Università di Pisa
Facoltà di scienze matematiche, fisiche e naturali
H
IGH-
RESOLUTION TIME-
TO-
DIGITAL CONVERTER FORS
IPM-
BASEDT
OF-PET
DETECTORSTesi di Laurea Magistrale
Relatore: Candidato:
Dott. Giancarlo Sportelli Pietro Carra
A
BSTRACTF
ollowing the recent trend of pushing the time resolution of positron emission tomography (PET) systems for time-of-flight (ToF) acquisitions, this thesis de-velops a model of time-to-digital converters (TDCs) based on tapped delay lines, suitable for generic FPGAs. The model allows to understand the time resolution limits of this kind of architecture, which is essentially given by the sum of the setup time and hold time of the registers in the FPGA, i.e., down to roughly 30 ps in state-of-the-art devices.An experimental setup has been realized with a custom-designed PET acquisition system able to acquire signals from two small detectors made of two LYSO crystals of 3 mm x 3 mm x 5 mm coupled to two SiPMs. The developed system features a two-channel TDC based on tapped delay lines, obtained using the carry chains of an Altera Arria 10 SoC-FPGA. This setup has been used to validate the model and to find the main causes of time resolution loss in the TDC. Early measurements made after an initial calibration provided an intrinsic time resolution of about 100 ps, which is sensibly worse than the best expected case. Results showed also that there are strong fluctuations in the propagation delay of the TDC buffers during long acquisitions, probably caused by small temperature instabilities. An ad-hoc firmware component has been then specifically designed to re-calibrate continuously the TDCs. With this new procedure, the best achieved time resolution for a single TDC channel becomes 38 ps. With the LYSO-based detectors, the system shows a coincidence time resolution of 116 ps FWHM, which is comparable with the best results found in literature for similar setups. The implemented data acquisition system uses a very small amount of resources in the FPGA and has a negligible dead time (5 ns) even for the most demanding PET applications.
M
OTIVATIONR
ecently, there has been a steady increase in the incidence of tumours and car-diovascular diseases, together with a substantial growth of cases with an un-favourable outcome. There is a wide consensus that an effective treatment of these conditions depends on an accurate and early diagnosis. One of the main instru-ments that oncologists and cardiologists have at their disposal to investigate these diseases is positron emission tomography (PET). PET is a functional imaging technique that quantitatively measures physiologic parameters detecting the spatial distribution of small amounts of biologic molecules labelled with a positron emitter (radiotracers). Thanks to this technique it is possible to reach a very high specificity (observing very specific molecules) and study biologically active substances without changing their behaviour with the labelling operation. Possible functional processes targeted by PET studies are glucose metabolism or protein synthesis. PET therefore permits precise evaluation of perfusion and metabolism in various organs and tissues and allows for an easy identification of the functional symptoms underlying disease development [1].The spatial distribution of the labelled molecule is constructed thanks to the unique properties of positron decay and annihilation. When a positron encounters an electron, they both annihilate, producing a pair of 511-keV photons that travel from the annihi-lation point in strictly opposite directions. The detection of a so-called coincidence event (i.e., the concurrent revelation of the two photons by two detectors) confines the position of the annihilation of theβ+particle along a line (the line joining the two detectors). This line is called line of response (LoR). By acquiring various events along different LoRs it is possible to reconstruct the distribution of the radiotracer and therefore form a PET image. This intrinsic collimation is what allows PET to be much more sensitive (over 100 times) than other nuclear imaging techniques that use physical collimators, like single photon emission computed tomography (SPECT).
There is now a huge amount of data regarding PET impact on healthcare and studies show that PET results alter patient treatment in more than 25% of cases [2]. Aside from what has been described, PET has also other interesting applications, like dose-monitoring in hadron-therapy and pharmacokinetics studies in small animals.
Finally, PET will play a crucial role in the most promising sector in healthcare: personalized medicine, i.e., medicine tailored specifically to every patient, based on his own genetic profile and his own attitude to drugs. Individual genetic predisposition
also very specific and sensitive functional data are necessary. In fact while many of the major diseases of the Western countries (cancer and heart disease for example) undoubtedly have a genetic component, they arise from highly complex and unpre-dictable interactions between genetic products and many external factors depending on environment and life style. Because of this, genetic profiles may help in determining a risk factor associated to each patient but this risk will not define neither whether this person will develop an illness, nor when such an illness will become evident to our diagnosis instruments.
This is the reason for the importance of PET in this field: while it cannot generate high resolution images like CT or MRI scans, its sensitivity and ability to detect and observe specific molecules is unrivalled (down to nm/ml or even to pm/ml) [3]. These characteristics therefore allow PET to detect the first signs of a specific disease, since they come in the form of an alteration at the molecular and functional level, not at the anatomical level, more easily accessible to the other techniques. Hence, early detec-tion of these phenomenons is dependent upon funcdetec-tional and highly sensitive imaging modalities. The synergy between functional imaging and genetic profiling is at the basis of personalized medicine: the tailoring of the treatment to the exact profile of the patient and to how he responds to different drugs.
However, conventional PET suffers from the fact that some of the information on the position of the annihilation is lost in the detection process. In fact PET can only detect the line in space in which the annihilation has occurred (the LoR), but not the precise point in that line. Time-of-flight PET (ToF-PET), i.e., PET that precisely measures the instant of arrival of the annihilation photons, is able to recover that information by measuring the difference between the time of arrival of the two photons to the detectors. This technique has several advantages: it creates better quality images with a lower amount of radiotracer injected into the patient (and thus a lower amount of radioactive dose administered) and allows to make PET scans shorter, decreasing costs. Another way of seeing this effect is that ToF is a sensitivity enhancer: ToF-PET images are, on average, more sensitive than standard PET images acquired with the same amount of radiotracer; this makes it possible to detect even smaller quantity of molecules. Also ToF information greatly reduces noise in images acquired with a non-complete angular coverage, as it happens in hadron-therapy. Thus ToF-PET is showing to be an incredibly valuable tool, especially considering the new developments in the detector and electronics field that are making the high precision measurements that ToF requires possible.
In this thesis, a series of instruments, models and techniques are developed to study the performances of electronics and detectors with respect to the specifics necessary for ToF-PET. Actually, the timing precision required for ToF information to be of practical
use are very hard to meet, and their assessment requires careful and thorough study, but it is now becoming of utmost importance. In fact, it is only thanks to the sensitivity enhancement brought by ToF, that PET will be able to represent a valid instrument of analysis in the field of personalized medicine, allowing for a precise and early diagnosis of the most subtle diseases. Furthermore, thanks to recent advantages in scintillator technology (with the discovery of new, fast materials like LSO and LaBr3) and the
development of new photodetectors (digital silicon photomultipliers), it is possible to reach timing resolutions in the order of the hundreds of ps, making ToF technically possible and powerful enough to provide that gain in sensitivity so much requested by personalized medicine.
