MASTER PROGRAMME
BIOMEDICAL ENGINEERING
Master Thesis
Title: Simulation studies on positron emission tomography spatial
resolution comparing monolithic and pixelated detectors
Name:
Verena Kettelhack
Matr.-No: 311384
For the degree: Master of Science in Biomedical Engineering
Institute of Experimental Molecular Imaging –
Physics of Medical Imaging Systems
Examiner(s): Prof. Volkmar Schulz
Institute/Company:
Town:
Date:
Aachen
02.03.2015
Simulation studies on positron
emission tomography spatial
resolution comparing monolithic and
pixelated scintillation detectors
submitted by
Verena Kettelhack
Master’s Thesis in Biomedical Engineering
presented to
Faculty of Medicine
RWTH Aachen University
in
March 2015
issued at
Institute of Experimental Molecular Imaging - Physics of
Medical Imaging Systems
supervised by
Prof. Dr. Volkmar Schulz
Statutory Declaration
I, Verena Sophie Kettelhack, declare that I have developed and written the enclosed
Master Thesis completely by myself, and have not used sources or means without declaration in the text. This thesis was not used in the same or in a similar version to achieve
an academic grading or is being published elsewhere.
Aachen, 03/02/1015
3
Acknowledgements
First of all, I would like to express my gratitude for Prof. Volkmar Schulz who provided
the opportunity for me to issue my Master’s thesis about this very interesting topic in
his work group. Second, I would also like to thank Prof. Fabian Kießling for agreeing
to be my second examiner.
Especially, I want to thank Patrick Hallen for the best support and supervision any
master student could hope for: thank you for all the time you invested in explaining,
discussing and helping me with any problem I had along the way.
Furthermore, I would like to thank the whole PMI group for making work very educational and once in a while a little amusing; thank you, too, for your support.
Lisa, thank you for proof-reading my thesis in such a detailed and thoughtful manner,
even though you did not have a lot of time. I really appreciate it.
5
Contents
1 Abstract
9
2 Introduction
2.1
11
Medical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1
Anatomical and Structural Imaging . . . . . . . . . . . . . . . . . 11
2.1.2
Functional Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3
Multi-modal Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Positron Emission Tomography
15
3.1
Physical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2
Technical Fundamentals of PET Imaging . . . . . . . . . . . . . . . . . . . 17
3.3
3.4
3.2.1
Radiation Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.2
Photo-detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.3
Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Image Quality and Spatial Resolution . . . . . . . . . . . . . . . . . . . . 20
3.3.1
Positron Range and Acollinearity . . . . . . . . . . . . . . . . . . . 23
3.3.2
Energy Deposition and Detector Response . . . . . . . . . . . . . . 24
3.3.3
Depth of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Simulations
29
4.1
GATE - Geant4 Application for Tomographic Emission
. . . . . . . . . . 29
4.2
PET Modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3
Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4
Simulation of Detector Response and DOI . . . . . . . . . . . . . . . . . . 30
4.5
Reconstruction Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6
Data Preparation for Evaluation . . . . . . . . . . . . . . . . . . . . . . . 33
7
Contents
5 Results and Discussion
5.1
Ideal Detector Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1
5.2
35
Resolution in Different Scanner Directions . . . . . . . . . . . . . . 37
Performance Comparison under Realistic Conditions . . . . . . . . . . . . 39
5.2.1
Visual Separability of Two Point Sources . . . . . . . . . . . . . . 41
5.3
Positron Range and Acollinearity . . . . . . . . . . . . . . . . . . . . . . . 43
5.4
Influence of DOI Determination . . . . . . . . . . . . . . . . . . . . . . . . 45
5.5
Phantom Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.6
Theoretical Determination of the Spatial Resolution . . . . . . . . . . . . 48
5.7
Influence of Reconstruction Software . . . . . . . . . . . . . . . . . . . . . 49
6 Conclusion
51
7 Outlook
53
8
1 Abstract
In this thesis, simulation studies are conducted in order to estimate the spatial resolution
of a positron emission tomography scanner with monolithic scintillators and to compare
them to conventional pixelated scintillators. A theoretical system with detectors providing a perfect response would allow a spatial resolution of 0.6 mm. This resolution is only
restricted by positron range and acollinearity and therefore marks the limit of possible
spatial resolution with a specific scanner geometry and the applied image reconstruction
software.
In order to simulate the response of the monolithic-scintillator-based system, a realistic response on detector level needs to be modeled. For this purpose, an experimentally
determined two-dimensional point-spread-function is used to smear the simulated interaction locations within the monolithic detector. To further provide a three-dimensional
point-spread-function, depth-of-interaction determination with a realistic resolution is
implemented. Under these conditions, a spatial resolution comparison between pixelated
and monolithic detectors is performed. The pixelated scintillator provides a resolution
of 0.78 mm and the monolithic scintillator one of 0.90 mm.
Additionally, the influence of physical effects, such as positron range and acollinearity,
are evaluated. Positron range and acollinearity combined degrade the spatial resolution by about 30 %. Due to not yet understood results of simulating a back-to-back
source with an implemented angular deviation representing the acollinearity, these two
effects could not be examined separately. Moreover, the influence of depth of interaction
determination in monolithic scintillators is analyzed. Without depth-of-interaction determination, the system’s spatial resolution degrades by 18 % in the center and by 26 %
at the edge of the field of view.
The last very influential factor affecting the spatial resolution of a system is the image
reconstruction software. The reconstruction software used is able to provide a 20 %
resolution recovery for pixelated detectors and a 30 % one for monolithic detectors.
9
2 Introduction
The importance of medical imaging becomes obvious in the daily routine of every hospital and it keeps increasing. Multiple different diseases show similar or even identical
symptoms and the diagnosis can be very inconclusive. Medical imaging allows the view
into the patient without risky diagnostic surgery and often provides either clear confirmation or disproof of a suspicion. Furthermore, pathological structures can be detected
and identified even before the disease breaks out.
2.1 Medical Imaging
The term medical imaging refers to multiple different technologies, which are able to
generate a visual representation of the anatomical, structural or functional interior of
a body. The different modalities can be roughly divided into two groups: anatomical
and functional imaging modalities. Anatomical and structural information of the body
is acquired with e.g. X-ray computer tomography (CT) or magnetic resonance tomography/ imaging (MRT/MRI). On the other hand, positron emission tomography (PET)
and single photon emission computed tomography (SPECT) pertain to the group of
functional imaging modalities. They can provide various functional information about
e.g. biological processes or molecular characteristics of tissue. Naturally, there are other
aspects that further subdivide the groups of modalities or even regroup them, but the
provided distinction is fully sufficient for the following thesis.
2.1.1 Anatomical and Structural Imaging
Radiography, the first imaging modality, became possible in 1895 due to Wilhelm Conrad Roentgen’s discovery of X-rays.1 The technology, which utilizes ionizing electromagnetic radiation in the range of up to 150 keV, is based on the passing of high energy photons trough the imaged object. Their attenuation depending on the density
and atomic structure of the material is measured and depicted in different gray values. Since the Calcium complexes in bones have a relatively high atomic number, X-ray
1
J. T. Bushberg, “The essential physics of medical imaging,” p. 4.
11
2 Introduction
imaging provides a very good contrast between bones and soft tissue; therefore, radiography is very applicable to bone fracture imaging. Another well-established application
of radiography is mammography; it is used for the diagnosis of abnormal mammary
tissue and therefore the early detection of breast cancer. The different tissues being
examined in mammography have a similar density and therefore, a softer X-ray beam
- about 30 keV - is needed to distinguish between the different structures.2 Since radiography only generates projection images, computer tomography is used to acquire
three-dimensional images of the body. This technology was first available in the early
1970s. CT is implemented by rotating an X-ray source and the corresponding detector
around the object and successively creating image slices. These slices are reconstructed
with Fourier-transform-based reconstruction algorithms.3 Drawbacks of this technology
are often based on the conflict between good image quality achieved by using a high
radiation dose and minimizing said dose in order to reduce the resulting risk for the patient. Furthermore, other imaging modalities provide a much better soft tissue contrast
and are therefore more applicable for soft tissue related examinations.
MRI however, completely forgoes photon radiation by acquiring images with the measurement of magnetization, which depends on the proton density and relaxation times.
A patient is placed in a static magnetic field of usually 1.5 - 3.0 T; then surrounding coils
generate radio wave pulses, which flip the spins of the protons within the patient. When
this excitation of the protons wears off, they emit the energy in form of radio waves,
which are then detected by receiver coils and processed to visualize an image. MRI
exhibits a very high spatial resolution of 0.1 - 1.0 mm depending on the field strength
and a good soft tissue contrast.4
2.1.2 Functional Imaging
MRI is a very versatile imaging modality and can also be used for acquiring some
functional information; it is then referred to as functional magnetic resonance imaging (fMRI). FMRI can visualize for example hemodynamics in the brain or in the heart
in order to recognize activity patterns in the tissue. However, its comparably low sensitivity can be a disadvantage in certain functional imaging applications. In general,
functional imaging uses molecular tracers or biomarkers of different kinds: fMRI examining haemodynamics uses the ferromagnetic haemoglobin in erythrocytes to visualize
2
Johns and Yaffe, “X-ray characterisation of normal and neoplastic breast tissues.”
J. T. Bushberg, “The essential physics of medical imaging,” p. 312 ff.
4
Ibid., p. 402 ff.
3
12
2.1 Medical Imaging
their movement and therefore the blood flow in the tissue.5
In order to obtain a functional PET image, a positron-emitting tracer is intravenously
injected. After a certain period of time depending on the molecular structure of the
tracer, the tracer molecules accumulate in certain physiological structures, e.g. in tumor
tissue or degenerative nervous tissue. Thus, the locations of interest can be located
clearly by detecting the radioactive decay and resulting annihilation events. A widely
used radioactive tracer is the glucose analogue [18 F]Fluorodesoxyglucose, which is very
convenient for the visualisation of high metabolic activity e.g. in the brain or tumor
tissue.
