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

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468 Nuclear Medicine Physics

Considering two spheres with radius and r

and r

, and using Equation

8.69, the ﬂuence ratio for the two spheres is given by

(8.70)

As previouslymentioned, the absorbed dose can be calculated from ﬂuence,

leadingto the replacementofΦ by Din Equation 8.70.Making r

= 1m,r

= r,

and D

= D and if the dose rate at one meter,

, is known, the relationship

becomes

D =

×t. (8.71)

The dose is proportional to the inverse of the square distance from the point

source to the point of interest.

Given the dose rate at one meter per unit activity, Equation 8.71 allows

simple calculations, for example, on radiation protection around radioactive

sources.

If we take into account scattered radiation by the buildup factor, this

becomes

D(r) =

×t ×B(μr) (8.72)

where r is the distance from source to the point of interest.

If there is a homogeneous medium between the radiation source and the

point of interest, the exponential attenuation must be included, leading to

D(r) =

×t ×B(μr) × e

−μ(E,Z)r

. (8.73)

For air, we often use B(μ(air)r) ×e

−μ(air)r

≈ 1, which simpliﬁes some radia-

tion shielding calculations. If the air attenuation is neglected and there exists

a slab shield of thickness x, then Equation 8.73 becomes

D(r) =

×t ×B(μx) ×e

−μ(E,Z)x

. (8.74)

To determine the value of

for radionuclides, we can use the air kerma

rate constant, Γ

. This factor is also called the k factor, speciﬁc γ-ray constant,

or exposure rate constant in the last case used, obviously, to calculate the

exposure instead of dose. It is deﬁned for each radionuclide.

Assuming the kerma approximation, then the dose rate

can be calculated

by the expression

D =

, (8.75)

Dosimetry and Biological Effects of Radiation 469

where A is the activity of the source and l is the distance from a point source.

The index δ means that only the photons with energy greater than δ are

considered. If A = 1 Bq and l = 1 m, then

= Γ

. (8.76)

Therefore, Γ

is the kerma rate at 1 m from a gamma source of activity 1 Bq.

The constant Γ

has to have units (m

Gy Bq

−1

) in order to give

units (Gy s

−1

Let us consider the geometry outlined in Figure 8.19.

To calculate the dose at the target point, t, due to the source at point s,we

can deﬁne the photon dose point kernel. The dose point kernel is a function

that represents the dose at a given distance from an isotropic point source, in

a homogeneous and inﬁnite medium, which can be written as

P(r, E) =

4πr





tB(μr)e

−μr

(8.77)

where r =|



−



| and

s is the source strength in photons per second. The

coefﬁcients in Equation 8.77, already earlier mentioned, are functions of the

energy and the atomic number of the medium. In general, a gamma source

emits photons of several energies so that it must be considered as an integral

in terms of energy:

P(r) =



P(r, E)dE. (8.78)

– r

→ →

→

FIGURE 8.19

Isotropic point source at point s and the point of interest t, within a homogeneous and inﬁnite

medium.

470 Nuclear Medicine Physics

After this integration, the point kernel can be deﬁned as a function of the

distance from the point source,

P(r) = P





−





. (8.79)

The point kernel is used in more general and complex geometry, such as

volume source and volume target. This is the case for NM, where dosimetric

calculations are made at the organ level or, in some cases, at the cellular level.

For the case of the organ level, the organ source and the organ target must be

considered. This geometry is outlined in Figure 8.20.

The concept of point kernel was introduced for use in the volume integra-

tion:

D(T ← S) =







−





(8.80)

where D(T ← S) means the dose in the target volume T, due to a source in

volume S.

In suitable models such as those involving spherical geometry, for example,

spherical source and spherical target, theoretically the integration (Equation

8.80) can be analytically solved.

The point kernel can be evaluated for any type of emission (photons,

electrons, α-particles, etc.) and any homogeneous medium.

The calculation of Equation 8.80 at an organ or tissue level, where the

geometry is rather more complex than the spherical case and the medium

is non homogeneous, is only possible by Monte Carlo simulation, which was

brieﬂy reviewed in this chapter.

Due to the importance of absorbed dose calculations during the admin-

istration of radionuclides in the human body, several years ago, a scheme

x y

→

– r

→ →

→

FIGURE 8.20

Volume source and volume target illustrating the general geometry for dose calculations, for

example, in nuclear medicine.

Dosimetry and Biological Effects of Radiation 471

known as MIRD (medical internal radiation dose) scheme was developed by

the MIRD committee.

