This chapter presents methods that constitute the theoretical background for digital signal and image processing proposed in this work. The algorithms described on the following pages are essential for formulation and solution of subsequent steps of signal and image processing in the J-PET detector. For clarity of the presentation, state-of-the-art data processing methods are presented separately. Details of the applications and modifications of algorithms described in this part will be introduced in chapters 4 and 5.

The methods described in this chapter are applied to different type of data, depending on the stage of data processing. Despite the fact that the details of the signal and image processing in the J-PET scanner will be explained and discussed in chapters 4 and 5, at this point I would like to underline a few important aspects. First of all, it should be stressed that Compressive Sensing (CS) theory is the key method in the proposed signal recovery scheme. The idea of the CS approach will be introduced in section 3.1. Moreover, one of the most important aspect of the proposed image reconstruction is the application of Total Variation (TV) regularization. The concept of the TV regularization will be discussed in section 3.2. Finally, the visualization of the feature space of the information acquired by the J-PET scanner is provided by using the Self-Organizing Maps (SOMs). The short introduction to unsupervised learning algorithms will be given in section 3.3.

Compressive Sensing

The CS [52, 53] is a signal processing method that exploits the sparsity of a signal to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. Consider the recovery of a finite signal y0 ∊ ℝN in a situation where the number M of available samples, denoted as measurement y Ω ℝM, is much smaller than the signal dimension N (yΩ is sampled on some partial subset , where the cardinality Ω = M). In the CS method, a sparse expansion x0 ∊ ℝN of signal y0, evaluated via linear and orthonormal transformation y0 = Ax0, is considered. In the following we assume that we are given a contaminated measurement y and hence:

where AΩ is a M ×N matrix modeling the sampling system, constructed from M rows of matrix A that corresponds to the indexes of y described in the subset Ω, and e is an error term.

Experimental and simulation scenarios

For the evaluation of the proposed data processing algorithms described in chapters 4 and 5, both experimental and simulation data of the J-PET scanner were used. The performance of the low-level signal processing methods, presented in chapter 4, is investigated by using a data set of reference signals registered in a single module scintillator strip of the J-PET detector. This experimental setup is described in section 6.1.1. The quality of the high-level data processing methods, introduced in chapter 5, is examined by using both experimental and simulation data. The experimental scenario describing the 3-layer prototype J-PET scanner is shown in section 6.1.2 and the simulated 1-layer J-PET detector is introduced in section 6.1.3.

Experimental scheme with 1 scintillator strip

In this section we introduce the scheme with single detection module of J-PET device that allows the investigation of low-level data processing described in chapter 4. The scheme of the experimental setup is presented in Fig. 6.1.

The single detection module is built from the BC-420 plastic scintillator strip, with dimensions of 5 × 19 × 300 mm, readout at two ends by Hamamatsu R4998 photomultipliers is denoted as PM1(2). Measurements are performed using γ photon from a 22Na source placed inside the lead collimator between the scintillator strip and the reference detector. The reference detector consists of a small scintillator strip with a thickness of 4 mm. A collimated beam emerging through a slit 1.5 mm wide and 20 cm long is used for irradiating desired points across the strip. In order to detect the event, coincident registration of signals from the PM1 and a reference detector is required. Such trigger conditions reduce the background from the deexcitation photon (1.27 MeV) to a negligible level [17]. The time of triggering by the reference detector is used to estimate the event arrival time. The constant electronic time delay between the true event time and the measured time of arrival at the reference detector does not influence the time and spatial resolutions and is shifted to zero. The full waveforms of signals on PM1 and PM2 are sampled using the Lecroy SDA 6000A oscilloscope with a sampling interval of 50 ps.

The first demonstration of PET technique for medical imaging use was given in early 1950s by Brownell and Burnham. This was an inspiration for the concept of emission tomography used to visualize functional processes in the body in the late 1950s. The first 3-dimensional PET detector, called PC-1, was developed at the Massachusetts General Hospital and completed in 1969. This PET device comprised two planar opposed arrays of crystal scintillators [23]. In 1973 Robertson and his co-workers built the first ring PET scanner, which consisted of 32 detectors [24]. The cylindrical array of detectors has soon become the prototype of the current shape of PET [25].