A
CKNOWLEDGEMENTSI
would like to thank David Gascón and David Sánchez (ICCUB, Barcelona), and Jesús Marín (CIEMAT, Madrid) for providing the flexToT prototype boards that have been used in the experimental setup of this thesis.T
ABLE OFC
ONTENTSPage
List of Tables xi
List of Figures xiii
1 Introduction 1
1.1 Basis of positron emission tomography . . . 1
1.2 Physics . . . 2
1.2.1 The beta decay . . . 3
1.2.2 Positron annihilation . . . 3
1.3 Type of events in PET . . . 4
1.4 Spatial resolution in PET . . . 6
1.5 Time of Flight PET . . . 8
1.5.1 Basic principles of Time of Flight PET . . . 8
1.5.2 Signal to noise gain due to ToF . . . 9
1.5.3 State of the art of ToF . . . 12
1.6 The benefits of ToF-PET . . . 14
1.7 Detectors used in PET . . . 15
1.7.1 Scintillators . . . 15
1.7.2 Photodetectors . . . 21
1.8 ToF grade acquisition electronics . . . 28
1.8.1 Non-linearity corrections . . . 32
1.9 Dead time . . . 33
1.10 Objectives of this thesis . . . 34
2 Materials and methods 37 2.1 Experimental setup . . . 37
2.2 Description of the FlexTOT ASIC . . . 41 ix
2.3 TDC: theoretical model . . . 43
2.3.1 Delay line model . . . 43
2.3.2 Differential and integral non-linearities . . . 44
2.3.3 Calibration . . . 45
2.3.4 Resolution . . . 46
2.4 TDC: functional description . . . 48
2.4.1 Hardware components . . . 49
2.4.2 Software components . . . 55
2.4.3 Key issues in the realization of the TDC. . . 56
2.5 Tests and procedures . . . 60
2.5.1 Calibration time . . . 60
2.5.2 DNL and INL measurement . . . 60
2.5.3 Time resolution . . . 61
2.5.4 Effects of periodic calibration . . . 62
2.5.5 Dead time . . . 62
2.5.6 Detector characterization . . . 63
3 Results and discussion 65 3.1 TDC characterization . . . 65
3.1.1 Calibration . . . 65
3.1.2 DNL and INL measurement . . . 68
3.1.3 TDC resolution . . . 74
3.1.4 Effect of periodic calibration on resolution . . . 78
3.1.5 Dead time and transfer bandwidth . . . 80
3.2 System characterization . . . 81
3.2.1 Energy resolution . . . 81
3.2.2 Coincidence time resolution . . . 82
4 Conclusions and future works 85
Bibliography 87
L
IST OFT
ABLESTABLE Page
1.1 Most common radiotracers and their characteristics [4]. . . 2 1.2 Most common scintillators and their characteristics, data taken from [5]. . . 20 1.3 Most common photodetectors and their characteristics, data taken from [5] 27 2.1 Specifics of the SiPMs used [6]. . . 38 2.2 Specifics of the scintillators used [7]. . . 38 2.3 Specifics of the FlexTOT ASIC. . . 42 2.4 Maximum theoretical data transfer bandwidth for different word sizes. . . . 63 3.1 Measured data transfer bandwidth for different word sizes. . . 80 3.2 Results of the time resolution measurements. . . 82
L
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IGURESFIGURE Page
1.1 Type of events in PET . . . 6 1.2 Origin of errors due to finiteness of the detector pixels (a) and to parallax (b) 7 1.3 Difference in the localization of the radioactive source using non-ToF PET
(a) and ToF-PET (b). . . 9 1.4 Structure of the energy levels of a scintillator material. . . 16 1.5 Curves representing two scintillation timing processes: an excitation process
characterized by a non-zero rise time constant and a simple decay process. The combination of these two processes represents a more realistic behaviour of the temporal evolution of a scintillation signal: it increases rapidly, but not instantaneously, up to its maximum and then slowly decreases. . . 17 1.6 CTR for different photoelectrons of different orders (first, third and fifth),
calculated with single- (dashed) and bi-exponential (continuous line) timing models. . . 18 1.7 Experimental validation of the model. Image and study from [8] . . . 19 1.8 Path of a photon inside a scintillator. . . 20 1.9 Effect of magnetic field on the crystal decoding accuracy of a conventional
PMT-based PET block detector (a) and of an APD-based PET detector (b), image from [5]. . . 21 1.10 Schematic representation of a depletion region. . . 22 1.11 Intensity of the current generated by photons versus the bias voltage in an
APD. . . 23 1.12 Structure of a SiPM composed by a matrix of SPADs (right). Each cell (left)
is a p-n junction with the quenching circuit attached in series. . . 24 1.13 Schematic of a passive quenching circuit, when a cell triggers some of
the current generated goes through the quenching resistor lowering the overvoltage. . . 25
1.14 Time measurements with a START pulse (pulse 1) and a STOP pulse (pulse
2). . . 28
1.15 Time stretching method. A capacitor is charged for a time T (the interval to be measured) with a current I1−I2and then discharged with a lower current I2. The total time taken for this process is Tm= (K + 1)T, with K = (I1− I2)/I2 31 1.16 Time-to-amplitude conversion. A capacitor is charged during the interval to be measured with a known current. The voltage is then held briefly to allow A/D conversion and then reset. . . 31
1.17 Vernier method. The START and STOP pulses enable two oscillators with different frequencies, two counters keep track of the number of cycles, when two cycle edges coincide the value of the counters is sampled. From these two values it is possible to find the duration of the time interval between START and STOP. . . 31
1.18 Tapped delay line. The START pulse is fed into a sequence of delay elements. The state of this elements is then sampled when the STOP pulse is reached and put in an array of registers. . . 32
1.19 Behaviour of the measured count rate as a function of the number of events reaching the system (log-log scale). . . 33
2.1 Schematic of the experimental setup. . . 38
2.2 PDE of the SiPMs used [6]. . . 39
2.3 Emission spectrum of the scintillators used [7]. . . 39
2.4 Schematic of the input stage of one channel of the FlexTOT ASIC. Image from [9]. . . 42
2.5 Plot of the quantization errorσ2as a function of the chosen calibration value. 46 2.6 Example of a condition that could give rise to metastability. . . 47
2.7 Schematic of the TDC architecture. . . 50
2.8 Schematic of an adder. . . 51
2.9 Block diagram of a Wallace Tree encoder. . . 52
2.10 Diagram of the coincidence processor state machine. . . 54
2.11 View of the LAB structure of an FPGA. . . 57
2.12 View of an algebraic module in the FPGA. The parts that are key for the functioning of the TDC are highlighted in green. Note that the adder cells are connected to the registers immediately next to them. Image taken from Chip Planner, Quartus Prime (Intel Corp. Santa Clara, CA). . . 58
LIST OFFIGURES
2.13 View of an algebraic module in the FPGA. The correct path the signal must follow is highlighted in green. A possible deviation is shown in the cell below. Image taken from Chip Planner, Quartus Prime (Intel Corp. Santa Clara, CA). 59 3.1 Standard deviation of the bin widths measured with different numbers of
events. . . 66
3.2 Duration of the calibration procedure when using different numbers of events. 66 3.3 Histogram of the event counts in each bin, the first and last ones are empty (250.000 total events), first TDC channel. . . 67
3.4 Plot of the measured DNL for each bin (first TDC channel). . . 68
3.5 Periodic structure of the DNL, obtained from averaging together groups of 20 bins. . . 69
3.6 Periodic structure of the bin width, obtained from averaging together groups of 20 bins in a 2.000 element long TDC. . . 70
3.7 Discrete Fourier transform of the bin width function in Fig. 3.6. . . 71
3.8 Most relevant frequencies in the Fourier transform in Fig. 3.7. . . 71
3.9 INL of the delay chain, calculated from the DNL (first TDC channel). . . 72
