Total variation regulated algorithm for PET image reconstruction

UNIVERSIT `

A DI PISA

Tesi di Laurea Magistrale

Total variation regulated algorithm

for PET image reconstruction

Candidato:

Tommaso Bettarini

Relatore:

Contents

Introduction 1

1 Positron Emission Tomography 3

1.1 Physical principles . . . 3 1.1.1 β+-decay . . . 4 1.1.2 Positron annihilation . . . 5 1.1.3 Photon interaction . . . 8 1.2 Photon detection . . . 11 1.2.1 Scintillators . . . 12 1.2.2 Photodetectors . . . 13 1.3 Data acquisition . . . 16 1.4 Spatial resolution . . . 18 1.5 Data corrections . . . 19

1.6 The IRIS PET/CT scanner . . . 20

2 Image reconstruction in PET 25 2.1 Analytic image reconstruction . . . 26

2.1.1 The Fourier Slice Theorem . . . 27

2.1.2 Filtered back-projection . . . 28

2.2 Iterative image reconstruction . . . 30

2.2.1 System model . . . 30

2.2.2 ML-EM algorithm . . . 32

2.2.4 MAP-EM algorithm . . . 36

3 A TV regulated algorithm for image quantification 39 3.1 Contrast recovery coefficient . . . 40

3.2 Total variation regulated algorithm . . . 42

3.3 Monte Carlo simulations . . . 43

3.3.1 Custom phantom . . . 44

3.4 Activity simulations and reconstruction . . . 46

3.5 Image analysis . . . 48

4 Results and discussion 51 4.1 Contrast recovery coefficients . . . 51

4.2 Signal-to-noise ratio and contrast-to-noise ratio . . . 57

4.3 In vivo validation . . . . 57

Conclusions 65

Introduction

Positron emission tomography (PET) is a nuclear medical imaging technique which allows non-invasive quantitative evaluation of biochemical and func-tional processes. It measures the spatial distribution of β+_-emitting

radioac-tive tracers inside the patient body and can be performed in combination with other techniques such as CT or MRI. Quantitative imaging is on of the key features of PET imaging and deals with the determination of the precise amount of radiotracer in each region of the body, other than its localization. In a PET scan, a positron-electron annihilation is followed by the emission of two 511 keV back-to-back photons that are detected using pairs of facing coincidence detectors, thus defining a line (called line of response) on which the annihilation is supposed to have happened. Image reconstruction con-sists in elaborating all the informations about the registered lines of response in order to obtain the original spatial distribution of the positron-emitter source. One class of reconstruction techniques are the iterative algorithms, which take in account the stochastic nature of the acquisitions and lead to estimated results, expressed as three-dimensional matrices of voxels. The most used iterative algorithm is called ML-EM (Maximum Likelihood Ex-pectation Maximization) and can be integrated with a priori information on the system.

The major limits in PET imaging and, more importantly, in quantitative imaging are represented by the partial volume effect, which causes an activity spill-over from the original radioactive object to the surrounding voxels, and the presence of noise in the reconstructed image.

The original contribution of this thesis consists in the implementation and validation of a total variation regulated ML-EM algorithm in order to im-prove the precision in quantitative PET measurements by smoothing out the image noise. To validate the algorithm a new custom micro phantom was designed and implemented inside the GATE simulating environment taking in account the specifications of the standard preclinical protocols.

The work presented here is structured as follows.

In chapter 1 the principles of positron emission tomography are described, along with data acquisition and early processing methods. The IRIS PET/CT scanner is also introduced and briefly described.

In chapter 2 the techniques of image reconstruction starting from the acquired data are discussed and the most popular iterative algorithm is com-pletely derived. The system model is also described.

In chapter 3 a regularization of the ML-EM reconstruction algorithm is presented, along with the description of the Monte Carlo simulations per-formed in this work and the design of a new micro phantom. The methods of analysis are then illustrated.

In chapter 4 the reconstruction results are shown and discussed, with comparisons between the new and the unregulated methods through several figures of merit. At the end of the chapter, results of a regularization of an in vivo acquisition are showed and discussed.

Chapter 1

Positron Emission Tomography

Positron Emission Tomography (PET) is a well-established nuclear medical imaging technique. It allows non-invasive quantitative evaluation of biolog-ical processes by measuring the activity distribution of a positron-emitting radionuclide inside the patient body.

When a β+_{-emitting radionuclide is injected in the patient, it follows}

its own peculiar physiological path according to its biochemical properties. Thus, mapping its spatial distribution is a way to gather informations about some of the patient’s biological features, such as cellular activity and its tem-poral evolution. PET is also used in pre-clinical studies on small animals to test new drugs and radiotracers [1].

1.1

Physical principles

In PET a β+_{-emitting radionuclide is injected into the patient’s body, where}

one of the emitted positrons annihilates with an electron in the patient bio-logical tissue. The annihilation event produces two almost back-to-back 511 keV photons, that eventually exit the body and are detected by one or more arrays of detectors surrounding the patient. Through an electronic circuit of coincidence detection, photons are registered and the line in which the annihilation event has happened can be pinpointed.

1.1.1

β

-decay

β+-decay is a type of radioactive decay in which a proton-rich nucleus with atomic number Z is converted in a nucleus with atomic number Z − 1 and a positron and an electron neutrino are emitted. The decay can be represented as follows:

A

ZX →AZ−1 Y + e++ νe. (1.1)

The daughter nucleus can be left in an excited state and reaches a stable configuration after a gamma decay. It is worth to notice that β+_{-decay is}

a three-particle decay where the kinetic energy of the recoil nucleus can be neglected due to its higher mass; therefore, the lighter products, the positron and the neutrino, share most of the kinetic energy. The positron emission spectrum is shown in figure 1.1: it is continuous up to a maximum of energy Emax. This is the maximum kinetic energy the positron can reach.

The decay is governed by the well-known exponential decay law:

N (t) = N0e−λt, (1.2)

where N0 and N (t) are the number of the non-decayed nuclei respectively at

time t0 and at time t > t0 and λ is the decay constant of the radionuclide.

The decay constant can be expressed in terms of the radionuclide mean-life τ through the relation λ = 1/τ .

The source activity A is defined as the number of disintegrations per second, that is:

A(t) = dN (t)

dt = −λN (t). (1.3)

The activity is measured in Becquerel (1 Bq = 1 disintegration/s) or in Curie (1 Ci = 3.7 × 1010 Bq).

Since positron sources are rare in nature, they have to be artificially pro-duced by accelerating charged particles in cyclotrons or linear accelerators and bombarding stable isotopes. The most used radionuclides in PET and

Figure 1.1: β+ spectrum of the most used positron emitters radioisotopes as a function of the positron kinetic energy [2].

their notable properties are listed in table 1.1. Sources with relatively short mean-life are to be preferred in order to reduce the activity in the patient after the examination.

