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

Подождите немного. Документ загружается.

448 Nuclear Medicine Physics

MediumVacuum

FIGURE 8.4

Outline of the concepts of track length, p, and greatest penetration depth, t

. Notice that t

is not

necessarily the depth of the ﬁnal point B.

8.3.7 Mean Energy Spent per Ion Pair Produced, W

In radiation dosimetry, the methods that use the ionization of matter are often

employed due to the very large number of ion pairs formed in a gas by only

one charged particle track. It will be useful to introduce a quantity named

mean energyspent in a gas per ion pair produced.This quantity is represented

by the symbol, W, and is given by

W =

(8.33)

where E is the kinetic energy of the charged particle and N is the mean

number of ion pair produced when the particle energy is totally dissipated

in a gas. As previously mentioned, the electrons can lose energy through

breamsstrahlung. The ions produced by bremsstrahlung radiation should be

included in N. The unit of this quantity is energy, J or keV.

Fortunately, in electron or photon dosimetry, the W value for electrons is

constant for the majority of gases for low-energy values.Aparameter of major

importance in dosimetry is the mean energy spent in a gas per unit electric

charge produced, whose value is





= 33.97 JC

−1

. (8.34)

8.4 Radiation Dosimetry Quantities

The fundamental concepts of classical dosimetry are nowadays well estab-

lished by ICRU [10,11]. However, new developments have been accomplished

Dosimetry and Biological Effects of Radiation 449

in a ﬁeld termed microdosimetry or nanodosimetry. Radiation dosimetry, some-

times called classical dosimetry, is based in the fact that absorption of radiation

energy in a medium irradiated with indirect ionization radiation, such as

photons (γ- or x-ray), can be described as a two-step process. In a ﬁrst step,

the energy of uncharged particles is transferred to electrons as kinetic energy,

through interactions with atoms, resulting in the ejection of those electrons

from atoms. In the second step, the released electrons impart their energy to

matter along their path.

In classical dosimetry, the quantity kerma, K (kinetic energy released to

matter per unit mass), is used to describe the ﬁrst step. The second step is

described by the quantity absorbed dose.

Kerma, K, is the quotient of the energy transferred to charged particles per

unit of mass at a point of interest, including radiative-loss energy, so that units

are J kg

−1

, with the special name gray (Gy). Kerma rate is the time derivative

quotient of kerma per unit time interval, so that the units are gray per second

(Gy s

−1

A more fundamental quantity can be deﬁned, which is the ﬂuence. Fluence

is deﬁned as the quotient of dN by da, where dN is the number of particles

incident on a sphere of cross-sectional area da, thus

Φ =

. (8.35)

The units are (m

−2

). It can be shown that this deﬁnition can be generalized,

by expressing as the lengths of the particle trajectories in a given volume [12]:

Φ =

. (8.36)

where dl is the sum of the particle trajectory lengths in the volume dV.

To develop classical dosimetry from ﬂuence to absorbed dose, let us con-

sider a narrow beam of photons, with energy E

, impinging in a half-extended

medium. Let us consider a point of interest at depth L, within the medium.

The number of primary or un-collided photons reaching depth L is

(L) = N

(0)e

−μ(E

,Z)L

(8.37)

where N

(L) is the number of particles at depth L, and N

(0) is the number

of incident particles. The coefﬁcient of linear attenuation, μ(E

, Z), is deﬁned

for a given energy E

and material with atomic number Z.

Using the deﬁnition of Equation 8.35, the ﬂuence equation for the beam is,

in units (m

−2

(L) = Φ

(0)e

−μ(E

. (8.38)

The energy ﬂuence is

Ψ = ΦE (8.39)

450 Nuclear Medicine Physics

in units of (J m

−2

). Therefore,

(L) = Ψ

(0)e

−μ(E

. (8.40)

Studies of the deposition of radiation energy in matter [13] led to the

deﬁnition of two components of kerma: collision kerma, K

, and radiative

kerma, K

K = K

(8.41)

As mentioned above, kerma describes the energy transferred from

uncharged particles (photons) to charged particles (electrons).

