Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics

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18 Nuclear Medicine Physics

the

C(p,n)

N reaction. However, in practice, the higher excitation func-

tion of this possible alternative reaction is not sufﬁcient, in itself, to make

it the main factor when compared with the price difference between H

and highly enriched carbon-13 (

C, only 1.10% in natural carbon) targets.

Ease of implementation and cooling during the irradiation of liquid targets

are added advantages in the use of a H

O target.

Once a nuclear reaction has been selected for production, the next step

is to choose the incident energy of projectiles on the target and the target

thickness (which determines the value to which the incoming energy will

be degraded). The resulting range of projectile energies across the target is

of extreme importance in achieving the best possible compromise in terms of

maximizing productionefﬁciencyin the desired radioisotope and minimizing

the percentage of radioisotopic impurities in the ﬁnal product.

The fact that, in any given nuclear reaction, the same entrance channel

(characterized by a given projectile and target) can lead to different exit chan-

nels, depending on the available energy, is an important physical justiﬁcation

for the need for an appropriate choice of energy range. The number of open

channels is generally low for low projectile energy values, but the number

of nuclear reactions increases as the projectile energy increases, to compete

with the reaction that generates the radioisotope and create impurities in the

ﬁnal product. Nonisotopic impurities can be removed chemically, whereas

the percentage of isotopic impurities can only be maintained at acceptable

levels by studying the range of energies that should be used. Any failure

to maintain these levels makes a production protocol based on a particular

nuclear reaction unviable, regardless of its physical yield. In this type of study,

the knowledge of the excitation functions of the nuclear reactions involved is

fundamental.

The process of choosing the parameters for the production of selenium-73

(

Se), a sulphur analog with potential use in PET [4,5], is illustrative of the

need to choose the incident energy and target thickness with a view to miti-

gating isotopic impurities and of the role played by the excitation functions

in this choice.

Se, whose half-life is 7.15 h, may be feasibly produced by

the reaction

As(p, 3n)

Se. However, the irradiation of Arsenic-75 (

As)

with protons leads to the formation of other selenium isotopes, such as

Selenium-75 (

Se; T

1/2

= 119.8 days) and Selenium-72 (

Se; T

1/2

= 8.4 days),

which can, due to their half-life, become signiﬁcant isotopic impurities. The

best compromise between maximization of

Se production and reduction of

these impurities can be inferred from a comparative analysis of the excita-

tion function for the

As(p,3n)

Se reaction and the excitation functions for

the reactions leading to the formation of isotopic impurities, namely

As(p,

Seand

As(p,4n)

Se.Since these two reactions showhigher cross-section

values for energies below 20 MeV and above 40 MeV and the reaction that

leadsto

Seshows anexcitation functionwith amaximum levelof 30–50MeV,

it can be concluded [5] that a proton beam of 40 MeV should be used and an

appropriate

As target thickness as well to degrade the beam to 30 MeV.

Cyclotron and Radionuclide Production 19

Apart from impurities due to a diversity of exit channels resulting from a

single entrance channel in a nuclear reaction, other impurities are common in

the ﬁnal product: the products of nuclear reactions induced in impurities in

the target material. Again, special importance should be given to isotopic

impurities (in both senses, since in most cases, isotopic impurities in the

target will lead to isotopic impurities in the ﬁnal product), which cannot

be chemically separated. To avoid this type of impurity, the logical solu-

tion is the use of isotopically enriched target materials. However, it is not

always possible (or affordable) to obtain 100% enrichment. In cases where

the target material is not a pure isotope, the choice of incoming and out-

going target beam energy, based on knowledge of the excitation functions,

can once again prove a major contribution toward minimizing levels of iso-

topic impurities in the ﬁnal product. A good example is the optimization of

the production parameters of carbon-10 (

C), an isotope used (as

) as

a tracer for regional cerebral blood ﬂow in PET, through the irradiation of

boron targets [2,6]. Natural boron consists of two stable isotopes,

B and

B, with 19.9% and 80.1% natural abundance, respectively. The irradiation

B with a proton beam of (at least) 30 MeV leads to the production of

through the

B(p, n)

C reaction whose excitation function is illustrated in

Figure 2.4. The irradiation of

B with the same beam would lead to the for-

mation of

C (unstable with β

decay and a 20.39 min half-life) through the

B(p,n)

