Characterization and Reduction of Noise in PET Data Using MVW-PCA

Transcription

1 IT 9 3 Examensarbete 3 hp Mars 29 Characterization and Reduction of Noise in PET Data Using MVW-PCA Per-Edvin Svensson Institutionen för informationsteknologi Department of Information Technology

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3 Abstract Characterization and Reduction of Noise in PET Data Using MVW-PCA Per-Edvin Svensson Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen Hus 4, Plan Postadress: Box Uppsala Telefon: Telefax: Hemsida: Masked Volume-Wise Principal Component Analysis (MVW-PCA) is used in Positron Emission Tomography (PET) to distinguish structures with different kinetic behaviours of an administered tracer. In the article where MVW-PCA was introduced, a noise pre-normalization was suggested due to temporal and spatial variations of the noise between slices. However, the noise pre-normalization proposed in that article was only applicable on datasets reconstructed using the analytical method Filtered Back-Projection (FBP). This study aimed at developing a new noise pre-normalization that is applicable on datasets regardless of whether the dataset was reconstructed with FBP or an iterative reconstruction algorithm, such as Ordered Subset Expectation Maximization (OSEM). A phantom study was performed to investigate the differences of expectation values and standard deviations of datasets reconstructed with FBP and OSEM. A novel noise pre-normalization method named "higher-order principal component noise pre-normalization" (HOPC noise pre-normalization) was suggested and evaluated against other pre-normalization methods on both synthetic and clinical datasets. Results showed that MVW-PCA of data reconstructed with FBP was much more dependent on an appropriate pre-normalization than analysis of data reconstructed with OSEM. HOPC noise pre-normalization showed an overall good performance with both FBP and OSEM reconstructions, whereas the other pre-normalization methods only performed well with one of the two methods. The HOPC noise pre-normalization has potential for improving the results from MVW-PCA on dynamic PET datasets independent of used reconstruction algorithm. Handledare: Pasha Razifar Ämnesgranskare: Ewert Bengtsson Examinator: Anders Jansson IT 9 3 Tryckt av: Reprocentralen ITC

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5 5 Preface This master thesis is the result of a study performed in Uppsala, Sweden, at GE Healthcare in co-operation with Centre for Image Analysis (CBA). The author is currently finishing his studies as a master student in Engineering physics and Electrical engineering at The Institute of Technology at Linköping University. During the early phases of the project the author cooperated with two students who simultaneously performed their master thesis work at GE Healthcare within the same area [, 2]. Therefore there are some similarities between the reports. This concerns the background chapter, 2, and parts of the synthetic study presented in sections and Apart from the work presented in this master thesis, work has been made on an application used to view and analyse dynamic Positron Emission Tomography (PET) data. The results of the studies have resulted in four manuscripts, listed below, that either have been or are to be submitted for publication in scientific journals. P Razifar, H H Muhammed, F Engbrant, P-E Svensson, J Olsson, E Bengtsson, B Långström, and M Bergström, Performance of principal component analysis and independent component analysis with respect to signal extraction from noisy positron emission tomography data a study on computer simulated images. Accepted for publication (29) P-E Svensson, J Olsson, F Engbrant, E Bengtsson, B Långström and P. Razifar, Characterization and reduction of noise in dynamic positron emission tomography data using masked volume-wise principal component analysis. Manuscript (29) J Olsson, R Oweinus, P-E Svensson, F Engbrant, B Långström, E Bengtsson, and P Razifar, Automated Method for Generation of Input Function in Positron Emission Tomography Studies Using Masked Volume-Wise Principal Component Images. Manuscript (29) F Engbrant, P-E Svensson, J Olsson, B Långström, E Bengtsson, and P Razifar, Application of Masked Volume-Wise Principal Component Analysis on In Vivo Animal Positron Emission Tomography Studies Using Flourine. Manuscript (29) I would also like to thank a few people for helping me with this master thesis: My supervisor, Pasha Razifar for his great commitment and support throughout the whole project. My friends and colleagues, Johan Olsson and Fredrik Engbrant with whom I have had a lot of interesting and fruitful discussions. My father, Per-Åke Svensson for proofreading and giving valuable feedback on the report.

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7 Contents Acronyms and abbreviations 9 Introduction. Setting Aim Structure of the thesis Background 3 2. Tomographical imaging modalities Overview Anatomical information Physiological information Integrated imaging modalities Differences and applications Positron emission tomography Overview Types of PET studies Acquisition Noise Corrections Reconstruction Principal component analysis Overview Algorithm Pre-normalizations Materials and methods Characterization of noise Motive The acquisition Selection of samples Expectation value Standard deviation Correlation between slices Correlation between frames Correlation between samples within the same slice

8 8 Characterization and Reduction of Noise in PET Data Using MVW-PCA 3..9 Correlation between samples within the same slice from one realization Reduction of noise Masked volume-wise PCA Reconstruction from selected principal components Background noise pre-normalization Higher-order principal component pre-normalization Synthetic images Clinical study Results Characterization of noise Frames Slices Correlation between samples within the same slice Correlation between samples within the same slice from one realization Reduction of noise Higher-order principal component pre-normalization Synthetic images Clinical study Closing remarks Discussion Interpretation of PC images Noise characteristics Reduction of noise in PET data using MVW-PCA Future work Conclusion Bibliography 56 List of Figures 6 List of Tables 62

