Biomedical Engineering Reference
In-Depth Information
The fraction in the small intestine is given by Eq. (16.17), with
λ
2
-
λ
1
= 0.289-1.01 =
-0.721 h
-1
and
1h
-1
:
λ
ST
=
q
SI
(
t
)
A
0
1
-0.721
(e
-1.01
t
-e
-0.289
t
)
1.39(e
-0.289
t
-e
-1.01
t
).
=
=
(16.30)
0.250 h
-1
,
Additionally, for Eq. (16.22) we need
λ
SI
=
6/24
=
λ
3
-
λ
1
=
0.0857 - 1.01
=
-0.924 h
-1
,and
-0.203 h
-1
. Thus,
λ
3
-
λ
2
=
0.0857 - 0.289
=
(0.250)(1)
e
-1.01
t
e
-0.289
t
(-0.203)(0.721)
q
ULI
(
t
)
A
0
=
(-0.721)(-0.924)
+
(16.31)
e
-0.0857
t
(-0.203)(-0.924)
+
= 0.375e
-1.01
t
- 1.71e
-0.289
t
+ 1.33e
-0.0857
t
.
(16.32)
Asacheck,wenotethat
q
ULI
(0)
=
-0.005, which, to within roundoff, is the required
initial condition,
q
ULI
(0)
0.
The fractional activities, (16.29), (16.30), and (16.32), are plotted in Fig. 16.9. With
the 1-h mean metabolic residence time in the stomach, the fraction of the radio-
nuclide there drops rapidly below unity following ingestion. Equation (16.26) shows
that radioactive decay adds only 1% to the clearance rate from the stomach. The ac-
tivity in the small intestine builds up to a maximum at 1.75 h; that in the upper large
intestine reaches a maximum at 7 h. About 70% of the originally ingested activity is in
these three compartments at this time. After about 24 h, the fractional activity in the
ULI is 0.17. Very little additional activity enters the ULI then, and so
q
ULI
(
t
) declines
exponentially thereafter, governed principally by the third term in Eq. (16.32). The
=
Fig. 16.9
Fractions of activity
A
0
,ingestedattime
t
=
0, in the
stomach (ST), small intestine (SI), and upper large
intestine (ULI) as functions of time. See Eqs. (16.29), (16.30),
and (16.32) in the example in text.
