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32
U
Th
Co Ni
Radioactivity
Naturally occurring radionuclides such as
235
and
232
form a chain of decay process,
as their daughter products are also radioactive.
Figure 5: Decay scheme of
226
Ra
(b) Fission
It is the spontaneous fragmentation of very heavy nuclei (when bombarded with neutrons)
into two lighter nuclei with the emission of two or three neutrons. A large amount of energy
(hundreds of MeVs) is also released in this process. Some of the fission fragments are found
to be very useful in nuclear medicine as they are carrier free radioisotopes with high specific
activity.
Isomeric transition
After beta or alpha emission, some daughter nuclei have excess energy, which they release
by emitting gamma photons. They are emitted almost the same time as beta or alpha rays are
emitted.
60 60
In some cases the daughter nuclei are not in their ground state. This happens when all
the energy associated with decay process is not carried away by the emitted particles. Such
nuclei can either be in an excited state or in a metastable (isomeric) state. In both the
situations the excess energy is often released in the form of one or more gamma photons.
Average lifetime of excited states is very short and energy is released within fraction of a
nanosecond, but the average lifetime of metastable states is very much longer and emission
may vary from few milliseconds to few days or even more. During this period nucleus
behaves as if it is a pure gamma emitter. Some of the metastable states have great clinical
+ – +
+
Radioactivity
Number of conversion electrons
Number of gamma rays emitted
33
application. The transition of a nucleus from a metastable state to a stable state is called an
99m
99m
Tc is good example of an isomer.
99
Tc
=
Tc +
99m
Tc is
isomeric transition.
The decay scheme of 99Mo is shown in figure 6 in which the metastable state of
represented.
Figure 6: Decay scheme of 99Mo
Internal conversion
There are situations when the excited nuclei instead of emitting a gamma photon utilise the
energy in knocking out one of the orbital electrons from the atom. This process is called
internal conversion and the emitted electron is called a conversion electron. The probability
of K conversion electron is more than L or M conversion electrons and the phenomenon is
more common in heavy atoms. The conversion coefficient is expressed as:
Conversion coefficient =
The internal conversion is followed by emission of characteristic x-rays and/or Auger
electrons.
Laws of Radioactive Decay
The radioactive decay has been found to be a spontaneous process independent of any
environmental factor. The radioactive decay is a random process. It is not possible to predict
34
N N t
N N t
or N
t
N No e
N
t
1/2
T
Radioactivity
the disintegration of a particular atom at a given time. However a random process can be
described in terms of probabilities and average constants.
In a sample containing a large number of identical radioactive atoms, the number of
decayed atoms (–N) during a short period of time (t), is proportional to the total number
of atoms (N) present at that time and also to the time duration. Mathematically it can be
expressed as:
N
(1)
t
In this equation the constant (known as decay constant) has a characteristic value for
each radioactive nuclide. Decay constant is the fraction of atoms undergoing decay per unit
time in a very large number of atoms with physical unit as inverse of time.
For simplicity decay constant can be defined as the probability of disintegration of a
nucleus per unit time. Thus =. 01 per sec means that the probability of disintegration of
each atom is 1 percent per second. It is important to note that this probability does not
change with time.
The exact number of parent atoms in a sample at any time can be calculated by integrating
equation-1, which takes the following form:
(2)
Where No is the initial number of atoms in the sample and N is the number present at
time t.
The term
, shows the number of disintegration per unit time and is known as activity.
The SI unit of activity is Bq (1 decay per second). The conventional unit of activity is Curie
(Ci) which is equal to 3.7x1010 disintegrations per second (dps).
Half life
The time after which 50% of the atoms in a sample undergo disintegration is called the halflife. The half life and decay constant are related by the following equation.
0.693 0.693
T or
1/2
(3)
Radioactivity
1/ 2
T T
1
T
1
1 0
( )
N
1
t
N N e
2 2
N
2
2
N
1 1 2 2
N N
1
t
N N e
1 1 0 2 2
N e N
1
t
or N N e
35
Average life
The actual lifetimes of individual atoms in a sample are quite different. Some have very
short and some have very long lifetime. The average lifetime, characteristics of the atoms, is
related to the half-life by:
1.44
av
(4)
The average life is a useful parameter for calculating the cumulated activity in source
organ in internal dosimetry. It is actually the time after which the parent would have
completely decayed to daughter with initial rate of decay
Radioactive equilibrium
In many cases the daughter element is radioactive and immediately starts disintegrating
after its formation. Although the daughter obeys the general rule of radioactive decay, its
activity does not follow the exponential law of decay (equation-1) while mixed with the
parent. The reason is that the daughter is produced (mono-exponentially) by disintegration
of its parent and at the same time it disintegrates (mono-exponentially) as a radioactive
element. So the activity of such elements changes bi-exponentially, first the activity increases,
reaches a maximum and then starts decreasing. The rate at which the activity changes in
such a mixture of radionuclides depends on the decay constant of both the parent and the
daughter.