A similar technique is used in SPECT imaging: a gamma-emitting tracer is injected
into the patient’s radial vein and then visualized with coaxially set-up detectors. The
differences between PET and SPECT are mainly marked by the necessity in SPECT to
use collimators. Photons inciding with an angle other than 90◦ cannot be correctly reconstructed in SPECT. Therefore, they are cut off with a lead collimator. A typical tracer
used is the metastable isotope
99m Tc;
it can be coupled to sestamibi, a pharmaceutical
agent used to examine myocardial perfusion or thyroid dysfunction.6
2.1.3 Multi-modal Imaging
Since the various imaging modalities show very different advantages, a combination of
them can allow for a broader and more profound patient treatment. For instance, it
is possible to combine the functional PET imaging with structural X-ray CT imaging:
Not only can tumorous tissue be localized using the functional PET information but the
image can directly be fused with the corresponding anatomical information provided by
the CT via software registration. The image fusion is facilitated due to the fact that
in most combination scanners the CT as well as the PET scanner are set up coaxially
with respect to the same scanner axis. Furthermore, a photon attenuation map can
be generated from the CT data, since the X-ray radiation and the PET annihilation
photons experience similar attenuation in the tissue. Another important advantage of
multi-modal imaging is the shorter patient set-up and actual image acquisition time,
which leads to a lower radiation dose and higher patient throughput.
Since PET focusses on function and activity of soft tissue, it is a modality often used
for oncological diagnostics or tissue perfusion studies. Despite of the good PET/CT
performance, this combined system is still limited by the relatively low soft tissue contrast
of the anatomical imaging device. One solution is the combination of a PET scanner
5
6
Friston et al., “Event-Related fMRI: Characterizing Differential Responses.”
Bihl and Brummer, Südwestdeutsches PET-Zentrum Stuttgart, PET and SPECT.
13
2 Introduction
with an MRI system, which leads to a better and more flexible soft tissue contrast and
an easier and more precise registration between the two modalities. Furthermore, during
the acquisition of images with this kind of combined system, only the radiotracer’s decay
contributes to the radiation dose affecting the patient. Nonetheless, the construction of
a PET/MRI system is more difficult than the one of a PET/CT scanner. The PET
disturbs the homogeneity of the magnetic field generated by the MR coils, and the
magnetic field affects the detectors and the electronics in the PET scanner.
Figure 2.1: Comparison of brain scans with PET, CT and T2-weighted MRI and their
respective fused images. [A. Boss and Stegger, “Hybrid PET/MRI of Intracranial
Masses: Initial Experiences and Comparison to PET/CT”]
Figure 2.1 shows an example of cranial images acquired with PET, MRI and CT. The
PET image clearly shows the different activity patterns of the neural tissue in the brain
on a color scale. CT and MRI show structural and anatomical information of the head.
CT hardly distinguishes between the different soft tissue regions inside the brain but
distinctly displays the cranium. MRI however, depicts the boundaries of the cerebral
architecture. The center images show the advantage of bimodal image acquisition, since
functional and anatomical information can be used complementary.
14
3 Positron Emission Tomography
3.1 Physical Basics
Positron emission tomography, an imaging technique of nuclear medicine, is based on
the detection of two coincident photons which result from the annihilation of an emitted
positron with an atomic electron. The emission of a positron is called beta decay, which
involves the conversion of a nuclear proton (p) into a neutron (n); the process results in
the release of a positron (β + ), which is the antimatter conjugate of the electron, and a
neutrino (ν).1
1 +
1p
−−→ 10n + 01β + + ν
The positron is initially emitted with a certain amount of kinetic energy, which can
assume a continuous range of values. After the emission, the positron loses this energy
by interacting with the surrounding matter. When the positron is effectively at rest, it
combines with an electron to form a positronium and after about 10−7 s the conglomerate
disintegrates. This annihilation process produces two photons with the energy of 511 keV
each. Due to the conservation of energy this is the same amount of energy contained
in the mass of the earlier positron and electron. Equation 3.1 assumes the positron and
electron to be at rest with no residual energy.
E = mc2 = 9.11 · 10−31 kg · (3 ∗ 108 )2 m s−1 = 8.210−14 J =
8.2 · 10−14 J
= 511keV (3.1)
J
1.6 · 10−19 eV
The high-energy photons can transfer their energy to the matter they are passing.
Certain different effects are possible, but a common result is the ionization or excitation
of the material’s atoms. There are three main mechanisms of high-energy photon interaction with matter:
1
Turkington, “Introduction to PET Instrumentation.”
15
3 Positron Emission Tomography
1. Photo-electric effect
2. Compton effect
3. Pair production
There are other mechanisms, but they are not of great relevance for PET imaging.
Furthermore, pair production requires photons of twice the energy of one annihilation
photon - 1022 keV - and will therefore not be discussed in detail.
Figure 3.1: Normalized photon energy spectrum generated in a simulation without energy cut-off.
Figure 3.1 shows a common photon energy spectrum with the general structures. The
maximum peak (photo-peak) rises around 511 keV and its full width at half-maximum
(FWHM) shows the fluctuations in the energy deposition of the otherwise mono-energetic
annihilation photons. The fact that not only 511 keV photons can be detected, but rather
a continuous spectrum of photons up to 511 keV, is due to the Compton effect (Compton
region). A photon colliding with a particle is scattered and changes the direction of its
path. If it is not scattered elastically, it loses kinetic energy and its wavelength increases.
Due to momentum conservation, the energy lost in Compton scattering correlates with
the deflection angle of the photon off its original path.2 The photons counted with
energies higher than 511 keV result from simultaneous events.
2
Bailey et al., “PET Basic Science,” p. 23.
16
3.2 Technical Fundamentals of PET Imaging
3.2 Technical Fundamentals of PET Imaging
Since gamma-radiation is very penetrating and cannot directly and easily be quantified,
an energy conversion is necessary. The energy of the gamma-photon is converted into
an electrical signal or charge, which is proportional to the total energy deposited by the
incident radiation. This conversion can be conducted with a varying number of steps
depending on the radiation detector used.
3.2.1 Radiation Detectors
There are three main categories of radiation detectors: proportional gas chambers, semiconductor detectors, and scintillation detectors.
The operating principle of a proportional chamber is the ionization of gas atoms by
the infrequently interacting gamma-radiation. A strong electric field is applied and
accelerates the electrons produced in the ionization. In turn, these collide with other
neutral gas atoms producing secondary ionization, which leads to a cascade. This cascade
is finally detected at the cathode and produces a signal proportional to the energy
deposited by the radiation. In a semiconductor radiation detector the incident radiation
excites and thereby frees the valence band electrons, so that an applied electric field will
cause a charge flow through the detector. This charge flow is detected and converted
into a signal.3
The most common radiation detector category used in PET is the scintillator, which
consists of an inorganic crystal. The high-energy photon enters the scintillator crystal,
where its energy is converted into optical photons by interaction with the crystal’s atomic
electrons. The two dominating interactions are Compton scatter and photo-electric
absorption; Compton scatter results in a scattered photon and a recoil electron, and
photo-electric absorption produces a photo-electron. The resulting particles in turn
interact with the crystal’s structure and elevate its atomic electrons to higher energy
levels. These exited states transition into their ground states under the emission of
optical photons. As visible light features much less energy than gamma-rays, one gammaray can give rise to about 2 - 46 optical photons per keV energy of the gamma-photon,
depending on the scintillator material.4
State of the art PET scanners usually use partitioned scintillator crystals of a certain
size, which make it somehow easy to identify the hit region, since the optical photons
mostly propagate along the separated crystal. However, there can be gamma-photons
3
4
Ibid., p. 29.
Ibid., p. 31.
17
3 Positron Emission Tomography
which are scattered and redirected into another crystal; consequently, they trigger a
scintillation event, which leads to a possible misidentification depending on the deposited
amount of energy. Also, the pixelation limits the spatial resolution to the size of one pixel,
since no smaller object can be represented with its actual size. In order to improve spatial
resolution, the size of the discrete detector crystals needs to be decreased; however, the
smaller their size is, the higher becomes the cost of production.
Another possible detector geometry is the continuous, monolithic scintillator crystal.
The optical light generated spreads isotropically throughout the crystal and generates
a photon count pattern at the read-out surface. This pattern varies depending on the
location of the scintillation event and is used to identify said location. In the case of
a monolithic crystal, the spatial resolution is not only constrained by the geometric
properties of the radiation detector but also by signal processing algorithms.
3.2.2 Photo-detectors
The optical photons propagating within the scintillator crystal are then processed by
photo-detectors, which finally convert the optical signal into an electrical one. There are
different technological approaches for this conversion: first, there is the photo-multiplier
tube (PMT), which is a reliable technique to amplify low light signals of scintillation light.
It typically consists of an evacuated tube with a photo-cathode at the entrance window.
Optical photons detach electrons in the cathode via the photo-electric effect, which are
then accelerated by an electric field. On their path they strike dynodes detaching more
electrons, which consequently leads to an amplification of about 106 of the electron
signal. Finally, they hit an anode causing a voltage drop over a resistor: this voltage
provides the measurable signal.5
The second conventional technology used is based on semiconductor-based photodiodes. The incident scintillation photons cause electron-hole pairs in the semiconducting material; then, the intrinsic voltage between the oppositely doted materials causes
charges to flow, which in turn can be measured with an external circuit. A special type
of photo-diode is the avalanche photo-diode (APD), which supplies an internal signal
amplification through avalanche multiplication. The free charge carriers are accelerated
by an additional electric field until they reach energies high enough to break free other
electrons, causing a current amplification. This effect is called avalanche effect.6
Based on the use of APDs is the silicon photomultiplier (SiPM), a parallel-connected
array of silicon APDs operated in Geiger-mode. In Geiger-mode, the external potential
5
6
Bailey et al., “PET Basic Science,” p. 32.
Ibid., p. 33.