1.1.2

Positron annihilation

After emission from the nucleus, the positron will interact with the surround-ing matter lossurround-ing its kinetic energy through multiple Coulomb interactions. Eventually (usually after a few millimeters in water [4]) it will reach

ther-Radioisotope Half-life (min) Emax (keV) Nuclear reaction 11_{C 20.38 960.2} 14_N(p,α)11_C 13_{N 9.97 1198.5} 13C(p,n)13N 16_O(p,α)13_N 15_{O 2.04 1732} 14_N(d,n)15_O 15_N(p,n)16_O 16_O(p,pn)15_O 16_O(3_He,α)15_O 18_{F 109.8 633.5} 20_Ne(d,α)18_F 16_O(3_He,p)18_F 16_O(4_He,pn)18_F 18_O(p,n)15_O

Table 1.1: Properties of the most used radioisotopes in PET [2, 3]. The half-life can be expressed as T1/2 = ln 2/λ.

Figure 1.2: Representation of an annihilation event. A positron is emitted by a β+-emitting radionuclide together with an electron neutrino. The positron interacts with the surrounding medium until it reaches thermal equilibrium and annihilates with an electron, thus producing two back-to-back photons [6].

mal equilibrium with the medium1 _{and it will annihilate with an electron}

(see figure 1.2). The distance between the positron emission point and the annihilation point is called range and contributes to the uncertainty in the localization of the decaying nucleus. Usually, for the so-called physiological radioisotopes, the positron range in water is about 1-2 mm [7]. In the ap-proximation that both the positron and the electron are at rest when the annihilation occurs, the reaction produces a pair of 511 keV γ-rays mov-ing in opposite directions (back-to-back photons) due to the momentum and energy conservation laws. However, most of the times, electron-positron sys-tems have a non-zero momentum, thus resulting in a loss of collinearity in photon production. Moreover, the positron and the electron can combine in a metastable intermediate species called positronium. Positronium is a non-nuclear, hydrogen-like bound state composed of the positron and the electron that revolve around their combined center of mass.

1_{At equilibrium, matter has a thermal energy of 3/2kT ≈ 1/40 eV at 27} ◦_{C. This}

In general2_{, a series of possibilities has to be considered [2]:}

• no bound state is formed (62% of the occurrences): the standard PET situation is a 2γ at rest annihilation with a lifetime of approximately 1 ns. A 3γ (or more) at rest annihilation is also possible, but it has a much lower probability (σ3γ = σ2γ/372) and is often neglected. In 2%

of the total occurrences an in-flight annihilation takes place before the positron thermalization with a life time of approximately 1 ps.

• positronium is formed (38% of the occurrences): in the singlet spin configuration, called para-positronium (1/4 of the bound states), the system self-annihilates with a lifetime of 100 ps; a pick-off event with another electron is also possible, but very improbable. In the triplet spin configuration, called orto-positronium (3/4 of the bound states), the self-annihilation via 3γ (lifetime: 140 ns) is usually anticipated by a pick-off event (lifetime: 1 ns).

All the processes that alter the positron range and the deviation from collinear-ity contribute to worsen the spatial resolution (see section 1.4). It can be shown that the distribution of the angular deviation in water is a Gaussian centered at zero and with a FWHM of ∆θ ≈ 0.5◦ [8].

1.1.3

Photon interaction

High energy photons interact with matter in three main ways, depending on the energy of the electromagnetic radiation: photoelectric effect, Compton scattering and pair production (see figure 1.3). In addition, there are other mechanisms of interaction, such as Rayleigh scattering or photonuclear reac-tions, which are far less likely to happen for photon energies around 511 keV and can be neglected.

In the photoelectric effect (figure 1.4, top), a photon interacts with an

2_{All the following considerations refer to a water environment, that represents the 70%}

Figure 1.3: Photon cross section in water [9].

orbital electron of an atom and part of its energy is used to overcome the electron binding energy, while the remaining part is converted in kinetic en-ergy for the electron. This effect has the highest cross section in human tissue at energies below 100 keV and has little impact at the energies of the annihilation radiation.

In Compton scattering (figure 1.4, middle), a photon undergoes an inelas-tic scatter with a loosely bound orbital electron, resulting in a photon with a lower energy, a change in its trajectory and the ejection of the electron from the atom. This effect has the highest cross section in human tissue between approximatively 100 keV and ∼2 MeV. Compton scattering is therefore the primary mechanism of interaction for annihilation photons.

In pair production (figure 1.4, bottom), a photon with at least an energy of 1.022 MeV (twice the energy equivalent to the rest mass of an electron) passes in the vicinity of a nucleus and is converted to an electron and a

Figure 1.4: Representations of the main radiation-matter interactions in the energy range of interest (<10 MeV) [10].

for 511 keV annihilation photons.

In the processes described above, an annihilation photon can disappear or change its direction in the patient body, introducing a blurring effect in the reconstructed image. A correction for attenuation can be adopted to ac-count for this effect. For a well-collimated monochromatic source of photons, attenuation takes the following form:

I(x) = I0e−µx, (1.4)

where I0 is the intensity of the unattenuated beam, I(x) is the intensity of

the beam after it has passed through a material for a length x and µ is the linear attenuation coefficient, its value depending on the material and the beam energy. Each of the three effects governing the interaction between photons and matter contributes to the global attenuation coefficient.

1.2

Photon detection

The aim behind a radiation detector is to measure the total energy lost or deposited by radiation upon passage through the detector itself, typically converting the deposited energy into a measurable electrical signal or charge. A PET detection system is usually composed of an array of scintillators coupled with photodetectors. After traveling through the patient body, a 511 keV annihilation photon reaches the scintillator and interacts with it. Then, the scintillator emits fluorescence photons that are eventually detected by a photodetector coupled to the scintillator through a light guide. The incident light is converted into an electrical current that can be acquired by an electronic read-out system.

Every detector component has to be optimized in order to accurately identify the position of the interaction of a 511 keV photon in the detector itself, measure the energy released in the interaction, and register the photon arrival time. Therefore, high spatial, temporal and energy resolutions and

high detection efficiency for 511 keV photons are the requirements for a good PET detector.

1.2.1

Scintillators

Scintillator detectors consist of a dense crystalline material that emits visible light isotropically when an incident ionizing radiation reaches the material and its energy is absorbed, in a process called scintillation. If the emission starts immediately after absorption (within approximately 10-8seconds), the process is called fluorescence. Scintillators can be divided in organic or inor-ganic ones, the latter being the most used in PET imaging. In this category of scintillators the fluorescence is due to the electronic band structure of the crystal lattice. The electrons bound at lattice sites are found in the lower energy band, the so-called valence band, whereas the electrons with enough energy to migrate throughout the crystal are found in the higher energy band, the conduction band. The gap between the two bands is called the forbidden region.

When a pure crystal absorbs the energy of a photon of an ionizing radia-tion, one of the valence electrons can jump to the conduction band leaving a hole in its place. The following disexcitation produces a photon of energy Eg,

equal to the energy gap between the two bands and in the ultraviolet range. In order to avoid photon self-absorption by the crystal itself, this process can be made more efficient by doping the scintillator, i.e., adding impurities in it, thus creating new energy levels in the forbidden region. In this case, the number of low-energy scintillation photons (in the visible or near-visible range) is generally proportional to the energy deposited within the crystal and they can be easily detected by a photomultiplier tube [11]. The process is summarized in figure 1.5.

