Biology Reference
In-Depth Information
so on. The concentration at time t is the sum of all partially eliminated
secretion events occurring before time t. The convolution integral in
Eq. (9-12) is the integral analogue of Eq. (9-16), accounting for the fact
that secretion events may occur at any moment. The secretion event
occurring at time t will undergo elimination for a time of t-
t
before
time t. The concentration at time t in this case is the integral
over [0,t] of all partially eliminated secretion events occurring
before time t.
In calculating the convolution integral, we assumed that the hormone
concentration before this time is zero. This assumption is not realistic,
because at least some basal concentration of the hormone will be
observed. The C(0) in Eq. (9-12) represents the concentration at time
t
0, and the term C(0)E(t) represents the residual of this concentration
at time t. Alternatively, if the integration is over the interval [
¼
1
,t] this
extra term will be absorbed by the integral as in Eq. (9-13). The
symbol * in Eq. (9-13) is the mathematical shorthand for a convolution
integral.
In principle, Eqs. (9-12) and (9-13) can be used with a set of experimental
data C(t
1
),C(t
2
),
and an assumed approximate value for the
elimination half-life, HL, to determine the characteristics of the secretion
rate as a function of time, S(t). This process is known as deconvolution;
that is, the inverse of a convolution.
...
B. Gold's Deconvolution Method
Gold's method is an example of one of the standard deconvolution
techniques described above. Gold's method is an iterative approach for
the solution of Eq. (9-13); that is, C(t)
E(t). As we saw in Chapter
8, an iterative method for S(t) starts with an initial estimate and
subsequently provides a better estimate based on the data. The process is
repeated until the iterations do not change significantly from one
cycle to the next. In Gold's method, the value of the secretion as
a function of time after k iterations,
¼
S(t)
∗
k
S(t), is expressed in terms of the
previous iteration as:
0
S
ð
t
Þ¼
C
ð
t
Þ
C
ð
t
Þ
(9-17)
k
S
k
1
S
ð
t
Þ¼
ð
t
Þ
Þ
:
k
1
S
ð
t
Þ
E
ð
t
In Eq. (9-17), the concentration as a function of time, C(t), is used as the
initial value for the secretion as a function of time,
0
S(t), to start the
iteration. The function E(t) is the exponential elimination function
from Eq. (9-10).
Figure 9-22 presents the Gold's deconvolution of the GH data shown in
Figure 9-6 assuming a 20-minute elimination half-life. It is clear there
are at least three large secretion events and maybe as many as 20 small