9 Contents 9 Acronyms and abbreviations ACF BN CT FBP FDG fmri FORE FOV FWHM LOR HOPC MRI MSE MVW OSEM PCA PC PET PIB ROI ROM SV SPECT TAC VOI WSS Auto-Correlation Function Background Noise Computed Tomography Filtered Back-Projection Fluorodeoxyglucose Functional Magnetic Resonance Imaging FOurier REbinning Field of View Full Width at Half Maximum Line of Response Higher-Order Principal Component Magnetic Resonance Imaging Mean Squared Error Masked Volume-Wise Ordered Subset Expectation Maximization Principal Component Analysis Principal Component Positron Emission Tomography Pittsburgh Compound-B Region of Interest Removal of Mean Standardized Variables Single Photon Emission Computed Tomography Time Activity Curve Volume of Interest Weak-Sense Stationary

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11 Chapter Introduction. Setting Positron Emission Tomography (PET) is a non-invasive imaging modality used to visualize the functionality in tissues and organs in vivo in medical and research applications [3]. PET is based on measuring the concentration of a molecule labelled with a radionuclide, known as a tracer, designed to follow a specific physiological or biochemical path. The scanner detects photons that are a result from positron-electron annihilation events, creating an image or a set of images showing the tracer concentration in the scanned object. PET data is acquired either as a static image volume from the whole scan or as a dynamic sequence of image volumes from different times of the scan. Whole body acquisition is mostly performed as static PET studies often used to detect tumours, whereas studies of the brain are performed as dynamic PET studies and used to detect neurological disorders such as Parkinson s disease, Alzheimer s disease, phobia and schizophrenia by studying the kinetic behaviour of the tracer. However, PET data suffers from noise and the different areas and tissues can be hard to discern. There are several methods for analysing PET data such as kinetic modelling, summation and multivariate image analysis methods. It has been shown that Masked Volume-Wise Principal Component Analysis (MVW-PCA) can be used as a multivariate method that without modelling assumptions can separate tissues and organs with different kinetic behaviours of the PET tracer in different components [4]. In this approach, a new prenormalization was suggested prior to application of PCA since noise variance varies, both temporally and spatially within a dataset. However the proposed pre-normalization approach is only applicable on datasets reconstructed using analytical methods such as Filtered Back-Projection (FBP)..2 Aim The aim of this project was to characterize noise in PET data reconstructed with FBP and Ordered Subset Expectation Maximization (OSEM), and to reduce the noise using MVW-PCA.

12 2 Characterization and Reduction of Noise in PET Data Using MVW-PCA.3 Structure of the thesis This thesis is divided into five chapters where the first is the introduction. The second chapter is a technical background to different tomographical imaging modalities with focus on PET. This chapter also contains an introduction to the multivariate analysis method PCA and a new application to dynamic PET data called MVW-PCA. The materials and methods used in this project are described in chapter 3. The results are presented in chapter 4 and discussed in chapter 5.

13 Chapter 2 Background 2. Tomographical imaging modalities 2.. Overview In the context of medical imaging, an imaging modality is any of the various types of equipment used to acquire images of the body. Tomography is a technique based on generating two-dimensional slices through a section of a three dimensional volume. Tomogram generating imaging modalities include Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Positron Emission Tomography (PET), Single Photon Emission Computed Tomography (SPECT), combinations and variations of these [3]. Each imaging modality has its own advantages, and is therefore used in different research areas and for diagnosing different disorders. These imaging modalities are used to obtain either anatomical information, physiological information or both by integrating two imaging modalities Anatomical information Computed tomography Computed Tomography (CT) uses x-rays to visualize thin slices through the human body [5]. An x-ray tube and a detector are placed in the CT cameras gantry on opposite sides of the patient. The patient is gradually passed through the gantry as the system is rotated creating images from different angles yielding a three-dimensional x-ray image of the patient. The images have low structural noise and high contrast between bone and soft tissue, see figure 2.(a) [3]. Magnetic resonance imaging Magnetic Resonance Imaging (MRI) is based on the relaxation properties of excited hydrogen nuclei (single protons) and generates images with high structural definition in soft tissue, see figure 2.(b). The patient is placed in a stationary magnetic field, causing a small fraction of the spinning protons to line up in parallel or anti-parallel direction compared to the field, this puts them in

14 4 Characterization and Reduction of Noise in PET Data Using MVW-PCA specific energy states. Radio frequency pulses are sent into the body, making the protons rotate in phase with the pulses and putting them in a state of higher energy. When the stimulation is turned off, the protons emit their excitation energy as a radio signal. This signal can generate a cross sectional image, where different proton densities are represented by brighter or darker areas in the image. Different relaxation times generate images with different contrast. In brain imaging T-weighted images (using the shorter spin-lattice relaxation time) are used to differentiate between white and grey matter of the brain. T2-weighted images (using longer spin-spin relaxation time) are used for investigation of diseased parts of the brain [3]. Applications in research include Diffusion MRI, which measures the diffusion of water molecules in biological tissues; Multinuclear Imaging, which uses relaxation properties of other molecules than hydrogen; and in field research, where portable instruments use the magnetic field of the earth. (a) (b) Figure 2.: Anatomical images. (a) is a coronal CT image of a torso whereas (b) is a sagittal MRI image of a head Physiological information Functional magnetic resonance imaging Functional Magnetic Resonance Imaging (fmri) is based on MRI and measures the haemodynamic response, the change in blood flow, related to neural activity in the brain. Haemoglobin has different magnetic properties depending on if it s oxygenated or deoxygenated, making the signal dependent of the level of oxygenation. This allows mapping of the functionality of different parts of the brain, see figure 2.2(a). The patient needs to lie still during the scan, Image courtesy of Dr Jens Sörensen, Uppsala University Hospital, Sweden 2 Department of Mathematics, Uppsala University,

