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42
4 Statistics ofRadiation Counting
Table 4.1
Degree of
freedom (N– 1)
10 2.56 3.94 4.87 9.34 15.99 18.31 23.21
11 3.05 4.58 5.58 10.34 17.28 19.68 24.73
12 3.57 5.23 6.30 11.34 18.55 21.03 26.22
13 4.11 5.89 7.04 12.34 19.81 22.36 27.69
14 4.66 6.57 7.79 13.34 21.06 23.69 29.14
15 5.23 7.26 8.55 14.34 22.31 25.00 30.58
16 5.81 7.96 9.31 15.34 23.54 26.30 32.00
17
18 7.02 9.39 10.87 17.34 25.99 28.87 34.81
19 7.63 10.12 11.65 18.34 27.20 30.14 36.19
20 8.26 10.85 12.44 19.34 28.41 31.41 37.57
21 8.90 11.59 13.24 20.34 29.62 32.67 38.93
22 9.54 12.34 14.04 21.34 30.81 33.92 40.29
23 10.20 13.09 14.85 22.34 32.01 35.17 41.64
24 10.86 13.85 15.66 23.34 33.20 36.42 42.98
25 11.53 14.61 16.47 24.34 34.38 37.38 44.31
26 12.20 15.38 17.29 25.34 35.56 38.89 45.64
27 12.88 16.15 18.11 26.34 36.74 40.11 46.96
28 13.57 16.93 18.94 27.34 37.92 41.34 48.28
29 14.26 17.71 19.77 28.34 39.09 42.56 49.59
Critical chi-square values
Probability
0.99 0.95 0.90 0.50 0.10 0.05 0.01
That the calculated chi-square value will be equal to or greater than
2 0.02 0.10 0.21 1.39 4.61 5.99 9.21
3 0.13 0.35 0.58 2.37 6.25 7.82 11.35
4 0.30 0.71 1.06 3.36 7.78 9.49 13.28
5 0.55 1.15 1.61 4.35 9.24 11.07 15.09
6 0.87 1.64 2.20 5.35 10.65 12.59 16.81
7 1.24 2.17 2.83 6.35 12.02 14.07 18.48
8 1.65 2.73 3.49 7.34 13.36 15.51 20.09
9 2.09 3.33 4.17 8.34 14.68 16.92 21.67
6.41 8.67 10.09 16.34 24.77 27.59 33.41
probability of its being smaller. Similarly, the probability of chi-square being as
large as 12.02 is only 10%, and being as large as 18.48 is only 1%. Typically, χ2
values that fall within 0.1–0.95 are acceptable. If the observed χ2 value falls outside
this range, it is an indication that the variation is beyond the statistical randomness
of the data and something is wrong with the experimental set-up, for example, measuring equipment, measurement technique, and so on.
2
In performing the χ
test, a number of measurements (a minimum of ten) are
made of the quantity, and the mean and χ2 of the measured values are calculated by
Eq. (4.9). The χ2 value is then compared with the value in Table4.1 corresponding
to the actual degree of freedom and for a particular p value.
227637
4511
Rt
RB
4.8 Minimum Detectable Activity
Problem 4.5
The following repeat counts of a radioactive sample were obtained in a wellcounter. Use the χ2 test to see if the variations in counts are due to statistical
variations of radioactivity or the counter is not working properly.
4580 4263
4635 4481
4625 4356
4578 4699
4525 4344
4668 4483
4391 4529
Answer
The average value of 14 measurements is N=4511.
Using Eq. (4.9),
43
2
50 5.
From Table 4.1, for degree of freedom 13 and the p-value of 0.01,
χ2=27.69. The computed χ2 far exceeds the theoretical value, so something in
addition to the statistical uctuation of the counts is operating. Most likely,
the well-counter is not functioning properly.
