Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics
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428 Nuclear Medicine Physics
squares; nonlinear least squares; generalized linear least squares; weighted
integration; or base function techniques [37,38].
Image realignment, nonlinear automatic spatial normalization, segmenta-
tion, coregistration, smoothing, development of statistical parametric maps,
development of posterior probability maps, and Bayesian estimates of param-
eters from generalized linear models (GLM) are steps that are included in
these methods’ data processing steps.
The major advantages of parametric images are the better low contrast
detection, a better ROI definition, and their allowing the use of powerful
SPM
∗
(statistical parametric mapping) algorithms.
Disadvantages that have been pointed out for this type of analysis are the
fact that it is similar to processing a black box, the distribution of regional
values is frequently distorted, and it is computationally more intensive.
7.2.10 Monte Carlo Simulation
In NM, the development of new imaging devices, reconstruction algorithms,
correction and optimization techniques, acquisition protocols, and descrip-
tion of time variable phenomena, such as detector or source movements, are
frequently based on appropriate simulations, particularly using Monte Carlo
techniques similar to those used in particle physics.
Quite a lot of Monte Carlo simulation packages have been used for either
SPECT or PET, with different advantages and disadvantages and varying
degrees of success.
Accurateand versatile general-purpose simulation packagessuch asGeant3
from CERN, EGS4 from SLAC, and MCNP from the Los Alamos National
Laboratory contain well-validated physics models, geometric modeling tools,
andefficientvisualization utilities,but theyrequirea major effortto be tailored
to PET and SPECT.
On the other hand, dedicated Monte Carlo codes developed for PET and
SPECT suffer from a variety of shortcomings. For example, SimSET, devel-
oped at the University of Washington, is one of the most powerful dedicated
codes for PET and SPECT simulations; it precisely and efficiently models
physical phenomena and basic detector designs [36]. However, the range of
detector geometries that can be simulated is limited: for instance, a detector
ring cannot be subdivided into individual crystals. In addition, neither Sim-
SET nor other dedicated codes explicitly account for time, which limits their
use for modeling time-dependent processes such as the movement of tracers.
Clearly, there is a need for a Monte Carlo tool that readily accommodates
complex scanner geometries while retaining the comprehensive physical-
modeling abilities of the general-purpose codes. To meet this demand, an
∗
SPM (statistical parametric mapping) is a statistical technique designed to examine the dif-
ferences in brain activity registered during functional studies in neuroimaging with PET and
MRI.
Systems in Nuclear Medicine 429
international collaboration of physicists at centers in different countries has
developed the simulation toolkit Geant4, which has recognized advantages
over its predecessors.
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8
Dosimetry and Biological Effects
of Radiation
Augusto D. Oliveira and J. J. Pedroso de Lima
CONTENTS
8.1 Introduction .........................................432
8.2 Interaction of Radiation with Matter: Photons ................434
8.2.1 Introduction ....................................434
8.2.2 Cross Section ...................................435
8.2.3 Compton Effect ..................................437
8.2.4 Thompson Scattering .............................438
8.2.5 Klein–Nishina Cross Section for the Compton Effect ......439
8.2.6 Corrections of the Bonding Energy of the Electrons .......440
8.2.7 The Photoelectric Effect ............................442
8.2.8 Pair Production ..................................443
8.2.9 Rayleigh Scattering ...............................443
8.2.10 Photonuclear Interaction ...........................444
8.2.11 Interaction Coefficients ............................444
8.3 Interaction of Radiation with Matter: The Electrons ............445
8.3.1 Introduction ....................................445
8.3.2 Radiative Energy Loss .............................445
8.3.3 Loss of Energy by Collision .........................446
8.3.4 Stopping Power .................................446
8.3.5 Linear Energy Transfer, L
Δ
,LET .....................447
8.3.6 Range and CSDA ................................447
8.3.7 Mean Energy Spent per Ion Pair Produced, W ...........448
8.4 Radiation Dosimetry Quantities ...........................448
8.5 Radiation Protection Quantities ...........................455
8.6 Quality Factor, Weighting Factors, and Relative Biological
Effectiveness .........................................455
