Modelling of high-energy contamination in SPECT
imaging using Monte Carlo simulation
Albert Cot, Member, IEEE, Enric Jané, Josep Sempau, Carles Falcón, Santiago Bullich, Javier Pavia,
Francisco Calviño, and Domènec Ros,
Abstract— 123 I is a commonly used radioisotope employed in
neurotransmitter SPECT studies. In addition to an intense line
at 159 keV, the decay scheme of this radioisotope includes a low
yield (∼3%) of higher energy photons which have a non-negligible
contribution to the final image when low-energy high-resolution
(LEHR) collimators are used. This contribution of high-energy
photons may achieve ∼28% of the total counts in the projections.
The aim of this work is to model each energy component of
the high-energy Point Spread Function (hPSF) for fan-beam
LEHR collimators in order to develop faster Monte Carlo (MC)
simulations of high-energy ray contamination. The modelling of
hPSF was based on the results of simulating photons through
the collimator-detector system using the MC code PENELOPE.
Since low-energy PSFs models for fan-beam collimators must
tend to a Gaussian distribution, we use the same function for
the hPSF modelling for high-energy photons. The parameters of
these Gaussian functions were obtained by minimizing the root
mean square (RMS) error between each simulated hPSF and the
function g(x, y) using the efficiency of the simulated hPSFs as a
constraint. The RMS attained with fit of g(x, y) to the simulated
hPSFs was always smaller than ∼2% of the mean efficiency per
pixel of the image. A very strong dependence of the efficiency
on the type and thickness of the backscatter material behind the
crystal was found. The hPSFs were parameterized for a wide
range of energies, ranging from 350 keV to 538 keV. Our results
indicate that Gaussian distributions approximate in a suitable way
the hPSF responses for fan-beam collimators. This model will be
an important tool to accelerate MC simulations of radiolabelled
compounds which emit medium- or high-energy rays.
have a non-negligible contribution to the final image when lowenergy high-resolution (LEHR) collimators are used [2]. This
contribution of high-energy photons may achieve ∼28% of the
total counts in the projections as our previous work indicated.
(see [4])
This high-energy contamination should be corrected in order
to achieve an accurate quantification. MC simulations provide
an accurate description of the high-energy contamination effect, simulating the full history of each emitted photon from
source to detector. However, tracking each photon through the
collimator is a very intensive computational task that is hardly
achievable in a reasonable time using available computers. An
alternative approach would be, given a specific collimator, to
model the PSF for all energies and positions of an isotropic
source, as is generally done for the low-energy photons. Then,
the MC simulation could be carried out only inside the object,
and the modelled hPSF would be used immediately after the
photon had suffered the last interaction before leaving the object
towards the collimator.
The aim of this work is to model each energy component of
the hPSF for fan-beam LEHR collimators in order to develop
faster MC simulations. The hPSF will serve as a general tool
for simulating radioisotopes when medium- and high-energy
photons are important.
Index Terms— High-energy contamination, SPECT quantification, PSF modelling, fan beam collimator, Monte Carlo simulation.
II. M ATERIALS AND M ETHODS
I. I NTRODUCTION
N
OWADAYS, most in vivo neurotransmitter Single Photon
Emission Computed Tomography (SPECT) studies employ pharmaceuticals radiolabelled with 123 I. In addition to an
intense line at 159 keV, the decay scheme of this radioisotope
includes a low yield (∼3%) of higher energy photons which
This work was partially supported by the grants MCYT (SAF2002/04270C02-01/02) and FIS (PI020485, G03/185, C03/06).
A. Cot and F. Calviño are in the Departament de Fı́sica i Enginyeria Nuclear,
Universitat Politècnica de Catalunya, Av. Diagonal 647 08028 Barcelona (email:[email protected]).
J. Sempau is in the Institut de Tècniques Energètiques, Av. Diagonal 647
08028 Barcelona.
E. Jané, C. Falcón and D. Ros are in the Unitat de Biofı́sica, Facultat de
Medicina, c/Casanova 143, 08036 Barcelona (e-mail:[email protected]).
J. Pavia is in the Servei de Medicina Nuclear, Hospital Clı́nic i Provincial
de Barcelona, c/Villarroel 170, 08036 Barcelona.
The modelling of hPSF was based on the results of simulating photons through the collimator-detector system using
the Montecarlo code PENELOPE [7], [8] for different energies
and locations over the LEHR collimator. This code tracks
each history and its secondary particles until their energy falls
below a threshold energy, which was set to 100 keV for both
photons and electrons. The PENELOPE package includes a
set of geometry routines (PENGEOM) which are capable of
handling any object formed by homogeneous bodies limited by
quadric surfaces (such as planes, cylinders, spheres, etc.). When
using this code, the user is responsible for writing a simple
steering main program from where some of the PENELOPE
and PENGEOM subroutines are called to perform the tracking
of the particles that are simulated.