15 Background 5 which usually takes between 5 minutes and 2 hours. Movement in excess of 3 millimetres will give unusable data. Images are usually taken every 4 seconds [6]. Single photon emission computed tomography Single Photon Emission Computed Tomography (SPECT) is based on counting single photons that are emitted by gamma emitting radiopharmaceuticals. The tracer is administrated intravenously, and the patient is positioned into a gamma camera. Projection data is acquired covering 36 degrees around the patient. The distribution of radiolabel molecules is measured and gives functional or biochemical information, see figure 2.2(b) [3]. Positron emission tomography Positron Emission Tomography (PET) is based on detection of positron-electron annihilations events. A PET scan is prepared by administrating a radionuclide to the patient. In more than 95% of the studies this is done intravenously by injection, but the radionuclide can also be administrated orally by taking a pill or by inhalation of gas. When the radionuclide decays, a positron is emitted. The positron travels a few millimetres until it has lost its energy by collisions and scatterings with the surrounding matter. When enough energy is lost the positron annihilates with an electron and generates two co-linear photons, each with 5keV energy, in anti-parallel directions. These photons are detected approximately at the same time. Interaction between the photon and the crystal in the detectors generates light flashes that are converted to electronic pulses that are in turn recorded by the cameras electronics. PET generates images with biological and functional information about the kinetic behaviour of the radiotracer [3], see figure 2.2(c). Since PET is the imaging modality used in this thesis, more details about PET will be discussed in section 2.2. (a) (b) (c) Figure 2.2: Functional images. (a) is an fmri image of a brain 3, (b) a coronal SPECT image of a torso 4 and (c) a coronal PET image of a torso 5. 3 Stanford Medicine, 4 Image courtesy of Dr Pasha Razifar, GE Healthcare, Sweden 5 Image courtesy of Dr Jens Sörensen, Uppsala University Hospital, Sweden

16 6 Characterization and Reduction of Noise in PET Data Using MVW-PCA 2..4 Integrated imaging modalities PET/CT PET/CT is a combination of PET and CT [7, 8]. The CT data replace the transmission data from a regular PET scan, which is used for corrections in the acquired data and helps with localization of the structures [3]. PET/MRI Compared to PET/CT systems, PET/MRI not only offers improved contrast in soft-tissue and reduced levels of ionizing radiation, but also MRI-specific information such as functional, spectroscopic and diffusion tensor imaging. PET/MRI has been successfully implemented for pre-clinical studies but combining PET and MRI for clinical use has proven to be a very challenging task. Technical advances in this area are expected in the near future [9]. SPECT/CT SPECT/CT is a device containing a CT system and a gamma camera on a single gantry []. The SPECT procedure is performed, and then complemented with a CT transmission scan. The transmission data is used for corrections in the reconstruction of the projections [3]. Finally the data from the two imaging modalities are also merged into a composite image Differences and applications CT scans are faster and more cost efficient than MRI and PET scans. CT images contain less structural information and detail than MRI but they have relatively low noise magnitude. The CT images have a high contrast between areas in the body with different densities, like bone and soft tissues, but nearly no contrast within the soft tissues. It can therefore be hard to distinguish pathological from healthy structures and different soft tissues from each other in the CT-images, and the high radiation dose limits the possibility of repeated scans [3]. CT is relatively unsuitable for diagnosing disorders in the brain. However it is widely used in oncology and for diagnosing heart diseases. MRI provides greater contrast between soft tissues than CT, and is therefore useful in neurological, musculoskeletal, cardiovascular, and oncological imaging. MRI provides little information about the functionality of the brain. Medical or bio stimulation implants such as pacemakers are considered a risk-increasing factor, towards MRI scanning because of the magnetic and radio frequency fields. However when using MRI or fmri, the patient is not exposed to radiation [3]. PET has a high level of statistical noise that limits its efficiency [3]. The information about the radio-chemicals used for particular functions is capable of giving important support to research and diagnosis. It is often used in oncology and drug development, and can also provide diagnosis in several neurological disorders such as schizophrenia, Alzheimer s disease, Parkinson s disease and phobia [ 4]. PET/CT has the advantages from both PET and CT. PET have high sensitivity when it comes to functional and biochemical information, whereas

17 Background 7 CT gives high quality structural images [7, 8]. The combination has proved to increase the diagnostic value compared to each imaging modality used separately [5]. PET/CT has an important role in whole body imaging in oncology. It is faster and more accurate than PET or CT alone for the depiction of malignancy. SPECT projections suffer from highly smoothed images and poor camera resolution. On the other hand it provides 3D information that can complement other studies. Therefore, it can provide information about localised function in internal organs, such as functional cardiac or brain imaging. Research areas include paediatrics [6]. SPECT/CT hybrid studies give accurate localization of tumours, measurement of invasion into surrounding tissues, and characterization of their functional status []. Two of SPECTs weaknesses are the long scanning time needed and the poor resolution in the resulting images. 2.2 Positron emission tomography 2.2. Overview PET is a non-invasive tomographic technique used to obtain anatomical and physiological information in vivo in healthy and pathological organs and tissues. PET has proven to be a useful tool in diagnosing cancer and cardiac diseases and has an increasingly important role in providing earlier diagnosis in several neurological disorders such as Alzheimer s disease, Parkinson s disease, phobia, epilepsy and cancer [ 3] Modern PET cameras mainly consist of a translating bed surrounded by a set of detector rings. The detector rings contains crystal detectors (over 8. detectors) capable of creating a large number of trans-axial images with a resolution depending on the scanner and image reconstruction algorithm. (a) (b) Figure 2.3: PET Camera GE/Advance Nxi 6, where (a) shows the gantry and translating bed, whereas (b) shows the detector ring. 6 General Electric,

