Download Notes On Physics For Residents In Radiation Oncology

1
Notes on Physics for Residents
In Radiation Oncology
Kenneth E. Ekstrand, Ph.D.
Professor
Department of Radiation Oncology
Wake Forest University School of Medicine
Winston-Salem, North Carolina
2
Copyright © 2002 by Kenneth E. Ekstrand
Kenneth E. Ekstrand
Department of Radiation Oncology
Wake Forest University Baptist Medical Center
Medical Center Boulevard
Winston-Salem, NC 27157.
3
Contents
I- Atoms, nuclei, and photons
Atomic structure
Electromagnetic radiation
Free radicals
The atomic nucleus
Characteristic X rays
Nuclear stability
Radioactive decay
Mathematics of radioactivity
8
9
10
11
11
11
13
19
II- Particle interactions and X ray production
Elastic and inelastic collisions
Heavy charged particles
Electrons
Bremsstrahlung
21
21
22
23
III- Cobalt teletherapy
A mathematical digression
The inverse square law
Cobalt 60 teletherapy
Source housing
Secondary collimators and penumbra
25
26
27
28
29
IV- Interactions of X rays and gamma rays with matter
The attenuation coefficient
Mass attenuation coefficient
Coherent scattering
Photoelectric absorption
Dependence of  on energy and atomic number
Compton scattering
Special cases for Compton scattering
Dependence of  on energy and atomic number
Pair production
Nuclear photo-disintegration
Mass attenuation coefficients vs. photon energy
31
32
32
32
33
33
34
34
35
35
36
4
Contents
V- Conventional X-ray machines
Diagnostic and therapeutic X-ray units
The line focus principle
X-ray circuitry
Half wave rectification
Single phase full wave rectification
Three phase full wave rectification
Filtration
37
37
37
40
41
41
42
VI- Megavoltage electron accelerators
Linear accelerators
Traveling wave linear accelerators
Standing wave linear accelerators
The treatment head
Multi-leaf collimators
Other types of electron accelerators
Quality assurance of medical accelerators
44
45
46
48
51
52
53
VII- Heavy particles and neutrons
Proton beam therapy
Neutron therapy
Boron neutron capture therapy
D-T reaction
Pi meson therapy
55
56
57
57
58
VIII- X-ray and gamma ray exposure and dose
X ray and gamma ray quantity
Practical ionization chambers
Absorbed radiation dose
Conversion of roentgens to rads
59
61
64
65
IX- The measurement of dose for beam qualities greater than 3 MV
The Bragg-Gray cavity theory
Determination of dose in a medium
Calibration of linear accelerators
The TG-21 method
The TG-51 method
Dose buildup
Kerma
Interface effects
Dose to soft tissue within bone
Dose equivalent
68
69
71
72
72
73
74
74
75
75
5
Contents
X- Dose measurement not involving ionization chambers
Thermoluminescent dosimetry
Silicon diode dosimetry
MOSFET dosimetry
Chemical dosimetry
Film dosimetry
Calorimetry
76
77
78
79
79
80
XI- Practical clinical dosimetry
Back scatter factor
Field size factors
Percent depth dose (PDD)
Factors the affect the PDD
Tissue air ratio (TAR)
Factors that affect the TAR
Tissue maximum ratio
Relationship between the TMR and TAR
Tissue phantom ratio (TPR)
Relationship between the TMR and the PDD
Arc therapy
81
83
84
84
87
87
88
89
89
90
91
XII- Isodose distributions
Single beam isodose distributions
Wedged fields
Combining isodose curves
Wedged pair
Perturbation of isodose curves
Compensating filters
Correcting isodose curves for irregular surfaces
Tissue inhomogeneity
Abutting fields-gaps
Irregular field dosimetry
Dose outside a field
Scatter vs. X ray beam quality
93
94
95
96
98
99
100
102
105
107
109
110
XIII- Electron beam therapy
The electron beam depth dose curve
Skin sparing
Producing a broad electron beam
Measurement of electron beam energy
Some practical electron beam formulae
Electron beam isodose curves
Virtual electron source
Electron field defining cut-outs
Corrections for tissue inhomogeneity
Oblique incidence
112
113
114
114
116
116
117
118
118
119
6
Contents
XIV- Dose reporting in radiation therapy
Definition of volumes
Dose reporting
Dose volume histogram
121
122
122
XV- Special topics in photon beam therapy
Intensity modulated radiation therapy
Stereotactic radiosurgery
Total body irradiation
125
128
130
XVI- Imaging in radiation therapy
Scattered photons and contrast reduction
Intensifying screens
Film technique
Fluoroscopy
Portal filming with linear accelerators
Electronic Portal Imaging Devices (EPIDs)
Digital Imaging
Computed tomography
Cone Beam CT with Linear Accelerators
Magnetic Resonance Imaging
Ultrasound Imaging
134
135
135
136
137
137
138
138
139
140
142
7
Contents
XVII- Brachytherapy
Radium and its decay
Secular and transient equilibrium
Specification of quantity if radionuclides
Exposure rate constant
Air kerma strength
Conversion of source strength to dose rate
Converting millicuries to mg-Ra equivalents
The radial dose function
Radium sources
Exposure rate from a line source
Isodose distribution from a linear source
The Manchester system
The Memorial system
The Paris system
Implant verification
Intracavitary implants for cancer of the cervix
The ICRU system
Radium substitutes
Permanent radioactive seed implants
Iodine 125 and Palladium 103
Isodose distributions from 125I and 103Pd
Summary of radionuclide properties
Non-photon sources
145
145
147
147
148
148
149
149
150
151
152
152
155
155
157
157
158
159
159
160
161
161
162
XVIII- Protection of individuals from ionizing radiation
Dose equivalent
Recommendations on annual exposure limits
Protection from brachytherapy sources
Protection from teletherapy sources
Primary protective barrier
Secondary protective barrier
Scattered radiation
Head leakage
Surveys of radiation levels
163
163
164
166
166
168
168
168
169
8
I- Atoms, nuclei, and photons
Atomic structure
An atom is not the most fundamental building block of matter. Atoms consist of a positively
charged nucleus, approximately 10-13 cm. in diameter, surrounded by negatively charged
electrons occupying orbitals whose size is approximately 10-8 cm.
The electrical force causes the electron to be attracted to the nucleus (opposite charges), but
quantum mechanics places restrictions on how small the orbital can become.
Each orbital has a specific energy of interaction associated with it, the binding energy, and an
energy quantum number n (n = 1,2,3 . . .). The smaller the number n is, the lower is the binding
energy (the more tightly the electron is bound to the nucleus).
In addition to the energy quantum number, each orbital has an additional quantum numbers l and
ml, associated with angular momentum. These quantum numbers affect the energy less than n
does but strongly affect the shape of the orbital. Two electrons can occupy each orbital. The
electrons pair off their spins (their intrinsic angular momentum) in opposite directions.
The group of orbitals having the same energy quantum
number (and therefore approximately the same binding
energy) is called a shell. The lowest energy shell (n=1) is
called the K shell. Next is the L shell (n=2), then come M,
N, etc.
Only differences between energy levels can be observed; therefore, where we take the zero
energy level to be is arbitrary. It is customary to take the zero level to be that of an electron that
is just barely free of the atom. Thus, the binding energies are negative numbers, with the lowest
level being the greatest in absolute magnitude.
Energy (eV)
-13.5
-3.4
-1.5
-0.9
.
.
0.0
n
1
2
3
4
.
.
unbound
Shell
K
L
M
N
.
.
The energy levels of a hydrogen atom.
9
Binding energy is usually specified in units of electron volts (eV). One eV is the amount of
energy an electron gains in crossing a potential difference of 1 volt (e.g., from a 1 volt battery).
Other common units of energy are:
1 erg
=
1 joule
=
1 calorie
=
6.24 x 1011 eV
6.24 x 1018 eV = 107 erg
4.18 joule
Hydrogen is the simplest of atoms, having only one electron orbiting the nucleus. For atoms
with many electrons, each shell with an energy quantum number n can have a maximum of 2n2
electrons. For example, the L shell can have 8 electrons, the M shell can have 18, etc. The
lowest energy levels are always filled first.
If an electron moves from a lower energy level to a higher one (excitation), the difference in the
binding must be supplied from outside the atom. If sufficient energy is supplied to remove the
electron from the atom (ionization), the positively charged atom which remains is called an ion.
If an electron jumps from a higher energy level to a lower one (de-excitation or relaxation), the
atom gives off the excess energy, frequently in the form of electromagnetic radiation.
Electromagnetic radiation
EM radiation is the propagation of energy through space in the form of waves of conjoined
electric and magnetic fields.
In free space the velocity of all electromagnetic (em) radiation is a constant, c.
c = 3 x 108 meter/second.
The wavelength and frequency of the wave are inversely proportional to one another, i.e.,
 = c/
 is the frequency in cycles/second or hertz
λ is the wavelength in meters
Quantum theory asserts that all em radiation is transmitted in discrete energy units called quanta
or photons. The energy of an individual photon is related to the wave frequency according to
E = h or E = hc/λ
where h is Planck’s constant (4.14 x 10-15 eV-second).
10
In the case of X rays, it is common to specify λ in units of angstroms (Å).
1 Å = 10-10 meter (about the size of an atomic orbital).
In this case, the above relationship becomes:
E (KeV) = 12.4/λ (angstroms)
The spectrum of observed electromagnetic radiation is enormous. For low frequencies (radio),
the quantum nature is not apparent; for high frequencies (X-ray), it is very apparent.
Free radicals
The fundamental properties of electrons are mass, electric charge, and spin. In atoms, the
electrons pair up with spins opposing so that the net (atomic electron) spin is zero.
A free radical is a molecule (usually electrically neutral) with an unpaired electron. A free
radical is very reactive chemically and usually has a short half life, but is longer lived than an
ion. Free radicals are produced by ionizing radiation. For example, the OHfree radical can be
produced in irradiated water according to (Hall 2000):
H2O+ (radiation)  H2O+ + eH2O+ + H2O  H3O+ + OH
(The reactions continue - but that’s radiobiology.)
11
The atomic nucleus
The atomic nucleus is made up of protons (positively charged) and neutrons (electrically
neutral). A nuclide (a species of atom with a specific number of neutrons and protons) is
represented by:
Z
A
X
for example:
184
74
W
where: X is the chemical symbol
A is the atomic mass number (the number of neutrons + protons)
Z is the atomic number (the number of protons)
As Z increases, the K shell binding energy increases in proportion to Z2.
hydrogen (Z=1):
tungsten (Z=74):
K shell binding energy = -13.5 eV
K shell binding energy = -69,500 eV (-69.5 KeV)
We frequently depict the energy levels in an atom by an energy level diagram, electrons moving
from one level to another are depicted by arrows.
n
1
2
3
4
5
shell
K
L
M
N
O
Binding
energy
(eV)
-69500
-12100
-2800
-590
-77
Average binding energies for
electrons in tungsten. The arrow
shows an electronic transition.
Characteristic X-rays
If an atom with a high Z is excited by an high energy (external) electron, a vacancy can be
created in the K or L shell. An outer electron will fill the vacancy and a photon with energy in
the X-ray region is created. The photon energy is equal to the difference of the two energy
levels and is characteristic of the particular atomic species involved (hence the designation).
Nuclear stability
Only certain combinations of neutrons and protons can form stable nuclei. A particular
combination of A and Z is called a nuclide. We call pairs of nuclides with the same Z (proton
number) or A (mass number) by special names.
1. Isobars - nuclides with the same A but different Z, e.g.,
27
60
Co and
28
60
12
Ni.
2. Isotopes - nuclides with the same Z but different A. (These atoms are chemically the
same but may have different nuclear properties), e.g., 816O and 815O.
3. Isotones - same number of neutrons (A - Z) but different Z, e.g.,
13
27
Al and
14
28
Si.
4. Isomers - same A, same Z, but different internal nuclear energy, different configuration
of the neutrons and the protons, e.g., 4399Tc and 4399mTc.
Because the mass of an atom is so small, we frequently measure it in units other than kilograms.
One unit is the atomic mass unit, AMU, is set to equal 1/12th of the mass of an ordinary carbon
atom which has a mass number A = 12 (6 protons and 6 neutrons). In more common units 1 AMU = 1.66 x 10-27 kg.
The Einstein equation E = mc2 tells us that mass and energy are different forms of essentially the
same thing. Thus, mass can be expressed in units of energy. If a mass is expressed in AMU, its
energy equivalence, expressed in MeV (million electron volts) is given by the relation:
Energy(MeV) = 931 x mass(AMU).
The masses of the subatomic particles in atomic mass units and energy units are listed below:
Mass (AMU)
Mass (MeV)
Electron
0.00055
0.51
Proton
1.00727
938
Neutron
1.00866
939
If we add the masses of 6 protons (6 x 1.00727) and 6 neutrons (6 x 1.00866) and 6 electrons
(6 x 0.00055), we would get 12.099 AMU for the mass of a carbon atom. However, by the way
the AMU has been defined, the carbon atom must have a mass of 12 AMU. The difference,
0.099AMU or 93 MeV, represents the energy that is released when the particles are bound
together, the nuclear binding energy.
For light nuclides (Z < 20), the most stable configuration occurs when the number of neutrons is
approximately equal to the number of protons. For heavier nuclides, stability occurs when the
number of neutrons is greater than the number of protons.
A graph of the stable nuclides in terms of their proton number (Z) and neutron number (A - Z)
shows a “line of stability.” Those nuclides lying above the line of stability have excess neutrons
(neutron rich) and those below have excess protons (proton rich).
13
Nuclides in which A is an even number and
A – Z is also an even number tend to be
most stable. In terms of evenness of the
numbers, the stability varies as follows:
Proton
number
Neutron
number
Most stable
even
even
Equally
stable
even
odd
odd
even
Least stable
odd
odd
Bismuth (Z = 83) is the stable nuclide
with the highest Z.
Radioactive decay
An unstable nuclide will, in time, transform itself into another nuclide with the emission of a
particle or particles. This is called radioactive decay. In addition to its own mass (which,
Einstein says, is a form of energy), the particle will carry off some energy in its motion, Kinetic
Energy.
The masses of the transformed nucleus (the daughter) plus the masses of all the particles that
were released plus the kinetic energy of all the particles must equal the mass of the initial
nucleus (the parent). One can determine if a specific decay mode is possible by adding up the
total mass of the final state. If that sum is less than that of the initial state, then the decay is
possible. There may, however, be other decay modes that are more likely to occur.
The modes of radioactive decay are characterized by the type of particle that is released. The
types of decay are: alpha decay, beta decay (both beta minus and beta plus), gamma decay
(isomeric transition), electron capture, and internal conversion.
Alpha decay (occurs only with high Z nuclides). In general terms, the decay proceeds as
follows:
Z
A
X
→
Z-2
A-4
Y + + Q
14
where  represents an alpha particle and Q represents the kinetic energy of the particle. An
alpha particle is the nucleus of a helium atom, A = 4 and Z = 2.
An example of this decay is:
226
Ra → 222Rn + α + Q
Q can be calculated by comparing the nuclear masses on both sides of the decay.
Mass of Ra atom = 226.0254
Mass of Rn atom = 222.0175
Mass of a helium atom = 4.0026
If we call mo the mass of an electron, then the mass of a radium nucleus = 226.0254 - 88mo
Masses of radon nucleus and the alpha particle are:
(222.0175 - 86 mo ) + (4.0026 -2mo ) = 226.0201 - 88mo
The difference is 0.0053 AMU. Multiplying by 931 (MeV/AMU), we find the kinetic energy of
the alpha particle to be 4.8 MeV. Notice all the electron masses cancel out, so we could have
ignored them in this calculation. This is not necessarily the case in other types of radioactive
decay.
We frequently depict this decay by an
energy level diagram similar to the one
we used in discussing X-ray transitions.
The arrow goes to the left indicating a
decrease in the atomic number of the
daughter nucleus.
The * indicates an excited state of the
radon (more about that later) which
occurs in 1% of all the radium decays.
The zigzag line represents a photon
emission.
Beta minus decay. A beta minus particle is just an ordinary electron that is emitted from a
nucleus. The general decay scheme is:
A
Z
X →
Z+1
Y+ β⎯ +
A
ν+
Q
The ν represents an anti-neutrino, a particle with a very small mass and no electric charge.
This particle is always present in beta decay. Q is shared between the β and the ν. Beta minus
decay occurs with neutron rich nuclides. In effect, a neutron converts itself into a proton and
emits the β in order to balance the electric charge.
15
A common nuclide that decays by beta minus decay is
phosphorus 32.
32
P → 32S + β⎯ + ν
Here we have omitted the Q (but it is always there). The
arrow goes to the right, indicating an increase in atomic
number, Z.
As in the case of alpha decay, we can get the kinetic energy involved in the decay by subtracting
the atomic mass of the daughter nuclide from that of the parent, neglecting all of the electron
masses (including the β ).
The transition energy is shared between the β and the  (which is not easily detected). The
energy of the β can have any value from 0 to 1.71 MeV. The average β is 0.69 MeV.
In many instances beta decay goes through an intermediate
state with the subsequent emission of a gamma ray. An
example of this is 137 Cs.
The decay goes through the excited state 95% of the time.
Thus 137 Cs is a good source for gamma ray photons. A
gamma ray is no different than an X-ray photon except for the
fact that it originates from a nucleus.
Beta plus decay. When a proton rich nucleus decays, a proton needs to be converted into a
neutron. One way of doing this is through the emission of a beta plus particle, also known as a
positron. A positron is just like an ordinary electron except it is anti-matter. It is positively
charged. When a positron comes in contact with an ordinary electron, the two particles
annihilate into two gamma rays. This generally happens soon after the nuclear decay has
occurred. Each gamma ray has an energy of moc2 or 0.51 MeV.
The general decay scheme is:
A
X→
Z
The
ν
Z-1
A
Y+
β+ + ν +
Q
is now called a neutrino (instead of anti-neutrino). An example of a beta plus decay is:
12
+
N → 12C + β + ν
In this case, to get the energy available for the decay from the atomic masses we need to be
careful in accounting for the electron masses. When we do this we find that the transition energy
(the kinetic energy shared by the positron and the neutrino) is less than the difference in atomic
masses of the parent and daughter. It is less by exactly 2 mo, i.e. two electron masses.
16
This apparent asymmetry in mass differences between beta plus and beta minus decay occurs
only because we are using the atomic masses. In terms of nuclear masses, the two processes are
symmetrical.
In our above example the mass of a 12N atom is 12.018703 AMU and the mass of a 12C atom is
12.000000 AMU. The mass difference is 0.018703 AMU. Multiplying this by 931 we have an
energy equivalent of 17.41 MeV. But the transition energy is only 16.39 MeV. The difference is
the energy equivalent of two electron masses (2×0.51 MeV).
If the atomic mass of the parent nuclide is not greater than the atomic mass of the daughter
nuclide by more than 2 mo, then beta plus decay cannot occur.
The decay is diagrammed as indicated. The arrow goes to
the left indicating a decrease in Z.
The zag in the arrow is to distinguish beta plus decay
from another possible decay mode (electron capture).
Some say that it represents the 2 moc2 in the atomic mass
difference, or the energy of the two annihilation gamma
rays which follow from the decay.
Nuclides that decay via beta plus decay are used in PET scanning (Positron Emission
Tomography). In PET scanning it is the two annihilation gamma rays that are detected. One of
the most common nuclides used in PET is 19F.
Gamma decay (Isomeric Transition) - As we saw in the case of 137Cs, beta decay frequently
produces an excited state of the daughter nucleus, which subsequently decays with the emission
of a gamma ray. Normally, this isomeric transition occurs almost instantaneously, but in a few
instances a metastable excited state occurs with a reasonably long time before the subsequent
gamma decay. In that instance, a pure gamma emitter can be chemically separated from the
parent nuclide.
The most common instance of this in medicine is
Technicium 99m (m for metastable). 99mTc
comes from the beta decay of 99Mo.
99m
Tc is useful in nuclear imaging studies because
the pure gamma emitter does not have any
contaminating beta particle associated with it.
17
Electron capture - Another mode of nuclear decay that can occur with proton rich nuclides is
electron capture. A proton changes into a neutron with the absorption into the nucleus of one of
the orbiting K electrons.
Electron capture can compete with β+ decay in proton rich nuclides, so a nuclide may decay by
more than one mode, with a probability of occurrence assigned to each mode. A single atomic
nucleus will decay by only one or the other mode.
Because a beta particle mass is not created in electron capture, there may be cases in which
electron capture is energetically possible, and β+ decay is not possible because of the 2 mo rule.
In this case we have “pure” electron capture.
An example of this is the nuclide Iodine-125. The decay
proceeds as follows:
125
I + e-
→
125
Te + ν
In electron capture, we write the energy level diagram
without the zag in the line with the arrow to distinguish
the mode of decay from beta plus decay.
Internal conversion - Sometimes instead of the excited state decaying by gamma emission, the
excess nuclear energy is released to an orbiting electron (usually a K electron) and the ejection
from the atom of the electron (now called a conversion electron). This will then result in
characteristic X-ray emission as an electron from a higher energy level drops down to fill the
vacancy in the K shell. In the case of 125I decay, two K shell vacancies are created, one due to
the electron capture and one due to the internal conversion.
Auger electrons - Sometimes instead of characteristic X-rays being produced, an electron fills a
K shell by giving its energy to another orbiting electron, which is ejected from the atom. This is
known as an Auger electron.
Nuclear fission - After absorbing a neutron, some nuclides will undergo nuclear fission. An
example is Uranium-235. The reaction is:
235
U + (slow neutron) → 236U (unstable)
236
U → 2 large chunks + neutrons +Q
For instance
236
U → 92Kr + 141Ba + 3 n + Q1
or
101
Mo + 133Sn + 2 n + Q2
( 3 neutrons)
(2 neutrons)
or many other possible reaction products.
18
The neutrons produced by the fission are called fast neutrons. They share the kinetic energy,
Q. They can be slowed down (moderated) and can cause further uranium fissions. This is the
principle of the nuclear reactor. A neutron that is slowed down to the point that it is in thermal
equilibrium with its environment is called a thermal neutron.
Nuclear reactors are useful for radiation therapy because they can supply:
1. Energy to power our accelerators
2. Neutrons for a form of neutron therapy
3. Fission Fragments-some of which are useful nuclides.
4. Neutrons for making useful radioactive isotopes through neutron activation
As an example of the latter cobalt-60 is made by putting natural occurring cobalt-59 in a reactor
and allowing it to absorb a neutron.
In general, isotopes produced in nuclear reactors are neutron rich and, therefore, tend to decay by
β- emission. A few, such as 125I decay by electron capture. Proton rich β+ emitters are not
produced in reactors. They are produced by bombarding materials with proton beams from
particle accelerators, such as a cyclotron. Positron emitters normally have short decay times.
Therefore, PET scanners must have a cyclotron nearby.
Nuclear fusion - Another type of nuclear transformation is nuclear fusion. In this case,
energetic nuclei are brought together to fuse into a single heavier nucleus with the production of
neutrons, and energy. The most common example involves the two heavy isotopes of hydrogen,
2
H (deuterium) and 3H (tritium). The reaction is:
2
H + 3H → 4He + n + Q
This reaction has been used in the past for neutron production in radiation therapy.
19
The mathematics of radioactivity
Activity (A) is a measure of a quantity of radioactive material in terms of the number of atoms
decaying per second. The units of activity are:
Bq (becquerel) = 1 nuclear disintegration/sec (the SI unit)
Ci (curie) = 3.7 x 1010 disintegration/sec (1 gram of radium has an activity of ~1 Ci)
The time it takes for a quantity of material to decay to one half its original activity is called the
half life, T1/2. The activity at any time, A(T), can be found from some initial activity A0
according to the equation:
Example:
Initial activity, Ao is 100 Bq. T1/2 = 3 hrs, current time t = 10 hrs,
A(t)
= 100 (0.5)3.333
what is A(t)?
= 100 (0.099) = 9.9 Bq
An equivalent form of the above equation for time, t, measured in hours (or minutes etc.) is:
A(t)= Aoe-λt
where λ is the decay constant (unit is 1/time)
e is Euler’s constant = 2.71828...
The relationship between  and the half life is
λ = 0.6931 T1/2
In the above example,
λ = 0.6931/(3 hours) = 0.231 hours-1 ;
λt = 2.31
A(t) = 100e-2.31 = 100(.099) = 9.9 Bq
The two forms of the equation for the decay are completely identical. Use whichever is most
convenient. One must be careful that the units for  are the inverse of the units for the time, t.
20
Activity is the negative of the rate of change of the number of atoms. The number of atoms
decays at the same rate as the activity decays.
N(t) =
number of atoms
Activity (Bq) = - dN/dt = -No(-λ)e
-λt
Activity (Bq) = λN
-λt
= Noe .
λt
= λ Noe-
= λN
-1
(λ is in seconds )
N = number of atoms = mass(grams) х (Avogadro’s Number)⁄(Atomic weight), therefore
Activity(Bq) = λ х mass х(Avogadro’s Number)⁄ (Atomic weight)
Activity(Ci) = Activity(Bq)⁄(3.7х1010)
Activity
-
exponential decay
When plotted on a semi-logarithmic scale the exponential decay curve appears as a straight line.
21
II- Particle interactions and X ray production
Elastic and inelastic collisions
Charged particles (electrons, protons, etc.) traveling through matter undergo collisions with the
atoms that make up the material. Collisions may be either elastic or inelastic in nature.
In an elastic collision the sum of the kinetic energies of the colliding particles is the same before
and after the collision.
Qincident particle = ∑Q(after collision)
In an inelastic collision some of the incident particle’s kinetic energy goes into changing the
state of the final particles so the kinetic energy is less after the collision.
Qincident particle > ∑ Q (after collision)
Examples - Bremsstrahlung (creation of photon, a new particle)
- Ionization or excitation of atoms
Heavy charged particles (α, protons, deuterons)
1.
-because of the large mass of these particles they undergo very little side scatter
- they travel in straight line
2.
- a monoenergetic beam of heavy particles has a well defined range
3.
- as they travel they lose energy via:
a. excitation - exciting an atomic electron into a higher energy shell
b. ionization - setting loose an electron from an atom, forming an ion pair
A charged particle’s range increases with the particle’s energy, but not in a linear fashion.
Range of Protons in Tissue
Energy (MeV)
range (cm)
10
20
40
100
150
0.12
0.43
1.51
7.87
16.10
For the same kinetic energy, a particle with a greater charge (e.g. an alpha particle) will have
shorter range.
An important measured quantity for radiation dosimetry is the W factor
22
W = average energy lost by the particle for every ion pair produced
= ~35 eV/(ion pair) in air for heavy particles with energies above a few MeV
Another important quantity for charged particles in matter is the LET (linear energy transfer).
The LET is the average energy loss per unit path length of the particle’s track.
The units of LET are (energy)/(distance) usually expressed as keV/micron(1 micron = 10-6m).
The LET varies considerably with the mass of the charged particle and with the particle’s
energy. The LET increases as the particle loses energy and slows down.
A third, related quantity is the specific ionization, SI, which is amount of ionization per unit path
length. The relationship between SI, LET, and W is:
LET(keV) = 0.001  SI(ions)  W(eVion pair)
Heavy particles show a dramatic increase in
SI (or LET) at the end of their range, at the
Bragg peak. In order to use such particles
for therapy the energy of the beam must be
varied so that the Bragg peak will be spread
out. (SOBP - spread out Bragg peak.)