The advantage of this scheme is that the integration (Equation 8.80) is solved

for several pairs of human organs, and the results are presented in tabu-

lar form or implemented in software packages. The integral equation 8.80 is

replaced by the equation

D(O

← O

) =

S(O

← O

). (8.81)

In this equation, the absorbed dose in the target organ,O

, due to cumulative

activity,

, in the source region O

is simply obtained by multiplying the

cumulated activity by an S-value, which is the absorbed dose at the target

organ per unit cumulative activity in the source region.

All the complexity of the integration (Equation 8.80) is hidden in the

simplicity of the S-value. All we have to do is calculate the cumulated

activity,



A(t)dt (8.82)

The S-value is deﬁned as

S(O

← O

) =



← O

)

(8.83)

where m

is the mass of the target region, Δ

is the mean energy emitted per

nuclear transition, and φ

← O

) is the fraction of energy emitted from

the organ source that is absorbed in the target region for the ith radiation

component.

Next there are deﬁned radiation protection quantities for application in

internal dosimetry.

The committed equivalent dose for an organ or tissue, T, is represent by

(τ).

Since the radionuclide irradiates the body organs after its incorporation,

this quantity is deﬁned as the integral in time τ of the equivalent dose rate in

the tissue of interest. It is deﬁned as the total committed equivalent dose in

the organ or tissue for the time τ after the incorporation:

(τ) =

+τ



(t)dt (8.84)

When the time of integration is not mentioned, a time of 50 years for adults

or 70 years for children is assumed.

472 Nuclear Medicine Physics

The committed effective dose, E(τ), is obtained, by multiplying the commit-

ted equivalent dose for each organ or tissue by the respective tissue weighting

factor. This quantity is deﬁned as the sum:

E(τ) =



(τ)w

(8.85)

To end this short review of the radiation protection quantities, it is often

useful to have a measure of the exposure of a population, which is made by

the collective effective dose.

The effective collective S, is deﬁned by the next integral:

S =

∞



dE (8.86)

or by the sum

S =



, (8.87)

where

is the average effective dose for the subgroup i of a population. The

time interval is not explicitly speciﬁed during which the population were

exposed, resulting in the corresponding effective dose. Therefore, that time

must be expressly speciﬁed in the presentation of effective dose values.

The requirement of homogeneous medium and point isotropic kernel has

already been mentioned. However, the MIRD scheme is not limited to organ

or suborgan dosimetry. In fact, a lot of useful applications have more recently

appeared in cellular dosimetry, for example, in targeted radiotherapy, where

dose at the cellular level is a major concern.

Adaptation of the formalism shortly summarized in Equations 8.81 through

8.83 for organs can be straightforwardly applied to cells or subcellular regions,

by simply replacing the reference of organs to a referenceof cells and adapting

the formalism for absorbed dose calculation to multiple sourceregions, so that

Equation 8.81 becomes

D(C

← C

) =



S(C

← C

). (8.88)

At the cellular level, the concepts of self-dose and cross-dose are often used,

so that the mean dose of a cell cluster is a sum of the self-dose and the cross-

dose. Similarly, we can deﬁne S-values for self-dose and cross-dose, S

self

and

cross

, respectively.

Differences between organ dosimetry, often called conventional dosimetry,

and cellular dosimetry are that in conventional dosimetry, no distinction is

made between the cells and the medium. The radioactivity is assumed to be

Dosimetry and Biological Effects of Radiation 473

homogeneously distributed throughout the volume considered. This implies

that the dose to the target cell nucleus is the same as that to the overall organ or

to any given subvolume. In cellular dosimetry, the approach of Equation 8.88

can be further extended to subcellular regions, such as cell membrane, cyto-

plasm, and cell nucleus. Conventional dosimetry predicts poorly absorbed

dose at the cell nucleus, which can be of great importance for small range

particles, such as Auger electrons.

A drawback of this cellular approach lies in the validation of cross-dose

due to difﬁculties in available data for comparison of results. In fact, calcu-

lations are made deﬁning a given cell cluster. Cell clusters can differ in the

number of cells, the cell and nucleus size, and the packaging of cells within

the cluster. Sometimes, cells are assumed to be spheres. A typical approach

at the cellular level assumes the last geometric details for a given cell cluster

[26], allowing the evaluation of S

cross

, S

selfNuc

, S

selfCyt

, and S

selfMb

, where the

index refers to cross, nucleus, cytoplasm, and cell membrane. Other authors

[27] calculate intracellular S-values for cell–cell, cell–cell surface, nucleus–

nucleus, nucleus–cytoplasm, and nucleus–cell surface, where the ﬁrst region

is the target and the second region is the source.