In an ideal scintillator all the photons are absorbed exactly in the same point where they reach the scintillator, the number of low energy photons emitted is proportional to the released energy3 and the emission spectrum

Conduction band hν1 ≥ Eg hν2 = Eg Valence band Conduction band hν1 ≥ Eg hν2 < Eg Valence band

Figure 1.5: The scintillation process in a pure scintillating crystal (left) and in a doped one (right). In the pure scintillator, an annihilation photon is absorbed and one of the valence electrons jumps to the conduction band. The photon emitted in the following disexcitation has the same energy Eg

separating the two bands and it will be reabsorbed. In the doped scintillator, impurities induce intermediates levels that can trap the excited electron. Thus, the following scintillation photon will have a lower energy and won’t be reabsorbed.

has a minimum overlap with the absorption spectrum. This is not true in real-world detectors as, for example, a photon can release its energy inside the scintillator in a single photoelectric interaction or in a multiple Compton scattering process, or even leave the crystal without being absorbed. The relative probability between photoelectric and total interaction in the mate-rial is called photofraction and is one of the key features to be considered when choosing a PET scintillator, along with detection efficiency, conversion efficiency, output spectrum, signal decay time and energy resolution. The most used inorganic scintillator crystals in PET and their characteristics are shown in table 1.2.

1.2.2

Photodetectors

The information carried by the low energy photons exiting the scintillating crystal is eventually converted into electric signal by a device called

pho-Material NaI(Tl) BGO LSO:Ce LYSO:Ce YAP:Ce Density (g/cm3_{) 3.67 7.13 7.40 7.10 5.37} Light Yield 41000 8200 30000 30000 21000 Effective Z 51 74 66 64 33 Decay constant (ns) 230 300 40 41 27 Photofraction at 511 keV 17% 40% 32% 33% 4.2% Attenuation coefficient at 511 keV (cm-1₎ 0.34 0.96 0.87 0.86 0.46

Table 1.2: Characteristics of the most commonly used scintillators in PET imaging [12, 13].

todetector. The most commonly used photodetectors are the photomultiplier tubes (PMTs), that basically consist in a vacuum glass envelope containing a series of electrodes called dynodes (see figure 1.6). The entrance of the tube is occupied by a photocatode, i.e., a thin layer or a material that easily liber-ates electrons through photoelectric emission. The photocatode is kept at a negative potential so the electrons are accelerated away from it and focused toward the first dynode. When an electron hit the dynode, it deposits its energy and secondary electrons are emitted toward the next dynode, kept to a higher potential than the previous one. At the end of the dynode series, the amplification through this structure will give rise to typically 106 _{– 10}8

electrons. The total charge is then collected at the anode at the end of the tube.

Several designs for scintillator-based photodetectors exist. The Anger camera (or gamma camera) consists of a large scintillating crystal read out by an array of PMTs. The block detector is another design in which a scintil-lator block is segmented into smaller elements and the empty space between

Figure 1.6: Scheme of a classic photomoultiplier tube coupled to a scintillator [14].

them is filled with a reflective material. A matrix of 2 × 2 PMTs reads out the light emerging from the block elements and the coordinates of the interaction point can be derived from the measurement of the ratios between PMT light outputs. In recent years, position sensitive photomultipliers (PS-PMTs) have been developed; they consist of PMTs with a special anode made of several crossed wires. If the pitch of the network created by the wires is small enough, the charge distribution generated across the dynode stages can be sampled with good precision. The final position can be deduced from the anode output [15].

Solid-state photodetectors are another group of devices for detecting low light levels. A recent type is represented by the SiPM (Silicon photomul-tiplier), which is built from an array of avalanche photodiodes (APDs) on a silicon substrate. The APDs in SiPMs operates in Geiger-mode (Geiger-mode APD or G-APD), i.e., it works with a reverse bias voltage above the breakdown voltage. In this mode the internal electric field becomes so high that a very high gain (105 _{– 10}6_{) is obtained, making the SiPM sensitive to}

very low intensity signals (down to the single photon) [16].

In general, a good photodetector for PET has to provide information on the spatial coordinates of the point where the photon has interacted, on the

total amount of light produced (proportional to the energy released by the 511 keV photon) and on the time of the interaction, for coincidences purposes.

1.3

Data acquisition

The interaction positions of the two annihilation photons with the detecting system define a line of response (LOR). In the ideal situation, the annihilation point lies along the LOR and the line of response coincides with the line of flight (LOF) of the two photons. When a pair of photons is detected, the coincidence event (or prompt) is recorded only if it satisfies the following rules4_:

• the second photon has to be detected within a predetermined timing window τ after the first one;

• the LOR conjoining the two detecting crystals is contained within an acceptable angle.

When the acquisition is complete, a further rule on the recorded events can be adopted: the energy deposited by both photons must belong to a pre-determined energy window. The purpose of these rules is to discard those photons that have undergone Compton scattering within the imaging field of view (FOV), or even those pairs of unrelated photons that enter opposing detectors within the coincidence timing window. Nevertheless, under some circumstances even a false coincidence can be recorded as true. In this case, coincidences are classified as scattered events and random events (see figure 1.7). In a scattered event, at least one of the annihilation photons undergoes Compton scattering within the patient changing its direction and losing en-ergy, but it still detected, thus leading to a mispositioned LOR. A random event is detected when two photons coming from different annihilations are detected within the coincidence window. The presence of these background

4_{This kind of collimation is called electronic in analogy with the passive collimation}

Figure 1.7: Representation of three types of coincidence events in PET. A scatter event (S) can be recorded although one of the two photons relative to annihilation 2 has undergone Compton scattering. In a random event (R), two photons originated from different annihilation events (3 and 4) are recorded as in coincidence. Both of these phenomena can be misinterpreted as true events (T) from the detection system, thus generating misplaced LORs (dashed lines) [5].

events is detrimental to the quality of the resulting images and several cor-rection techniques have been developed (see section 1.5) [17].

The first of the above-mentioned rules points out the importance of a high temporal resolution for the detection system. When an annihilation photon is detected, the electrical pulse of the PMT generates a trigger pulse and two photons are said to be in temporal coincidence if the respective pulses are separated in time by less than τ . The value of τ must be chosen taking in account the finite time resolution of the detector and the potential difference in the distance traveled by the two photons. If it is too small com-pared to the temporal resolution of the detection system, true coincidences will be missed. If it is too large, a high number of random coincidences will be recorded.

1.4

Spatial resolution

The non-collinearity of annihilation photons and the finite distance traveled by the positron before annihilation place a physical lower limit to the spatial resolution that can be achieved in positron emission tomography. The con-tribution of the non-collinearity to the worsening of the spatial resolution in the center of a detector ring can be expressed as:

F W HM = tan ∆θ 2

!

·D

2 ' 0.0022D, (1.5)

where D is the diameter of the detector ring and ∆θ is the deviation from collinearity seen in section 1.1.2. The positron range contribution is defined as [4]:

r = 2.35 · rms, (1.6)

where rms is the root mean square of the range distribution.

Other contributions are due to the detection system and the reconstruction technique. A detector size factor can be calculated to take in account the finite dimensions of the detectors. In this case, the region of response between two opposite detectors is not a line but rather an extended tube and in the midpoint of the LOR the coincidence response function is triangular in shape. Moving towards one of the detectors, the response function acquires a trapezoidal shape. From geometrical considerations one can see that the minimum FWHM of the coincidence response function is equal to half the size of the crystal (d) in the same direction [5].