The fundamental requirement that comes together with the development of the PET technology was to create radiopharmaceutical tracers that could be administered safely to the patient’s body. Currently, the most important radiopharmaceutical in PET examinations is fludeoxy- D-glucose (18F-FDG) [26]. 18F-FDG is an analog of glucose used for cellular metabolism having the hydroxyl group replaced by radionuclide 18F. Radionuclide 18F is produced in a cyclotron and has a relatively long half-life, about 110 min, that allows its supply to remote places. Similarly as glucose, 18F-FDG is absorbed by brain or kidney cells, and what is most important, by the cancer cells presenting abnormally high metabolism in comparison to the healthy organs. Therefore, PET imaging presents the distribution of glucose consumption by the cells and an overall cellular activity in the patient body.

Radionuclides are unstable due to the unsuitable composition of neutrons and protons and, therefore, decay by emission of radiation. When a radionuclide is proton rich, as in the case of 18F, it decays by the emission of a positron (β+) along with a neutrino (v):

For instance, the scheme for positron decay from 18F is:

The stability of radionuclide is achieved by converting a proton (11 p+) in the nucleus to a neutron (1 0 n). Since a daughter nucleus is one atomic number smaller than a parent nucleus, one of the orbital electrons has to be ejected from the atom. As both an electron and β+ are emitted in the decay described in Eq. (2.1), the right-hand side of this formula has to be at least two electron mass more than the left-hand side, i.e., 2 × 511 keV = 1022 keV. The energy beyond 1022 keV is shared as kinetic energy by β+ and a neutrino.

Results constituting the basis for this dissertation have been published in seven articles [1–7] and presented at eight conferences. The research has been realized at the Jagiellonian University by means of the Jagiellonian PET scanner. Ideas of proposed signal and image processing have been positively judged and approved for application by the Jagiellonian PET collaboration.

A.1 Derivation of error resulting from signal recovery procedure

In the following we prove the theorem for average value of the recovery error (σ 2x ) introduced in section 4.2, in Eq. (4.20). We assume, for the sake of simplicity, that the matrix A, transforming the sparse expantion x into the signal y, has normally distributed elements with zero means and 1/N variances.These values of the parameters of normal distribution ensure that the matrix A is orthonormal. Hence, based on Eq. (4.17), the covariance matrix Sr is given by:

where M is the number of acquired samples, N is the number of samples in complete signal, σ is standard deviation of noise in signal and P denotes the covariance matrix of a prior distribution of sparse expantion x. The σ2x is equal to the trace of the matrix Sr and hence:

The sum in the last term in Eq. (A.2) may be approximated by a definite integral. In the following we will use for the calculations a basic rectangle rule, and:

where h = 1/2. At the very beginning, see Eq. (4.19), we assumed that the function P(k) has the form:

We will perform the integration using the substitution t = e−Tk. Without any significant loss of precision, we change the integration limits from [1 − h, N + h] to [0, N]. The calculations of the integral I will be as follows:

Finally, substituting the integral I in Eq. (A.5) into formula in Eq. (A.2), gives the average value of the recovery error:

A.2 Derivation of error resulting from limited number of photoelectrons

As mentioned in section 4.4.1, the function yk, describing the kth signal from a single photoelectron is assumed to be a Gaussian function with standard deviation σp. The function ˜ yk, given in Eq. (4.37), may be approximated with:

where tkr is a random variable with ftr distribution, which denotes the kth λ photon’s registration time, λ contributes to the signal width and n = 1, 2, …,N. The probability that the random variable ˜yk(n) is equal to the specified value may be calculated based on the previously introduced function Ф see Eq. (4.46).

This chapter describes the subsequent steps of high-level data processing in the J-PET scanner. The goal of high-level reconstruction is an estimation of radioactive tracer distribution injected to the patient’s body. The image reconstruction is based on information on registered set of events of positron-electron annihilations. As it was discussed in the previous chapter, a single event is derived based on the detection of both γ photons in two scintillators in a narrow time window.

In recent years, PET data acquisition has been shifted to list-mode [10], where each registered event is saved individually. Storing coincidence events individually, instead of accumulated counts in sinogram, is preferred for efficient data processing for whole-body PET scanners that additionally record TOF information. Moreover, in listmode data, each event may be described with additional information such as, e.g., deposited energy in the scintillators, detection time difference. Therefore, each event may be described with preselected set of features, creating high-dimensional feature space.