18 8 Characterization and Reduction of Noise in PET Data Using MVW-PCA Types of PET studies PET images used for investigations of the body are usually acquired as a set of stationary images across the body called static imaging. PET studies of the brain are often performed dynamically, meaning that the acquired data is a set of images from the same volume but from different time sequences. This makes it possible to analyse the kinetic behaviour of the used tracer in different parts of the brain. These multivariate image sets can be used to obtain physiological, biochemical and functional information of the brain using analysis methods such as kinetic modelling, compartment modelling, summation or multivariate analysis [7 2] Acquisition A complete PET study consists of three different scans; blank scan, transmission scan and emission scan. The blank scan is performed every day to normalize the detectors of the camera, which are highly sensitive and therefore have different efficiency. The blank scan is performed with no patient in the camera. Instead of a tracer a radioactive rod source rotating around the cameras gantry yield the detectable photons [3]. The transmission scan is performed using the same source as in the blank scan but with the object in the Field of View (FOV). The data from the transmission scan and the blank scan is used for a so-called attenuation correction to compensate for the scanned objects geometry, and the fact that the photons from the decay site have to travel through different amounts of tissue when going in different directions. In an integrated system such as a PET/CT camera the CT data can replace the transmission scan [3]. The emission scan is based on detection of positron-electron annihilation events. When performing a PET emission scan a molecule, labelled with a short-lived positron emitting radionuclide such as C or 8 F, called a tracer is administered to the patient prior to or during the scan. There is a wide range of different tracers used in PET. A radionuclide is created using a cyclotron or generator and is incorporated into a compound designed to follow a specific physiological path. The radionuclide used when creating PET tracers is short lived. Hence reducing the amount of radiation exposed to the patient, see table 2.. This also makes it possible to perform several scans on the same patient using the same or different kinds of tracer (multi-tracer study) [3]. Nuclide β + energy [MeV] β + range [mm] Half life [min] C O F Ga Table 2.: Commonly used isotopes in clinical PET studies. When the substance decays it emits a positron and a neutrino. After travelling a short distance, typically a few millimetres, the positron annihilates with

19 Background 9 an electron in the surrounding tissue. This annihilation event yields two colinear, anti-parallel, photons with 5 kev energy each. If detectors detect the photons on opposite sides of the cameras FOV within a timing window of about 2 ns (a photon travels 3 m in ns) the event is registered as a true coincidence. When a 5 kev photon hits one of the crystals a light flash is emitted and registered by a photo detector. The light flash is converted to an electrical pulse that is registered by the camera s electronics. After a detector has detected a photon it is paralysed for a short period of time. During this so-called dead-time the detector cannot detect any photons. Apart from dead-time there are a number of different factors that effect the precision of the scan result, two major factors are random coincidences and scattered coincidences. Random coincidences are detections originating from two different annihilations, but whose generated photons hit opposite detectors within the time-window. Scattered coincidences are detections in which one or both of the photons have been scattered in the tissue before hitting the detector, resulting in a Line of Response (LOR) (the line through the detectors detecting the coincidence) that does not correspond to the position of the annihilation [3] Noise In traditional PET scanners, the main sources of noise are in decreasing order of magnitude: emission, transmission and blank scan [22]. With newer attenuation correction modes, e.g. CT, the noise from emission is clearly dominating. The detector system only affects the magnitude of the noise whereas the recording system, various corrections, the image reconstruction method and its parameters also affect the distribution and correlation of the noise [3]. Radioactive decay, measured by PET detectors, obey the Poisson distribution. With an expected number of counts µ during a given time interval this distribution will have the standard deviation µ. The deviation from µ for each sample from this distribution is defined as noise. Apart from the statistical noise in the signal there are many factors that affect the noise both during the acquisition and the reconstruction. Some factors that affect noise during the acquisition are the choice of acquisition mode, scan duration, amount of administered tracer, geometry of tracer distribution, detector efficiencies, attenuation, dead-time, random coincidences and scattered photons pairs that falsely have been registered as true coincidences. When reconstructing the PET data the applied correction methods as well as the choice of reconstruction algorithm have a heavy impact on the statistical properties of the noise [3] Corrections Acquisition data from PET scans contain several errors that need to be compensated for prior to the reconstruction procedure. These compensations include corrections for differences in detector efficiencies, random coincidences, scattered coincidences, dead-time and attenuation correction, where attenuation correction is the main factor to affect the measured counts in PET acquisition. Since a scanned object usually is not symmetric, the photons need to travel through different amounts of tissue. By using the results from the blank scan and the transmission scan these differences in attenuation can be compensated for [3].