Another effect that contributes to worsen spatial resolution is the parallax effect, related to the lack of information on the depth of interaction inside the crystal. If an annihilation photon is not normally incident on the detector ring, it may interact in a crystal different from the one it impinges upon and so get assigned to the wrong crystal. This term can be approximated by the following equation:

p = α√ r

where r is the radial position of the source, R is the radius of the PET ring and α is a term that depends on the scintillating crystal.

It is also possible to have a coding error, i.e., an inaccuracy in the iden-tification of the crystal where the first interaction occurs due to multiple interactions in more than one crystal element (inter-crystal scatter ). Finally, the lower limit of the spatial resolution can be expressed as [18]:

F W HM = 1.25q(d/2)2_{+ b}2_{+ (0.0022D)}2 _{+ r}2_{+ p}2_{, (1.8)}

where the contributions in the quadrature sum are respectively given by the detector size, the coding error, the non-collinearity of annihilation photons, the positron range and the parallax effect. The 1.25 factor is related to the image reconstruction technique [19].

Although several techniques have been introduced to reduce the effect of the technology-related terms, the relatively low spatial resolution remains one of the major drawbacks of PET, when compared to other molecular imaging techniques such as Magnetic Resonance Imaging (MRI) or X-ray Computed Tomography (CT). This has led to the development of hybrid imaging modalities, e.g., PET/CT or PET/MR [5].

1.5

Data corrections

Acquired data are usually stored in a list-mode format, where each entry contains the coordinates of the LOR, energy, time and any other information that may be appropriate. At this level, data can be reformatted in order to slim them down before reconstruction and corrections can also be applied. Corrections are necessary when the restoration of the true activity concen-tration in every point of FOV is required, e.g., in quantitative imaging.

For positron emission acquisitions it can be shown that the position of the source along a line of response conjoining two detectors does not affect the count rate relative to that line of response, thus the attenuation correction

takes the form of −µD, where D is the total thickness of the medium and µ is the linear attenuation coefficient [17]. More accurate attenuation corrections can be obtained with a transmission scan [20, 21].

A decay correction is necessary to ensure that the measurement of the ra-dioisotope activity density is relative to a precise point in time (t0), usually

the start of acquisition. The correction for an event collected at a certain time point t is a multiplicative factor e(t−t0)/τ_{, where τ is the decay constant} of the radioisotope. A correction for the detector dead time can also be per-formed [22].

A normalization of the acquired data is requested to compensate unavoid-able non-uniformities in the scanner FOV. These non-uniformities are due to different detection efficiencies from LOR to LOR that can result from a number of reasons, such as variations in crystal size, light yield or electronics behavior. Normalization values are experimentally derived from long scans of a dedicated phantom.

Apart from the rules mentioned in section 1.3 in order to reduce the num-ber of registered scatter and random events, post-acquisition corrections can be performed. The scatter correction consists in estimating the number of scatter counts contributing to a given LOR as a result of a Compton in-teraction and removing the unwanted events from that line line of response [23–25]. The random correction is based on a statistical estimation of the random count distribution using, for example, the so-called delayed window technique [11]. The obtained distribution is then subtracted from the prompt events.

1.6

The IRIS PET/CT scanner

IRIS PET/CT scanners (see figure 1.8, left) are produced by Inviscan SAS and are used for preclinical studies on mice and rats. One of these scanners is currently installed at the Laboratory of Imaging Biomarkers of the CNR Institute of Clinical Physiology in Pisa [26].

Figure 1.8: On the left, rendering of an IRIS PET/CT scanner [27]. On the right, picture of a IRIS PET ring with 16 modules arranged in two octagonal rings attached to the scanner gantry [26].

The PET component of the scanner consists of 16 modules arranged in two octagonal rings (see figure 1.8, right). Each detector module can acquire coincidences with the six opposing modules, three belonging to the same ring and three belonging to the other, and comprises a matrix of 702 lutetium-yttrium orthosilicate crystals doped with cerium (LYSO:Ce, see table 1.2). The matrix is coupled to a 64 anodes PMT (H8500C, Hamamatsu Photonics K.K., Hamamatsu, Japan). Each module is completely independent from the others, with no light guide between scintillator and PMT. A gap of ap-proximately 6.8 mm divides the two rings. A summary of the PET system specifications is reported in table 1.3.

The output signals from each PMT pass through a front-end condition-ing stack made of a codcondition-ing board, a pulse shape preamplifier and a timcondition-ing board. The coding board consists of a Symmetric Charge Division (SCD, [28, 29]) resistive network that reduces the 8 × 8 signals into 8x + 8y sig-nals. A passive resistive chain further reduces the number of signals to four (XA, XB, YA, YB) and a standard "Anger logic" is used for reconstructing

event locations. Both position and last dynode signals are conditioned by fast pre-amplifiers. The amplified last dynode signal passes through a

con-Module

Crystal material LYSO:Ce

Crystal pixel size (mm3_{) 1.6 × 1.6 × 12}

Crystal pixel pitch (mm) 1.69

No. of crystals 702 (27×26) System No. of modules 16 (8×2) No. of crystals 11232 Inner diameter (mm) 110.8 Gantry aperture (mm) 110 Axial FOV (mm) 95 Transaxial FOV (mm) 80 Dataset No. of LORs 23,654,592 Coincidence scheme 1 vs 6 No. of module pairs 48 Coincidence window (ns) 6.2 (2τ )

Table 1.3: Specifications of the PET component of the IRIS PET/CT scan-ner.

stant fraction discriminator (CFD) to produce the timing signal.

The position signals generated by the front-end are sent to a back-end data acquisition system that 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 [30]. The main processing tasks carried out by the Control FPGA are the data transfer management, the event tracking, the run-time configuration and status control.

Digital timing signals are fed to the Control FPGA for coincidence pro-cessing [31]. Events from two modules are accepted when their arrival time is within a maximum difference of 3.1 ns, corresponding to an actual

coinci-dence window of 6.2 ns. When a coincicoinci-dence is detected in an allowed module pairs, the corresponding DAQs are triggered to perform signal digitalization. Events are sent to the local data acquisition PC in the form of data packets and are stored in a list-mode format, which contains the four position signals, the module identifier, and other event tags (e.g., gating signals or random coincidence flag).

Chapter 2

Image reconstruction in PET

The aim of reconstruction is to recover tomographic images of the radio-tracer distribution using the acquired projections. The measured counts on a particular LOR (called projection data) are proportional to the integrated activity along that line. Projection data measured at many different angles are then converted into a three-dimensional distribution. The reconstruction methods are divided into analytical and iterative approaches. In the ana-lytical approach, it is assumed that the data are deterministic and that it is possible to find an exact solution, whereas in the iterative approach the intrinsically stochastic nature of the projection data is taken in account and estimation techniques tend to approximate solutions.

