**1. PRINCIPLES OF PRODUCTION OF RADIOISOTOPES USING CYCLOTRONS**

**1.1 Introduction**

The development of nuclear technology is one of the most significant
achievements of the 20^th^ century. The pioneering work of Marie and
Pierre Curie in uncovering substances with previously unrecognized
properties, for which they coined the term, radioactive, opened up many
new fields of opportunity. The Curies\' discovery was the result of
Marie\'s belief that the ore pitchblende contained another more active
substance than uranium. Within a few months of starting to analyze
pitchblende in 1898, Marie Curie had isolated two previously unknown
elements. She named the first, Polonium, after her native Poland: the
second she called Radium, in response to its intense radioactivity.
Practical applications in scientific research for radioisotopes followed
from these discoveries in the period 1920 to the early 1930s. However,
the few naturally occurring radioisotopes that were available severely
limited the scope of what was possible. The full potential was not
realized until radioisotopes could be produced artificially.

The first major advance occurred in 1934 with the invention of the
cyclotron by Ernest Lawrence in Berkeley, California. With this
electrical machine being used to accelerate deuterons to very high
speeds, it became possible to create the nuclear instability that we now
know is a pre-requisite of radioactivity. By directing a beam of the
fast-moving deuterons at a carbon target, Lawrence induced a reaction
which resulted in the formation of a radioisotope with a half life of 10
minutes.

Particle accelerators and in particular cyclotrons, were very important
in the preparation of radioisotopes during the years of 1935 to the end
of World War Two (WWII). After WWII, reactors were used to produce
radioactive elements and the use of accelerators for this purpose became
less common. For a discussion of reactor produced radioisotopes see IAEA
TECDOC 1340 "Manual for Reactor Produced Radioisotopes". As the
techniques for using radiotracers became more sophisticated, it became
clear that reactor produced radionuclides could not completely satisfy
the growing demands and therefore, accelerators were needed to produce
new radioisotopes which could be used in new ways in both industry and
medicine. Most of these applications use the radionuclides at tracer
concentration to investigate some process or phenomenon. In industry,
these applications have taken the form of embedding the tracer into a
system, or actually inducing radioactivity in the system by charged
particle or neutron activation directly. The IAEA publication TRS 423
"Radiotracer Applications in Industry - A Guidebook" provides an
excellent overview of recent research in the use of radiotracers in a
variety of applications.

The other way these radionuclides are used is in the medical application
or what has become to be known as nuclear medicine. Although nuclear
medicine traces its clinical origins to the 1930s, the invention of the
gamma scintillation camera by American engineer Hal Anger in the 1950s
brought major advances in nuclear medical imaging and rapidly increased
the use of radioisotopes in medicine. The medical use of radionuclides
can be broken down into two general categories; imaging and
radiotherapy. Imaging can be further divided into Positron Emission
Tomography (PET) and Single Photon Emission Tomography (SPECT).

The production of radionuclides for use in biomedical procedures, such
as diagnostic imaging and/or therapeutic treatments, is achieved through
nuclear reactions in reactors or from charged particle bombardment in
accelerators. In accelerators, the typical charged particle reactions
utilize protons although deuterons and helium nuclei (^3^He^++^ and α
particles) play a role.

One clear advantage that accelerators possess is the fact that, in
general, the target and product are different chemical elements. This
makes it possible to:

-   find suitable chemical or physical means for separation,

-   obtain high specific activity preparations due to the target and
    product being different elements, and

-   produce fewer radionuclidic impurities by selecting the energy
    window for irradiation.

**1.2 Cyclotrons for Radioisotopes Production**

The production of radionuclides with an accelerator demands that
particle beams are delivered with two specific characteristics. The beam
must have sufficient energy to bring about the required nuclear
reactions, and sufficient beam current to give practical yields.