20 2 Characterization and Reduction of Noise in PET Data Using MVW-PCA Reconstruction The two most commonly used methods for reconstructing tomographic data are Filtered Back-Projection (FBP) and Ordered Subset Expectation Maximization (OSEM). FBP is an analytical and computationally efficient inversion algorithm for the two-dimensional radon transform that is both fast and easy to implement. OSEM on the other hand is an optimized iterative expectation maximization algorithm, which iteratively maximizes a target function in order to reconstruct the tomographic data. In practice FBP produces relatively high noise variance in regions with low signal compared to OSEM. OSEM on the other hand produces noise that is more dependent on the signal, with high noise levels in high signal regions and low noise levels in low signal regions. 2.3 Principal component analysis 2.3. Overview Principal Component Analysis (PCA), also known as Hotelling or Karhunen- Loève (KL) transform, was discovered by Karl Pearson in 9 [23]. It is a method that explaines the variance-covariance structure through linear combinations of the original variables. Each linear combination, known as Principal Component (PC), is picked in such a way that it maximizes the variance, which is the same as minimizing the Mean Squared Error (MSE). This is done under the constraint that the norm of its weight vector equals one and that the new PC is uncorrelated to all previous PCs. Even though there may be many variables, a large part of the systems total variability can often be accounted for by a small number of PCs. The PCs can then replace the variables without any significant loss of information. PCA s general objectives are data reduction and interpretation [24] Algorithm Each observation x, x 2,..., x p is stored as a row vector in the input matrix x x 2... x n X = x 2 x = [x, x 2,..., x p ] T. (2.) x p x p2... x pn The unbiased estimate of the covariance matrix associated with X is s s 2... s p S X = s 2 s s p s p2... s pp (2.2) where s ik = n n (x ij x i )(x kj x k ). (2.3) j=

21 Background 2 If S X has the eigenvalue-eigenvector pairs (λ, e ), (λ 2, e 2 ),..., (λ p, e p ) where λ λ 2... λ p and the eigenvectors are stored in the matrix E = [e, e 2,..., e p ] T the principal components are defined as where the variance is given by and covariance by Pre-normalizations Y = EX = [y, y 2,..., y p ] T (2.4) V ar(y i ) = e T i S X e i = λ i, i =, 2,..., p (2.5) Cov(y i, y k ) = e T i S X e k =, i k. (2.6) Depending on the application it is often preferred to normalize data prior to PCA. The most commonly used pre-normalizations prior to PCA are the Removal of Mean (ROM) and Standardized Variables (SV) pre-normalization. When pre-normalizing a dataset X the variables are mapped to a new set of pre-normalized variables Z. Removal of mean If observations are known to have a non-zero expectation value it should be subtracted from the observation to avoid PC to always point in that direction. In many datasets the expectation value is unknown but can be estimated by the arithmetic mean. Removal of the arithmetic mean is the default option in most implementations of PCA. The pre-normalized variable with removed mean is defined as z ik = x ik x i (2.7) where x ik is the kth input variable in observation i. Standardized variables Observations may have different variations in the measurements and PCs from non-normalized data are in general not invariant to this. A common approach to handle this is to standardize the variables, giving each variable zero expectation value and unit variance. This is done by removing the arithmetic mean from the observations and then scale them with their standard deviation. The standardized variable to x ik is defined as z ik = x ik x i s i (2.8) where s i is the estimated standard deviation of the observation i. When using pre-normalization to standardized variables PCA will choose eigenvectors based on correlation instead of covariance [24].

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23 Chapter 3 Materials and methods 3. Characterization of noise 3.. Motive A phantom study was performed in order to characterize the noise in PET data reconstructed with FBP and OSEM The acquisition The study was performed on an explore VISTA Dual-Ring small animal PET scanner (GE Healthcare) seen in figure 3.. The unit contains 2 rings of 8 phoswich detector modules capable of performing 3D data acquisition with an axial Field of View (FOV) of 48 mm and an effective trans-axial FOV of 67 mm. The spatial resolution is.8. mm for reconstructions made with OSEM and.5.8 mm for reconstructions made with FBP [25]. Figure 3.: explore VISTA small animal PET scanner In the experiments a phantom with two cylindrical inserts was used. Both inserts was 5 mm in diameter. The insert in the upper part of the gantry was filled with 223 kbq/cm 3 of 8 F and the insert in the left part of the gantry with 73 kbq/cm 3 of 8 F. The duration of the emission scan was 9 minutes. General Electric,

24 24 Characterization and Reduction of Noise in PET Data Using MVW-PCA Acquired data was reconstructed from list mode to a dynamic dataset by first applying the FOurier REbinning (FORE) algorithm to produce 6 two dimensional sinograms with a spatial resolution of 75 for 28 angles spanning the axial FOV for 8 frames. Corrections were made to compensate for the decay of 8 F. Two datasets were then reconstructed from the sinograms using OSEM and FBP in 2D mode. The dimensions of the reconstructed data were Selection of samples Three circular Region of Interests (ROIs) of equal size were selected. One for each insert and one for a region where there were no radioactive substance present. Each ROI had a diameter of mm which gave N ROI = 633 sample points per ROI. The 4 mm reduction in diameter compared to the inserts was chosen to avoid most of the spillover effects since the spatial resolution measured with Full Width at Half Maximum (FWHM) is less than 2 mm for the explore VISTA scanner [25], see figure 3.2. Figure 3.2: An FBP reconstruction of slice 3 in the first frame, showing a cross section of the two inserts. The outline of the three ROIs is drawn in white Expectation value The expectation value was estimated for each ROI in all slices and frames for both FBP and OSEM using the arithmetic mean 3..5 Standard deviation x = N ROI x i. (3.) N ROI The sample standard deviation was calculated for each ROI in all slices and frames for both FBP and OSEM using the square root of the unbiased sample i=