Electrons (and β+ and β-)
A fundamental difference between the way electrons
and heavy particles interact with matter is that the
electrons suffer many lateral deflections as they
traverse the material. In addition, at any given energy
the electron’s LET is much less than that of a proton.
23
A monoenergetic beam of electrons in matter shows:
1
-significant lateral scatter
2
-a less well defined penetration depth compared to that of protons
3
-no Bragg Peak at the maximum penetration depth
4
-electrons lose energy via:
a. excitation and ionization interactions with the atoms
b. interactions with an atom’s nucleus
5
-the interactions with the nucleus are of two forms
a. elastic scattering - e.g. back scattering of electrons from a metal surface
b. inelastic - bremsstrahlung (X ray production)
Bremsstrahlung
When an electron undergoes bremsstrahlung all or
part of its kinetic energy can be converted into a
photon’s energy. The result is that when an
electron beam strikes a target, X rays are produced
with a spectrum of energies, with a maximum
energy equal to the energy of the electrons striking
the target.
A bremsstrahlung spectrum from 90 keV electrons striking
a tungsten target is shown in the diagram.
In addition to the continuous bremsstrahlung spectrum
there are characteristic X-ray peaks at energies
corresponding to electron transitions within the atoms of
the target.
24
Few X rays are produced at the maximum energy and, in a vacuum, many X rays are produced
at low energies. For X rays from an evacuated tube the spectrum goes to zero for low photon
energies because the very low energy photons can not penetrate the glass of the X ray tube.
They are filtered out.
Using the relationship between photon energy and
wavelength
Wavelength(Angstroms) = 12.4/Energy(keV)
we can transform the above curve to give the
spectrum in terms of the photon wavelength.
The spectrum shows two K characteristic X rays
Kα - electron from L shell fills K shell vacancy
Kβ - electron from M shell fills K shell vacancy
The L characteristic X rays are at ~9 keV (1.4 Å) and are not shown in the spectra.
In the energy range of diagnostic X rays and
superficial therapy X rays photons are produced
equally in all directions.
At higher energies the X rays are more likely to be
produced more forward to the direction of the
incident beam of electrons.
All of the above interactions of electrons also apply to positrons. Positrons, however, undergo
annihilation when they come in proximity with an ordinary electron, producing 2 gamma ray
photons, each of energy 0.51 MeV, i.e. the electron’s and positron’s masses are transformed into
the energy of the photons.
e+ + e- 
2 photons (Energy: 0.51 MeV each)
total 1.02 MeV of kinetic energy.
25
III- Cobalt teletherapy
A mathematical digression (Similar triangles)
Many problems of radiation therapy are solved using geometrical analysis. One of the most
common theorems used is that concerning similar triangles.
Two triangles are called similar if their corresponding angles are the same. If two triangles are
similar then the corresponding sides are all in the same proportion.
If α = δ and β = ε ,
then γ = ζ and
A/D = B/E = C/F
As an example we consider the length of a radiation field, diverging from a point source, at
two different source to surface distances (SSDs).
In the diagram the point source is at A. The triangles ABC and
ADE are obviously similar, therefore the sides are all in
proportion.
(1/2L1)/ (1/2L2) = SSD1 / SSD2
Thus the lengths of the field vary directly as the SSDs.
L1/L2 = SSD1/SSD2
Example - An old 4 MV linear accelerator for which the standard treatment SSD is 80 cm with a
maximum field length of 32 cm. What SSD is needed for 50 cm long field?
32/50 = 80/X
X = 125 cm
An SSD of 125 cm is needed.
Area
If the field length varies directly as the SSD and the field width also varies as the SSD, the area,
which is (length)X(width), varies directly as the SSD2.
26
The inverse square law
In free space (or air) the number of particles (or
X rays) traversing area A1 is the same as the
number traversing A2.
The intensity of the beam is the number of
particles traversing per unit area.
Intensity at SSD1 = (number of particles)/A1
Intensity at SSD2 = (number of particles)/A2
From this we infer the inverse square law for
radiation intensity
(Intensity at SSD2) / (Intensity at SSD1) = (SSD1)2 / (SSD2)2
Intensity varies inversely as the square of the distance from the source.
This rule comes about solely because of geometry and the fact that we have a point source for
the radiating particles or photons.
There are situations where the photons are not emitted from a point source. One case in which
the mathematics is simplified is that of a very long line source.
If we can neglect the ends of the cylinders (which is
possible for an infinitely long line source) then the
number of photons passing through the surface of the
larger cylinder (2) is the same as that through the smaller
cylinder (1).
Area of cylinder 1 = 2πR1h
Area of cylinder 2 = 2πR2 h
Intensity at R1 = (number of photons)⁄2πR1h
Intensity at R2 = (number of photons)⁄2πR2h
Therefore:
(Intensity at R2)⁄(Intensity at R1) = R1⁄ R2
For a long line source the intensity varies inversely as the first power of the distance.
For a realistic line source the situation lies somewhere in between the inverse square law and the
inverse first power.
27
For distances close to a radium needle the
dose follows closely the 1⁄R law.
For distances far from the needle the source
begins to look like a point, and the dose is
close to the value given by the inverse square
law.
Cobalt 60 teletherapy
Prior to the development of nuclear reactors and the production of artificial radioactive isotopes
Ra was the only source available for isotopic teletherapy. It was not an ideal source for
teletherapy because its long half life (1620 years) meant that the specific activity, the activity
per gram of material, was low, approximately 1 Curie per gram.
226
With radium the dose rates were low and the treatment distances were necessarily short.
In 1951 Harold Johns developed a 60Co
teletherapy unit. 60 Co with its short T1/2 and high
specific activity produced an intense gamma ray
beam.
Remember the specific activity for radium is ≈ 1 curie/gram
To calculate the activity of 1 gram of pure 60 Co we use:
Activity (Bq) = λN
= (decay constant)(# of atoms)
For 1 gm 60 Co the number of atoms are:
N = 1/60  (6.02 x 1023)
λ
= 1022 atoms
= 0.6931 =
0.6931
= 4.2  0-9 sec-1
5.26365246060 (seconds)
T1/2
22
-9
Activity = 10  4.2  10 = 4.2  10
13
28
decay/sec (Bq)
1 Curie= 3.7  1010 Bq
Specific Activity = 4.2  1013 = 1,128 Curie/gram
3.7  1010
We can never get pure 60 Co, but sources have been produced with specific activities of 150
Curie/gm, which is still much higher than that of radium.
60
Co is produced by bombarding pellets of natural 59 Co with neutrons in a reactor. The reaction
is:
59
Co + n  60Co +
γ
A typical teletherapy unit will house ≈ 10,000 Ci of cobalt in a cylindrical source with a diameter
of 1.5 to 2.0 cm. The source diameter has a significant bearing on the penumbra of the photon
beam.
The 60 Co is encapsulated in stainless steel to prevent Cobalt from leaking out into the
environment and also to absorb the β particle produced in the decay.
Source housing
Cobalt sources are housed in containers of lead or other heavy metals, with an aperture, the
primary collimator, through which the source is exposed during the treatment of the patient.
The exposure mechanism must have a fail-safe system so that the source returns to its shielded
position in the case of a loss of power.
Two examples of source exposure mechanisms are shown.
Source is positioned by air pressure.
Source is positioned by a motor
which is opposed by a spring.
29
Secondary collimators and penumbra
The penumbra is the region at the edge of a photon beam, at the transition from a fully exposed
source to a fully blocked or collimated source.
There are a number of phenomena that can
contribute to the size of beam penumbra. Two of
the contributions which depend on the design of
the cobalt unit are :
Transmission penumbra
Geometric penumbra
Transmission penumbra
Geometrical penumbra
Geometric penumbra occurs because the diameter of the source is not zero (not a point source).
In the diagram s is the source and p is the
penumbra. The ABC and CDE are similar
triangles, therefore their sides are in proportion.
From this we can conclude:
p ⁄ s = (SSD - SDD) ⁄ SDD
or
p = s (SSD - SDD) ⁄ SDD
This is the formula for the geometrical penumbra
on the patient’s surface. At depth the penumbra
increases as triangle CDE increases.
30
To decrease the penumbra on a cobalt teletherapy unit we can:
1. Decrease the source diameter - but a 1.5 - 2 cm source diameter is needed to have
sufficient activity.
2. Decrease the SSD - poorer depth dose and beam divergence
lose skin sparing due to electrons generated by collimator
and blocking tray
3. Increase SDD - again lose skin sparing due to electrons from blocks and tray
With some cobalt units the final vane of the collimator can be moved closer to the patient to
improve the penumbra in special situations. These collimator vanes are called “penumbra
trimmers”.
31
IV- Interactions of X rays and gamma rays with matter
The attenuation coefficient
X-rays and gamma rays (photons) interact with matter in a manner quite different from that of
charged particles. An individual photon can penetrate a significant thickness of material before
it interacts with an atom. With the interaction it may lose all of its energy or just part of its
energy with a lower energy scattered photon as a result.
A narrow monoenergetic beam of photons is attenuated by matter in an exponential fashion.
For a narrow beam few scattered
photons will hit the detector.
Thus a broad photon beam will
appear to be attenuated less than a
narrow beam.
For any absorber thickness the
detector will read higher for a
broad beam because of the greater
number of scattered photons
reaching the detector.
The narrow beam will be attenuated according to the
equation:
I(x) = I0e-μx,
where I(x) is the intensity through a thickness x,
I0 is the initial beam intensity, and μ is the linear
attenuation coefficient.
The broad beam will deviate somewhat from the simple
exponential attenuation.
The half value layer (HVL) is that thickness of an absorber that attenuates a photon beam to ½
of its original intensity. HVL normally refers to the case of narrow beams.
The HVL and the attenuation coefficient, μ, are inversely related to each other.
HVL = 0.6931/μ
The units of the HVL are usually millimeters or centimeters of a particular material. The units of
 will therefore be (millimeters)-1 or (centimeters)-1.
32
Mass attenuation coefficient
The linear attenuation coefficient is a measure of the fractional attenuation of the beam per
centimeter. It will be useful to consider a new quantity called the mass attenuation coefficient,
μρ. It is defined by:
μ⁄ρ = μ ⁄ (density of the material)
The units of μ⁄ρ will be (cm)-1 ⁄(g⁄cm3) = cm2⁄g.
Photons can interact with matter by a number of different processes. It is useful to separate the
attenuation coefficient into parts corresponding to the different modes of interaction.
μ = ω +τ + σ + π +
or:
σph,n
μ⁄ρ = ω⁄ρ + τ⁄ρ + σ⁄ρ + π⁄ρ + σph,n⁄ρ
where the components of  correspond to the following processes:
ω
-
coherent scattering
τ
-
photo-electric absorption
σ
-
Compton scattering
π
-
pair production
σph,n
-
nuclear photo-disintegration
Coherent (classical) scattering (ω)
Coherent scattering is elastic scattering of photons by atoms. It is important only for low energy
photons, below the X ray energy range. Coherent scattering is responsible for the blue color of
the sky.
Photoelectric absorption (τ)
In photoelectric absorption:
The photon disappears and an electron is
ejected from the atom
The kinetic energy of electron equals the
photon’s energy minus the electron’s binding
energy
An ion is produced.
33
The ejected electron produces many other ions as it travels through the material.
Characteristic X rays will also be produced because of the vacancy in the atomic shell.
The probability of photoelectric absorption greatest if the photon’s energy is just slightly greater
than the binding energy of the electron -- thus favoring interaction with K shell (if the photon has
enough energy). If the photon energy is less that K shell binding energy the K electrons can not
participate in photoelectric absorption (but L, M... etc. electrons can.)
Dependence of  on energy and atomic number
/ρ is proportional to (Energy)-3
therefore τ/ρ decreases rapidly as the
photon energy increases.
The jump in τ /ρ occurs when the
photon energy becomes greater than the
binding energy for that shell, allowing
the electrons in that shell to participate
in the absorption process.
3
/ρ is proportional to Z ,i.e., (atomic number)3. Therefore τ/ρ increases rapidly as the Z of the
attenuator increases. This is evident in the case of diagnostic X rays where bone (Zbone = 13.8)
attenuates much more than soft tissue (Zmuscle = 7.4).
bone    
muscle    
bone is 6.5 times greater than muscle.
Compton scattering (σ)
Compton scattering is elastic scattering of photon
and a “free” electron. We assume that the binding
energy of the electron is small compared to the
incident photon’s energy and can be neglected.
The scattered photon’s energy is equal to the
incident photon’s energy minus the energy
imparted to the Compton electron.
hνscatt = hνinc - Eelectron
34
Scattered photons can interact again further downstream.
There is a complex relationship between hνscatt , the scattering angle φ, and the parameter α.
α = hνinc /moc2 , where moc2 is the energy of an electron’s mass (0.511 MeV).
The equations are simplified for very high energy photons, α » 1 and for low energy photons,
α « 1.
Special cases for Compton scattering
Direct hit:
Maximum energy transferred to the electron. Photon is back scattered (φ = 180 0),
electron is forward scattered (θ = 0 0)
Eelectron = hνinc2α /(1 + 2α)
 hνinc − (moc2 )/2 (for α » 1)
hνscatt = hνinc (1 + 2α)
 (moc2 )/2 = 0.256 MeV (for α » 1)
Grazing impact:
Minimum energy transferred to the electron. Photon is forward scattered (φ = 00).
Electron is scattered at θ = 900.
Eelectron = 0;
hνscatt = hνinc.
900 photon scattering (φ = 900 ):
Eelectron = hνincα (1 + α)
2
 hνinc − moc (for α » 1)
hνscatt = hνinc (1 + α)
 moc2 = 0.511 MeV (for α » 1)
Dependence of  on energy and atomic number
 / ρ is proportional to (Energy)-1. Since τ /ρ decreases more rapidly with photon energy,
(Energy)-3, at some energy Compton scattering will become a more probable interaction than
photoelectric absorption.
35
/ρ is proportional to Z A, i.e. (Atomic number /Atomic mass number). For most biological
materials Z A is about 1/2 , σ/ρ is said to be independent of Z, except for hydrogen for which
Z/A is 1.0.
The quantity
ρe = ρNA(Z/A) is called the electron density, where NA is Avogadro’s
number. ρe describes the number of electrons per cm3 of material. For most biological
materials NA( ZA) is about 31023 electrons per gram, except for hydrogen, where it is twice
that number. We say that σ is proportional to the electron density and tissue that has a lot of
hydrogen will exhibit increased Compton scattering.
Pair Production ( π )
In pair production the photon interacts primarily
with the nucleus.
The photon disappears and an electron-positron
pair is produced.
The energy of the photon becomes the masses of
the two particles plus their kinetic energies.
The photon must have an energy greater than 1.02
MeV (= 2moc2) for pair production to occur.
π is proportional to Z, i.e., it is more probable for high Z materials.
ρ increases with energy (above the threshold of 1.02 MeV) and will be the dominant
interaction for very high energy photons, above the energy range normally used for therapy.
Nuclear photo-disintegration (σph,n)
If a photon has sufficient energy it may be absorbed by a nucleus, causing the nucleus to emit a
neutron (usually) or other particle. The resulting transformed nucleus is radioactive.
An example of this is:
γ +
16
O 
15
O + neutron
For 16O the photon energy must be greater than 15.6 MeV.
T1/2 of 2 minutes.
Another example is:
γ +
Energy threshold = 10.5 MeV.
13
14
N 
15
13
O under goes beta+ decay with a
N + neutron
N has a T1/2 of 10 minutes.
36
σph,n is always less than π (pair production) so nuclear photo-disintegration is not important
for radiation dosimetry purposes. Although patients can become slightly radioactive from
photons with energies above 10 MeV, the dose from the induced radioactivity is negligible. The
neutrons which are produced by high energy photons are significant for radiation protection
considerations.
Mass Attenuation Coefficients vs. Photon Energy
The mass attenuation coefficients for muscle and bone are very similar except at the low energy
region, where the photoelectric effect is the dominant process (Hubbell, 1969). For very high Z
materials such as lead the photoelectric effect remains dominant at energies up to 1 MeV.
37
V- Conventional X-ray Machines
Diagnostic and therapeutic X-ray units
Conventional x-ray machines accelerate electrons across a potential difference (voltage) of up
to 300 kilovolts. A tungsten target (high Z) is the anode (positive). The cathode (negative) is a
hot filament which is the source of electrons. The X-ray tube is evacuated. Medical x-ray
machines are of two types, diagnostic and therapeutic.
In simplest form both types of X-ray units are
similar.
Intensity of X rays depends on the voltage
(kV, kilovolts) and the tube current (mA,
milliamps).
The kV determines the energy of the
electrons striking the target.
The mA determine the number of electrons
per second hitting the target.
Note: The direction of the current flow is opposite to the flow of the electrons.
In addition to radiation (bremsstrahlung), electrons lose energy in the target by the collision
processes, ionization and excitation. This energy eventually ends up as heat in the target. The
difference in diagnostic and therapeutic tubes is related to how the problem of heat is dealt with.
The ratio of radiation to collision energy loss increases as the electron energy increases and as
the Z of the target increases (hence tungsten targets).
At 100 keV 99% of the electrons’ energy goes into heat.
Diagnostic units are required to produce
sharp images. This means short exposure
times and small electron focal spots at the
target. (Our discussion of penumbra versus
spot size can also be applied to image
sharpness.)
The heat is deposited in the target for a
short time in a small area. For this reason
diagnostic tubes employ rotating anodes to
distribute the heat over a wider area.
Diagnostic X-ray (e.g. simulators)
tube voltage 50 to 140 kV
38
tube current 125 mA maximum
Dual focal spot 0.6 and 1.2 mm
Darkness of image (i.e. exposure) depends on kV, mA and exposure times. Frequently, we just
set the product of current and time, i.e. mAs (milliamp-seconds) since exposure is directly
proportional to the product of the two.
Therapy units are of two types: Superficial- 50 to 150 kV.
Orthovoltage- 200 to 300 kV.
focal spot about 5 mm.
tube current 10 to 20 mA.
For therapy the mA is usually set for the particular operating conditions, i.e. is not variable.
Exposure is determined by setting the time, usually minutes (instead of seconds). Heat buildup
is not instantaneous, but would become very large over a long exposure. For this reason heat in
the target is removed by circulating oil through the anode.
In this energy range the useful X-rays emerge perpendicular to the electron beam. This is a
reflection target in contrast to a transmission target for higher energy beams.
The line focus principle
The filament is actually a line source.
Because of the angle of the anode, the apparent focal spot
size is smaller than the length of the focal spot on the
anode.
The two lengths are related by the equation:
a = A  tan θ
X-ray circuitry
In order to produce the necessary high voltages, we
use the principle of the transformer. A transformer
consists of a pair of coiled wires which are electrically
insulated from each other but which are coupled
magnetically, frequently by an iron core.
One set of wires is called the primary (the input) and
the other set is the secondary (the output).
39
The transformer only works with time varying or alternating current, AC. The ratio of secondary
voltage to that of the primary depends on the ratio of the number of loops of wire in the primary
and the secondary.
The relationship between the primary voltage, V1, and the secondary voltage, V2 is
V2 V1 = N2 N1
where N is the number of turns of wire.
If N1 < N2 we have a step up transformer, V2 > V1 ;
If N2 < N1 we have a step down transformer, V2 < V1
Simplified schematic diagram of the main components of an X ray circuit
Key to the diagram:
A xfmr:
Autotransformer- Allows the input voltage (line voltage) to the primary of the High
Voltage transformer to be varied. In effect, sets the kV at the tube.
F:
Filament of the X ray tube (Cathode)
FC:
Filament Control- Controls the current to the filament. Sets the mA
in the X ray tube
F xfmr:
Filament transformer (step down)
HV xfmr:
High Voltage transformer (step up). Provides the kV to the tube.
mA:
Meter that reads the mA in the tube.
sw:
Switch to turn the kV on and off. Controls the X ray exposure time.
T:
X ray target (Anode)
V:
Meter that reads the input voltage at the HV transformer primary. In effect, reads the
kV at the tube
40
The current in the filament circuit determines the temperature of the filament (cathode) and,
therefore, how many electrons go to the anode, i.e., the tube current.
In a self-rectified X-ray unit (e.g. some dental X-ray units), the tube itself allows the current to
flow in only one direction (from the anode to the cathode). In more powerful units, the hot
anode itself can “boil off” electrons which can then flow to the cathode during part of the AC
cycle.
To avoid this, rectifiers are put into the high voltage circuit.
Rectifiers are devices, symbolized by
, which allow current to flow in
one direction, the direction indicated by the “arrow”, but which prevent the current flow in the
opposite direction.
Half Wave Rectification
When the voltage at the anode side of the transformer (H) is negative the rectifier prevents this
voltage from acting on the anode of the tube. The X-ray tube is, in effect, disconnected from the
transformer for half of the AC cycle.
Voltage vs. time at transformer (H)
Voltage vs. time at Anode (D)
41
Single Phase (1 Ø) Full Wave Rectification
With the help of a full wave rectification circuit the X-ray tube is on for both halves of the AC
cycle. X-ray production still exhibits peaks and valleys.
When A is positive and H is negative, the
current flow is A-B-C-D-E-F-G-H.
When A is negative and H is positive, the
current flow is H-G-C-D-E-F-B-A. In
both cases, the current flow and the
voltage across the tube is always positive.
Voltage versus time at transformer
Voltage versus time at Anode
Three Phase (3 Ø) Full Wave Rectification
By summing together three separate full wave rectified voltages that are each 120O out of phase
with the others, the voltage (kV) at the anode becomes nearly constant in time. The amount of
variation in the kV is called ripple.
Because the high voltage at the anode of
the X ray tube varies with time, the
average voltage is less than the
maximum or peak kilovoltage. We
usually denote the peak kilovoltage by
kVp.
42
Filtration
Because the bremsstrahlung process produces an X ray
beam with a spectrum of energies, with a maximum
energy (in keV) given by the kVp, the average energy
is always significantly less than the maximum. In the
diagnostic and orthovoltage energy range one can
increase the average photon energy by placing a sheet
of metal, called a filter, in the beam. This is called
hardening the beam. With high Z materials, where the
photoelectric process is dominant, low energy photons
are more likely to be absorbed by the filter than high
energy photons resulting in an increase in the average
energy of the beam.
Depending on the kV of the X ray unit, different metals are used to harden the beam.
Voltage (KV)
Filter material
10-120
100-250
> 250
A1
Cu + A1
Sn + Cu + A1 (Thoraeus)
A Thoraeus filter is a compound filter of tin, copper, and aluminum.
The attenuation coefficient decreases sharply at a photon energy just
below the K edge (29 KeV). Thus the tin is not a good filter at these
energies. In addition, characteristic X rays are produced by the tin.
The copper will absorb the X rays at energies below 29 KeV. The
aluminum will absorb characteristic X rays produced in the copper.
When using a compound filter the higher Z material is always
placed closest to the source of the X rays. Otherwise the
characteristic X rays produced in the material will not be absorbed.
Quality (penetrating power) of an X ray beam will depend on:
1. kVp
2. 3 vs 1 (for a given kVp 3 will have a greater average voltage than 1)
3. Amount and type of filtration
Therefore, kVp is not a good descriptor of the penetrating power of an X ray beam.
43
The HVL (half value layer) is the best single measure of quality for conventional X-rays.
As described earlier, to measure HVL you need “good geometry”- minimum scattered X-rays
reaching the detector.
- narrow beam
- absorber away from detector
It is important not to confuse the beam filtration, which is a piece of metal inserted into the
beam, with the HVL, which is the result of a measurement on the beam.
Typical X-ray beam qualities used in therapy are:
Accelerating Voltage
Filtration
80 KVp
120 KVp
180 KVp
200 KVp
250 KVp
300 KVp
2 mm A1
2 mm A1
0.2 mm Cu
0.5 mm Cu
1.0 mm Cu
Th III*
2.5 mm A1
4.0 mm A1
0.5 mm Cu
1.0 mm Cu
2.0 mm Cu
4.0 mm Cu
4 MV
6 MV
10 MV
18 MV
none
none
none
none
11 mm Pb **
12 mm Pb **
14 mm Pb **
14 mm Pb **
*
Half Value Layer
There are three different types of Thoraeus filters with differing thickness of Sn, Cu, and Al.
The HVL (using a metal such as Pb) is not a particularly useful measure of beam quality for
megavoltage X ray beams.
**
44
VI- Megavoltage electron accelerators
Linear Accelerators
Linear accelerators for protons or heavier particles have been around since 1928.
An early linear accelerator for proton beams
Protons are sent down a series of tubes with alternating positive and negative high voltages. The
protons experience acceleration at the gaps between the tubes. The gaps alternate between
accelerating and decelerating voltages. However, the AC voltage varies in time so that by the
time the protons reach the next gap the polarity of the voltage switches. Thus the protons
experience only acceleration as they traverse the gaps.
At megavoltage energies electrons are move at a speed very close to the velocity of light,
whereas protons travel much more slowly. With the faster moving electrons the voltage at the
accelerator sections must change much more rapidly. Consequently the frequency of the AC
voltage must be much higher for an electron accelerator, around 3000 megahertz (3 gigahertz).
Electric fields which oscillate at this high a frequency are called microwaves. Microwave
technology became practical with the development of radar.
Two different types of microwave power sources have been developed, the magnetron and the
klystron. Every electron linear accelerator uses one or the other of these devices.
There are two types of electron linear accelerators, travelling wave and standing wave. Of the
two the travelling wave design is simpler and more easily understood. It is illustrated below.
45
Traveling wave linear accelerators
Instead of tubes lined up on an axis we have a single tube, called a
wave guide, divided up into sections by discs with central holes.
The sections are called cavities.
Microwaves are injected into one end of the accelerator wave guide. At this end electrons are
also injected. Those electrons which encounter the accelerating portion of the electric field will
travel down the tube in synchrony with the field. Microwaves are dumped at the end of the
waveguide into a sink where they are absorbed and their energy is converted into heat.
Standing wave linear accelerators
In a standing wave accelerator the microwaves are not dumped at the end but instead are
reflected back. If the accelerator is “tuned” to the frequency of the microwaves, a standing wave
pattern is set up as illustrated below.
46
The standing wave pattern alternates in time between accelerating and decelerating phases and
also alternates in space down the wave guide. Separating the accelerating/decelerating cavities
are null cavities which do not provide acceleration at any time. Since the null cavities (or
coupling cavities) do not affect the electron beam a better design has them moved off to the side
of the structure. This side coupled standing wave accelerator, illustrated below, is a highly
efficient accelerator, capable of producing a 6 MeV electron beam in a length of 30 cm.
A 6 MeV accelerator of the standing wave
design can be mounted vertically on a gantry
with the X rays emerging from a target at the
end of the accelerator. The gantry can rotate the
accelerator around the patient as illustrated. The
center of rotation is called the isocenter.
Usually the isocenter is 100 cm from the X ray
source.
For a traveling wave accelerator, or a standing wave
accelerator above 6 MeV, the length of the tube
makes a vertically mounted accelerator impractical.
Therefore the accelerator is mounted horizontally in
the gantry. The electron beam is bent 90o, or more
often 270o, after leaving the accelerator wave guide.
47
A block diagram of the major components of a standing wave electron linear accelerator is
shown below.
Accelerator structure- accelerates electrons using an electric field oscillating at microwave frequency.
AFC (automatic frequency control) system- senses the optimum operating frequency of the accelerator
structure and tunes the klystron or magnetron to this frequency.
Bending magnet- directs a horizontal electron beam towards the patient, e.g., vertically down if the
accelerator is above the patient.