The reconstruction methods may also vary depending on the acquisition mode, that can be 2D or 3D. Two-dimensional PET imaging considers LORs lying within a specified plane orthogonal to the scanner axis and their data are reconstructed in two-dimensional images called slices, that are eventually stacked to form a 3D volume. The 2D image elements are called pixels. In three-dimensional PET imaging, both direct and oblique planes are acquired and reconstructed. In this case, the 3D image elements are called voxels [2]. In the reconstructed image, the value relative to each voxel is proportional to the number of β+_{-emitting nuclei contained in the volume spanned by the}

Figure 2.1: An object f (x, y) and its projection pθ(s) for an angle of θ [33].

boundaries [32], but their inherent complexity restricts their usage. It is worth noting that the spatially discrete nature of the data can only allow a spatially discrete reconstruction, with an inherent limited spatial resolution.

2.1

Analytic image reconstruction

In analytical reconstruction methods it is assumed that measurement noise can be ignored, thus allowing a simplified reconstruction and a direct math-ematical solution. Generally, in a tomographic technique, a line integral (or projection) pθ(s) represents the integral of some parameters of the object

along a line. More precisely, in PET imaging, the parameter is the total activity f (x, y) of the β+-emitting radionuclide in the entire LOR (see figure 2.1) and the projection can be written as:

pθ(s) =

Z +∞

−∞ f (x, y)dt. (2.1)

pθ(s) =

Z +∞

−∞

Z +∞

−∞ f (x, y)δ(x cos θ + y sin θ − s)dxdy = R{f (x, y)}, (2.2)

which represents the Radon transform, or sinogram1_{, of the function object}

f (x, y) and maps the object space (x, y) in projection space (or Radon space) (s, θ) [34]. Every point in Radon space corresponds to an integral line in the spatial domain. The equation x cos θ + y sin θ = s represents a LOR that crosses the object with an angle of incidence θ and at a distance s with re-spect to the origin of the axes.

Knowing the Radon transform of an object allows us to reconstruct its ra-diotracer distribution: if a infinite number of one-dimensional projections of the object for an infinite number of different angles could be acquired, a per-fect reconstruction of the original object f (x, y) would be computed and the reconstruction process would consist in calculating the inverse Radon trans-form. Regrettably, this is not possible, as only a finite number of projections can be acquired.

2.1.1

The Fourier Slice Theorem

The most commonly used analytical method for a discrete sampling of pro-jection data is the filtered back propro-jection algorithm (FBP), based on the Fourier slice theorem [35]. The theorem states that the one-dimensional Fourier transform of the projection p(s, θ) to a given angle θ coincides with the central slice section of the two-dimensional Fourier transform of f (x, y) inclined at the same angle:

P (ν, θ) = F (u, v)|u=ν cos θ,v=ν sin θ, (2.3)

where P (ν, θ) = F1D[p(s, θ)] and F (u, v) = F2D[f (x, y)], where F is the

Fourier transform operator [36]. This important equivalence shows that, in principle, once a set of projections p(s, θ) has been acquired, f (x, y) can be

1_{The term sinogram is due to the fact that a point in real space traces a sinusoidal}

Figure 2.2: Due to the fact that the one-dimensional Fourier transform P (ν, θ) of the sinogram is defined on a polar coordinates system while F (u, v) is defined on a Cartesian coordinates system, the interpolation at high fre-quencies is inaccurate [37].

recovered by applying a two-dimensional inverse Fourier transform to the one-dimensional transform of p(s, θ):

f (x, y) = (F−1F )(x, y) =

Z Z +∞

−∞ F (u, v)e

2πi(ux+vy)_{dudv. (2.4)}

Unfortunately, due to the fact that the F (u, v) sampling is discrete, this pro-cedure generates an oversampling around the origin in the frequency domain and a loss of precision at high frequencies, where the information on the image details is stored (see figure 2.2).

2.1.2

Filtered back-projection

An equal sampling through the Fourier space can be achieved by changing the rectangular coordinate system in the frequency domain (u, v) into a polar

coordinate system (ν, θ):

u = ν cos θ

v = ν sin θ. (2.5)

Rewriting also the differentials as dudv = νdνdθ, equation 2.4 becomes:

f (x, y) = Z 2π 0 dθ Z +∞ −∞ P (ν, θ)e

2πiν(x cos θ+y sin θ)_{νdν, (2.6)}

where the relation 2.3 has been used. This integral can be split into two by considering θ from 0 to π and from π to 2π. Then, using the property of Fourier transformations F (ν, θ + π) = F (−ν, θ), it becomes:

f (x, y) = Z π 0 dθ Z +∞ −∞ P (ν, θ) |ν| e2πiνsdν, (2.7) The integration over ν may be seen as a filtering operation, where the fre-quency response of the filter is given by |ν|. Hence, defining h(s) = F (|ν|), we finally obtain:

f (x, y) =

Z π

0

pθ(s) ⊗ h(s)dθ. (2.8)

The integrand pθ(s) ⊗ h(s, θ) is therefore called filtered projection and

equa-tion 2.8 expresses that each filtered projecequa-tion is back projected. The resulting projections for different angles are then added to form the estimate of f (x, y). Interestingly, the filter h(s), also known as ramp filter because of its shape in the frequency domain, helps to enhance image edges by increasing high frequencies intensity and dampening low frequencies that carry no informa-tion. Unfortunately, this filter also worsens image noise that lies at high frequencies.

A method analog to the two-dimensional FBP can be generalized in three-dimensional image reconstruction, though facing the consequences due to the finite axial extent of the scanner that causes some of the projections to be truncated [38, 39].

2.2

Iterative image reconstruction

With iterative algorithms data acquisition and noise structure can be real-istically modeled, thus allowing a higher image quality with respect to the analytical methods. Moreover, iterative algorithms have the potential to ac-count for the stochastic nature of the PET events, reducing the effect of the noise in the reconstructed image. Modeling sources of noise causes the ob-tained images to be approximate solutions (also called estimations) of the actual image.

Iterative algorithms do not convolve the data by a filter that amplifies high-spacial frequency components such as noise. Therefore, a better image quality is obtained when compared with analytical methods. On the other hand, modeling complex phenomena, such as the detection process or photon transport in tissue, requires a greater computing power and only an estimate of the solution can be found. Thanks to the latest computational resources and to the introduction of faster algorithms, iterative reconstruction is now a feasible technique.

2.2.1

System model

The system model is represented by a probability matrix P which correlates positron emission and photon detection:

y = P f , (2.9)

where fj represents the radiotracer activity in voxel j (j = 1, . . . , N ) and

yi represents the i-th (i = 1, . . . , M ) pair of detectors (or LOR). Each

el-ement pij contains the probability of detecting an emission from voxel site

j in LOR i. An ideal model will include all the parameters describing the imaging process, such as system geometries, detector properties, radiation-matter interactions, and positron range. The aim of reconstruction is to use the projections y to find the image f .

