Laurea Magistrale in Scienze Fisiche
An accurate system model for the
PET/CT IRIS pre-clinical scanner
Tesi di Laurea in Fisica Medica
Anno Accademico 2016/2017
Autore:
Introduction 1
1 Positron Emission Tomography 3
1.1 Review of the physics principles . . . 4
1.1.1 Positron emission and annihilation . . . 4
1.1.2 Photons interaction with matter . . . 6
1.1.3 Radiation detection . . . 8
1.2 The PET system . . . 11
1.2.1 Data acquisition . . . 13
1.2.2 Spatial resolution . . . 15
1.2.3 Noise in PET measurement . . . 16
1.2.4 Data Representation . . . 17
1.2.5 Data Correction . . . 18
1.2.6 Detector Normalization . . . 20
2 Image Reconstruction in PET 21 2.1 Analytical Methods . . . 22
2.1.1 Filtered Back-Projection . . . 24
2.1.2 Three dimensional Analytic Reconstruction . . . 24
2.2 Iterative Methods . . . 26
2.2.1 MLEM . . . 27
2.3 The System Models . . . 30
2.3.1 Geometric and Multi-ray Models . . . 32
3 The IRIS Scanner 33 3.1 Scanner description . . . 33
3.2 Scanner readout system . . . 33
3.3 Reconstruction methods . . . 35 i
4.2 Monte Carlo simulation . . . 38
4.2.1 Parameters of the simulation . . . 40
4.2.2 Events distribution . . . 43 4.3 Detector Matrix . . . 46 4.3.1 Limits Eects . . . 47 4.4 System Matrix . . . 48 5 Reconstruction Results 51 5.1 Model comparison . . . 51
5.1.1 Uniformity and noise . . . 55
5.1.2 Recovery coe cients . . . 57
6 Conclusions 63
Positron Emission Tomography (PET) is a nuclear medicine technique which al-lows determining the in-vivo spatial and temporal distribution of a radio-emitting compound inside the examined subject. The principle at the basis of a PET sys-tem is the use of a positron emitting radioactive tracer. The emitted positron annihilates with an electron in the surrounding tissue producing a back-to-back couple of 511 keV photons that could be detected.
Using tomographic reconstruction techniques it is possible to build a three-dimensional image of the location of the tracer inside the subject.
The reconstruction techniques can roughly be subdivided into two major groups, analytical and iterative methods. The iterative methods are usually pre-ferred over the analytical ones as they yield to a better quality of the images. This results is achieved by employing realistic models of the system at the expense of a high computational cost, concerning calculation time and memory requirements, and can be performed only on modern workstations. The system model (or sys-tem matrix) used by the iterative methods de nes the relationship between the object space and the measurement space, and it is often subdivided into fac-tors that can be computed separately to reduce the time needed to perform its construction.
The original contributions of this thesis consists in the creation and the vali-dation of an accurate factorised system model for the INVISCAN PET/CT IRIS scanner. The creation is performed using GEANT4 Monte Carlo simulation with particular attention on the optimization of the simulation parameters in order to obtain reasonable simulation time. The validation is carried out comparing the quality of the reconstructed images obtained with the new models and those avail-able with the commercial scanner. The measures are performed following NEMA NU 4-2008 standard for small animals, using real data acquired with the IRIS scanner installed at the Istituto di Fisiologia Clinica - CNR Pisa. The new models are made of two components, an analytical component called geometric matrix and a second element, called detector matrix, which models the interactions of the photons inside the detector modules to take into account the penetration and inter-crystal scatter eects that can occur in the detector modules.
Chapter 1 presents the positron emission tomography with an overview of the physical principles and key aspects of the data acquisition process.
Chapter 2 deals with the techniques used to reconstruct the images starting from the acquired data. It includes a general introduction to the problem, a brief description of the analytical methods and a description and the formulation of the most common iterative reconstruction algorithm. The last part of the Chapter explanis the methods used to create the system model.
Chapter 3 describes the hardware and software characteristics of the IRIS scanner.
Chapter 4 presents the creation of the IRIS factorised system model, the description of the Monte Carlo simulation used to construct the detector matrix and all the aspects of the construction of the nal system matrix.
Chapter 5 shows the results of the comparison between the new models and the models currently used to perform the images reconstruction within the IRIS scanner. This analysis is based on the NEMA NU 4-2008 protocol that prescribes the measurements to accomplish to evaluate the performance of a small animal scanner.
1
POSITRON EMISSION TOMOGRAPHY
Positron Emission Tomography (PET) is one of the most common molecular imaging technique used for the visual representation, characterization and quan-ti cation of biological processes that take place in a living being at the cellular and sub-cellular level [1].
It has applications both in clinical and pre-clinical studies. The clinical PET imaging is used mainly in three medical areas:
cancer diagnosis and management (localisation of tumours and metastases); cardiology and cardiac surgery (measurements of myocardial perfusion and
viability);
neurology and psychiatry (management of brain tumours, pre-surgical eval-uation of epilepsy, diagnosis of dementia).
In the eld of pre-clinical studies, PET is used as a research tool for drug dis-covery and development on small animals subjects such as mice and rats. The principle at the basis of a PET imaging system is the use of a positron emit-ting radioactive tracer (radiotracer) that is a chemical compound in which one or more atoms have been replaced by a radioactive isotope. The radiotracer follows the usual biochemistry of the compound and, by the study of the distribution of the activity of the radioisotope, it is possible to gather biochemical information about the tissue where the tracer has been localised. All the radioisotopes used in PET studies undergo β+-decay thus emitting a positron that annihilates in the
biological tissues with the emission of two photons. The detection of the photons is the key to determining the distribution of the radiotracer inside the patient.
The radiotracer to be used depends on the pathology and organ of interest, one of the most common is the Fluorodeoxyglucose (FDG). The FDG has a chemical structure close to the glucose one and a similar behaviour, it is taken up by high-glucose-using cells such as brain, kidney and by cancer cells. This
compound contains 18F that is a β+ emitter, so it could be used in combination
with a PET scanner to determine the location of cancer cells inside the patient body.
1.1 Review of the physics principles
The fundamental principle of a PET imaging system is the detection of the two photons that arises from the annihilation of a positron emitted inside the patient body by the decay of a radioactive tracer.
1.1.1 Positron emission and annihilation
The β+ decay is common when an atom has an excess of protons with respect to
the number of neutrons so that the nucleus can gain stability converting one of the exceeding protons into a neutron[2]:
ZX →Z−1Y + β++ νe
The kinetic energy of the recoiling nucleus can be neglected due to its mass hence the released energy is mostly shared between the positron and the antineutrino. The β+ spectrum of the most commons positron emitters radioisotopes used in
PET is shown if Fig. 1.1.
Figure 1.1: β+ spectrum of most used positron emitters radioisotopes as a
func-tion of the positron kinetic energy [1]
The beta emission is governed by the standard exponential law where the number of nucleus at a given time is
N (t) = N0e−
t τ
where N0 is the number of the nucleus at time t=0 and τ is the mean lifetime of
the isotope. The activity is de ned as the number of disintegration per second:
A(t) = dN (t)
dt = −λN (t) = 1 τN (t)
The time that is necessary to halve the number of atoms is called half-life and is obtained by:
T1/2 = τ · log(2)
The positron sources used in nuclear medicine are arti cially produced by target-ing stable isotopes with positively charged particles.