The first cyclotron dedicated to medical applications was installed at
Washington University, St. Louis in 1941, where radioactive isotopes of
phosphorus, iron, arsenic and sulfur were produced. During World War II,
a cyclotron in Boston also provided a steady supply of radionuclides for
medical purposes. In the mid 1950\'s a group at Hammersmith in the
United Kingdom put into operation a cyclotron wholly dedicated to
radionuclide production. The major change occurred in the early and mid
1960's where the work in hot atom chemistry (e.g. the in situ chemistry
of nucleogenic atoms occurring in a target being bombarded) laid the
foundation for the synthesis of organic compounds labeled with positron
emitters. A 1966 article by TerPogossian and Wagner focused on the use
of carbon‑11 (Ter-Pergossian and Wagner 1966). As the field of Nuclear
Medicine has progressed, the number of available types of particle
accelerators with varying characteristics dedicated to radionuclide
production for nuclear medicine has also expanded. The major classes of
accelerators are the positive and negative ion cyclotrons. More recent
innovations include superconducting magnet cyclotrons, small low energy
linacs, tandem cascade accelerators and helium particle only linacs.

Cyclotrons come in many sizes depending on the function for which they
are intended. Some examples are shown in Figure 1. The cyclotron on the
left is a deuteron machine designed to produce only oxygen-15 for PET
studies. The machine on the right is the 650 MeV cyclotron at TRIUMF in
Vancouver Canada where a wide variety of radionuclides are produced and
other experiments are carried out.

![](media/image1.jpeg){width="2.7098425196850395in"
height="2.791732283464567in"}![](media/image2.jpeg){width="4.016141732283464in"
height="2.8007874015748033in"}

*Figure 1. Comparison of cyclotrons from the small single isotope
machine to the large multipurpose research machine*.

The basic characteristics of the cyclotron are the same. There is an ion
source to produce the ions, an acceleration chamber to accelerate the
ions, and a magnet to contain the ions to a circular path. The
construction of a modern cyclotron is shown in Figure 2.

![](media/image3.png){width="3.875in" height="3.115972222222222in"}

*Figure 2 Internal working parts of a modern cyclotron.*

**1.3 Nuclear Reactions**

When a charged particle and nucleus collide, the energy of the collision
can be divided into two parts. These are the energetic threshold and the
excitation energy. In the classic sense, a reaction between a charged
particle and a nucleus cannot take place if the center of mass energy of
the two bodies is less than the coulomb barrier. In the case which
applies to the production of radionuclides with a cyclotron, this
implies that the charged particle must have the energy greater than
(Evans 1955) :

*B=Zze^2^/R*

where, Z and z = the atomic numbers of the two species

> e^2^ = the electric charge, squared
>
> R = the separation of the two species in cm.

The values of the coulomb barrier for the four major particles used in
cyclotrons (p, d, ^3^He, ^4^He) are plotted in Figure 3 as a function of
the Z of the material. Actually, these reactions take place at energies
well below this barrier due to the effects of quantum tunneling.

![](media/image4.wmf){width="6.252083333333333in"
height="4.2965277777777775in"}

Figure 3 Plot of the coulomb barrier as a function of the atomic number
of the nucleus for the four major particles used in cyclotrons

If the reaction is endoergic, then energy of an amount greater than this
must be supplied in order for the reaction to proceed. The magnitude of
this difference is called the Q value. If the reaction is endoergic,
then the threshold will be the coulomb barrier plus this difference. If
the reaction is exoergic, Q values are positive and the threshold energy
will just be the coulomb barrier. In reality, as a result of quantum
mechanical tunneling, nuclear reactions start to occur when the tails of
the energy distributions overlap and so they will occur at energies
below the coulomb barrier. An example of the possible reaction pathways
is shown in Figure 4 along with their corresponding Q values. The energy
changes in a nuclear reaction are large enough that changes in the mass
of the reactants and products are observable.

![](media/image5.wmf){width="5.126388888888889in"
height="2.8604166666666666in"}

Figure 4. Q values and thresholds of nuclear decomposition for the
reaction of a deuteron with a nitrogen-14 nucleus after forming the
compound nucleus, ^16^O.

