Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics
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458 Nuclear Medicine Physics
from sources. Since Shannon’s work [21], we know that entropy is the physi-
cal quantity suitable to describe such complex systems. Shannon defined the
entropy of a probability distribution, which emerges from information the-
ory. This can be applied in radiation physics, dosimetry, and protection in a
straightforward fashion [16,22].
Let us consider a Gaussian distribution, with null average and standard
deviation σ:
p(x) =
1
√
2πe
exp
−
x
2
2σ
2
(8.63)
In Figure 8.6, 4 distributions are shown with different values of the standard
deviation.
Defining the entropy of a probability distribution, p(x),as
S =−
+∞
−∞
p(x) In p(x)dx (8.64)
and using Equation 8.63, it follows that
S(x) = In(kσ) with k =
√
2πe (8.65)
which means that the entropy of a Gaussian distribution is a function of the
standard deviation. In Figure 8.6, the entropy values for p1, p2, p3, and p4
are S1(σ) = 2.11, S2(σ) = 2.52, S3(σ) = 3.03, and S4(σ) = 3.72, respectively. In
40 30 20 10 0
x
p(x)
–10 –20 –30 –40
0
0.05
0.1
0.15
0.2
0.25
p1(x)
p2(x)
p3(x)
p4(x)
FIGURE 8.6
Graphical representation of 4 Gaussians, p1(x), p2(x), p3(x), and p4(x), with standard deviations,
σ1 = 2, σ2 = 3, σ3 = 5, and σ4 = 10, respectively, and μ = 0.
Dosimetry and Biological Effects of Radiation 459
20 1510
Standard deviation
Entropy
50
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
FIGURE 8.7
Entropy as a function of the standard deviation for a Gaussian with zero average.
Figure 8.7, the entropy, S(σ, μ = 0), is shown as a function of σ, for a proba-
bility distribution Gaussian with.
In the case of discrete probabilities, we have
S =−
n
i
p
i
In p
i
for p
1
, ..., p
i
, ..., p
n
(8.66)
The concept of entropy allows us to obtain new insights in radiation physics
and to seek links between physics and biology.
Let us assume that a monoenergetic narrow photon beam impinges on a
half-extended water medium. If all photons have the same energy value, then
the probability of finding a photon with that energy is, obviously, one. Appli-
cation of Equation 8.66 with p
i
= 1, ∀i, gives the entropy value S = ln1 = 0.
This is expected for such a highly ordered set of photons, which presents no
spread in energy distribution of the primary or un-collided photons.
Using Monte Carlo simulations, the set of energy deposition points due
to the interaction of radiation with matter was determined at the kerma
approximation, which means that only the photon interactions are consid-
ered; this assumes that the energy transferred to electrons is locally deposited
at their points of origin. The narrow beam simulated has 100,000 photons of
energy 50 keV, with three interactions being allowed: coherent and incoher-
ent scattering and photoelectric absorption. At this energy, pair production is
negligible.
The points of the first interaction for each photon of the beam are shown in
Figure 8.8.
In each of the points shown in Figure 8.8, one of the three interactions
can occur: photoelectric, Compton, or Rayleigh. For energies large enough,
460 Nuclear Medicine Physics
40 30 20 10 0
x (cm)
z (cm)
–10 –20 –30 –40
0
5
10
15
20
25
30
35
40
45
50
55
60
FIGURE 8.8
First points of interaction in water for a 50 keV photon beam in water.
pair production can also occur. The Rayleigh interaction, also called coherent
scattering, is elastic and does not deposit energy. For the photoelectric inter-
action, all of the photon energy is deposited at the point of interaction. In the
Compton interaction, also termed incoherent scattering, the photon deposits
a small fraction of its energy. This leads to the energy spectrum shown
in Figure 8.9. This type of spectrum shows the degradation of the energy
of primary photons.
In the spectrum of Figure 8.9, two groups of lines can be identified: a sin-
gle line at 50 keV, due to the photoelectric effect (total absorption of photon
energy) and a group of lines due to the Compton interaction (small amounts
of energy absorbed). The Compton interaction mainly results in small values
of energydeposited, and it is one of the principal causes of image degradation.