18
3.2 Technical Fundamentals of PET Imaging
applied to the photo-diode is strong enough to amplify one electron-hole pair, produced
by a single photon, to a maximum output voltage. The SiPMs provide an analogue signal, which can be converted into a digital one depending on the placement of a following
analogue-to-digital-converter (ADC). If every single APD’s signal is directly digitized,
the SiPM is called digital (dSiPM).7
Furthermore, there are different ways of coupling the scintillator crystals to the photodetectors. First, there is one-to-one coupling, where one single crystal is coupled to an
individual photo-detector. The hit crystal can be clearly identified if scattered events are
rejected. However, it is very difficult to realize in crystal sizes below 4x4 mm since very
small photo-detectors are needed; instead, APDs can be used, as they can be manufactured in arrays. Despite the good performance of one-to-one coupling, its complexity of
electronics and its costs limit their use, especially in pre-clinical systems. Another coupling method, which circumvents these problems, uses larger PMTs without the intrinsic
ability to determine the interaction’s position. The PMTs are glued to the crystals via
a light guide. A weighted positioning algorithm is used to reconstruct the position of
interaction within the detector considering a weighted sum of the PMTs’ signals.8
3.2.3 Data Processing
Figure 3.2: Schematic representation of different possible event detections. A: True coincidence. B: Random coincidence. C: Scattered coincidence. [Turkington,
“Introduction to PET Instrumentation”]
In order to locate the origin of the annihilation event and thus the distribution of the
radio-tracer, a coincident photon pair has to be identified. However, to avoid the pairing
7
8
Dam et al., “A Comprehensive Model of the Response of Silicon Photomultipliers.”
Bailey et al., “PET Basic Science,” p. 34.
19
3 Positron Emission Tomography
of coinciding events not resulting from the same annihilation, certain restrictions have
to be made:
• Both events have to be detected within a certain time window, the coincidence
window.
• The reconstructed line of response (LOR) between the two events must lie in a
certain angular range.
• The deposited energy of the photons needs to range in the selected energy window.9
Nevertheless, true coincidences (figure 3.2 A) with a 180 ± 0.5◦ angle and identical
timing are not the only events being detected. A so-called random coincidence (figure
3.2 B) results from the simultaneous annihilation of two positrons emitting four photons.
Two opposing photons from different annihilation events are detected as one coincident
event, and the other two photons are lost due to the criteria mentioned above. Figure
3.2 C shows a true coincidence, whose photon is scattered. If the deflection angle is still
in a certain range, the two emitted photons will be recognized as one coincident event.
However, the reconstructed LOR will lead to a deviation of the actual annihilation
location. To avoid events which are strongly scattered in the object, energy cuts are set
in order to filter events outside of the photo-peak.10 Further restrictions in the acquisition
of the single events can be made. In order to decrease the probability of a false crystal
identification in pixelated crystals caused by detector scatter, scatter rejection can be
applied.
3.3 Image Quality and Spatial Resolution
The image acquisition and processing quality of an imaging system in general can be
quantitatively evaluated by using certain parameters. Signal-to-noise ratio (SNR) e.g. is
one of them; in order for an object to be visible, its signal must be stronger than the image
noise.11 Noise increases and consequently, SNR decreases, if more random coincidences
are recorded or due to intrinsic detector noise. However, the focus of this thesis will be
put on another parameter - the system’s spatial resolution. Spatial resolution describes
the ability of the imaging system to display the smallest discernible details of an object.
Nevertheless, it needs to be distinguished from the term image resolution, since image
9
Bailey et al., “PET Basic Science,” p. 41.
Turkington, “Introduction to PET Instrumentation.”
11
Smith, “The Scientist and Engineer’s Guide to Digital Signal Processing.”
10
20
3.3 Image Quality and Spatial Resolution
resolution only refers to the quality of the final image; it gives a measure of how many
pixels, lines or dots are contained in a unit of length. The digitization settings that
provide the actual output image determine the final image resolution.12
Normalized Activity
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.0030
32
34
36
38
40
42
44
46
40
42
44
46
Profile axis, x-direction /mm
(a) Point source
0.12
Normalized Activity
0.10
0.08
0.06
0.04
0.02
0.0030
32
34
36
38
Profile axis, x-direction /mm
(b) Two separable point sources
Figure 3.3: Projection profiles of an image of a single point source (a) and two resolvable
point sources (b).
In PET scanner performance evaluation, the quantification of spatial resolution is often
defined via the width of the peak at half the maximum (FWHM) and the full width tenth
maximum (FWTM) of the reconstructed image line profile of either a gamma-photon
12
Gonzalez and Woods, “Digital Image Processing, Second Edition.”
21
3 Positron Emission Tomography
beam or a positron-emitting point source. Since this measure is common practice, it will
be used in this thesis. Another measure - usually applied in the field of chromatography
- is the peak-valley ratio (PVR), which quantifies the separability of two peaks. Since
the line profile of two point sources also creates two peaks, this is another measure that
will be utilized in the further course of this thesis. For the detector and scanner design
Figure 3.4: Model of various successive factors causing a degradation of the spatial resolution. [Stickel and Cherry, “High-resolution PET detector design: modelling components
of intrinsic spatial resolution”]
an entire chain of events degrading the spatial resolution by imposing an uncertainty on
the positioning of annihilation events need to be considered. The first effects, positron
range and acollinearity (figure 3.4 A, B), are of physical nature and are explained in section 3.3.1. The following two aspects, energy deposition and detector response (figure
3.4 C and D), directly correlate with the properties of the scintillator, namely crystal
size in pixelated scintillators, general detector and scanner geometry, and material properties (Section 3.3.2).13 The influence of the first four points on the spatial resolution
is explained and examined further in this thesis. However, the effect of different readout appliances and processing algorithms is neglected, since no optical simulations are
conducted. Even though some of the response functions of the degrading factors do not
follow a perfectly Gaussian distribution (positron range and detector response), all of the
effects are added up quadratically in order to calculate the system’s spatial resolution.
Therefore, the spatial resolution Γ in mm FWHM for a point source, which is located
at radius r from the center of the detector ring, has been determined by Moses as:14
Γ = krecon ·
s
(0.5 ·
d)2
+
s2
+ (0.0044 ·
R)2
+
b2
+
12.5 · r
r 2 + R2
2
(3.2)
d describes the crystal width, s is the positron range, R is the detector ring radius, and
k depends on the image reconstruction algorithm. The factor b represents the crystal
decoding error factor and amounts to d/3 for optical decoding. Equation 3.2 considers
the crystal width d and therefore refers to systems using discrete detector crystals.15
13
Stickel and Cherry, “High-resolution PET detector design: modelling components of intrinsic spatial
resolution.”
14
Moses, “Fundamental limits of spatial resolution in PET.”
15
Ibid.
22
3.3 Image Quality and Spatial Resolution
3.3.1 Positron Range and Acollinearity
Figure 3.5: Schematic representation of an annihilation process showing positron path,
positron range and acollinearity.
As mentioned previously, the annihilation does not take place immediately after the
beta decay. The positron is emitted with a certain amount of kinetic energy, which causes
it to travel a tortuous path until most of its energy is lost by Coulomb interactions with
the surrounding matter. The distance between the location of the actual radioactive
decay and the annihilation is called positron range.16 Table 3.1 shows that compared to
other isotopes 18 F has the shortest positron range due to the smallest energy. The larger
the positron range is, the higher the inaccuracy of the reconstructed location, which
naturally leads to a degradation of spatial resolution. As shown by Derenzo (1979),
the positron range follows a non-Gaussian distribution.17 Another effect which is caused
by the positron’s residual momentum is the so-called acollinearity. For a positron at rest,
the two gamma-photons are released back-to-back with an 180◦ angle, but the principle of
momentum conservation causes an angular deviation. Since the momentum varies within
a range, the angular deviation also takes on a certain range of values, which follow a
Gaussian distribution.18 The deviation ranges in the order of v/c rad with v being the
velocity of the atomic electron and c the speed of light. If the annihilation process takes
place in water, the angular deviation is about 0.5◦ ; and since the human body consists of
ca. 65 % water, this approximation is reasonably applicable.19 The effect of acollinearity
16
Badawi, “Aspects of Optimisation and Quantification in Three-Dimensional Positron Emission Tomography.”
17
Derenzo, “Precision measurement of annihilation point spread distributions for medically important
positron emitters.”
18
Levin and Hoffman, “Calculation of positron range and its effect on the fundamental limit of positron
emission tomography system spatial resolution.”
19
Steén and Uhlén, “Development of a Time-of-Flight and 3D Demonstration Set-up for PET.”
23
3 Positron Emission Tomography
on spatial resolution increases with an extending diameter between opposing detectors
with a factor of 0.0022 (equation 3.2).
Table 3.1: Half life (t1/2 ), maximum positron energy and positron range (FWHM in
H2 O) of different β + -emitting isotopes. [Cal-Gonzalez et al., “Positron range effects
in high resolution 3D PET imaging”]
Isotope
t1/2
(min)
Maximum positron
kinetic energy (MeV)
Mean positron range
(mm)
11 C
20.3
0.96
1.03
13 N
10.0
1.19
1.32
15 O
2.0
1.70
2.01
18 F
109.8
0.64
0.64
68 Ga
68.0
1.89
2.24
82 Rb
1.3
3.15
4.29
3.3.2 Energy Deposition and Detector Response
Events that occur from multiple interactions involving scatter within the crystal produce
a blurred energy deposition, which in turn leads to a more inaccurate positioning of the
event. The minimization of the spatial extent of the energy deposition distribution in
the detector (figure 3.4 C) is the first aspect that can be influenced with the detector
design. A slimmer distribution facilitates the localization of the interaction within the
crystal. The detector response to an infinitesimally narrow gamma beam is described by
a point-spread function (PSF); it describes the probability distribution of the difference
vector between the true and the reconstructed positions of the gamma interaction.
Ideally, the PSF of one scintillator crystal is represented by a uniform distribution with
the width of one pixel. However, inter-crystal scatter and pixel identification uncertainty
distort the probability function. Detectors with one-to-one-coupled read-out avoid these
problems, since certain signals associated with scatter can be rejected and the pixel
identification is very exact. Due to the limitations of one-to-one coupling explained in
section 3.2.2, the light sharing configuration with a light guide can be used. In this
case, light patterns are read out and scatter rejection can only be implemented with
calibration data. The PSF describing the response of the whole scanner in the image
24
3.3 Image Quality and Spatial Resolution
space converges to a Gaussian function, which results from the convolution of the single
uniform distribution functions.