8.7 Energy Degradation in Matter ............................456
8.8 Monte Carlo Simulation ................................465
8.9 Conventional and Cellular Dosimetry for Radionuclides ........467
8.10 Physics of the Biological Effects of Radiation .................474
8.10.1 The Cell as a Target of Ionizing Radiation ..............481
8.10.2 Composition, Shape, and Size .......................481
431
432 Nuclear Medicine Physics
8.11 Action Modes. Target Theory Models: Survival Curves .........483
8.11.1 Water Radiolysis .................................486
8.11.2 Effects in Aqueous Solutions ........................487
8.11.3 Ionizing Radiation, Ionization Mechanisms, and
Biological Action .................................488
8.12 Mathematical Models of Cellular Survival in Ionizing
Radiation Fields ......................................494
8.12.1 Multiple Target Models (Target Theory Models) ..........495
8.12.1.1 One Sensitive Region n Hits ..................495
8.12.1.2 Multiple Sublethal Sensitive Zones/One-Hit
Model ..................................497
8.12.1.3 Mixed Model .............................499
8.12.1.4 The Linear–Quadratic Model .................501
8.12.1.5 Comparison of the L–Q and Target
Theory Models ............................504
8.13 Nontargeted Effects Complementary to Direct Action of
Ionizing Radiation for Low Doses .........................505
References ..............................................506
8.1 Introduction
In living matter, atoms and molecules maintain their individual and collec-
tive configurations as a result of very specific energy interactions, which are
subject to complex physical–chemical and biological rules. Living matter at
all levels can be understood as a set of active processes that organize matter
and energy within ordered complex systems.
Radiation interactions with living matter releaseenergy in biological tissues
through processes that increase the entropy of the biological system, often
beyond the limits that the organisms can tolerate, leading to the harmful
effects of radiation.
This perturbation in the organization of energy and matter can lead to a
great variety of biological consequences, ranging from local recoverable situ-
ationsto complexirreversiblemodifications. Suchbiological effectsarerelated
to the energy of molecular bonds and of the reactions in living processes.
In living matter, the orders of magnitude for molecular and atomic binding
energiesare from a fraction to tens of eV; those for molecular vibrational states
aretenthsof eV;and, formolecular rotationalstates, theyarehundredthsof eV.
The lowest ionization potentials of elements, molecules, and radicals with
biological interest lie between 11 and 14 eV. Examples include 11.24 eV for C,
13.54 eV for H, 13.57 eV for O, 14.24 eV for N, 13.12 eV for CH
4
, 11.42 eV for
C
2
H
2
, 10.56 eV for C
2
H
4
, 11.62 eV for C
2
H
6
, and 9.96 eV for −CH
3
.
Radiation acts on matter by transferring its energy to the medium (kinetic
energy in the case of particles and electromagnetic energy in the case of
Dosimetry and Biological Effects of Radiation 433
photons). In the case of photons, the ionization results mainly from secondary
electrons generated by photoelectric, Compton, and pair production effects.
Atomic and molecular energy levels in living tissues can be perturbed by
energy delivered by radiation; and as a consequence, this can initiate a chain
of various possible events, depending on the type and the energy transferred,
which can include energies below the minimum ionization potential (11 eV),
down to the thermal energies.
Some characteristic examples of biological interest are the energies neces-
sary to break covalent (ca. 5 eV) and hydrogen bonds (0.1–0.2 eV); the energies
necessary to separate molecular dipoles (0.01–0.03 eV); the dielectric relax-
ation energies of proteins and water (0.4 and 0.2 eV, respectively, at 25
◦
C);
and the energy values for inducing photo-conduction (1–4 eV).
Below the minimum molecular rotation levels, the quantized energy levels
in living tissues are so numerous and close to each other that energy absorp-
tion fromradiation can be considered a continuum process,unlike the discrete
events which occur at high energies (such as the photoelectric effect,Compton
effect, etc.).
Energy from the absorption in tissue of photons having thermal energy is
shared by a huge number of electrons of the medium and directly results in
a bulk increase in temperature.
Particularly important types of radiation, relevant to biological action, are
the so-called ionizing radiations that are those which are capable of induc-
ing ionization in biological structures. Within this group, ionizing radiations
include short wavelength electromagnetic radiations (remote UV rays, x- and
γ-rays, and cosmic photons), particles, and heavy ions.