4.8 Minimum Detectable Activity
The efciency of different detectors is limited by the dead time at high count rates
and by statistical uctuations at low count rates of the backgrounds. In the latter
situation, the minimum detectable activity (MDA) that gives a statistically signicant count is given by
MDA
33/
where σR is the standard deviation of the background count rate RB obtained by
counting over a period of time t. Equation (4.10) requires that the sample count rate
must be at least three standard deviations of the background to be signicant.
(4.10)
44
TP
P
TN
A
A
TP TN
TP TN FP FN
Positive
TP
TP FP
N
TN
TN FN
4 Statistics ofRadiation Counting
4.9 Evaluation ofDiagnostic Tests
It is often required to evaluate the usefulness of a new diagnostic test to determine
the presence or absence of a particular disease. This aspect of the test is commonly
described by two entities: sensitivity and specicity. The sensitivity of a test is the
probability of being able to identify correctly those having the disease in a diseased
population (true positive, TP). The specicity of the test is the probability of being
able to identify correctly those who do not have the disease in a healthy population
(true negative, TN). By these denitions, it is obvious that a given test may not
identify all patients correctly whether or not they have the disease. This results in
false- positives (FP) in the healthy group and false-negatives (FN) in the diseased group.
It should be noted that when sensitivity is assessed for a diseased population or
specicity for a healthy group, the disease or healthy status of the group must be
assessed by an established standard diagnostic test. This test is called the “gold
standard” and is considered the best method available for comparison. As determined by the gold standard in a group of N patients, if NP is the total number of
persons with disease, and NA is the total number of persons without disease, then the
results of the new test can be summarized as in Table4.2. The following parameters
can be obtained from data in Table4.2.
ensitivity
pecificity
TP FNTPN
TN FPTNN
ccuracy
predictive value
egative predictive value
Table 4.2 Distribution of data obtained by a new diagnostic test
T Disease present Disease absent Totals
+ TP FP N +
– FN TN N–
Total N
p
N
A
N
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
780 60
840
.%
145 15
160
.%
780 145 60 15
1000
.%
780 15
795
.%
145 60
205
.%
4.10 Questions
Problem 4.6
In a group of 1000 patients, 840 patients (Group A) had brain tumor and 160
patients (Group B) did not have tumor by biopsies. The SPECT study diag-
nosed 780 patients having tumor in Group A and 15 patients having tumor in
Group B.Calculate the sensitivity, specicity, accuracy, positive predictive
value, and negative predictive value of the SPECT study.
Answer
True positive=780
True negative=160–15=145
False-negative=840–780=60
False-positive=15
45
145
780
145
780
100 92 6
100 90 6
925
100 92 5
780
145
100 98 1
100 70 7
Sensitivity
Specificity
Accuracy
780
145
780 145
Positive predictive value
Negative predictive value
4.10 Questions
1. Dene accuracy and precision.
2. Do systematic errors give an accurate measurement? Can systematic errors give
a precise measurement?
3. A radioactive sample gives 15,360 counts in 9min:
(a) What are the count rate of the sample and its standard deviation?
(b) If the sample contained a background count rate of 60cpm obtained from a
2-min count, what would be the net count rate of the sample and its standard
deviation?
4. How many counts of a sample are to be collected to have a 1% error at the 95%
condence level?
5. Within how many standard deviations of a mean count of 62,001 is 730?
46
4 Statistics ofRadiation Counting
6. To achieve an estimated percent standard error of 3%, how many counts must be
collected?
7. The χ2 value of 11 measurements are due to statistical variations of measure-
ments of a quantity is 4.2. What is the probability that the variations of measurements are due to statistical variations of the quantity?
Suggested Readings
Harris M and Taylor G.Medical Statistics Made Easy. CRC Press. 2003
Martin PM. Nuclear medicine statistics. In: Rollo FD, ed. Nuclear Medicine Physics,
Instrumentation and Agents. St. Louis: Mosby; 1977:479–512.