Even if the matrix P is highly sparse, it may well occupy several giga-bytes of memory. Exploiting symmetries is an useful method to compress the system matrix, but the computation of P remains a challenging opera-tion. A possible approach to overcome this problem is to decompose P into components which are separately calculated and stored [40, 41]. One of the possible decomposition is:

P = Pdet.sens· Pdet.blur· Pattn· Pgeom· Ppositron, (2.10)

where:

• Pdet.sens is a M × M diagonal matrix that models the detection

effi-ciency of each detector pair and it is used to evaluate the normaliza-tion values (see secnormaliza-tion 1.5). It is usually obtained from experimental measurements [40];

• Pdet.blur is a M ×M matrix which contains the detector blurred response

to the incident radiation; more precisely, it models phenomena such as inter-crystal scatter or parallax effect. The matrix Pdet.blur can be

evaluated through analytical calculations or Monte Carlo simulations [42];

• Pattn is a M × M diagonal matrix which contains the attenuation

cor-rection factors due to photon interaction within the patient body (pri-marily Compton scattering) for each LOR. It can be computed from a CT scan or by forward projecting a reconstructed attenuation image [43, 44];

• Pgeom is a M × N matrix which models the detector geometry and

accounts for the different intersection lengths between each LOR and each voxel . It can be calculated using a ray tracing algorithm such as the Siddon method [45];

Photon-pair non-collinearity is another factor that can also be included when simulating the system [41]. The flexibility of this approach allows the calcu-lation of each component with the desired accuracy and it has been shown that the quality of the reconstructed images can be significantly improved [47, 48].

2.2.2

ML-EM algorithm

The Maximum Likelihood Expectation Maximization algorithm (ML-EM) was first introduced by Dempster et al. in 1977 [49] and first applied to PET by Shepp and Vardi in 1982 [50]. It is the most popular iterative reconstruction method for PET and it is based on the maximization of a cost function.

The aim of reconstruction is to determine the value fj of the radiotracer

activity f (x, y, z) in each voxel j. This value can be defined as follow:

fj =

Z

j-th voxel

f (x, y, z)dxdydz. (2.11)

Due to the random nature of radioactive disintegration, it seems appropri-ate to use statistical tools when approaching the reconstruction problem. The idea is to find the activities fj (j = 1, 2, 3, . . . N ) that maximize a likelihood

function of the measured projections yi (i = 1, 2, 3, . . . M ). In other terms,

ML-EM search the estimate of the radiotracer distribution which maximizes the probability of observing the measured data.

Assuming that emissions from each voxel are completely random and with rate constant in time, the number of events cij generated in the j-th voxel and

contributing to the i-th projection are Poissonian variables [36, 51]. Their mean values can be written as E [cij] = pijfj, being pij the ij-th element of

the system matrix P , i.e., the mean probability of an emission from the j-th voxel to be detected by the i-th pair of detectors (or LOR). The i-th total

projection can be written as:

yi =

X

j

cij. (2.12)

If the dead time effects in the counting can be neglected, the detection of each photon pair by the system is independent and the projections follows the Poisson distribution as well.

Given the distribution f , the conditional probability, or likelihood, for the projections y is: p(y | f ) = Y i p(yi | f ) = Y i e− ¯yiy¯i yi yi! , (2.13)

where ¯yi = E [yi] = PjE [cij]. The value of f that maximizes the objective

function likelihood can be found deriving equation (2.13). Since the loga-rithm is a monotonic function, taking the logaloga-rithm of the previous equation simplify the calculation:

L(y | f ) = ln p(y | f ) =

M

X

i=1

yiln ¯yi− ¯yi− ln yi!. (2.14)

The last term is a constant and can be neglected when maximizing. Finally we obtain: L(y | f ) =X i X j cijln pijfj− pijfj, (2.15)

Since L(y | f ) is not known, the idea behind the Expectation Maximization is to find the f that maximizes its current expectation give the data y and the current fit f(k). Thus, in the "E" step, the conditional expected value of the log-likelihood is evaluated, obtaining:

EhL(y | f ) | y; f(k)i= E   X i X j cijln pijfj − pijfj | yi; f(k)  = =XXEhcij | yi; f(k) i ln pijfj − pijfj. (2.16)

When deriving the mean value of cij conditioned to yi, one must take into

account that the condition is given by the relation yi =Pjcij (see equation

2.12). According to the probability theory, given j independent Poissonian variables and their total sum, the conditional probability is a binomial dis-tribution with the following parameters2_:

 P jcij, E[cij] P jE[cij]  . Thus, recalling that the expected value of a binomial distribution with parameters (a, b) is ab, it follows that:

Ehcij | yi; f(k) i =X j cij E [cij] P jE [cij] = yi pijfj(k) P lpilf (k) l . (2.17)

Substituting in equation 2.16, we obtain:

EhL(y | f ) | y; f(k)i=X j   X i yi pijfj(k) P lpilf (k) l ln pijfj− X i pijfj  . (2.18)

In the "M" step the aim is to find the f(k+1) maximizing the conditional expected value: ∂E[L(y | f ) | y; f(k)] ∂fj = f_j(k)X i pij yi P lpilf (k) l pij pijfj −X i pij = 0. (2.19)

Solving for fj, we finally get the recursive form:

f_j(k+1) = f (k) j P ipij X i pij yi P lpilf (k) l . (2.20)

The initial guess f_j(0) is usually chosen uniform and non-negative. Since image noise increases after each iteration, the stopping criterion is based on the best trade-off between spatial resolution and noise itself, i.e., the number of iterations is to be determined empirically.

The ML-EM cycle can be summarized in the following steps (see figure

1)

P

p

f

2)

P

3)

P

p

P

4) Update image

f

f

Figure 2.3: Flow diagram of the Maximum Likelihood Expectation Mini-mization algorithm. Starting from an uniform and non-negative initial guess f_j(0), the algorithm iteratively computes new image estimates based on the measured projections yi

2.3):

1. forward-project current image values f_j(k) into projection domain;

2. compare projection with measured data yi, obtaining a correction

fac-tor;

3. back-project the correction factor into image domain for each LOR;

4. update current image estimate weighted by pij.

2.2.3

OS-EM algorithm

A slight modification of the ML-EM algorithm, named Ordered Subset Ex-pectation Maximization, was introduced by Hudson and Larkin in 1994 [52]. In this algorithm, the projection data are grouped in equipollent subsets and each iteration of the standard ML-EM is applied on one subset at a time (subiteration) in a specified order. The resulting image of each subset

be-is used to start the following OS-EM iteration. Each subiteration can be defined as follows: f_j(k,b+1) = f (k,b) j P i∈Sbpij X i∈Sb pij yi P lpilf (k,b) l , (2.21)

where Sb (b = 1, 2, 3, . . . B) are the collections of the indices of the

projec-tions in the b-th subset of the data. If B = 1, OS-EM obviously reduces to ML-EM.

Usually projection data are divided so that each subset contains data ac-quired from a different angle of view. The main advantage of OS-EM when compared to ML-EM is the reduction of computational time; a reconstruc-tion gets roughly faster than the convenreconstruc-tional algorithm by a factor B, as the image is updated more frequently. On the other hand, the asymptotic convergence to the maximum likelihood estimator is no longer guaranteed and the OS-EM algorithm may cycle through different solutions, i.e., one solution for each subset. As a consequence, a high number of subsets might compromise image quality.