C reaction [7–9]. In the production of

C for the above mentioned

PET application, the simultaneous production of

C is highly undesirable,

given the comparatively longer half-life of this isotope (with inevitable con-

sequences for the dose administered) and the impossibility of its chemical

separation from

C. The use of 100%

B targets would be the solution to the

problem, thus eliminating

C production. However, such targets are impos-

sible to obtain, at least commercially. In a real-life situation involving the

use of highly enriched targets of

B, but with some percentage of

B, the

production of

C can be mitigated by careful choice of the irradiation param-

eters, including the proton beam energy and target thickness. Indeed, with

an increase in beam energy (at least in the range where both the excitation

functions are known) followed by proper (but not complete) target beam

energy degradation, the relative percentage of

C can be reduced by more

than 50%, while still maintaining a reasonable yieldin the absolute production

C [2].

2.1.5 Ion Kinetic Energy Degradation in Matter Interaction

In the previous sections, the possibility of ions accelerated in a cyclotron

interacting with a target nucleus to produce nuclear reactions (which, in a

broad sense, includes the phenomena of elastic and inelastic scattering) was

demonstrated. However, these projectiles may also interact with the electrons

in the electronic cloud of the target material or, in general terms, the matter.

Coulomb interaction then plays such a key role that the degradation of the

20 Nuclear Medicine Physics

beam of particle energy from interaction with the nucleus becomes negligible

in comparison with the result of interaction with the electronic cloud. Thus,

given the use of ion beams, especially hydrogen ion beams, in the produc-

tion of radionuclides in PET-dedicated cyclotrons, a knowledge of how beam

energy degrades in crossing the target becomes very relevant.

The degradation of the energy of a beam of ions is almost entirely due to

inelastic collisions with the bound electrons of the atoms forming the material

crossed by the beam. The atoms of the target material will be ionized or

excited with the energy that comes from the kinetic energy of the beam, which

consequently, decreases. The decrease in beam energy during this process is

almost continuous as it crosses the target material. The energy reduction rate

decreases when it becomes small enough for the beam particles to capture

electrons, but it continues until the energy of the beam is equivalent to the

thermal energy of the atoms in the medium.

2.1.6 Stopping Power: Bethe’s Formula

For ion velocities that are much greater than the typical velocities of electrons

in the electronic cloud of atoms in the medium traversed by a projectile beam,

it is possible to calculate the average energy degradation of the beam along

its path in this medium. This degradation reﬂects the stopping power of this

matter, for this beam. Under these conditions (up to a tenth of MeV for protons

and one MeV for α particles), in which it is unlikely that the particle beam will

capture electrons, it has been demonstrated that for nonrelativistic velocities

[10,11]:

−

4πz

NZ log





. (2.6)

The left side of the equation is called the linear stopping power of the mate-

rial, corresponding to the degradation of the average beam-energy per unit

length of its path through the medium. In accordance with the international

system, it should be expressed in joules per meter.

In nuclear physics, it is common to use MeV as the energy unit and CGS

units, multiplied by the density of the medium material, for distance. In this

context, the stopping power is expressed in MeV g

−1

. It, therefore, cor-

responds to the linear stopping power divided by the density of the material

(and thus only makes sense when the material is speciﬁed) and should be

termed massic stopping power.

On the right-hand side of the equation, m

and q

are the mass and

charge of the electron, z and v are the atomic number and velocity of

the particles constituting the beam, and Z and N are the atomic number

and number of atoms per volume of the material crossed by the beam,

respectively. I is the average excitation potential of an atom of the material

crossed by the beam. It represents the average energy, extended to all con-

nected electrons, that can be transferred to an electron during the process

Cyclotron and Radionuclide Production 21

of excitation, including ionization. It is substantially larger than the ioniza-

tion potential of the atom and should not be confused with it. The value

for most elements can be found in the literature that has emerged from

experimental measurements. The precise theoretical calculation is complex,

but some semiempirical expressions produce a good approximation, for

example, [11]:

= 9.1(1 +1.9Z

−2/3

) [eV]. (2.7)

The expression for the linear stopping power presented above can be

demonstrated and is, therefore, valid [12–14] in quantum mechanics or clas-

sical mechanics. For relativistic velocities of the beam, it should be corrected

[15,16]:

−

4πz



log





−log

(1 −β

) −β



. (2.8)

This expression, in which β is the ratio between the speed of the beam

projectiles and the speed of light in the vacuum, is known as the Bethe formula.