25 Materials and methods 25 variance s = N ROI (x i x) N ROI 2. (3.2) i= 3..6 Correlation between slices Before estimating the correlation between slices all samples within each ROI were divided into 9 groups where every third sample in u and v direction was put in the same group. This was done in order to reduce correlation between samples within the same group since this would alter the correlation estimate. The distance between the samples was set to three since a shorter distance would result in considerably more correlation between the samples within each group and a longer distance would lead to too few samples in each group. It shall also be mentioned that even though the data was divided into groups the arithmetic mean of all samples within a ROI was used to estimate the expectation value for all samples within that ROI. For each group the correlation matrix of the correlations between the m slices is r 2... r m R Slices = r r m r m2..., r ik = where the covariances are calculated with the sample covariance s ik = p N Group p n= N Group j= s ik sii skk (3.3) (x ijn x i )(x kjn x k ). (3.4) N Group is the number of samples within the current group and p is the number of frames. x ijn is the j:th sample in the current group and ROI in slice i (or k for the corresponding variable x kjn ) at frame n. x in (or x kn ) is the mean of all samples in all groups in the current ROI in slice i (or k) and frame n. The mean of the 9 correlation matrices was calculated, and the two diagonals closest to the main diagonal was plotted for each ROI and dataset Correlation between frames The estimation of the correlation between frames was done in much the same way as for slices. The samples in each ROI were divided into 9 groups, the sample correlation matrix of the correlations between the p frames is r 2... r p R Frames = r , r s ik ik = (3.5) sii skk r p r p2...

26 26 Characterization and Reduction of Noise in PET Data Using MVW-PCA where the covariances are calculated with the sample covariance s ik = m N Group m w= N Group j= (x ijw x i )(x kjw x k ). (3.6) N Group is the number of samples within the current group and m is the number of slices. x ijw is the j:th sample in the current group and ROI in frame i (or k for the corresponding variable x kjw ) at slice w. x iw (or x kw ) is the mean of all samples in all groups in the current ROI in frame i (or k) and slice w. The mean of the 9 correlation matrices was calculated and the diagonal closest to the main diagonal was plotted for each ROI and dataset Correlation between samples within the same slice A PET slice can be seen as a stochastic process with properties that can be estimated if enough independent realizations are available. One way of acquiring independent realizations of a slice or a whole scan is to perform gated scans [26]. In this study only one scan was available and the different slices in each observation had to be used. Since 2D reconstruction was used the differences in the statistical properties of the different slices were small enough to treat the slices as separate realizations of the same stochastic process of a slice. The calculations are similar to those for the correlation between slices and frames but there is one sample correlation matrix with the lags (k, k 2 ) for each choice of coordinates (u, v), i.e. the correlation matrix is four dimensional with elements given by s u,v,u+k,v+k r u,v,u+k,v+k 2 = 2 su,v,u,v (3.7) su+k,v+k 2,u+k,v+k 2 with covariances estimated from the sample covariance s u,v,u+k,v+k 2 = p m p n= w= m (x u,v,w,n x u,v )(x u+k,v+k 2,w,n x u+k,v+k 2 ). (3.8) x u,v,w,n is the sample at coordinate (u, v, w, n) and x u,v is the mean of the samples at coordinate (u, v) over all frames and slices. The correlation at different coordinates (u, v) within a slice for FBP and OSEM reconstructed data was investigated, both inside the three ROIs and outside the inserts. The differences in extent, orientation and isotropy of the correlation in datasets reconstructed with FBP and OSEM where studied Correlation between samples within the same slice from one realization Auto-Correlation Function (ACF) is a property of a stochastic process that describe its correlation between different points in time, or space depending of the unity of the dimensions in the stochastic process. So far the different slices have been treated as different realizations of a stochastic process of a slice when estimating the ACF between points within a slice. If the stochastic process is Weak-Sense Stationary (WSS) the ACF can also be estimated from a single realization. WSS is a weaker form of stationarity that requires mean and ACF

27 Materials and methods 27 to be independent of time, or coordinate within the slice in this case. If X[u, v] is the stochastic process of a slice these two criterion are written as and E{X[u, v]} = m (3.9) E{X[u + k, v + k 2 ]X[u, v]} = E{X[k, k 2 ] 2 }. (3.) If a region A in a slice is WSS the samples within that region can be used to estimate the ACF, that is the same for every point within this region. To do this a smaller region B inside region A is used. An estimate used to estimate the auto-correlation coefficient (normalized auto-correlation) for coordinate (u, v) is defined as (x[u, v] x k,k 2 ) ( b[u k, v k 2 ] b ) r k,k 2 = (u,v) A (u,v) A (x[u, v] x k,k 2 ) 2 (u,v) A ( b[u k, v k 2 ] b ) 2 (3.) where b is the template region within B with the arithmetic mean b, x is the realization of the slice and x k,k 2 the mean of x in a region of size B placed on the coordinate (u, v). Even though only a single realization x is used, the estimate will approach the autocorrelation function as the size of A and B is increased. The estimate has at least two applications. If used on WSS signals it estimates the ACF and if used on non-wss signals it identifies similarities and periodicities within the signal (how well B is correlated with A). This estimate was used in different parts of a trans-axial slice and the results were compared to the ACF estimates in section To compare this estimate to the sample correlation the region A was set to 2 2 pixels, and it was then placed on the centre of each ROI in the centre slice in the middle frame in the dataset. This estimated the ACF for the 5 neighbouring samples in each direction for each ROI (B was set to an pixel region).