When considering ionization within biological tissues, it is important to
consider the fact that ionizing radiation has to deliver an average of ∼34 eV
in tissue to produce an ion pair. This means that ionizing radiation together
with ionization induces other atomic and molecular processes that do not
involve an electrical charge, and these events are the most probable. In fact,
for every ionization event that occurs, about 20 eV, that is, nearly twice the
biological ionization potential, are dispensed in excitations of energy levels,
either atomic or molecular, and other processes, all requiring energy below
the ionization potential. This corresponds to the delivered 34 eV minus the
11–14 eV necessary to produce ionization.
Ionization creates an electrical imbalance, which, in the first stage, causes
the newly created ion to attempt to regain electrical stability by combining
with the nearest available opposite charge. This generally causes alterations
incompatible with the surrounding cell structure and leads to cell damage or
death.
The most sensitive target for radiation effects in tissues is the deoxyribonu-
cleic acid (DNA) molecule, and it is accepted that the effects on this molecule
are the end process of a chain of physical and chemical events initiated by
the ionizing interaction and followed by biological changes of various types.
However, recently, several nontargeted effects have been identified and are
434 Nuclear Medicine Physics
under intense research, such as the bystander effect, in which cells that are
not directly exposed to radiation show the same effects as exposed cells.
The timing of the most important events that take place when ionizing
radiation impinges on biological tissue (at time t = 0) is accepted to be about
10
−15
s for ejection of secondary electrons (ionization) or atomic excitation,
about 10
−10
s for formation of radical ions, about 10
−5
s for formation of
free radicals, and 10
−15
–10
−3
s for breaking molecular bonds. The species
produced free radicals, molecules, and ions; diffuse in the medium; and can
disturb the normal chemical intracellular reactions. Over time, varying from
seconds to hours, cellular lesions may appear, whereas stochastic biological
effects may only become evident years later [1].
In NM, the main interests are in the ionizing and excitation effects of high-
energy photons (γ- and x-rays) or of the resulting secondary electrons.
8.2 Interaction of Radiation with Matter: Photons
8.2.1 Introduction
There are five types of interactions that must be considered with regard
to applications of radiation physics in medicine and radiological protection
problems:
1. Compton effect
2. Photoelectric effect
3. Pair production
4. Coherent or Rayleigh scattering
5. Photonuclear interaction
The first three of these are the most important, because they lead to energy
transfer to electrons, which, by Coulomb interactions along their track, impart
the energy to matter. Rayleigh scattering is elastic; the photon only changes
the direction of their track by a small angle, without loss of energy. The
photonuclear interaction is only significant for energies above a few MeV,
where it can cause problems in radiation protection resulting from the
production of neutrons and subsequent induction of radioactivity.
The photoelectric effect is dominant at low energies, whereas the Compton
effect is more important at intermediate energy values, and, finally, pair pro-
duction is the main interaction at higher energies. For materials of low atomic
number, Z, for example, carbon, air, water,human tissue,etc., the domainzone
of the Compton effect is very wide, extending from approximately 20 keV to
30 MeV.
In Figure 8.1, the relative cross sections are shown, as an example, for water
for energies up to 200 keV. Water is often used as a material whose properties
are equivalent to those of biological tissue.
Dosimetry and Biological Effects of Radiation 435
0.0
0 0.02
Coherent
Photoelectric
Incoherent
To ta l
0.04 0.06 0.08 0.0 0.12 0.14 0.16 0.18 0.2
0.1
0.2
0.3
0.4
0.5
Relative cross section
Energy (Mev)
0.6
0.7
0.8
0.9
1.0
FIGURE 8.1
Graphical representation of cross-sections, normalized to unity, which includes the photoelectric
effect, the coherent scatter, the incoherent or Compton scattering, and the total sum. The pair
production exists only for energies above 1022 keV.
8.2.2 Cross Section
Within the context of ionizing radiation, the word interaction is applied to pro-
cesses in which the energy and/or the direction of the radiation are changed.
These processes are stochastic; therefore, we can only speak in terms of prob-
abilities of the occurrence of interaction. These probabilities can be expressed
in terms of cross-sections or interaction coefficients.