If we start with a pure sample of a parent with a half life of
contains
atoms initially. The decay of this parent can be expressed by:
( )
1 1 0
and decay constant
and
(5)
The rate of decay of the parent is the rate of formation of the daughter. Let the daughter
decays at the rate
where
is the decay constant of the daughter and
is the number
of atoms present in the daughter. The net rate of formation of the daughter can be given by:
N
2
We know that
t
( )
1 1 0
N
2
t
N
thus equations 6 can be written as:
t
( )
2
t
1
2 2 1 1 0
( )
(6)
(7)
(8)
36
2
t
e
N e N e C
2 2 0
( )
N N
0
2 0 1 0
( ) ( )
C N N
1 2 2
t t t
N N e e N e
( ) ( )
t t
2 2 2
1
t
N N e
( )
( ) (1 )
t t t
0.693( )
(1 )
t
(1 )
t
2 2
( );
t T A
/2
Multiply throughout by
2 2 1
2 1 0
Where C is the constant of integration
1
2 1
( )
( )
Radioactivity
tt
(9)
at
t
thus
1
2 1
Substituting the value of C and re-arranging:
1
2 1 0 2 0
( ) ( ) ( )
2 1
Neglecting the second term on the right hand side of the equation
N N e e
2 1 0
A N
1
2 1
and
1 2
( )
1 1 0
Equation-10 can therefore be expressed in terms of activity as:
A N e e
2 1 1 0
A A e
2 1
T
1
T T
1 2
2
1 2 1
2 1
T T
1 2
T T
1 2
This equation is for a simple parent-daughter mixture. In general three different situations
arise from equation-13.
(10)
(11)
(12)
(13)
(a) Secular equilibrium
When the half-life of the parent (T1)is too long in comparison to that of the daughter (T2)
equation-12 or equation-13 may be expressed as:
After one half-life of the daughter
half-lives the daughter may grow up to 3/4 of the parent and after four half lives (of the
daughter) it rises to about 94% of the parent activity. Thus activity of the daughter gradually
increases and after it’s few half-lives, the activity of the parent and daughter become almost
equal (Figure 7). The parent and daughter are then said to be in secular equilibrium.
A A e
2 1
0.693
T
2
will become nearly
and after two
A
1
(14)
Radioactivity
1/2
T
1/2
T
1 2
T
T T
2
t T
( ,2 ,3 ,4 )
T T T T T etc
37
Figure 7 : Secular equilibrium
113
One example of secular equilibrium is the generator system
parent nucleus
113
Sn has a half-life of 118 days, and the daughter
113m
113m
Sn/
In, where the
In, a half-life of 1.7
days. A second example is that of the 68Ge/68Ga generator with parent-daughter half-lives of
270 days and 68 min, respectively. The importance of such a generator system is that the
daughter has an apparent half-life equal to that of the parent when it is with the parent in
equilibrium. The decay of
226
Ra to
222
Rn is another example of secular equilibrium.
(b) Transient equilibrium
Half-life of the parent is few times (by a factor of 10 or more) longer than that of the
daughter but not as much long as in secular equilibrium. One example is the technetium
radionuclide generator where the parent 99Mo (
(
= 6 h). In this case the activity of daughter increases and eventually exceeds the activity
= 67 h) decays to the daughter
of parent slightly, reaches a maximum and then decreases and decays with the half-life of
the parent (Figure 8). Theoretically the activity of the daughter is slightly more than the
parent after maximum growth of the daughter. For t>> T2 the exponential term in equation13 approaches zero, equation-13 can therefore be written:
A A
2 1
The growth of the daughter for multiples of its half-life
1
for
2 2 2 2 2
nearly 50%, 75%, 87.5% and 94% respectively of the activity of the parent. It is therefore
advisable to elute the activity from technetium generator after every 24 hrs.
99m
Tc
(15)
will be
38
(c) No equilibrium
Radioactivity
Figure 8: Transient equilibrium
When the half-life of the daughter is longer than the half-life of the parent there would be
no equilibrium between them as can be seen in figure 9.
Figure 9: There is no equilibrium when the physical half-life of daughter is longer than that of
the parent
Radioactivity
X n Y p
39
Production of radionuclides
Radionuclides used in nuclear medicine are either reactor produced or Accelerator (Cyclotron)
produced. There is one more method that makes use of radionuclide generators. The parent
radionuclide in the generator is also produced by any one of the above two main methods.