The most common used atoms for PET imaging are called physiological ra-dioisotopes because their corresponding stable atoms are main constituents of the human body or can easily replace some functional group. All of them have a short lifetime. This a favourable aspect because limits the decay of the compound to a narrow temporal window (in which the PET acquisition is performed) beyond which the decay is practically negligible. The amount of residual activity is thus reduced and it is possible to minimise the amount of tracer needed to perform the analysis. However, for very short lifetimes (as for15O) it is necessary to produce
the radiotracer just before the experiment, with an in-situ cyclotron.
After the emission, the positrons lose their energy mainly through multiple Coulomb interactions within the biological tissues and reach thermal equilibrium before annihilation. The range of the particle is the distance between the point of emission and the point of annihilation and depends on the density and the Z of the medium. In Table 1.1 are shown some physical properties of typical physiological radioisotopes. Half-Life (min) Positron average kinetic energy (MeV) Positron endpoint kinetic energy (MeV) Positron average range in water (mm) 11C 20.4 0.385 0.960 1.2 13N 10.0 0.491 1.198 1.6 15O 2.0 0.735 1.732 2.8 18F 109.8 0.242 0.633 0.6
Table 1.1: Physical properties of physiological radioisotopes. [1]
In the simplest approximation, where the annihilation takes place at-rest, the result is the emission of two back-to-back 511 keV photons to guarantee the conservation of energy and momentum. However, in practice, the annihilation is never at rest because the electron is bound to the atom and his energy cannot be neglected. In the reference frame of the centre of mass of the system the
process always leads to two back-to-back γ-rays, but in the laboratory reference frame is present a non-collinearity that brings to smaller angle of emission. It is also possible to have an annihilation in ight, for radioisotopes in water this process occurs ∼ 2% with respect to the total. Another possible process is the annihilation via 3γ but it is usually neglected because of its small cross section (σ3γ = σ3722γ σ2γ).
An extensive study [3] showed that the distribution of the non-collinearity of the annihilation process it is a convolution of two Gaussian curves, with dif-ferent σ. The narrower component is consistent with the annihilation of the free positron whereas the broader part can be explained by the formation of a metastable bound-state between a positron and an electron, called positronium [4]. This system has two minimum energy con gurations: para-positronium (sin-glet state) and ortho-positronium (triplet state). Both systems can decay via a self-annihilation or via pick-o, that is the annihilation of the positron with a free electron. A summary of the annihilation processes is shown in Tab. 1.2.
The angular deviation from collinearity together with a presence of a not null range are fundamental physical limits that degrade the spatial resolution in the reconstructed images.
State Annihilationprocess Comments Lifetime Ang. dev. non-bound
in- ight via 2γ
emission of order of 2% ∼ 1 ps narrow at rest via 2γ
emission standard PET situation ∼ 1 ns narrow at rest via 3γ
emission improbable
positronium
para-positronium self-annihilation
1/4 of the bound state preferred annihilation
for para-positronium
∼ 100 ps narrow para-positronium
pick-o improbable ∼ 1 ns narrow
ortho-positronium
self-annihilation via 3γ, it is anticipatedby pick-o ∼ 100 ns narrow ortho-positronium
pick-o 3/4 of the bound state ∼ 1 ns large
Table 1.2: Summary of the annihilation processes. [1]
1.1.2 Photons interaction with matter
The emitted photons can interact with matter in several ways like photoelectric absorption, Compton and Rayleigh scattering and pair production([5]).
The photoelectric eect is the interaction of the photons with an absorber atom in which the γ-ray completely disappear. In its place an energetic electron is emitted by the atom, called photoelectron, from one of its bound shells (typically K). This is a threshold eect and the electron is emitted only if the incident photon has energy (hν) greater than the electron binding energy (EB). The energy of the emitted particle is:
Ee = hv − Eb
A rough approximation of the probability of interaction is: σ ∝ Z
n
E3.5 y
n = 4 − 5
The Compton scattering is the interaction between a photon and an electron of the medium in which a portion of the gamma energy is transferred to the electron and the direction of the photon is changed. All angles of de ection are possibles, so the energy transferred could vary from 0 to a large amount of the energy of the incident photon. The probability of interaction depends linearly on Z and falls o with increasing energy.
The Rayleigh scattering is the elastic scattering of the photon o an atom, so there is no change in the energy of the incident particle but only a change in his propagation direction. The probability of the interaction is :
σ ∝ Z
2
E2
The pair-production is the creation of electron-positron pair by a photon in proximity of a nucleus. This could happen only if the energy of incident photon is greater than 1.022 MeV that is the sum of the rest mass of the two particles.
The interactions that play an important role in a PET system are only the pho-toelectric absorption and the Compton scattering, as the pair-production eect cannot happen because the photons emitted in an annihilation process have en-ergy below the threshold and the cross-section of the Rayleigh scattering is never dominant respect to the cross-sections of the others interactions (see Fig. 1.2 and 1.3).
Interaction inside the patient
In the biological tissues, for energies of 511 keV, the most important interaction that can occur is the Compton scattering, as shown in Fig 1.2, assuming the body as made of water. The mean free path for a photon of 511 keV in water is of the order of 10cm so there is a fraction of particles that are de ected from their initial direction. This fraction can be a consistent part of the total number of photons in some experiment (as brain PET). The interactions inside the patient degrade the nal image resolution as it is di cult to discriminate between scattered and unscattered photons (especially for elastic scattering).
100 101 102 103 Energy (keV) 10-6 10-4 10-2 100 102 104 Cross-section (cm2/g) Rayleigh scattering Compton scattering Photoelectric absorption
Figure 1.2: Photon cross-sections in water. [6]
Attenuation
If the photons are scattered to a direction outside of the Field of View of the scanner or are absorbed, the number of γ-rays that reach the detector is less than the number of emitted particles. This eect is called attenuation and for a collimated beam travelling along the x direction can be expressed by the formula:
I(x) = I0e−µx
where I(x) is the intensity after a distance x from the source, I0 is the intensity
at the source and µ is called linear attenuation coe cient and accounts for all kinds of interactions of the photon.
In most cases it is necessary to correct the registered photon counts taking into account the attenuation eect.
Interaction inside the detector
The detector is usually made of a high Z and high density material aiming to maximise the number of photons that interact with photoelectric absorption. If the energy deposition is localised to a small volume, it is possible to better determinate the position where the photons interacted. This aects both the spatial resolution and the reconstructed image noise. In Fig. 1.3 are shown the cross-sections of the interactions for two typical detector materials.
1.1.3 Radiation detection
After escaping from the biological tissue, the photons travel through the air and then reach the detector where they can be revealed. A single PET detector is usually composed of a matrix of scintillators coupled to photodetectors and an electronic read-out system.
100 101 102 103 Energy (keV) 10-3 10-2 10-1 100 101 102 103 104 Cross-section (cm2/g) Rayleigh scattering Compton scattering Photoelectric absorption (a) NaI(Tl) 100 101 102 103 Energy (keV) 10-3 10-2 10-1 100 101 102 103 104 Cross-section (cm2/g) Rayleigh scattering Compton scattering Photoelectric absorption (b) LYSO(Ce)
Figure 1.3: Photon cross-sections in typical scintillation materials.[6]
Scintillators
A scintillator consists of a dense crystalline material that emits visible light when an incident photon interacts with its constituents, the emitted light can be mea-sured by the coupled photodetector and is proportional to the deposited energy. Scintillators are available in the form of organic or inorganic and can be in solid or liquid state. The most common form used in nuclear medicine is a solid inorganic material.