Circulator- Allows microwave power to pass through to the accelerator but prevents microwaves that are
reflected back from reaching the klystron or the magnetron.
Electron gun- the source of electrons. A high voltage pulse injects electrons from a hot cathode into the
accelerator structure.
Klystron or magnetron- the source of microwaves for the accelerator.
Modulator- provides the high voltage pulse to activate the electron gun and the klystron or magnetron.
The modulator converts ordinary AC voltage into a high voltage pulse of ~ 4 microsecond
duration. The dose rate at the patient can be controlled by varying the number of pulses per
second (e.g., 30 to 300 pulses per second).
48
The Treatment Head
After the beam leaves the bending magnet it impinges on the target to produce the X ray beam.
If electron beam is desired the target is retracted. Treatment head for photon beams and for
electron beams are shown below.
Primary collimator- The x ray target is surrounded by lead or other heavy metal shielding to
minimize radiation exposure outside the beam area. The primary collimator is an opening to
allow the beam to come through. The opening, usually circular, defines the maximum field size
at the standard treatment distance. In many accelerators this is a circular field with a diameter of
50 cm. The moveable collimators confine the beam further to an adjustable rectangular field.
Ion chambers- Parallel plate ion chambers monitor the output (dose rate) from the accelerator.
They can be set to turn off the machine whenever the prescribed dose is received by the patient.
A pair of chambers is used in order to provide redundancy in case one of the chambers fails to
act properly. The chambers can also monitor the symmetry of the beam profiles along the two
axes perpendicular to the beam direction.
The readings from the chambers are usually set so that 1 monitor unit equals 1 centigray at the
standard treatment distance at d-max for a 10 cm X 10 cm field.
Flattening filter- At megavoltage energies the X rays are produced primarily in the forward
direction, along the beam axis. The flattening filter is a cone shaped object which preferentially
absorbs photons on the central axis, producing a more uniform beam profile at the treatment
distance.
49
If the beam is uniform at the patient’s surface (at d-max), then at a depth within the patient it
will be somewhat non-uniform due to the greater number of scattered photons on the central axis
of the beam compared to off axis points and also because the beam on the central axis is a little
harder (more penetrating) than that on the periphery. Therefore flattening filters are designed to
produce a higher dose at d-max on the periphery of the beam compared to that on the central
axis. This will result in better uniformity at depth. These high dose regions are called the horns
of the beam.
Illustrating the horns for a 32 cm X 32 cm photon beam from a 4 MV linear accelerator.
50
Moveable collimators- The moveable
collimators (or jaws) can be set to project a
rectangular treatment field. The collimators
may be either symmetric, in which case the
two jaws maintain the same distance from the
beam central axis, or asymmetric.
The collimators must tilt to align with the
divergence of the beam edge. The tilt angle
varies from 0o for x2 = 0. to 11o for a fully
opened field, (projecting 20. cm. at 100.0 cm
SSD.)
X ray beam shaping- Most often a non-rectangular field is required. An irregular field may be
shaped by a blocks made of Lipowitz's metal which has a high density (~9 g/cm3) and a low
melting point (70O C). This metal is an alloy consisting of 50% bismuth, 27% lead, 13% tin and
10% cadmium. The shaped blocks are mounted down stream from the moveable collimators.
The blocks are 7.6 cm thick and should transmit no more than 5% of the X ray beam.
Scattering foils- When electron beams are used the flattening filter is replaced by scattering
foils. As the beam traverses the foils, some electrons suffer an angular deflection from the
incident beam direction. This will produce a broadened beam at the treatment distance.
In lieu of scattering foils, some accelerators use magnetic fields to deflect the beam laterally and
scan the beam across the patient.
Electron applicator- Because of scattering, the edges of a collimated electron beam become less
well defined as the distance from the collimator to the patient increases. For this reason the final
collimator for an electron beam, the electron applicator, is placed close to the patient’s skin.
51
Multi-leaf collimators
An alternative method of creating shaped X ray fields is through the use of multi-leaf collimators
(MLC). The MLC consists of two opposing banks of bars made of high density tungsten alloy
(density ~ 18 g/cm3).
The bars, or leaves, can move in and out to shape
the field, approximating the desired profile. The
various accelerator manufacturers utilize very
different designs for their MLCs. In one case the
MLC replaces the upper jaws, in another the
lower jaws, while a third adds the MLC as a
tertiary collimator, downstream from the lower
jaws.
In this discussion X is the distance that an
individual leaf edge projects from the beam
central axis and Y is the projection of the position
of the leaf in the array.
No matter which configuration a manufacturer chooses certain factors must be incorporated into
the design.
1-Interleaf X ray transmission. In order to prevent X rays
traversing the spaces between leaves a tongue and groove
(as illustrated) or similar design must be employed. The
amount of interleaf transmission is an important parameter
in the evaluation of MLC designs.
2. Beam divergence- This must be accounted for in two ways. The
individual leaves must be tilted (vertically) so that the face of the leaf
follows the beam divergence. The angle will depend on the position of
the leaf in the array, a greater angle the further the leaf is away from the
beam central axis.
If Yi is the distance the leaf projects at 100 cm then the angle of tilt will
be
tan-1(Yi⁄100)
This angle will vary from 0O for Yi = 0, to 11O for Yi = 20 cm.
52
To take care of divergence in the X dimension a leaf needs to tilt horizontally as it is moved
away from the beam center. MLCs which perform this motion are called doubly focused
collimators, since they follow divergence in both X and Y directions.
An alternative design is to have a curved end to the
leaf so that the beam edge is always tangential to
the leaf edge. This design results in somewhat
greater transmission penumbra but simplifies the
construction of the MLC.
Multi-leaf Collimators are useful in two ways. First, they dispense with the need to fabricate
field shaping blocks. Secondly, if the leaves are programmed to move during a treatment, a
varying X ray intensity profile will be created. Treatments delivered in this manner are called
intensity modulated radiation therapy or IMRT.
Other types of electron accelerators
Betatron – The betatron as a megavoltage X ray unit preceded the electron linear accelerator.
As the name indicates, the betatron can only be used to accelerate electrons. The electrons
circulate in a evacuated toroidal tube called the “doughnut”. The doughnut is in a time varying
(AC) magnetic field.
The electrons are accelerated according to
Faraday’s law, the same principal that causes the
electric current to flow in an electrical generator.
As the magnetic field varies from zero to its
maximum value the electrons are accelerated to ever
higher energies. When the magnetic field reaches its
maximum the electrons are made to strike a target, to
produce X rays, or are extracted as an electron beam.
Betatrons require very large and cumbersome electro-magnets and are no longer commonly used
53
in radiation therapy departments.
Microtron – A microtron uses a large DC (not time varying) magnet to keep electrons in
circular orbits. The radius of the orbit increases as the energy of the electron beam increases.
The acceleration is provided by a microwave cavity (or cavities) as in a linear accelerator. The
electrons make many passes through the cavity until they reach the desired energy.
After the electron beam is extracted
the beam is steered (again by
magnets) to a gantry in a treatment
room. A single microtron can be
made to supply beam to more than
one room.
Quality assurance of medical accelerators
Each megavoltage therapy unit (Cobalt-60 unit, linear accelerator, or microtron) must be
thoroughly tested prior to the initiation of patient treatments. Many parameters are measured
during acceptance of the unit. Some of the more basic parameters are:
Machine output: The dose per monitor unit (or per unit time for cobalt-60) for a standard
operating condition, usually a 10 cm X 10 cm field at 100 cm SSD and a standard depth.
This must be determined for all X ray and electron modes.
Variation of the output with field size.
Variation of the dose with depth in a phantom.
The symmetry and flatness of the beam at points off the central axis of the beam.
Transmission factors of the wedge filters.
Congruence of the localizing light field with the radiation field.
Stability of the field isocenter with gantry, collimator, and couch rotation.
Coincidence of the localizing lasers with the accelerator isocenter.
These parameters are also checked on a routine basis, either daily, weekly, monthly depending
on the particular parameter being tested. In addition a full calibration of the unit, repeating all
the measurements of the initial acceptance, is performed on a yearly basis.
The AAPM (American Association of Physicists in Medicine) has published recommendations
54
for the QA of medical accelerators (Kutcher et al, 1994). Some of the recommended
QA measurements along with the tolerance (which will require corrective action if exceeded) are
listed below.
Frequency
Daily
Parameter
X ray and electron output constancy
Localizing lasers
Optical distance indicator
Tolerance
3%
2 mm
2 mm
Monthly
X ray and electron output constancy
X ray depth dose constancy
Electron depth dose constancy
Light/radiation field coincidence
X ray and electron symmetry constancy
X ray field flatness constancy
Electron field flatness constancy
2%
2%
2mm for 80% of max dose
2 mm
3%
2%
3%
Yearly
Full calibration of all machine parameters
55
VII- Heavy particles and neutrons
Proton beam therapy
Beams of protons have some features which make them attractive for radiation therapy. High
energy protons are considered to be low LET particles except at the end of their range, in the
Bragg peak, with kinetic energy less than 1 MeV, where the LET is high (ICRU, 1998).
LET (keV/micron)
30
25
20
15
10
5
0
1
4
16
64
250
Proton Energy (MeV)
Another feature of proton beams is the dose distribution. Beyond the Bragg peak the dose is
zero. Also, the tissue upstream from the Bragg peak is spared with a lower dose.
The Bragg peak, however, is too narrow in depth to be useful in delivering dose to most tumors.
The Bragg peak is usually “spread out” in depth by varying the energy of the proton beam and
therefore varying the range (Hall, 2000). This “spread out Bragg peak” is illustrated below.
Protons undergo very little lateral scatter. Thus they can be useful when high precision
radiotherapy is required.
56
Beams of protons and heavier particles such as deuterons ( 2H ions) are produced in a cyclotron.
In a cyclotron the beam is kept in a circular orbit by a constant or DC magnetic field. The
acceleration occurs when the beam crosses the gap between “D” shaped electrodes called “dees”.
An AC voltage, varying at radio
frequency, is maintained between the
electrodes. The RF is chosen so that the
beam will always experience a positive
acceleration as it crosses the gap. As the
energy of the protons increases, the radius
of their orbit increases, and the beam
spirals outward from the center until the
beam is extracted from the machine.
Another type of accelerator which can
be used to produce a proton beam is
the proton synchrotron. In the
synchrotron the beam is kept in a
fixed orbit by magnets whose field
increases as the proton energy
increases. Acceleration is provided by
microwave cavities. The protons are
injected into the ring from a linear
accelerator.
Neutron therapy
Neutron beams can also be produced in a cyclotron. A proton beam or a deuteron beam of
energy greater than 30 MeV is directed against a beryllium target and a neutron beam is
produced. Because neutrons are not electrically charged the attenuation of a neutron beam in
matter is similar to that of a photon beam. A 66 MeV deuteron beam produces a neutron beam
with an attenuation in tissue similar to a 6 MV X ray beam.
However, unlike photons neutrons do not release electrons as they interact with tissue. Most of
the interactions are elastic scattering with the nuclei of the elements which make up tissue, i.e.,
carbon, oxygen, nitrogen, and hydrogen. In elastic scattering the maximum energy is transferred
when the masses of the interacting particles are the same (i.e., hydrogen nuclei). Thus, the
majority of the neutron beam’s energy is converted to energetic protons.
57
We can therefore make the comparison:
Photon beams  electrons  ionization
Neutron beams  protons  ionization
A neutron beam will deposit more energy in tissue with a greater hydrogen content, e.g. fat, than
in other types of tissue.
Other types of neutron interactions- These are relatively unimportant for most neutron beams but
do increase in importance for higher energy neutrons.
Inelastic scattering- A neutron can transfer energy to a nucleus by excitation. The excited
nucleus will subsequently release a  ray. Example
n +
12
12
C  n +
C* 
12
12
C* (excited)
C + 
Neutron capture. Example
n +
1
2
H 
H + 
Spallation- A neutron can chip off “chunks” like alpha particles from a nucleus. Example
n +
12
C  3 alphas
+
n
Boron neutron capture therapy
Neutrons which can be obtained from a reactor at a very low energy, less than 0.5 eV, are called
thermal neutrons. The element boron has a very large probability for capturing thermal neutrons.
The reaction is:
n + 10B 
7
Li +  + (2.3 MeV of energy)
The alpha particle and the lithium nucleus which emerge from this reaction are both high LET
particles. Thus this process has a potential advantage over conventional radiation therapy. In
practice the patient is given compounds with boron which go to the tumor, e.g., in the brain. The
tumor is then exposed to a beam of thermal neutrons which produces the high LET radiation
where the boron has been taken up.
A major difficulty is the low penetrability of the thermal neutrons.
58
DT (deuteron-tritium) reaction
An alternative to using a cyclotron to produce a fast neutron beam. Bombarding a tritium ( 3H )
target with deuterons ( 2H ) will produce helium and a 14 MeV neutron.
2
H + 3H  4He
+ n
The energy of the deuteron needs only to be around 100 keV for the reaction to proceed. These
devices, however, have not proven to be practical for radiation therapy purposes.
Pi meson therapy ( π )
Pi mesons are sub-atomic particles that exist in the nucleus but are not found free in nature
because of their short half life (T1/2 = 2.6 x10-8 sec ). They are called mesons because their mass
is between that of an electron and a proton.
Mass of a pi meson = 140 MeV (e- 0.51 MeV, proton 937 MeV)
Mesons come in three charge states π+, πo(neutral), and π . The negative charge state has been
proposed for radiation therapy.
To an atom a π resembles a heavy electron. It can form an orbit around a nucleus, but will be
quickly absorbed by the nucleus because of the nuclear force. The nucleus then has an excess
energy of 140 MeV and explodes into high LET particles. The high LET particles at the end of
the track of a pi minus meson are called “stars”.
Pi meson therapy has not yet proved to be practical.
59
VIII- X ray and gamma ray exposure and dose
X ray and gamma ray quantity
Beam quality refers to the average energy of an X ray or gamma ray beam, specified by the half
value layer or the penetration in water for megavoltage beams.
Beam quantity refers to the number of X ray photons or the amount of energy flow in an X ray
beam. A number of different measures of beam quantity can be used.
1.
Photon Flux density , a measure of the number of photons traversing a unit area in
a unit time (photons/(area-time)).
2.
Photon Fluence:
 =  (t)dt
3.
Energy Flux density :
 = h
i.e., # photons/area.
(energy x photon flux density) = (energy/area-time)
4.
Energy Fluence

=
 (t)dt
energy/area
Although all of the above quantities are important for fundamental X ray physics, they are not
used much in practice because of the difficulty in measuring them. A more practical unit of X
ray quantity is the roentgen.
The roentgen is defined by a measurement of ionization in air by X or gamma rays.
Specifically it is that quantity of radiation that produces an ionization charge (either + or -) of
2.58X10-4 coulombs/(kg of air) when all the electrons liberated within that volume have been
completely stopped in air.
Units:
coulomb – a measure of electric charge. The amount of charge that a current of 1 ampere
produces in 1 second.
Thus, 1 ampere = coulomb/sec.
The charge of one electron is 1.6 x 10-19 coulomb.
The definition of the roentgen is intimately tied to how it is measured, which is with the free air
ionization chamber. This is a device for measuring the amount of ionization produced in a
defined volume of air. This device is illustrated below.
60
Ions produced in the air volume will be attracted to the collecting plates (positive ions to the
negative plate and vice versa). The electrometer will integrate the ionization current to
determine the charge produced in the volume during an exposure.
In order that the free air chamber properly measures all the ionization certain conditions must
hold. These conditions are:
1. Electronic equilibrium- Some of the electrons which are liberated will escape the
sensitive volume, and the ionization they are producing will be lost. If electronic
equilibrium holds, the loss of this ionization will be off set by electrons that enter the
volume having been liberated outside it. This is illustrated below.
2. Plate separation- The separation between the air volume and the collecting plate must
be greater than the range of the most energetic electron in order that all the electron’s
energy be deposited in the air.
61
Both of the above conditions can not be satisfied for very high energy X ray beams. For this
reason the roentgen is not defined for X ray energies above 3 MV.
3. A third condition for measuring X ray quantity with a free air ion chamber (or any ion
chamber for that matter) is that the voltage between the collecting surfaces be high
enough that all the ion pairs produced in the air are collected. The chamber is then
said to be operating in the saturation region. At a voltage below the saturation
voltage some of the ion pairs may recombine before the can be collected. This is
illustrated by the graph below.
60
50
40
saturation region
30
20
recombination
10
0
Voltage between collecting surfaces
Above the saturation region additional ionization is produced by the ions themselves as they are
accelerated to the collecting plates.
Practical ionization chambers- The thimble chamber
The free air ionization chamber is too delicate and bulky for use in the field. It is only used by
the National Institute of Standards and Technology (NIST) to provide a national calibration
standard. In order to reduce the size for a practical ionization chamber electronic equilibrium
must be established in a small volume. This is done by surrounding the ion collecting surfaces,
the electrodes, with material with a density much greater than that of air. The composition of the
wall material must “air equivalent” as described below.
The most common form for a practical chamber is the
thimble chamber which consists of an inner electrode,
an aluminum wire, surrounded by an outer electrode,
which in turn is surrounded by a sufficient amount of
“air equivalent’ material to ensure electronic
equilibrium.
Electrons that are liberated in the chamber wall produce
ionization in the air volume. The ion pairs are collected
by the inner and outer electrodes, which are oppositely
charged.
62
For the purpose of the thimble chamber, the wall material is considered to be air equivalent if its
effective atomic number is close to the effective atomic number of air. A formula for the
effective atomic number has been proposed by Mayneord.
Zeff =
2.94
[ a1(Z1)
2.94
+ a2(Z2)2.94 + …]
ai = fraction of all the electrons contributed by the ith element
ZI = atomic number of the ith element in the material
From this equation we find that the Zeff for air is 7.67. Common materials for the walls of
thimble chambers are graphite (Z = 6) or various types of plastics.
Example: A thimble chamber has an air volume of 0.6 cm3. This is typical of many ion
chambers in use today. How much charge will be produced within the air volume if the chamber
is exposed to 1 roentgen of X-ray photons?
1 roentgen = 2.58 x
-4
10 coul/kg (air)
Density of air at room temperature and standard pressure = 1.196
gm/liter
= 1.196 x 10
-6
-6
kg/cm3
-6
Mass of air in chamber = 0.6 cm3(1.196 x 10 ) kg/cm3 = 0.718 x 10 kg.
(2.58 x
-4
-7
-10
10 ) (7.18 x 10 ) = 1.85 x 10
coulomb
This is a very small charge to be detected by the electrometer. Very sensitive electrometers are
needed to calibrate X ray units.
If the air temperature or pressure deviates from the standard conditions the mass of air in the
chamber will change and the charge produced will be different for the same X ray exposure.
Thus temperature and pressure must be accounted for in any X ray unit calibration.
Generally we can not determine the air volume in a thimble chamber to the degree of precision
necessary for X ray dosimetry (uncertainty less than a fraction of a percent). In addition the
presence of the aluminum electrode can cause the chamber to deviate from strict air equivalence.
Therefore a chamber must be calibrated at an ADCL (Accredited Dosimetry Calibration
Laboratory). The laboratory determines a calibration factor, Nx, which converts the electrometer
reading (Rdg) to exposure in roentgens. The calibration factor is valid for a certain atmospheric
temperature and pressure; usually the standard conditions are:
o
o
Temperature: 22 C or 295 K
Pressure: 760 mm Hg or 1 Atm
63
For the conditions under which the X ray unit is calibrated the exposure is determined by:
Exposure(roentgens) = Rdg  Nx  PTP
Nx = chamber calibration factor, converts reading to roentgens
PTP = temperature & pressure correction factor
=
o
760
. ( 273 + temperature C )
pressure
295
If ion chamber is sealed (fixed air mass), no need for PTP
Electrometer cannot be in or near the X ray field because radiation will affect its electronics.
To overcome this, we have:
1.
Condenser chamber - Early Victoreen
- a capacitor (condenser) is connected to the thimble.
- charge the condenser which charges the thimble.
- expose the thimble to radiation.
- ionic current in the thimble discharges the condenser.
- measure the lost charge with an electrometer .
- condenser chambers suffer from “stem effect”, i.e. the
calibration of the chamber will vary depending on the amount of
the condenser exposed. Therefore a stem correction factor must
be determined.
2.
Farmer chamber -
- thimble connected by a 10 meter
coaxial cable to electrometer located outside
the therapy room.
- thimble made of graphite (Zavg close to air)
60
- plastic cap for use with Co to create
electronic equilibrium, necessary for “in air” measurements
Farmer Chamber
64
Ion chambers with a configuration different from the thimble are sometimes used.
1. Parallel plate chambers. Two parallel
conducting surfaces separated by a small gap. These
chambers have very good spatial resolution in the
gapped direction. They are useful in electron beam
dosimetry.
very good up and down spatial resolution
less side to side spatial resolution
2. Extrapolation chamber - A parallel plate chamber in which the plate separation can be
varied. The measured charge can be extrapolated to zero separation. This can provide very
accurate measurements of surface doses.
Absorbed radiation dose
As a measure of radiation quantity exposure, measured in roentgens, is applicable only to X rays
with energies less than 3 MeV. To overcome these limitations we introduce a new measure of
radiation quantity, that of radiation dose, measured in rads or gray.
Radiation dose is a measure of energy absorbed per unit mass of some material. It is only
appropriate for energy absorbed from ionizing radiation, but is not restricted to photon radiation.
The types of radiation that dose can be applied to are:
X rays and gamma radiation
Electron beams
Heavy charged particles, protons, alpha particles, etc.
Neutron beams
Exotic particle beams, pi mesons, mu mesons, etc.
Since it is a direct measure of energy absorbed by a material object electronic equilibrium is not
necessary. For a given X ray exposure, however, the amount of absorbed dose will be dependent
on the composition of the material that is irradiated.
There are two units of absorbed dose in current use. The older unit is the rad. This unit was
chosen because for muscle tissue a rad is numerically close to a roentgen.
1 rad = 100 ergs/gram of material
The newer unit of absorbed dose is the gray.
1 gray = 1 joule/kilogram of material.
65
Since 1 Joule = 107 ergs and 1 kilogram = 103 grams
1 gray = 104 ergs/gram = 100 rads.
or
1 rad = 1 centigray
Conversion of roentgens to rads
In order to understand how an X ray exposure, in roentgens, can be converted into a dose, in
rads, we will do the conversion in steps, as follows:
Roentgens in air  rads in air
Rads in air  rads to a small mass of unit density air (smuda)
Rads to smuda  rads to a small mass of unit density tissue (smut)
Rads to smut  entrance dose to tissue
To convert roentgens to rads in air we have to know how to convert the measured charge into the
energy that went into creating that charge.
The conversion factor is the W factor for electrons, the average energy expended by an electron
for every ion pair created. This factor is W = 33.97 eV/(ion pair).
The charge of an ion is 1.6  10-19 coulomb. The same factor is used to convert eV to joules, i.e.,
1 eV = 1.6  10-19 joules.
1 Roentgen = (2.58  10-4 coulomb/kg)  (1.6  10-19 coulomb/ion )
= 1.613  1015 ions/kg
= (1.613  1015 ions/kg)  33.97eV/ion
= 54.79  1015 eV/kg
= (54.79  1015 eV/kg)  (1.6  10-19 joule/eV)
= 87.6  10-4 joule/kg = 0.876  10-2 Gy = 0.876 rad to air.
We now consider a smuda (small mass of unit density air, frozen air if you wish). The mass is
just large enough to bring about electronic equilibrium at is center (i.e., dose buildup). For 60Co
gamma rays this mass will be a ball with a radius of 0.5 cm. For lower energy photons the mass
will be even smaller. The only difference between this situation and that of normal density air is
the small attenuation of the photons by the mass. We therefore have to add an attenuation factor,
Aeq, so that:
Dose to smuda(cgy or rads) = Exposure(roentgens)0.876Aeq.
Aeq has been determined for various photon energies to be
60
.985
-
Co
.99
1.00
Cs
- orthovoltage or lower energy.
137
66
Now we must tackle the more difficult task of determining the dose to smut (small mass of unit
density tissue). To do this we introduce the concept of the energy absorption coefficient, µen.
en is the fraction of the incident photon energy that is absorbed locally by the material per unit
length of material.
Remember µ = attenuation coefficient (µ =0.6931/HVL).
 measures photons absorbed and scattered out of the material.
en =  x (fraction of photon’s energy absorbed locally)
Just as in the case of  we define the mass energy absorption coefficient, (en) by dividing by
the density of the material, .
In the photon energy range where the photoelectric effect is dominant µen  µ since almost
all of the photon’s energy is absorbed locally.
In the megavoltage energy range where the dominant process is Compton scattering
µen  µ since much of the incident photon’s energy goes to the scattered photon which is
not absorbed locally.
Dose to tissue is proportional to (µen)tissue and dose to air is proportional to (µen)air
therefore
dose to smut = (µen )tissue (

en
)air
 dose to air
= (µen )tissue (
 en )air .876  Aeq  exposure
= f  Aeq  exposure
where f, the Roentgen to rad factor, is (µen )tissue (
)  .876

en

air
67
The variation of the f factor for common materials versus photon energy is shown on the two
graphs. A few things can be noted about the f factor and its variation with beam quality:
For air, f = 0.876, a constant value
For soft tissue, f is fairly constant for different beam qualities and is a bit greater than fair .
This is because soft tissue has hydrogen, which has a greater probability of Compton
scattering. Otherwise soft tissue has a Z close to that of air, hence fsoft tissue  fair .
For bone, f is close to that for soft tissue for high energy photons. Compton scattering is the
dominant process and the probability of Compton scattering is independent of the Z of the
material. In the lower energy region, where photoelectric absorption is important, f for bone
increases dramatically.
It must be noted that the f factor is only defined up to a beam quality of 3 MV. Above that the
roentgen is not defined.
If we are measuring exposure using an ionization chamber with an ADCL determined correction
factor NX and a temperature-pressure correction factor PTP, and the chamber gives a reading
Rdg, then the dose to a smut is given by
Dosesmut = f  Aeq  Exposure = f  Aeq  PTP  NX  Rdg
The Dosesmut can be converted to dose to the patient using the back-scatter factor or the tissue-air
ratio. We will discuss these factors in detail later.
68
IX- The measurement of dose for beam qualities greater than 3 MV.
The Bragg-Gray cavity theory
We consider X rays incident on some
material (the phantom) in which a air filled
cavity has been placed. The cavity is small
enough that the flux of electrons in the
medium is not disturbed. The energy absorbed
(per unit mass) in the medium is related to the
energy absorbed (per unit mass) in the air by
means of the Bragg-Gray relation.
The dose to the gas in the cavity is:
Dose to the gas = Eg = Jg  (W/e)
Eg = energy absorbed per unit mass of gas (air) - unit is joule/kg or gray
Jg
= ionization per unit mass of gas (air) – unit is coulomb/kg
W/e = 33.97 joule/coulomb
The dose at the same point when the cavity is replaced by the medium of the phantom is
Dosemedium
= Eg  (S)medgas
or
Dosemedium = Jg  We  (S)medgas
This is the Bragg-Gray relationship for determining dose to a medium from ionization in a small
gas filled cavity.
(S)medgas =
average mass stopping power ratio (medium to air). The stopping power ratio is
averaged over the energies of the electrons traversing the cavity.
The stopping power is the rate (per cm) that the medium removes energy from an electron. The
units are MeV/cm.
The mass stopping power is the stopping power divided by the density () of the material. The
units are MeV-cm2/gm.