The relativistic correction terms express the effect of the increased rate of

beam energy degradation with increasing beam particle speed for relativistic

beam velocities. This effect [17] comes from the increase in the maximum

energy that can be transferred to an electron (1st parcel) and from the Lorentz

contraction of the particle beam Coulombian ﬁeld, which allows energy to be

transferred to electrons located further away (2nd parcel).

In the upper limit of relativistic velocities, when the beam particle energy,

beam

, veriﬁes the condition

beam

≥

proj

(2.9)

the spin of beam particles with mass m

proj

becomes important [18,19], and

even the relativistic expression for the linear stopping power loses validity.

The energy degradation rate becomes increasingly dependent on the spin of

the particle, and considerably higher energy degradation can be observed for

spin-1 particle beams than for spin-

(of protons) or spin-0 particle beams. In

the same energy range, the density and dielectric properties of the medium

should be considered. The electric ﬁeld of the projectile felt in an atom that

is off course decreases due to the polarization of the atoms in the medium,

causing a decreasein the transfer of energy[20–24], an effectthat becomes pro-

gressively more important as the effect of the interaction distance increases,

due to the Lorentz contraction.

In the range of energies used in the production of radionuclides for biomed-

ical use, if all these ultra-relativistic effects are disregarded, the relativistic

expression for the linear stopping power or Bethe formula will remain valid.

22 Nuclear Medicine Physics

In fact, for a proton beam, for example, the ultra-relativistic effects only

become signiﬁcant for energies in the order of 10

MeV.

In the lower limit, as previously stated, the expressions for the stopping

power lose their validity for projectile velocities low enough for signiﬁcant

electron capture and the consequent neutralization of the beam. The probabil-

ity of electron capture by completely ionized beam nuclei depends strongly

on the ratio between their speeds. The physical principles involved were dis-

cussed in relation to a proton beam by Niels Bohr [25], who showed that the

process can be completely different depending on the medium crossed by the

beam and also calculated the probability of its occurrencein media with a high

atomic number. He found, in this case, a proportionality to the sixth power

of the ratio between the speed of the electron in the medium (considered the

ﬁrst Bohr orbit) and the speed of the proton beam, as opposed to the probabil-

ity corresponding to media with low atomic numbers, which is proportional

to the 12th power of the ratio of the velocities [26]. As a consequence of the

neutralization process, the effective charge of the beam is reduced, leading to

a reduction in the energy degradation rate over distance. Therefore, both the

Bethe formula and the classic expression for linear stopping power overes-

timate the rate of spatial energy degradation in the low-energy range of the

incident beam.

Given the assumptions contained in the deﬁnition of linear stopping power,

it seems logical that the linear stopping power of a compound is given by

the sum of the linear stopping power of each of the individual atoms that

constitute it. This principle, which reﬂects Bragg’s rule, simpliﬁes to a large

extent the calculation of the linear stopping power of compounds. However,

the results are only an approximation (although usually very good). When

a compound is formed, there is a change in the electron wave functions of

the more peripheral layers. Therefore, the excitation potential will change,

generally corresponding to a greater binding of the electrons involved. An

increase in the value of the electron excitation potential (which is expected to

be in the same order of magnitude as the energies corresponding to chemical

bonds, i.e., a few units of electron volts) will, of course, resultin a slight change

in the value of the mean excitation potential for all electrons, I. This can be

up to 2%, corresponding to a linear decrease in stopping power in the order

of tenths of a percentage. The effect is less evident in a compound made of

heavy elements, where the percentage of electrons in the element involved in

chemical connections is smaller.

2.1.7 Range

The distance that a beam of a given energy has to travel through a given

material, if this energy is to degrade completely, is called range. The range

R of a beam of given energy in this material is, therefore, the integral of the

inverse of the linear stopping power for the beam in this material, where the

Cyclotron and Radionuclide Production 23

integration limits are the value of the energy of the beam when entering the

material, E

, and zero:

R =



−

dE. (2.10)

2.1.8 Straggling: Statistical Fluctuation in Energy Degradation

Bethe’s formula calculates only an average rate of linear energy degradation.