2.2.4

MAP-EM algorithm

When some kind of a priori information about the β+_{-emitter distribution}

is known or one or more features of the resulting image are required (for example, a certain grade of smoothness or the non-negativity of the tracer concentration), it is possible to enforce this knowledge in the ML-EM al-gorithm. Applying Bayes’ theorem, this information is added as a penalty term to the objective function. Therefore, algorithms that include prior in-formation are called Bayesian methods, penalized methods, or Maximum a posteriori Expectation Maximization (MAP-EM) algorithms.

the likelihood p(y | f ) and the prior probability p(f ):

p(f | y) = p(y | f )p(f )

p(y) . (2.22)

Taking the logarithm, the new objective function L0(y | f ) arises:

L0(y | f ) = ln(p(f | y)) =

= ln(p(y | f )) + ln(p(f )) − ln(p(y)). (2.23)

The first term corresponds to the log-likelihood defined in equation 2.14, while the second term contains the prior information about the image. The introduction of a priori information in the objective function enforces the desired properties on the image estimate at each iteration. The last addend of equation 2.23 is independent of f and can be neglected when deriving.

The selection of the prior function can be made in various ways. For example, it can be formulated according to known pixel distributions [53] or it can be given as anatomical image data, e.g., a CT or MRI image [54, 55]. The simplest form of image prior assumes statistical independence between voxels [56, 57], whereas more refined priors, as the Gibbs distribution, model spatial dependence through a energy function U (f ). Gibbs distribution has the general form

p(f | β) = 1 Ze

−βU (f )

, (2.24)

where Z is a normalizing constant and β is a Bayesian parameter setting the weight of the prior. U (f ) is expressed as a sum of potential, each of which is a function of a neighborhood (or clique) of voxels [51].

Substituting the Gibbs prior into the MAP objective function 2.23 pro-duces:

and the recursive form 2.20 becomes: f_j(k+1) = f (k) j P ipij + β_∂fj∂ U (f )|_{f =f}(k) X i pij yi P lpilf (k) l . (2.26)

The partial derivatives of U (f ) should be evaluated at the next estimate

f(k+1), which is not yet available, and are replaced with derivatives at the current estimate f(k). The slow convergence os the ML-EM algorithm per-mits such a substitution (Green’s one-step-late algorithm [58]), thus giving a closed form for equation 2.26. If the parameter β is sufficiently small, the one-step-late approximation generally converges more quickly than the EM algorithm [59]. It is clear from equation 2.26 that, when β = 0, the MAP-EM algorithm reduces to a classic ML-EM.

In the next chapter, the work of this thesis is presented and a proper energy function is chosen in order to improve quantification in PET acquisi-tions.

Chapter 3

A TV regulated algorithm for

image quantification

Besides physiological applications, PET imaging, when performed in combi-nation with other techniques as CT or MRI, can also be used for quantitative studies. In other words, one could find interesting not only where the radio-tracer has accumulated, but also how much of it is located in each voxel. A proper method of quantification of PET images can reduce variabilities be-tween two, or more, different observers and help assessing patient response to therapy [60].

The standardized uptake value (SUV) is a parameter that shows the rel-ative tissue uptake in a region of interest (e.g., a tumor). In its general formulation it is defined as:

SU V = C

A/w, (3.1)

where C is the tissue activity concentration, measured in kBq/mL, A is the injected activity, measured in MBq, and w is the body weight of the patient, measured in kg. In a PET acquisition, C is basically the measured voxel intensity. Other definitions consider w as the total body surface or the lean body mass. In some cases, for each region of interest (ROI), the maximum

value SU Vmax is taken instead of the total activity concentration. Being A

and w independent from the acquisition system, it is evident that a higher precision when measuring concentration C leads to a better image quantifi-cation. Precision in quantitative imaging is of major importance in dynamic studies, where the same region of interest is measured at different time frames in order to extract kinetic parameters of the subject through mathematical fit procedures based on predetermined models. Hence, the precision of ex-tract parameters depends on the precision of the single measurement.

The aim of this thesis was then to show that a proper choice of the energy function in a MAP-EM reconstruction algorithm can improve PET images quantification by smoothing out image noise while preserving object edges.

3.1

Contrast recovery coefficient

Every PET image is obtained by the convolution of the actual radiotracer distribution with the point spread function (PSF) of the system. Due to the finite nature of the PSF, an infinitely small point source is smeared out and appears in the final image as a finite-size blob with lower activity. For the same reason, small finite objects appear larger and with lower activity than the original source. This phenomenon is called partial volume effect (PVE) and is illustrated in figure 3.1. Taking the same volume of interest (VOI) on an ideal image and on an image affected by PVE, a spill-over of activity happens, i.e. a decrease on the average (and maximum) activity concentra-tion C is measured, while higher activity can be found in the surrounding voxels.

An important figure of merit to evaluate the PVE for an object with a known object-background contrast ratio is the contrast recovery coefficient (CRC), defined as

CRC = aobj/abkg − 1

Figure 3.1: Partial volume effect. In the acquired image, the actual source (on the left) undergoes a spill-over of radioactive signal (on the right) due to the finite nature of the system PSF [61].

where aobj and abkg are the measured activity concentrations in the region

of interest of the object and in the background respectively and OC is the object contrast, defined as the ratio of the true activity concentrations of the object and the background. In an ideal case, the CRC would be equal to the unity, but, due to the presence of PVE, lower values are obtained in real-life acquisitions. Hence, the CRC can be used as a multiplying factor to recover the actual concentration C (see equation 3.1) in a region of interest of the reconstructed image. A more precise CRC evaluation leads to a better activity uptake estimation and one way to achieve this result is to develop a proper regularized reconstruction algorithm.

Other important parameters to evaluate the precision of an image are the signal-to-noise ratio (SNR) and the contrast-to-noise ratio (CNR), that were chosen in this work to estimate the effectiveness of noise reduction techniques.

3.2

Total variation regulated algorithm

Total variation (TV) is generally defined as the norm of the gradient of a differentiable function f . When f is a 2D or 3D image, TV is indicative of the sparsity of the image gradient [62]. In three dimensions, it takes the following form:

T V (f ) =

Z

|∇f | dxdydz, (3.3)

where the integral domain coincides with the entire image domain.

Over the years, several methods of image regularization have been pro-posed [63–65] and a wide range of them involves TV [66, 67]. Its use in a constrained minimization problem was first introduced by Rudin, Osher and Fatemi in 1992 [68]. In 1998 Johnson, Huang and Chan first showed that a TV-based regularization suppresses noise effectively while capturing sharp edges for PET images [69]. A year later, Panin, Zeng and Gullberg first described TV norm as an energy function in a maximum a posteriori OS-EM algorithm, showing that it smooths the image while preserving the edges [70]. Since then, it has been demonstrated in a variety of applications that TV as a prior for image reconstruction provides low-noise images, while retaining sharp details [62, 71–74].