The mass stopping power ratio is the mass stopping power of the medium divided by the same
quantity for air. The quantity is unitless. The table below illustrates how this quantity can be
determined. The density of air at 20oC is 0.0012 g/cm3.
69
Stopping Power
Electron
energy
(MeV)
0.1
0.5
1.0
5.0
10.0
Water
(MeV/cm)
Air
(MeV/cm)
4.115
2.028
1.844
1.900
1.976
.00436
.00216
.00199
.00220
.00237
Mass Stopping Power
Water
(Mevcm2/g)
4.115
2.028
1.844
1.900
1.976
Air
(MeVcm2/g)
3.633
1.802
1.661
1.833
1.979
Mass
Stopping
Power Ratio
Water to Air
(S)medair
1.133
1.125
1.110
1.037
0.998
Note that even though the stopping power varies with electron energy, the stopping power ratio,
medium to air, does not vary to a great degree.
Example – What is the dose to a point in a water filled phantom irradiated with 10 MV X
rays if at that point the ionization in a 0.6 cm3 air filled cavity is 10-8 coulomb? The density
of air is 0.0012 g/cm3 . The average mass stopping power ratio for electrons crossing the
cavity is 1.117.
The mass of the air in the cavity is 0.6  0.0012 = 0.00072 g = 7.210-7 kg. Thus
Jg = 10-87.210-7 = 1.3910-2 coulomb/kg.
Dose = Jg  W/e  (S) = 1.3910-233.971.117 = 0.527 joule/kg = 0.527 Gy
Determination of dose in a medium
We now identify the Bragg-Gray cavity with an air filled ionization chamber placed in a medium
(a water phantom, for example). In order to use the Bragg-Gray theory, we need to know the
mass of air in the chamber exactly. Since the density of air is a known quantity (which depends
on temperature and atmospheric pressure) we can determine the mass of air if we know the air
volume exactly. This is difficult to measure for an individual ionization chamber. We must
therefore resort to an indirect method of determining the air mass. We can use Nx, the chamber
calibration factor provided by the ADCL, specifically the Nx determined for 60Co radiation.
A highly simplified explanation of how this is done will be presented.
The situation is illustrated below
70
The first two steps we have already discussed.
We recall that
Dosein air = fAeqExposure
or written out in all its glory
Dosein air =
0.876(en)medium( ) AeqPTPNXRdg
en
air
We then make the assumption, though it is not
strictly true, that the identical equation applies to step three, i.e. the dose to medium under
Cobalt-60 irradiation.
To get to step four we utilize the Bragg-Gray relation and note that for the same Jg, i.e. for the
same chamber reading, the relationship between the dose from high energy X rays and the dose
from Cobalt-60 is:
DoseHi E = DoseCobalt(Smedair)Hi E(Smed
)Cobalt
air
Where here the S refers to the ratio of mass stopping power medium to air, i.e. the above
formula has a ratio of ratios. Combining the above two equations we arrive at the equation
connecting the dose in the medium with the ion chamber reading.
DoseHi E =
[0.876( )
en
(en)airAeq(S
medium
med
]P
)Hi E(Smedair)Co
air
TPNXRdg
The complicated expression in the square brackets has been tabulated as a single quantity, C.
Thus the equation simplifies to:
DoseHi E = CPTPNXRdg
To summarize in words, C is the factor that takes a reading from an ionization chamber which
has been calibrated in air with 60Co gamma rays and converts it to dose at depth in a phantom
that is irradiated with high energy X rays.
Note that for 60Co gamma rays C is simply fAeq.
For electron beams there is a similar factor, CE, which is used for dose measurements using a
Coblat-60 calibrated ionization chamber. Values for these factors are listed below.
71
Radiation
60
Co
6 MV
10 MV
20 MV
Cλ
.95 (f x Aeq)
.94
.93
.90
Electron
energy
6 MeV
10 MeV
20 MeV
CE
.93
.92
.88
For electron beams of a given energy CE increases with depth in the phantom, since the energy
of the beam decreases with depth. For photon beams the energy dose not change appreciably
with depth and, therefore, C remains constant.
Calibration of linear accelerators
Linacs are calibrated so that 1 monitor unit (MU) equals 1 cGy under the following conditions:
1.
2.
3.
The point at which the dose is specified is d-max in a phantom. No “in air”
calibrations.
The field size is 10 cm. x 10 cm. (usually)
The SSD is at the isocenter of the unit (usually 100 cm.)
The depth in the phantom at which the ion chamber is placed for calibration is usually not dmax.
At many institutions, the ion chamber is at 5 cm depth. In that way, there is no question whether
it is at a point of electronic equilibrium. One can easily relate the dose at the chamber depth to
the dose at d-max by the known percent depth dose.
Example : A Farmer chamber is being used to check the calibration of a 6 MV linear accelerator.
The chamber is placed at a depth of 5 cm in a water phantom. The water surface is 100 cm from
the X ray source. The X ray field size is 10 cm  10 cm. The temperature is 20 oC and the
atmospheric pressure is 722 mm Hg. The calibration factor for the Farmer chamber with the
electrometer has been determined to be 47.9108 roentgens/coulomb.
For a 200 monitor exposure the electrometer reads 3.5110-8 coulombs. The ratio of the dose at
5 cm depth to the dose at d-max is 0.860.
Using the factor C = 0.94 determine if the calibration of the unit is correct.
The temperature/pressure correction is PTP = (760722)(273+20)(273+22) = 1.045
The measured dose at 5 cm is given by the equation
Dose = CPTPNXRdg = 0.941.04547.91083.5110-8 = 165 cGy.
The expected dose is 0.86200 = 172 cGy.
Therefore the calibration of the accelerator is low by 4%.
The TG-21 method (Task Group 21 of the AAPM).
72
An improvement of the Cλ and CE method..
Two major refinements
1.
Replace (unrestricted) stopping power ratio (S)medair with the restricted stopping
power ratio (L ρ)medair.
This takes into account the fact that some ionizations result in energetic secondary
electrons (greater than 20 KeV kinetic energy) which do not deposit energy locally in
the air cavity. These interactions should not be counted in calculating the mean
stopping power.
2.
Consider separately the effects of electrons generated in the medium and those
generated in the chamber wall.
The TG-51 method (Task Group 51).
The TG-51 protocol is a departure from the previous calibration protocols in that the chamber is
calibrated in a water phantom and the resulting calibration factor, ND,w converts the chamber
reading, M, to dose in gray. The chamber reading of course must be corrected for temperature
and atmospheric pressure. A 60Co beam is still the reference for the calibration. To calibrate
beams other than 60Co a correction factor, kQ, which depends on beam quality (Energy), must be
applied to the calibration factor. The beam quality is determined by a measurement of the
percent depth dose at a depth of 10 cm for a 10 × 10 cm2 field. The X ray beam is calibrated at a
depth of 10 cm in a water phantom. The dose at d-max can then be determined using the known
PDD. At this point the ion chamber integrator can be adjusted to set 1 MU equal to 1 cGy as is
the conventional setting. The process is illustrated in the diagram below.
73
Dose buildup
Although photon interactions (and therefore attenuation of the
beam) begin immediately as the beam
crosses the
entrance surface of the patient or phantom, the dose is
not at a maximum on the surface for X-rays in the MeV
energy range. The electron fluence has to build up.
After electronic equilibrium is established, the dose begins
to decrease because of attenuation of the photons by the
medium.
d-max is the depth in phantom (or patient) at which maximum
dose buildup occurs.
d-max increases with photon energy, because the range of the
Compton electrons increases.
The surface dose is not zero because:
1. electrons are produced in the air before beam reaches the patient
2. electrons produced by collimators
3. electrons from blocks, trays, or wedges
4. some electrons back scattered in the patient.
Surface doses can vary a lot from situation to situation. If you really need to know the surface
dose, it should be measured for the specific setup of your treatment.
d-max depends on:
1.
2.
3.
photon energy, greater photon energy means greater electron energy, therefore,
greater depth for equilibrium.
density of medium, greater density means equilibrium will be established at a
shallower depth.
field size, especially for X-rays above 10 MV. As field size increases, d-max
decreases. Greater surface dose means equilibrium will be established at shallower
depth.
The relationship between X-ray energy and d-max is shown below.
ENERGY
200 KV
Cobalt-60
4MV
6MV
10 MV
18 MV
22 MV
d-max (cm. of water)
0.0
0.5
1.2
1.5
2.5
3.3
5.0
Kerma: Kinetic Energy Released in Matter.
74
Kerma is the energy released from a photon beam in the material as opposed to the energy
absorbed (dose). The units are the same as dose (gray). The relationship between Kerma and
dose is illustrated in the figure below.
The dose that is absorbed at a point in a phantom comes
from the kerma that is released upstream from the point.
Air kerma is the kerma that is released in air.
Exposure is directly related to air kerma according to the
relation
Air Kerma(gray) = 2.5810-4WeExposure(R)
Since W/e = 33.97 joule/coulomb
We have:
Air Kerma(gray) = 0.00876Exposure(roentgens)
Interface effects- Changes in dose levels at the interface between tissues of different atomic
number (Z) or density ().
Fano Theorem- If electronic equilibrium is established in a phantom (or patient) then at the
boundary of a change in density of the material the fluence of electrons and therefore the dose
does not change.
Example: Lung in a patient. Density of lung is 1/3rd that of normal tissue but the dose is
approximately the same as the tissue around it.
In the energy region where Compton scattering is dominant, even a change in the Z of the
material does not bring about a change in dose.
Example: Bone tissue interface. For high energy X rays bone receives the same dose as
surrounding soft tissue.
Cases where dose does change at an interface are:
1 - Dose Build-up - High energy X rays. Air/tissue interface, dose at interface increases until
electronic equilibrium is established. For very high energy X rays can have dose loss at
tissue/lung or tissue/air interface, especially for small fields.
2 Dose Build-up - For low energy X rays for which the photoelectric is important there is
dose buildup at the tissue/bone interface.
3
- Dose Build-up - For very high energy X rays for which pair production is significant
there is dose buildup at the tissue/bone interface because of the Z dependence of pair
production.
75
Dose to soft tissue within bone
The dose to the bone mineral is of no significance. It is the soft tissue elements within the bone
that are of concern. For irradiation with low energy X rays, the average dose to a soft tissue
cavity within the bone depends on the size of the cavity, because of the limited range of the
electrons generated in the bone mineral.
The determination of the
dose to soft tissue elements
in bone is very complex
problem. It is highly
dependent on the photon
energy.
For high energy X rays the
dose to the soft tissue is the
same as the dose to the bone
mineral.
Dose Equivalent -
used in radiation protection calculations for different types of radiation
DE (Sievert or rems) = Dose(Gy or rads) x Q(quality factor)
Radiation
X-ray, γ, ethermal neutron
other neutrons, p, charged nuclei, α
Q (quality factor)
1
5
20
Maximum permitted occupational exposure is 5 rem/yr.
Maximum permitted exposure to general public is 0.1 rem/yr
Relative Biological Equivalent
RBE dose = dose (rads) x RBE
RBE - used in radiobiology, depends on beam type and biological end point desired (i.e. 50%
cell kill)
RBE of 250 KV X rays = 1.0
RBE of 1-10 MV X rays = .8-.85
X- Dose measurement not involving ionization chambers.
76
Thermoluminescent dosimetry (TLD)
Background information- When large numbers of atoms coalesce to form a solid the electrons
interact to form energy “bands” which the electrons occupy, instead of the discrete energy levels
found with isolated atoms. As with single atoms the lowest energy bands are occupied until all
the available states are full, after which the next lowest is filled. If an energy band is not filled
then electrons are free to migrate from place to place.
The regions between available energy bands are called
forbidden bands. No electrons can occupy these energy
states.
The highest energy band that is full is called the valence
band. Electrons in the valence band can not move.
The next band above the valence band is the conduction
band. Electrons in the conduction band can move.
Electrical insulator - no electrons in the conduction band.
Electrical conductor - electrons exist in the conduction band and are free to flow.
For an insulator (such as LiF) chemical impurities can cause isolated energy states to appear in
the forbidden region. These are called electron traps that can hold electrons between the
conduction and valence bands.
How TLD works
1- Ionizing radiation strikes TLD crystal.
Radiation ejects electrons from the
valence band into the conduction band.
Electrons are caught in the traps.
2- We then heat the crystals. The
electrons are released to the conduction
band and then fall to the valence band
emitting light.
3- The light is converted into electrical
current by a photomultiplier tube.
The current from the photomultiplier is amplified by an electrometer. If the current is displayed
as a function of the temperature of the crystal we see a curve with a maximum at a specific
temperature. This is called the “glow curve”. Its height is proportional to the radiation dose.
77
Crystals used in TLD - CaSO4, CaF, LiF
Although CaSO4 and CaF2 give more luminescence than LiF, LiF is preferred because it has a Z
similar to tissue and thus its response does not change much with photon energy.
The glow curve
The peaks in the glow curve at the
higher temperatures are from electrons
in the deeper traps, therefore they are
the most stable with respect to time.
The lower temperature peaks tend to
fade with time.
Disadvantages of LiF
1. Supra-linear relationship above 1000 cGy, that is, as the dose is increased the response
increases proportionally a greater amount. We like to limit the dose to less than 200 cGy.
2. Glow curve - low temperature peaks gradually fade with time. Area under the glow curve
will change with time. To avoid this we can anneal the TLD material at 100 oC for 10
minutes before reading; or let the TLD’s rest for a day before reading.
3. TLD’s give only a relative dose. Must be compared against a known dose of radiation.
Silicon diode dosimetry
Semiconductor - a material with a small energy gap between valence and conduction band.
Thermal energy allows some valence electrons to go to conduction band. Semiconductors have
electrical conductivity somewhere between that of an insulator and a conductor. Silicon is an
example.
Impurities in the silicon can increase the conductivity. If we add atoms with an extra valence
electron (example: phosphorus), an electron donor, we get an n type material (negative). If we
add atoms with one less valence electron (example: boron), an electron acceptor we get a p type
material (positive). The “hole” (electron deficit in the valence band) can carry current as if it
were a positive electron.
Diode- pn junction. If we adjoin p type silicon with n type silicon we have a pn junction at the
interface. The junction turns the material into a diode (rectifier), a device that allows current to
flow in one direction but not the other.
78
If the pn junction is irradiated a current is produced. The magnitude of the current is
proportional to the dose rate.
Advantages of diodes for dosimetry are:
Small size: Sensitive volume is much smaller than an ionization chamber
Useful for small fields (radiosurgery) and fields with narrow dose gradients.
Useful for electron beam depth dose measurements.
Disadvantages: Hi Z material. Response is dependent on photon energy.
Relative measurement (not absolute). Must be compared against a known radiation
dose.
Radiation can damage the diode and affect the sensitivity, especially for high energy
electron and photons.
MOSFET dosimetry
MOSFET is an acronym for Metal Oxide Silicon Field
Effect Transistor. It is a transistor with two n doped (or p
doped) regions called the source and the drain that are
imbedded in a p (or n) type silicon substrate. An
insulating layer of metal oxide (the gate oxide) connects
the source and the drain. A gate contact is layered above
the gate oxide. If the gate voltage exceeds a threshold
value, VT, current will flow.
Ionizing radiation alters VT. The change in VT can be related to the dose to the gate oxide. The
change in VT is permanent and thus the dose is can be stored indefinitely. MOSFET dosimeters
can be single use devices or multiple use devices with limited number exposures possible.
Chemical dosimetry
The most common chemical dosimeter is the Fricke or Ferrous Sulfate dosimeter. It consists
of a solution of FeSO4 in water (with a little bit of H2SO4).
When the solution is irradiated the ferrous ion is converted to the ferric ion, i.e.,
Fe
Fe
+++
++
+ irradiation 
Fe
+++
absorbs strongly UV light in the wave length range 3040 Angstoms or 304 nm.
79
The amount of UV 304 nm light absorption is a measure of the dose. In order to convert to dose
one needs to know the G factor.
G = number of molecules affected per 100 eV absorbed of ionizing radiation.
= 15.4 for FeSO4
Advantages: The solution is mostly water, i.e., Z same as water/tissue.
This is an absolute measure of radiation dose.
Disadvantages: Insensitive (good for 1,000 - 50,000 rad)
Film dosimetry
Radiographic film consists of a polyester film base upon which is laid an gelatin emulsion
containing silver halide crystals, primarily AgBr.
Ionizing radiation (or light) will neutralize some of the silver ions in the crystal, i.e.,
+
Ag  Ag to form metallic silver. The pattern of metallic silver on the film is the latent image.
Developer interacts with the crystal reducing more silver ions to metallic silver in those crystals
which have some metallic Ag. The unaffected AgBr crystals are then dissolved away. The
filmis transparent where the crystals have been washed away, but black where the metallic silver
has been left. The degree of blackness can be related to the radiation dose.
Advantages: Ease of use, can measure an entire depth dose curve from a single exposure.
Very high spatial resolution, useful for radiosurgery measurements and electron beams.
Disadvantages: High Z, (Ag and Br), response depends on photon energy.
Relative measurement, will not give an absolute dose.
Film density vs. dose is non-linear. As the dose gets large the density tends to saturate.
Calorimetry
80
Calorimetry depends on the fact that the energy absorbed by a phantom eventually is converted
to heat. If the phantom is thermally insulated then the temperature of the phantom will rise. Very
sensitive thermometry must be used since the temperature rise, T, is small. The relationship
between T and dose is:
Dose(Gray) = 4.186  C  T (oC)
where C is the specific heat of the phantom material. Examples of specific heats are:
Cwater = 1000 calorie/(kg-oC)
Cgraphite = 170 calorie/(kg-oC)
Advantage: Absolute dosimetry. Most fundamental of all the dosimetry methods.
Disadvantage: Insensitive. Temperature rises are very small.
81
XI- Practical clinical dosimetry – Dosimetry on the beam central axis.
Back scatter factor
For photon beams at energies of 60Co and below, calibrations are commonly done in air. To
convert from a dose in air to a dose in phantom at d-max depth and at the same distance from
the source we employ the back scatter factor (BSF).
The dose in phantom will increase relative to the dose to
smut in air because of photons being scattered back from the
depths of the phantom.
Backscatter factor is tabulated in many places. A good
source is British Journal of Radiology Supplement #17
The dose at d-max is called the entrance dose (also applied dose, given dose).
In terms of exposure (in roentgens) we have:
Entrance dose(cGy) = Exposure  f  Aeq  BSF
f = Roentgen to rad conversion factor
Aeq = attenuation factor (1 for orthovoltage, 0.985 for 60Co)
For 60Co one has to be careful in applying this formula. Frequently the exposure is measured at
some standard distance (100 cm) from the source. The entrance dose is at d-max (0.5 cm) depth
with the phantom surface at the same standard distance, the SSD.
Entrance dose = Exposure  f  Aeq  BSF 
SSD(SSD + d )2
max
BSF not applicable above 3 MV. Above 3 MV we do not measure doses in air and exposure is
not defined.
With the use of the back scatter factor it is assumed that the patient (or phantom) is sufficiently
thick that all of the potential back scattering occurs (full backscatter). If the volume being
exposed is thin then the entrance dose will be lower.
BSF depends on X ray beam quality in a complex way.
1. At low photon energies where the photoelectric effect is the dominant interaction there are
no scattered photons, therefore BSF is close to unity.
2. As the photon energy increases BSF increases due to Compton scattered photons.
3. For megavoltage photons BSF is close to unity because scattered photons tend to go in
the forward direction, i.e., back scattering decreases.
BSF depends on the X ray field size. A larger field produces more backscatter.
82
This figure illustrates the
dependence of the backscatter factor
on field size and X ray beam quality.
BSF strongly dependent on field
size.
Maximum BSF at HVL ~0.5 to ~1.0
mm Cu.
Note: BSF  1.0 always.
BSF depends on field shape.
At the center of the field a square field will produce more scattered photons than a rectangular
field of the same area.
For every rectangular field there is a square field, the equivalent square field, that has the same
amount of scattered photons as the rectangular field.
The same equivalent square field applies to both forward scattering and back scattering.
Therefore the same equivalent square can be used for determining the back scatter factor and the
percent depth dose.
Tables of equivalent squares exist in the literature (BJR supplement #17). A frequently used
approximation for determining the EQSF (equivalent square) is the area/perimeter method.
EQSF = 4AreaPerimeter
Example: For
60
Co radiation what is the Backscatter factor for a rectangular field 5 cm20 cm?
EQSF = 8cm  8 cm
BSF(88) = 1.029
Therefore BSF(520) = 1.029.
Note: BSF(1010) = 1.035
1010 is the same area as 520.
83
It is important to remember that the 4A/P formula only applies to rectangular fields, not to
random irregularly shaped fields. Also the equivalent square of a circular field is a square of the
same area.
BSF is independent of SSD. It doesn’t matter where the point is in relationship to the source of
photons, the BSF is the same.
Field size factors
Cobalt Teletherapy Units and Linear Accelerators are usually calibrated for a 10 cm X 10 cm
field. For larger or smaller fields one must apply Field Size Factors to the calibrated output.
Field size factors are most easily understood for the case of Cobalt units, since we can talk about
“in air” dose rates.
Suppose our Cobalt unit has an in air dose rate at 100.5 cm (isocenter + d-max) of
100 cGy/min for a 10 cm X 10 cm field. For fields less than 10 X 10 there is a smaller dose rate
in air, for fields greater than 10 X 10 the dose rate will be more due scatter from the collimator
faces. At d-max in a phantom there is also a change in the backscatter factor with field size. The
output factor (the dose rate at d-max in the phantom) will be the product of the two.
For example, if the “in air” dose rates are:
6 X 6: 97 cGy/min
Field Size
6X6
10 X 10
20 X 20
10 X 10: 100 cGy/min
Dose rate in air
(cGy/min)
97
100
103
20 X 20: 103 cGy/min
Backscatter Factor
1.021
1.035
1.061
Output Factor
(cGy/min)
971.021=99
1001.035=103.5
1031.061 = 109.3
For linacs we don’t have any in air measurements, but we do separate the change in output with
field size into collimator scatter and phantom scatter (backscatter) factors. Below is an example
for 6 MV X rays.
Field Size
Collimator Scatter
Factor
(Sc)
Phantom Scatter
Factor
(Sp)
Field Size Factor
CGy/MU
Sc,p = ScSp
6X6
10 X 10
15 X 15
20 X 20
0.973
1.000
1.018
1.029
0.987
1.000
1.013
1.016
0.960
1.000
1.031
1.045
We separate the field size dependence into the factors because Sc depends only on the
collimators whereas Sp depends on the blocked field size.
Percent depth dose (PDD)
84
Now that we now know how to determine the dose at d-max on the central axis, we would like to
know the dose at other depths in the patient. There are essentially two methods of
accomplishing this. One involves use of the percent depth dose (PDD).
The PDD can be used in two ways. If you know the dose at d-max and you want to know the
dose at d then
Dose at d = (1/100)PDDDose at d-max.
If you know the dose at depth = d and you want to know the dose at d-max then
Dose at d-max = 100(Dose at d)PDD
To find the PDD for 60Co and X rays < 4.0 mm Cu HVL use tables in BJR Supplement 17.
The PDD for X rays >4 MV are usually measured for each individual machine.
Factors that affect the PDD
1.
2.
3.
4.
5.
Field Size: At a given depth as the field size increases the PDD increases, due to
increased scatter dose at depth. The field size that defines the PDD is the field size on
the patient surface.
Beam quality: For a given depth as the beam quality increases the PDD increases, due to
the increased penetration of the primary (i.e., unscattered) photons.
Depth: For a given beam quality as the depth in the patient increases the PDD decreases.
This is due to the increased absorption of primary photons with depth.
Elongation of field: For a given field area, an elongated field has a lower PDD than a
square field. The number of scattered photons is less. The PDD for a rectangular field is
the same as that for its equivalent square.
SSD: Everything else being equal, as the SSD increases the PDD increases. This is
sometimes hard to grasp initially. As the SSD increases, the output (dose per MU)
decreases very significantly. But the PDD is not about dose per MU. It is just the ratio of
two doses.
85
The PDD is made up of three factors.
1. The inverse square factor. As the point at depth moves away from the source the dose
decreases.
2. The attenuation factor. As the depth increases, the attenuation of the primary photons
increases.
3. The factor due to scattered photons generated by interactions upstream.
Therefore:
2
PDD = 100.{(SSD + d-max)(SSD + depth)} (attenuation factor)(scatter factor)
The attenuation factor, AF, and the scatter factor, SF, do not depend on the SSD. For constant dmax and depth the first factor, the inverse square factor increases as SSD increases. In fact for
very large SSD it is close to 1.0, independent of the depth.
Using the above expression we can derive a formula for the PDD at one SSD in terms of the
PDD at another SSD
therefore we have:
86
Mayneord’s F factor, has the following properties:
F >1 if SSD2 > SSD1, i.e. PDD2 > PDD1
F= 1 if SSDs are equal (obviously)
F< 1 if SSD2 < SSD1, i.e. PDD2 < PDD1
Representative PDD are listed below for 60Co (British Journal of Radiology, Supplement #17)
and for 6 MV X rays.
Note how the PDD changes with SSD.
60
Co, SSD = 80. cm
55
100.0
75.1
51.2
23.3
Field Size / Depth (cm)
0.5 (d-max)
5
10
20
10  10
100.0
78.6
56.0
27.4
20  20
100.0
81.0
60.6
32.4
60
Co, SSD = 100. cm
55
100.0
76.7
53.3
24.7
Field Size / Depth (cm)
0.5 (d-max)
5
10
20
10  10
100.0
80.2
58.7
29.3
20  20
100.0
83.0
63.0
34.4
6 MV X rays, SSD = 100. cm
Field Size / Depth (cm)
1.5 (d-max)
5
10
15
20
55
100.0
84.1
62.5
46.0
33.8
10  10
100.0
85.9
66.2
50.2
37.7
15  15
100.0
86.9
68.3
52.7
40.4
20  20
100.0
87.5
69.6
54.4
42.1
25  25
100.0
87.7
70.4
55.6
43.4
Tissue air ratio (TAR)
87
The PDD is useful for calculating treatment times or monitor units when the SSD is fixed,
usually at 100 cm (isocenter). For non-standard SSDs we have to correct the PDD using the
Mayneord factor. This adds extra steps to the calculation. For isocentric treatments where the
source to tumor distance is fixed (the SAD or source to axis distance, 100 cm) it is more
convenient to use TARs or TMRs in the calculations.
If we know the dose rate to smut (in air), then multiplying it by the TAR gives us the dose rate at
depth in the patient or phantom. From there we can determine the time to deliver the intended
dose. This type of calculation would be used primarily for Cobalt-60. For higher energy X rays
we need to use the TMR because there are no “in air” dose measurements.
Factors that affect the TAR
1.
2.
3.
4.
Field Size: At any depth as the field size increases the TAR increases due to increased
scatter. The TAR is defined using the field at depth, usually the field at isocenter.
Beam quality: At depth as beam quality increases the TAR increases.
Depth: For a given beam quality as the depth increases the TAR decreases.
Field elongation: An elongated field has a lower TAR than a square field of the same area.
Use the equivalent square to determine the TAR.
The TAR is independent of the SSD or SAD. The TAR is the ratio of doses at the same
distance from the source. As the SAD increases the individual dose rates will decrease (due to
inverse square law) but the ratio will stay the same.
88
Note that the TAR at a depth equal to d-max is the BSF.
If you recall we said that
2
PDD = 100.{(SSD + d-max)(SSD + depth)} (attenuation factor)(scatter factor)
We can now identify what the (attenuation factor  scatter factor) is. It is just the TAR BSF.