In fact, the energy of a charged projectile does not continuously decrease as

the formula may suggest; instead, there are a large number of small but ﬁnite

decreases, corresponding to interactions with the electrons in the medium.

There are two implications of the consequent ﬂuctuation around the average

value that occurs for each projectile-medium pair. First, for a given trajectory

of the projectile in the medium, the degradation of energy ﬂuctuates around a

mean value. In addition, the path of the beam that corresponds to the energy

degradation also oscillates around a statistical average. The phenomenon that

is manifested by these statistical ﬂuctuations is called straggling.

It is possible to theoretically study the variance in the amount of energy

degradation in projectiles with the same initial energy crossing a thickness x

of a given medium,











−ε



, (2.11)

where ε

denotes the average energy degradation and ε

 denotes the

expected value for the squared value of energy degradation in a single

event (note that the expected energy value for a single event ε will cor-

respond to the average energy ε

). Livingston and Bethe [27], using quantum

assumptions, arrived at the expression:





= 4πq







log



. (2.12)

The notation is consistent with the one used in Bethe’s formula. Z



denotes

the number of electrons in atoms that are not present in the more internal

layers for which the average energy of excitation is greater than 2m

. The

sum is performed for all layers not excluded by this criterion, weighted by

the constant k

, with Z

being the number of electrons in the nth layer, whose

average excitation energy is I

For high energies (compared with 2m

), the sum can be ignored and Z



can be replaced by the total number of electrons in the atom, to arrive at the

expression deduced by Niels Bohr [17] using only classical assumptions [28]:





= 4πq

NZ. (2.13)

24 Nuclear Medicine Physics

Consider, moreover, that a beam of projectiles from the same initial energy

experiences the energy degradation when crossing a given medium. The vari-

ance in the value of the corresponding distance, using a notation similar to

the one above, is,











−χ



. (2.14)

Once again, the expected value of the distance travelled in a particular

instance χ will correspond to the average distance χ

. Thus, if dx and dE

are an inﬁnitesimal distance and energy degradation, respectively, and if dE

is the mean energy degradation corresponding to the distance dx:













. (2.15)

If this expression is integrated, the value of σ

for the degradation of the

ﬁnite energy E can be obtained:















−3

dE. (2.16)

Using the Livingston and Bethe expression and integrating from the initial

beam energy E

to total energy degradation, the variance in range value, σ

is obtained:



4πq







−3



1 +





log



dE. (2.17)

This expression can also be simpliﬁed for high energies in the incident beam

leading to



4πq





−3

dE. (2.18)

The study of linear stopping power based on Bethe’s formula only considers

interactions with the electrons in the medium, resulting in an energy transfer

that is small in comparison with the total energy. The energy degradation

experienced by each particle of a monoenergetic beam crossing the same dis-

tance in the medium can be seen as an independent statistical phenomenon.

Considering only interactions with the electronic cloud, it is expected that

the distribution of the energy degradation value in each of the large number

of beam projectiles shows a Gaussian distribution around the mean value.

Cyclotron and Radionuclide Production 25

However, the energy degradation in each nuclear interaction, although so

comparatively rare that its contribution to the stopping power of the medium

can be ignored, is signiﬁcantly higher than that which is found in every

electronic interaction, which has a considerable inﬂuence on the statistical

ﬂuctuations under discussion. A perhaps more important consequence is that

the distribution of the energy degradation value is not Gaussian around the

average, but it shows a “tail” toward the higher degradation values. This is

particularly true for heavy projectiles (e.g., ﬁssion products) at distances close

to the range value [25].

2.1.9 Energy Degradation versus Ionization

An ion passing through a given medium may interact with an electron, trans-

ferring enough energy to overcome the ionization potential of the atom to

which it is connected. In this case, it causes an ionization known as primary

ionization, as it is the direct result of projectile–electron interaction. In most

cases, the electron is ejected from the atom with relatively low kinetic energy

in comparison with the ionization potential. However, some primary ion-

izations result in electrons ejected with energies greater than or equal to the

ionization potential of the medium. These electrons, called delta rays, can lead

to the secondary ionization of most atoms in the medium. Total ionization is the

sum of primary ionization, in which the number of ions produced equals the

number of ejected electrons, and secondary ionization, in which the electron

resulting from primary ionization ionizes one or more atoms.