3.10 INL can be viewed as the difference between the timestamps associated to the real bins widths and the timestamps associated to ideal bins (those of a chain with DNL=0, thus perfectly linear). The red dots are the real timestamps, the blue line represent perfect linearity. . . 73
3.11 Zoom of Fig. 3.10. . . 73
3.12 Histogram of the time delay measurements obtained with the method of the two clocks (first TDC channel). . . 74
3.13 Accuracy of the TDC, assessed with the method of the two clocks.µis the real value to be measured, m is the measured value (as the position of the peak of the gaussian that best fits the distributions in Fig. 3.12 (first TDC channel). . . 75
3.14 Histogram of the 14 different resolution values measured as the FWHM of the gaussian that best fits the distributions in Fig. 3.12 (first TDC channel). 76 3.15 Histogram of the 138 different resolution values measured with the method of the delayed TDC clock (first TDC channel). . . 77
3.16 Histogram of the ToF difference measured by the two TDC channels (CTR). The events are generated splitting one signal with two cables introducing different delays. The gaussian that fits the distribution of delays has an FWHM of 54.9 ps . . . 77
3.17 Series of measurements taken without calibration. Expected (real) value is 729.9 ns (yellow line). First TDC channel. . . 78 3.18 Series of measurements taken with calibration. Expected (real) value is
729.9 ns (yellow line), in this case the experimental results are much more accurate. First TDC channel. . . 79 3.19 Plot of the energy spectrum, the 511-kev peak is fitted with a gaussian (blue
line). . . 81 3.20 Plot of the energy resolution as a function of the overvoltage in the SiPM. . 82 3.21 Histogram of the timing distributions for different source positions; from
left to right: -1 cm, 0 cm and 1cm from the center of the field of view of the instrument. . . 83 3.22 Gaussian fitting the distribution with 100 ps expected mean. Measured
value is 100.2 ps, FWHM=115 ps. . . 83
C
H A P T E R1
I
NTRODUCTIONI
n this chapter, first the physical principles of the PET imaging are introduced, then, the various factors that contribute in defining the spatial resolution in a PET image are discussed, with a particular focus on how the time-of-flight (ToF) technique could be used to improve the quality of the images produced with PET. Finally, the technology of PET detectors and their characteristics and performances are described, together with a review of the most important methods used for high-resolution time measurements.1.1
Basis of positron emission tomography
PET imaging is based on the detection ofγphotons generated during the annihilation of positrons. These particles are emitted by a radioactive tracer injected inside the object under examination. In the clinical practice, the tracer is administered as an intravenous injection of a chemical compound, labelled with a short-living β+ emitter. The tracer will then distribute in the various organs, following the path of the biologically active molecule to which it is bound. By measuring the activity distribution inside the patient, a PET scan allows to study the metabolism of a particular organ or tissue, its physiology (functionality) and its biochemical properties. In this way, PET may detect biochemical changes in an organ or tissue that can identify the onset of a disease process before anatomical changes related to the disease can be seen with other imaging processes, such as computed tomography (CT scan) or magnetic resonance imaging (MRI). Various
Radiotracer Half life (min)β+ mean range in H2O (mm)
18F 109.8 1.4
15O 1.7 2.7
11C 20.4 1.7
13N 10.0 2.0
Table 1.1: Most common radiotracers and their characteristics [4].
tracers can be used and attached to various biochemical molecules to study different organs and diseases. The most common are listed in table 1.1.
The most widely used tracer is 18F – FDG (fluorodeoxyglucose). Chemically, the FDG behaves like a glucose molecule and thus it can be used to map the glucose consumption in the body. This characteristic is helpful to diagnose cancer, since tumoural cells are characterized by a very high glycolytic activity [10], but also to image those regions that are normally using high quantity of glucose, like brain and kidney [11],[12].
PET is used both in clinical and research applications, the most important of which are oncology, for cancer diagnosis and management and cardiology, for the study of miocardial perfusion. PET also finds applications in neurology for the diagnosis of neurological pathologies like dementia and in pharmacokinetics. In fact in pre-clinical trials it is possible to radiolabel a new drug and inject it into animals. The uptake of the drug, the tissues in which it concentrates, and its eventual elimination, can be monitored far more quickly and cost effectively than with the older technique of killing and dissecting the animals to discover the same information ([13]).
Lately, scanners that combine multiple imaging techniques like PET and computed tomography (PET-CT) [14],[15] and PET with magnetic resonance imaging (PET-MRI) [16],[17] have been developed. PET-MRI in particular poses new challenges in the devel-opment of PET scanners, given that the high intensity magnetic fields used in MRI could interfere with the rest of the electronic system.
Another relatively new field of application of PET is the dose (the quantity of radiation that the patient receives) monitoring in the treatment of cancer with high energy hadrons (hadron-therapy) [18], [19].
1.2
Physics
PET is an emission based imaging technique, meaning that the source of radiation that will produce the image is inside the scanned body. This is in contrast with transmission techniques such as CT or X-RAY, in which the image is produced by radiation that is
1.2. PHYSICS
generated outside the scanned body and passes through it. Emission-based techniques have the advantage of being able to image functional processes, but have to deal with a problem that is not present when using transmitted radiation: the position of the radioactive source is not known a priori. The simplest way to solve this problem is using collimators, but this comes at the cost of losing the majority of the radiation. PET, however, mitigates this loss detecting the photons emitted during the annihilation of positrons. Positron decay involves the simultaneous emission of photons; these photons are two in the vast majority of cases, and in first approximation co-linear; thus the detection of a so-called coincidence event (i.e., the concurrent revelation of the two photons by two detectors) confines the position of the annihilation of theβ+particle along
a line (the line joining the two detectors). This line is called line of response (LoR). By acquiring various events along different LoRs it is possible to reconstruct the distribution of the radiotracer and therefore form a PET image.
1.2.1 The beta decay
Positrons are particles produced in theβ+decay, in which a proton inside a nucleus is
converted into a neutron, releasing a positron and an electron neutrino. The decay of a radioactive source follows the well-known exponential law:
(1.1) N(t) = N0e−λ(t−t0)
Where N0 and N(t) respectively represent the number of radionuclides present at
time t0 and t andλis the decay constant of the process. The activity of the radioactive
source, which is equal to the rate at which positrons are generated, is defined as the number of decays per second:
(1.2) A(t) = −dN
dt =λN(t)
This quantity is proportional to the number of positrons generated by the source and, in first approximation, to the total intensity of the PET image and to the radioactive dose received by the patient.