Therefore we can say that
PDD(depth, As) = 100. {(SSD + d-max)(SSD + depth)}
2
{TAR(depth, Ad)BSF(As)}
Where As is the field size on the surface and Ad is the field size at depth. The factor 1BSF is
necessary to insure that PDD(depth = d-max) is 100.
One can use the above formula to generate a table of TAR from the PDD table.
Representative TARs for Cobalt-60 are listed below.
Field Size /
55
10  10
20  20
1.018
0.849
0.640
0.347
1.035
0.905
0.718
0.410
1.062
0.957
0.790
0.500
Depth (cm)
0.5 (d-max)
5
10
20
Tissue maximum ratio (TMR)
Since the TAR cannot be used for photons above 60Co we introduce a new quantity, the TMR or
tissue maximum ratio. It is just like the TAR except the dose in air is replaced by the dose at dmax in a phantom.
The same factors that influence the TAR (field size, depth, quality, etc.) affect the TMR. Like
the TAR the field size for the TMR is specified at depth. Just as the TAR, TMR is independent
of the SAD.
89
At a depth of d-max the TMR is 1.0.
Representative TMRs for 6 MV X rays are listed below.
Field Size /
55
10  10
15  15
20  20
25  25
1.000
0.898
0.727
0.580
0.458
1.000
0.918
0.770
0.632
0.511
1.000
0.929
0.797
0.665
0.547
1.000
0.935
0.813
0.688
0.573
1.000
0.938
0.823
0.705
0.593
Depth (cm)
1.5 (d-max)
5
10
15
20
Relationship between the TMR and the TAR.
If d is the depth in the patient, Ad is the field size at depth, d, then:
TMR(d, Ad) =
TAR(d, Ad) /BSF (Ad).
Tissue phantom ratio (TPR) A generalization of the TMR concept.
For the TPR we use a reference depth that can be greater than d-max. Frequently the reference
depth is a standard depth such as 5 cm.
Obviously, when the reference depth is d-max, then TPR = TMR.
90
Relationship between the TMR and the PDD
Recalling the relationship between the TAR, the BSF, and the percent depth dose we find the
following formula to be approximately true, sufficiently accurate for most applications.
PDD(depth, As) = 100. TMR(depth, Ad) {(SSD + d-max)(SSD + depth)}
2
We can use this relationship to calculate the percent depth dose from a TMR table.
Example: Calculate the PDD for a 6 MV X-ray beam using the TMR table and compare
it to the PDD in above table. The depth is 10 cm, the surface field size is 20 x 20, and the
SSD is 100. cm.
The field width at 10 cm depth is 20(110/100) = 22 cm. The TMR for
a depth of 10 cm and a 22x22 field size is 0.817.
Then PDD(10.,20x20) = 100.0.817 {101.5110} 2 = 69.6
Monitor unit calculations can be solved using either the PDD or the TMR. It is usually easiest to
use:
PDD for standard SSD setups
TMR for isocentric setups
Example: Standard 100 cm SSD. Calculate monitor units using both techniques. We want to
deliver 150 cGy at a depth of 15 cm using a 12x12 surface field. 6 MV X-rays, no blocking.
Using percent depth dose:
PDD(15, 12X12) = 51.3
Output(Dose/MU) for a 12X12 = 1.014 cGy/MU
Therefore dose/MU at 15 cm depth = 0.5131.014 = 0.520. To deliver 150 cGy we need
150/0.52 = 288 MU
Using TMR:
Field size at 115 cm = (115/100)12, i.e. 13.8cm X 13.8 cm TMR(15,13.8X13.8) = 0.657
Sc(12X12) = 1.007, Sp(13.8X13.8) = 1.010
Dose/MU to a phantom at d-max = (101.5/115)21.0071.01 = 0.792 cGy/MU
Dose/MU at depth = 15 cm is 0.6570.792 = 0.520 cGy/MU
Required MUs are 150/0.52 = 288 MU
Note that in this case the calculation is simpler using the PDD.
91
Example: Isocentric set up, 100 cm SAD. We want to deliver 100 cGy to a 15X15 field at
isocenter at a depth of 12 cm using 6 MV X Rays, no blocking. Calculate using both
techniques.
Using PDD:
Surface field size is 1588/100 = 13.2, i.e. 13.2 cm X 13.2 cm
SSD = 100 – 12 = 88 cm.
PDD(12, 13.2X13.2) = 61.2% at an SSD of 100 cm. We have to convert this to an SSD of 88
cm using the Mayneord F factor.
F =
{(88 + 1.5)(88 + 12)}2{(100 + 1.5)(100 + 12)}2
= 0.975
So the PDD at SSD = 88 is 61.2 %0.975 = 59.7 %.
Sc(15X15) = 1.018
Sp(13.2X13.2) = 1.008
Entrance dose rate at 88 cm SSD =
Therefore the dose rate at d-max is
(101.589.5)21.0181.008
= 1.320 cGy/MU
Dose rate at depth = 1.3200.597 = 0.788 cGy/MU.
To deliver 100 cGy we need 100/0.788 = 127 MU.
Using TMR:
TMR(12,15X15) = 0.744
Sc(15X15) = 1.018
Dose rate at d-max at 100 cm SAD =
Sp(15X15) = 1.013
(101.5100)21.0181.013
= 1.062 cGy/MU
Dose rate at depth at 100 cm SAD = 1.0620.744 = 0.790.
To deliver 100 cGy we need 100/0.790 = 127 MU.
In this case the calculation is simpler using the TMR.
Arc Therapy (Rotation)
In arc therapy the treatment unit rotates around the patient while the beam remains on. The unit
is set so that the tumor is at the center of rotation (the isocenter). The treatment can be a full 360
degree rotation or a partial rotation (e.g. 1800 arc or 1200 arc). To calculate the monitor units we
approximate the rotation by a series of fixed fields separated by fixed degree increments. The
greater the number of fixed fields (the smaller the degree increments) the greater the accuracy of
the calculation.
92
For each of the fixed fields we determine the TMR, which will depend on the tissue depth, and
calculate the average TMR. From the average TMR the required MUs can be calculated.
We illustrate this procedure with an example.
Example: Illustrated below.
Deliver 200 cGy to isocenter by 3600
rotation, 6 MV X rays
8 cm X 8 cm field
Accelerator delivers 1 cGy/MU for a 10
cm X 10 cm field at d-max for an SSD
of 100 cm.
Field size factor is 0.981
The “dose rate” is 250 MU/minute.
We divide the patient contour into 12
segments with the depth to isocenter as indicated and we calculate the average TMR.
Field
A
B
C
D
E
F
G
H
I
J
K
L
Tissue Depth
8
10
12
15
14
11
11
14
16
15
11
8
Average TMR =
TMR
0.818
0.756
0.696
0.614
0.642
0.726
0.726
0.642
0.587
0.614
0.726
0.818
0.697
2
The average dose/MU at the isocenter is (101.5100) 0.9810.697 = 0.704 cGy/MU.
Therefore the required MUs are 200/0.704 = 284 MU.
At 250 MU/minute the rotation will take 1.14 minutes to complete. The treatment unit must
adjust its rate of rotation so that the rotation will be completed in exactly 1.14 minutes.
Alternatively, the unit might rotate at a set rate (say 1 rotation/min) and the MU/min will be
adjusted so that the 284 MUs will be delivered in a single rotation.
93
XII- Isodose Distributions
An isodose distribution is a two dimensional representation of photon beam or a combination of
photon beams.
Single beam isodose distributions
The single beam distribution is usually in a plane containing the central axis of the beam and
parallel to one of the collimator faces. They sometimes may be supplied by the accelerator
manufacturer, but more frequently are measured directly for each individual unit using a motor
driven probe in a water tank.
Isodose distribution depends on many factors including: Beam quality, field size, elongation,
penumbra, SSD, depth, distance off axis.
The central part of the set of curves is just a graphical
representation of the central axis depth dose.
The edges represent the penumbra.
The bowing of the isodose curves, i.e. the fact that the
curve is not flat off axis (especially for 60Co) is due to
diminished scatter off axis. For linacs we can
compensate for this by the flattening filter, with the
accompanying “horns” at shallow depths.
94
The figures below compare the isodose distributions for Cobalt-60, 6 MV, and 18 MV X rays.
Wedged fields
A wedge is a wedged shape piece of metal that is placed
in a photon beam in order to tilt the isodose curves by a
specified angle, the wedge angle. The wedge angle refers
to the degree of tilt and not to the physical angle of the
metal piece. A wedged field is a simple example of
intensity modulation of the photon beam.
The wedge angle is only approximate, since the tilt of the
curves varies somewhat with depth.
A wedged field can also be achieved without a physical
piece of metal in the beam. This is done by sweeping a
single collimator jaw across the field as the exposure is
taking place. Such a wedge is called a “dynamic
wedge” or a “virtual wedge”.
95
Two important effects need to be taken into account when calculating monitor units for wedged
fields.
1. The wedge attenuates the beam on the central axis of the beam and therefore a wedge
attenuation factor alters the dose/MU.
Dose/MU(wedged) = (Wedge Factor)  Dose/MU(unwedged).
For doses off the central axis a different wedge factor must be employed.
2. The wedge “hardens” the beam and therefore the central axis depth dose tables need
to be modified. The wedged beam is somewhat more penetrating than the unwedged. An
advantage of the virtual wedge is that this beam hardening is not present.
Combining isodose curves
Single field isodose curves are generally not of great interest clinically. Isodose distributions are
most often used to determine the result of combining two or more fields.
Parallel opposed ports (POP) are used to produce a uniform dose within a volume. Frequently
the fields are equally weighted, i.e., 50% of the dose from one field 50% from the opposed field.
The figures below show the results of POP for 60Co gamma rays and 4 MV X rays with two
different patient thickness (or IFDs, interfield distances) 14 cm and 24 cm.
96
From these figures we can observe some general rules about isodose curves for POPs.
1. The dose is fairly uniform throughout the treated volume. With cobalt gamma rays
there is a “bowing in” or an “hour glass” shape to the 95% isodose line. This bowing in
is more pronounced with the greater IFD. With linacs, the flattening filter produces a
flatter beam at depth, which decreases the bowing in.
2. The maximum dose is at d-max, and the max dose increases as the IFD increases.
With linear accelerators the max dose occurs at d-max, but lateral to the central axis.
This is due to the “horns” of the linac beam.
Wedged pair
Wedged fields are frequently used for two fields
whose axes are not parallel, i.e., so that they meet at
an angle.
The angle made by the central axes of the two
beams is called the hinge angle.
Obviously, for POPs the hinge angle is 180O.
97
The figures below illustrate this situation with a 90O hinge angle.
In both cases the each field is delivering 100% to d-max and ~70% at isocenter. The isodose
curves for the combined fields are indicated by the dotted line. For the non wedged fields the
summed fields create a “hot spot” (high dose area) in one corner of the overlap area and a cold
spot in the opposite corner. The dose ranges from 120% to 160% (dotted lines). For the wedged
fields we have a large uniform dose region at 140%.
We note that the dose uniformity comes about when the wedges are arranged such that the
isodose lines for the two fields are roughly parallel to each other. This leads to the following
relationship between the wedge angle and the hinge angle for the best dose homogeneity.
wedge angle = 90O  1/2(hinge angle)
Examples of the use of this formula.
hinge angle
120O
90O
60O
wedge angle
30O
45O
60O
Another common use of wedged fields is in a three field technique, for example a single anterior
field with two opposed laterals. The anterior field produces a dose gradient in the region that is
covered by the three fields. The use of wedges with the lateral fields can result in dose
uniformity as shown below.
98
There are many more uses for wedged fields that are out of the scope of this discussion.
Perturbations of isodose curves
The single field isodose curves that we have shown so far are strictly only applicable to a flat
patient surface. This generally does not hold for real patients. One way to correct this situation
is to fill in the irregular surface with unit density material, called bolus.
The disadvantage to the use of bolus is the loss of the skin sparing effect of megavoltage X ray
beams. Bolus can be used in this way with orthovoltage and superficial X rays, where skin
sparing is non-existent, and with electron beams which have very little skin sparing.
99
Compensating filters
One way to achieve dose uniformity while preserving skin sparing is to move the “bolus”
material away from the patient’s surface, by at least 20 cm. Electrons generated in the bolus,
which produce the skin dose, will scatter out of the field.
In order to account for the beam divergence the lateral dimensions of the filter must be
diminished by the ratio of the source to filter distance (SFD) and the SSD, i.e.
y / = y(SFD/SSD)
If the thickness (along a given ray emanating from the source) of the filter is equal to the missing
tissue along the ray then the filter will overcompensate due to the loss of scattered photons
produced in the filter. The filter’s thickness needs to be decreased by between 12% to 30%
depending the beam quality.
The filter may be made of material that is not unit density. In that case the thickness of the filter
is scaled by the density of the material.
Compensating filters are another example of beam intensity modulation. They can be
constructed (with computer guidance) to produce desired dose distributions that are not
homogeneous.
There are a number of different techniques to measure the patient’s surface contour. Some of
them are:
1 Rod Box -plastic rods slide in and out from a frame to fit the contour
2 Moire Camera - projects lines on the surface to get topographical map
3 Mechanical feeler interfaced to a router which cuts styrofoam mold
4 Film Densitometry
5 CT Scans
100
In some instances wedges can be used as a simple one dimensional compensator.
The use of a wedge in this case
flattens the isodose lines that have
been tilted due to the sloping patient
surface.
Correcting isodose curves for irregular surfaces
When an irregular surface is present but a compensator is not being used the isodose curves must
be corrected for the contours. Modern planning computers employ sophisticated algorithms for
this purpose. We shall illustrate three simple methods that are suitable for manual calculations.
These methods are:
1- The effective SSD method
2- The TAR method
3- The isodose shift method
1- Effective SSD method
Assume we wish to find the isodose percent at
point P . A ray is drawn towards the source
through the patient’s contour and the tissue
deficit, h, is determined. The effective SSD for
that ray is SSD + h.
The isodose curve is shifted downward (dotted
lines) a distance h from its normal position. The
isodose percent is read from the shifted curves.
The isodose percent is multiplied by a correction
factor which is
CF =
{(SSD + d-max)(SSD + d-max + h) }2 .
In the above example, if SSD = 100, h = 1.5 cm, d-max = 0.5 (for Cobalt 60) and the shifted
isodose percent is 72, then the corrected percent is
2
72{100.5102} = 69.9
101
2- TAR (or TMR) method
In the TAR method the isodose curves are not
shifted, i.e. the phantom surface for the isodose
curves and the patient surface coincide on the
central axis.
For the point P the depth d and the tissue
deficit h are noted and the percent isodose is
read from the unshifted curve. The isodose
percent is multiplied by the correction factor
{

}
CF = [TAR(d,Fld size)] [TAR(d+h,Fld size)]
In the above example, for 60Co, 15 X 15 field size, d = 6.5 cm and h = 1.5 cm, then
CF =
{0.8870.834} =
1.064
The isodose percent from the curves is 65%.
The corrected isodose percent is 651.064 = 69.1% which is close to our previous value.
3- Isodose Shift Method
With this method the uncorrected isodose curve
and the patent contour are set so that they
coincide at the central axes. A series of rays
are traced and the tissue deficit h is noted.
Along each ray the isodose curve is shifted by a
fraction of h, the fraction depending on the
beam energy.
Isodose shift(in cm) = f  h(cm),
where f is approximately 2/3 for 60Co to 6 MV
X rays. f decreases to ~0.5 for very high X ray
energies.
Thus the isodose curves are shifted in the
direction of the shape of the contour, but not
quite as much as the contour.
102
Tissue Inhomogeneity
Up to now we have assumed that the patient
was homogeneous in composition with a
density of 1.0 gm/cm3. The presence of large
bones or of air in the lungs makes this
assumption invalid. The effect of these
inhomogeneities on X ray or gamma ray
beams is complex.
The presence of air filled lung in the beam
path will decrease the attenuation of the
primary photons, i.e. will increase the dose,
but it will also decrease the amount of
scattered photons.
Modern planning systems have sophisticated methods for correcting the dose for inhomogeneity.
We will discuss some of the simpler methods suitable for manual calculations.
1-Effective path length (or Ratio of TMRs)
Consider the situation illustrated below. After traversing a depth d1 of normal density tissue the
beam travels through d2 of tissue with an relative electron density different from that of water (or
soft tissue). Finally a thickness d3 of normal density is encountered.
Typical values for densities are:
for lung
 = 0.25 to 0.35
for bone
 = 1.65.
Although this specifically refers to the
electron density ratios, these are close to the
physical density.
103
The actual depth in the patient is:
d = d1 + d2 + d3.
The effective depth in the patient is:
deff = d1 + d2 + d3
This is sometimes called the radiological path length.
If we are calculating dose using TMRs or TARs we use the TMR or TAR for deff. If we are
using PDDs then we look up the PDD for the depth d and then multiply it by a correction factor,
CF = TMR(deff, Ad) TMR(d,Ad)
or the equivalent with TARs.
As an example, for the diagram above let us assume 6 MV X rays with d1 = 2 cm, d2 = 6 cm,
d3 = 4 cm, the field size is 10 cm X 10 cm , 100 cm SSD. Let us assume a lung density  = 0.3.
The depth in the patient is
d = d1 + d2 + d3 = 12 cm.
The uncorrected percent depth dose is
PDD(12, 10X10) = 59.3%
deff = 2 + (0.3)6 + 4 = 7.8
Field size at 12 cm is 11.2 X 11.2
CF = TMR(7.8, 11.2X11.2) TMR(12,11.2X11.2) = 0.840 0.720
= 1.167
Thus the PDD increases by 16.7%. PDD = 1.16759.3% = 69.2%.
2-Batho (or Power Law) Correction Factor
An alternate correction factor has been proposed. It is just a ratio of two TMRs taken to the
power 1-.
CF =
{ TMR(d ,A ) TMR(d +d , A )}
3
d
2
3
1-
d
Notice the overlying tissue (d1) has no influence on this correction factor.
Going back to our previous example with 6 cm lung, d1 = 2, d2 = 6, d3 = 4, and  = 0.30. The
uncorrected PDD is still 59.3%.
{
TMR(4, 11.2X11.2) TMR(10,11.2X11.2)
CF =
= 1.148
}
1 - 0.3
=
{0.94740.7778}
104
0.7
In this case the PDD increases by 14.8%. PDD = 1.14859.3 = 68.1%.
3- Equivalent TAR Method
In the equivalent TAR (or TMR) method both the depth and the field size are scaled by the
existence of the inhomogeneity. For example, in lung the field size is scaled downwards because
the quantity of scattered photons will be diminished because of the lower density.
CF = TMR(d,A)TMR(d,A)
The method for calculating d and A depends on the geometry of the inhomogeneity and is more
suitable for computer rather than manual calculations.
4- The Isodose shift method
A variation of the isodose shift method for irregular contours.
The isodose curve is placed on the contour as
usual. Beyond the inhomogeneity (usually lung
for this method) the curve is shifted downward by
some fraction n of the total thickness of the
inhomogeneity.
The fractional shift n depends on the energy of
the beam.
All of the above mentioned methods only correct for changed in the photon fluence. The low
density of the lung also has an effect on the Compton electrons. Their range is much greater in
lung than in unit density tissue. This is evident in an increase in the penumbra width in lung.
Above 10 MV the worsening of the penumbra is particularly severe.
Other, more sophisticated, methods for correcting for inhomogeneities are employed by modern
treatment planning computers.
105
Abutting fields – gaps
It is not uncommon to be confronted with the problem of treating a patient with fields which abut
one another. This comes about when it is required to treat a region of length greater than can be
encompassed in a single field.
Since the field lengths increase as the beam travels
through the patient a gap must be provided on the
patient’s surface in order to prevent an overlap occurring
at a critical structure (e.g. spinal cord).
Gaps are arranged so that the overlap begins at a
depth, d.
One way to correctly determine the surface gap makes use of the concept of decrement lines.
Decrement lines are lines which at any
depth are a constant percentage of the
central axis value.
Obviously the central axis is the 100%
decrement line.
The diagram illustrates decrement lines
(solid) in relation to the isodose lines
(dotted).
By matching the 50%
decrement lines for the abutting
fields at the depth of concern,
the skin gap can be measured as
the distance between the 50%
lines on the surface.
106
A more common method of gap determination rests on the assumption that the geometrical
field edge (as indicated by the edge of the light field) corresponds to 50% decrement line. This
is a good assumption in most cases. The surface gap is then calculated using the rule of similar
triangles.
The general case, in which the SSDs of the two fields are not the same, is illustrated in the diagram
below.
The shaded triangles are similar as
are the corresponding triangles of
the other field.
Thus we get the relations:
g1 (
= d SSD1
½ L1)

g2 (
 ½L )
2
=

dSSD
2
Since gap = g = g1 + g2
we get the well known gap formula:
g = d(½L1SSD1 + ½L2SSD2)
Notice that it is the half-lengths of the fields that appear in the formula. If the fields are
collimated asymmetrically then we substitute the edge-to-central axis distance for ½L2 and ½L1.
When the fields are irregularly shaped then the calculation of gaps becomes even trickier.
In the diagram we are looking down onto the patient’s surface.
For the y-gap, gy, Y2 is the full collimated length, but Y1 is smaller
than the full length, due to the presence of the block.
In this case for the x-gap, gx, we must use a negative X1 in the
above equation,
i.e.,
gx = d(X2SSD2 – X1 SSD1)
gy = d(Y2SSD2 +Y1 SSD1)
107
Irregular field dosimetry
Earlier we used approximation methods for arriving at the equivalent square for an irregularly
shaped field. A more accurate method of calculation uses the concept of Scatter-Air Ratios
(SAR) or Scatter –Maximum Ratios (SMR).
First we must determine the Zero Area TMR (or TAR) i.e., the TMR0 (or the TAR0). From a
plot of the TMR versus Field Size we can extrapolate the TMR to what it would be for an
infintesimally small field. This is the TMR0, which depends only on depth and beam quality.
The TMR0 represents the attenuation of the primary beam, i.e., a beam without scattered
photons.
To get the SMR we simply subtract the TMR0 from the TMR for a given field size.
SMR(depth, field size) = TMR(depth, field size)  TMR0(depth).
The SMR represents the component of the TMR from the scattered photons. For convenience
the table of SMRs is given as a function of depth and field radius, instead of field area. The
radius is that for a circular field of a given area. A table of SMR (along with the TMR0) for a 6
MV X ray beam is shown below.
Table of SMR – 6 MV X rays
radius
depth (cm)
5
8
10
12
15
20
TMR0
0.835
0.719
0.651
0.589
0.507
0.395
radius
5
8
10
12
15
20
1
2
3
4
5
6
7
8
0.027
0.027
0.026
0.025
0.024
0.021
0.046
0.055
0.061
0.059
0.055
0.049
0.065
0.077
0.081
0.078
0.077
0.067
0.072
0.092
0.098
0.098
0.097
0.087
0.079
0.105
0.112
0.114
0.115
0.107
0.085
0.113
0.124
0.127
0.131
0.121
0.091
0.121
0.135
0.139
0.142
0.134
0.093
0.128
0.143
0.149
0.153
0.146
9
10
12
14
16
18
20
0.095
0.133
0.151
0.157
0.162
0.157
0.097
0.138
0.156
0.163
0.171
0.166
0.101
0.146
0.166
0.176
0.185
0.183
0.103
0.149
0.173
0.186
0.198
0.198
0.105
0.154
0.178
0.192
0.206
0.208
0.107
0.158
0.183
0.198
0.213
0.216
0.109
0.161
0.188
0.203
0.218
0.223
We illustrate how we use the SMR table by calculating the TMR for the field illustrated below.
108
The calculation point (central axis) is the center
of the diagram. We break the irregularly shaped
field into, in this case, 18 pie piece shaped
sections, each with an angle of 20O. The
sections are labeled A to R.
The distance from the central axis to the field
edge is noted. The primary beam component,
TAR0, will be the same for each of the sections.
The scattered beam component (SMR) will be
the average of the SMRs for each of the
sections.
Sections B and C are cut by an intervening
blocked area. This will require special
consideration. This will be illustrated below.
For our example, the depth will be 10 cm and the beam quality is 6 MV. We then average the
SMR as follows:
section
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
radii
10
3.5, 8, 11.5
3,7,11
6
5.5
6
4
4
10
10
10
6.5
5
5
5.5
11
11.5
10
SMR(10,10)
SMR(10, 3.5)  SMR(10,8) + SMR(10,3.5)
SMR(10,3) – SMR(10,7) + SMR(10,11)
SMR(10,6)
SMR(10,5.5)
SMR(10,6)
SMR(10,4)
SMR(10,4)
SMR(10,10)
SMR(10,10)
SMR(10,10)
SMR(10,6.5)
SMR(10,5)
SMR(10,5)
SMR(10,5.5)
SMR(10,11)
SMR(10,11.5)
SMR(10,10)
SMR
0.156
0.163 – 0.143 + 0.089 = 0.109
0.161 – 0.135 + 0.081 = 0.107
0.124
0.118
0.124
0.098
0.098
0.156
0.156
0.156
0.129
0.112
0.112
0.118
0.153
0.163
0.156
AVERAGE =
0.130
The TMR0 at 10 cm depth is 0.651. From this we can find the total TMR by reversing the above
equation for the SMR
Average TMR(depth, field) = TMR0(depth) + Average SMR(depth)
Thus the average TMR = 0.651 + 0.130 = 0.781.
109
We can use this TMR to calculate Monitor Units just as one would do so for a square or
rectangular field. We can also find from a TMR table the square field that had this TMR at the
depth of 10 cm. In this case it is a 12X12 field. This is therefore the true equivalent square field
for the irregular shown above.
The type of calculation shown above is sometimes called the Clarkson Method or the Scatter
Integration method. For greater accuracy a computer calculation would have broken up the
field into many more thinner slices than the 18 used in this manual calculation.
Dose outside a field
The Clarkson Method can be used to calculate dose outside of field. Consider the situation
shown in the diagram below.
The calculation point is located 3 cm
outside the edge of a 10 cm X 15 cm 6 MV
X ray field. The calculation depth is 8 cm.
Since the point is outside the field there
will be no primary photon beam
contribution (except for the transmission
through the collimator, which we will
ignore).
As before, we sum the SMRs over the
eighteen 20O angular intervals. However
only seven, sections A to G, contribute to
the SMR. For each section we subtract the
SMR for the smaller radius from the smr
from the larger.
section
A
B
C
D
E
F
G
radii
8.5, 6
12.5, 4
14, 3.5
13, 3
14, 3.5
12.5, 4
8.5, 6
Average
=
SMR
0.133 – 0.125 = 0.008
0.147 – 0.092 = 0.055
0.149 – 0.085 = 0.064
0.142 – 0.077 = 0.065
etc. 0.064
0.055
0.008
0.319  18 = 0.018
The average SMR is 0.018. In the center of the field the TMR at 8 cm depth for a 10X15 field is
0.838. Therefore we expect the point 3 cm outside the field to receive 0.018  0.838 or 2.1% of
the dose that the center of the field receives. Transmission of the primary beam through the
collimators will contribute an additional 1 to 3 percent depending on the specific treatment unit.
The Clarkson method can also be used to calculate the dose from scattered photons under a
block. Let us estimate the dose by transforming a complicated situation into a simpler one.
