Biomedical Engineering Reference
In-Depth Information
where
c
1
is the constant of integration. With the initial condition,
q
SI
(0) = 0
,we
find
c
1
= -
A
0
λ
ST
/(
λ
2
-
λ
1
)
.Thus,
A
0
λ
ST
λ
2
-
q
SI
e
λ
2
t
λ
1
[e
(λ
2
-λ
1
)
t
-1],
=
(16.16)
or
A
0
λ
ST
λ
2
-
λ
1
[e
-λ
1
t
-e
-λ
2
t
].
(16.17)
q
SI
(
t
)
=
[This solution is similar to Eq. (4.40).]
For the activity in the upper large intestine,
d
q
ULI
d
t
=
λ
SI
q
SI
-
λ
3
q
ULI
,
(16.18)
λ
ULI
. Using the integrating factor
e
λ
3
t
and Eq. (16.17) for
q
SI
,we
write in place of (16.18)
e
λ
3
t
d
q
ULI
d
t
where
λ
3
=
λ
R
+
λ
3
q
ULI
A
0
λ
SI
λ
ST
λ
2
-
[e
(λ
3
-λ
1
)
t
-e
(λ
3
-λ
2
)
t
].
+
=
(16.19)
λ
1
Integrating both sides yields
e
(λ
3
-λ
1
)
t
λ
3
-
+
c
2
,
-
e
(λ
3
-λ
2
)
t
λ
3
-
A
0
λ
SI
λ
ST
λ
2
-
q
ULI
e
λ
3
t
=
(16.20)
λ
1
λ
1
λ
2
where
c
2
is the constant of integration. With the initial condition,
q
ULI
(0)
=
0
,it
follows that
c
2
= -
A
0
λ
SI
λ
ST
λ
2
-
1
λ
3
-
1
λ
3
-
A
0
λ
SI
λ
ST
λ
1
-
=
λ
1
)
.
(16.21)
λ
1
λ
2
(
λ
3
-
λ
2
)(
λ
3
-
Substituting this value for
c
2
and multiplying both sides of Eq. (16.20) by
e
-λ
3
t
,we
obtain
A
0
λ
SI
λ
ST
e
-λ
1
t
e
-λ
2
t
q
ULI
(
t
)
=
λ
1
)
+
(
λ
2
-
λ
1
)(
λ
3
-
(
λ
3
-
λ
2
)(
λ
1
-
λ
2
)
.
e
-λ
3
t
+
(16.22)
(
λ
3
-
λ
2
)(
λ
3
-
λ
1
)
We could proceed in similar fashion to solve for the activity in the lower large
intestine as a function of time, but will not do so.
The general decay and growth functions, such as those described by Eqs. (16.11),
(16.17), and (16.22), are often referred to as the Bateman equations. They date from
the early days of study with the naturally occurring radioactive decay chains.
2)
2 H. Bateman,
Proc. Cambridge Philos. Soc.
15
,
423 (1910).