During the PET examination, an event is regarded as valid if: two γ photons are registered within a predefined time window, and the energy deposited in the scintillator by both γ photons exceeds the selected threshold. However, a number of events registered as having met the above criteria are undesirable. Different types of events in PET measurement are:

• • true. Pair of γ photons derive from a single positron-electron annihilation and reach the scintillators without interacting with the atoms in detector volume.

• • random. Pair of γ photons derive from two different positronelectron annihilations occur at approximately the same time.

• • scattered. Pair of γ photons derive from a single positron-electron annihilation when one or both of them have undergone a Compton interaction.

As the true events are essential for the PET imaging, the random and scattered events distort the reconstructed distribution of radiotracer.

This chapter is organized as follows. First, in section 5.1, the method for classification of events will be proposed. For this purpose we apply a self-organizing map (SOM) network to analyse the high-dimensional feature space created by a set of three types of events. In particular, we provide a visualisation of high-dimensional feature space and we investigate the class distributions on a 2-dimensional SOM.

Positron Emission Tomography (PET) is at present one of the most technologically advanced diagnostic methods for non-invasive imaging in medicine [8,9]. It plays a unique role both in medical diagnostics and in monitoring effects of therapy, in particular in oncology, cardiology, neurology and psychiatry. In PET measurement the patient is injected with radiotracer, containing a large number of metastable atoms of positron emitting radionuclide. Since the rate of assimilation of radiopharmaceuticals depends on the type of the tissues, sections of the diseased cells can be identified with high accuracy, even if they are not yet detectable via morphological changes. Therefore, PET is extremely effective in locating and diagnosing cancer metastases.

As the result of positron annihilation, two photons travelling off with nearly opposite directions are produced. The detection system is usually arranged in layers forming a ring around the diagnosed patient. In the basic PET measurement scheme, the information about the single event of positron annihilation is collected in the form of a line joining the detected locations that passes directly through the point of annihilation, i.e. the Line-of-Response (LOR). The set of registered LORs forms the basis for PET image reconstruction. New generation of PET scanners utilizes not only information about the LORs but also takes advantage of the measurement of the time difference between the arrival of the two photons at the detectors, referred to as the Time-Of-Flight (TOF) difference [10]. State-of-the-art TOF-PET scanners use scintillation crystals and operate at a time resolution, defined hereafter as the coincidence resolving time (CRT), of the order of 300−400 ps [11, 12]. The fundamental improvement brought by TOF is an increase in signalto- noise ratio (SNR); in the first approximation the SNR improvement due to TOF application is inversely proportional to the square root of the CRT [13]. For example, if a time resolution of 400 ps is applied, this yields, for a patient of about 40 cm average transaxial size, an SNR about three times better than for non TOF information measurement. The time resolution achievable with the scintillator detectors is limited by the optical and electronic time spread caused by the detector components, and by the time distribution of photons contributing to the formation of electric signals.

This chapter describes the subsequent steps of low-level data processing in the J-PET scanner. The goal of low-level reconstruction is an evaluation of time and position of each particular event of positron emission. A simple setup with two detection modules that permits the reconstruction of information about an event is shown in Fig. 4.1.

A single detection module consists of a long scintillator strip and a pair of photomultipliers (PMs) attached to opposite ends of the strip (see Fig. 4.1). The subsequent steps of the reconstruction take place in the reverse order than the physical processes of interactions in the detector. Light pulses produced in the strip propagate to its edges where they are converted via photomultipliers into electric signals. Measurement of electric signals results in timestamps from both sides of each scintillator, allowing the extraction of timing, position and energy information for each γ photon interaction marked with gray circle in Fig. 4.1. The time and position of the γ photon interaction in the scintillator strip is calculated based on times at left (t(L)) and right (t(R)) side of the strip. In the first approximation, the time of interaction may be estimated as an arithmetic mean of t(L) and t(R) and the position of interaction along the strip may be calculated as (t(L) − t(R)) vsc /2, where vsc denotes the speed of light signals in the scintillator strip. The energy deposited in the scintillator strip may be expressed in terms of the number of photoelectrons registered by the photomultipliers and is proportional to the arithmetic mean of a number of photoelectrons registered at the left and right sides of the scintillator; the value of energy calibration factor was evaluated in [17]. The registration of single event of positron emission, used for the image reconstruction, is based on the detection of both γ photons in two modules in a narrow time window. Therefore, a single image-building event, marked with black star in Fig. 4.1, includes information about four times of light signals arrival to the left and right ends of the two modules that register in coincidence.

