t)is
exponential, as in Eq. (9-10). At time t, because of hormone elimination,
the residual amount from the secretion event at time
t
0
will be C
0
e
k
ð
t
t
0
Þ
:
The secretion event at time
t
1
will contribute, by time t, a residual
concentration of C
1
e
k
ð
t
t
1
Þ
;
and so on. Thus, under the assumptions
made, the hormone concentration in the blood at time t will
be given by:
C
C
0
e
k
ð
t
t
0
Þ
þ
C
1
e
k
ð
t
t
1
Þ
þ þ
C
n
1
e
k
ð
t
t
n
1
Þ
:
ð
t
Þ¼
(9-14)
Equation (9-14) is not quite exact because it assumed the secretion events
at the specified moments are instantaneous. Graphing the function
C(t) from Eq. (9-14) will result in a function with jumps, similar to
Figure 1-23 in Chapter 1 (the only difference is that the heights of the
jumps, corresponding to the secretion amounts C
0
, C
1
,
C
n
are now
different). In reality, the hormone concentrations change in a continuous
fashion, as the secretion events can never be instantaneous. As an
approximation, however, a sharp increase in the hormone concentration
can be considered the result of a secretion event with a secretion rate S(t)
that is constant over a very short interval of time and zero outside of this
interval, as in Figure 9-20. The upper panel depicts such a secretion rate
function S(t) and the lower panel represents the corresponding
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