An inorganic crystal scintillator is usually formed by adding impurity to a pure crystal with the aim of changing its energy levels. Without the impurities the electrons of the material can be only bound to the lattice sites (like the valence band) or free to move throughout the crystal in the conduction band. With the absorption of energy, an electron can be elevated from the valence band to the conduction band and then the gap in the lattice is rapidly lled be another electron with the emission of another photon. The value of the energy gap usually falls in the range of energies of the U.V. light, so it cannot be revealed by the photodetector that is capable of detecting only light in the visible portion of the e.m. spectrum. Moreover, the crystal is not transparent to that wavelength. The eect adding impurities is to create new energy levels in the forbidden gap to allow the process of uorescence. In this way the emitted photons have typical energy in the visible spectrum and could travel across the scintillator without being reabsorbed until they reach the coupled photocathode.
An ideal scintillator for PET measurements should have:
good stopping power for photons of energy of 511 keV to maximise the number of photons that interact and deposit energy in the detector. short decay time from the excited state to let possible to discriminate
high number of photons emitted per unit energy (light output) to achieve good energy and timing resolution.
The properties of typical scintillators are shown in Tab. 1.3.
As the photon interaction is a probabilistic event, the particle may penetrate the scintillator and, if the direction of incidence is not orthogonal to the detector surface, an interaction may occur in a dierent crystal respect the one where the γ-ray rst entered. This issue is called penetration eect (Fig. 1.4). Moreover, the photoelectric absorptions are just a fraction of the total number of interactions and, due to Compton scattering, a photon can deposit its energy in several points of the detectors. This phenomenon is called inter-crystal scatter (Fig. 1.4). A high number of photons are thus misplaced due to these eects.
Figure 1.4: Graphical representation of the inter-crystal scatter and the pene-tration eect. These eects are important blurring parameters, causing errors when a photon does not deposit the most of its energy in the crystal where it has entered the scintillator.
Photodetector
The detection of the scintillation photons is usually performed with the use of a PhotoMultiplier Tube (PMT). A PMT is a device capable of converting visible light in an electrical pulse. In the simplest form, the PMT is made of a vacuum glass envelope containing a series of electrodes called dynodes. The inner part of the entrance window is coated with a thin layer of a material that easily emits electrons when energy is deposited on it, via the photoelectric eect. This part of the tube is called photocathode because it is kept at a negative potential to accelerate the electrons away from it. The probability of the electronic emission is called quantum e ciency and it is of the order of 15-25%. The electron is then accelerated toward a dynode, striking it and freeing more electrons. At each
Material (g/cmDensity3) Light yield Decay time (ns) µ511keV (cm−1) Photofrac-tion at 511 keV
Sodium iodide (NaI:Tl) 3.67 41000 230 0.34 17%
Bismuth germanate (BGO) 7.13 8200 300 0.96 40%
Lutetium oxyorthosilicate (LSO:Ce) 7.40 30000 40 0.87 32%
Lutetium yttrium oxyorthosilicate
(LYSO:Ce) 7.10 32000 40 0.82 30%
Gadolinium oxyorthosilicate (GSO:Ce) 6.71 8000 60 0.70 25%
Yttrium aluminum perovskite
(YAP:Ce) 5.37 ∼21000 27 0.46 4.2%
Lutetium aluminum peroovskite
(LuAP:Ce) 8.3 12000 18 0.95 30%
Barium uoride (BaF2) 4.89 1400 (fast)9500(slow) 630(slow)0.6(fast) 0.43
Lanthanum bromide (LaBr3: Ce) 5.08 63000 16 0.47 15%
Table 1.3: Properties of scintillation material used in PET. Data from [7, 8]
stage from the cathode to the anode more and more electrons are liberated at each dynode interaction, resulting in a current pulse that can be extracted from the glass envelope and measured.
In a PET detector, it is necessary to know the position in the crystal matrix where the initial photon had interacted, this information can be retrieved by the use of a special PMT con guration (block detector [9]) or with the use of more complex devices like Position sensitive PhotoMultiplier Tubes (PsPMT). In recent years the detection of scintillation photons along with the determination of the interaction position has been performed with the use of Silicon PhotoMultiplier (SiPM) [10].
1.2 The PET system
A standard PET system is made of a set of detectors positioned around the object under study to detect the pairs of emitted annihilation photons. The tomographic acquisition requires the collection of a full set of line integrals de ned by the possible line of ight obtained sampling the object along spatial and angular directions.
Two fundamentals concept are those of Line of Flight (LOF) and Line of Response (LOR). The LOF is the line in which the annihilation event happened ant it is de ned by the two γ-rays ight path. The LOR instead is the line
connecting the two crystal pixels in which the photons are eectively detected. In an ideal situation the two lines should be the same but in the real situation the LOR may be misplaced due to penetration and inter-crystal scatter eects (1.5).
Figure 1.5: Schematic representation of a PET scanner composed of two facing detector modules made of a matrix of scintillating crystals. The concept of pixel, LOR and LOF are highlighted.
There exist several detector arrangements that can properly scan the object. Similarly to SPECT, the detector could rotate around the object, so at least a pair of facing detectors is needed to acquire a full set of line integrals. A much more convenient arrangement is a ring geometry (or a polygonal ring approxima-tion) in which many LOR could be sampled simultaneously without any detector movement.
The intersection of the possible LORs between all detectors de nes the Field of View (FOV) of the scanner: in a ring geometry the FOV is a cylinder centred on the scanner axis. To increase the FOV size along the axial dimension, modern PET scanners have more than one ring of detectors creating a so-called multi-ring geometry.
Multi-rings PET systems are divided into two categories: 2D and 3D scanners. In a 2D PET, the acquisition of the coincidences between detectors belonging to dierent rings are forbidden, so the events recorded come from a single slice of the object. This simpli cation makes the acquisition system and the image recon-struction process easier and to further reduce the number of recorded unwanted single events the rings are often physically separated from each other by a sep-tum of high Z material. A 3D scanner instead can record inter-ring coincidences leading to an increased sensitivity despite a more complex image reconstruction and acquisition process.
1.2.1 Data acquisition
Exploiting the back-to-back emission, the non-collinearity is not considered for simplicity, it is possible to determine the line in which the annihilation of the positron occurred.
Every time a photon reaches a detector and interact with it, it is recorded as a single event along with a timestamp. As the two γ-rays are generated at the same time they should reach the detectors simultaneously. The dierence between the two timestamps is due to the dierent path length they travel before reaching the scintillators and to the nite timing resolutions of the detectors. To take into account this eects a pair of single events is identi ed as a coincident event if the timestamps dierence of the two photons is less than a de ned constant or, in other words, if it lies inside a so-called time window.
The time window is usually taken at least twice the timing resolution of the scanner, that is the ability of a pair of detectors to determine the time dierence in arrival of the annihilation photons. The timing resolution depends mostly on the properties of the scintillator and it is usually of the order of few nanoseconds [11].
The determination of the line where the annihilation event occurred exploiting the time coincidence window is called electronic collimation, for the analogy with the passive collimation used in others nuclear medicine imaging techniques such as scintigraphy or SPECT.
Moreover, a coincidence event is regarded as valid and recorded only if: the LOR is within a valid acceptance angle of the tomograph;
the energy deposited by each one of the detected events is inside a selected energy window.
The energy window is needed to reject object scattered events that have en-ergies dierent than 511 keV.