Application of Equation 8.66 gives the entropy value S(j = 0) = 2.25.
From the point of view of image formation, the component of the primary
energy, 50 keV, obtained by the photoelectric effect, is the desirable process
leading to a signal suitable for the formation of the image. The low energy
group, due to Compton interactions, corresponds to noise in the image, which
must be reduced as far as possible.
The data of Figure 8.9 can be normalized to obtain a probability distribution
of the energy deposited in the first interaction. It is often named the primary
interaction of the beam in the target medium. We can apply the entropy to
the normalized distribution of energy deposited as a statistical parameter to
Dosimetry and Biological Effects of Radiation 461
4945 41 37 33 29
E
(
keV
)
j = 0
No. of photons
25 21 17 13 9 5 1
0
2000
4000
6000
8000
10000
12,000
14,000
FIGURE 8.9
Energy spectrum of the energy deposited in water due to the first interactions of a photon beam
with energy 50 keV. The number of photons is related to the total number of photons simulated,
which was 100,000. Entropy of the spectrum, S( j = 0) = 2.25.
describe the spread of the energy values. Obviously, a Gaussian is not a good
probability distribution to fit the data in Figure 8.9. How do we statistically
describe a distribution like this? Gaussian parameters, such as mean and stan-
dard deviation, are not enough. The entropy is a suitable statistical descriptor
of the complexity of the energy deposited in matter.
In spite of the wide use of absorbed dose, new approaches have been intro-
duced. One such new approach results from the photon track and studies the
probability, P(E, j), that j secondary interactions will occur during the degra-
dation of a photon of energy E [23]. The probability, P(E, j), can be estimated
by the relative frequency of the number of secondary interactions. The prob-
ability P(E, j) introduces the step j in the energy degradation of the primary
photon. In Figure 8.10, Monte Carlo results are shown for the estimation of
this probability for a 50 keV photon beam in water. Note that the probability
of a photon to suffer more than 16 or 18 interactions vanishes.
The pattern of the points of energy deposited for the second interaction is
rather different from that for the first interactions shown in Figure 8.8, as can
be seen in Figure 8.11.
The energy spectrum for the points of Figure 8.11 is shown in Figure 8.12.
At step j = 0 (Figure 8.8), there is a significant component in the energy dis-
tribution at 50 keV, meaning that some of the incident photons lose all of their
energy in the first interaction, by the photoelectric effect. For other values of
j, such as in Figure 8.11, whose energy distribution is shown in Figure 8.12,
there also exists a small component at 50 keV due to the photoelectric absorp-
tion of photons that previously suffered only coherent scattering, which does
not modify the energy of the photons in the interaction. Obviously for j = 0
(primary interactions), all the interactions have x = y = 0; whereas for j = 0,
some complexity appears in the pattern of energy deposition. After the first
interaction, a third group appears in the energy spectra just to the left of
462 Nuclear Medicine Physics
24 22 20 18 16 14 12
j (number of secondary interactions)
10 8 6 4 2 0
0
0.05
0.1
0.15
Relative frequency
0.2
0.25
FIGURE 8.10
Relative frequency of the number of secondary interactions, which is an estimative of the prob-
ability P(E, j), obtained for a Monte Carlo simulation of a narrow beam of 100,000 photons with
energy of 50 keV, in water.
the photoelectric peak. It can be shown that the origin of this group is the
backscattering from deeper layers in the medium irradiated [24]. Applying
Equation 8.66 to the data of Figure 8.12 gives S( j = 1) = 2.53, which is higher
than S( j = 0).
Amazingly,after nineinteractions, the lineof theprimary beamis still visible
(Figure 8.13), meaning that some photons suffered 8 elastic collisions before
total absorption by photoelectric effect.
The energy spectrum corresponding to Figure 8.13 is shown in Figure 8.14.