1.2.2 Positron annihilation
Positrons travel through matter, losing energy in inelastic collisions and Coulomb in-teractions. The energy loss stops when the positron has reached thermal equilibrium with the environment surrounding it and annihilates with an electron. The annihilation usually produces twoγphotons, that obey to the energy and momentum conservation rules. Let us first consider a simplified case, in which the electron and the positron
are both at rest when they collide. The energy released by this reaction will then be equal to the sum of the masses at rest of the two particles: 1022 keV. In this case, the emitted photons have an energy of 511 keV each and are co-linear. The assumption of co-linearity of the photons is fundamental for PET reconstruction algorithms, however it is an approximation: the physics of the positron decay is more complex and puts a limit in the spatial resolution achievable. What happens is that the positron and the electron do have a residual momentum before annihilation. Furthermore, when the positron reaches thermal energies, it can interact with the electrons surrounding it, forming bound states in which the positron and one electron revolve around their center of mass [20]. This state is called positronium and can exist in two forms: a triplet state with spin S=1, named ortho-positronium, and a singlet state with spin S=0, named para-positronium. To fully model theγemission in the positron-electron annihilation these two states must be taken into account:
• para-positronium state, in water, is formed 25% of the times and has a half-life of 0.1 ns. Given that in water the annihilation process takes place in approximately 1.8 ns, this state almost always decays via self-annihilation;
• ortho-positronium state, in water, is formed 75% of the times and has a half-life of 140 ns; therefore it is much more likely that the positron decays by interacting with yet another electron. This process is called pick-off annihilation and is responsible for the majority part of the angular deviation between the twoγphotons.
It is also important to note that sometimes the annihilation can emit three photons, however the cross section of this process is negligible compared with the standard two-photon emission:σ3γ'372σ2γ [21].
1.3
Type of events in PET
In PET, both annihilation photons must be detected without having previously interacted for an event to be considered as a true one. Every single hit registered by the detectors is analyzed and its time and energy are estimated. When two single events are validated for energy (i.e. their energy is found to be compatible with that of a 511-keV photon) and occur within a predefined coincidence time window, a coincident event is counted. The precision with which energy and time of an event can be determined is limited by the performances of the detectors and the electronic system. As a consequence, there is a background of undesired events being counted as true, that degrades image contrast and signal to noise ratio (SNR). These events can be classified in three groups:
1.3. TYPE OF EVENTS IN PET
• Scatter coincidences. These occur when one or both photons undergo Compton scatter interactions, losing the information regarding the position of the annihi-lation process that generated them. The Compton scatter can happen both in the patient and in the detector. These events can be recognized and rejected with an accurate measurement of the photon energy, given that the Compton scatter is inelastic [22].
In PET-CT it is also possible to use CT data to estimate the scattering events [23]. The energy window for the acceptance of a true event should not be made too narrow, otherwise the detection efficiency of the system would be very poor. Typically, photons with energy in the range 350-650 keV are accepted [5]
• Random coincidences. A random coincidence event happens when two positrons annihilate approximately at the same time and one photon from each decay is detected by the scanner within the coincidence time window. The random event rate (Nrandom) between two detectors A and B can be estimated as a function of the single event rate of the two detectors (NA, NB) and of the amplitude of the
coincidence window (2τ):
(1.3) Nrandom= 2τNANB
A direct measurement of the random coincidences can also be performed with the so-called delayed window technique: events registered at a certain time by a detector are put in coincidence with those of another detector registered at a later time, with a delay much larger than the coincidence window. Being the events recorded in these two time frames completely uncorrelated, the coincidences detected are all random.
• Multiple coincidences. These are events in which more than two photons are detected within the coincidence time window. They can easily be characterized and rejected.
All the events that are registered as coincidences (regardless of what they really are) will be referred to as "prompt" events throughout this text.
Sometimes also one of the two annihilation photons is attenuated by the scanned body. In this case the coincidence is lost. However it is possible to correct for this factor by appropriately multiplying the number of counts registered in each LoR for the correct attenuation coefficient. These coefficients can be calculated as such:
(1.4)
Z
LoRµ
Scatter coincidence Random coincidence
Detector ring
Figure 1.1: Type of events in PET
Whereµ(x) is the attenuation coefficient of the material in position x along the LoR. µcan be extrapolated from previously acquired CT or MR images.
1.4
Spatial resolution in PET
From the description of the physical process that is at the basis of PET imaging, we can identify two fundamental limits to the maximum achievable spatial resolution. The first one comes from the finite positron range, i.e. the finite distance that the positron travels before annihilation. The second one is the deviation from co-linearity that results from the effects described in section 1.2.2. Spatial resolution (R) in PET is therefore a combination of these two physical factors and of the intrinsic resolution of the detection system. Resolution can be described as a quadrature sum of independent contributions and summarized in the following formula [24]:
(1.5) R = a
q
b2+ c2+ (d/2)2+ p2+ r2
Now let us analyse all the elements of this equation in detail:
• a is a multiplicative degradation factor that accounts for the reconstruction algo-rithms used to form the image. Its value is normally in a range between 1 and 1.3.
• b is the decoding error. In a pixel detector, a group of pixels can be multiplexed to the read-out electronics. This analogue multiplexing may introduce possible uncertainties on the position of the event that triggers the signal [25].
• c is the error caused by the acollinearity of the two annihilation photons. A good approximation of the entity of this effect is considering a 0.25°standard deviation
1.4. SPATIAL RESOLUTION IN PET
(a)
(b)
Figure 1.2: Origin of errors due to finiteness of the detector pixels (a) and to parallax (b)
from perfect collinearity (in water), thus the impact on resolution is given by:
(1.6) c = 0.5 D × tan(0.25°) = 0.0022 · D
where D is the distance between the detectors that revealed the coincidence. This error essentially depends on the detector ring diameter, giving a 2 mm FWHM for whole body scanners [5].
• d/2 is an error due to the finiteness of the detector pixel size (d). In fact the reconstruction algorithms always consider the events measured in a certain pixel to be at its center and this introduces an error proportional to the pixel size (Fig. 1.2a).
• p is the effect of parallax: it is caused by the fact that the photon can interact with the detector at different depths, while the event is always assumed to have happened at the surface (Fig. 1.2b). The entity of this error is a function of the position of the annihilation point with respect to the center of the field of view (l), the radius of the field of view (L) and a coefficient depending on the type of detector used and its physical properties (α) [26]:
(1.7) p =αp l
l2+ L2
• r is the uncertainty given by the positron range. The quantity that PET aims to determine is the distribution of theβ+ emitter inside the field of view, however what it actually measures is the position of the positron annihilation, which has travelled a finite distance from where it originated. The entity of this effect depends on the maximum emission energy of the positron and consequently on the radionuclides used. The overall impact of this factor is a blurring of '0.5 mm for18F, '1.1 mm for 11C, '1.5 mm for 13N, '2.4 mm for15O and ' 6 mm for82Rb [27].
1.5
Time of Flight PET
1.5.1 Basic principles of Time of Flight PET
SNR can be enhanced by confining the position estimation of the positron annihilation along the LoRs using precise Time-of-Flight information. Time information is used in conventional (or non-ToF) positron emission tomography to determine if two detected photons are in time coincidence and therefore belong to the same positron annihilation event. Each detected photon is tagged with a detector position and detection time: if the detection time difference between two photons is smaller than a pre-set coincidence window (traditionally 5-10 ns), the two events are considered physically correlated to the same annihilation event. Such measured difference in detection time is directly related to the actual photon time-of-flight difference, blurred by a measurement uncertainty, the time resolution, which depends on several instrumental factors.