An electron released in matter imparts its kinetic energy in a sinuous

path with many more collisions than photons with the same energy. These

interactions can be of two types:

1. Collision interactions. Coulomb-force interactions with atomic elec-

trons resulting in the local dissipation of energy as ionization or

excitation in or near the electron track.

2. Radiative interactions. Bremsstrahlung or braking radiation is emit-

ted, and the photons produced will carry their energy far away from

the charged-particle track.

Since we are only interested in the energy deposited at the point of interest

near the charged-particle track, the radiative component of kerma can be

neglected.

The value of kerma and collision kerma can be obtained using appropriate

coefﬁcients

∗

K = Ψ ×



(E, Z)



(8.42)

and

= Ψ ×



(E, Z)



(8.43)

The coefﬁcients are, for K, the mass energy (transfer coefﬁcient) and, for K

the mass energy (absorption coefﬁcient), and ρ is the density of the medium.

Both are functions of the energy of the photons and the atomic number of the

material.

Collision kerma is deﬁned as the quotient of the energytransferred to charged

particles per unit mass at the point of interest, excluding radiative loss energy.

We should note that the difference between kerma and collision kerma is

the bremsstrahlung. In fact, the relationship between mass energy (transfer

coefﬁcient) and mass energy (absorption coefﬁcient) is

(1 −g) (8.44)

∗

Tables of this coefﬁcients can be found in the site: http://physics.nist.gov.

Dosimetry and Biological Effects of Radiation 451

where g represents the average fraction of secondary-electron energy which

is lost in radiative interactions, that is, bremsstrahlung production and (for

positrons) in-ﬂight annihilation.

For the absorbed dose, it is not possible to write a mathematical expression

directly related to the energy ﬂuence of photons, such as Equations 8.42 and

8.43. Since the absorbed dose is the energy deposited by electrons liberated in

matter, this energy is not directly related to the ﬂuence of the incident radia-

tion, such as photons. The so-called kerma approximationis often used, which

means that it is assumed that electrons are in equilibrium or can be consid-

ered as if the energy of the photons transferred to electrons is deposited at the

point of origin of the electrons: in other words, electrons do not carry away

any energy; they deposit their energy at their site of origin. It can be shown for

the kerma approximation that the collision kerma equals the absorbed dose

[13]. In fact, absorbed dose is the quotient of the energy imparted to matter

per unit mass. The units of absorbed dose are the same as those for kerma

(Gy). It must be recognized that absorbed dose represents the energy per unit

mass that remains in the matter at the point of interest, which produces any

effects attributable to the radiation. Often, it is assumed that biological effects

are proportional to the absorbed dose; however, sometimes this assumption

is not applicable. Consequently, the absorbed dose is viewed as the most

important quantity in radiological physics. Later in this section, the concept

of absorbed dose will be further discussed.

Returning to the photon beam, if the kerma approximation is valid, then

the collision kerma equals the absorbed dose

cpe

Kc, (8.45)

in which cpe means charged particle equilibrium.

From Equation 8.40 and considering Equations 8.43 and 8.45, we can write,

(L) = D

(0)e

−μ(E

. (8.46)

The term D

(L) means the absorbed dose at a point of depth L for a narrow

beam of photons with energy E

. Actually, Equation 8.46 only refers to the

attenuation of un-collided photons, often called primary radiation and does

not consider the scattered component, often termed secondary radiation, which

can be very important from both dosimetric and image quality points of view.

In short, as the photons of the primary beam (un-collided photons) move

forward in depth, they suffer interactions and will be renamed secondary pho-

tons. Likewise, electrons eventually ejected from atoms are also considered

as secondary radiation. To quantify the transition from primary to secondary

photons, it is necessary to use an exponential, whose behavior depends on

the linear attenuation coefﬁcient.

For the purpose of dosimetry and, more speciﬁcally, for measurements in

radiation ﬁelds, it may be often convenient to use the kerma rate in free space,

called the free-in-air kerma rate, with units of (Gys

−1

452 Nuclear Medicine Physics

In the 1920s, the ﬁrst studies led to the description of a unit, called the

roentgen, which is the “electrostatic unit of charge in 1 cm

of air.” It must

be noted that at the time of the deﬁnition of the unit roentgen, there was no

special name yet for the related physical quantity. Over the years, this became

the exposure, which is now deﬁned as the total electrical charge of ions of

one sign, produced per unit mass. It is important to notice that exposure is

not a dose quantity but a descriptive parameter for the characterization of

radiation ﬁelds consisting of x-rays and γ- rays only. The deﬁnition of the

concept of exposure in air imposes a limitation on its application. It only

refers to x-ray and low-energy gamma-charged particles. It cannot be used

for the description of neutron ﬁelds.