110
The situation is indicated below. A 26 cm X 26 cm field has a kidney shaped blocked region of
approximate size 5 cm X 10 cm (figure a). We transform the field and the blocked regions into
circular regions with the same areas and transpose the blocked region to the center of the field
(figure b). The field has a radius of 14.7 cm and the blocked region has a radius of 4 cm.
We will calculate for 6 MV X rays at a depth of 12 cm. Under the block the SMR will be:
SMR = SMR(12,14.7)  SMR(12, 4) = 0.188  0.098 = 0.090
Since the point is under the block there is no contribution from the TMR0. To estimate the TMR
for the unblocked area we might simply use the 26 X 26 cm2 , or subtract the blocked area (50
cm2) and use a 25 X 25 cm2 field.
TMR(12, 25X25) = 0.776
Therefore the ratio of doses under the block to that of the open area will be 0.090/0.776 = 0.116,
or 11.6%. Transmission of the primary beam through the block can add as much as 5%. Thus
we see the dose under a block of this sort can be substantial.
Scatter vs. X ray beam quality
As we have seen photon scatter, as represented by the SMR, increases as the field size increases.
The SMR decreases as the X Ray energy increases. This can be seen in the figure below.
111
Since the amount of scattered photons decreases with energy, we would expect that high energy
beams have a sharper penumbra compared to lower energies. However we must also consider
the effect of secondary (Compton) electrons on the penumbra. Because higher energy photons
produce more energetic electrons with a greater range and therefore greater lateral scatter, we
find that, above 6 MV, penumbra actually increases as the beam energy increases.
XIV- Electron beam therapy
112
The electron beam depth dose curve
Electron beams can be produced from linacs, microtrons, and betatrons by removing the X-ray
target from the machine. Radiobiologically electrons are identical to photon beams. The LET
and RBE are the same as photons (except for an increased LET and RBE with very low energy
electrons).
The depth dose characteristics are very different from photons.
Unlike photons, each electron in a beam begins to interact immediately as it enters a material
body. Electrons lose energy at a constant rate with depth until they run out of energy and
become incorporated with the material, i.e. a charge is built up. In practical terms this charge is
usually insignificant.
Unlike heavy charged particles (like protons), electrons are
deflected in random directions as they interact.
Thus a narrow electron beam (pencil beam) will rapidly
broaden and diminish in intensity with depth.
For a broad beam this diminishment is evident only at the
edges of the beam, where it appears as a very broad
penumbra.
The diagram shows the difference in the depth dose curves of
broad and narrow beams.
Electrons interact with material in two distinctly different ways, by collision and radiatively.
1.- Collision. Part of the electron’s energy is transferred to an atom resulting on either
excitation or ionization of the atom. It is the ionization component of the interaction that
contributes to the radiation dose.
2.- Radiative (bremsstrahlung) The electron interacts with the nucleus of an atom, losing
some energy to produce an X ray photon. This photon can interact and deposit dose at
depths greater than the range of the electrons. This works against the purpose for which
electron breams are used.
The ratio of radiative to collision energy loss is directly proportional to both the energy of
the electron and the atomic number of the material, i. e.
radiativecollision Energy  Zmaterial
Thus the higher energy electron beams have a greater radiative contamination than the lower.
113
The main characteristics of an electron beam depth
dose curve are illustrated in the diagram. They are:
1.
2.
3.
4.
Skin sparing (some or none, depending on energy)
Broad maximum dose region
Sharp fall off to Rp (practical range)
Tail due to Bremsstrahlung
The practical range is the depth at which a straight line
representing the beam fall off intersects the straight
line representing the bremsstrahlung tail.
For a particular accelerating unit and for a given beam energy, broad electron beams have
basically the same central axis percent depth dose curve. For small field sizes the depth dose
deteriorates, becoming like the narrow beam shape for very small fields.
The criterion used to separate broad beams from narrow beams is:
If the beam width > Rp then the broad beam percent depth dose holds.
If the beam width < Rp then at any depth the PDD will be less than that of the broad
beam, i.e. somewhere between the broad beam and the pencil beam.
Skin sparing
Skin sparing for electron beams is fundamentally different from that of photon beams. As a
beam enters the phantom (or patient) normal to the surface the electrons all travel in a uniform
direction. With increasing depth the motion picks up a lateral component which results in a
greater amount of energy deposited per unit volume at depth compared to the surface.
Unlike photon beams the ratio of
surface dose to the dose at d-max is
greater for higher energy beams than
for lower energy beams.
The curve shows the surface doses for a
particular linear accelerator. Other
machines may have different values for
the surface doses, but the overall trend
is common.
114
Producing a broad electron beam
The electron beam as it leaves the bending magnet of the accelerator is a very narrow pencil
beam. It is converted into a useable broad beam either by scattering foils or by scanning the
beam across the patient with deflection magnets.
The scattering foils have an effect the PDD curve.
1. X ray photons can be produced in the foils.
Therefore the radiative tail is greater in a scattered beam
than in a scanned beam.
2. Some energy is lost in the foil. The energy of the
beam leaving the foil is less than the energy entering the
foil. In addition the beam is less monoenergetic as it
leaves the foil, i.e., there is a spread in energies. This
results in a more gradual fall off of the dose at the end
of the range.
Measurement of electron beam energy
In the accelerator the energy of the electron beam is very
precisely defined. The electrons, however, interact with the
exit window, scattering foils, and even the air through
which they travel.
As the beam strikes the surface of the patient there will be a
significant spread in the energy, and the average energy
will be reduced.
It is therefore important to be able to determine the energy
of the beam striking the patient or phantom. A number of
different techniques have been devised for doing this.
1. Nuclear Reaction Thresholds - Materials can undergo nuclear transformations when
bombarded with electrons. The threshold energy for each material is precisely defined.
For example:
63
Cu + e-
 62Cu
+n
115
The threshold energy is 10.855 MeV. The resultant 62Cu is radioactive.
By varying the beam energy in small increments and measuring the radioactivity produced one
can determine when the beam energy has reached the reaction threshold.
2. Threshold for Cerenkov Radiation- The velocity of light in a vacuum, c, is a universal
constant. However, in a transparent medium, such as a gas, the velocity of light will be less
than c.
velocitymedium = cn
where n is the index of refraction of the medium.
Megavoltage energy electrons will be travelling very close to c and therefore can be
travelling faster than light in the medium. When this occurs Cerenkov light is produced.
(This is the blue glow that one can see in water moderated nuclear reactors.) By varying the
index of refraction of a gas in a container being irradiated by an electron beam the
threshold for Cerenkov radiation can be determined. From this the velocity, and
therefore the energy, of the electrons is determined.
3. Energy-Range relationships- From the central axis depth dose curve one can determine
the practical range, RP, which can be related to the energy of the beam at the phantom
surface, EP,0. The relationship is
EP,0 = 0.22 + 1.98RP + 0.0025(RP)2
where EP,0 is in MeV and RP is in centimeters.
An alternative formula uses R50, the depth at which the
dose is 50% of the maximum. In this case the
relationship is
EP,0 = 2.33R50.
Once the energy at the phantom surface is known the energy at any depth, d, can be found by the
formula
EP(d) = EP,0(1.0  dRP)
i.e., the energy decreases linearly with depth to zero at the practical range. The dose beyond RP
is due to X ray contamination.
116
Some practical electron beam formulae
When the electron beam energy at the patient surface is known, there are some approximate
formulae that can be used in clinical situations. These relate RP and R80 (the depth of the 80%
isodose line on the central axis) to the incident beam energy. They are:
RP(cm) 
½Energy(MeV)
and
R80(cm)  1/3Energy(MeV)
Clinically RP represents the depth beyond which the dose is negligible. R80 represents the limit
depth of the useful dose.
Electron beam isodose curves
The main characteristics of electron beam isodose curves are the broad high dose region,
followed by the rapid fall off. At depth the isodose percents less than 50% bulge out and the
percents greater than 50% contract inward. Thus the penumbra at depth deteriorates. This
deterioration is worse with the higher energy beams.
The figure at the left shows typical broad
beam central axis depth dose curves of
electron beams from a linear accelerator.
These curves illustrate many of the points
that have been previously discussed, i.e.,
the increase in surface dose with energy,
the increase in bremsstrahlung with
energy, and the practical formulae
mentioned above.
It must be noted that the standard PDD curves apply only to broad electron beams, i.e., for
field sizes greater than RP . For smaller fields the R50 the depth dose will be degraded.
117
Special considerations are necessary
when a dimension of the beam (length or
width) is less than Rp . Use with caution.
The curves illustrate the degradation of the
depth dose for small 20 MeV electron
beams for which Rp is 10 cm.
Virtual electron source
The electron beam is not diverging from the X ray target, but is scattered and broadened by the
scattering foils and other material in the beam. The effective point of the divergence is called
the virtual source.
The virtual source is usually downstream from the X
ray target.
When treating with electrons at something other
than the standard distance (usually 100 cm from the
X ray target), an inverse square correction must be
applied. But the distance must be measured from
the virtual source, not the X ray source.
Example: Suppose the standard treatment distance is 100 cm from the X ray source ( the
conventional 100 cm SSD), but a treatment must be given with the patient surface 10 cm beyond.
The virtual source is 15 cm downstream from the X ray target, i.e. the effective SSD is 85 cm.
The dose point is at d-max, 2 cm. The dose per MU will be decreased by the factor:
2
= 0.804
Correction factor = (85 + 2)(85 + 10 + 2)
[
]
118
Electron field defining cut-outs
Linear accelerator electron applicators generally define square, rectangular, or circular fields. If
an irregular field shape is required a cut-out of lead or Lipowitz’s metal (low melting point
alloy). The thickness of these metals that is required to stop an electron beam can be estimated
by considering the electrons’ practical range, RP.
We have said that in water RP(cm) = ½Energy(MeV). The physical density of Pb is 11.3
times that of water, and the electron density is about 8 times that of water. Thus we would
expect we would expect it to take about 1/10th the amount of lead to stop the beam. This has
been shown to be true, i.e.,
Pb thickness (cm)  (1/20)Energy(MeV).
Lipowitz’s metal is not quite as dense as Pb and it generally takes a 20% greater thickness.
If Pb or other heavy metal is used as an internal shield then back scattering of electrons must be
taken into account. Back scattering can increase the dose of the tissue adjacent to the Pb by as
much as 70%, depending on the beam energy. This can be mitigated by covering the Pb with low
Z material such as plastic or aluminum.
The percentage back scatter decreases as the energy increases and increases as the Z of the
material increases.
Corrections for tissue inhomogeneity
When an electron beam passes through a region that is not of unit density the depth dose curve is
altered. If the inhomogeneity has  < 1.0, such as lung, then the beam will penetrate to a greater
depth.
Within the inhomogeneity the depths on the PDD
curve can be changed to the effective depth using
the coefficient of equivalent thickness or the
CET.
If d1 is the thickness of unit density tissue through
which the beam has passed, and d2 is the depth in
the non-unit density tissue, the geometric depth is
d = d 1 + d2 .
The effective depth is
deff = d1 + CETd2.
The dose at point P at depth d is the dose in a unit
density phantom at depth deff.
The CET for lung is usually taken to be the density which is ~ 0.25. For bone the CET is taken
to be the electron density ratio, 1.65.
A constant value for the CET is only an
approximation; it can vary with depth in the
inhomogeneity.
119
The figure shows the effect of lung density on
the depth dose curve of a 9 MeV electron beam.
The beam first passes through 3 centimeters of
normal density tissue, i.e. d1 = 3.
The dotted line indicates the depth dose for a unit
density phantom.
Suppose we want to know the dose at 5.5 cm
depth.
The effective depth is deff = d1+CETd2 = 3 cm + 0.252.5 cm = 3.63 cm.
Therefore the dose at 5.5 cm is that which would occur at the effective depth in unit density
tissue, i.e. 40% of the maximum.
Hot spots and cold spots can occur at the edges of high
density inhomogeneities due to the increased scattering
that occurs, as shown in the illustration.
Even greater departure from normal dose distributions
will occur when metallic internal shields are used with
electron beams.
Effect of oblique incidence
When electron beams are incident on the surface of a patient at angles other than the
perpendicular the depth dose curve is distorted.
We can understand this phenomenon by thinking of a broad
electron beam as being made up of from a number of pencil beams
adjacent to each other. As the pencil beam spreads the intensity of
the beam decreases.
With normal incidence adjacent pencil beams combine to counteract the dose decrease and
produce dose uniformity.
With oblique incidence there is an enhancement of the dose at shallow depths and a
corresponding diminishment of dose at greater depths.
The change in the depth dose curve for a 9 MeV electron beam with a 45 degree angle of
incidence is shown in the diagram.
120
XV- Dose reporting in radiation therapy
121
Source is ICRU Report 50 (International Commission on Radiation Units and Measurements)
Prescribing, Recording, and Reporting Photon Beam Therapy (The concepts are also commonly
used for electron beams.)
Definitions of volumes - Illustrated in the diagram
GTV (Gross Tumor Volume): The palpable or visible extent and location of the tumor (can be
evaluated by different imaging modalities). If the tumor has been surgically removed there is no
GTV.
CTV (Clinical Target Volume): This contains the GTV and suspected sub-clinical disease. This
is the volume that must receive the prescribed dose to achieve cure or palliation
PTV (Planning Target Volume): This volume is the CTV plus a margin which takes into
account all inaccuracies due to organ motion, patient motion, and set up errors. It is the volume
defined for treatment planning in order to ensure that the CTV receives the prescribed dose.
(Note- The PTV margin does not include beam penumbra.)
Treated Volume: This is the volume enclosed by the prescription isodose level (in the above
example the 95% isodose level). It should be equal to or greater than the PTV; it depends on
beam arrangement.
Irradiated Volume: A volume that receives a dose that is significant with respect to normal
tissue tolerance (in the above example the 20% isodose level).
122
Dose reporting
The ICRU recommends that the reported dose be the dose at a point, the ICRU reference point.
The point is chosen to satisfy the following:
The dose at the point is clinically relevant and representative of dose in PTV
The point can be clearly and unambiguously defined.
The dose can be accurately determined (physically).
There is no steep dose gradient at the point.
To satisfy these requirements the ICRU recommends the reference point be:
1- at the center of the PTV and where the tumor cell density is greatest
2- on or near the beam central axis, or axes for multiple beams
If both conditions cannot be simultaneously satisfied, the first takes precedence.
In addition to the dose to the ICRU reference point, the report should include the maximum
and minimum dose in the PTV.
The ICRU report recognizes three levels of dose reporting.
Level 1: Basic- Reference point dose and estimates of maximum and minimum doses in the
PTV using central axis dose tables.
Level 2: Advanced- Level 1 plus dose distribution (isodose curves) in a plane or planes.
Level 3: 3D- Level 1 plus volumetric dose distributions, non-coplanar beams, Dose volume
histograms (DVHs).
Dose Volume Histogram (DVH). The dose-volume histogram is a mathematical device for
evaluating a three dimension dose distribution over a volume of interest. The volume may be the
PTV (planning target volume), the volume of an organ of interest, or even the entire patient
volume.
The starting point for constructing a DVH is a three dimensional
dose distribution over a set of images of a patient. Within this
image set the external contour of the object of interest must be
delineated (segmented).
As an illustration we consider the case of a stereotactic
radiosurgery plan for treating a brain tumor. In the diagram the
tumor is outlined with a bold line. Isodose lines are indicated
with a lighter line. In this case the prescription dose will be 1500
cGy to the 60% isodose line. The maximum dose will be 2500
cGy.
123
To calculate the DVH a three dimensional grid is
constructed which encompasses the object of interest.
(In our example this grid is shown for four
representative slices.) Each point on the grid
represents a single volume element (voxel). For each
point on the grid the dose is calculated.
Only the doses at points within the contoured object
of interest are counted in the histogram.
A histogram is constructed representing the frequency
that a volume element within the object receives a
particular dose value.
The horizontal axis displays the dose values and the
vertical shows the volume receiving that dose.
This form of the DVH is often called the differential dose
volume histogram. A more common display of the DVH
data is through the cumulative, or integral, DVH. This is
derived from the differential DVH as follows.
Differential DVH
124
Integral DVH
Each dose bin in the integral DVH represents the volume receiving that dose plus the volumes
receiving any greater dose. In the example, the bin at 1500 cGy contains the volumes in the
shaded area in the differential DVH.
An alternative method of displaying the data is to divide the volume at each dose by the total
volume of the object of interest (i.e., normalization). This allows a display the DVHs for
different organs with widely differing volumes on a single chart as is shown in the example
below.
XV- Special topics in photon beam therapy
125
IMRT –Intensity Modulated Radiation Therapy
IMRT is a technique using X-ray beams which are modified so that the X ray fluence is nonuniform across the beam. This is done conform the dose to the target volume and to spare
sensitive normal tissue.
For the example indicated in the diagram the objective
would be that the prescription dose is to conform to the
PTV and the dose to the organ at risk (OAR), the cord,
is minimized. Five beams are chosen for irradiation with
the fluence profiles for each beam as indicated.
The fluence profiles for other slices through the patient’s
CT volume will be different since the PTV and the
patient anatomy will be different. The result will be a
two dimensional fluence map for each beam.
IMRT treatment planning. The task is to determine
the photon fluence maps which will satisfy the treatment planning goals. The planning is
normally done by dividing the photon beams into a large number of beamlets with a 5 mm × 5
mm or 10 mm × 10 mm cross section. Each beamlet has an adjustable weight. The beamlet
weights, which determine the fluence profile, are optimized iteratively so as to minimize a “cost
function” or objective function. The objectives may be defined in terms of deviations from the
desired dose to the PTV and deviations from zero or some minimum dose to normal tissue
outside the PTV. In addition doses may be constrained so as not to exceed defined permissible
points on the dose-volume histograms for organs at risk. Input data for the planning process are
the number and the orientation of the beams, the target dose goals, and the constraints.
Once the optimal beam arrangements and fluence maps are determined the technique for
delivering the beams needs to be determined. There are a number of different methods for
accomplishing IMRT. They can be divided into arc based and fixed field techniques.
Arc based:
MIMiC (Multileaf Intensity-modulating Collimator)
Uses a single slice multileaf collimator attached to the linac. Achieves IMRT on
a slice by slice basis.
Helical tomotherapy
Uses a linac mounted on a CT style gantry which rotates around the patient as
he/she moves through the gantry.
IMAT (Intensity Modulated Arc Therapy)
Dynamically shapes an MLC field as the linac rotates around the patient.
Uses a conventional linear accelerator.
126
Fixed field: A linear accelerator is used with a set number of fixed fields. The photon
fluence is shaped to a desired pattern. The fluence shaping may be done by one of three
techniques:
Compensator
A filter similar to a missing tissue compensator can be used to shape the photon
fluence.
Multi-segmented Multi-Leaf Collimator technique (SMLC) – “step and shoot”
For each static gantry location the field is formed by summing a set of subfields.
The beam may be switched off as the collimators move from one field segment to
another. This mode is often called “step and shoot” method.
The SMLC technique is illustrated below.
The desired photon fluence is indicated in the picture. The
darker areas indicate greater photon fluence. This fluence map
can be created by combining the six field segments below.
Each segment is weighted by an appropriate factor which
determines the relative number of monitor units for the
segments.
Dynamic Multi-Leaf Collimator technique (DMLC) – “sliding window”.
In this technique the MLC leaves sweep across the field with the opening
between the leaves determining the photon fluence
distribution.
127
We use the fluence map in the diagram to
illustrate this technique.
The situation begins with the a leaves set to define the outer edge of the field, and the b leaves
adjoining the a leaves to form a closed field. As the leaves move from left to right a gap opens,
letting the X rays through. In the final position the b leaves define the outer edge.
We will track the shaded leaf set which needs to create the illustrated profile. We have divided
the profile into five regions, three with uniform dose (I, III, V) and two transition regions (II and
IV).
In region I the two leaves sweep across at the same velocity exposing that region to a
constant photon fluence. In region II the a leaf slows, allowing the gap between a and b
to widen. In region III both leaves again move at the same velocity, with a wider gap. In
region IV the b leaf slows, to bring the gap back to the setting for region I. For optimal
efficiency in the transition regions one of the leaves is always at the maximum leaf speed.
128
Stereotactic radiosurgery (SRS)
SRS is done by targeting a lesion by means of a
fixed reference frame instead of marks on the
patient's skin. In the case of brain treatments
this frame is affixed to the patient using pins
which are made to pierce the skin and anchor to
the skull. A number of different reference
frames are available commercially. Here we
describe the Leksell system. After the frame is
firmly attached to the patient an appliance
containing fiducial markers is fastened to the
frame. The fiducial markers are radio-opaque
crosses (for angiography), radio-opaque rods
(for CT), or channels containing a CuSO4
solution, which are visible on MR scans.
When the patient undergoes a CT or MR scan the resultant images appear with dots which
indicate the position of the fiducials.
The planning computer transforms the X, Y, and Z coordinates of the image from a system
based on the CT (or MR) unit to one based on the stereotactic frame. The coordinates in the
center of the frame are set to (X,Y,Z) = (0.,0.,0.) in some systems or to (100.,100.,100.) in
others. The X and Y for any target within the image can be read as displacements from this
origin. The Z (axial) position is determined by the location of the spot created by the diagonal
rod.
129
The planning computer provides the target locations in terms of the frame coordinate system.
Treatment can be accomplished by a number of different techniques.
The Leksell Gamma Knife is a unit which has
201 separate 60Co sources, each with an activity of
~30 curies. The sources are arranged in an
annular array and are indiviually collimated and
focussed on a single point in the center of the
annulus. Four separate collimator sizes are
available providing fields of 4 mm, 8 mm, 14
mm, and 18 mm in diameter at the focus.
Linac radiosurgery is frequently done with
multiple arcs from an X ray beam with
collimators of dimension 5 mm up to 40 mm.
Alternatively radiosurgery can be done with
small static fields which are made to conform to
the lesion shape by the use of multi-leaf
collimators
130
Total body irradiation (TBI)
In order to treat the entire body in a
single field an extended treatment
distance is necessary, at least 400
cm from the X ray source.
A 40 cm  40 cm at 100 cm
becomes 160 cm  160 cm at 400
cm.
TBI can be delivered either in an
AP/PA mode or with opposed
lateral fields.
The AP/PA technique allows one to limit the dose to sensitive organs such as lung, kidneys, and
liver.
Lateral treatments, with the patient in a semi recumbent
position with knees raised, offers greater patient
comfort.
Because of the large variation in the lateral thickness
throughout the patient's body, compensators are
necessary for lateral treatments in order to keep the
dose variation within  10%.
Choice of X ray energy- For large patient
thicknesses the sum of the entrance and
the exit dose is greater than the midplane
dose. This effect is greatest for the lowest
energy photons. For this reason opposed
lateral TBI treatments should be done with
the highest photon energy available.
For AP/PA treatments, the patient thickness rarely exceeds 30 cm. 6 MV X rays are adequate
since the entrance & exit does not exceed the midplane dose by 10%.
131
Surface dose- Since megavoltage photon beams do not deliver the
full dose to the patient surface, beam "spoilers" are needed to bring the
surface to full dose. 1 cm acrylic plastic is sufficient for 6 MV, but 2
cm is required for 18 MV X rays. The spoiler should be as close to the
patient as possible for maximum effect.
Spoilers are needed for both AP/PA and lateral treatments.
Dose limitation to critical organs- Lungs, kidneys, and liver are
radiation sensitive organs. The dose to these organs can be limited
by partial shielding blocks (e.g. 50% transmission) made of lead or
Lipowitz's metal.
The appropriate
thickness of metal
can be determined from an attenuation curve.
In the case of lung shielding the dose to the
blocked area of the anterior and posterior chest
wall can be increased by boosting with electron
beams.
Monitor Unit Calculations. The TBI dose is conventionally specified at the mid point of the
patient at the level of the umbilicus. For a linear accelerator that is calibrated in the conventional
manner to produce 1 cGy per Monitor Unit (MU) at d-max at a source-to-calibration point
distance SCD the dose per MU at the reference point can be determined by:
Dose ⁄ MU = Sc×Sp(FS)OAF(SCD⁄SAD)2TMR(d,FS)TraySpoiler
where: Sc is the scatter factor for the maximum collimator setting (usually 4040)
Sp(FS) is the phantom scatter factor for the field FS. The effective field of the patient must be
estimated, but frequently 3030 cm2 will do.
OAF is the off-axis-factor which is necessary if the calculation point is not on the central axis of
the beam
SCD is the source-to-calibration distance
SAD is the distance from the source to the patient's mid-plane
132
TMR(d,FS) is the TMR for the midplane depth, d, and field size FS
TRAY is the attenuation factor of the tray supporting the shielding blocks or compensators
Spoiler is the attenuation factor of the spoiler.
Example: A patient who is 30 cm wide at umbilicus level is to be treated at an SAD of
400 cm with an 18 MV X ray beam which has been calibrated at d-max at an SSD of 100
cm. Typical vales would be: Sc = 1.10, Sp = 1.01, TMR(depth-15, FS=302) = 0.815.
Tray = 0.98, Spoiler = 0.96, OAF = 1.00(central axis).
Dose/MU = 1.101.01(103.3⁄400)20.8150.980.96 = 0.0568 cGy/MU.
Typically the dose for one of the lateral fields would be 75 cGy.
Therefore the MUs would be
MU = 75⁄0.0568 = 1320
Alternatively one can measure the Dose/MU and the TMR at the TBI treatment distance with the
attenuators and spoiler in place. The two methods for determining the MUs usually agree within
one percent.
Compensators For lateral treatments the large variation in
depth to the midplane necessitates the use of compensators,
specifically for the head, neck and the legs.
Compensators can be fashioned out of many materials;
aluminum is most commonly used. First one must
determine the broad beam attenuation coefficient of the
metal, . If point A is the dose point in question (e.g. the
head) then one calculates the dose a point A and compares
to the reference dose (e.g. at the umbilicus).
the
it
133
One can then calculate the required attenuator thickness, t, according to
-t
Dose(pt. A)e
= Dose(reference)
or
t = (1⁄)ln(Dose(pt. A)⁄Dose(reference))
where ln refers to the natural logarithm.
The lateral dimensions of the compensator must be
demagnified if the compensator is placed upstream
from the patient.
Otherwise bolus type compensator can be applied
directly on the patient.
134
XVI- Imaging in radiation therapy
Scattered photons and contrast reduction
An image on the film is formed when objects
absorb X rays and cast a “shadow” on the
film.
The quality of an image is described by:
Density – the darkness of the film. This
depends on the total number of photons
striking the film.
Contrast- the difference in density between
one region of the film and an adjacent region.
Sharpness- how blurred is the change in
density from one region to another.
X rays scattered in the object being imaged
will decrease overall contrast in the image.
Scattered radiation increases with field area.
Use the minimum field area that provides the
diagnostic information.
Scatter radiation increases with the kV.
This is very evident in images made with
megavoltage X rays.
Image contrast can be increased with the imposition of an anti-scatter grid between the patient
and the film.
135
A grid consists of lead strips separated by low Z, low density material. Unscattered primary
photons travel parallel to the Pb strips and pass through the grid. Scattered photons travelling at
an angle to the strips are absorbed by the Pb. Some primary photons are absorbed too.
Grids increase the image contrast but decrease the sensitivity of
the image receptor (the film-grid system). This will require a
greater exposure to get the same density as image made without
the grid.
The grid strips are usually not exactly parallel but instead are
focused at the X ray source.
Grid ratio = height of grid strips/distance between strips = h/d.
Scatter reduction increases but receptor sensitivity decreases as
the grid ratio increases.