Let us consider a layer of a given material inside of which N scatter centers
exist per unit of volume. Let us consider an infinitesimal layer thickness dx
andconsider a frontsurface withareaA. The total numberof scattering centers
within the layer is NA dx.
Since the atoms and molecules are not points, we can consider that a colli-
sion or scattering occurs when the incident particles collide inside a spherical
surface, of radius R, centered on the atoms or molecules. Therefore, there will
be an effective collision surface, σ = πR
2
, for each of the scatter centers, and
the total collision surface for the layer dx will be NA dx.
The probability of collision for an incident particle with a scatter center in
the layer dx, of frontal surface A, is defined as
dw =
total collision surface
frontal surface
, (8.1)
that is,
dw =
NAσ dx
A
= Nσ dx. (8.2)
436 Nuclear Medicine Physics
The interaction probability, per unit length of the particle track, is
dw
dx
= Nσ. (8.3)
The quantity σ is termed the cross-section, and from the above discussion,
we can express the probability of the occurrence of a given process in terms of
the cross-section. If the process is one of absorption, scattering, etc., we obtain
a cross section for absorption, scattering, etc.
The quantity μ = Nσ is termed the linear attenuation coefficient and has units
of (m
−2
), because N is the number of atoms per unit volume (m
−3
); whereas
σ has the units (m
2
) per atom.
The quantity, λ, defined by
1
λ
= Nσ, (8.4)
is called the mean free path (the mean value of the space traveled by the
particle between two consecutive collisions).
Let us consider a layer of a material with thickness x, and let n be the
number of particles with a given energy incident in the layer. For a given
layer thickness, dx, the variation of the number of the particles will be dn and
leads to
dn =−μn dx. (8.5)
By integration, we obtain
∗
n = n
0
exp(−μx). (8.6)
Sometimes, it is convenient to use the expression
n = n
0
exp
−
μ
ρ
ρ ×x
. (8.7)
The coefficient (μ/ρ)is termed the linear mass attenuation coefficient and has
the units (m
2
g
−1
).
∗
Resolution of Equation 8.5
dn =−μn dx
dn
n
=−μ dx
ln n =−μx +ln C
n = C exp(−μx)(x = 0),
n
0
= C exp(0) = C
n = n
0
exp(−μx)
Dosimetry and Biological Effects of Radiation 437
In general, the cross section depends on the scattering angle, and we can
also define the differential cross-section,
dσ = σ(θ)dΩ, (8.8)
which considers the scatter within a solid angle dΩ, which forms an angle θ
with the incident direction. The quantity σ(θ) = dσ/dΩ gives us the angular
distribution of the cross section and, therefore, a measure of the anisotropy
of the process.
One can calculate σ, the total cross section, from σ(θ), if the phenomenon is
azimuth independent, with regard to the incident direction, hence
dΩ = 2π sin θ dθ. (8.9)
We have
σ = 2π
π
0
σ(θ) sin θ dθ, (8.10)
expressing σ in (m
2
) per atom or nucleus.
∗
8.2.3 Compton Effect
The description of the Compton effect can be divided into two parts: kinemat-
ics and cross section. The first relates to energies and angles of the particles
involved in a Compton interaction. In both cases, it is usually assumed that
the electron in a collision is a free and stationary electron. This assumption
is not rigorous, because the electrons, which can occupy several energy lev-
els, are moving around and are bonded to the atomic nucleus. However, the
resulting errors are of little consequence in radiation physics as applied to
medicine under conditions of high atomic number and low energy due to
the competing preponderance of the photoelectric effect, where the electron
bonds are more important for the Compton interaction.
A photon of energy w and linear momentum
k can collide with a free and
stationary electron (Figure 8.2).
Theelectron is ejected with energyTand angle θ in relationto the directionof
the incident photon. The scattered photon has energy w
, linear momentum
w
/c, and angle φ in the plane defined by the direction of the incident photon
and the direction of the ejected electron.
The assumption of a free electron means that the kinematic relationship is
independent of the atomic number of the medium.
∗
The unit often used in nuclear physics for cross-section is 1 barn= 10
−24
cm
2
.