For monolithic crystals, the PSF mainly depends on the detector geometry, its surface characteristics and, as for the pixelated detector, on the photo-detector, and the
reconstruction algorithm. If a detector is assumed to respond perfectly, the PSF can be
described as a Dirac impulse, which means that every interaction event can be located
perfectly at the correct position.
Figure 3.6: Experimentally determined point-spread-function of a monolithic LYSO scintillator crystal. [Maas et al., “Monolithic scintillator PET detectors with intrinsic depthof-interaction correction”]
The realistic PSF in the xy-plane of a monolithic detector using a LYSO scintillator
has been experimentally determined by Maas et al.20 In order to measure the PSF of a
detector, a gamma-photon pencil beam is directed at the detector, and a lead collimator
is used for creating a beam with a certain width. The detector response is recorded
and subsequently corrected for the beam diameter. A slimmer PSF implies a better
intrinsic detector resolution, which in turn contributes to a better spatial resolution of
the complete system.
20
Maas et al., “Monolithic scintillator PET detectors with intrinsic depth-of-interaction correction.”
25
3 Positron Emission Tomography
3.3.3 Depth of Interaction
Another desirable performance characteristics of a scintillator crystal is a high stopping
power leading to a high scanner sensitivity. Sensitivity describes the ability of a system
to detect coincident photons per activity. A high stopping power not only leads to a
more efficient event detection but also reduces the gamma-photons’ propagation path
inside the crystal. The interaction depth of a gamma-photon entering a scintillator is
distributed exponentially with the attenuation coefficient µ depending on the material
properties. LYSO-scintillators have an attenuation length µ of about 0.77 cm−1 .21
Ix = I0 · e−xµ
(3.3)
I0 represents the unattenuated photon beam intensity and Ix the intensity measured
through a material of the thickness x.22 If the detector set-up does not allow the determination of the depth of interaction (DOI), the energy deposition of the gamma-photon
gets projected onto the crystal surface. This does not affect the location determination
if the incidence angle is orthogonal to the entrance surface; but as soon as the angle is
oblique, a parallax error occurs. Figure 3.7 schematically explains this case.23
Figure 3.7: Schematic visualization of the parallax error occurring without DOI determination. [Bailey et al., “PET Basic Science”]
21
Standards and Technology, Physical Measurement Laboratory.
Bailey et al., “PET Basic Science,” p. 26.
23
Ibid., p. 37.
22
26
3.4 Objective
The parallax error causes further decrease of spatial resolution. One way to avoid it
before the image reconstruction, is to reduce the depth of the crystals. España et al.
presented a DigiPET system with sub-millimeter spatial resolution by using a monolithic
crystal with only 2 mm depth.24 However, the sensitivity obtained in the performance
evaluation of the DigiPET was about 0.3 % compared to the nanoScan PET/MRI system of Nagy et al., which provided 8 %.25 Hence, with varying crystal depth, there is
always a compromise between a good system resolution and a high sensitivity. In order
to solve this conflict between sensitivity and resolution caused by crystal depth, DOI determination can be implemented. This can be achieved by using monolithic scintillators,
since the light pattern at the read-out surface is also affected be the interaction depth.
3.4 Objective
In order to specify the objective of examining the characteristics of PET spatial resolution, different positron emission tomography scanners with different applications need
to be clearly distinguished. First, there are clinical scanners used mainly in oncological
diagnostics. They are utilized to diagnose and locate tumors and metastases in human
patients. A gantry of a certain size is needed in order to fit a human patient inside,
and a high sensitivity is advantageous to enable minimization of the radiation dose affecting the patient. However, spatial resolution is not such a critical factor due to the
dimensions of a human body and the respective tumor size; a spatial resolution of 3 4 mm seems to meet the demands. Also, multi-modal imaging systems combining PET
and CT or MRI compensate for mediocre spatial resolution. Nevertheless, most cancer types are much better treatable if they are detected early. Therefore, better early
detection of small nodules and primary tumors or metastases is beneficial. Since much
smaller structures are to be visualized, higher spatial resolution could be a supporting
aspect. Second, there are preclinical scanners, which are used for oncological tumor
studies mainly in small animals like mice and rats. Naturally, these animals grow much
smaller tumors and the difficulty of detecting and distinguishing them from natural
physiological spots with high activity increases. Thus, better image quality and spatial
resolution are necessary to successfully conduct experiments. These preclinical scanners
will lie in the focus of this thesis. Therefore, the aims of the conducted simulation studies
are the quantification and comparison of spatial resolution provided by either scanners
24
S. Espana and Holen, “DigiPET: sub-millimeter spatial resolution small-animal PET imaging using
thin monolithic scintillators.”
25
K. Nagy and Gulyás, “Performance Evaluation of the Small-Animal nanoScan PET/MRI System.”
27
3 Positron Emission Tomography
with conventional pixelated or monolithic detector crystals. Additionally, the impact of
different factors influencing the spatial resolution from the annihilation event up to the
image reconstruction are to be evaluated (see figure 3.4).
28
4 Simulations
4.1 GATE - Geant4 Application for Tomographic Emission
GATE is a simulation tool kit developed to numerically simulate scenarios in the field of
nuclear medicine and radiotherapy. It is based on the Geant4 platform, which provides
Monte-Carlo simulations of particles passing matter. GATE contains packages, which
are tailored to the specific use of PET and SPECT, and therefore, is highly beneficial for
designing new scanners, optimizing acquisition protocols and developing and evaluating
image reconstruction algorithms. The scanner geometry is defined within the program,
and physical values and simulation parameters can be adapted.
4.2 PET Modality
Figure 4.1: Sketch of the scanner geometry with respective coordinate system.
The preclinical scanner geometry used for the simulations is based on the Hyperion IID
system.1 Ten single detection modules are mounted on a gantry and make up a PET ring
1
Weissler et al., “Design concept of world’s first preclinical PET/MR insert with fully digital silicon
photomultiplier technology.”
29
4 Simulations
with a diameter of 209.6 mm (smallest distance between two opposing detector surfaces)
and an axial field of view of 30 mm per detector ring. One module is made up of six
sensor stacks in a 2x3 arrangement. Each sensor stack contains 30x30 LYSO scintillator
crystals, each with the dimension of 1x1x12 mm3 . Since no optical simulations are
conducted, the processing units are neglected in the simulation. They consist of 4x4
Philips Digital Photon Counting DPC sensors, and in turn every DPC comprises 2x2
pixels consisting of 3200 single-photon avalanche photo diodes (SPAD).2
4.3 Simulation Parameters
For the determination of the spatial resolution, an
18 F
point source with no spatial
expansion and an activity of 100 kBq is placed in the center of the detector. An attenuation sphere with a radius of 5 mm consisting of water is placed around the source;
it enables the annihilation after a certain positron range and provides a physiologically
comparable setting. The simulated time amounts to 120 s neglecting the half-life of
the element. Furthermore, four simulation runs are conducted and their mean and the
statistical uncertainty are determined and shown.
4.4 Simulation of Detector Response and DOI
No optical simulations are conducted and therefore the simulated measurement stops at
the gamma-photon’s absorption within the scintillator. In case of the pixelated detector, intrinsic detector scatter is considered in the simulation, and GATE provides the
crystal’s ID, in which the weighted mean of multiple interactions takes place. For the
monolithic detector crystal, local crystal coordinates of the first interaction without any
further simulated detector scatter are documented. However, the uncertainty of localizing the gamma-photon’s absorption also needs to be taken into account. Therefore, the
experimentally measured response to a pencil beam (Maas et al.3 ) is used to smear the
exact crystal coordinates. For parametrization, the one-dimensional projection of the
PSF (figure 4.2) is fitted with a bimodal Gaussian function (equation 4.1). Accordingly,
the x- and y-coordinates are smeared individually with the same function. Its tails,
which are broader than the ones of a regular Gaussian function, are ascribed mostly to
detector scatter. The FWHM of the fitted PSF is 1.05 mm and its FWTM 2.2 mm.4
2
Wehner et al., “MR-compatibility assessment of the first preclinical PET-MRI insert equipped with
digital silicon photomultipliers.”
3
Maas et al., “Monolithic scintillator PET detectors with intrinsic depth-of-interaction correction.”
4
Ibid.
30
4.4 Simulation of Detector Response and DOI
Figure 4.2: Line profile of experimentally determined PSF [Maas et al., “Monolithic scintillator
PET detectors with intrinsic depth-of-interaction correction”]







2
2
(x − µ1 )
(x − µ2 ) 
1
1
exp − q
exp − q
+√
P SFmonolithic = √


2σ1 π
2σ2 π
2σ 2 
2σ 2 
1
µ1 = −0.020, σ1 = 0.995, µ2 = −0.002, σ2 = 0.418
(4.1)
2
(4.2)
µ is the mean and σ the standard deviation of the function. Unfortunately, no further
information is given about the properties of the PSF varying with the location on the
surface of the detector. Thus, the smear of the coordinates is assumed to be consistent
over the entire area of the detector. This is not an entirely realistic assumption; however,
it is sufficient and assumes the best case scenario. Furthermore, for the simulation
of monolithic scintillator performance, a realistic depth of interaction determination is
implemented. For that, the depth coordinate determined by GATE is smeared with a
Gaussian function, which is defined by a given FWHM depending on the depth, whose
values can be taken from figure 4.3. Even though a slightly adapted function, like the bimodal Gaussian function, is rather likely to define the DOI resolution, this approximation
is satisfactory. The values used for the smearing were determined by Dam et al. and
provide a weighted DOI resolution of 4.1 mm.5
5
Dam and Schaart, “A practical method for depth of interaction determination in monolithic scintillator
PET detectors.”