40302010–10–20–30–40
0
5
10
15
20
25
30
35
40
45
50
55
60
40
30
20
10
–10
–20
–30
–40
–40 –30 –20 –10 0 10 20 30 40
0
0
x (cm)
z (cm)
y (cm)
x (cm)
FIGURE 8.11
Patterns of the points of the second energy interactions in water for a photon beam in water with
primary energy of 50 keV.
Dosimetry and Biological Effects of Radiation 463
494541373329
E (keV)
j = 1
No. of photons
25211713951
0
2000
4000
6000
8000
10000
FIGURE 8.12
Energy spectrum of the energy deposited in water due to the second interactions of a photon
beam with energy 50 keV. The number of photons is related to the total number of photons
simulated, which was 100,000. The entropy value is S(j = 1) = 2.53.
Obviously, for dose calculation, all the steps of the photon track have to be
simultaneously considered, which means that the set of points obtained for
all values of j has to be considered. From Figure 8.10, it can be concluded that
the probability of having photons with more than 18 secondary interactions,
j = 18, is vanishingly small. Equation 8.66 leads to S( j = 8) = 2.88.
The results of the calculation of the entropy of all spectra for each step of
the photon track evolution lead to the results shown in Figure 8.15.
The energy spectrum obtained as a superimposition of all the values of
j is shown in Figure 8.16. In fact, the spectrum of this figure is the total
40 30 20 10 –10 –20 –30 –40
0
5
10
15
20
25
30
35
40
45
50
55
60
40
30
20
10
–10
–20
–30
–40
–40 –30 –20 –10 0 10 20 30 40
0
0
x (cm)
z (cm)
y (cm)
x (cm)
FIGURE 8.13
Patterns of the points of the ninth interactions (the first primary interaction followed by 8
secondary interactions, j = 8) in water for a photon beam in water with primary energy of 50 keV.
464 Nuclear Medicine Physics
49 45 41 37 33 29
E (keV)
j = 8
No. of photons
25 21 17 13 9 5 1
0
200
400
600
800
1000
1200
FIGURE 8.14
Energy spectrum of the energy deposited in water due to the ninth interactions of a photon
beam with energy 50 keV. Note: The number of photons is related to the total number of photons
simulated, which was 100,000. The entropy value is S( j = 8) = 2.88.
3.1
3
2.9
2.8
2.7
2.6
2.5
Entropy
2.4
2.3
2.2
2.1
2
012345678
j (Number of secondary interactions)
9 10111213141516
FIGURE 8.15
The entropy of the energy deposited in water as a function of the photon track step.
50,000
45,000
40,000
35,000
No. of photons
30,000
25,000
20,000
15,000
10,000
5000
0
1 5 9 13 17 21 25
E (keV)
29 33 37 41 45 49
FIGURE 8.16
Energy deposition spectrum for 50 keV primary photons deposited in water in a half-extended
geometry with a narrow beam, due to the first 8 steps of the photon track.
Dosimetry and Biological Effects of Radiation 465
20
0
500
1000
1500
2000
No. of photons
2500
3000
3500
4000
22 24 26 28
j = 8
Total = j0 +…+j8
j = 7
j = 6
j = 5
j = 4
j = 3
j = 2
j = 1
j = 0
30 32 34
E (kev)
36 38 40 42 44 46 48 50
FIGURE 8.17
Details of the high energyregion of the spectrum in Figure 8.16. The component j = 0has nonzero
value exclusively at 50 keV, meaning total deposition of primary energy in the first interaction of
the photon by photoelectric effect. The remainder of the spectrum is a sum of the several steps j.
Each component j suffers a displacement toward the low energy as j increases.
energy deposited due to a primary beam of photons with 50 keV. However,
the concept of photon track with steps j allows us to conclude that there exists
a specific energy and spatial distribution associated with each j value.
Figure 8.17 shows only the high energy group; the lower energy group from
Compton scattering is omitted. It can be seen how the total energy spectrum is
decomposed into several components associated with each step of the photon
track, each one having a well-defined displacement toward the lower energy
region.