Conventional PET reconstruction uses the time information only to identify the line along which the annihilation occurred. It is unable, though, to determine which point along the line is the source of the two photons; therefore all the points along the line are given the same probability of emission (1.3a). Analytical or iterative reconstruction algorithms are used to estimate the activity distribution most consistent with the measured projection data. ToF-PET uses the difference of the time of arrivals of two coincidence photons to better locate the annihilation position of the emitted positron (1.3b).
The time-of-flight difference t is directly related to the distance x of the annihilation point from the center of the field of view (FOV) (x = ct/2) along the LoR identified by the two detectors in coincidence. The time of flight of a certain photon is proportional to the distance between the point it originated from, and the detector that revealed it. Using this information it is possible to localize the events along the LOR identified by the two detectors that revealed them. The limitation in our ability to localize the annihilation point is due to the finite time resolution ∆t of the coincidence system, therefore, a probability function and not an exact position will be assigned to each event. The events can therefore be positioned along the LOR with an uncertainty equal to the full width at half maximum (FWHM) of the probability function∆x =c∆t/2, where c=30 cm/ns is the speed of light. According to this formula, a timing resolution of 500 ps would confine the position of the annihilation with a precision of 7.5 cm. It is evident that a precision smaller than the imaged object size would result in a significant reduction of statistical noise in the image.
1.5. TIME OF FLIGHT PET
(a)
dx
(b)
Figure 1.3: Difference in the localization of the radioactive source using non-ToF PET (a) and ToF-PET (b).
1.5.2 Signal to noise gain due to ToF
The mechanism through which ToF improves quality in PET images can be described as follows, starting from the approach by Brownell and Strother [28], who equate the SNR in conventional PET with the square root of the noise equivalent count rate (NECR). Their work has been further expanded by Conti in [29].
With the assumption of a uniform activity distribution located in a cylindrical object of diameter D, at the center of the field of view, assuming that Nrandom and Nscatter are
known without uncertainties, the relation between SNR and NECR can be expressed by:
(1.8) SN R =pN ECR = n−1/2
µ N
true2
Ntrue+ Nscatter+ kNrandom
¶1/2
With n being the number of image elements along a LoR and thus contributing to form the image and k being equal to 1 if randoms are measured from singles and 2 if randoms are measured with the delayed coincidence window technique [28]. This expression derives from the fact that the SNR in a specific point e (SNRe) of the image is proportional to
the variance V ARe in the counts of the true events attributed to that specific point as
follows:
(1.9) SN Re= const · Ntrue,eV ARe−1/2
Where Ntrue,eare true coincidence counts in e and V ARe is the variance which weights
the variance of the samples from each LoR passing from e for their contribution to the LoR. If we assume a uniform cylinder of diameter D and an image element size of d × d, we can estimate the number of true events in the element at the center of the cylinder as
a function of the total number of trues in the image:
(1.10) Ntrue,e' Ntrue d
2
π(D/2)2
This expression being valid for both ToF and non-ToF PET. Thus the value of the SNR can be found from the value of V ARe, which can be expressed as:
(1.11) V ARe=
X
LoR | e∈LoR
wLoRV ARLoR
With wLoR being a weight factor that takes into account the contribution of that specific LoR to the element e. Given that Ntrue,LoR= Nprom pts,LoR− Nscatter,LoR− Nrandom,LoR,
that we can assume that Ntrue,LoR follows Poisson statistics and that the variances of
the estimates of randoms and scatters are negligible [28], we can conclude that the variance of the prompt events is equal to the variance of the true ones, and thus V ARLoR
will be proportional to Nprom pts,LoR= Ntrue,LoR+ Nscatter,LoR+ Nrandom,LoR. Assuming
a cylindrical symmetry, the weights attributed to each LoR are all equal and therefore V AReis simply proportional to the number of prompts:
(1.12) V ARe∝ Ntrue,LoR+ Nscatter,LoR+ Nrandom,LoR
But Ntrue,LoR, Nscatter,LoR, and Nrandom,LoRare different in the case of ToF and non-ToF
PET. Before continuing, let us introduce the following definitions: • DFOV: diameter of the scanner field of view;
• D: diameter of the object to be images; • d: side of a square image element; • ∆t: time resolution of the ToF-PET;
• ∆x: distance of the annihilation from the center of the field of view ∆x = ∆tc/2.
Variance in conventional PET. The three terms on the right side of equation 1.12
can be calculated as such:
• Ntrue,LoR: average number of trues in an image element (equation 1.10) times the
number of object image elements along the LoR:
(1.13) Ntrue,LoR= Ntrue
d2 π(D/2)2·
D d
1.5. TIME OF FLIGHT PET
• Nscatter,LoR: average number of scatters in an image element times the number of object image elements along the LoR:
(1.14) Nscatter,LoR= Nscatter
d2 π(D/2)2·
D d
This assumes that the scatter is uniform in the object and null outside it;
• Nrandom,LoR: average number of randoms in an image element times the number of image elements (NOT only those in the object) along the LoR:
(1.15) Nrandom,LoR= Nrandom
d2 π(DFOV/2)2·
DFOV
d This assumes that the randoms are uniform in the whole FOV.
Variance in ToF-PET. The three terms on the right side of equation 1.12 can be
calculated as such:
• Ntrue,LoR: average number of trues in an image element (equation 1.10) times the
number of object image elements along the LoR that contribute to form the image in e: (1.16) Ntrue,LoR= Ntrue d2 π(D/2)2· ∆x d
This time not all the elements in the LoR are counted, since ToF restricts the number of elements contributing to each point in the image.
• Nscatter,LoR: average number of scatters in an image element times the number of
object image elements along the LoR that contribute to form the image in e:
(1.17) Nscatter,LoR= Nscatter
d2 π(D/2)2·
∆x d
Assuming that the scatter is uniform in the object and null outside it;
• Nrandom,LoR: average number of randoms in an image element times the number
of image elements (NOT only those in the object) along the LoR that contribute to form the image in e:
(1.18) Nrandom,LoR= Nrandom d
2
π(DFOV/2)2·
∆x d This assumes that the randoms are uniform in the whole FOV.
Substituting these values in equation 1.9, we can find the two expressions for SNR in case of conventional and ToF-PET:
(1.19) SN R2(conv) = 4 π d D 3 N true2
Ntrue+ Nscatter+ (D/DFOV)Nrandom
(1.20) SN R2(T oF) =4 π d D 3 D ∆x Ntrue2
Ntrue+ Nscatter+ (D/DFOV)2Nrandom
Rewriting these equations in terms of the random fraction Rf = Nrandom/(Ntrue+Nscatter)
and introducing an auxiliary variableβ= D/DFOV (patient size over FOV diameter), we can express SNR(T oF) as a function of SNR(conv):
(1.21) SN R(T oF) = s D ∆x · s 1 +βRf 1 +β2R f · SNR(conv)
From this last formula we can identify two contributions that ToF information brings in the PET image reconstruction: (1):q∆xD , a reduction of statistical noise in the image caused by the reduced number of image elements along each LoR that contribute to form the image in a specific point (D/∆x), and (2):
r
1+βRf
1+β2R
f ·, a reduction in noise caused
by random events. This last contribution is often neglected in ToF models and is more important when Rf is high or, equally, when the field of view of the scanner is large. Given
that one of the recent trends in PET imaging is the enlargement of the scanner FOV, to increase sensitivity, the importance of including ToF information in future devices is growing.