31
4 Simulations
6
DOI resolution FWHM /mm
5
4
3
2
1
00
2
4
6
DOI /mm
8
10
12
Figure 4.3: DOI resolution for different depths in a 12 mm deep monolithic crystal. [Dam
and Schaart, “A practical method for depth of interaction determination in monolithic
scintillator PET detectors”]
4.5 Reconstruction Software
The reconstruction software used to generate images from the pre-processed data deploys
an iterative reconstruction algorithm; this algorithm approximates the solution of an
inverse problem consisting of a set of linear equations:6
hpj i =
P
X
aj,i fi
j = 1, ..., NLOR
(4.3)
i=1
hpj i is the measured data, fi represents the estimated vector describing the image
model, and aj,i describes the system matrix elements. The system matrix provides the
system’s response to an input function. An important step in solving equation 4.3 is
the approximation of said system matrix a. An approximation is used, since an exact
calculation of the whole matrix is inhibited by its large size. The system’s geometry
and characteristics are known and taken into account as well as the assumption that
the system’s measurements are limited by a finite resolution. The precision the system
matrix is approximated with is described by an adjustable parameter in the reconstruction. This parameter provides the number of samples that are randomly generated for
the estimation of every system matrix element. The shape defining the distribution of
6
Bailey et al., “PET Basic Science,” p. 71.
32
4.6 Data Preparation for Evaluation
the random samples differs depending on the scintillator geometry: for the pixelated detector, the shape is defined by a uniform probability density function with the width of
one crystal, clearly restricting the possible range of the gamma-absorption. For a monolithic scintillator, the uncertainty is given by the PSF describing the detector response
(equation 4.1).
4.6 Data Preparation for Evaluation
Section 3.3 explains that FWHM and FWTM values are measured in the line profile
of the image peak of a point source. As it is difficult to choose a representative and
reproducible section for the line profile, not only single line profiles but rather projections
of the field of view volume onto one axis are generated. FWHM and FWTM values
are always taken from the one-dimensional projection profiles if not explicitly stated
otherwise and are indicated with their respective uncertainties.
33
5 Results and Discussion
5.1 Ideal Detector Resolution
In order to determine the physical limits of PET spatial resolution, the response of a
scanner with ideal monolithic detectors is simulated, i.e. the detector response is described by a Dirac-PSF. For modeling the realistic detector response, the experimentally
determined PSF in equation 4.1 is applied. The point source is located in the detector
center and surrounded by an attenuation sphere of water (radius of 5 mm), which enables
the annihilation of the positron with an electron.
Table 5.1: FWHM and FWTM (with respective statistical uncertainies) of the spatial
resolution of a scanner with monolithic detector crystals comparing an ideal
and a realistic detector response.
Ideal detector
Scanner direction
Realistic detector
FWHM
(mm)
FWTM
(mm)
FWHM
(mm)
FWTM
(mm)
x
0.605 ± 0.001
1.090 ± 0.001
0.920 ± 0.002
1.702 ± 0.002
y
0.609 ± 0.002
1.090 ± 0.003
0.922 ± 0.003
1.705 ± 0.003
z
0.583 ± 0.001
0.993 ± 0.001
0.881 ± 0.004
1.648 ± 0.004
If a monolithic detector was designed to perfectly localize a gamma-photon interaction
within the scintillator crystal, a spatial resolution of about 0.6 mm FWHM in x- and
y-direction and 0.5 mm in z-direction could be achieved (table 5.1). In this case, only
the physical effects, positron range and acollinearity, and the image reconstruction software influence the spatial resolution. Object scatter occurring in the attenuation sphere
consisting of water can be neglected.
Figure 5.1 shows a step-wise approximation of the spatial resolution limit and its
dependence on the quality of the detector response. The same simulation data is smeared
35
5 Results and Discussion
with different detector response FWHM values from 0.0 mm up to 1.2 mm. However,
the probability density function, which is used to smear the data is a simple Gaussian
function and not a bi-modal one. Therefore, the results at ca. 1.05 mm FWHM detector
response are not completely compatible with the ones provided by a realistic detector
with the empirical detector PSF.
Spatial resolution FWHM /mm
0.85
0.80
0.75
FWHM x-direction
FWHM y-direction
FWHM z-direction
0.70
0.65
0.60
0.55 0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.8
1.0
1.2
Detector resolution FWHM /mm
Spatial resolution FWHM /mm
1.5
1.4
1.3
FWTM x-direction
FWTM y-direction
FWTM z-direction
1.2
1.1
1.0
0.9 0.0
0.2
0.4
0.6
Detector resolution FWTM /mm
Figure 5.1: Dependency of the system’s spatial resolution (FWHM and FWTM) on the
varying width of the detector response function.
36
5.1 Ideal Detector Resolution
As the width of the detector response function tends to 0.0 mm, the spatial resolution
of the whole system converges to its physical limits of about 0.6 mm. Enhancing the
detector response from 1.0 mm to 0.6 mm could increase the spatial resolution by 0.12 mm
(15 %). This could be beneficial depending on the actual effort and cost of improving the
detector response. However, a further improvement of the detector response to less than
a width of 0.6 mm does not pay off considering the increasingly smaller advancement of
the spatial resolution.
5.1.1 Resolution in Different Scanner Directions
It is unusual to achieve a better spatial resolution along the z-axis than the radial xand y-axis, even though this effect can be observed in all the provided results. When
acquiring the activity of a point source in the center of the scanner, every LOR through
the source without z-component (z = 0) meets the two opposing detector surfaces in
an almost orthogonal angle with at most ±8.2◦ angular deviation; this leads to an
interaction localization with almost no parallax error. The resolution along the z-axis
however, is influenced by a stronger parallax error, since the LORs with a z-component
(z 6= 0) meet the crystal surfaces with a deviation of up to ±24.7◦ (given a scanner
length in z-direction of 96.6 mm). This mainly affects the pixelated detector due to the
missing DOI determination.
For the monolithic detector, the better z-resolution could be caused by a binning
error from the voxel binning of the field of view in the image reconstruction. A voxel
size of 0.25 mm is chosen and the reconstruction software sets the grid the image is
reconstructed on. Since spatial resolutions of up to 0.6 mm are achieved, the location of
the bins can affect the measured values substantially. To evaluate the possible binning
error, the FWHM of a binned Gaussian function with variable starting points for the
bins is measured and compared to its actual FWHM. Figure 5.2 shows the probability
density of certain FWHM values caused by differing binning; the standard deviation of
this distribution amounts to 0.011 mm. The following values will all be corrected for the
binning bias 0.017 mm, which is the difference between the mean FWHM and the actual
FWHM given by this distribution. The fact, that the binning is always equal for all
the reconstructed measurements could explain the permanently deviating resolution in
z-direction. However, the respective uncertainties provided with each value arise from
the statistical fluctuations of the simulations.
37
5 Results and Discussion
700
600
Probability density
500
400
300
200
100
0 0.710 0.715 0.720 0.725 0.730 0.735 0.740 0.745
0.705
FWHM values /mm
Figure 5.2: FWHM values of a binned Gaussian function (FWHM = 0.7 mm) with varying starting points for the binning.
In order to further examine this hypothesis, another simulation is conducted, shifting
a point source off-center in z-direction for z = 0.1 mm, a fraction of one voxel. The
results in table 5.2 show the considerable effect of voxel binning, if the voxels are larger
than a quarter of the FWHM-based resolution. Nevertheless, the image reconstruction
computation time scales with the power of six (cubically with the number of voxels and
additionally the need for a cubically increased statistic); in order to save computation
time, these effects were accepted and regarded as systemic errors.
5.2 Performance Comparison under Realistic Conditions
The performances of monolithic and pixelated detectors are simulated under realistic
conditions and compared. Physical aspects, such as positron range and acollinearity,
are simulated, and DOI determination is possible in the monolithic crystal. Signal readout effects, which occur due to light sharing in the pixelated scintillator are neglected
since no optical simulations are conducted; the monolithic scintillator however, partially
38
5.2 Performance Comparison under Realistic Conditions
Table 5.2: FWHM and FWTM (with respective statistical uncertainies) of the spatial
response provided by monolithic detectors with a point source shiftet in z =
0.1 mm.
Scanner direction
FWHM
(mm)
FWTM
(mm)
x
0.905 ± 0.003
1.698 ± 0.003
y
0.895 ± 0.002
1.687 ± 0.002
z
0.887 ± 0.002
1.651 ± 0.002
considers signal read-out effects, since the experimentally determined PSF is used to
smear the coordinates.
Table 5.3: FWHM and FWTM (with respective statistical uncertainies) of the spatial
resolution comparing monolithic to pixelated detector crystals.
Pixelated
Scanner direction
Monolithic
FWHM
(mm)
FWTM
(mm)
FWHM
(mm)
FWTM
(mm)
x
0.782 ± 0.004
2.351 ± 0.007
0.903 ± 0.002
1.685 ± 0.002
y
0.779 ± 0.007
2.292 ± 0.007
0.905 ± 0.003
1.688 ± 0.003
z
0.692 ± 0.006
2.241 ± 0.006
0.864 ± 0.004
1.631 ± 0.004
Comparing the spatial resolution performance of the different detectors shows clear results regarding the FWHM of the projection profile (table 5.3). The pixelated detector
provides a better resolution of about 0.78 mm compared to the monolithic one, which
achieves 0.90 mm. Interestingly, the FWTM values of the projection profiles follow the
opposite trend. The pixelated detector shows an FWTM of about 2.35 mm and the
monolithic detector one of 1.69 mm. In order to further examine these effects, the line
profiles are compared.
The line profile of the pixelated detector shows broader tails than the fitted Gaussian
function, unlike the monolithic detector, whose line profile matches the fit quite well.
One reason for this might be the differences in handling detector scatter. Using pixelated scintillator detectors with one-to-one coupling, the hit crystal is directly identified
and detector scatter blurs the acquired information. As explained previously, there are
39
5 Results and Discussion
different coupling methods in order to detect the optical photons, but as no optical
simulations are conducted, the model resembles a one-to-one-coupled read-out.