Different number of cells, cell size, or cell arrangement (package) make any

comparison difﬁcult. It seems apparentthat there is a need to deﬁne some fun-

damental quantities at the cellular level in order to allow comparison between

different calculation methods and clusters. With this scope, and considering

that the cell is the basic unit of life, the basic unit of cellular dosimetry can be

deﬁned as a pair of cells, one of them with radionuclide incorporated and the

other clean (Figure 8.21).

For the basic unit of cellular dosimetry, we can deﬁne several S-values:

In the earlier deﬁnitions of all the S-values, only the cell and the nucleus are

considered as targets. The source can be the cell (C1), the cytoplasm (Cy1),

the cell surface (CS1), and the cell nucleus (N1). Additional factors affect-

ing the S-values are RC1, RN1, RC2, RN2, and R12, which are the radius of

cell 1, radius of cell nucleus 1, radius of cell 2, radius of cell nucleus 2, and

Source

Tar ge t

FIGURE 8.21

The basic unit of cellular dosimetry can be deﬁned as a pair of cells. Acontaminated one as source

and a clean one as target.

474 Nuclear Medicine Physics

TABLE 8.1

S-Values between Two Cells Including

Subcellular Regions

Self-dose Cross-dose

S(C1 ← C1) S(C2 ← C1)

S(C1 ← Cy1) S(C2 ← Cy1)

S(C1 ← CS1) S(C2 ← CS1)

S(C1 ← N1) S(C2 ← N1)

S(N1 ← C1) S(N2 ← C1)

S(N1 ← Cy1) S(N2 ← Cy1)

S(N1 ← CS1) S(N2 ← CS1)

S(N1 ← N1) S(N2 ← N1)

intercellular distance, respectively. The intercellular distance can be deﬁned

as the distance between the cell centers. As can be seen, cellular dosimetry

involves a lot of parameters that contribute to the difﬁculties already men-

tioned in comparisons between different methods and cell clusters. In Table

8.1, the need appears for a rigorous deﬁnition of all the parameters involved.

8.10 Physics of the Biological Effects of Radiation

Radiationdosimetry quantiﬁesthe energydeposited byradiation inbiological

tissue through the main quantity in radiation protection: the absorbed dose.

Deterministic or stochastic effects appear, depending on its value. From the

physical point of view, the absorbed dose can be determined independently

of the biological situation. From the biological point of view, the studies of the

effects in the living matter, for a given localization and absorbed dose value,

are made independently of the physical action.

Very often, the biological effects of ionizing radiation are correlated with

absorbed dose values in tissues or cells at risk. In spite of that, other physical

factors are also related, such as the radiation type, the energy spectrum, the

spatial and time distribution of the energy deposited, the total number of cells

irradiated, or different biological responses to radiation qualities.

In the attempt to estimate biological effects, for example, carcinogenesis,

it is often assumed that the risk is simply proportional to the dose received.

However, it is well known that biological experiments reveal a much more

complex relationship [20].

In a simpliﬁed approach, we can consider that radiation protection studies

the results of the interaction between a physical agent (radiation) and man

(biological medium) or the environment. If we seek a ﬁeld of knowledge

that is multidisciplinary, embracing a vast set of areas, we can arrive at, for

example, the characterization of the biological effects of radiation.

Dosimetry and Biological Effects of Radiation 475

It is considered that two distinct types of biological effects exist: deter-

ministic and stochastic. Edwards and Lloyd [28] and Doll [29] publish two

review papers about deterministic and stochastic effects, respectively. From

a pragmatic point of view, the effects that can be diagnosed in a medical

examinationwithin the few weeks after radiation exposurearecertainly deter-

ministic effects, such as skin burns or reduction in the levels of the blood cells,

which, in extreme situations, can be fatal within weeks or months.

Perhaps, the best distinction between deterministic and stochastic effect

is to mention the mechanisms of their production. In general, the stochastic

effects (mainly cancer) are caused by nonlethal mutations in cells, whereas

deterministic effects are caused by cell death.

The scheme shown in Figure 8.22 describes the processes related to the

effects of ionizing radiation in living tissues.

The occurrence and proliferation of modiﬁed cells can have inﬂuences from

modiﬁcations through other agents that act beforethe irradiation. These inﬂu-

ences are common and can include exposure to other carcinogenic agents.

As already indicated, in general, there are two types of radiation effects:

stochastic and deterministic (Figure 8.23).

The replacement of dead cells is a natural process of maintenance of tissues.