28 28 Characterization and Reduction of Noise in PET Data Using MVW-PCA 3.2 Reduction of noise 3.2. Masked volume-wise PCA PCA can be performed on a set of images of any dimension, for example images of an object from different points in time. PCA will then separate the images into PC-images where early PC-images describe most of the variance within the input set of images. This approach has been used on dynamic PET data resulting in images with high contrast between structures with different kinetic behaviours of a tracer. PCA can be performed with either slices or volumes as observations, and either on the whole dataset or on selected parts [4]. Since it is common that only a limited part of a dataset contains information that is actually of interest, the dataset can be masked to only include data within this Volume of Interest (VOI). This procedure reduces memory usage, computation time and also has the advantage that the directions of the eigenvectors only are dependent of data inside the VOI and not influenced by noise or other disturbing signals in the background. Masked Volume-Wise PCA (MVW-PCA) is performed in a number of steps: A mask representing the scanned object is created from either transmission images from PET, or CT if a PET/CT study is performed. The mask is used to extract the VOI from the background and can also be used to perform background noise pre-normalization, described in section PCA is performed on the data within the VOI. PCs created with MVW-PCA is referred to as MVW-PCs[4]. To view the MVW-PCs they are placed back into the mask, see figure 3.3. Pre-normalize and mask PCA Unmask Frame Frame 2 Frame p Mask PET data Masked data volumes MVW-PCs Unmasked MVW-PCs Figure 3.3: Illustration of the MVW-PCA procedure.

29 Materials and methods 29 In this report a slightly different approach is used when creating the unmasked MVW-PCs. Instead of projecting the masked pre-normalized data onto the eigenvectors retrieved from the PCA, the pre-normalized non-masked PET data is projected onto the eigenvectors. This gives an identical result inside the mask as seen in figure 3.4, the memory usage and speed is the same, but the removed background will still be visible instead of the sharp mask border and the padded zeroes seen in figure 3.4(a). Another advantage of using this method is that no information in the PET dataset is lost during the MVW-PCA and the whole original dataset can therefore be reconstructed using the method described in section (a) 5 (b) (a) (b) Figure 3.4: A slice from MVW-PC showing the differences between masked data (a) and non-masked data (b) projected onto the first eigenvector. The study was performed on the brain of a patient with alzheimers disease injected with the tracer C-Pittsburgh Compound-B Reconstruction from selected principal components Since the signl in PET datasets is temporally correlated whereas the noise is not, MVW-PCA can be used to reduce the dimensionality of datasets. In the space spanned by the MVW-PCs the signal is mostly described by lower-order MVW-PCs whereas noise is described by higher-order MVW-PCs. It is therefore usefull to be able to separate data spanned by the low-order MVW-PCs from data spanned by the higher-order MVW-PCs in the original frame-space. Since PCA and MVW-PCA with non-masked data merely perform a change of basis of the pre-normalized data, no quantitative information is lost during the MVW-PCA and the full original dataset or parts of it can be reconstructed from the MVW-PCs. As described in the section PCs are created from the equation Y = EZ, where Z is the pre-normalized data. To retrieve the pre-normalized data from all MVW-PCs one can simply use the inverse of E. Z = E Y

30 3 Characterization and Reduction of Noise in PET Data Using MVW-PCA In order to calculate a selected number of MVW-PCs from the pre-normalized data, rows that correspond to unwanted MVW-PCs are removed from E, creating Ẽ and Ỹ. In the same way columns that correspond to unwanted MVW-PCs are removed from E in order to calculate the modified pre-normalized data Z from the selected number of MVW-PCs in Ỹ. Something to notice is that E is an orthogonal eigenvector matrix that have the property E T = E which saves computation time. MVW-PCs may now efficiently be removed from Z without any unnecessary calculations using Z = E -Ỹ = E -ẼZ = E TẼZ = AZ, where A = E TẼ. The modified input matrix X is then retrieved from Z by doing the inverse normalization. A flow chart of the procedure is shown in figure 3.5. X Z Y Pre-normalization MVW-PCA X Z Ỹ Un-normalization Change of basis Removal of MVW-PCs Figure 3.5: Removal of MVW-PCs from X Background noise pre-normalization When performing 2D reconstructions of PET data, each slice tends to have varying levels of noise. Since PCA cannot separate variance due to signal from variance due to noise, it is desirable to have each slice scaled with the standard deviation of the noise to get unit noise variance in every observation. In PET data reconstructed with FBP the background contains a large amount of noise. Background noise pre-normalization use a mask to separate the background from the VOI in order to estimate the standard deviation of the background in each slice. The pre-normalization is performed using the equation z ik = x ik x i s w (3.2) where s w is the estimated standard deviation for the samples within the background of slice w where the sample x ik is located. Background noise prenormalization is only used on PET data reconstructed with FBP since the noise in data reconstructed with OSEM is too signal dependent, which results in noise magnitudes close to zero outside the VOI [4] Higher-order principal component pre-normalization Higher-Order Principal Component (HOPC) pre-normalization is a novel method presented for the first time in this report. Much like Background Noise (BN) pre-normalization it retrieves an estimate of each slice s standard deviation, which is then used for pre-normalizing the slices. Since the reconstruction of early MVW-PCs is a good approximation of the expectation value in a dataset, it can be removed and the standard deviation of the reconstruction of the rest of the MVW-PCs should approximately be that of the noise.