Production ofRadionuclides
As mentioned in Chap. 1, currently more than 4040 nuclides are known, of which
nearly 3760 nuclides are radioactive, and the remainder are stable. The majority of
radionuclides are articially produced in the cyclotron and reactor. Some shortlived radionuclides are available from the so-called radionuclide generators in
which a long-lived parent radionuclide is loaded and decays to a short-lived daughter radionuclide. The latter is separated from the parent nuclide for use in nuclear
medicine. The following is a brief description of the sources of different
radionuclides.
5.1 Cyclotron-Produced Radionuclides
5
Charged particles can be accelerated under an electromagnetic eld in cyclotrons or
linear accelerators to have high kinetic energy, which are then allowed to react with
stable nuclides to cause nuclear reactions producing different radionuclides. For
radionuclide production in nuclear medicine, cyclotrons are commonly used. Both
positively charged (protons, α-particles) and negatively charged (H
be accelerated in cyclotrons, and their construction and specic use are discussed below.
A typical cyclotron consists of two hollow D-like copper structures called “dee”
separated by a small gap (Fig.5.1). The dees (A and B) are kept in a high-vacuum
tank, and an electromagnetic eld is applied between them. Positive or negative ions
are produced in an arc ion source at the gap, which are then attracted toward the
oppositely charged dee. The magnetic eld then bends them in a circular path
instead of letting them in a straight path. When the charged particles arrive at the
gap, the electrical polarity is changed, by which the particles are repelled by the like
charges and attracted by the opposite charges, thereby gaining further acceleration.
This scenario happens every time the particles cross the gap between the two dees
and approach toward the periphery with increasing energy. Ultimately, the particles
are deected outside in the form of a beam by a deector D through a window
© The Author(s), under exclusive license to Springer Science+Business Media, LLC,
part of Springer Nature 2025
G. B. Saha, Physics and Radiobiology of Nuclear Medicine,
https://doi.org/10.1007/978-1-0716-4816-2_5
−
) particles can
47
48
W
KE
m
222
2
Fig. 5.1 Schematics of a
cyclotron. A and B dees
with vacuum, D deector,
S ion source, V alternating
voltage, W window
D
5 Production ofRadionuclides
V
~
B
S
A
W.The kinetic energy of the particles depends on the charge (e) and mass (m) of the
particle, the magnetic eld (H) in Gauss, and the radius (r) of the cyclotron, as
given below:
Her
..=
(5.1)
The beam energy may range from a few keV to several billion electron volts
(BeV or GeV) depending on the design of the cyclotron. Charged particles such as
protons, deuterons, α-particles, H−, etc., can be accelerated in a cyclotron. In some
special cyclotrons, heavy ions like 32S are also accelerated.
There are several features that distinctively characterize positive ion or negative
ion cyclotrons. Negative ion cyclotrons need to be run under relatively higher vac-
–7
uum than positive ion cyclotrons (~10
Torr vs 10–5 Torr), because H− can lose
electrons by encounters with any molecule during acceleration. The beam extraction efciency in positive ion cyclotrons is ~ 80% and the remaining 20% is lost by
interaction with the cyclotron housing inducing radioactivity. This warrants more
shielding around these cyclotrons. In contrast, since negative ions are extracted with
~ 100% efciency and do not interact with the nucleus of any nuclide, the cyclotron
housing is not activated, thus requiring less shielding. Targets can be bombarded
internally at any radius of the positive ion cyclotrons for radionuclide production,
whereas this is not possible in negative ion cyclotrons for lack of nuclear interaction
of negative ions with target nuclei. In latter cyclotrons, the particle beam is intercepted with a thin carbon foil (~5μm) toward the exit of the beam to strip two
electrons from H− forming an external proton (positive) beam to cause nuclear reactions in target nuclei. In positive ion cyclotrons, a deector routes the particle beam
outside for external irradiation of targets. A unique advantage of negative cyclotrons
111
In
5.1 Cyclotron-Produced Radionuclides
49
is that the particle beam can be split into two beams by placing one carbon stripping
foil to cover only a part of the beam area and another foil at some distance away
covering the remaining area. Thus, two same targets or two different targets can be
irradiated simultaneously with the two beams to produce radionuclides.