Reactor produced radionuclides
Most of the radionuclides used in nuclear medicine are produced in a nuclear reactor
either by bombardment of stable isotopes with neutrons or obtained as a fission fragments of
236
U.
Bombardment of stable isotopes in a rector (neutron activation)
The samples of stable isotopes that need to be activated are placed in reactor core with
high neutron flux. The stable isotope accepts neutrons depending upon its cross section and
becomes unstable or radioactive. There are two types of possible nuclear reactions :
1 1A A A
(a) (n, ) reaction :
X n X X
Z Z Z
(b) (n, p) reaction:
A A
Z Z
1
In (n, ) reaction the product emits gamma photon on de-excitation. An example of this
type is
130
Te(n, )
131
Te.
131
Te is unstable and decays to
131
I by beta emission. Since all the
atoms in the sample do not undergo nuclear reaction so the product will have both radioactive
and non-radioactive atoms. Such radionuclides cannot be carrier free and therefore will have
low specific activity. Another example is when 98Mo on accepting a neutron becomes 99Mo*,
which is unstable and decays to
67 h and that of
99m
Tc is 6 h there will be transient equilibrium between them. This is the
99m
Tc with – emission. Since physical half-life of 99Mo is
most commonly used radionuclide generator in nuclear medicine.
The (n, p) reaction on the other hand may produce carrier free radionuclides with high
specific activity. A good example of (n, p) reaction is 32S(n,p)32P.
Fission fragments
235
U on accepting a neutron becomes
spontaneously by fission reaction into various radionuclides. However a given atom of
will be fragmented into two radionuclides. A large number of radionuclides are produced as
fission fragments of which
131
I and 99Mo are of great help in nuclear medicine. These
radionuclides can be produced both by neutron activation as well as by fission reactions.
The fission products are always preferred as they are carrier free and therefore available
with high specific activity.
236
U*, which is very unstable and gets fragmented
236
U*
40
Radioactivity
Cyclotron produced radionuclides
The method of production and important radionuclides that are relevant in nuclear medicine
are described in a separate chapter on “Cyclotron” in this book.
Radionuclide generators
When there is huge difference in the physical half lives of parent and daughter. The half-life
of parent is much more than the daughter then there is radioactive equilibrium between them
as has been described earlier. The parent (mother) radionuclide is either reactor produced or
cyclotron produced. The best example of reactor produced parent is 99Mo. The example of
cyclotron produced parent is 68Ge which is produced either by 69Ga (p,2n)68Ge or
66
Zn(a,2n)68Ge reaction. It decays to 68Ga by electron capture and is in secular equilibrium
with the daughter. Some of the important radionuclide generators that are commonly used in
nuclear medicine are given in the table 1.
Table 1: Commonly used generators in nuclear medicine.
Daughter Decay mode Half life (T
99m
Tc IT 6 h
113m
In IT 100 min
87m
Sr IT 2.8 h
68
Ga +, EC 68 min
82
Rb +, EC 78 sec
) Parent Half life (T
1/2
99
Mo 67 h
113
Sn 120 d
87
Y 80 h
68
Ge 278 d
82
Sr 25 d
1/2
)
Further reading
1. Physics in Nuclear Medicine by J.A. Sorenson and M.E. Phelps, Grune and Stratton Inc. (Publisher)
New York, 1980
2. Nuclear Physics by D.C. Tayal Himalaya Publishing House, Mumbai, 1980
3. Physics of Radiology by H. E. Johns and J. R. Cunningham, Charles C. Thomas Publisher
4. Introductory Physics of Nuclear Medicine by R. Chandra, Lea & Febiger Publisher
Counting Statistics
A.K. Pandey
The acquired data in nuclear medicine may have two types of error-systematic or random.
The systematic error is primarily due to improper calibration of instruments or manufacturing
defect in the instrument whereas random error is due to random nature of physical processes
involved (radioactive disintegration, interaction of radiation with the detector material etc).
The variations in measurement lead to error in the final result. The random nature of nuclear
disintegration can be expressed in terms of Poisson’s statistics.
Poisson’s Distribution
The Poisson’s distribution can be applied in the following conditions:
1. The events are discrete or discontinuous
2. The events are independent i.e. the first event has no influence on the second event.
3. In a given interval any number of events can occur.
4. The probability of a single occurrence of the event in a given interval of time is
proportional to the length of the interval.
5. In an infinitesimally small interval of time the probability of an event to occur more
than once is negligible
The Poisson’s probability distribution (for sample counting) can be expressed as:
m
Where
is m. The term
number of counts can be estimated if the average number of counts is known.
is the probability of obtaining a count when the average (true) count
),( mP
is the Poisson probability density. Probability of getting a given
),( mP
memP
4 1
!/)(),(
(1)
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