1.2
1.0
Normalized activity
Line profile
Gaussian Fit
0.8
0.6
0.4
0.2
0.0
3
4
5
6
Profile axis, z-direction /mm
7
(a) Pixelated
1.2
1.0
Normalized activity
Line profile
Gaussian Fit
0.8
0.6
0.4
0.2
0.0
3
4
5
6
Profile axis, z-direction /mm
7
(b) Monolithic
Figure 5.3: Line profiles of a point source and their Gaussian fits.
In contrast, the complete monolithic-scintillator-based detector generates different
light patterns depending on the interaction location. This detector response first needs
to be calibrated, meaning a specific light pattern for each known location is recorded.
In the actual data acquisition, these patterns are compared to the calibration data via
40
5.2 Performance Comparison under Realistic Conditions
the k-nearest-neighbour classification algorithm.1 The most similar pattern is assumed
to be most probable to represent the actual location. This method already provides an
intrinsic detector scatter reconstruction, as detector scatter already affected the calibration patterns. Therefore, this could be the reason the resulting line profile provided by
the monolithic detector does not display the broad tails caused by detector scatter.
5.2.1 Visual Separability of Two Point Sources
The following measurements ought to provide a different perspective on the term resolution. The two peaks appearing in the projection profiles are considered visually
separable, if at least two data points lie in the valley between the two peaks. The separability of two point sources around the center of the scanner is evaluated in all three
scanner directions.
The resolution provided by the system with pixelated detectors - the two sources were
separable at 0.9 mm - sufficiently correlates with the determined FWHM values of the
single source image (0.8 mm). The scanner with monolithic detectors however, only
resolves two point sources at a distance of 1.2 mm instead of 0.9 mm as it is expected
regarding the FWHM of the single point source. This effect leads to the assumption
that a single point source does not necessarily provide a precise measure for the spatial
resolution of a PET system.
Table 5.4: Comparison of the peak-to-valley ratios at the same distance 1.2 mm of the
two point sources using pixelated and monolithic detectors.
Pixelated
Monolithic
Peak-valleyratio
Peak-valleyratio
x
1.731 ± 0.012
1.195 ± 0.045
y
1.893 ± 0.018
1.124 ± 0.016
z
2.396 ± 0.016
1.229 ± 0.051
Scanner direction
In order to further compare the peak-to-valley ratios, the values at equal distance
1.2 mm are shown in table 5.4. The system with the pixelated detectors shows a higher
peak-to-valley ratio than the one with the monolithic detectors. This correlates with the
results about the separable distances.
1
Dam and Schaart, “A practical method for depth of interaction determination in monolithic scintillator
PET detectors.”
41
5 Results and Discussion
8
7
Peak-to-valley ratio
pixelated
monolithic
6
5
4
3
2
1
0.8
1.0
1.2
1.4
1.6
1.8
Source distance in x-direction /mm
2.0
8
7
Peak-to-valley ratio
pixelated
monolithic
6
5
4
3
2
1
0.8
1.0
1.2
1.4
1.6
1.8
Source distance in y-direction /mm
2.0
8
7
Peak-to-valley ratio
pixelated
monolithic
6
5
4
3
2
1
0.8
1.0
1.2
1.4
1.6
1.8
Source distance in z-direction /mm
2.0
Figure 5.4: Peak-to-valley ratios as a measure of the scanner’s ability to resolve two
point sources depending on the sources’ distance, comparing pixelated and
42
monolithic detectors.
5.3 Positron Range and Acollinearity
Regarding the developing separability of the two different detector kinds however, a
trend emerges. The previously explained effect of detector scatter causing broader tails
in the line profiles of the pixelated detectors is represented in the peak-to-valley ratios
of two sources with varying distances. Starting from a source distance of 2 mm, the
monolithic detector is able to separate the peaks more clearly in x- and z-direction than
the pixelated detector, despite its worse resolution limit. Interestingly, the trend of the
comparison in y-direction strongly deviates from the other two directions; unfortunately
this phenomenon can currently not be explained. It could be conclusively assumed, that
the separability of two point sources gives a realistic measure of the spatial resolution
of a system, but the peak-to-valley ratio loses its validity when the sources are farther
apart.
5.3 Positron Range and Acollinearity
In order to examine positron range and acollinearity in a system with realistic detectors,
they are completely omitted in the simulations by introducing a perfect back-to-back
gamma-ray-emitting source in the scanner center. The isolated influence of these physical
effects on the spatial resolution has been shown in table 5.1, where a perfect interaction
localization in the detector is given.
Table 5.5: FWHM and FWTM (with respective statistical uncertainies) of the spatial
response with perfect 180◦ γ-rays comparing pixelated and monolithic detectors.
Pixelated
Scanner direction
Monolithic
FWHM
(mm)
FWTM
(mm)
FWHM
(mm)
FWTM
(mm)
x
0.540 ± 0.001
1.107 ± 0.001
0.697 ± 0.001
1.227 ± 0.001
y
0.537 ± 0.001
1.167 ± 0.001
0.704 ± 0.002
1.235 ± 0.002
z
0.432 ± 0.002
0.854 ± 0.002
0.675 ± 0.003
1.168 ± 0.003
The simulated scanner with a perfect gamma-ray-emitting source provides a spatial
resolution of about 0.54 mm for the pixelated detectors and 0.70 mm for the monolithic
detectors. In case of the pixelated detector, the resolution is about 0.24 mm better than
the one provided under realistic conditions and for the monolithic detector, the value
is similar (compare to table 5.3). This leads to the conclusion that positron range and
acollinearity altogether lead to a degradation of the spatial resolution of about 30 - 40%
43
5 Results and Discussion
for the investigated scanner.
In order to individually examine these two effects, a gamma-ray-emitting point source
is simulated with an angular deviation representing realistic acollinearity. In order to
simulate acollinearity, the direction of the emitted gamma-rays is smeared by a Gaussian
with an FWHM of 0.58◦ by the GATE simulation.
Table 5.6: FWHM and FWTM (with respective statistical uncertainies) of the spatial
response of the pixelated and monolithic scintillator with back-to-back γrays with a Gaussian 0.58◦ FWHM acollinearity distribution neglecting the
positron range.
Pixelated
Scanner direction
Monolithic
FWHM
(mm)
FWTM
(mm)
FWHM
(mm)
FWTM
(mm)
x
0.797 ± 0.002
2.120 ± 0.002
0.878 ± 0.004
1.612 ± 0.004
y
0.772 ± 0.003
2.042 ± 0.003
0.888 ± 0.003
1.632 ± 0.003
z
0.496 ± 0.003
1.104 ± 0.003
0.695 ± 0.002
1.695 ± 0.002
As shown in table 5.6, completely neglecting the positron range gives a spatial resolution comparable to the one given by the simulation of a regular beta-emitting point
source leading to the conclusion that acollinearity is the more dominating effect. This
seems unexpected since the simulations are based on a preclinical system and the influence of acollinearity increases with a larger scanner radius. In contrast, the mean
positron range of
18 F
taken from literature is 0.64 mm.2 Regarding the empirical res-
olution formula 3.2, the positron range of an isotope ought to have a stronger effect,
or frankly any effect at all, on the spatial resolution.3 A reason for this might be an
unrealistic simulation of the positron range in GATE. However, it needs to be considered that a variety of theoretical values for the positron range of
18 F
are denoted in
literature. One possible reason for this deviation from the expected values could be an
erroneously assumed positron range. Furthermore, a discrepancy in the resolution only
in z-direction is very unlikely due to the fact that acollinearity and positron range are
isotropic phenomena.
Another possible reason for this discrepancy could be an unidentified filtering process
in the image reconstruction software, which drowns out these smaller effects.
2
3
Cal-Gonzalez et al., “Positron range effects in high resolution 3D PET imaging.”
Moses, “Fundamental limits of spatial resolution in PET.”
44
5.4 Influence of DOI Determination
5.4 Influence of DOI Determination
The simulation of a scanner with monolithic scintillators assumes that a determination
of the depth of interaction of the incident gamma-photon is possible with a resolution
shown in figure 4.3. The performance of monolithic detectors with and without DOI
determination is compared.
Table 5.7: FWHM and FWTM (with respective statistical uncertainies) of systems with
monolithic scintillators, with and without DOI determination.
Monolithic with DOI
Scanner direction
Monolithic without DOI
FWHM
(mm)
FWTM
(mm)
FWHM
(mm)
FWTM
(mm)
x
0.903 ± 0.002
1.695 ± 0.002
1.074 ± 0.004
2.117 ± 0.004
y
0.905 ± 0.003
1.698 ± 0.003
1.060 ± 0.004
2.104 ± 0.004
z
0.864 ± 0.004
1.631 ± 0.004
1.078 ± 0.007
2.443 ± 0.007
As presumed, the spatial resolution provided by the monolithic detectors deteriorates by
about 18 % without any DOI correction. Also, the degrading resolution in z-direction
fulfills the expectations, which is explained in section 5.1.1. The fact that the effect in
z-direction is not more apparent could be caused by the previously explained binning
error.
Table 5.8: FWHM and FWTM (with respective statistical uncertainies) at the edge of
the FOV (at z = 38 mm) provided by the monolithic scintillator, both with
and without DOI determination
Monolithic with DOI
Scanner direction
Monolithic without DOI
FWHM
(mm)
FWTM
(mm)
FWHM
(mm)
FWTM
(mm)
x
0.911 ± 0.007
1.724 ± 0.007
1.155 ± 0.004
2.589 ± 0.004
y
0.908 ± 0.003
1.707 ± 0.003
1.241 ± 0.020
2.543 ± 0.020
z
0.878 ± 0.006
1.632 ± 0.006
0.909 ± 0.007
2.502 ± 0.007
45
5 Results and Discussion
The effect of resolution degradation is also simulated at the edge of the field of view. As
expected, the resolution without DOI determination at the edge of the FOV is impaired
by about 26 %. This effect is caused by the greater parallax error. Interestingly, the
resolution of the system with DOI determination is able to compensate this parallax
error and shows almost no resolution degradation.