One remarkable experimentally demonstrated property of the ionization

caused by the energy deposition of charged projectiles is that the energy

degradation per ion formed, w, is practically independent of the energy,

charge, and mass of the particle that gives rise to ionization, whether primary

or secondary. Considering the total energy available for ionization, regardless

of whether the carrier particle is a beam particle or a secondary electron, this

property can be qualitatively explained. On average, each excitation leads

to a W

exc

reduction of available energy. Similarly, each ionization resulting

in a low kinetic energy electron (unable to produce secondary ionizations)

reduces, on average, the energy available by a W

ion1

amount. If an electron

with enough kinetic energy to induce secondary ionizations results from this

ionization, the accountable reduction in available energy will be equal to the

ionization potential of the medium, V

, as the remaining energy will be avail-

able for secondary ionizations (in which the respective degradation will be

accounted for). With σ

exc

, σ

ion1

, and σ

, the cross-section values for each of

these processes, the average amount of energy required to form an ion will

be given by

w =

exc

+σ

ion1

+σ

ion1

+σ

. (2.19)

26 Nuclear Medicine Physics

The approximate constant value of w is then explained by the low (com-

pared to the other cross sections) value of σ

and the fact that the energies

exc

and W

ion1

and the ratio between the σ

exc

and σ

ion1

cross-section values

are virtually independent of the ionizing particle energy [29].

The almost constant ion beam energy degradation value per ion formed

in the medium crossed and the consequent analogy between the distribution

of total ionization along the path of the projectile and the energy degrada-

tion rate have been widely used [30,31] to determine the energy of charged

particles, using the number of ions created in a known medium as a ﬁrst

approximation.

2.1.10 Thick Target Yield

The amount of radionuclide that is expected to be produced by a nuclear

reaction in a target of a given thickness can be calculated by integrating the

reaction excitation function over the range of beam energies on target. The

limited practical interest in the absolute quantity of radionuclides produced

without taking into account the decay during irradiation paved the way for

a generalized measure of the efﬁciency of a nuclear reaction using the so-

called thick target yield, the radioisotope activity expected to be induced by

the irradiation of a thick target, per unit of beam current, in saturation.

Saturation condition is reached when there is a balance between the number

of nuclides being produced and decaying in constant rate production. Given

the large number of target nuclei and the relativelylow cross section of nuclear

reactions involved, the reduction in the number of target nuclei during a pro-

duction process (such as the one carried out in a PET-dedicated cyclotron) can

be considered negligible. The rate of production will be constant if the projec-

tile beam intensity is constant, which happens in most protocols for routine

production. If P represents the constant rate of production of a radionuclide

with a decay constant λ, the temporal variation of the number N of nuclides

in the target is given by

dN(t)

=−λN(t) +P. (2.20)

Considering the activity A is nil at the moment production begins, the

solution for this differential equation is

A(t) = λN(t) = P(1 −e

−λt

). (2.21)

As time t goes on, the activity and the rate of production tend, asymptoti-

cally, to equalize, as shown in Figure 2.5, where time is counted in units of the

half-life of the radionuclide in question. It should be observed that although,

strictly speaking, saturation is only achieved for an inﬁnite time, after a time

Cyclotron and Radionuclide Production 27

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t/T

A/P

01 2345678910

FIGURE 2.5

Evolution of the relationship between the activity and the rate of production, assumed constant,

of a radionuclide, in units of its half-life.

equal to about a dozen half-lives it does not make sense to extend irradia-

tion any longer (at this rate of production), because the number of nuclides

produced will have already practically compensated for those which have

decayed. Thus, in this condition, the activity of the radionuclides produced

at the end of irradiation does not depend on irradiation time.

The thick target yield Y can be calculated from

Y =

Mzq



inc





−1

σ(E) dE, (2.22)

is the Avogadro number, z the projectile atomic number, and q

is the

electron charge. M and H are the atomic mass and the target material isotopic

enrichment, respectively. The inverse of the linear stopping power multiplied

by the excitation function σ(E) is integrated along the beam energy values,

from entering (E

inc

) to emerging from the target (E

The convenient practical units of thick target yield used in daily pro-

duction of radionuclides in PET-dedicated cyclotrons are MBq/μA sat,

where sat is an explicit indication that the measurement refers to saturation

condition.

It is also common for the thick target yield to refer not to a range of energies

but only to an energy value. This value indicates the energy from the incident