The nuclear reaction cross-section represents the total probability that
a compound nucleus will be formed and then decomposes along a particular
channel. This is often called the excitation function. This function
determines the amount of a radionuclide that may be made on a given
cyclotron, and the levels of contamination of other radioisotopes which
can be present in the target material. In the "touching spheres" model
of nuclear reactions, we can visualize two spheres coming toward each
other. If the spheres touch, then there will be a reaction and if they
don't touch, there won't be a reaction. In this visualization, the
reaction probability is proportional to the cross-sectional area of the
two spheres. Then the total reaction cross section is given by the
equation:

$\sigma_{R} = {\pi r}_{0}^{2}(A_{p}^{\frac{1}{3}} + A_{T}^{\frac{1}{3}})^{2}$
where r~0~ ≈ 1.6 fm

The unit of this measure is the barn where 1 barn = 1 x 10^-24^ cm^2^.
The expression barn comes from the fact that the probability for a
neutron to interact with a target is proportional to the area of the
nucleus, which compared to the size of the neutron appeared as big as a
barn.

The nuclear reaction cross-section represents the total probability that
a compound nucleus will be formed and that it will decompose in a
particular channel. A nuclear reaction will not occur except by
tunneling effects if the minimum energy is below that needed to overcome
the Coulomb barrier and a negative Q of the reaction. Particles with
energies below this barrier have a very low probability of reacting. The
energy required to induce a nuclear reaction increases as the Z of the
target material increases. For many low Z materials it is possible to
use a low energy accelerator, but for high Z materials, it is necessary
to increase the particle energy (Deconninick 1978).

**1.4 Calculation of Radioisotope Yield**

The rate of radionuclide production is dependent on a number of factors
including the magnitude of the reaction cross-section as a function of
energy, the incident particle energy, the thickness of the target in
nuclei per cm^2^ which will determine the exit particle energy, and the
flux (related to beam current) of incoming particles. The rate of
production is given by:

$- \frac{\text{dn}}{\text{dt}} = R = \text{nI}(1 - e^{- \lambda t})\int_{E_{s}}^{E_{0}}{\frac{\sigma(E)}{\frac{\text{dE}}{\text{dx}}}\text{dE}}$

where:

> **R** is the number of nuclei formed per second
>
> **n** is the target thickness in nuclei per cm^2^
>
> **I** is the incident particle flux per second and is related to beam
> current
>
> **λ** is the decay constant and is equal to ln2/t~1/2~
>
> **t** is the irradiation time in seconds
>
> **σ** is the reaction cross-section, or probability of interaction,
> expressed in cm^2^ and is a function of energy (see FIGURE3)
>
> **E** is the energy of the incident particles, and
>
> **x** is the distance traveled by the particle
>
> $\int_{E_{s}}^{E_{0}}{}$is the integral from the initial to final
> energy of the incident particle along its path

This relationship can be approximated where the number of reactions
occurring in one second is given by the relation (Deconninick 1978):

dn = N~A~ I~0~ ds ~ab~

where:

> **dn** is the number of reaction occurring in one second
>
> **I~0~** is the number of particles incident on the target in one
> second
>
> **N~A~** is the number of target nuclei per gram
>
> **ds** is the thickness of the material in grams per cm^2^
>
> **~ab~** is the parameter called the cross-section expressed in units
> of cm^2^

In practical applications, the thickness ds of the material can be
represented by a slab of thickness s thin enough that the cross-section
can be considered as constant. N~A~ Δs is then the number of target
atoms in a 1 cm^2^ area of thickness s. If the target material is a
compound rather than a pure element, then the number of nuclei per unit
area is given by the expression:

$$N_{A} = \frac{F_{A}C\Im}{A_{A}}$$

where:

> **N~A~** is the number of target nuclei per gram
>
> **F~A~** is the fractional isotopic abundance
>
> **C** is the concentration in weight
>
> **** is Avogadro's Number
>
> **A~A~** is the atomic mass number of nucleus A

The above equations lead to one of the basic facts of life in
radioisotope production. It is not always possible to eliminate the
radionulidic impurities even with the highest isotopic enrichment and
the most precise energy selection. An example of this is given below in
Figure 5 for the production of Iodine-123 with a minimum of I-124
impurity (Guillaume et al. 1975; Lambrecht and Wolf 1973; Clem and
Lambrecht 1991; Qaim and Stöcklin 1983).