5.5 Phantom Simulations
The measurement of FWHM and FWTM of a point source peak is a very simplified
method to determine spatial resolution. To get a visual impression of the image quality,
it is sensible to scan or simulate a phantom. The phantom consists of spherical sources,
each with a different diameter. The sources’ activity density is kept constant in order to
simulate hot spots with high activity next to smaller structures. The point sources each
have an activity of 100 kBq and the sources with spatial extent have 2985 kBq /mm3 .
The images acquired with monolithic and pixelated scintillators are compared. In order
to get a quantitative statement, the peak-to-valley ratios are determined for each of the
different diameter regions.
Figure 5.5: Sketch of a structure phantom, which consists of spherical sources with different diameters and point sources.
Figures 5.7 (a) and (c) show the line profiles of the image through the x-axis of the
scanner, the spheres of the diameter 2.41 mm on the left and the point sources on the right
46
5.5 Phantom Simulations
(a) Monolithic, transversal
(b) Monolithic, coronal
(c) Monolithic, sagittal
(d) Pixelated, transversal
(e) Pixelated, coronal
(f) Pixelated, sagittal
Figure 5.6: Image of a structure phantom viewed in different planes, (a)-(c) acquired
with monolithic detectors and (d)-(f) with pixelated detector, all based on
an averaged gray scale.
side. Comparing the results of the pixelated with the ones of the monolithic detector,
a slight difference in separability is visible. The monolithic detector is able to separate
the peaks more clearly, which can be confirmed quantitatively by the disposition of the
peak-to-valley ratios. For the 2.41 mm-sources, the monolithic peak-to-valley ratio is
about factor 10, and for the point sources about factor 3 higher than the pixelated one.
As previously explained, this effect could result from the difference in handling detector
scatter, leading to slimmer tails in the line profiles. Comparing figures 5.7 (b) and (d),
the line profiles though the y-axis and their respective peak-valley ratios do not show
a clear disposition. Neither the pixelated detector nor the monolithic detector are able
to better separate the peaks representing the sources. Nevertheless, it is important to
consider the cleanness of the entire simulation setting; this means, object scatter is almost
non-existent leading to very low noise levels, which are not realistically comparable to
normal scanner features. The sources in a realistic phantom are normally embedded in
47
5 Results and Discussion
plastic surroundings to fixate them, which causes much more object scatter than a water
sphere. In regard of these aspects, not the absolute peak-to-valley ratios are considered
but rather the tendencies.
1400
3000
1200
2500
Activity, a.u.
3500
Activity, a.u.
1600
1000
2000
800
1500
600
400
1000
200
500
00
5
10
15
20
25
30
Profile axis, x-direction /mm
35
00
40
(a) Pixelated, x-direction
15
20
25
30
35
40
35
40
1600
1400
800
1200
Activity, a.u.
Activity, a.u.
10
Profile axis, y-direction /mm
(b) Pixelated, y-direction
1000
1000
600
400
800
600
400
200
00
5
200
5
10
15
20
25
30
Profile axis, x-direction /mm
(c) Monolithic, x-direction
35
40
00
5
10
15
20
25
30
Profile axis, y-direction /mm
(d) Monolithic, y-direction
Figure 5.7: Line profiles of the phantom image through the x- and the y- axis of the
scanner (average of the four simulation runs).
5.6 Theoretical Determination of the Spatial Resolution
As introduced previously, equation 3.2 gives an estimation of the spatial resolution of
a PET scanner. This equation is mentioned in literature repeatedly, but it is partially
based on empirical assumptions, e.g. the factor 0.5 scaling the pixel width d of the
detector. The theoretical detector response of a pixelated detector has been explained
in section 3.3.2. The variance V of a uniform distribution with the boundaries a and b
48
5.7 Influence of Reconstruction Software
√
amounts to (b − a)2 /12 and accordingly, its standard deviation to σ = (b − a)/ 12. For
a pixelated detector, b − a is equal to its width d. The quadratically added summands
are given as FWHM values. Consequentially, the scaling factor of d should rather be
chosen as:
√
2.35
2 2 ln 2 · σ = √ = 0.68
12
(5.1)
In the further course of the discussion, the equation to approximate the spatial resolution Γ of a PET system is given by:
Γpixelated = krecon,pix ·
q
(0.68 · d)2 + s2 + (0.0044 · R)2
(5.2)
Additionally, this formula ought to be customized for the use of monolithic scintillators since its application is currently limited to the use of pixelated-scintillator-based
detectors. For the monolithic scintillator, the FWHM of its PSF is implemented as
contribution of the detector:
Γmonolithic = krecon,mono ·
q
(FWHMPSF )2 + s2 + (0.0044 · R)2
(5.3)
5.7 Influence of Reconstruction Software
Equations 5.2 and 5.3 give an estimation of the spatial resolution taking into account
the performance of the reconstruction software (rrecon ). Different sources and simulation settings are considered and the respective rrecon is determined by comparing the
measured FWHM and the expected FWHM, the latter without being affected by reconstruction. Unfortunately, the consideration of the dissociated impact of positron range
and acollinearity has to be omitted due to not yet understood results shown in table 5.6.
Table 5.9: Reconstruction factor krecon for different simulation settings determined with
equation 3.2 for pixelated and monolithic detectors.
Detector
type
Simulation and
source properties
Measured
FWHM (mm)
Expected
FWHM (mm)
krecon
Pixelated
Realistic
0.78
1.04
0.75
Pixelated
Back-to-back
0.54
0.68
0.82
Monolithic
Realistic
0.90
1.31
0.69
Monolithic
Back-to-back
0.70
1.05
0.67
Monolithic
Ideal detector
0.60
0.788
0.76
49
5 Results and Discussion
The image reconstruction software for the pixelated-scintillator-based scanner shows
an average krecon,pix of 0.795 ± 0.025. This means the software provides a so-called
resolution recovery of about 20 % from the estimated value to the measured one. For
the monolithic-scintillator-based scanner, a krecon,mono of 0.713 ± 0.024 is determined
and hence a resolution recovery of almost 30 %. Due to the very small amount of data,
these values have to be treated with caution. For both, simulations with pixelated
and monolithic detectors, in principle equal approaches are used to reconstruct the
images from the acquired data. Still, the simulations of the two different detectors, both
based on different scintillator geometries, record different data and thus, two slightly
varying versions of the reconstruction software need to be applied. The one version
reconstructing monolithic data takes into account the probability of falsely localized
interactions within the crystal. This probability is given by the true detector response
(the PSF) of the monolithic detector and is implemented into the approximation of the
system’s matrix. In contrast, the version reconstructing the pixelated data receives and
processes discrete crystal IDs for each interaction. The software does not incorporate
any information about the probability of falsely identified crystals. The calculation of
the system matrix only takes into account the possible range within the one determined
crystal in order to avoid binning artefacts. Therefore, the data processing during the
image reconstruction of the monolithic detector provides a more realistic model than
the reconstruction of the data from the pixelated detector. Hence, this might lead
to a better image reconstruction and a stronger resolution recovery. Conclusively, an
implementation of scatter reconstruction into the software reconstructing the pixelated
data could be beneficial for further improving the spatial resolution.
50
6 Conclusion
In this thesis, simulation studies are conducted in order to estimate the spatial resolution
of a positron emission tomography scanner with monolithic scintillators and to compare
them to conventional pixelated scintillators. A theoretical system with detectors providing a perfect response would have a spatial resolution of 0.6 mm. In this case, the
resolution is only restricted by positron range and acollinearity and therefore marks the
limit of possible spatial resolution with the specific scanner geometry and the applied
image reconstruction software.
In order to simulate the response of the monolithic-scintillator-based system, a realistic response on detector level needs to be modeled. For this purpose, an experimentally
determined PSF is used to smear the simulated interaction locations within the monolithic detector. To further provide a three-dimensional PSF, DOI determination with a
realistic resolution is implemented. Under these conditions, a realistic resolution comparison between pixelated and monolithic detectors is performed. The pixelated scintillator
provides a resolution of 0.78 mm and the monolithic scintillator 0.90 mm. Regarding the
modelling of the monolithic detector’s response, it needs to be kept in mind, that the
implemented PSF only describes the response on the surface center of the scintillator
crystal. This is an optimistic approximation, since the detector response most likely
deviates at the margin of the crystal.
Additionally, the influence of physical effects, such as positron range and acollinearity,
are evaluated. Positron range and acollinearity combined degrade the spatial resolution
by about 30 %. Unfortunately, these two effects could not be examined separately due
to questionable results simulating a back-to-back source with an implemented angular
deviation representing the acollinearity. Moreover, the influence of DOI determination
in monolithic scintillators is analyzed. Without DOI determination, the system’s spatial
resolution degrades by 18 % in the center and by 26 % at the edge of the FOV, which is
caused by a greater parallax error. Interestingly, the implemented DOI determination is
able to almost entirely compensate for this effect at the edge of the FOV.
The last very influential factor affecting the spatial resolution of a PET system is
the image reconstruction software. The reconstruction software used is able to provide
51
6 Conclusion
a 20 % resolution recovery for pixelated detectors and a 30 % recovery for monolithic
detectors. Consequently, a realistic comparison of the mere detector performance is
difficult, since effects caused by the detector are not easily distinguishable from effects
caused by the reconstruction software. However, the simulation results of a scanner based
on monolithic scintillators serve well as an approximation for possible spatial resolution.
With respect to all results, we could draw the conclusion that the effort of calibrating
the monolithic detectors is not worthwhile regarding the spatial resolution of the system.
Pixelated detectors with a pixel size of 1 mm provide a better resolution and improving
the image reconstruction would probably be more beneficial.
52
7 Outlook
As previously described, the simulations in this thesis are conducted with approximated
models. Even though they provide quite realistic results in most cases, the models could
be further refined. For the detector response of the monolithic scintillator, a locationdependent PSF could be implemented for smearing the coordinates in order to include
the deviating response at the crystal margin. Concerning the actual read-out of the light
patterns formed in the monolithic scintillator, there is also still room for improvement:
the detector response currently degrades the spatial resolution by 0.32 mm (about 35 %).