For a stationary grid, each lead strip will create a fine line on the image. Diagnostic imaging
units frequently employ moving grids (Potter-Bucky) to blur out the grid lines. This is not
practical in a therapy simulator. A simulator commonly uses a 6:1 grid ratio, focused at 150 cm
(useable from 100 - 180 cm).
Intensifying screens
Diagnostic X-ray films require an exposure of 30 to several hundred mR at the film for an
image. Since most of the incident radiation is absorbed by the patient, this would mean a
high exposure to the patient. In addition, X-ray tubes are limited in the amount of exposure they
can deliver because of heat.
To overcome this problem the film is placed between X-ray absorbing fluorescent screens. A
typical screen material is CaWO4. The screen absorbs the X-rays and produces many more
photons of light. This can produce a receptor sensitivity of 0.25 - 0.5 mR. Some newer type
screens (“rare earth” screens) have some advantages over CaWO4.
Film technique
The darkness of the image on the film depends on:
A. The thickness of the object being imaged.
B. The X-ray focus to film distance (FFD).
C. The tube voltage (kV) being used.
D. The tube current (mA).
E. The exposure time (seconds or milliseconds). Usually the product of current and time
are set (mAs).
F. The grid ratio.
G. The type of intensifying screen.
H. The type of film. The screen and film combination is usually set.
136
Typical film techniques used on a simulator.
Abdomen (medium AP/PA), SAD 100, FFD 140, regular cassette, 80 kV, 80 mAs.
Chest (medium, AP/PA), SAD 100, FFD 140, reg. cassette, 80 kV, 40 mAs.
Kidney-IVP, extended distance, regular cassette, 80 kV, 200 mAs.
Pelvis (lateral, large) SAD 100, FFD 140, “Rare Earth” cassette, 100 kV, 300 x 2 mAs.
Fluoroscopy
Fluoroscopy on simulators is achieved by
electronically intensifying an image on a
fluorescent screen and converting it to a
video image with a video camera. The video
image can be displayed on an analog
monitor, or can be processed digitally.
The heart of a fluoroscopy system is the
image intensifier or I I.
In an I I X-rays strike a fluorescent screen which
converts X-ray photons to light photons (about
2,000-3,000 light photons for each X-ray photon).
The light photons then strike a photocathode, which
emits electrons. These electrons are accelerated and
focused on a second fluorescent screen, which form
an image with greatly increased brightness.
The increase in the image brightness is due to
1. minification of the image
2. acceleration of the electrons.
Although there is a gain in brightness, each stage of the image conversion adds noise and
geometrical distortion. Hence, the definitive imaging needs to be done with plane films.
Modern computer equipped simulators, however, can digitally correct the distortion inherent in
the II.
With an image intensifier, there can be a choice of 6 inch, 9 inch, or 12 inch input field. The
larger input field results in poorer resolution and greater distortion.
A typical simulator operates in two modes:
137
1. Normal mode: 12 inch diameter field
2. Magnified mode: 9 inch diameter field.
Portal filming with linear accelerators
Because in the megavoltage energy range almost all of the photon interactions are Compton
scattering, the images show much poorer contrast than diagnostic films. Anti-scatter grids are of
little value because the high energy photons can penetrate the grid.
Because the photons have a high energy, instead of an intensifying screen a cassette is needed
simply to provide electron buildup. A centimeter of plastic might do, but a better quality image
will be achieved with a thin sheet of high density material like copper. The buildup will occur in
a shorter distance with less blurring due to lateral scatter of the electrons.
Lateral electron scatter is greater in 1 cm
of plastic than in 1 mm of Cu.
Electronic Portal Imaging Devices (EPIDs)
The use of EPIDs dispenses with the film (and film development) in creating portal images,
providing for real time image acquisition. Although different devices have been used in the past,
the current technology uses Active Matrix Flat Panel Imagers (AMFPIs). An AMFPI
operates in a manner similar to an LCD (Liquid Crystal Display) except it produces an electronic
image instead of displaying one.
The device consists of a large array of pixilated imaging elements and associated control
circuitry. Each element consists of an electrical switch, a capacitor, and a photo sensor. An X ray
converter lies immediately in front of the array. The X ray converter consists of a copper plate to
produce electrons and a phosphorescent screen to convert electrons into light photons. The light
138
stimulates the photo sensor of the array which stores the charge produced until the charge on the
element is read.
The size of each pixel (picture element) can be determined from knowledge of the area of the
panel and the size of the array matrix. Imaging areas as large as 410 X 410 mm2 with an image
matrix of 1024 X 1024 elements have been produced.. In this case the size of each pixel is
obtained from
Pixel width = (Image width)/(Matrix size) = (410 mm)/1024 = 0.40 mm.
This is the smallest width that can be resolved by this particular EPID.
As with radiographic film there is no geometrical distortion in an image produced by an AMFPI.
The signal from each photo sensor is converted to a digital number. The maximum size of the
number can vary from 12 bits (212 = 4096 individual signal levels) to 16 bits (216 = 65536 signal
levels). These represent the number of gray levels which can be shown when the information
from the EPID is displayed by the computer.
Digital Imaging
These notions of pixel size, matrix size, image area, and the depth of pixel digitization are not
confined to EPIDs, but are relevant to all digital imaging modalities, i.e. computed tomography
(CT), magnetic resonance imaging (MR), and emission tomography.
Example: How much computer storage, in megabytes, is required for an image with a 1024 X
1024 matrix digitized to a depth of 12 bits. One byte is equal to 8 bits.
Image size = 1024 X 1024 X 12 = 12582912 bits.
Storage = 12582912
/8 = 1572864 bytes ≈ 1.6 megabytes.
One can reduce image storage requirements by decreasing the matrix size, which will degrade
spatial resolution, or the depth of digitization, which will decrease gray level discrimination.
Computed tomography
CT uses an X-ray tube to produce a collimated X-ray
beam. The intensity of the beam is measured by a
detector after the beam passes through the patient.
Modern CT machines utilize a X-ray fan beam which
rotates around the patient within a circular array of
detectors
The X ray source is rotated around the patient who is
surrounded by a ring of detectors. An X-ray
transmission profile is obtained from each angle of view.
The computer then inverts the transmission profiles into
a two-dimensional array of attenuation coefficients, µ. A
139
value of µ is assigned to a region of the scanned space called a pixel. The numerical values for
the attenuation coefficients are normalized according to the formula
H = ( pixel  water)water 1,000
These H units are called Hounsfield numbers or CT numbers. An image density value is then
assigned to each CT number with black representing the smallest or most negative number and
white the highest number. These density values are then displayed on a video screen to form the
image.
The contrast within the image can be adjusted by selecting what range of Hounsfield numbers
will be represented by the range of density values or gray scale. This is called windowing.
By selecting which Hounsfield number represents the central value of the window, one can
emphasize the visualization of different structures within the image. For instance, in order to
visualize bone, one would center the window on the high H numbers. In contrast, an image
emphasizing lung would be windowed around the low numbers.
The size of an individual pixel can be varied, depending on the total area of the CT slice. The
thickness of the slice in the axial direction can also be varied from ~ 1 mm to ~ 1 cm. The pixel
area multiplied by the slice thickness is a 3 D volume element, called a voxel.
A 3-dimensional image of the patient can be obtained stacking together contiguous CT scans.
The highest resolution volumetric image will be obtained with the smallest slice thickness.
Cone Beam CT with Linear Accelerators
CT volumetric image can be obtained using a
flat panel imager mounted on the gantry of a
linear accelerator. The source of X rays can be
either the megavoltage treatment beam, or
preferably a kilovoltage X ray generator
mounted at 90 degrees from the central axis of
the treatment beam. The KV X ray unit is
preferred because of the lesser degradation of
the image caused by Compton scattering.
The technique is called cone beam CT because the entire volume of the patient is imaged by a
single rotation of the accelerator gantry in contradistinction to a conventional CT in which slices
are imaged with a narrow fan beam. The use of cone beam CT with a linear accelerator permits
image guided radiation therapy (IGRT). With this procedure a patient is imaged in position prior
to treatment. The cone beam CT is then compared to the planning CT and adjustments to the
patient position are made to align the treatment CT with the planning CT.
140
With some IGRT systems the treatment table can be rotated
in roll, pitch, and yaw in addition to the three translational
movements to permit complete alignment of the patient.
Magnetic Resonance Imaging (MRI)
Magnetic moment of a proton. Magnetic resonance
imaging is possible because protons and some other atomic
nuclei have magnetic moments, usually denoted by μ, that
are related to the particle’s spin. Many common elements in
biological tissue do not have a nuclear magnetic moment
(16O, 12C, 32S) because they have even numbers of protons
and neutrons whose spins pair off in the nucleus. Electrons
also have a magnetic moment, but the electrons spins pair
off within the atomic orbitals and in the formation of
chemical bonds.
In the presence of an external magnetic field, denoted by B0, the proton’s
magnetic moment will have a tendency to align itself with the field
although thermal jostling of the molecules will prevent complete
alignment. The spins will be aligned either parallel or anitparallel with
B0.
If the proton’s magnetic moment is displaced from the
normal alignment the spin will precess around the B0 axis
in the same way that a spinning top precesses under the
influence of gravity. The frequency at which it precesses
is called the Larmor frequency, denoted by ω, the value
of which is determined by the field strength according to :
ω = γ×B0
where γ is called the gyromagnetic ratio. Each nuclear spin has a different value for γ and
therefore a different Larmor frequency. For protons the Larmor frequency (in Megahertz) is
ω = 42.58 × B0
where B0 is in Tesla.
As the proton precesses around the B0 axis it will emit RF (radio frequency) electromagnetic
141
radiation at the Larmor frequency. A coil of wire surrounding the object being sampled can
detect the RF being generated. In addition, if RF is applied externally to the protons at the
Larmor frequency it will cause the spins to tip away from alignment with B0.
The amount of tipping will depend upon the duration of the pulse the RF.
Thus a particular duration of RF will tip the spin
900 from the B0 axis. This will result in the
maximum RF detected from the precessing spins
when the excitation RF is turned off. A pulse of
twice that duration will tip the spin 1800 and no
RF will be detected.
Magnetization - The magnetization of a sample,
denoted by M, is the net magnetic moment per unit
volume due to all the spins. (We confine ourselves to
the proton magnetic moment.) After the RF
excitation the magnetization at any point in the
sample will have two components, one transverse to
the B0 axis and the other longitudinal to it.
Relaxation times. With time the magnetization realigns itself with the B0 axis. The unique thing
about the magnetization is that the characteristic relaxation times are different for the
longitudinal and the transverse components. The longitudinal relaxation time is called T1 and the
transverse is T2. T1 is also called the spin-lattice relaxation time and T2 is called the spin-spin
relaxation time.
Creating an MR Image of a Patient. The MR
image is created by measuring the MR signal
at a matrix of points within the magnet. This
can be done by applying gradient magnetic
fields superimposed on the primary magnetic
field. By gradient fields we mean fields whose
strength varies throughout the object being
studied. With this the Larmor frequency varies
throughout the object. The excitation RF is then applied with a range of frequencies. The
resulting detected signal can then be decoded to provide the needed spatial information.
By using different sequences of RF pulses one can make images that are representative of either
the density of protons in the sample, or the relaxation times, T1 or T2, or by combinations of the
three characteristics of the object. Thus different pulse sequences can produce strikingly
different images of the same patient.
142
The MR image is primarily generated by protons attached to water molecules within the patient.
The molecular environment of other hydrogen atoms causes small shifts in the protons’ Larmor
frequency. The presence of these molecules can be detected by means of spectroscopic
frequency analyses of small regions of the patient. There are techniques to suppress the signal
from the water protons, which would otherwise overwhelm the very weak signals from these
other protons.
MRI contrast agents. Some atoms have unpaired electrons in their normal state. The magnetic
moment of an electron is 660 times greater than that of a proton. The presence of these unpaired
electrons will tend to shorten the T1 of water protons. Gadolinium (Gd) is one element which
has a large magnetic susceptibility (tends to increase the magnetic field locally). Gd is a useful
contrast agent for T1 weighted images. It is, however, a toxic metal and therefore must be
chelated for use as a contrast agent.
Ultrasound imaging
Ultrasound images are created using a piezoelectric ceramic or crystal attached to a mechanism
that establishes the position and direction of the ultrasound beam. A piezoelectric material is a
substance that changes shape when a voltage is applied across it and, in reverse, produces a
voltage when a mechanical stress is applied to it. The ceramic is both the producer of the
ultrasound waves and the detector of sound waves that are reflected back to it.
The ultrasound probe, or transducer, must be applied directly to the surface of the patient, since
ultrasound does not travel through air at the frequencies used in imaging. These frequencies are
between 1 megahertz to 10 megahertz (MHz). Most often frequencies between 3 to 5 MHz are
used.
When a pulse of electricity is applied to the piezoelectric
element ultrasound waves are generated. An object below
the surface may reflect the wave back to the transducer.
The reflected wave will be detected after a time, t, which
is
t = 2d/v
where d is the depth of the object below the surface and v
is the velocity of sound in the tissue.
143
The velocity of sound in tissue is assumed to be 1540 meters per second. This is only
approximately true for the various tissues in the body.
Example: An object lies below a patient’s surface. Sixty-five microseconds after the initial
ultrasound pulsation the transducer picks up the reflected sound wave. What is the depth of the
object?
d = vt/2 = 1540×65×10-6/2 = 0.05 meters = 5 cm.
An image is made by scanning the probe across the patient’s surface and also by varying the
angle that the sound beam makes with the surface. This can be done either mechanically or by
electrically steering the beam from a linear array of piezoelectric elements.
Ultrasound waves are reflected back to the transducer whenever there is a change in the acoustic
impedance, Z, of the body tissues. The ratio of the reflected intensity to the incident intensity is
the intensity reflection coefficient, R, and is given by
where Z1 is the acoustic impedance of the first tissue and Z2 is that of the second..
The units of Z are kg/m2sec.
The values of Z for some tissues are:
Muscle:
Fat:
Bone:
Air:
1.66×10-6 kg/m2sec
1.37×10-6
7.8×10-6
0.0004×10-6
Thus the intensity reflection coefficient for muscle-fat is 0.009, whereas for muscle-bone is 0.42,
i.e., bone is strongly reflecting whereas fat is weakly reflecting. The muscle-air interface is even
more highly reflecting. Very little ultrasound will be transmitted through lung.
One major difficulty with using ultrasound for imaging in the human body is the fact that the
sound waves are highly attenuated. In acoustics attenuation is usually measured in dB (decibels)
where 3 dB represents an attenuation factor of ½. The attenuation coefficient, α, in tissue
increases linearly with the ultrasound frequency according to the relation
α(dB/cm) = 0.5×frequency(MHz).
144
From this we see that the half value thickness varies with frequency as:
HVT(cm)
6
3
1.2
0.6
Frequency(MHz)
1
2
5
10
Thus there is a tradeoff in the choice of ultrasound frequency. A lower frequency results in
greater penetration, whereas a higher frequency produces an image with better spatial resolution.
145
XVII- Brachytherapy
Radium and its decay
Radium as a brachytherapy source is of interest for historical reasons. Much of our current
knowledge is based on studies using radium. Radium-226 is a naturally occurring radioactive
nuclide. It is produced in the decay of 238U (T1/2 = 4.5 x 109 yr) and can be extracted from
uranium ore.
The decay of radium and its daughters is:
226
Ra T1/2 = 1620 years
  decay
222
Rn T1/2 = 3.82 days
  decay
218
Po (called RaA) T1/2 = 3.05 minutes
T1/2's very short until 210Pb
  decay
214
Pb (called RaB) T1/2 = 26.8 minutes
 β- decay plus some  rays
214
Bi
(called RaC) T1/2 = 19.7 minutes
 β- decay: multiple isomeric (excited) states, producing most of the γ rays
214
Po (called RaC’) T1/2 = 164  10-6 seconds
  decay
210
Pb (RaD)
T1/2 = 21 years
 β decay
210
Bi (RaE)
T1/2 = 5 days
 β decay
210
Po (RaF)
T1/2 = 138.4 days
  decay
206
Pb (RaG) stable
The useful gamma rays come from the decays of
214
Pb and especially 214Bi.
Secular and transient equilibrium
Because radium decays into radon, an inert gas, and because alpha and beta particles are
produced, radium sources are encapsulated in a Pt (platinum) tube (or needle).
When a radium capsule is first constructed there is initially no gamma rays emitted because it
takes time for the activity of the radon to build up. When the buildup has approaches the
maximum we have a condition called secular equilibrium. Secular equilibrium occurs when
the T1/2 parent >> T1/2 daughter.
146
Equilibrium is reached in ~5-8 T1/2 of daughter
( ~ 1 month for Ra capsule).
When equilibrium is reached the activity of
radon equals the activity of radium (i.e., the
number of radon atoms decaying equals the
number of radon atoms produced).
Since the half lives of all the daughters of radon
up to 210Pb are much shorter than the half life of radon, we have at equilibrium activity parent =
activity daughter and
Activity(226Ra) = Activity(222Rn) = Activity(218Po) = Activity(214Pb) = etc. etc.
Activity =   Number of atoms present
Remember in SI units
therefore
λRaNRa = λRnNRn = λPo NPo = etc.
since λ = 0.6931T we therefore have at equilibrium
1/2
NRa  T
1/2 Ra
= NRn T
1/2 Rn
=
NPo T
1/2 Po
= etc
i.e., at equilibrium the number of atoms of each nuclide are inversely proportional to their half
lives.
Transient equilibrium occurs when T1/2 of the parent is greater, but not much greater, than the
T1/2 of the daughter.
In transient equilibrium there is an initial
buildup of the daughter nuclide on a time scale
dependent on the daughter’s half life. After
transient equilibrium is established the
activities of both the parent and daughter
decay with the half life of the parent.
At equilibrium the following relation holds:
Activity(daughter) = Activity(parent)  [T1/2 parent
(T1/2 parent T1/2 daughter)]
147
Specification of quantity of radionuclides
Initially radionuclide quantity was simply milligrams of radium. When other sources became
available the quantity of was specified in terms of the curie or millicurie, where 1 curie is the
activity of 1 gram of radium. The curie is a specification of activity (number of decays per
second), not a measure of a mass of material.
1 Ci = 3.7 x 1010 disintegration/sec = 3.7 x 1010 Bq
Radium equivalent mass
Another unit for specifying a quantity of a radionuclide that is a substitute for radium is the
radium equivalent mass or milligram radium equivalent (mg-Ra eq). This is the quantity of a
radionuclide that has the same exposure rate at 1 meter in air as 1 milligram of radium. This
exposure rate is 0.825 mR per hour.
Note that the mg-Ra eq is not a measure of mass, nor is it really a measure of activity, but rather
a measure of exposure rate.
Note that 1 mCi of radionuclide X does not have the same exposure rate as 1 mCi of Ra.
Exposure rate constant ( factor)
For a 1 mg (milligram) point source of radium (or 1 mg-Ra eq of something else) the exposure
rate at 1 cm is 8.25 R/h. This is the exposure rate constant for Ra, denoted by . The exposure
rate in air for an activity A(mg) at a distance r(cm) is given by
Exposure rate(R/h) =   Ar2
The units of  are (R-cm2)(mg-h).
If radionuclide X is measured in mCi instead of mg-Ra eq then the  factor is something
different from 8.25 and the units are (R-cm2)(mCi-h) .
Since a radium source is always encapsulated, the  factor assumes a standard thickness of
encapsulation material, which is 0.5 mm of platinum. For thicker material the exposure rate is
reduced by 2% for every additional 0.1 mm Pt.  for other radionuclides is often given for
unfiltered or “bare” sources. In this case the exposure rate for an encapsulated source is
Exposure rate = (Ae-t)r2
where t is the thickness of the material and  is the attenuation coefficient. However, frequently
the strength of an encapsulated source is specified in terms of apparent activity in which case the
exponential factor is left off. Apparent activity is the activity of a bare source that produces
the same exposure rate at a reference distance (1 meter) as the encapsulated source in question.
148
The  factors for commonly used sources are listed below.
226
Ra
8.25
Radionuclide
 (R-cm2)
(mCi-h)
137
Cs
3.26
192
Ir
4.69
Air kerma strength
In the SI system of units the quantity air kerma takes the place of the quantity exposure.
Air kerma is related to exposure by:
Air Kerma(cGy) = Exposure(R)  We
where We = 0.876 cGy/R.
The Air Kerma Strength, denoted by SK is just the Air Kerma Rate at 1 meter from the source.
The usual units for SK are Gy-m2h or cGy-cm2h which are numerically the same value. For
simplicity the units of SK are often given as U, i.e.
1 U = 1 Gy-m2h = 1 cGy-cm2h
Using the above equations for exposure rate and air kerma we arrive at a relationship between
air kerma rate and activity
SK = AWe
Conversion of source strength to dose rate
The earliest systems of dose calculations from point sources ignored the absorption of the
photons by the patient. In that case the variation of dose rate with distance was totally
determined by the inverse square law, just as if the source were in air. The dose rate would then
be obtained from the exposure rate using the f factor.
Dose rate (cGy/h) = fAr2
Example: What is the dose rate in tissue 5 cm away from a 10 mCi radon seed? The  factor for
a radon seed is the same as the  for radium, i.e. 8.25. The f factor is 0.96. Therefore the dose
rate would be
Dose rate = 0.968.251052 = 3.17 cGy/h.
To get the dose rate for sources specified in terms of Air Kerma Strength recall from the
definition of the f factor
f = (We)(en)tissue
(en)air.
149
We therefore find
Dose rate(cGy/h) = SKr2
where the constant  is simply
(en)tissue(en )air which has the numerical value 1.12 for

photon energies above 200 keV. For nuclides with lower energy photons, such as Iodine-125, 
has a different value which depends on source construction.
We see that the simplification in using Air Kerma Strength over Activity for source specification
lies in the fact that we need not know the  factor for the particular nuclide. In spite of this the
use of activity (or apparent activity) persists.
Converting millicuries to mg Ra equivalents
Since the calculated dose rate must be the same for a particular source whether its strength is
specified in mCi or mg Ra eq the conversion from one to the other for some nuclide X is:
Strength(mg Ra eq) = (X  )Apparent Activity(mCi)
Ra
Example: What is the mg Ra eq of a 25 mCi Cesium 137 source?
Strength(mg Ra eq) = (3.268.25)25 = 9.88 mg Ra eq
The radial dose function
The simple inverse square law formula for the dose from a point source is sufficiently accurate
for simple manual calculations. Better accuracy can be obtained by accounting for the absorption
of the photons by the tissue. The radial dose function, g(r), is the factor that accounts for tissue
absorption and expresses the deviation from the inverse square law.
The equation for the dose rate then becomes
Dose rate (cGy/h) = f(Ar2)g(r)
or
Dose rate(cGy/h) = (SKr2)g(r)
As we can see in the graph, for 137Cs g(r) is close to unity for small distances from the source,
and is within 10% of unity as far out as 8 cm from the source. This is true for many
radionuclides with photon energies above 200 keV.
150
Since brachytherapy is about small distances
from the sources we can see that ignoring g(r)
will not lead to great inaccuracies. This is not
true for low energy sources such as 125I or
103
Pd.
Radium sources
Initially all brachytherapy was done with radium. The sources come in two forms. They are:
1- Tube sources for mold treatments (where the radium was placed in some carrying material)
or intracavitary treatments (where the radium is placed in an applicator that is then placed in a
body cavity).
2- Radium needles for interstitial treatments (where the needles are inserted directly into the
tissue being treated).
In order to determine dose rates around radium sources the following quantities must be known:
1- Activity of radium (in mg). For tubes sources the total activity is stated. For needles the
linear density is frequently given. The linear density, , is the amount of radium per cm of
active length.
2- Filtration of source (in mm Pt). 0.5 mm Pt is standard.
3- Active length (the length over which the activity is distributed). This is always less than the
physical length of the tube or needle.
4- Distribution of the radium throughout the active
length.
For tube sources the radium is distributed uniformly
across the active length. For needles there are three
possible distributions as shown in the diagram:
uniform, dumbbell, and Indian club.
The dumbbell and Indian club needles would be used in the case of “uncrossed ends” in an
interstitial implant.
151
All sealed brachytherapy sources must be periodically checked for leakage. Leakage is
particularly hazardous for radium sources because radon gas can spread contamination far from
the leaking source. Because of this hazard radium has been replaced with safer substitutes.
Exposure rate from a line source
The exposure rate can be determined by breaking up a
line source into segments and treating each segment as a
point source. In the diagram we have broken up the
source into three segments. If A is the total activity then
the exposure rate at point P can be estimated as:
Notice that the encapsulation thickness t is different for
each of the segments.
A more accurate exposure rate will be calculated if we use a larger number of shorter segments.
For a very large number the summation then turns into an integration, which is known as the
Sievert integral. Tables of the Sievert integral can be found in the literature.
It must be noted that the use of the Sievert integral to determine the dose rate in tissue is only an
approximation, since it does not take into account the absorption of photons by the tissue, i.e.,
the radial dose function, g(r), is set to unity. Nevertheless, for small distances from the source,
the Sievert integral is sufficiently accurate.
Quimby (Goodwin, Quimby, and Morgan, 1970) has
produced extensive tables of the dose rate in tissue from
a linear radium source. The dose rate depends on the
active length, the filtration thickness, and the distances
“along” and “away” from the center of the source as
shown in the diagram. An example of these tables are given below.
Dose rate (cGy/hr) for a 1 mg Radium source, 1.5 cm active length
distance “along” (cm)
distance
“away” (cm)
0.5
1.0
2.0
3.0
4.0
5.0
0.
20.6
6.75
1.90
0.86
0.49
0.31
1.0
8.12
4.14
1.57
0.78
0.47
0.30
2.0
1.63
1.56
0.98
0.61
0.39
0.27
3.0
0.57
0.68
0.58
0.43
0.31
0.23
4.0
0.25
0.35
0.36
0.29
0.25
0.19
5.0
0.10
0.21
0.24
0.22
0.19
0.15
152
Example: What is the dose rate at point P from two radium sources
situated as in the diagram?
Source A: along = 3 cm, away = 4 cm, from table 0.31cGy/hr for a 1
mg source. 6.2 cGy/hr for a 20 mg source.
Source B: along = 3 cm, away = 2 cm, from table 0.58 cGy/hr for a 1
mg source. 5.8 cGy/hr for a 10 mg source.
Dose rate at P = 6.2 + 5.8 = 12.0 cGy/hr.
Isodose distributions from a linear source
The figure shows the isodose distribution from a
radium tube source. Instead of being spherically
symmetric, as we would expect for a point
source, the isodose surfaces have an ovoid shape.
Along the axis of the source the dose is reduced
and the isodose lines are constricted due to the
excess filtration by the encapsulation. This has
significance when using these sources in gyn
intracavitary implants. The sources in the
colpostats, or equivalently the ovoids, should
align with the antero-posterior direction in order
the reduce the dose to the bladder and the
rectum.
Systematic approaches to implants
Much of the early work in brachytherapy was done without the help of treatment planning
computers. In order to facilitate brachytherapy planning various systems have been developed.
The best known of these are:
Manchester system (Paterson-Parker)
Memorial system (Quimby)
Paris system
The Manchester (Paterson-Parker) system
The Manchester system (Meredith 1947) includes planar, two-plane, and volume implants. In
each of these the concept is the same, a non-uniform distribution of the radium in order to
produce a uniform distribution of dose. More of the radium is in the periphery of the implant
than in the center. The radium is distributed according to specific rules. The rules are designed
to insure the dose over the treated surface or volume dose not vary by more than  10%.