![](media/image6.png){width="4.250694444444444in" height="3.625in"}

*Figure 5 Nuclear reaction cross-sections for production of ^123^I and
^124^I from ^124^T*

As can be seen from the above graph, it is not possible to eliminate the
*^124^I* impurity from the *^123^I* because the *^124^I* is being made
at the same energy. All that can be done is to minimize the *^124^I*
impurity by choosing an energy where the production of *^124^I* is near
a minimum. In this case a proton energy higher than about 20 MeV will
give a minimum of *^124^I* impurity.

**1.4.1 Saturation Factor**

The rate of production is, of course, affected by the fact that the
resulting nuclide is radioactive and is thus undergoing radioactive
decay. For short-lived nuclides, the competition between formation and
decay will come to equilibrium at sufficiently long bombardment times.
This point is called saturation, meaning that no matter how much longer
the irradiation occurs the production rate is equal to the rate of decay
and no more product will be formed.

The rate of formation in this case is given by:

$$R = \frac{N\lambda}{(1 - e^{- \lambda t})}$$

Where;

> **R** is the rate of formation of nuclei (dN/dt)

**N** is the number target nuclei present at the end

**λ** is the decay constant

**t** is the time of bombardment

The term in the denominator is often referred to as the saturation
factor and accounts for the competition of the production of nuclei due
to the particle reaction and the radioactive decay of the nuclei which
have been produced.

**1.5 Nomenclature**

The nomenclature for nuclear reactions as it is usually used throughout
this document needs to be defined. If a carbon-13 nucleus is irradiated
with a proton beam to produce a nucleus of nitrogen-13 with a neutron
emitted from the compound nucleus, this reaction will be written as
^13^C(p,n)^13^N. In a similar manner, if a neon-20 nucleus is bombarded
with a deuteron beam to produce a nucleus of fluorine-18 with the
concomitant emission of an alpha particle, this reaction sequence will
be abbreviated as ^20^Ne(d,α)^18^F.

**1.6 Cyclotron Targetry**

There are literally several hundreds of radioisotopes which can be
produced with charged particle accelerators. The cyclotron is the most
frequent choice, but the linac and other accelerators may become more
common with the development of smaller, more reliable machines. The goal
of cyclotron targetry is to get the target material into the beam, keep
it there during the irradiation and to remove the product radionuclide
from the target material quickly and efficiently. The specific design of
the target is what allows one to achieve this goal. The efficiency of
radionuclide production will depend to a great extent on having a good
design for the cyclotron target. For production of radionuclides, the
target material may be either gas, liquid or solid. Targets are,
consequently, designed to accommodate the material being irradiated. The
design of the target will also depend on whether the target is placed
inside (internal) or outside (external) to the cyclotron. A great deal
of useful information about targets and target chemistry can be found in
a book by Clark and Buckingham (Clark 1975) and in the proceedings of
the International Workshop on Targets and Target Chemistry. These
proceedings are available on line through the courtesy of TRIUMF at
<http://www.triumf.ca/wttc/proceedings.html>. These are very valuable
resources for all who are concerned with the production of
radioisotopes.

[Targets in General]{.underline}

In any target there must be an area for containment of the material to
be irradiated in the beam, a water cooling jacket, helium cooling flow
in the front foil, and a vacuum isolation foil leading to the cyclotron
vacuum chamber or beam line.