As pointed out earlier [24], the components shown in Figure 8.17 mainly
include the photons backscattered in deeper layers.
8.8 Monte Carlo Simulation
The interaction of radiation with matter can be simulated using the well-
known method of Monte Carlo. Monte Carlo simulation can be seen as a
black box to which we have to provide some input data, such as details of
geometry of radiation source, target and medium, type of radiation, energy,
and direction of radiation flight. The black box diagram of a Monte Carlo code
is shown in Figure 8.18.
Monte Carlo methods of radiation transport for photons will be summa-
rized next following the ideas of Chan and Doi [25].
As mentioned earlier, we must provide a set of input data, which in some
codes have the form of a file and in others are introduced by dialog windows.
466 Nuclear Medicine Physics
Input file
Interaction within matter and scoring
Output file
FIGURE 8.18
Black box of a Monte Carlo code.
Once we have defined the geometry of the problem with a source and a
detector, the simulation can start.
1. A primary photon is emitted from the source. Its initial direction of
flight is randomly chosen from the spatial distribution of the beam
or source. The photon energy is chosen according to the spectrum
emitted by the photon source.
2. The free path length of the photon is sampled from the exponential
distribution function; and a potential collision site is determined from
the free path length, the previous direction of flight, and coordinates.
3. If the point is outside the scattering medium and the radiation detec-
tor, the photon history ends and the computer returns to step 1 for
other photon history.
4. If the collision site lies within the medium, the type of interaction is
chosen from the possible processes already mentioned earlier.
5. For the photoelectric effect, all of the photon energy is absorbed, the
photon history ends, and the computer returns to step 1.
6. If coherent scattering takes place, the photon is scattered with its orig-
inal energy at an angle sampled from Thomson equation modified
by the form factor. No energy is released. The next free path length
is determined.
7. If incoherent scattering occurs, the new direction of flight is sam-
pled from the Klein–Nishina formula modified with the incoherent-
scattering function, using a uniform distribution of the azimuthal
angle. The energy of the recoil electron is absorbed at the collision
site (meaning the kerma approximation). The free path length of the
photon with the remainder energy is determined. If the energy of
the scattered photon is less than a preset cut-off energy, then the
scattered photon is assumed to be absorbed at the collision site in a
photoelectric effect. A new photon history starts from step 1.
8. Eventually, characteristic x-rays of all elements in the medium have
to be considered. In that case, those characteristic photons start a new
Dosimetry and Biological Effects of Radiation 467
history. The energy of the previous photon subtracted by the energy
of the characteristics photon is absorbed in the collision point.
9. If the kerma approximation is not valid, the electron tracks are also
simulated.
The development of Monte Carlo methods is outside of the scope of this
book.
8.9 Conventional and Cellular Dosimetry for
Radionuclides
The number of photons emitted by a radioactive source is termed the source
strength, which may be different from the number of atoms that disintegrate.
The source strength, S, and the activity, A, are related by the expression
S = A ×F, where the factor F is the number of photons emitted per decay,
which can be less than one. For Cesium-137, 94.1% of the β-decay results
in Barium-137 m and 5.9% of the decays results in the fundamental state of
Barium-137. Competition between internal conversion electrons and γ-ray
emission for Barium-137m transition to fundamental state leads to 89.9% of
γ-ray emission. Thus, for each decay of Cesium-137, the photon emission of
0.662 MeV has a frequency of F = 0.899 ×0.941 = 0.846.
Considering dose calculation, we have previously mentioned the concept
of radiation attenuation and the linear attenuation coefficient (see Equation
8.37). However, time and distance must also to be considered. For the time,
there is a direct proportionality. In fact, if we consider the dose rate at a given
point,
˙
D =
dD
dt
, (8.67)
then the dose at the same point, for an exposure of time t is
D =
˙
D ×t (8.68)
In relation to the distance, consider a sphere of radius r and a radioactive
source at the center of the sphere emitting S photons.
The fluence (photons per unit area) at the surface of the sphere is
Φ =
S
4πr
2
(8.69)