Biomedical Engineering Reference
In-Depth Information
Fig. 11.8
Relationship between observed count rates
r
c
and
true event rates
r
t
for nonparalyzable and paralyzable counters
with dead time
τ
.
Unlike Eq. (11.92), one can only solve Eq. (11.96) numerically for
r
t
in terms of
r
c
and
τ
. For low event rates or short dead time (
r
t
τ
1), Eq. (11.96) gives
(11.97)
r
c
=
r
t
(1 -
r
t
τ
).
This relationship also leads to the same Eq. (11.93) for nonparalyzable systems
when
r
t
τ
1
(Problem 61).
Figure 11.8 shows plots of the measured count rates
r
c
as functions of the true
event rate
r
t
for the nonparalyzable and paralyzable models. For small
r
t
,bothgive
nearly the same result (Problem 61). For the nonparalyzable model, Eq. (11.92)
shows that
r
c
cannot exceed
1/
τ
. Therefore, as
r
t
increases,
r
c
approaches the as-
symptotic value
1/
τ
, which is the highest possible count rate. For the paralyzable
counter, on the other hand, the behavior at high event rates is quite different. Dif-
ferentiation of Eq. (11.96) shows that the observed count rate goes through a maxi-
mum
1/
eτ
when
r
t
= 1/
τ
(Problem 63). With increasing event rates, the measured
count rate with a paralyzable system will decrease beyond this maximum and ap-
proach zero, because of the decreasing opportunity to recover between events. With
a paralyzable system, there are generally two possible event rates that correspond
to a given count rate.
Example
A counting system has a dead time of 1.7
10
4
s
-1
is observed,
what is the true event rate if the counter is (a) nonparalyzable or (b) paralyzable?
µ
s. If a count rate of 9
×
Solution
(a) For the nonparalyzable counter with
r
c
=
10
4
s
-1
10
-6
9
×
and
τ
=
1.7
×
s,
Eq. (11.92) gives for the true event rate
10
4
s
-1
9
×
10
5
s
-1
.
r
t
=
10
-6
s)
=
1.06
×
(11.98)
1-(9
×
10
4
s
-1
)(1.7
×