In the absence of any physical processes that could lead to shifting from the ideal response of the scanner, the total number of coincidence events detected will be proportional to the total amount of tracer contained in the line of response.
Despite this criteria, some of the recorded events are actually unwanted be-cause they descend from scattered photons or two photons coming from dierent annihilations.
Summarising, the various kind of events that could occur during a PET ac-quisition are (see Fig. 1.6):
true coincidence event: it derives from the detection of two photons, arising from a single annihilation event, which reach two facing detectors within the timing and energy windows without interacting with any other medium apart from the detector;
random coincidence event: it consists in the detection of two photons aris-ing from dierent annihilation events that enter the detector within the coincidence timing window;
(a) True (b) Scatter (c) Random Figure 1.6: Dierent kinds of coincidences.
scattered event: it occurs when one or both photons originated by a single annihilation event undergo scattering before entering the detector within the timing coincidence window. This event leads to a misplaced LOR. Since Compton scattering involves a loss of energy, some events may be neglected using a narrower energy window.
The sum of true, scattered and random events is usually called prompt counts, where only true counts contains useful information for the image reconstruction. The acquisition in PET can be performed in 2D or 3D mode (Fig. 1.7). In 2D mode, the coincidences events between detectors belonging to dierent rings are physically avoided by the interposition of absorber material between the adjacent rings or just discarded at acquisition time. The acquired data is thus limited to a set of direct planes, that are the imaging planes perpendicular to the scanner or patient axis. This has the advantage of restricting the number of recorded scatter and random events at the cost of a remarkable loss of sensitivity. To exploit all the potential of the electronic collimation it is possible to remove the septa and perform a fully 3D acquisition where all kinds of coincidences between opposing detectors are allowed, namely the acquisition spans on both direct planes as well as the LORs lying on oblique imaging planes that cross the direct ones.
It is important to note that both 2D and 3D PET imaging lead to 3D images, in the case of a 2D acquisition the various planes are stacked together to form a 3D volume (Multi-sliced Imaging or 3D-MS). The increased sensitivity of the 3D mode has a lot of advantages as it contributes to a higher SNR and so to decrease the radiotracer amount needed to perform the scan, but goes to the detriment of the complexity of the reconstruction task, both in terms of data representation and storage and in terms of the image reconstruction process.
Figure 1.7: Comparison of fully 2D and fully 3D PET acquisition modes. Figure taken from A. Alessio and P. Kinahan, PET Image Reconstruction [12].
1.2.2 Spatial resolution
The spatial resolution of a system represents its ability to distinguish between two points after image reconstruction. In a PET scanner it is usually expressed as the FWHM of the three-dimensional Point Spread Function (PSF) obtained as the quadrature combination of several terms. In general, the spatial resolution is not constant along the whole FOV and the three-dimensional PSF is not isotropic. This is caused by the fact that the scanner does not sample all the LOR in the same way. The degradation of the spatial resolution is essentially related to the uncertainty in the determination of the LOR that depends on the detector geometry, the physics of the β+ and on others aspects related to the detection
process ant the technology used.
The β+ aspects are related to the positron range in tissues and the
non-collinearity of the emitted photons.
Being rms the root-mean-square of the range distribution, the contribution to the FWHM of the spatial resolution is
r = 2.35 ∗ rms
and the contribution of the non collinearity is represented by:
F W HM ∼ ∆θD
4 ' 0.0022D
where ∆θ is the mean angle of non collinearity (in water ∆θ ∼ 0.5◦ [1]) and D is
To investigate the geometrical aspects lets consider a ring geometry and a pixellated detector module. When a coincidence is detected the LOR is de ned by the pair of crystals that detected the singles events. Due to the nite size of the crystals, the LOR is represented by a region named tube of response. From simple geometrical consideration, there is a higher probability for the γ-rays to have been generated in the centre of the tube of response. The position of the annihilation is then known with an uncertainty that has a FWHM equal to half the size of the crystal. Exactly the distance refers to the crystal pitch which is the distance between the centre of two consecutive crystals (considering the thickness of the optically re ective material between two scintillators).
F W HM = d 2
When the crystals are not facing each other, the parallax error arises and it is usually indicated with p.
Assuming that both crystals elements have been correctly identi ed as the region of the detector where the rst interaction of the two photons occurred, the geometrical and parallax errors are the only two contributions to the spatial resolution that do not arise directly from the physics of the annihilation process. However, some errors could also occur in the identi cation of the crystals. The crystals are usually identi ed by the photodetector via a light sharing tech-nique that calculates the centroid of the light spot emerging from the crystal. Penetration eects and the eect of multiple interactions in more than one crys-tal (inter-cryscrys-tal scattering, ICS) are included in the so-called coding errors term that is usually indicated by the letter b.
Summarizing the best achievable spatial resolution in a PET system can be expressed as: F W HM = rec · s d 2 2 + (0.0022D)2+ p2+ b2+ r2
where rec is a term related to the further degradation of the PSF due to the image reconstruction process.
1.2.3 Noise in PET measurement
The estimation of the FWHM of the spatial resolution does not include eects from noise. The detection e ciency refers to the e ciency with which a radiation measuring instrument converts the emission from the radiation source to a useful signal. To obtain maximum information with a minimum amount of injected radioactivity is desirable the maximum possible detection e ciency.
The detection e ciency can be de ned as D = g · · f · F where
g: geometrical e ciency, that is the solid angle coverage of the PET ring. It can be increased by reducing the ring diameter or increasing the axial dimension. The former is limited by the patient size and the latter by the building cost.
: intrinsic detection e ciency, it is obtained as the product of the detectors detection e ciency of the two detectors involved in the 511 keV photons detection process. This could be elevated by increasing the crystal thickness but it goes to the detriment of the parallax error.
f: electronic recording e ciency.
F: scatter and absorption into the object.
The noise of a PET scanner is usually measured in terms of the standard deviation of the uniformity measurement, that is the measurement performed on a region of uniform activity.
In addition to the statistical noise the uniformity is limited also by the presence of the scattered and random coincidences.
1.2.4 Data Representation
With the term data representation is indicated the way the information obtained from the scanner is stored into a digital form to be used for the image reconstruc-tion.
Raw data from the scanner are usually stored in a format called list-mode. This format includes for each measurement coordinates of the pixels de ning the LOR, the energies and the timestamps. Additionally, more information can be stored, like depth-of-interaction (DOI) in crystals.
The list-mode format is the natural way to store PET data but it is not convenient in terms of occupied computer space disk and of using the data for the image reconstruction process, especially in 3D mode acquisitions where the sensitivity is very high and many events are recorded.
The events, so, are processed and those meeting certain criteria, falling inside the energy and timing windows, are stored. To further reduce the amount of saved data, the list-mode format is substituted with the matrix-based format, often called histogram, in which the only information stored is the LOR and the number of events (counts) belonging to that LOR that meet the criteria. All other information is lost.
The count number stored for each LOR is proportional the line-integral of the activity along the LOR itself, being i and j the crystals de ned the LOR:
Nij = k
Z
LORij
ρ(x, y, z)dL (1.1) This equation is of fundamental importance for the process of image reconstruc-tion as it will be shown in Chapter 2.
A common way of storing the histogram is the so-called sinogram in which, apart from storing the counts, each LOR is represented by a set of physical coordinates. In the simpli cation of a 2D scanner, in a single plane ad a distance z from the centre of the scanner, a LOR is identi ed by two coordinates, s and φ. The s coordinate represents the geometrical distance from the axis and φ represents the inclination. So, in 2D PET each LOR is identi ed by the set (s, φ, z), while in 3D PET a fourth coordinate is needed to account for the inter-rings LORs.