The quantity exposure is the ionization equivalent to the collision kerma, in

air, for γ- and x-ray radiation. The exposure, X, can be calculated from ﬂuence

by the expression:

X = Ψ ×



(E, Z)





−1

air

(8.47)

where

w is the mean energy expended, in a gas, per ion pair formed and e is

the charge of the electron, so that (

w/e)

air

is the mean energy expended, in a

gas, per unit electric charge formed.

From Equations 8.43 and 8.47, we can determine the relationship between

collision kerma and exposure:

(

)

air

= X ×





air

(8.48)

Equation 8.48 means that the energy imparted to matter is obtained from

the charge produced by radiation times the energy expended per unit charge.

The constant (

w/e)

air

can be used as a constant, independent of the photon

energy above some keV. For air, it is usual to use the value

(

w/e)

air

= 33.97 JC

−1

(8.49)

so that

(

)

air

= X ×33.97 (8.50)

The kerma can be measured with high precision by the so-called free-

in-air chambers, which are speciﬁcally designed ionization chambers ﬁlled

with atmospheric air. However, in practical applications, such as radiation

protection, radiobiology, or radiation therapy, the absorbed dose in other

nongaseous materials such as tissue or water is often needed, although these

cannot be easily directly measured. Therefore, a method is required to con-

vert air kerma into absorbed dose in other media. This can be done by the

so-called kerma approximation in which the absorbed dose is assumed to be

Dosimetry and Biological Effects of Radiation 453

equal to the collision kerma. If the collision kerma in a medium A equals the

absorbed dose in the same medium, then, the kerma approximation can be

used; hence, for a given constant energy ﬂuence of photons, the absorbed

dose in a medium B is related to that of a medium A by

= D

(μ

/ρ)

(μ

/ρ)

(8.51)

Equation 8.51 reﬂects the special importance of the mass energy-absorption

coefﬁcients μ

/ρ in applied photon dosimetry [14].

An alternative way to deﬁne absorbed dose is [15]:

D = lim

m→0

(8.52)

The absorbed dose is a very well-deﬁned quantity. However, some restric-

tions apply to its use. Absorbed dose is, from Equation 8.52, a measure of

energy imparted, ε, to a medium divided by the mass, m, of the medium, ε/m

[8]. Let us consider Figure 8.5. “When m is large enough to cause signiﬁcant

attenuation of the primary radiation (e.g., photons), the ﬂuence of charged

particles in the mass element under consideration is not uniform. This causes

the ratio ε/m to increase as the size of the mass m is decreased. As m is fur-

ther reduced we will ﬁnd a region in which the charged particle ﬂuence is

sufﬁciently uniform that the ratio ε/m will be constant. It is in this region

that the ratio ε/m represents absorbed dose. Thus expectation value, ¯ε,of

the energy imparted over an appropriate size mass element must be used to

determine absorbed dose. If m is further decreased from the region of constant

ε/m, we will ﬁnd that the ratio will diverge… Hence, the determination of

Absorbed

dose

ε/m

Speciﬁc

energy

Speciﬁc

energy

Absorbed

dose

FIGURE 8.5

Left: Energy density as a function of the mass for which energy density is determined. The curve

showed is one of the possible curves obtained if we choose a mass element at a random site

inside the exposed medium. Right: Conjecture about the effect of decreasing dose. The zone with

a pattern is where absorbed dose has physical meaning.

454 Nuclear Medicine Physics

absorbed dose also requires that the mass element m be large enough so that

the energy deposition is caused by many particles and many interactions…

It must be realised that the macroscopic quantities deﬁned using the dif-

ferential notation imply that a limiting process, as described above, has

occurred” [8].