1.5.3 State of the art of ToF
The system time resolution is a key parameter that qualifies the performances of a ToF-PET scanner. If only two detector elements A and B are considered, the time resolution can be measured by placing a point source between the detectors, and measuring the FWHM of the distribution of time of flight difference between detector A and B. This value is often called coincidence time resolution, CTR.
In the present and past generations of ToF scanners, the detector is based on an inorganic crystal that converts the high energy photon into light photons, coupled with a light sensor and associated to an electronic circuit. The next sections describe the factors that influence time resolution.
Scintillators The intrinsic material characteristics of the scintillator are the first and
1.5. TIME OF FLIGHT PET
the time resolution is determined by the timing characteristics of the scintillator and its absolute light output. Materials with high light output and fast decay and rise time, for example LaBr3 and LSO, are good candidates for ToF-PET. A more thorough discussion
on scintillators follows in section 1.7
The crystal geometry and surface finish have also a direct and indirect effect on the time resolution. A surface finish with smaller reflectivity decreases the light output of the crystal, thus having an indirect degrading effect on time resolution. A long crystal implies a path length dispersion and consequently a transit time dispersion of light in the detector on the order of 50 ps/cm for a material with a refractive index of 1.7 [30].
Photosensors The photosensors commonly used in commercial PET scanners are
PMTs. While PMTs are a very sound and diffused technology in PET, a one-to-one cou-pling with scintillator crystals is often difficult, due to the challenges in the reduction of PMTs dimensions [5]. Also the incompatibility of PMTs with MR technology ([31],[32]) and the complexities behind the realization of a great number of PMTs with consistent properties may come in favour of future ToF implementations based on SiPMs [33].
Electronics The signal from the light sensor is fed into the electronic chain for time
triggering and stamping, which usually comprises a discriminator and a time-to-digital converter (TDC). Time jitter from electronics components, time slewing due to pick up of smaller amplitude signals and non optimal TDC design can degrade the time resolution of a PET scanner. While the electronics could be a major limitation for time resolution, at the state of the art, its contribution can be made negligible with respect to the other factors like scintillator properties with a careful design [34]. However, with the need of always-higher timing resolutions [35],[18], coupled with the demand of high-bandwidth data acquisition systems [36], there is a great pressure on developing flexible electronics that has better timing properties, high maximum acquisition rates and low development and realization costs.
Different companies in the domain of medical imaging have now all introduced time-of-flight technology in their whole body PET/CT and some in their PET/MRI systems. These commercially available scanner use LYSO as a scintillator and their timing properties range from the 540 ps CTR of Discovery 690 (GE) to the 316 ps of Vereos Digital (Philips) [37].
Prototype systems based on LaBr3 are reaching better time resolutions. The
read by PMTs, reaching a 375 ps CTR [38] and one read by SiPMs (245 ps CTR) [39]. Both these scanners do not estimate depth-of-interaction (DOI) information and thus ToF measurement suffer from an added uncertainty since the length of the photon path inside the crystal is not known. A LaBr3 PET detector with good ToF resolution and
two-level DOI discrimination was constructed and timing resolutions in the range of 150–200ps were obtained [40].
1.6
The benefits of ToF-PET
There are different ways to use the improved SNR associated with ToF-PET, but there are also other advantages deriving from the spatial and temporal information embedded in ToF data. First of all, the gain in SNR can be viewed as a sensitivity amplifier, i.e. the image reconstructed with ToF information is less noisy, as if a higher number of events had been acquired. This virtual increase in sensitivity can be useful in different ways [41]:
• given a fixed scan time and a fixed amount of injected radiotracer, ToF can provide better quality images, improving diagnosis and lesion detection. At the moment scanners do not make use of the full spatial resolution available because it is preferred to use a larger image pixel size (about 4 mm) to increase the number of counts per pixel and thus reduce noise. The lower noise of ToF reconstruction allows the full exploitation of the resolution limit of PET scanners (2 mm). • The scan time can be shortened preserving the image quality, improving patient
comfort and clinical workflow and reducing artefacts caused by patient movements; • the amount of radiotracer administered to the patient can be reduced maintaining scan time and image quality unaltered. Presently there is great interest in reducing the dose of a combined PET-CT study. Reducing the dose of a PET exam would not only reduce cancer risk induced by radiation in the patient, but also the occupational radiation exposure of clinical staff performing the exam.
However there are also less immediate advantages in the use of time of flight, in particu-lar the additional information provided by ToF allows to overcome missing or inconsistent data [42]. This is the case for example of incomplete angular coverages caused by the use of specific scanner architectures like that of in-beam systems for hadron-therapy treatment facilities. Also there is evidence that ToF information tends to compensate for inaccurate scatter and attenuation corrections [43]. Finally, a longer acquisition time is typically necessary for larger patients, however this does not usually compensate for the
1.7. DETECTORS USED IN PET
poor quality of the data. Given that the increased sensitivity gain provided by ToF is higher for larger patients, ToF acts as an equalizer, bringing the image quality in larger patients closer to that in patients of average size.
1.7
Detectors used in PET
Different types of detectors can be used in PET scanners, the most utilized being scintil-lating materials coupled to photosensors.
Scintillators are materials able to absorb light at a high frequency and emit it at a lower frequency, usually in the visible or UV range. This process is very important since most photosensors are insensible to 511 keV photons.
1.7.1 Scintillators
Scintillation process. Scintillator materials have the property of emitting visible or
UV photons when they interact with radiation. Inorganic scintillators are crystals of salts containing a small amount of impurities that activate the scintillation processes. Their luminescence is due to the particular structure that the electrons energy levels have in crystals.
When a photon interacts with an electron (either one in the valence band, VB, or in a deeper band) the latter absorbs energy from the former and can be promoted in the conduction band (CB), leaving a hole behind. The electron can subsequently undergo Coulomb collisions with other electrons, exciting them and gradually losing energy. In this way the energy of a single photon is transferred to a great number of electrons that can populate the CB (Fig. 1.4). After a certain time, the electrons de-excite, filling the holes in the VB. During the de-excitation process they can emit energy in various forms, one of this is the emission of a photon.
The emitted light, however, has an energy to which the material is very sensible, since it is equal to the energy difference between the CB and the VB, thus it can be reabsorbed very easily. By introducing impurities (i.e., other chemical elements) in the crystal it is possible to create allowed states (activator states) in the forbidden band; facilitating the emission of visible light from the scintillator.
The number of emitted scintillation photons (Nph) following the absorption of a
photon of energy E (photoelectron yield) can be expressed by the following formula:
(1.22) Nph=
E
Conduction band Donor levels Valence band Energy Radiative emission Eccitation
Figure 1.4: Structure of the energy levels of a scintillator material.
Where Ega pis the energy gap between CB and VB,βis a coefficient (>1) that takes into
account the fact that the electrons can de-excite without emitting radiation,Ψc−a is the efficiency with which the electrons are transferred from the CB to the activator states andΨa−vis the efficiency of the transfer between the activator states and the VB.
The energy of the emitted photons depends on the gap between the activator states and the VB and is usually in the order of a few eV (visible or UV range). The decay timeτsof the scintillator -the time it takes for a scintillator to produce light following the interaction with a high energy photon- depends on the half life of the excited state involved in the emission and ranges between 40 ns and 500 ns in the most common cases [44].
