Biomedical Engineering Reference
In-Depth Information
and 2 are registered by both counters. After event 2 is registered, events 3 and then
4 restart the dead period for the paralyzable detector, which misses both. Event 3
is ignored by the nonparalyzable counter, which recovers in time from event 2
to register 4. Events 5 and 7 are recorded by both, but 6 is missed. Of the seven
events in this example, the paralyzable counter registers four and the nonparalyz-
able, five. Such instruments thus actually count the number of
intervals
between
events to which they respond, rather than the number of events themselves. In
practice, counting systems often exhibit behavior intermediate to the two extremes
illustrated in Fig. 11.7.
Dead-time corrections can be made to convert a measured count rate
r
c
into a
true event rate
r
t
. With a nonparalyzable system, the fraction of the time that the
instrument is dead is
r
c
τ
. Therefore, the fraction of the time that it is sensitive is
1-
r
c
τ
, which is also the fraction of the number of true events that can be recorded:
r
c
r
t
= 1-
r
c
τ
.
(11.91)
Thus, the true event rate for a nonparalyzable counter is given in terms of the
recorded count rate and the dead time by the relation
r
c
1-
r
c
τ
r
t
=
(Nonparalyzable)
.
(11.92)
When the count rate is low or the dead time short (
r
c
τ
1
),
r
t
=
(11.93)
r
c
(1 +
r
c
τ
).
With a paralyzable counter, on the other hand, only intervals longer than
τ
are
registered. To analyze for the dead time, we need the distribution of time intervals
between successive random events that occur at the average rate
r
t
. The average
number of events that take place in a time
t
is
r
t
t
. If an event occurs at time
t
=
0
,
then the probability that no events occur in time
t
immediately following that event
is given by the Poisson term,
P
0
=
exp
(-
r
t
t
)
. The probability that an event will occur
in the next time interval d
t
is
r
t
d
t
. Therefore, given an event at time
t
0,the
=
probability that the next event will occur between
t
and
t
+d
t
is
P
(
t
)d
t
=
r
t
e
-
r
t
t
d
t
.
(11.94)
The probability that a time interval larger than
t
will elapse is
∞
r
t
e
-
r
t
t
d
t
=
e
-
r
t
τ
.
(11.95)
τ
The observed count rate
r
c
is the product of the true event rate
r
t
and this probabil-
ity:
r
t
e
-
r
t
τ
r
c
=
(Paralyzable)
.
(11.96)