1.2.5 Data Correction
During the acquisition the count number of the dierent LORs is altered by the presence of scattered and random coincidences. Therefore the data need to be corrected to recover the right count number.
Scatter Correction
The scatter correction consists in estimating the number of scatter counts con-tributing to a given LOR as a result of a Compton interaction. One of the sim-plest modellable situation is under the single scattering approximation in which it is assumed that only one Compton scattering occurs to only one of the two coincidences detected photons along their path.
Many LOFs may generate a scatter count to a given LOR and, in addition, the LOFs could either correspond to an acceptable LOR (emission point within the FOV) or may be generated from activity outside the FOV.
Random Correction
Apart from limiting the occurrence of random events by reducing the time win-dow, it is also possible to apply a correction for random coincidences in the re-constructed image. This process is based on an estimation of the random counts distribution that may be either stored in a separate histogram or subtracted from the prompt counts.
The estimation can be direct or indirect. In the latter case the count can be derived from the single count rate of each detector with the use of the formula:
Rij = Ci· Cj · 2τ
where C is the count rate in the detector and τ is the duration of the timing window.
A better estimation that is not aected by the systematic error in the a priori estimation of τ is obtained with the method of the delayed window [19]. In this approach, the logic pulse from one detector is delayed in such a way that it cannot be correlated with the other photon of the annihilation pair. With the use of another coincidence processor the coincidences between the delayed pulse from detector i and the prompt pulse from detector j are counted in Rij. At the end
of the acquisition process, this count represents a good estimation of the random coincidences occurred for each LOR.
Attenuation Correction
For photons of 511 keV, the attenuation coe cient µ in soft tissues is ∼ 0.096cm−1
while in bones is ∼ 0.17cm−1, there is so a relatively high probability for a γ-ray
to interact inside the patient before reaching the detectors. The line-integral in eq. 1.1 should be modi ed in order to include the eect of the attenuation:
Nij = k
Z
LORij
ρ(x, y, z)dL · Pi· Pj (1.2)
where
i and j represents the crystal indexes Nij is the number of counts in LORij
Pi = e −R
Liµ(x)dxis the probability of one of the photons to reach the detector
i, along the path Li between the annihilation point and detector
Pj = e −R
Ljµ(x)dx is the probability of the other photon to reach the detector
j, along the path Lj between the annihilation point and detector
The probability P for both the γ −rays to reach the detector could be written as:
P = Pi· Pj = e− R
Lµ(x)dx with L = L
i+ Lj
and it does not depend on the position of the annihilation point along the LOR but only on the line integral of the attenuation coe cient along the LOR itself. The quantity
1 P = e
R
Lµ(x)dx
is usually called the attenuation correction factor (ACF). For example, a LOR crossing 10 cm of soft tissue has an ACF value of ∼ 3.
The attenuation has two eect on the nal image: counts are underestimated, so highly aecting quantitative imaging, and image artefacts could arise due to the dierent length of tissue crossed or to the presence of strong attenuation factors like bones.
In the past years, the most common approach used to derive the ACFs in a PET scanner for each LOR is to directly measure µ(x, y, z) in the so-called transmission scan that can be performed with 511 keV coincidence photons. This method consists in comparing the counts recorded in a certain line in the presence of the object (transmission scan) to those obtained without the object (blank scan). Most modern scanners, in addition to the PET acquisition, have the possibility to acquire and process the information of a CT transmission scan (Computed Tomography) and the CT data provide a fast source for the correction of photon attenuation in PET emission data [13].
1.2.6 Detector Normalization
The lines of response of the scanner have dierent sensitivity for a variety of rea-sons, such as variations in the detector electronics e ciency (as detector thresh-olds), crystal size and light yield and geometric factors (as solid angle subtended). It is necessary to know this variation to reconstruct artifact-free images and to perform quantitative studies.
The process of correcting for these eects is known as normalization and the individual correction factors of each LOR are known as normalization coe cients [1, 14]. These coe cients are experimentally derived from the scan of a geomet-rically uniform positron source, like a planar object or a cylinder. The simplest possible approach to correct the data is known as direct normalization: after a correction for non-uniform radial illumination has been applied, the normaliza-tion coe cients are assumed to be propornormaliza-tional to the inverse of the counts in each LOR [14].
The main problems of this approach are:
to obtain a low statistical noise normalization factors, it is necessary to collect a high number of counts in each LOR. As the source used usually has a relatively low activity density to limit excessive dead time and high random count rate, the scan times are quite long, typically several hours; the source used must have a very uniform activity concentration or the
resultant coe cient will be biased;
the amount of scatter events and its distribution in the normalization scan may be substantially dierent from that encountered in normal imaging and this can result in bias and possible artefacts.
2
IMAGE RECONSTRUCTION IN PET
The image reconstruction algorithms in PET are divided between two categories: analytic and iterative [1, 12, 15]. Each category has its advantages and, for practical use, the choice of the algorithm to use depends on the application and often on the resources used to accomplish the task.
The dierence between the methods is deeper and relies on the way the data values are modelled, either as deterministic or stochastic (random) variables [12]. The simplest approach is to consider the data as deterministic, so containing no statistical noise. Within this assumption the noise is a deterministic number and if known it is possible to recover the exact original image from the data. The analytic methods exploit this premise using direct mathematical tools to calculate the original image.
In reality, the measured data are intrinsically stochastic due to the physical aspects of PET like positron decay process, attenuation, interaction within the patient (scatter and random events) and photon detection process. The noise in the image is thus more accurately represented as a random variable making impossible to nd an exact solution for the reconstruction problem. The iterative methods aim to solve the problem through a series of successive approximation using a more realistic model of the system.
Projections
Most image reconstruction algorithms start from the concept of projections. For simplicity let ρ(x, y) be a density function in the x-y plane, the projection is de ned as the line integral of the density distribution ρ along a particular line:
p(s, φ) = Z +∞
−∞
ρ(s · cosφ − t · sinφ, s · sinφ + t · cosφ)dt (2.1) For each angle the projections correspond to a line pro le. Putting together all the pro les for every angle leads to the sinogram, that is usually represented as
an image with the s coordinate along the abscissa and φ along the ordinate where each row of the image represents a projection.
Figure 2.1: Example of a projection of a density function f(x,y) and its respective sinogram. Figure taken from A. Alessio and P. Kinahan, PET Image Recon-struction [12].
The analytical process of transforming the object into its projections to create a set of sinograms, one for each plane, is called Radon transform. The links between the Radon transform and the PET acquisition are:
the data recorded by the scanner, after the appropriate corrections, corre-spond exactly to the projections
the integration lines along which the projections are computed, correspond to the lines of response of the scanner.
2.1 Analytical Methods
The analytical methods are the fastest and very suitable for the reconstruction of data from a 2D-mode acquisition, but the limiting factors are that the exact solution has to be discretized and that, as said before, the measured data are treated as noiseless.