Through sophisticated feedback mechanisms, the rates of production of new

cells are changed to compensate the rate of loss of mature cells. For example,

Radiation

Ionization

Modification of the structure of molecules

Cellular damage

DNA damage

(most serious damage)

The cell don’t survive

of doesn’t reproduce

The cell repairs the damage

Imperfect repair

(modiﬁed viable cell)

Perfect repair

(without consequences)

Chemical modification in molecules

which are not critical for the working of the cell

(production of free radicals)

Chemical modification in molecules

which are critical for the working of the cell

FIGURE 8.22

Radiation effect in living tissues.

476 Nuclear Medicine Physics

Radiation

Modification of cells in gonads,

with function of transmission

of information to the descendants

of the exposed individual

Transmission of incorrect

hereditary information

Severe damages to

some of the descendants

Eventually cancer

Somatic effect

Hereditary effect

Stochastic effect

Modified viable cell

Modiﬁcation of a somatic cell.

Maintain the

reproductive capacity

It can originate

clones of modified cells

Deterministic effect

Loss of the function of the organ.

This loss is greater for

a high number of cells affected

Dead cell

FIGURE 8.23

Stochastic and deterministic effects of radiation.

the external layer of the skin is replaced approximately every 6 weeks; and

blood cells, depending on the type, are replaced with half-lives varying from

some days to some years. The body has the capacity to tolerate the death

of some additional cells due to external agents, such as radiation or chemi-

cals, because it is able to replace cells at a higher velocity than under normal

conditions.

The deterministic effects occur when the number of dead cells exceeds the

capacity for replacement, that is, when the equilibrium state between produc-

tionand death isperturbed byexcess ofthe latter. Thetargetorgansandtissues

then stop working properly, eventually leading to biological modiﬁcations

with serious consequences.

One important parameter is the dose rate. For low LET radiation, the dose

received at low dose rate or in fractions of time has very different conse-

quences when the same dose is received at high dose rate, in a short period

of time.

How do we relate the biological effects with the radiation that impinges on

the tissues? There is a need to quantify, somehow, the energy deposited in the

irradiated medium. Radiation dosimetry is the discipline of radiation physics

which allows that quantiﬁcation. The absorbed dose is the fundamental

quantity, in such a way that it is used as the name of the discipline.

Dosimetry and Biological Effects of Radiation 477

The absorbed dose is the expected value of the deposited energy in matter,

per unit of mass, at a given point of interest,

D =

dε

. (8.89)

Even today, it is common practice to measure the radiation effects as a

function of absorbed dose, in which the dose rate is used as a parameter.

However, there is an assumption in this approach that is sometimes ignored

orneglected. Theenergydeposited has a constant valuein allthe biological tis-

sue. Nowadays, it is well known that this assumption is, in general, incorrect,

particularly for low doses, which are important in radiation protection [30].

Any model concerning the biological response should consider the physi-

cal, chemical, and biological factors. The physical factors are, among others,

associated with the spatial pattern of the points of energy deposition, at

a scale of the order of the nanometer (dimensions of the DNA molecule).

Beyond the direct action, the chemical factors are related with the capacity

for the production of free radicals, mainly in water, in the vicinity of the DNA

molecule. Biologically, we should consider the repair and the probability of

lethality of DNA lesions [31].

Nevertheless, research at biological and molecular levels still maintains

the practice of expressing biological effects as a function of absorbed dose.

Many of the models of biological response are deﬁned through a dose–effect

relationship. Some studies with cultures of cells of mammals indicate that the

survival of the cells is a function of the absorbed dose, described by survival

curves.

As an example, for the deterministic effects, it is accepted that the risk of

a radiation exposure has a sigmoid relationship (see ICRP-60 [32], ICRP-103

[19] or [28]). Edwards and Lloyd [28] proposed a relationship between the

risk, R, and absorbed dose, D, as follows:

R = 1 −exp(−H), (8.90)

H = ln 2





, (8.91)

where H is named Hazard function and is related with the dose through the

parameters D

and V. The parameter D

is the average lethal dose and

signiﬁes that it is expected that 50% of the exposed individuals shows the

effect in question; V is related with the speed of rise.

An artiﬁcial threshold of dose is deﬁned from 10% of the value of the sig-

moid function. Note that this threshold value is deﬁned for deterministic

effects and is not considered for stochastic effects.

Radiation dosimetry is of fundamental importance for the study of biologi-

cal effects and can then be developed in accordance with biological needs. The

following ﬁgures were obtained from the data of Edwards and Lloyd [28].