In our case, the complexity associated with the description of
the collimator geometry (∼4 · 104 hexagonal holes) prevented
us from defining the whole collimator geometry. Instead, the
whole collimator was reduced to a one-hole structure whose
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position and geometry were changed on the fly following the
tracked particles. This new methodology greatly reduced the
amount of geometry-related computing burden compared with
the full description of the whole collimator. The acceleration
of this new methodology was ∼5 times faster than previous
methods [1].
The LEHR collimator used is a fan-beam collimator with
focal length F=355 mm, hole side w=0.866 mm, septal thickness s=0.2 mm, thickness L=40 mm and a field of view (FOV)
of 50×25cm2 . The detector was modelled as a 0.95 cm NaI
crystal with a 0.12 cm thick aluminium layer between the
collimator and the crystal, which forms the protective casing.
The detection energy resolution was described by means of
a Gaussian distribution with a full width at half maximum
(FWHM) of 11% at 159keV, which was assumed to be proportional to the square root of the deposited energy. The detection
window was set from 143 to 175 keV. Behind the crystal an iron
layer of 5 cm was introduced. This layer models the elements
behind the detection crystal (photomultipliers, electronics, and
other elements) which are responsible for the backscattering of
the photons into the crystal where they can be detected. The
material and thickness of this layer were adjusted to match
the simulated efficiencies with those obtained experimentally
using a 511 keV photon source at several locations, with an
agreement of ∼5%. Simulations were performed on a Linux
workstation with two Intel Xeon processors at 2.8GHz and
2MB of RAM. The GNU Fortran 3.3.4 compiler was used with
the optimization option –O3, and a typical simulation for 529
keV photons and 104 counts took around 1h.
The simulation code was used to obtain the PSFs for several
photon energies and source locations. The photon emission
of 123 I is dominated by a low energy line (relative yields
are given in parenthesis) of 159 keV (96.5%) and also has
several high-energy lines with the following energies: 346
(0.15%), 440 (0.50%), 505 (0.37%), 529 (1.62%) and 538
(0.44%). Only yields higher than 0.1% were considered. The
set of source energies was chosen taking into account that
photons with energies lower than 350 keV have a negligible
effect with respect to photons with higher energies, which have
higher yields and efficiencies. In particular the simulations were
performed for photon energies E(keV) = 346, 400, 440, 505,
529, 538 and for (isotropic) sources at z0 (cm) = 5, 10, 15,
20, 25, where z is the distance of the source to the collimator
front plane and off-axis distances x0 (cm) = 0, 5, 10, 15 (the
x-axis is defined as the fan-beam direction, i.e. the direction
perpendicular to the focal line on the collimator front plane).
Given that the fan-beam focal line is parallel to the y-axis, all
simulations were performed at y0 =0, since the change of the
PSF would have been a simple translation (apart from boundary
effects, which are not considered due to their small contribution
to the total efficiency and the subsequent modelling).
In order to model the hPSF, we used the function g(x, y)
g(x, y) = A exp
−
0
( x−x
bx )
2
+
y−y0
by
2n
, n ∈ [0.5, 1]
(1)
where x and y are cartesian coordinates on the collimator
front plane and the parameters A, bx and by are functions of
(E,x0 ,z0 ), the energy and the x-z coordinates of the source
(as before, we regard the dependence on y0 as negligible).
The maximum of g(x, y) was assumed to be located in the
point (x0 ,y0 ), owing to the zero effect of the fan-beam hole
configuration in the hPSF peak. This function was chosen
because of the negligible content of high frequency information
and its good agreement with the experimental hPSF shapes.
We considered the cases when n was equal to 0.5 and 1: the
former corresponding to the exponential case and the latter to
the Gaussian case.
The modelling was attained minimizing the RMS between
each simulated hPSF and the function g(x, y) using the efficiency of the simulated hPSFs as a constraint. This constraint
was used because the main objective is to quantify the highenergy photon contribution to the image, and thus a good
estimation of the efficiency is of great importance. Furthermore,
this constraint gave more stability to the minimization of the
RMS. In order to increase the statistics of the hPSF data, the
original 512 × 512 detection matrix was reduced to 64 × 64
points. The downhill simplex method routine [6] was used to
solve the RMS minimization problem, which allowed us to
obtain the values of the parameters (A,bx ,by ) for each source
location and energy.