Back-projections
A way to nd the original image is to use the back-projection operator, that is de ned as:
b(x, y) = Z π
0
This operation is not the analytic inverse of the projection but consists, in prac-tice, in re-distributing the activity, recorded by the line integral, along with the integration path. Because the exact location of the source is not known the ac-tivity value is uniformly distributed along the whole path. By repeating this procedure for all the angles between 0 and 2π, using the superposition eect, the result obtained is an image close to the original one with a lot of blurring in correspondence to the activity spikes. It can be demonstrated that for an in nite number of angular projections and in nite spatial sampling, the back-projected image b(r, φ) (in polar coordinates) is equivalent to the original image ρ(r, φ) convolved with a 1/r function:
b(r, φ) = ρ(r, φ) ⊗ 1 r (2.3)
Central Slice Theorem
This eect can be simply understood by introducing the central slice theorem. In two dimensions it states that 1D Fourier transform of the projections p(s, φ), respect to the variable s for a xed angle φ, is equal to a slice through the centre, with the same angle, of the 2D Fourier transform of the distribution function ρ(x, y).
F1[p(s, φ)] = P (vs, φ) = F (vscosφ, vssinφ) = F2[ρ(x, y)] (2.4)
where vs is the conjugate Fourier variable of s.
Figure 2.2: Illustration of the two-dimensional central-section theorem, where f (x, y) stands for ρ(x, y). Figure taken from A. Alessio and P. Kinahan, PET Image Reconstruction [12].
φ < π, it is possible to reconstruct the whole F (vx, vy) and then the inverse 2D
Fourier transform of F (vx, vy)will give back the ρ(x, y).
2.1.1 Filtered Back-Projection
One of the most common algorithm that use this approach is the so called Filtered Back-Projection, or FBP, that can be summarized by these steps:
Unidimensional Fourier Transform of each projection.
Filtering each projection in the Fourier space by multiplying by the ramp function.
Inverse unidimensional Fourier transform of each ltered projection. Projection backward the ltered projections.
Mathematically its expression is:
ρ(x, y) = Z π
0
pF(s, φ)dφ (2.5) where pF represent the ltered projection:
pF(s, φ) = F1−1[F1[ρ(s, φ)] · |vs|] (2.6)
In the real situation, the results of the FBP reconstruction are not exact due to limited angular and spatial sampling and the presence noise. When angular sampling is not in nite, the image is aected by a common star artefact since, during the back-projections, counts are accumulated preferably along angles of projections.
The ramp lter ampli es the higher frequencies, causing noise ampli cation. To reduce these eect low pass lter are used.
2.1.2 Three dimensional Analytic Reconstruction
In the 2D acquisition mode, the data acquired can be stored in the form of a sinogram p(s, φ) that is a set of 1D parallel projections of the object f(x, y), orthogonal to the scanner axis, for a set of orientation φ ∈ [0, π].
In the same way, the LORs measured in a 3D acquisition can be grouped into a set of lines parallel to a direction speci ed by a unit vector
~n = (−cosθsinφ, cosθcosφ, sinφ)
where the angle θ represents the angle between the LOR and the trans-axial plane, so that the data for θ = 0 correspond to a 2D acquisition.
The set of line integrals parallel to ~n de nes the 2D parallel projection of the tracer distribution
p(~s, ~n) = Z
f (~s + t~n)dt
where the position of the line is speci ed by the vector ~s ∈ ~n⊥ where ~n⊥ is the
projection plane orthogonal to ~n.
In the case of a cylindrical scanner with more than one ring (see Fig. 2.3), assuming continuous sampling, the scanner samples all the LORs such that the line de ned by (~s, ~n) has two intersections with the lateral surface of the scanner that correspond to two detectors in coincidence.
For each θ 6= 0, not all the LORs parallel to ~n that cross the FOV of the scanner are measured because they have only one intersection with the lateral surface of the scanner and cannot be measured in coincidence. In this case, a parallel projection is measured only for a subset of LORs, and it is called a truncated projection[14]. The existence of this problem is sometimes referred to the lack of shift-invariance of the scanner in the 3D acquisition mode.
(a) 2D parallel projections, same as 3D
parallel projection for θ = 0. (b) 3D parallel projection for θ 6= 0. The pro-jection is truncated because there exist lines intercepting only one of the border crystals (dashed lines).
Figure 2.3: Graphical representation of non-truncated and truncated projection; a ring is de ned by each pair of facing crystals.
A positive aspect of this acquisition method is instead the fact that from an analytical point of view the parallel planes are enough to perform the reconstruc-tion and so the data are redundant.
A formulation of the FBP algorithm is possible also for a 3D data set, but its application can be performed only in the presence of non-truncated projections. Several dierent analytical techniques have been developed to solve this prob-lem, the main approaches are those of data re-binning and the re-projection al-gorithm. The simplest of the re-binning methods is the Single-Slice Re-binning
Algorithm (SSRB)[16], in which, after having detected a coincidence event, the average of the two photons axial coordinates is calculated and the event is placed in the closest sinogram. This method oers a good image quality only if the scanned object occupies a small fraction of the FOV.
In some application, where the object occupies a small fraction of the FOV, it is possible to restrict the axial angle range to use only non-truncated projections and perform a direct 3D-FBP. However, this is usually not helpful for clinical and pre-clinical application as the subject occupies the entire axial length of the scanner [12, 14]
2.2 Iterative Methods
Iterative algorithms oer a dierent approach respect to the analytical methods improving the image quality by the use of a realistic model of the system. This model accounts of a lot of physical aspects like detector geometry and properties, photon interaction within patient and detectors, positron range and 2γ's non-collinearity and the stochastic nature of the PET events. The iterative techniques are thus able to reduce the noise in the reconstructed image, but disadvantaging the computational complexity of the whole procedure.
All iterative methods comprise some basics components:
model for the image: this is usually a discretization of the image domain into a series of pixels (2D) or voxels (3D). Usually, the shape of these objects is rectangular but also models with spherical elements (blobs) have been proposed [17].
An object used to relate the image (in the form of activity distribution) to the data (in the form of projections). This object is usually called system model, and it is related to the other variables by the linear expression:
n = S · ρ (2.7)
where n are the measured data, ρ are the activity values and S is the system model.
As we're dealing with a linear expression, the various objects can be ex-pressed within standard algebra tools, such as vectors and matrices. The projections are modelled as a vector n ∈ RN where N is the number of
LORS of the scanner. The activity is represented by a vector ρ ∈ RM
where M is the number of voxels of the FOV. The system model is repre-sented as a matrix S ∈ RN ×M. This fact explains why the terms system
model and system matrix are usually used as synonyms.
The element ni ∈ n represents the number of counts in the LOR i (with
i = 1, 2...N), the element ρj ∈ ρ represents the activity of the voxel j
probability that two γ-rays emitted by the annihilation of a positron in the voxel j are recorded into the LOR i.
Model for the data: it is a statistical relationship between the measured values and the expected values of the measurements. In other words, this model describes how the measured values vary around their mean value and it is derived from the physical understanding of the acquisition process. In most algorithms, as photons detection is Poisson distributed, a Poisson model is used, although after the scatter, random and attenuation correction the data is no longer distributed in this way.
2.2.1 MLEM
The most popular of these algorithms is the Maximum Likelihood Expectation Maximisation (ML-EM), it was introduced by L. A. Shepp and Y. Vardi in 1982 [18] and it also represents the foundation for many other iterative algorithms.