[Solid Targets]{.underline}

Because the density of solids is typically higher than that of liquids
or gases, the path length of the beam is shorter, and the target
somewhat smaller. The solid can be in the form of a foil or a powder. If
the solid is a good heat conductor, then the beam can be perpendicular
to the solid. A typical solid target for conductive powders is shown in
Figure 6.

![](media/image7.wmf){width="4.624305555555556in"
height="2.188888888888889in"}

*Figure 6 Typical solid powder target for use with low beam current or
with thermally conductive solids*

A picture of a typical external solid target is shown in Figure 7. The
powder is held in the small cavity. The cover foil is shown next to the
cavity.

![](media/image8.wmf){width="3.9334645669291337in"
height="2.7417322834645668in"}

*Figure 7 Picture of solid powder target used at Brookhaven National
Laboratory*

If the solid is not a good thermal conductor or when very high beam
currents are used, it is typical to form the solid on an inclined plane.

[Liquid Targets]{.underline}

In the case of liquids, the target has similar dimensions to the solid
target since the target material occupies a specific volume unless the
liquid volatilizes. The difference is that the liquid is typically added
and removed from the target while it is in place on the cyclotron. A
typical liquid target for the production of fluorine-18 from oxygen-18
water is shown in Figures 8a and 8b.

![](media/image9.wmf){width="4.376388888888889in"
height="2.2263888888888888in"}

*Figure 8a Schematic diagram of a typical external target used for
irradiating liquids. This particular target is constructed of silver and
is used to produce fluorine-18 from oxygen-18 enriched water.*

![](media/image10.png){width="2.28125in" height="1.7916666666666667in"}

*Figure 8b Picture of the oxygen-18 water target from the Pulsar linear
accelerator from AccSys. Note the grid covering the front foil to enable
the front foil to withstand higher pressures and beam currents.*

[Gas Targets]{.underline}

Gas targets are widely used and are usually some type of cylinder to
hold the gas under pressure with a thin beam entry foil usually referred
to as a window. The principal constraint on the gas target is to remove
the heat from the gas since gases are not very good heat conductors and
the targets must be quite large in comparison to solid or liquid targets
in order to hold the necessary amount of material. A schematic diagram
of a typical gas target is shown in Figure 7.

![](media/image11.wmf){width="4.376388888888889in"
height="3.542361111111111in"}

*Figure 9 Schematic diagram of a typical gas target showing the chamber
for the gas cut at an angle to account for the multiple scattering.*

![](media/image12.jpeg){width="4.25in" height="3.191732283464567in"}

*Figure 10 Picture of a gas target used for the production of ^123^I
from ^124^Xe. The coldfinger on the bottom of the target allows the gas
to be transferred into the target more efficiently. The large volume in
the front of the target was used to capture the xenon if the front foil
ruptured.*

The design and construction of cyclotron targets is a multidisplinary
field. Engineering, Physics and Chemistry each plays an important part
in the way cyclotron targets are constructed. In almost all cases, it is
necessary to make some compromises in the design, since very often the
best solution of a problem with one aspect of the target will introduce
an unacceptable constraint on another aspect of the target.

**1.7 Generators**

Radionuclide generator systems consist of a parent radionuclide, usually
a relatively long-lived nuclide which decays to a daughter nuclide,
itself radioactive but with a shorter half-life. The system requires an
efficient separation technique of the daughter nuclide from the parent.
Conventionally, the parent is adsorbed onto a solid support and decays
by particle emission. A solvent in which the daughter complex is
soluble, is employed to elute (i.e. separate) the desired radionuclide.
The generator systems currently finding application to PET studies
remain as primarily research sources for pharmaceutical development.

For those research centers and clinical facilities without the luxury of
a cyclotron, several generator systems for production of positron
emitting radionuclides have been proposed. Their production routes have
been reviewed (Lambrecht 1983; Finn et al. 1983; Knapp and Butler 1984;
Guillaume and Brihaye 1986; Qaim 1987; Welch and McCarthy 2000). Of the
systems proposed, copper- 62, gallium-68, iodine-122 and rubidium-82
radionuclides continue to find applications. The decay characteristics
of these generator systems are included in Table 1. A great deal of
effort has been expended upon the production and construction of these
generator systems, including investigations into solid support materials
and elution characteristics.