153
A rectangular single plane interstitial
or mold the implant consists of lines of
radium around the periphery along
with lines parallel to the long side of
the rectangle. The spacing of the lines
is twice the treatment distance, i.e. 2h.
In the case of a mold h is the distance from the plane of the implant to the surface being treated.
In the case of an interstitial implant h is 0.5 cm and the lines are spaced 1 cm apart.
The linear density, , is the amount of radium per centimeter of length.
Note that for a implant width less than 2h there are no internal lines, all the radium is on the
periphery. If there is only a single internal line then the linear density for the internal line is ½
of the linear density on the periphery. For more than one internal line the linear density is 2/3
that of the periphery.
For circular molds there are different distribution rules which depend on the ratio of the implant
diameter to h.
Having arranged the radium according to the Manchester rules, one calculates the quantity of
radium  the implant duration (i.e. mghours) required using a set of tables. The entries in the
table, which depend on the implant area and the treatment distance, give the mg-hrs for 1000
Roentgens exposure. An abbreviated table is given below(from Meredith 1947).
\
area distance
10 cm2
16 cm2
20 cm2
25 cm2
30 cm2
40 cm2
50 cm2
milligram-hours of Radium for 1000 R
0.5 cm
1 cm
2 cm
3 cm
235 mg-hrs
315
368
430
490
603
705
433
566
641
722
795
934
1072
923
1113
1225
1362
1487
1732
1958
1590
1830
1979
2152
2320
2620
2897
4 cm
5 cm
2500
2790
2965
3238
3348
3695
4018
3580
3883
4080
4312
4534
4942
5327
Note that the entries in the tables are for exposure in roentgens. For dose in cGy the entries will
be a little different.
Example: A 5 cm  5 cm skin lesion is to be treated by a planar mold placed 2 cm above the
surface. 6000 roentgens are to be applied over 120 hours. What is the total amount of radium
needed and how is it to be distributed?
154
For h = 2 cm and the area = 25 cm2 we have 1362 mg-hrs for 1000R. For 6000 R we will
need 8172 mg-hrs. For a total time of 120 hours 8172120 = 68 mg of radium.
Since the width is greater than 2h there will be a
single line of radium through the center. If  is
the linear density around the periphery, the inner
line will have a density of ½ .
Thus we have
45 + 5½  = 68 mg
Therefore
 = 3 mg/cm.
Often when one is doing an interstitial implant,
needles can not be inserted across one or both of the
ends, perpendicular to the internal lines. Such a
situation is called an “uncrossed end”.
For a single uncrossed end, Indian club needles
might be used. For two uncrossed ends, dumbbell
needles are appropriate. In the absence of these
special needles one extends the length of the implant
by 10% for each uncrossed end. The treated area is
less than the implanted area by the same 10%.
Example: A 4  4  1 cm3 tumor is to be treated with an interstitial radium implant. 6000 R will
be delivered in 100 hours. Both ends will be uncrossed. How much radium is needed and how
will it be distributed?
The treated area is 16 cm2 and the treatment distance h is 0.5. From the table we get 315
mg-hrs for 1000 R or 1890 mg-hrs for 6000 R. For 100 hrs we need 18.9 mg of radium.
Since the ends can not be crossed the radium will extend 10% of the 4 cm length for each
end for a total length of 4.8 cm. There will be two external lines of linear density  and
three internal lines of density 2/3. Thus we have
24.8+ 34.82/3 = 19.2 = 18.9
Therefore  1 mg/cm. The two internal lines will be 0.67 mg/cm.
In the Manchester system single plane interstitial implants are acceptable for tumors whose
thickness is not greater than 1 cm. For tumors with thickness greater than 1 cm but no greater
than 2.5 cm a two plane implant is used.
155
For tumors with a thickness greater than 2.5 cm a volume implant is used. The tables and
distribution rules differ from those for planar implants, but the basic idea is the same. The
radium is loaded non-uniformly with a greater part of the radium on the periphery.
The Memorial (Quimby) system
In the Quimby approach the sources are distributed uniformly, spaced 1 cm apart, throughout the
plane of a planar implant or throughout the tumor volume. This leads to a non-uniform dose
distribution with the maximum dose at the center of the implant and the minimum at the
periphery. If the dose is specified at the maximum then fewer milligrams of radium are used
than in the Paterson-Parker system. If the dose is specified at the minimum the more radium is
used in the Quimby system.
The Paris System
The basic rules of the Paris system are as follows (Pierquin et al, 1987):
The activity per cm is uniform for all the sources.
The sources are place parallel with their centers cutting through a single plane.
The spacing between the line sources is constant but not necessarily 1 cm. It can be
between 0.8 cm and 1.5 cm for “short” lines of sources and between 1.5 cm. and 2.2 cm
for “long” lines (10 cm or longer).
For planar implants the centers of the sources lie in a line.
For volume implants the centers of the sources lie at the corners of squares or equilateral
triangles.
There are no crossed lengths. Therefore the active lengths of the lines must extend
outside the tumor volume. The treated length = 0.7  the active length.
156
The prescription isodose (reference isodose) is take to be a percentage of the basal dose rate.
The basal dose rate is defined as follows:
For planar implants the elemental basal dose rates are calculated
in the central plane, midway between each line.
The basal dose rate, BD, is the average of the elemental basal
dose rates. For the four line implant illustrated
BD = 13( BD1 + BD2 + BD3)
For volume implants the elemental basal dose rates are at the intersections of the
perpendicular bisectors of the lines connecting the intersections of the sources with the
central plane, as shown below.
The reference isodose line is that which is 85% of the basal dose rate.
157
Implant verification
No matter which system one uses after the implant is done the dosimetry must be verified. This
is normally done by reconstructing the sources locations using a pair of orthogonal radiographs,
usually antero-posterior and lateral. One must be aware that the distances that are measured on
the radiograph are magnified by the factor:
Magnification Factor = (source to film distance)(source to implant distance)
If the distances can not be determined, a ring of known diameter imaged on the film can provide
the requisite factor, provided the ring can be placed exactly in the location of the implant.
Most frequently the source locations are digitized into a treatment planning computer and the
dose distribution is calculated. Alternatively dose rates at specific points can be calculated. If
one is using the Manchester or the Memorial tables, one can make adjustments to the pre-implant
plan by measuring the actual dimensions of the implant area or volume.
Example: The diagram shows the arrangement of a single plane radium implant as seen
on AP and lateral radiographs. The active length is 4.5 cm and one end is uncrossed.
What is the actual area of the implant?
The length of the implant is 4.5 cm, reduced 10% by the uncrossed end, or 4.0 cm.
The width can be determined from the radiographs.
In the AP view the width is 4.31.2 = 3.6 cm.
In the lateral view the width is 1.651.4 = 1.18 cm
The width of the implant is therefore (3.62 + 1.182)
The implant area is 4.0 cm 3.8 cm = 15.2 cm2.
½
= 3.8 cm.
Intracavitary implants for cancer of the uterine cervix
Cancer of the uterine cervix is treated with radium tubes (or radium substitutes) held in
applicators which are inserted in the uterine canal (the tandem) and two ovoids in the vaginal
fornices. Originally the applicators were made of soft rubber; most often now rigid metal
applicators are used.
158
The rubber ovoids were ovoid in shape which conformed
to the isodose surfaces of a single radium source. More
modern “ovoids” (colpostats) are cylindrical, with tungsten
pieces in the top and bottom faces of the cylinder to
provide partial shielding to the bladder and the rectum.
Various methods of dose specification have been used.
One common method calculates the dose to point A
(Meredith 1947).
Point A is 2 cm from the axis of the tandem and 2 cm up
from the lateral fornices. Point B is 5 cm from the tandem.
The doses to points A and B as well as doses to points in the patient’s rectum and bladder are
calculated, either by a computer calculation routine or “by hand” as illustrated below.
Example:
Using the “along and away” tables presented previously, calculate the dose rate to point
A with the following loading of the sources: S1-15 mg, S2-10 mg, S3-10 mg, S4-15 mg,
S5-15 mg. The distance from S4 to A is 5 cm, from S5 to A is 3.25 cm. The sources
have a 1.5 cm active length and a 2 cm physical length.
Source
S1
S2
S3
S4
S5
distance
“along”
3 cm
1
1
0
0
distance
“away”
2 cm
2
2
5
3.25
dose rate
per mg
0.58 cGy/hr
1.57
1.57
0.31
0.70
mg per
source
15 mg
10
10
15
15
Total =
dose rate
(cGy/hr)
8.7 cGy/hr
15.7
15.7
4.65
10.5
55.3 cGy/hr
If it is desired to deliver 3000 cGy to point A, then the implant should remain in place for
3000/55.3 or 54.25 hours.
An alternative method of “dose” specification is the total milligrams of radium implanted
multiplied by the implant time. The units are mg-hrs. In the above example the result is
65 mg  54.25 hrs = 3526 mg-hrs.
The ICRU (International Commission on Radiation Units) system.
In the ICRU system (ICRU 1985) the dose is specified to the isodose surface that just surrounds
the target volume. Additional information about the implant also needs to be recorded. The
added information is:
159
Total reference air kerma, which is the air kerma
strength  number of hours implanted
(similar to mg-hrs).
Dimensions of the implant reference volume:
height, width, and thickness (as shown in the
diagram)
In addition the doses at various reference points
(bladder, rectum, pelvic wall, lymphatics) need to be recorded.
Radium substitutes
Because of the hazards associated with radium sources (gaseous radon from a leaking source can
cause wide spread contamination) they are no longer in common use. Various nuclides, which
produce penetrating gamma radiation, can be used as substitutes.
If the required milligrams of radium is determined using some standard tables (Paterson-Parker
for example) then the equivalent millicuries for a radium substitute, say nuclide A, can be
determined using the ratio of the  factors:
ActivityA = (RaA)mg of Radium
Permanent radioactive seed implants
Certain nuclides, with short half lives, can be permanently implanted into a tumor and left to
decay. Two which have been used in this way are:
Γ = 8.25 R-cm2mCi-hr (filtered by 0.5 mm Pt equivalent)
Radon-222,
T½ = 3.83 days,
Gold-198,
T½ = 2.70 days,  = 2.38 R-cm2mCi-hr
To calculate the required activity for a Paterson-Parker type implant one uses the following
steps.
1. Calculate the mg-hrs of radium using the
usual Paterson-Parker techniques.
2. Calculate the mCi-hrs of the nuclide using
the ratio of the  factors.
The activity implanted decays in time as shown
in the diagram.
160
The total mCi-hrs is the area under the decay curve, which is:
Total mCi-hrs = Initial Activity1.44T½(hrs)
3. The required initial activity is (mCi-hrs)(1.44 T½).
Example: A patient is to receive a single plane implant with gold seeds using the PatersonParker technique. The area of the implant is 10 cm2 and a total dose of 5000 cGy is to be
delivered. What is the activity of 198Au required?
From the Paterson-Parker tables we see that 235 mg-hrs are required for 1000 roentgens.
1000 R = 960 cGy (using the f factor of 0.96). So for 5000 cGy we need
235 5000960 = 1224 mg-hrs of radium.
The mCi-hrs for gold is 1224Ra = 12248.252.38 = 4243 mCi-hrs
Au
Since T½ = 2.7 days = 64.8 hrs the required activity is
4243 (1.44  64.8) = 45.5 mCi.
Brachytherapy planning computer programs can display the total dose for a permanent implant
with a given initial activity, taking into account the decay of the source.
Iodine-125 and Paladium-103
125
I and 103Pd decay via electron capture with the subsequent release of low energy characteristic
X rays (27 keV for 125I and 20 keV for 103Pd). These nuclides can not be treated as radium
substitutes because the X rays are strongly attenuated in tissue. Schemes have been devised for
rapidly determining the total millicuries required depending on the average dimension of the
tumor to be implanted (Anderson et al, 1985).
For 125I the required activity is given by:
Activity of
125
I(mCi) = 5da(cm)
For this activity the dose at the periphery of the
tumor is:
Peripheral dose (cGy) = 48,000da.
For tumors of average dimension greater than 3
cm a different equation for the activity is used in
order to keep the peripheral dose at 16,000 cGy.
After the required activity is determined a nomogram is used to calculate the seed spacing for a
given seed activity.
161
For 103Pd the technique is the same; da is calculated. However the conversion to total activity
differs.
Activity of 103Pd = 21da
for da < 2.4 cm.
This is based on a total dose to decay of 11,500 cGy.
Isodose distributions from 125I and 103Pd
Because of the low energy X rays 125I and 103Pd
sources are encased in a thin titanium (low Z) tube.
Nevertheless both types of sources show significant
anisotropy along the tube axis due to absorption by
the source itself and the welds at the end of the tube.
Some planning computers do account for the
anisotropy in the isodose calculations. Others treat
the sources as an isotropic point source but add a
dose reduction factor to account for anisotropy.
Typical anisotropy factors are:
0.95 for
125
0.90 for
103
I
Pd.
Summary of radionuclide properties
Nuclide
226
Ra
Rn
137
Cs
192
Ir
198
Au
125
I
103
Pd
222
*
Half life
(T1/2)
1600 years
3.83 days
30 years
74 days
2.7 days
60 days
17 days
Depends on source construction
Average
photon
energy
(keV)
800
800
660
370
410
29
21

R-cm2
mCi-hr
mCi to AirKerma
conversion

8.25
8.25
3.26
4.69
2.36
1.45
1.48
7.23
7.23
2.86
4.11
2.07
1.27
1.3
1.12
1.12
1.12
1.12
1.12
0.88*
0.74*
162
Non-photon sources
Californium-252
252
Cf (a transuranium element) T1/2 = 2.6 years
97% of the time it undergoes alpha decay.
3% spontaneous fission
252
Cf is a source of fast neutrons. One microgram yields 2.8106 neutrons per second. It has
been used as a neutron brachytherapy source.
Strontium-90
90
Sr is a nuclear reactor product.
Strontium-90 decays to Yttrium-90
The decay is pure .
Rangemax = 1.8 mm
T1/2 = 28.6 Years
Transition energy = 0.54 MeV
Yttrium-90 decays to Zirconium-90
Average  Energy = 0.19 MeV
T1/2 = 64 Hours
Again the decay is pure . Transition energy = 2.23 MeV
Rangemax = 11.0 mm
100
Most of the dose is provided by the 90Y
decay.
90
80
Percent Depth Dose
Average  Energy = 0.93 MeV
70
60
Used for the treatment of pterygium
and other eye conditions.
50
40
30
A 55 mCi applicator has a surface dose
rate of 48 cGy/second.
20
10
0
0
1
2
3
4
5
6
depth (mm)
Depth dose of a 90Sr applicator
Strontium-89
Like 90Sr, 89Sr is a pure  emitter. It decays to 89Y.
Transition energy = 1.46 MeV
T1/2 = 52 days.
Ionic Sr is a bone seeker. 89Sr is used to treat wide spread bone metastases.
163
XVIII- Protection of individuals from ionizing radiation
Radiation protection standards for nuclear reactors and reactor by product radionuclides are set
(in the U.S.) by the Nuclear Regulatory Commission (NRC).
Standards for naturally occurring radionuclides, e.g. radium, and for X-ray units and accelerators
are set by the state radiation protection offices.
The National Council on Radiation Protection and Measurements (NCRP) issues
recommendations that are generally adhered to by the state offices.
Dose equivalent
The dose equivalent for the various types of radiation is:
Dose Equivalent(sieverts or rems) = QDose(gray or cGy)
Q is called the quality factor.
For exposures to a mixture of radiation types is
Dose Equivalent = Q1Dose1 + Q2Dose2 + Q3Dose3 + . . .etc.
The quality factors for various types of radiation are listed below (NCRP 1987).
Radiation Type
Q
Photons, electrons, positrons
1
Neutrons < 10 keV
5
Neutrons 10 keV to 20 MeV
10 to 20 depends on energy
Neutrons > 20 MeV
5
Protons > 2 MeV
2
Alpha particles, heavy particles
20
Recommendations on annual exposure limits
Based on the recommendations of the NCRP the annual whole body exposure limits for
stochastic effects are:
Individual
Maximum annual exposure Maximum annual exposure
Occupational exposure
50 mSv
5 rem
Occupational cumulative
10 mSv  age in years
1 rem  age in years
Public: frequent exposure
1 mSv
0.1 rem
Public: infrequent exposure
5 mSv
0.5 rem
Education (<18 yrs of age)
1 mSv
0.1 rem
Fetus
0.5 mSv per month
0.05 rem per month
Negligible risk
0.01 mSv
0.001 rem
164
Special limits for individual organs (deterministic or non-stochastic effects) are:
Occupational
Occupational
Public
Public
Eye lens
Skin, hands and feet
Eye lens
Skin, hands and feet
Maximum annual
150 mSv
500 mSv
15 mSv
50 mSv
Maximum annual
15 rem
50 rem
1.5 rem
5 rem
For stochastic effects the probability of occurrence is directly proportional to the dose, but the
severity is independent of the dose (e.g. cancer).
For deterministic effects the severity increases with the dose absorbed (e. g. eye lens
opacification).
In addition to the above limits, radiation exposures should be in accordance to the ALARA
principle, that is As Low As Reasonably Achievable.
Protection from radiation involves consideration of the two “D’s” and a “B”. They are Distance,
Duration, and Barrier. One minimizes exposure by:
1- Maximizing distance. For radiation protection purposes the source can be considered
a point source so that the inverse square law applies.
2- Minimizing the duration of exposure.
3- Imposing a barrier between the source of radiation and the exposed individual. Often
the task in a protection problem is the calculation of the appropriate barrier thickness for
a given distance and exposure time.
Protection from brachytherapy sources
The exposure from a brachytherapy source can be
found from the following expression
Exposure(Roentgens) = (AtB)d 2
 is the exposure rate constant in R-cm2mCi-hr
A is the activity in mCi
t is the time in hours at the location
d is the distance (in cm) from the source to the
individual
B is the barrier transmission factor, the fraction of
the incident radiation that gets transmitted.
165
The exposure is assumed equal to the dose equivalent in rems, i.e. we set the roentgen to rad
conversion factor to unity in protection calculations.
Once the barrier factor is determined, the appropriate
thickness of the material (usually lead or concrete)
can be found using the known tenth value layer or
from published transmission curves.
Example:
A brachytherapy technologist works at a location that
is 1 meter from a place where 100 mg of radium is
stored. What thickness of a lead barrier is required in
order to not exceed the maximum annual exposure to
the worker?
The dose equivalent must not exceed 5 rem per year
or
0.1 rem per week. We can assume the worker is at
the location for 40 hours per week
From the above equation we have
weekly exposure = 0.1 rem = (8.2510040B)1002 = 3.3B
Therefore B = 0.03. From the graph we find that a transmission of 0.03 occurs for 7 cm of lead.
We can also determine the thickness knowing that the tenth value layer, the thickness required
to reduce the exposure by 1/10th. The TVL is 3.3HVL.
The TVL for radium is 4.6 cm.
For B = 0.03 we calculate 0.03 = (0.1)X or X = -log10(0.03) = 1.5
Thus we need 1.5 TVLs or 1.54.6 cm = 7.0 cm.
The barrier thickness, which we have determined, satisfies the requirement for the maximum
permissible exposure; however, it does not satisfy the ALARA principle. In this case one can
easily decrease the exposure by adding more lead to the barrier, or by moving the work station
further from the radium.
For low dose rate brachytherapy the implant patients are kept in ordinary hospital beds and
radiation protection concerns are directed to the nursing staff. Portable shields can be used to
minimize the exposure rates in areas adjacent to the patient’s room.
High dose rate brachytherapy must be carried out in a room which has adequate shielding in the
walls.
166
Protection from teletherapy sources (X ray and 60Co)
Because teletherapy units have collimated
beams, only certain areas outside the treatment
room will require protection from the direct
beam. These areas will require a primary
protective barrier. Other areas will receive
only scattered radiation and head leakage.
These areas require a secondary protective
barrier, a lesser thickness than that for the
primary. A maze wall is frequently used in
order to minimize the shielding requirements for
the door.
Primary protective barrier
For a point outside the treatment room that would be in the primary beam, the required barrier
factor can be determined from the equation:
DE = (WUTB)d 2
where:
DE is the weekly maximum permissible dose equivalent (rems)
W is the weekly workload in cGy per week at 1 meter from the source
U is the use factor, the fraction of time the beam is directed to the location
in question
T is the occupancy factor, the fraction of time an individual spends at the
location in question
d is the distance (in meters) from the source to the point
B is the barrier factor
Typical workloads are 100,000 cGy per week for megavoltage X ray units and 60,000 cGy per
week for 60Co.
Suggested use factors are:
U = 1 for the floor
U = ¼ for the walls
U = ¼ to ½ for the ceiling
For occupational exposure the occupancy factor is always 1.
For public exposure the value for the occupancy factor depends on the type of room adjacent to
the treatment room.
167
Suggested values for the occupancy factor are:
Full occupancy, T = 1, for work areas, offices etc.
Partial occupancy, T = ¼, for corridors, rest rooms, etc.
Occasional occupancy, T = 1/16, for waiting rooms, toilets, stairways, etc.
For high energy photons, where Compton scattering is dominant, the barriers can be either lead,
steel, or concrete. For cost considerations concrete is most frequently used, except where space
is at a premium. After the barrier factor is calculated the thickness of the material can be
determined from transmission curves or from the TVLs.
Example:
A corridor next to a 6 MV linear accelerator (see
figure) has an assumed occupancy factor of ¼. The
point in question is 5.5 meters from the X ray source.
The use factor for the accelerator pointing at that wall
is also ¼. Using a weekly workload of 100,000
cGy/wk at 1 meter, what is the thickness of concrete
that provides adequate protection to individuals using
that corridor?
For frequent public exposure the yearly dose
limit is 0.1 rems/yr. This means the weekly
dose should be no more than 0.002 rems/wk
Therefore we have
DE = 0.002 = (100,001/41/4B)
Solving for B we find B = 10-5
Therefore we need 5 TVLs.
(5.5)2
The TVL for 6 MV X rays is 33.5 cm of
concrete; so 5 TVLs are 168 cm.
This can also be read directly from the graph.
Secondary protective barrier
For walls that do not receive primary radiation, barriers for scattered radiation and head leakage
must be in place. To determine the appropriate thickness scatter and head leakage are considered
separately.
168
Scattered radiation
The amount of scattered radiation will depend on the
size of the patient’s treatment field.
For a standard 2020 cm2 field the dose equivalent to
an individual beyond the barrier is:
DE =(WTB) d d )2
( 1 2
where:
d1 is the distance from the X ray source to the patient
d2 is the distance from the patient to the individual
 is the fraction of the incident radiation that is scattered at the appropriate angle
(90 in the situation depicted in the diagram).
For 6 MV X rays  is 610-4. For other beam qualities  is approximately the same magnitude.
Since the scattered radiation is lower in energy, the barrier transmission is less than that of the
primary. For these reasons the thickness of barriers for scattered radiation are much less than
those for the primary beam.
Head leakage
For modern linear accelerators the leakage of the beam
from the treatment head at 1 meter from the source is
kept at or below 0.1% of the useful beam. Therefore the
dose beyond the barrier is:
DE = (0.001WTB)d2
For leakage the use factor is always unity.
Head leakage for cobalt teletherapy units is treated a little differently, since leakage is present
even when the unit is not “on” . The Nuclear Regulatory Commission has set the requirement
that leakage in the off position is, on the average, no more than 2 mR/hr at a distance of 1 meter
from the source.
169
Surveys of radiation levels
To verify the integrity of the radiation barriers outside
a treatment room a survey of the radiation levels must
be done. Various instruments are used for this
purpose. An ionization chamber can be made to
accurately measure photons. Because of the low level
of radiation present a large chamber volume is
necessary. In addition the device must be portable and
battery operated. This type of unit is commonly
referred to as a “cutie pie”.
More sensitive meters can be made with a detector that is more responsive than a standard
ionization chamber. The variation in response of a gas filled chamber as the electrode voltage is
increased is shown in the diagram.
As the voltage is increased above the saturation
region (or the ionization plateau) the current
increases as additional ionization is produced in the
gas. This voltage region is called the proportional
region. The current detected is proportional to the
initial ionization produced.
Above the proportional region is the GeigerMueller region. A single particle traversing
the chamber causes all the gas in the chamber
to ionize, resulting in a large signal. A
detector operating in this region is responsive
to very low levels of radiation. A GeigerMueller counter can be used to quickly scan
around a room to find radiation hot spots; an
ionization chamber is then used to obtain a
more accurate reading of the exposure level.
170
For accelerators operating above 10 MV readings of neutron levels outside the room must be
done. A neutron survey meter consists of a tube operating in the proportional region. The tube
is filled with BF3 gas that has been enriched with 10B, which has a high probability for absorbing
a thermal neutron and emitting an alpha particle. To detect fast neutrons, the tube is centered in
a polyethylene sphere. The fast neutrons are slowed through collisions with the hydrogen in the
sphere.
A neutron survey meter
171
References
Primary references
Hendee W R. Medical Radiation Physics, Second Edition. Chicago: Year Book Medical
Publishers, 1979.
Khan F M. The Physics of Radiation Therapy. Second Edition. Baltimore: Williams & Wilkins,
1994.
Additional References
Anderson L L, Nath R, Weaver K A, Nori D, Phillips T L, Son Y H, Chiu-Tsao S, Meigooni A
S, Meli J A, Smith V. Interstitial Brachytherapy, Physical, Biological, and Clinical
Considerations. New York: Raven Press 1990.
Central Axis Depth Dose Data for Use in Radiotherapy. British journal of Radiology
Supplement No. 17. London: British Institute of Radiology 1983.
Goodwin P N, Quimby E H, and Morgan R H. Physical Foundations of Radiology. Fourth
Edition. New York: Harper and Row, 1970.
Hall E J. Radiobiology for the Radiologist. Fifth Edition. Philadelphia: Lippencott Williams &
Wilkins, 2000.
Hubbell J H. Photon Cross Sections, Attenuation Coefficients, and Energy Absorption
Coefficients From 10 keV to 1000 GeV Washington, D.C.: Nat. Stand. Ref. Data Ser., Nat. Bur.
Stand.(U.S.) 29, 1969.
International Commission on Radiation Units and Measurements (ICRU) Clinical Proton
Dosimetry Part I: Beam Production, Beam Delivery and Measurement of Absorbed Dose. ICRU
Reoprt No. 59. Bethesda, Maryland: ICRU 1998.
International Commission on Radiation Units and Measurements (ICRU) Dose and Volume
Specification for Reporting Intracavitary Therapy in Gynecology. ICRU Report No. 38.
Bethesda, Maryland: ICRU 1985.
International Commission on Radiation Units and Measurements (ICRU) Prescribing,
Recording, and Reporting Photon Beam Therapy. ICRU Report No. 50. Bethesda, Maryland:
ICRU 1993.
Kutcher G C, Coia L, Gillin M, Hanson W F, Leibel S, Morton R J, Palta J R, Purdy J A,
Reinstein L E, Svensson G K, Weller M, and Wingfield L. Comprehensive QA for Radiation
Oncology: Report of AAPM Radiation Committee Task Group 40, Medical Physics 21; 581,
1994.
Meredith W J. Radium Dosage, The Manchester System. Edinburgh: E & S Livingstone 1947.
172
National Council on Radiation Protection and Measurement. Recommendations on Limits for
Exposure to Ionizing Radiation. NCRP Report No. 91. Bethesda, Maryland: NCRP 1987.
Pierquin B, Wilson J F, Chassagne D. Modern Brachytherapy. New York: Masson, 1987.