The two main features of the data acquired in the J-PET scanner that have the greatest impact on the TOF resolution are: (i) a very short rise-time and duration of the signals and (ii) a relation between the shape and amplitude of the signals and the hit position.

The J-PET collaboration brings together scientists from different disciplines in order to develop a more affordable whole-body PET scanner. The application of the plastic scintillators entails the desing of unique detector geometry, dedicated electronics and reconstruction algorithms. Studies presented in this dissertation cover the analysis of the data processing in the J-PET scanner.

The operational principles of the J-PET scanner are similar to stateof- the-art PET detectors, except that the extremely precise time information is of paramount importance. In the J-PET, the time resolution influences not only the uncertainty of position reconstruction along the LOR, as in case of the conventional PET systems, but also has an impact on the uncertainty of position reconstruction along the scintillator strip. Therefore, the J-PET detector requires a development of novel methods at each step of data processing. The problem of data processing has been split into two separate parts in order to distinguish the low-level signal recovery and reconstruction from the high-level image processing. The goal of the low-level data processing is an evaluation of the information about each event of positron-electron annihilation based on the raw signals acquired during the PET examination. This information includes the estimated values of position and time of annihilation and deposited energy by two γ photons in the scintillators. In the next stage, denoted in this work as high-level data processing, the list of reconstructed events of positron-electron annihilations is subjected to further analysis with the primary goal of estimating the radioactive tracer distribution after injection into the patient’s body. Main achievements related to both stages of data processing in the J-PET tomography are shortly summarized in the next two sections.

Summary of low-level data processing

In this dissertation a novel scheme of signal recovery in plastic scintillators in the J-PET scanner was introduced. The idea of signal recovery is based on the Tikhonov regularization theory, which uses the training data set of signals. The compact representation of these signals was pro vided by the PCA decomposition. One of the most important aspects of this part of research is the statistical analysis of the error level of recovered signals.

Hypothermia is defined as a drop of core temperature (usually measured in oesophagus or rectum) below 35°C. Impact of hypothermia onto electrical conduction system of heart was first described in 1892. The first observations concerning changes in ECG of a patient in accidental hypothermia date back to 1938. Hypothermia is associated with presence of characteristic changes in ECG, depending on hypothermia stage.

Motion artefacts related to shivering and tremors are characteristic of mild hypothermia (Figure 1) together with sinus bradycardia resulting from compensatory stimulation of sympathetic nervous system.

Prolonged exposure to low temperature significantly impairs both electric as well as mechanic activity of the heart. As a result of catecholamines release, advanced stages of hypothermia are distinguished by decreased peripheral vascular resistance and noticeable drop in cardiac output. Prolonging of action potential duration and decrease in conduction velocity in conduction system of the heart are related to delay in activation/inactivation of cellular ionic (sodium, potassium and calcium) currents. Hypothermia causes prolonging of both depolarisation duration as well as repolarisation of sinus node cells. Decrease of body temperature below 32°C causes substantial reduction of conduction velocity. In such situation, serious sinus bradycardia develops, with PR and QT/QTc intervals prolongation as well as prolongations of P wave, QRS and T wave durations (Figure 2). There is a non-linear relationship between reduced heart rate and decrease in core temperature. Prolongation of QT/QTc interval is caused both by delayed ventricular repolarisation and well as presence of so-called Osborn wave (J wave). In some patients, reduction of conduction velocity in atrio-ventricular node leads to development of conduction blocks.

Osborn wave (J wave) is certainly the most evident anomaly in ECG of hypothermic patients. Occurrence of the wave in hypothermic patients was first recorded in 1938 (Tomasjewski), but the full description of of the phenomenon was published in 1951 by J. Osborn (hence eponym). Osborn wave is a positive deflection appearing at junction between QRS complex and ST segment, resembling “camel hump” (Figure 3).