The ML-EM method merges the concepts of maximum likelihood estimation and expectation maximization. Let Y1...Yn be n independent random variables
[19] with probability density function fi(yi; θ) depending on a vector parameter
θ. The joint probability density of n independent observations y = (y1, ..., yn) is:
fi(y; θ) = n
Y
i=1
fi(yi; θ) = L(θ, y)
This expression, viewed as a function of the unknown parameter θ given the data y is called the likelihood function. Often, instead of this relation, is used the so-called log-likelihood function:
logL(θ; y) =
n
X
i=1
logfi(yi; θ)
A sensible way to estimate the parameter θ given the data y is to maximize the likelihood function (or the log-likelihood), choosing the parameter values that makes the data actually observed as likely possible.
It is often di cult to manage the likelihood function and the solution of the problem is obtained with the iterative expectation maximization algorithm. It is composed of two steps:
Expectation step (E step): calculate the expectation value of the likelihood (or log-likelihood) function, given the data y under the current estimate of the parameter θ.
Maximization step (M step): nd the parameter value that maximize the expected likelihood.
For an imaging system the activity value ρ is the unknown parameter and the projections n represent the measured data. Each projection ni, along the LORi,
is the summation of the number of recorded photons arising from the voxels ρj
crossed by the LORi:
ni =
X
j
nij (2.8)
The value nij is the number of recorded photons for each voxel and is a random
variable with a Poisson distribution wwith expectation values given from the de nition of the system matrix 2.7:
E(nij) = Sijρj (2.9)
The probability mass function for a Poisson distribution of parameter λ is P (p; λ) = e−λλ
p
p! (2.10)
where λ is the expected values of the random variable.
Thus, the probability of detection of nij photons expecting a value E(nij)is:
P (nij; E(nij)) = e−E(nij) E(nij)nij nij! = e−Sijρj(Sijρj) nij nij! (2.11) It is possible to set-up the log-likelihood function as the logarithm of the joint probability mass function of all the random variables nij:
log L(ρ) = logY i,j e−Sijρj(Sijρj) nij nij! =X i,j [nijlog(Sijρj) − Sijρj] − X i,j log(nij!) (2.12) the last term does not contain the parameter ρj to be estimated and can be
neglected without changing the maximization problem.
The objective function contains the random variable nij, the E step of the
algorithm consists in replacing the random variable by its expected values using the measured values ni and the current estimate of the parameter ρ:
E(nij) = ni
Sijρcurj
P
kSikρcurk
(2.13) The expected likelihood becomes:
log EL(ρ) =X i,j ni Sijρcurj P kSikρcurk log(Sijρj) − Sijρj (2.14)
The M step of the algorithm is the maximization of the EL function by taking the derivative respect to the parameter ρj and setting the derivative to zero:
∂EL ∂ρj =X i ni Sijρcurj P kSikρcurk Sij Sijρj − Sij = 1 ρj X i ni Sijρcurj P kSikρcurk −X j Sij (2.15)
Solving for ρj and writing it in terms of iterations: ρ(k+1)j = ρ (k) j PN i=1Sij N X i=1 Sij ni PM j=1ρ (k) j Sij (2.16)
Figure 2.4: Schematic representation of a single iteration of the ML-EM algo-rithm. Figure adapted from A. Alessio and P. Kinahan, PET Image Reconstruc-tion [12].
The single iteration of the ML-EM algorithm (see Fig. 2.4) starts with an initial distribution ρ(0)
j usually chosen uniform and non-negative and then is
com-posed of four steps:
forward projection of the current image: gives an estimation of values of the projections on the basis of the current image;
comparison between the estimated projections and the measurements: gives a multiplicative correction factor for each projection;
back-projection of the corrected projections into the image domain: gives a multiplicative correction factor for each voxel;
correction of the image by a weighting term based on the system matrix. The process is then repeated until the estimate reaches the maximum likeli-hood solution. The stopping point of the reconstruction is usually chosen empir-ically when the obtained image is the best trade-o between spatial resolution and noise.
2.3 The System Models
The aim of the image reconstruction process is to obtain a nal image as close as possible to the object under investigation. Using an iterative algorithm, the system model represents one of the most critical components and must be as accurate as possible by including all physical aspects of the image formation and acquisition. There are three main methodologies used to directly obtain the whole model: experimental, Monte Carlo (MC) and analytical [20].
The empirical approach is the most accurate way under ideal experimental con-ditions [21]. Each element of the system matrix could be obtained by positioning a radioactive point source at every voxel of the FOV and recording the scanner response. Unfortunately, this approach is also the most challenging one due to complicated experimental setup and time needed to perform the full acquisition. For example, [22] make a temporal estimation of 2.6 years to complete the system matrix acquisition of a clinical scanner, even after assuming scanner symmetries. An alternative way to the direct measurement of the system matrix is to re-produce the experimental source scanning process with a Monte Carlo (MC) simulation. Although the feasibility and bene ts of this method have been shown in several works [20, 23], the simulated acquisition has some issues that need to be discussed:
current MC software packages provide a very accurate modelling of the par-ticle transport in the object and the scanner. However some aspects of the acquisition process can not be correctly modelled (light propagation eect, electronic noise, light guides, magnetic shields..). Therefore the accuracy of the MC model will be less than the experimental one.
the time needed to perform the simulation is proportional to the number of events and a high number is required to reduce the inherent statistical noise of the method. As the number of elements of the system matrix is usually considerable, with standard computation power, the total calculation could be unworkable.
The last methodology relies on the analytical implementation of the system model that, unlike the MC models, is noise free and usually a lot faster, as fast that often the elements of the matrix can be calculated on the y during the reconstruction process without the need to store them in memory. However, these methods disregard important physical aspects of the PET acquisition chain. A well-known analytical method is the Siddon's ray-tracer [24] where the value of each system matrix element is calculated as simple geometric consideration like the length of the intersection between a speci c LOR and voxels of the FOV. The geometric approximation can be further improved by using multiple-ray-tracing schemes [25] [21].
For those approaches in which the calculation is slow, it is a better choice to store the matrix into the computer memory to faster retrieve it when needed. The size of the whole model can exceed easily hundred of TB and often it is necessary to load it entirely into the central memory of the computer to perform fast image reconstruction. The size is reduced using the symmetries of the scanner and the sparseness of the matrix. The number of non-null elements is usually very low with respect to the total number (order of 0.1%). Therefore it is possible to save a lot of space saving only these entries to disk. The whole reconstruction framework thus relies on sparse matrix algorithms [21]. Even using this expedient the size of the matrix can exceed tens of GB and additional compression methods that take advantage of the quasi-symmetries of the system have been proposed [26].
Hybrid models
Instead of computing the whole system matrix with a single technique it is possi-ble to model each aspect of the PET acquisition chain with dierent approaches and later assembly the various sub-models into the nal model.
The bene ts of the eects separation rely on the fact that a homogeneous treatment for all eects that can occur during a PET acquisition may result in a non-optimal calculation. By modelling each eect with the approach that best ts its requirements, it is possible to reach a high accuracy at a reasonable computation eort.
For example, it is not worth to perform extensive experimental measures or simulation for those aspects which responses have accepted analytical noise-free model available. This category includes the modelling of the geometrical fac-tors but also other physical aspects such photon non-collinearity which angle of deviation is well known to have a Gaussian distribution.
In cases where the eects are di cult to model with analytical approaches, i.e. scattering and penetration within the detectors, the empirical model should be preferred. Usually, the empirical models are substituted by MC simulation due to the complexity of the measurements.
The approach of eects separation is gaining more and more attention due to its simplicity relative to full experimental or full MC models and its exibility in terms of dierent scanner con gurations. Furthermore, with this approach, it is possible to concentrate the eort on the aspects that are most important for the system under study. The full body human studies usually are more concerned with attenuation and scatter in the object whereas in small animal or dedicated studies, like brain PET imaging, the greater eort is typically made to model the detector aspects responsible for the penetration and inter-crystal scatter eects.