Timing properties of scintillators. The time resolution of a scintillator is usually
mathematically derived from two parameters: total photoelectron yield and decay time [45],[46],[47]. This model is sufficiently accurate for scintillators with a decay time greater than 100 ns, with coincidence timing resolution larger than 2 ns [45],[46]. How-ever, for faster scintillators, like the ones recently developed (LaBr3or LSO), that allow
for coincidence timing resolution reaching 1 ns or less, this way of calculating time resolution leads to significant errors [48],[49]. Several studies suggest that the effect of scintillation rise time should be included to correctly estimate the timing properties of these fast scintillators [48],[49],[50].
It can easily be understood that any complex luminescence phenomenon leading to scintillation introduces a non-zero rise time. Its effect on the emission process are
1.7. DETECTORS USED IN PET
Figure 1.5: Curves representing two scintillation timing processes: an excitation process characterized by a non-zero rise time constant and a simple decay process. The com-bination of these two processes represents a more realistic behaviour of the temporal evolution of a scintillation signal: it increases rapidly, but not instantaneously, up to its maximum and then slowly decreases.
illustrated in Fig. 1.5. Clearly, the rise time can have a profound impact on the timing property of a scintillator detector and therefore should be considered in the modelling of the time resolution.
The emission of photoelectrons in a photon-electron conversion process is regulated by the Poisson distribution [47]: between time 0 and t, the number of generated photo-electrons is equal to:
(1.23) P(t)N= f (t)Ne− f (t)/N!
where f(t) represents the mean number of photoelectrons that can be observed be-tween the time 0 and t, and f(0) = 0. The physical interpretation of the scintillation phenomenon is inserted into this model with the function f(t), which is conventionally de-fined neglecting rise time (τr) and considering only decay time (τd) as a single-exponential
model:
(1.24) f (t) = a
Z t
0
e−s/τdds = Yh1 − e−t/τdi
With Y = f (∞) = aτdbeing the total number of photolectrons emitted in the whole
pro-cess (total photoelectron yield). Note that Y can be seen as Y =R∞
0 I(t)dt =
R∞
Figure 1.6: CTR for different photoelectrons of different orders (first, third and fifth), calculated with single- (dashed) and bi-exponential (continuous line) timing models. The difference between the two models is evident: the single exponential wrongly estimates a better timing resolution.
τr= 0.5 ns,τd= 40 ns, Y = 1000 (values typical of LSO scintillators).
I0τd and thus a = I0is the initial photoelectron emission rate.
This model can accurately predict the time resolution of slow scintillators, like NaI, in which the the rise time is negligible compared to its long decay time constant [45], [47]). However, whenτrbecomes comparable toτd(which is the case of the newly developed
scintillators) the single-exponential prediction are highly inaccurate. Furthermore, recent studies have shown that a smallerτrcan significantly improve resolution [48],[49],[50].
This suggests that a bi-exponential model, taking both rise and decay time into account, can better describe the luminescence process of fast scintillators:
(1.25) f (t) = a Z t 0 e−s/τd³1 − e−s/τr´ds = Y · 1 −τr+τd τd e−t/τd +τr τd e−t/τe f f ¸ Withτe f f=ττrτd r+τd, Y = f (∞) = a τd τrτe f f.
The theoretical Poisson distribution PN(t) is the resolution of a single detector, not
the actual CTR. The value of the CTR can be derived from PN(t), convolving two
single-detector timing distributions corresponding to a pair of photoelectrons of the same order, (assuming identical detectors [46]). The coincidence timing distributions that correspond to three pairs of photoelectrons (first, third and fifth) are shown in Fig. 1.6. These distributions confirm that the scintillation rise time has a significant impact on the CTR between two fast detectors.
1.7. DETECTORS USED IN PET
Figure 1.7: Experimental validation of the model. Image and study from [8]
Important characteristics of a scintillator Given that the light emitted
isotropi-cally by the scintillator has to reach the photosensor to be detected, photons must be reflected by the lateral walls of the scintillator crystal, and transmitted at the basis of it (as in Fig. 1.8). For this reason, scintillators are usually covered with a reflective material such as teflon and are coupled with optical grease to the photosensor. From what has been described about the scintillation process, it is possible to understand the desired characteristics of a scintillator:
• a high initial photon emission rate (high a in the model presented before), following the scintillation event. This parameter in fact influences the timing performance of the detector.
• a high total photoelectron yield, meaning a high number of visible photons emitted per unit energy deposited. This parameter is related to energy resolution and to the timing performances of the scintillator;
• lowτd and τr, since the decay and rise time limit the maximum countrate the
scintillator can sustain and impact its timing resolution;
• a low attenuation length for 511 keV photons. The attenuation length is the mean distance that a photon travels before being absorbed, a low attenuation length implies that the detector can be shorter and still stop the majority part of the photons it receives.
Scintillator
Photosensor Reflective walls
Photon
Figure 1.8: Path of a photon inside a scintillator.
NaI BGO LYSO LaBr3
Initial photon emission rate (photons/ns) 90 21 380 1900
Light output (103photons/MeV) 41 9 30 60
Decay time (ns) 230 300/60 40 16
δE/E at 662 keV 0.06 0.10 0.10 0.03
Effective Z 50 73 64 46
Attenuation length at 511 keV (mm) 25.9 11.2 12.6 22.3
Table 1.2: Most common scintillators and their characteristics, data taken from [5].
Other important parameters for the ease of use of scintillators include the refractive index, the emission wavelength, which should match the spectral sensitivity of available photodetectors, the hygroscopicity and the magnetic susceptibility, which should both be as low as possible. Also the intrinsic radioactivity can be an issue because it creates noise and lowers the SNR. Given the wide range of desired characteristics, the perfect scintillator does not exist, however there is a long list of materials suitable for PET design concepts, the most used ones are described in Table 1.2.
Progress in photodetectors and electronics, but also in the light output of LSO and LYSO, which currently offer the best combination of luminosity and coincidence detection efficiency, have also made it possible recently to implement ToF technology in whole-body PET with system-wide timing resolution in the order of 600 ps FWHM [34],[51]. The search for the ideal scintillator has recently brought another promising material that is currently raising interest: lanthanum bromide (LaBr3: C e)[52], which has one of the
highest light outputs, and the best initial emission rate of all known scintillators. This material not only achieves the best reported energy resolution (<3% at 511 keV), but also appears as the most suitable scintillator to date for ToF-PET, potentially achieving a
1.7. DETECTORS USED IN PET
Figure 1.9: Effect of magnetic field on the crystal decoding accuracy of a conventional PMT-based PET block detector (a) and of an APD-based PET detector (b), image from [5].
time resolution of ∼300 ps [53] in full scanners. 1.7.2 Photodetectors
Photodetectors are devices that convert light into an electric signal, there are two main classes of photodetectors used in PET: photomultiplier tubes (PMTs) and their solid state counterpart, such as avalanche photodiodes (APDs) and Silicon Photomultipliers (SiPMs).
Photomultiplier tubes. PMTs are composed by a photocatode, a material that emits
electrons when hit by light, and a series of charge amplifying elements called dynodes. The electrons produced are accelerated by an electric field and hit a series of dynodes causing the emission of multiple secondary electrons. The initial signal is in this way amplified and can be read with an appropriate electronic from the last dynode, called anode.