Medical cyclotrons are compact negative ion cyclotrons that are commonly used
for production of short-lived radionuclides such as 18F, 11C, 13N, 15O, 68Ga, and so on
used in positron emission tomography (PET) imaging. These nuclides are produced
with low-energy particles and hence the small size of the cyclotron that can be
installed in a small room. In contrast, several medically useful radionuclides such as
111
In, 67Ga, etc., require high-energy particles and so larger cyclotrons, normally
positive ion cyclotrons, are used for their production. The typical energy of the
medical cyclotrons ranges between 3 and 18MeV.
Some medical cyclotrons have provisions for acceleration of deuteron and
H−interchangeably. Whereas high-energy cyclotrons are shielded mostly by thick
concrete walls, medical cyclotrons are self-shielded with lead blocks for reasons of
compact nature. Four lead block quadrants are mounted on casters that wheel them
in and out for closing and opening for easy access to service the cyclotron. Siemens,
GE Healthcare, Best Cyclotrons, and Advanced Cyclotron Systems are the major
manufacturers of medical cyclotrons in the USA.
When targets of stable elements are irradiated by placing them in the external
beam of the accelerated particles or in the internal beam at a given radius inside a
cyclotron, the accelerated particles interact with the target nuclei, and nuclear reactions take place. In a nuclear reaction, the incident particle may leave the nucleus
after interaction with a nucleon, leaving some of its energy in it, or it may be completely absorbed by the nucleus, depending on the energy of the incident particle. In
either case, a nucleus with excitation energy is formed and the excitation energy is
disposed of by the emission of nucleons (i.e., protons and neutrons). Particle emission is followed by γ-ray emission when the former is no longer energetically feasible. Depending on the energy deposited by the incident particle, several nucleons
are emitted randomly from the irradiated target nucleus, leading to the formation of
different nuclides. As the energy of the irradiating particle is increased, more nucleons are emitted, and therefore a wider variety of nuclides is produced.
111
An example of a typical cyclotron-produced radionuclide is
duced by irradiating
111
Cd with 12-MeV protons in a cyclotron. The nuclear reaction
In, which is pro-
is written as follows:
111
where
Cd is the target nuclide, the proton p is the irradiating particle, the neutron
n is the emitted particle, and
Cd p,n
111
In is the product radionuclide. In this case, a second
nucleon may not be emitted, because there is not enough energy left after the emission of the rst neutron. The excitation energy that is not sufcient to emit any more
nucleons will be dissipated by γ-ray emission.
50
5 Production ofRadionuclides
As can be understood, radionuclides produced with atomic numbers different
from those of the target isotopes do not contain any stable (“cold” or “carrier”) isotope detectable by ordinary analytical methods, and such preparations are called
carrier-free. In practice, however, it is conceivably impossible to have these preparations without the presence of any stable isotopes. Another term for these preparations is no- carrier- added (NCA), meaning that no stable isotope has been added
purposely to the preparations.
The target material for irradiation must be pure and preferably monoisotopic or
at least enriched isotopically to avoid the production of extraneous radionuclides.
Because various isotopes of different elements may be produced in a target, it is
necessary to isolate isotopes of a single element; this can be accomplished by appropriate chemical methods such as solvent extraction, precipitation, ion exchange, and
distillation. Cyclotron-produced radionuclides are usually proton-rich and therefore
decay by β+-emission or electron capture.
5.2 Reactor-Produced Radionuclides
A nuclear reactor is constructed with fuel rods made of ssile materials such as
enriched
235
U and
239
Pu. These fuel nuclei undergo spontaneous ssion with
extremely low probability. Fission is dened as the breakup of a heavy nucleus into
two fragments of approximately equal mass, accompanied by the emission of two to
three neutrons with mean energies of about 1.5MeV.In each ssion, there is a concomitant energy release of ~200MeV that appears as heat and is usually removed
by heat exchangers to produce electricity in the nuclear power plant.