The question that arises is: What is the appropriate sized mass that can

be used to calculate absorbed dose? Even without a quantitative answer to

this question, we can expect that a limited zone in the space {m, ε/m} exists,

where we can speak about absorbed dose (Figure 8.5 on the right, zone with

a pattern). Another problem is: What happens when the dose decreases? Our

conclusion is that it is not possible to deﬁne absorbed dose below a given

value, because we fall into a stochastic zone as is shown in Figure 8.5 on

the right-hand side, for example, with the shadowed zone labeled as C. An

alternative way to enunciate this problem is: What dose value is high enough

to fulﬁl the condition of continuity required by the dose concept?

These conclusions are relevant to the discussion of the linear no-threshold

(LNT) hypothesis. Our conclusion is that dose–effectrelationship at low doses

is meaningless due to the failure of the absorbed dose concept [16]. The

stochastic quantity that must be used is the speciﬁc energy (imparted), z,

which is the quotient of ε by m, thus

z =

(8.53)

The units are also (J kg

−1

≡ Gy) previously mentioned and named gray

[11]. This quantity is fundamental in microdosimetry.

Radiation scattering has already been mentioned. In a radiation ﬁeld, this

can be deﬁned: the primary radiation, which is the un-collided component

and the secondary radiation, which means those particles resulting from col-

lisions. Therefore, absorbed dose at a given point is the sum of primary, D

(p)

and secondary, D

(s)

, components:

(t)

= D

(p)

(s)

(8.54)

To describe secondary radiation, the buildup factor is often used, which can

be deﬁned for ﬂuence, exposure, absorbed dose, etc.

For absorbed dose, the deﬁnition of the buildup factor is

B(μL) =

(t)

(p)

(s)

(p)

= 1 +

(s)

(p)

(8.55)

where μ is the attenuation coefﬁcient and L is the depth in the medium.

Equation 8.46 refers only to primary photons; therefore, to include the

buildup factor from Equation 8.55, it can be shown that

(L) = D

(p)

(0)B(μL)e

−μ(E

(8.56)

Dosimetry and Biological Effects of Radiation 455

Let us note that if L = 0, then this equation becomes

(0) = D

(p)

(0)B(0) (8.57)

meaning that B(0) is the backscatter factor, BSF = B(0).

8.5 Radiation Protection Quantities

In this section, we will review the radiation protection quantities.

The dose equivalent is deﬁned as follows:

H = QD. (8.58)

This quantity is the product of the absorbed dose, D, with the quality

factor, Q.

The equivalent dose is deﬁned as



T,R

(8.59)

and D

T,R

is the average of absorbed dose in a tissue or organ, T, for a radiation

type, R, and w

is a radiation weighting factor.

The effective dose is

E =



. (8.60)

in which w

is the tissue weighting factor. The values of the weighting factors

can be found in ICRP-103 [19].

There are also other quantities, termed operational quantities, and they are

used in individual monitoring [10,11,17].

8.6 Quality Factor, Weighting Factors, and Relative

Biological Effectiveness

The ejected electrons may have sufﬁcient energy to cover distances in tissue

that can reach a few millimeters and leave a trail of ions. By deﬁnition, speciﬁc

ionization is the number of ion pairs formed per unit distance traveled by the

particle.

In radiobiology and radiation protection, when radiation traverses an

absorbing tissue, an important quantity is the linear rate of energy loss of

charged particles (or the density of energy absorption by the medium) due to

ionization and excitation processes.

456 Nuclear Medicine Physics

LET is a measure of the spatial density of energy absorption. It is the mean

energy lost by the radiation particle, due to collisions with electrons, per unit

distance traversed in material.

In radiation dosimetry, LET is used as the basic physical quantity in terms

of radiation biological effectiveness. This may not be absolutely true, as there

is evidence that other factors, such as pH variation in the medium or dose

rate, may also be important.

In radiobiology, radiation is often referred to as low LET (e.g., photons or

electrons) and high LET (e.g., α-particles). Recognizing the importance of the

LET (often given only as L), the parameter quality factor was introduced, and

this is a function of the radiation LET.