Moreover, the PSF of a pixelated detector including detector scatter and crystal identification errors could be experimentally determined. Even though these effects are
partially simulated by GATE, they are not incorporated into the image reconstruction.
Using the PSF to determine a more realistic approximation of the system’s matrix could
obliterate the discrepancy between the performance of the reconstruction software versions for pixelated and monolithic detectors. This would potentially lead to a greater
resolution recovery during the image reconstruction. Furthermore, a more realistic simulation of the pixelated detectors could be provided, which would also allow a better
performance comparison between the two different detector types.
53
Abbreviations
ADC
Analogue-to-digital converter
APD
Avalanche photo-diode
CT
Computed tomography
DOI
Depth of interaction
DPC
Digital photon counter
FOV
Field of view
fMRI
Functional magnetic resonance imaging
FWHM
Full width half maximum
FWTM
Full width tenth maximum
LOR
Line of response
MRI
Magnetic resonance imaging
MRT
Magnetic resonance tomography
PET
Positron emission tomography
PMT
Photo-multiplier tube
PSF
Point-spread-function
PVR
Peak-to-valley ratio
SiPM
Silicon photo-multiplier
SNR
Signal-to-noise ratio
SPAD
Single-photon avalanche diode
SPECT
Single photon emission computed tomography
55
List of Figures
2.1
Comparison of brain scans with PET, CT and T2-weighted MRI and their
respective fused images. [A. Boss and Stegger, “Hybrid PET/MRI of Intracranial
Masses: Initial Experiences and Comparison to PET/CT”]
3.1
. . . . . . . . . . . . . . 14
Normalized photon energy spectrum generated in a simulation without
energy cut-off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2
Schematic representation of different possible event detections. A: True
coincidence. B: Random coincidence. C: Scattered coincidence. [Turkington, “Introduction to PET Instrumentation”]
3.3
. . . . . . . . . . . . . . . . . . . . 19
Projection profiles of an image of a single point source and two resolvable
point sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4
Model of various successive factors causing a degradation of the spatial
resolution. [Stickel and Cherry, “High-resolution PET detector design: modelling
components of intrinsic spatial resolution”]
3.5
. . . . . . . . . . . . . . . . . . . . . 22
Schematic representation of an annihilation process showing positron path,
positron range and acollinearity. . . . . . . . . . . . . . . . . . . . . . . . 23
3.6
Experimentally determined point-spread function of a monolithic LYSO
scintillator crystal. [Maas et al., “Monolithic scintillator PET detectors with intrinsic depth-of-interaction correction”]
3.7
. . . . . . . . . . . . . . . . . . . . . . . . . 25
Schematic visualization of the parallax error occurring without DOI determination. [Bailey et al., “PET Basic Science”] . . . . . . . . . . . . . . . . . 26
4.1
Sketch of scanner geometry with coordinate system. . . . . . . . . . . . . 29
4.2
Line profile of experimentally determined PSF [Maas et al., “Monolithic scintillator PET detectors with intrinsic depth-of-interaction correction”]
4.3
. . . . . . . . . 31
DOI resolution for different depths in a 12 mm deep monolithic crystal.
[Dam and Schaart, “A practical method for depth of interaction determination in monolithic scintillator PET detectors”]
. . . . . . . . . . . . . . . . . . . . . . . . . . 32
57
List of Figures
5.1
Dependency of the system’s spatial resolution (FWHM and FWTM) on
the varying width of the detector response function. . . . . . . . . . . . . 36
5.2
FWHM values of a binned Gaussian function (FWHM = 0.7 mm) with
varying starting points for the binning. . . . . . . . . . . . . . . . . . . . . 38
5.3
Line profiles of a point source and their Gaussian fits. . . . . . . . . . . . 40
5.4
Peak-to-valley ratios as a measure of the scanner’s ability to resolve two
point sources depending on the sources’ distance, comparing pixelated
and monolithic detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5
Sketch of a structure phantom, which consists of spherical sources with
different diameters and point sources. . . . . . . . . . . . . . . . . . . . . 46
5.6
Image of a structure phantom viewed in different planes, (a)-(c) acquired
with monolithic detectors and (d)-(f) with pixelated detector, all based
on an averaged gray scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.7
Line profiles of the phantom through the x- and the y- axis of the scanner
(average of the four simulation runs). . . . . . . . . . . . . . . . . . . . . . 48
58
Bibliography
A. Boss S. Bisdas, A. Kolb M. Hofmann U. Ernemann C. D. Claussen C. Pfannenberg
B. J. Pichler M. Reimold and L. Stegger (2010). “Hybrid PET/MRI of Intracranial
Masses: Initial Experiences and Comparison to PET/CT.” In: Journal of Nuclear
Medicine.
Badawi, R. D. (1998). “Aspects of Optimisation and Quantification in Three-Dimensional
Positron Emission Tomography.” Thesis. University of London.
Bailey, D. L. et al. (2005). Positron emission tomography: basic science. Suhrkamp.
Bihl, H. and H. Brummer (2010). Südwestdeutsches PET-Zentrum Stuttgart, PET and
SPECT. url: https://sites.google.com/site/germanpetcenterstuttgart/petand-spect-what-is-the-difference.
Cal-Gonzalez, J. et al. (2009). “Positron range effects in high resolution 3D PET imaging.” In: Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE,
pp. 2788–2791. doi: 10.1109/NSSMIC.2009.5401950.
Dam, H.T. van et al. (2010). “A Comprehensive Model of the Response of Silicon Photomultipliers.” In: Nuclear Science, IEEE Transactions on 57.4, pp. 2254–2266. issn:
0018-9499. doi: 10.1109/TNS.2010.2053048.
Dam S. Seifert, R. Vinke P. Dendooven H. Löhner F. J. Beekman H. T. van and D.
R. Schaart (2011). “A practical method for depth of interaction determination in
monolithic scintillator PET detectors.” In: Physics in Medicine and Biology 56.13,
p. 4135. url: http://stacks.iop.org/0031-9155/56/i=13/a=025.
Derenzo, S. E. (1979). “Precision measurement of annihilation point spread distributions for medically important positron emitters.” In: 5th International Conference on
Positron Annihilation.
Friston, K.J. et al. (1998). “Event-Related fMRI: Characterizing Differential Responses.”
In: NeuroImage 7.1, pp. 30 –40. issn: 1053-8119. doi: http://dx.doi.org/10.1006/
nimg.1997.0306. url: http://www.sciencedirect.com/science/article/pii/
S1053811997903062.
Gonzalez, R. C. and R. E. Woods (2002). Digital Image Processing, Second Edition.
59
Bibliography
J. T. Bushberg J. A. Seibert, E. M. Leidholdt Jr. J. M. Boone (2012). The essential
physics of medical imaging.
Johns, P C and M J Yaffe (1987). “X-ray characterisation of normal and neoplastic breast
tissues.” In: Physics in Medicine and Biology 32.6, p. 675. url: http://stacks.iop.
org/0031-9155/32/i=6/a=002.
K. Nagy M. Tóth, P. Major G. Patay G. Egri J. Häggkvist A. Varrone L. Farde C.
Halldin and B. Gulyás (2013). “Performance Evaluation of the Small-Animal nanoScan
PET/MRI System.” In: Journal of Nuclear Medicine.
Levin, Craig S and Edward J Hoffman (1999). “Calculation of positron range and its
effect on the fundamental limit of positron emission tomography system spatial resolution.” In: Physics in Medicine and Biology 44.3, p. 781. url: http://stacks.iop.
org/0031-9155/44/i=3/a=019.
Maas, Marnix C et al. (2009). “Monolithic scintillator PET detectors with intrinsic
depth-of-interaction correction.” In: Physics in Medicine and Biology 54.7, p. 1893.
url: http://stacks.iop.org/0031-9155/54/i=7/a=003.
Moses, William W. (2011). “Fundamental limits of spatial resolution in PET.” In: Nuclear
Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers,
Detectors and Associated Equipment 648, Supplement 1.0. NIMA4 th International
Conference on Imaging techniques in Subatomic Physics, Astrophysics, Medicine, Biology and Industry, S236 –S240. issn: 0168-9002. doi: http://dx.doi.org/10.1016/
j.nima.2010.11.092. url: http://www.sciencedirect.com/science/article/
pii/S0168900210026276.
S. Espana R. Marcinkowski, V. Keereman S. Vandenberghe and R. Van Holen (2014).
“DigiPET: sub-millimeter spatial resolution small-animal PET imaging using thin
monolithic scintillators.” In: Physics in Medicine and Biology 59.13, p. 3405. url:
http://stacks.iop.org/0031-9155/59/i=13/a=3405.
Smith, S. W. (1997). The Scientist and Engineer’s Guide to Digital Signal Processing.
Standards, National Institute of and Technology (2009). Physical Measurement Laboratory. url: http://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html.
Steén, M. and P. Uhlén (1998). “Development of a Time-of-Flight and 3D Demonstration
Set-up for PET.” Thesis. The Royal Institute of Technology.
Stickel, Jennifer R and Simon R Cherry (2005). “High-resolution PET detector design:
modelling components of intrinsic spatial resolution.” In: Physics in Medicine and
Biology 50.2, p. 179. url: http://stacks.iop.org/0031-9155/50/i=2/a=001.
Turkington, T. G. (March 2001). “Introduction to PET Instrumentation.” In: Journal of
Nuclear Medicine Technology.
60
Bibliography
Wehner, J et al. (2015). “MR-compatibility assessment of the first preclinical PET-MRI
insert equipped with digital silicon photomultipliers.” In: Physics in Medicine and
Biology 60.6, p. 2231. url: http://stacks.iop.org/0031-9155/60/i=6/a=2231.
Weissler, B. et al. (2012). “Design concept of world’s first preclinical PET/MR insert
with fully digital silicon photomultiplier technology.” In: Nuclear Science Symposium
and Medical Imaging Conference (NSS/MIC), 2012 IEEE, pp. 2113–2116. doi: 10.
1109/NSSMIC.2012.6551484.
61