Neutrons emitted in each ssion can cause further ssion of other ssionable
nuclei in the fuel rod, provided the right conditions exist. This obviously will initiate a chain reaction, which causes the number of ssions to increase exponentially
ultimately leading to a possible meltdown of the reactor core. This chain reaction
must be controlled, which is in part accomplished by the proper size, shape, and
mass of the fuel material and other complicated and ingenious engineering techniques. To maintain a self-sustained chain reaction, only one neutron is needed for
each of successive ssions, and excess neutrons (more than one) are removed by
cadmium rods, called control rods. which are positioned in the reactor core, where
control rods capture excess neutrons, preventing them from being captured by other
uranium nuclei to induce yet another uranium ssion (cadmium has a high probability of absorbing a thermal neutron). General Electric, Hitachi and Westinghouse
Electric are the major manufacturers of nuclear reactors in the USA, which are
mostly used for production of electric power. The University of Missouri has a
small reactor, which produces a limited amount of radionuclides for use in nuclear
medicine.
The fuel rods of ssile materials are interspersed in the reactor core with spaces
in between. Neutrons emitted with a mean energy of 1.5MeV from the surface of
5.2 Reactor-Produced Radionuclides
51
the fuel rod have a low probability of interacting with other nuclei and therefore do
not serve any useful purpose. It has been found, however, that neutrons with thermal
energy (0.025eV) interact with many other stable nuclei efciently, producing various radionuclides. To make the high-energy neutrons, or so-called fast neutrons,
more useful, they are thermalized or slowed down by interaction with low molecular
weight materials, such as water, heavy water (D2O), beryllium, and graphite (C),
which are distributed in the spaces between the fuel rods. These materials are called
moderators. The ux, or intensity, of the thermal neutrons so obtained ranges from
1011 to 1014 neutrons/cm2s, and they are useful in the production of many radionuclides. When a target element is inserted in the reactor core, a thermal neutron will
interact with the target nucleus, with a denite probability of producing another
nuclide. The probability of formation of a radionuclide by thermal neutrons varies
from element to element.
In the reactor, two types of interaction with thermal neutrons occur to produce
various radionuclides: ssion of heavy elements and neutron capture or (n, γ) reac-
tion. These two nuclear reactions are described next.
5.2.1 Fission or (n, f ) Reaction
When a target of heavy elements is inserted in the reactor core, heavy nuclei absorb
235
thermal neutrons and undergo ssion. Fissionable heavy elements are
237
233
Np,
232
U,
Th, and many others having atomic numbers greater than 92. Fission
U,
239
Pu,
of heavy elements may also be induced in a cyclotron by irradiation with highenergy charged particles, but ssion probability depends on the type and energy of
the irradiating particle. Nuclides produced by ssion may range in atomic number
from about 28 to nearly 65. These isotopes of different elements are separated by
appropriate chemical procedures that involve precipitation, solvent extraction, ion
exchange, chromatography, and distillation. The ssion radionuclides are normally
carrier-free or NCA, and therefore radionuclides of high specic activity are available from ssion. The ssion products are usually neutron-rich and decay by
−
-emission.
β
Many clinically useful radionuclides such as
duced by ssion of
235
U.An example of thermal neutron ssion of
131
I, 99Mo,
133
Xe, and
137
Cs are pro-
235
U follows,
showing a few representative radionuclides:
235
1922365313139102
Un IY n
92
0
U
99
Mo Sn n
42
117
66
4
133
54
137
55
155
Sm Zn n
62
156
62
Pd Pd n
Xe Sr n
Cs Rb
Sm Zn n
117
97
37
135
101
1
3
0
1
2
50
46
38
78
30
77
30
0
1
2
0
1
2
0
1
2
n
00
1
3
0
1
3
0