Gallium-68 finds significant application in assessment of blood-brain
barrier integrity as well as for tumor localization. It is widely used
as a source for the attenuation correction of most positron emission
tomographs. The parent germanium-68 is long-lived (t~1/2~ = 275 d) and
is generally not attempted on medium energy accelerators due to the low
production yields (Pao et al. 1981; Loc'h et al. 1982). The primary
source for the parent radionuclide is the spallation processes available
at large energy accelerators where parasitic position and operation are
available (Grant et al. 1982; Robertson et al. 1982). The use of this
generator is somewhat limited by the chemistry. It is an ideal PET
radiotracer due to its non-halogenated and non-volatile chemical
properties and its 68 minutes half-life, which permits chemical
manipulation for the production of many PET radiopharmaceuticals.
However, little progress towards using Ga-68 in the clinic has been made
due to the long synthesis times required by manual production of
Ga-68-labeled PET imaging agents.Other tracers and diagnostic agents can
be developed which could prove very useful in the clinic. This is an
area where the Agency could play an important role in supporting studies
on gallium organic chemistry and on new labeling techniques or linking
agents.

Rubidium-82 is a myocardial blood flow agent and has found clinical
application. The application of rubidium-82 chloride in diagnosis of
ischemic heart disease and location of myocardial infarcts is an active
area of application for this generator system (Gould 1988). The short
half-life (t~1/2~ =1.27 m) of rubidium-82 and its similarity to
potassium in biologic transport and distribution suggest that this
generator-produced radionuclide might find a clinical role in
thrombolytic therapy monitoring.

**Table 1. Examples of Generators Yielding Positron Emitting Daughter
Radionuclides of Clinical Interest**

+-----------+-------------+------------+--------------+--------------+
| *         | > **Decay   | > **       | > **Decay    | > **Char     |
| *Parent** | > Mode      | Daughter** | > Mode**     | acteristic** |
|           | > (%)**     | >          | >            | >            |
| **(Half   |             | > **(Ha    | > **(%)**    | > **γ        |
| Life)**   |             | lf-Life)** |              | -Energy(%)** |
+-----------+-------------+------------+--------------+--------------+
| Sr-82     | > EC(100)   | > Rb-82    | > B^         | 0.78 MeV (9) |
|           |             | >          | +^(96),EC(4) |              |
| (25 days) |             | > (76      |              |              |
|           |             | > seconds) |              |              |
+-----------+-------------+------------+--------------+--------------+
| Ge-68     | > EC(100)   | > Ga-68    | > B^+        | 1.078(3.5)   |
|           |             | >          | ^(88),EC(12) |              |
| (278      |             | > (68      |              |              |
| days)     |             | > minutes) |              |              |
+-----------+-------------+------------+--------------+--------------+
| Zn-62     | > B+        | > Cu-62    | > B^+^(98)   | 1.17(0.5)    |
|           | (18),EC(82) | >          |              |              |
| (9.13     |             | > (9.8     |              |              |
| hours)    |             | > minutes) |              |              |
+-----------+-------------+------------+--------------+--------------+
| Xe-122    | > EC(100)   | > I-122    | > B^+        | 0.56(18.4)   |
|           |             | >          | ^(77),EC(23) |              |
| (20.1     |             | > (3.6     |              |              |
| ours)     |             | > minutes) |              |              |
+-----------+-------------+------------+--------------+--------------+

**1.7.1 Radionuclide generator equations**

A synopsis of the equations to allow the calculation of the maximal
concentrations of daughter nuclide from a particular generator, or the
determination of the appropriate time to elute a generator is given
through the following expressions (Finn et al. 1983).