More than 80 years after its development, the (PMT) remains the photodetector of choice to convert scintillation photons into electric signals in most radiation detection applications. This is due to its unique combination of very high gain (> 106), low noise and fast response relative to other photosensing devices (Table 1.3).
However, significant drawbacks of PMTs are (i) their extreme sensitivity to magnetic fields, which renders them unsuitable for use in combination with MR [5], (ii) their intrinsically high cost, (iii) their big dimensions, which make them unsuitable for one-to-one coupling to scintillator crystals.
Solid state photodetectors. Solid-state photodetectors have several inherent
Figure 1.10: Schematic representation of a depletion region.
can be adapted to individual crystals, ruggedness, demonstrated insensitivity to magnetic fields up to at least 9.4 T [5], and potentially inexpensive mass production. By allowing direct coupling to discrete crystals and independent parallel signal processing of every electronic channel, most detrimental effects due to the so-called decoding errors can be avoided [54].
Avalanche Photodiodes. Avalanche photodiodes (APDs) are semiconductor
de-vices, usually made of silicon, that convert incident light (coming from a scintillator in PET applications) in a number of electron/hole pairs proportional to the energy of the incoming photons. In semiconductors, the energy gap between valence band and conduction band is narrow enough (1-2 eV) to allow the CB to be thermally populated by a small number of electrons (and thus a small number of holes are in the VB). Thus electron-hole pairs naturally exist in semiconductors, therefore constitute a source of noise, the so-called dark noise. This noise can be lowered by lowering the temperature and by doping the semiconductor. Doping a semiconductor means introducing impurities in its lattice structure so as to create an abundance of electrons (n-doping) or of holes (p-doping). Interfacing a p-doped region with an n-doped region creates a p-n junction in which the excess of holes in the p region recombines with the excess of electrons in the n region. Since the two regions are electrically neutral, the recombination of electron-hole pairs creates a region of fixed spatial charge in the semiconductor and thus an electric field that stops the recombination from proceeding further. In this region there are no free charge carriers (hence noise is greatly reduced) and thus it is called depletion region. This area is usually too small to be used to detect ionizing radiation and therefore it must be extended applying a reversed potential, called bias voltage (Fig. 1.10), to the sides of the junction, so as to lower the electric field created by static charges and widen the depletion region.
1.7. DETECTORS USED IN PET
Figure 1.11: Intensity of the current generated by photons versus the bias voltage in an APD.
below) the breakdown voltage (the voltage at which a single electron-hole pair would create a self-sustaining discharge in the APD). APDs are usually operated with a gain of 50–150 where the noise can be kept at a relatively low value and optimum performance can be achieved (above 200V in Fig. 1.11).
Given their low gain, they need to be connected to an amplifier to produce a signal strong enough to be detected, introducing an additional source of noise. APDs are not used anymore in state-of-the-art PET systems because of their relatively low gain and SNR, their thermal instability and their poor timing properties [55],[56].
Silicon Photomultipliers. When a potential higher than the breakdown voltage is
applied to an APD, the charge carriers generated by ionizing radiation can trigger a self-sustaining avalanche due to the impact ionization mechanism. When operated in this mode (geiger mode) the device is insensitive to the energy of the incoming radiation, since it will always output the maximum possible current. However it can be arranged in energy-sensitive devices as will be explained below, with gains in the order of 105− 107 [33].
One geiger-mode APD is called Single Photon Avalanche Diode (SPAD). A Silicon Photomultiplier (SiPM) is formed by a great number of SPADs (tipically with linear dimensions in the order of tens of µm) arranged in a matrix (Fig. 1.12). Each SPAD operates independently from the others and therefore, if the rate of incident light coming from the scintillator is uniform and low enough, it is possible to count the photons one by one, assuming that they will hit different SPADs [57], [58]. The number of photons detected by the SiPM can thus give an estimate of the energy of the high-frequency
Figure 1.12: Structure of a SiPM composed by a matrix of SPADs (right). Each cell (left) is a p-n junction with the quenching circuit attached in series.
photons that interacted with the scintillator.
When an avalanche is triggered, the current continues to flow until the avalanche is quenched by lowering the bias voltage down to or below the breakdown voltage. In order to be able to detect another photon, the bias voltage must be raised again above breakdown. This whole process requires a suitable circuit, called quenching circuit [59]. It should be capable to sense the leading edge of the avalanche current, quench the avalanche by lowering the bias down to the breakdown voltage and restore the photodiode to the operative level at a later time. The most common kind of quenching is called passive quenching: it is done with a single resistor in series to the SPAD (Fig. 1.13). The avalanche current self-quenches simply because it develops a voltage drop across a high-value resistor (about 100 kΩ or more). After the quenching of the avalanche current, the SPAD voltage bias slowly recovers to its normal value (1-100 ns are needed [59]). A more advanced form of quenching is the active one. In this case a fast discriminator senses the steep onset of the avalanche current across a 50Ω resistor and provides a digital output pulse, synchronous with the photon arrival time. It then quickly reduces the bias voltage to below breakdown, then relatively quickly returns bias to above the breakdown voltage ready to sense the next photon. This method allows for a faster recovery, but is obviously more complex to realize [59] and requires a lot of space in the SPAD cell. This in turn reduces the sensible area of the SPAD, which means that the single SPADs have to be bigger to keep the same active area in the SiPM and therefore their density has to be lower. A lower SPAD density creates saturation problems, since one cell can only detect one photon at a time.
Given that the SPADs are connected in parallel, the total output of a SiPM is the sum of the outputs of its SPADs, therefore it is proportional to the number of cells activated and, if saturation has not occured, to the number of photons incident on the SiPM active
1.7. DETECTORS USED IN PET
Figure 1.13: Schematic of a passive quenching circuit, when a cell triggers some of the current generated goes through the quenching resistor lowering the overvoltage.
area. The efficiency with which photons are detected is called photodetection efficiency (PDE) and is defined as:
(1.26) P DE = FF · QE · Pavalanche
Where FF is the fill factor, the fraction of the SiPM area that is actually sensitive to the photons, QE is the quantum efficiency, the probability of a photon to interact with the SiPM (depends on photon wavelength) and Pavalancheis the probability that a free
charge triggers an avalanche [57]. Typical values of these quantities are:
• FF: 0.6-0.8. In fact the remaining area is needed for the quenching circuit and to optically isolate the SPADs;
• Pavalanche 0.5–1, increasing with with the increase of the overvoltage (Vbias− Vbreakdown);
• QE: 0.8-0.9, depending highly on the wavelength of the photons and on the structure of the SPADs.
For PET applications it is important that SiPMs have a high dynamic range, because the light flux on the device can be very high. In a SiPM this characteristic is determined by the number of SPADs that it contains and by the dead time they require to recharge after an event. Saturation happens when the rate of photons is too high with respect to the number of cells, thus increasing it can be a solution.
SiPMs have a much lower noise rate than APDs, however it still is an important parameter. Noise is mainly caused by three factors in these devices:
• Thermal counts. As photons can produce free carriers and trigger avalanche discharges, so does thermal energy. The rate with which these thermally produced avalanches occur is called dark count rates (DCR). Typical values of DCR range