A common solution in image denoising is the application of a gaussian ker-nel in post-processing, i.e., at the end of the reconstruction [75, 76]. However, unlike TV, gaussian filtering smooths out both uniform regions and edges, thus worsening the already existing PVE [77]. For this reason, the TV norm was chosen in this work as an energy function for a maximum a posteriori OS-EM reconstruction algorithm (see equation 2.26), in order to improve the precision on the measured CRCs. The three-dimensional discrete TV norm for an image f ∈ <N ×M ×L can be written as:

T V = N X i=1 M X j=1 L X k=1 q

(fi+1,j,k− fi,j,k)2+ (fi,j+1,k− fi,j,k)2+ (fi,j,k+1− fi,j,k)2

and the derivative with respect to the single voxel becomes:

∂T V ∂fi,j,k

=

= q fi,j,k − fi−1,j,k

(fi,j,k − fi−1,j,k)2+ (fi−1,j+1,k− fi−1,j,k)2+ (fi−1,j,k+1− fi−1,j,k)2

+

+q fi,j,k− fi,j−1,k

(fi+1,j−1,k − fi,j−1,k)2+ (fi,j,k − fi,j−1,k)2+ (fi,j−1,k+1− fi,j−1,k)2

+

+q fi,j,k− fi,j,k−1

(fi+1,j,k−1− fi,j,k−1)2+ (fi,j+1,k−1− fi,j,k−1)2+ (fi,j,k − fi,j,k−1)2

+

+q 3fi,j,k− fi+1,j,k− fi,j+1,k− fi,j,k+1

(fi+1,j,k − fi,j,k)2+ (fi,j+1,k− fi,j,k)2+ (fi,j,k+1− fi,j,k)2

.

(3.5) Starting from the ML-EM algorithm developed by the Functional Imaging and Instrumentation Group (FIIG) of the Department of Physics of the Uni-versity of Pisa, the previous equation was derived and added to the code by using the Insight Segmentation and Registration Toolkit (ITK) [78]. The reconstruction algorithm used in this work then becomes:

f_j(k+1)= f (k) j P ipij + β ∂ ∂f_j(k)T V f (k) X i pij yi P lpilfl(k) . (3.6)

3.3

Monte Carlo simulations

In order to study the effectiveness of the regularizing maximum a posteri-ori OS-EM algposteri-orithm, Monte Carlo simulations were performed with GATE (Geant4 Application for Tomographic Emission [79, 80]), a Monte Carlo simulator commonly used by the PET community. The underlying Geant4 toolkit is a general purpose Monte Carlo simulation package [81, 82]. The whole IRIS scanner geometry was reproduced inside the Geant4 environment as shown in figure 3.2. Along with the geometry, the electronic read-out of the system and the main radiation-matter interactions were also simulated.

Figure 3.2: Visualization of the IRIS geometry built inside the Geant4 envi-ronment.

3.3.1

Custom phantom

The standard for scanner performance evaluation in small animal PET is the National Electrical Manufacturers Association (NEMA) NU 4-2008 proto-col [83]. According to the protoproto-col, the image quality phantom1 _{is designed}

such that small rods of different diameters can be filled with a radioactive solution and acquired on a cold background, in order to simulate small hot lesions. Using the measurements, one can get recovery coefficient (RC) val-ues that, basically, are the ratios of the apparent activity concentration to the true activity concentration and can be seen as a measure of the partial volume effect. However, such a geometry cannot properly reproduce actual lesions that are usually encountered when studying small animals, i.e., mice and rats. Typical lesions, such as tumors, have a quasi-spherical shape and phantoms for preclinical studies should take this fact into account. More-over, in most practical cases, activity concentrations have to be detected

1_{NEMA PET small animal phantom NU4:}

in a uniform non-zero background [84] and the NEMA NU-4 2008 quality phantom does not allow the determination of CRC values for hot regions on a hot background (see section 3.1). The NEMA NU 2-2012 [85] phantom2

fulfills both of these requirements by providing small hollow spheres on a fillable background chamber for the purpose of evaluating lesion detection performance of PET systems. Unfortunately, the NU 2-2012 protocol has been designed for clinical, i.e., human, studies and the required phantom has prohibitive dimensions for small animal PET systems. The micro hollow sphere phantom3 _{could offer a valid solution, however, the smallest sphere of}

this phantom has still a large diameter (3.95 mm) when compared to some of the typical lesions encountered in small animal imaging [86–89].

For these reasons, the need for accurate lesion detection in small animal scanners has been the primary motivation to design and evaluate a new micro hollow sphere phantom during this work. The phantom is designed such that the cylindrical background chamber has an internal radius of 20 mm and a height of 15 mm. The higher and the lower caps are 2 mm high and the phantom wall is 1.75 mm thick, as in the NU 4-2008 phantom. Six hollow spheres with internal diameter of 2 mm, 3 mm, 4 mm, 5 mm, 6 mm and 7 mm are placed at half of the phantom height and have a wall thickness of 0.15 mm. Although the smallest commercially available hollow sphere for micro phantoms has an internal diameter of 3.95 mm and a wall thickness of 1 mm4_{, in the last years smaller spheres with thinner walls have been easily}

fabricated, with an internal diameter down to 1.25 mm and walls of 0.1 mm [90].

The six spheres are arranged in such a way that each sphere is at least 2 mm distant from the others, in order to minimize spill-over effects. The complete separation between the background and the spheres environments

2_{NEMA IEC PET body phantom: www.spect.com/pub/NEMA_IEC_Body_Phantom}

_Set.pdf

3_{Micro hollow sphere phantom: www.spect.com/pub/Micro_Hollow_Sphere_Phantom}

.pdf

allows the preparation of different spheres-to-background contrast ratios by simply choosing the most adequate activity concentrations. The phantom geometry was then simulated in GATE (see figure 3.3).

3.4

Activity simulations and reconstruction

The starting activity in the phantom was fixed to 3.7 MBq, as suggested by the NEMA NU 4-2008 protocol. 18_{F activity concentrations for an}

ob-ject contrast (OC) of 2:1 were calculated taking in account the proportions between spheres volumes and background volume and is reported in table 3.1. In the GATE environment, the background and the four smallest hollow spheres were filled with the obtained concentrations, whereas the two largest spheres (6 mm and 7 mm in diameter) were left empty so to evaluate pos-sible spill-over effects from the hot background. The phantom was placed at a quarter of the FOV axis. This was necessary in order to avoid known IRIS simulation artifacts due to the lower sensitivity at the center of the FOV. A Monte Carlo simulation of 1200 s (approximately 5·108 _{events) was}

performed with a 250-750 keV energy window. The same procedure was then repeated for two more OC values (4:1 and 8:1). Each simulation took ∼40 hours on an eight-core Xeon CPU.

Images were reconstructed using the TV regulated OS-EM algorithm with different values of the regularizing parameter β (see equation 3.6). The num-ber of subsets was fixed to 6 and the reconstructions were stopped after eight iterations, while the values β = 10−3, β = 10−4 and β = 10−5 were chosen. As a matter of fact, it is known that relatively high values of β significantly decrease the noise but also introduce bias in the images [62]. A reconstruc-tion with β = 0 was also performed, in order to compare results with the standard OS-EM algorithm. The voxel size for the reconstruction was set to 0.420 mm × 0.420 mm × 0.855 mm. Normalization was implemented dur-ing reconstructions, while no corrections for dead-time, decay, attenuation or random counts were applied.

Figure 3.3: GATE simulated phantom. The higher and the lower caps were rendered transparent for a better view of the inside. The central chamber of the phantom can be filled with a radioactive source, as well as the six hollow spheres. In this work, the two blue-rendered largest spheres were left cold.

Contrast ratio 2 4 8

Background activity (MBq/ml) 0.20 0.20 0.19

Spheres activity (MBq/ml) 0.40 0.80 1.52

Table 3.1: Calculated background and spheres activity concentrations in order to obtain the wanted contrast ratios. If each value is multiplied to the associated volume, global activity would be equal to 3.7 MBq.