Q = Q(L) (8.61)

Q(L) =

⎧

⎪

⎨

⎪

⎩

1 (L ≤ 10)

0.32L −2.2 10 < L < 100

300/

√

LL≥ 100

(8.62)

The units of L are (keV μm

−1

), whereas the quality factor is dimensionless.

In order to simplify the radiation protection quantities, the ICRP introduced

weighting factors for radiation and tissues [32], and these have been recently

revised [19]. The determination of the values of the weighting factors was

guided by the concept of Relative Biological Effectiveness (RBE).

In 1931, a determination was reported for the ﬁrst time of the RBE of x-

and γ-rays. In radiobiology, RBE equals the ratio of the absorbed doses of two

types of radiation that produce the same speciﬁc effect. The RBE was ﬁrst

used, in radiation protection, as a weighting factor; but complications arose

due to their dependence on dose, dose rate, fractionation, and the cells or

tissues in which the effect is being assessed. Consequently, in 1959, the ICRU

recommended that the term RBE be used only in radiobiology and the quality

factor was proposed [18].

8.7 Energy Degradation in Matter

Physical interactions occur in the ﬁrst 10

−15

s. Between 10

−11

s and 10

−6

s, free

radicals, molecules, and ions will be produced and distributed, which will

change the normal chemical reactions within cells exposed to radiation.After

several years, biological effects can appear [1].

Radiation dosimetry aims at quantifying the energy imparted or the energy

deposited by radiation within biological tissues through the main radiologi-

cal protection quantity: the absorbed dose. Biological effects can be stochastic

Dosimetry and Biological Effects of Radiation 457

or deterministic, depending on its value. In their new fundamental recom-

mendations [19], ICRP is proposing that deterministic effects will be named

tissue reactions.

Biological effects are often correlated with absorbed dose in tissues or cells

at risk. However, other factors are also important, such as type of radia-

tion, energy spectra, spatiotemporal distribution of energy deposited, the

total number of cells exposed, or radiosensitivity of cells. As an attempt to

foresee biological effects, such as carcinogenesis or cell killing, the assump-

tion is often accepted that risk is proportional to absorbed dose. How-

ever, some radiobiological experiences very often reveal rather complex

relationships [20].

Radiation dosimetry and protection are directly related to the hierarchy of

life: from molecules, passing through cells up to organisms and continuing to

population, community, ecosystems, and biosphere. As far as the ecosystem

level is concerned, studies on environmental radioactivity have to be carried

out; whereas for smaller-sized systems, for example, absorbed dose at human

organ levels, we are mainly concerned with medical applications of radiation,

such as radio-diagnostics or radiotherapy. Radiobiology deals with effects in

tissues, cells, or subcellular structures and the main target of radiation, DNA.

Molecular biology is also very important in helping to seek the biophysical

mechanism behind radiation effects. From the dosimetry point of view, we

will be concerned with microdosimetry or nanodosimetry. The cell is the fun-

damental unit of life and can be seen as a fundamental unit for radiation

dosimetry if we are interested in cellular dosimetry.

Considering the several orders of magnitude of size on going from

molecules to organisms, several questions arise about the deﬁnition of dose,

if we consider that dose is the expectation value of energy imparted over an

appropriate size mass element. In fact, if we increase the dimensions of a cell

to the dimensions of the moon, then a portion of the DNA is equivalent to a

person walking on the moon. What is the relationship between a tissue dose,

a cell dose, and the DNA dose? The answer to this question is not trivial.

Biological cells, tissues, organs, or living beings are complex systems. Let

us look at life as a process that organizes matter and energy in an orderly

system. From that point of view, we can see life as a complex set of processes

that decreases the entropy of the system. The interaction of radiation with

matter supplies energy through interaction processes that the biological sys-

tem cannot handle, leading to the lethal effects of radiation. The effects due to

the exposure of radiation are changes in the structure and behavior of these

biologically complex systems. The exposure of organic or inorganic matter to

radiation leads to changes in the target, due to the enormous statistical com-

plexity that arises from the point of view of radiation physics. Any radiation

beam can also be regarded as a complex system due to its interaction with

matter, which leads to statistical complexity at the level of energy deposi-

tion. We are dealing with two complex entities: a biological entity, the target

(which can be also inorganic), and a physical entity, the radiation beam or ﬁeld