System Matrix factorization The most common way to combine the mod-els is to express the system matrix as a multiplication of independent matrices,
each one standing for one or a set of eects, that can be calculated and stored separately [27].
A general decomposition is:
S = N · A · D · G · R (2.17) where
N is a N × N diagonal matrix that contains the normalization factors for each LOR.
A is a N ×N diagonal matrix with the attenuation correction for each LOR. D is a N × N matrix which models the detector response to the incident radiation; precisely it contains the correlation between the LOFs and the LORs. The element dij of the matrix is the probability that the photons
that travel along the LOF j are assigned to the LOR i.
G is a N × M matrix that contains the correlation between the emission in a given voxel and the LOFs of the scanner. The element gjk represents
the probability that the photons of a back-to-back pair emitted by the annihilation of a positron in the voxel k reach the surface of the crystals which identify the LOF j
R is a M × M matrix which models the positron range eect. The element rkl is the probability that the positron which annihilates in the voxel l was
emitted in the voxel k. This component could be calculated using a Monte Carlo simulation [28].
Multi-ray Model As an alternative to the matrix factorization, several works combine the dierent models in a scheme of multivariate integrals [25].
In this work, a factorised model for the IRIS scanner has been computed, and it has been compared with a simple analytical geometric model and the analytical multi-ray model currently used by the IRIS reconstruction software.
2.3.1 Geometric and Multi-ray Models
The multi-ray method incorporates an accurate geometric model along with crys-tal depth eects. The method is based on the tracing of multiple rays from inte-gration points inside one detector crystal through the image volume space toward the integrations points in the opposite detector to model the LORs.
If integration points are taken only on the pixel surface, discarding the crys-tal depth and interaction eects (black detector), the object obtained is a pure analytical geometric model, hereafter indicated as G or simply geometric ma-trix/model.
The full multi-ray model will be instead denoted simply as Multiray. For a complete derivation of the method refer to Moehrs et al - 2008 [25].
3
THE IRIS SCANNER
The scanner used in this thesis is the IRIS PET/CT distributed by the Inviscan s.a.s (France) and it is commercially available as a solution for pre-clinical studies on mice and rats (see Fig. 3.1). The scanner comprises a full ring PET and a high resolution CT system placed sequentially like in clinical PET/CT scanners [29].
3.1 Scanner description
The PET component [30] of the scanner consists of 16 modular detectors (from now on called modules) arranged in two octagonal rings. Each module can ac-quire coincidences with the six opposing modules, three on the same ring and the other three belonging to the adjacent one. Each module is composed of a matrix of 702 lutetium-yttrium orthosilicate crystals doped with cerium (LYSO:Ce) of 1.6x1.6x12mm3 arranged in a matrix of 27x26 pixels (trans-axial vs axial
direc-tion) with a crystal pitch of 1.69 mm. Each matrix is directly coupled to a 64 anodes PMT (H8500C, Hamamatsu Photonics K.K., Hamamatsu, Japan) and is completely independent of the others with no light guide between scintillators and PMT. The two rings are divided by a gap of 6.84 cm. A summary of the PET system speci cation is reported in Tab. 3.1 [30].
3.2 Scanner readout system
The output signals from each PMT pass through a front-end conditioning stack made of a coding board, a pulse shape preampli er and a timing board. The coding board consists of a Symmetric Charge Division (SCD) [32, 33] resistive network. The SCD reduces the 8 × 8 signals of each PMT into 8 x + 8 y signals. These signals enter a passive resistive chain that further reduces the number of signals to four (XA, XB, YA, YB). A standard Anger logic is used
(a) External rendering of the scanner. [31] (b) Picture of the PET ring with 16 modules arranged in two octagonal rings attached to the scanner gantry.
Figure 3.1: Examples of the reconstructed images of the phantom.
Module
Crystal material LYSO:Ce Crystal pixel size(mm3) 1.6 × 1.6 × 12
Crystal pixel pitch 1.69 mm No. of crystals 702 (27x26) System No. of modules 16 (8x2) No. of crystals 11232 Inner diameter 110.8 mm Gantry apeture 100 mm Axial FOV 95 mm Transiaxial FOV 80 mm Dataset
No. of lines of response 23654592 Coincidence scheme 1 vs 6 No. of modules pairs 48 Coincidence window (ns) 5.2 (2τ)
for reconstructing event locations. Both position and last dynode signals are conditioned by fast pre-ampli ers.
The ampli ed last dynode signal passes through a constant fraction discrim-inator (CFD) to produce the timing signal. The position signals generated by the front-end are sent to a back-end data acquisition system, speci cally designed for this scanner. It features 16 Data Acquisition (DAQ) boards based on peak-sensing 12-bit A/D converters, which are hosted on a FPGA-based mainboard. The mainboard hosts a Control FPGA, powers the whole system and provides data connection to PC via USB 2.0 [34]. The main processing tasks carried out by the Control FPGA are the data transfer management, the event tracking, the run-time con guration and status control.
Digital timing signals are fed to the Control FPGA for coincidence processing [35]. Events from two modules are accepted when their arrival time is within a maximum dierence of 2.6 ns, corresponding to an actual coincidence window (2τ) of 5.2 ns. When a coincidence is detected in an allowed module pairs the corresponding DAQs are triggered to perform signal digitization. In this way, only coincidences are acquired. The coincidence processor rejects multiple coin-cidences. Events are sent to the local data acquisition PC in the form of data packets and are stored in a list-mode format. The list-mode format contains the four position signals, the module identi er, and other event tags (e.g., gating signals or random coincidence ag).
3.3 Reconstruction methods
Images can be reconstructed with a multi-core line of response (LOR) based 3D Maximum Likelihood Estimation Maximization (ML-EM). The statistical models used for the reconstruction are evaluated with the Siddon multi-ray based algo-rithm described in [25]. A direct normalisation as described in 1.2.6 is applied during the ML-EM reconstruction.
The reconstructed image consists of a grid of N = 1224120 pixels which cor-respond to 101 × 101 × 120 FOV voxels along the x, y and z directions. The size of each voxel is 0.855 × 0.855 × 0.855 mm3
4
SYSTEM MATRIX COMPUTATION
4.1 IRIS Factorized Model
The model proposed in this work is in the form of a factorised model described in the relation 2.17 where only the analytical geometrical component and the detec-tor component have been included. The normalization facdetec-tors are incorporated directly into the ML-EM algorithm and the attenuation and the positron range components have not been included. Due to computer implementation reasons we worked on the transposed system and the system matrix can be written as:
S = G · D (4.1)
where:
M is the number of VOXELS N is the number of LOFS/LORS
G ∈ RM ×N is the geometric matrix. The element g
ik represents the
prob-ability that the photons of a back-to-back pair emitted by the annihilation of a positron in the voxel i reach the surface of the crystals which identify the LOF k.
D ∈ RN ×N is the detector matrix. The element d
kj represents the
prob-ability that the photons that travel along the LOF k are assigned to the LOR j. In the case of a black-detector, where the photons deposit all their energy right into the crystals where they entered, the detector matrix is equal to the identity matrix and the system model is equivalent to the geo-metrical. In the real situation the photons can be assigned to others pixels due to penetration and inter-crystal scatter eects and the D matrix models precisely the detector response to the incident radiation.