Considering a simple radionuclide generator system of parent-daughter in
which the half-life of the parent is longer than that of the daughter,
the pair will eventually enter a state of transient equilibrium. This
can be represented schematically as:

A→B→C

where A is the parent radionuclide which decays to the radioactive
daughter B which in turn decays to the daughter nuclide C.

The ratio of decay of each radionuclide is described by the equation:

$\frac{\text{dN}}{\text{dt}} = \lambda N$ or $N = N_{o}e^{- \lambda t}$

Where N is the number of radioactive atoms at a specific time t and λ is
the decay constant for the radionuclide and is equivalent to
(ln(2)/t~1/2~).

Considering a generator system, the parent is generally adsorbed onto a
solid support and serves as the sole source for the daughter
radionuclide production. However, the number of daughter atoms present
at any time t is described in a slightly more involved expression:

$\frac{\text{dN}_{B}}{\text{dt}} = \lambda_{A}N_{A} - \lambda_{B}N_{B}$

Since the daughter is decaying as well as being produced. The net rate
of change on N~B~ with time is therefore indicated by the decay of A to
B minus the decay of B to C. Substitution of the integral of the
expression for A yields the net rate of change for B as:

$\frac{\text{dN}_{B}}{\text{dt}} = \lambda_{A}N_{A}^{0}e^{- \lambda_{A}t} - \lambda_{B}N_{B}$

Integrating this equation to calculate the number of atoms of B at ant
time t gives:

$N_{B} = \left( \frac{\lambda_{A}}{(\lambda_{B} - \lambda_{A})} \right)N_{A}^{0}(e^{- \lambda_{A}t} - e^{- \lambda_{B}t}) + N_{B}^{0}e^{- \lambda_{B}t}$

The first term on the right side of the equation represents the growth
of the daughter nuclide B from the parent A decay and the loss of B
through decay. The second term represents the decay of B atoms but since
the parent A is generally considered a pure parent radionuclide upon
generator manufacture, this term is zero. The equation can be rewritten
in terms o f activities and results in:

$A_{B} = \left( \frac{\lambda_{A}}{(\lambda_{B} - \lambda_{A})} \right)A_{A}^{0}(e^{- \lambda_{A}t} - e^{- \lambda_{B}t})$

Consideration of the general conditions for parent/daughter pairs, the
cases are of transient equilibrium in which the parent half-life is
greater than the daughter, or secular equilibrium in which the parent
half-life is much greater than the daughter. Naturally if the decay
should involve branching ratios, the equation must be appropriately
modified.

Further, in the case of the PET generators, it is often useful to
calculate the time when the daughter activity is at the maximum value,
t~max~. Differentiation of the equation with respect to time gives the
result

$t_{\text{max}} = \text{ln}\left( \frac{\left( \frac{\lambda_{A}}{\lambda_{B}} \right)}{(\lambda_{B} - \lambda_{A})} \right)$

The role of generators for the future of clinical PET remains uncertain
at this time. The initial supposition that the generators hold potential
for PET imaging at sites without a cyclotron or accelerator are being
re-evaluated due to the costs associated with the procurement and
scheduled availability of the parent radionuclide. Further, any
supplementary equipment, such as that of the infusion system required
for the strontium/rubidium generator, may result in low demand or choice
of alternative radionuclides (Welch and McCarthy 2000).

8.  **References**

Clark, JC and Buckingham, PD, *Short-Lived Radioactive Gases for
Clinical Use.* Butterworths, London, 1975.

Clem RG, and Lambrecht RM (1991) Enriched ^124^Te targets for production
of ^123^I and ^124^I. *Nuclear Instruments and Methods* **A303**,
115-118.

Deconninck, G. (1978) Introduction to Radioanalytical Physics, Nuclear
Methods Monographs No.1 Elsevier Scientific Publishing Co. Amsterdam.

Evans, Robely D., The Atomic Nucleus. McGraw-Hill Book Co. New York
1955.

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