biodex ATOMLAB 960 - Decay Factor Method

E-8 APPENDIX E Biodex Medical Systems, Inc. © 2014
DECAY FACTOR METHOD
If you do not have a calculator with the ex function, the above expression can be rewritten as
follows:
A
f
= X A
i
where
X = X
n
(N) x X
r
(R)
N = integer number of half lives in time t; and
R = decimal remainder after dividing t by T
1/2
. To verify this, remember three important
relations regarding exponentials:
e
(A + B)
= e
A
x e
B
;
e
In(A)
= A ; and
e
AB
= (e
A
)
B
.
Now t / T
1/2
= N + R
where N is an integer, and R is the remainder less than 1.
For t = 32.5 hours and T
1/2
= 6.007 hours,
t / T
1/2
= 32.5 / 6.007 = 5.410
so that N = 5 and R = 0.410.
Now, to simplify calculations, we can write the above relation as:
A
f
= A
i
e
-λt
= A
i
e
-ln (2)t /T
1/2
A
f
= A
i
e
-ln (2)(N + R)
= A
i
e
-ln (2)N
x e
-ln (2)R
= A
i
x X
n
(N) x X
r
(R)
X
n
(N) = e
-
In (2)N
= (e
In (2)
)-N = 2
-N
This expression is used to generate the values in Table 1.
X
r
(R) = e
-ln (2)R
This expression is used to generate the Decay Factor Chart for values of R from 0.001 to 0.999.
For the above example
X
n
(5) = 0.03125
X
r
(0.410) = 0.75262
A
f
= X A
i
= [X
n
(N) x X
r
(R)] A
i
= [0.03125 x 0.75262] 43 mCi
A
f
= 1.01 mCi