III. R ESULTS
A very strong dependence of the efficiency on the type and
thickness of the backscatter material behind the crystal was
found. The contribution of the backscattered photons to the
image was ∼60% for a centered source at z0 = 15 cm with
the same high-energy spectra as that of 123 I. Thus, more than
half of the high-energy photons detected in the 143 to 175 keV
window go through the collimator/detector system and, after
backscattering in the material behind the detector, reenter the
crystal before depositing their energy.
The simulated hPSFs were modelled using the function
g(x, y) for n = 0.5, 1. Figure 1 shows the smooth hPSF profile
of a centered 529 keV source at z0 =15 cm from the collimator
front plane together with the profiles of the modelled hPSFs.
The centered peak of the collapsed profile in the x-axis hPSF
profile is partially due to a geometric effect. The configuration
of the hexagonal holes of the collimator is such that photons
have to cross a smaller amount of lead to reach the detector in
the directions perpendicular to the hexagon walls. This feature
is responsible for the so-called ”star effect”, which can be seen
in Figure 2 corresponding to a 529 keV source centered at 15
cm from the collimator front plane.
The RMS attained with the fit of g(x, y) to the simulated
hPSFs was always smaller than ∼2% of the mean efficiency
per pixel of the image, for both n = 0.5 and n = 1. Figure
?? shows the plot of the RMS attained with n = 1 versus that
obtained choosing n = 0.5. Note that both options allow a
modelling of the hPSFs with a similar RMS. Since low-energy
PSFs models for fan-beam collimators must tend to a Gaussian
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6
5
5
4
4
3
2
n=0.5
−9
7
5
3
2
1
x 10
6
RMS for n=0.5
n=1
Arbitrary units
Arbitrary units
6
distribution [5], [3], using a Gaussian function (i.e. choosing
n = 1) for the hPSF modelling would enable us to use the same
description for both high and low energy photons. Therefore,
a Gaussian function was chosen to model the hPSF.
4
3
2
1
1
0
0
200
400
Distance (mm)
0
0
200
400
Distance (mm)
Fig. 1. The hPSF collapsed profiles through the x-axis (long side of the
collimator) the y-axis (short side of the collimator). The n=0.5 (right) and n=1
(left) models are shown with a continuous line. The source energy is 529 keV
and it is located at 15 cm from the collimator front plane.
0
0
1
2
3
4
RMS for n=1
5
6
7
−9
x 10
Fig. 3. RMS obtained with n=1 (horizontal line) versus RMS attained choosing
n=0.5. Each dot corresponds to a particular source energy and location.
Furthermore, simulations showed that off-axis hPSFs are
very similar to those obtained with centered sources. For this
reason, we tried to fit the off-axis hPSFs using the values of
bx and by obtained with their centered counter-parts. Figure 4
shows the relation between the RMS attained by the general
fit (i.e. without fixing any of the parameters bx , by and A)
and that obtained constraining the values of bx and by to those
given by the fit of the centered hPSFs for n = 1. Note that the
points lay close to the line x = y (and obviously all the points
are in the x ≥ y region), indicating that the approximation
bx (E, x0 , z0 ) = bx (E, z0 ) and by (E, x0 , z0 ) = by (E, z0 )
may be reasonable when trying to estimate the high-energy
contribution in SPECT studies using 123 I.
Thus, the simulated hPSFs can be modelled using a Gaussian
distribution depending on three parameters. Of these parameters, bx (E, z0 ) and by (E, z0 ) depend on the source energy
and the distance from the collimator front plane, whereas
A(E, x0 , z0 ) depends on one more variable, that is, the offaxis distance.
IV. C ONCLUSION
Fig. 2. Image of a 529 keV source located at 15cm from the collimator front
plane, where the star effect can be appreciated. The fan-beam focal line is
parallel to the y-axis (vertical direction).
The hPSF were parameterized for a wide spectrum of energies. This model included most of the high-energy emission of
123
I. Our results indicate that Gaussian distributions adequately
approximate the hPSF responses for fan-beam collimators. This
model could be an important tool to accelerate MC simulations
of radiolabelled compounds that emit medium- or high-energy
rays with the ultimate goal of correcting high-energy contamination.
0-7803-8701-5/04/$20.00 (C) 2004 IEEE
−9
7
x 10
6
RMS general fit
5
4
3
2
1
0
0
1
2
3
4
RMS constrained fit
5
6
7
−9
x 10
Fig. 4.
RMS with the general fit (vertical axis) versus that obtained
constraining the bx and by values of off-axis hPSFs fits to those found with the
centered hPSFs (horizontal axis) for n=1 (Gaussian PSF). RMS for centered
PSFs fall in the x = y line while the rest lie in the x ≥ y region.
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