31 Materials and methods 3 The pre-normalization can be divided into three steps. The first is to do MVW-PCA on the whole FOV (can also be performed on the VOI if speed is of high importance) without any pre-normalization and set the first MVW- PC to zero. The second step is to reconstruct the MVW-PCs and estimate the standard deviation in each slice in the reconstruction. The third step is to perform the prenormalization described in equation 3.2 but with the new estimated noise standard deviation s w. X (x ik x i )/s w Z Removal of lower-order PCs X s w Estimation of std Figure 3.6: Late Principal Component Pre-normalization The advantage with HOPC pre-normalization compared to background noise pre-normalization is that it is not dependent on the presence of noise in the background and that the background noise is proportional to the amount of noise in the rest of the slice. In order to decide the number of early MVW-PCs to use in the reconstruction of late MVW-PCs the standard deviation of the estimated noise was compared to the standard deviation of the background in datasets reconstructed with FBP. To retrieve a quantitative measure of how well the two curves fit, one of the curves was multiplied with a scalar value that minimizes the MSE. This was done since PCA picks the same eigenvectors no matter the scale of the input data. The MSE for the two curves was then calculated. The comparisons were made on two clinical datasets reconstructed with FBP, where the first was a dataset retrieved from a full body scan with the tracer Fluorodeoxyglucose (FDG) described in section 3.2.6, and the second was a brain study with the tracer Pittsburgh Compound-B (PIB) Synthetic images Motive To get an understanding of how PCA acts on actual dynamic PET data, synthetic images were produced. This was chosen as a starting point to the project since realizations of noise and signal are known prior to the analysis and can be used to validate the results. Another advantage of using synthetic images is the possibility to modify and study one parameter at a time to get a better understanding of how the analysis method react to different input. Signal In this study MATLAB (The Mathworks Inc., Natick, Massachusetts) was used to create various datasets of dynamic synthetic images. The synthetic images had one slice per observation with a size of pixels that included four geometric structures. The spatial size and the kinetic behaviour of the structures was chosen to resemble the cerebellum (CBL), occipital cortex (Occip), frontal

32 32 Characterization and Reduction of Noise in PET Data Using MVW-PCA cortex (FrntCx) and white matter (WhitM) in PET studies with the tracer C labelled PIB as seen in figure 3.7. CBL WhitM FrntCx Occip (a) Spatial structures Concentration of activity [Bq/cm 3 ] (b) Temporal values Noise CBL FrntCx WhitM Occip expectation Concentration of activity [Bq/cm 3 ] Noise CBL FrntCx WhitM Occip (c) Temporal standard deviations Figure 3.7: Spatial and temporal behaviour of the signal structures. The spatial background mask used in background noise pre-normalization is shown in a black colour. The temporal behaviour of the four Time Activity Curves (TACs) was calculated with the kinetic function k(t) = αe βt ( e γt ), (3.3) using the parameters found in table 3.. Spatially all regions had a constant value. The signal was then convoluted with a point-spread function to create a spillover effect similar to that of images retrieved from a PET scanner. The time interval between the different frames was set to an interval used in 24-frame (6 min) PET studies with the tracer PIB and the values in the time vector t was set to the centre of each interval, see table 3.2. Structure Function α β γ CBL (mean) k (t) FrntCx (mean) k 2 (t) 3... WhitM (mean) k 3 (t) Occip (mean) k 4 (t) Noise (std.) k n (t) Table 3.: Parameters for the temporal functions. Noise Raw PET data is usually considered to be Poisson distributed, but after reconstruction the noise is often approximated as normal distributed. The standard deviation, k n, of the noise was calculated with equation 3.3 with parameters from table 3.. The standard deviation of the added noise was spatially constant, which is a simplification of the noise typically seen in data reconstructed with FBP. Both spatially correlated and uncorrelated noise was studied. To correlate the noise, it was convoluted with a low pass filter followed by a multiplication with a scalar for each frame to compensate for the loss of standard

33 Materials and methods 33 Frame Length [min] Time, t [min] Table 3.2: Time protocol for a 24-frame scan (6 min) with the tracer PIB. deviation. The same realization of the noise was used for both uncorrelated and correlated noise. The synthetic datasets, x(t), were defined with the model ( 4 ) x(t) = k i (t) v i ρ v + c t (k n (t) n) ρ e, (3.4) i= where v i is a vector defining the spatial structure i, seen in figure 3.7(a). k i (t) and k n (t) also shown in figure 3.7 are the functions defined by equation 3.3 with parameters from table 3.. n is a noise vector for the entire slice that has zero mean and unit variance. ρ v and ρ n are point spread functions (low pass filters) and c t a constant used to compensate for the reduction in standard deviation caused by the convolution. Generated datasets Three synthetic datasets were generated using the kinetic functions and structures described in section and The first set consisted of nothing more than the signal, which is the four regions convoluted with a point-spread function that gave the edges a blurry appearance, see figure 3.8. The second and third dataset had the same signal components as the first dataset, but with

34 34 Characterization and Reduction of Noise in PET Data Using MVW-PCA added Gaussian noise. The second dataset has uncorrelated noise that can be seen in figure 3.9, and the third dataset had correlated noise that is shown in figure Concentration of activity [Bq/cm 3 ].5 Figure 3.8: Montage of dataset without any noise. The sequence should be seen along rows starting from the upper left corner and ending in the lower right. Pre-normalization All datasets described in were pre-normalized with the ROM pre-normalization, pre-normalized to SV, BN pre-normalized with the background mask seen in figure 3.7(a) and Higher-Order Principal Component (HOPC) pre-normalized. Estimates Since the signal in the synthetic datasets was known in advance, it was used to retrieve accurate estimates of the noise. The standard deviation of the noise in the pre-normalized noisy datasets was calculated by subtracting the non-noisy dataset that had been pre-normalized with the same coefficients as the two noisy datasets. In order to get a quantitative evaluation of the performance of the different pre-normalizations, reconstructions of early PCs was used and compared to the correct signal. To measure the error in the reconstructions the MSE was calculated with MSE = N p (u,v,n) S (x[u, v, n] x[u, v, n]) 2, (3.5) where N is the number of samples in an observation, p is the number of observations, (u, v, n) are the coordinates within a synthetic dataset S, x[u, v, n] is the signal dataset and x[u, v, n] is the reconstructed dataset.

35 Materials and methods Concentration of activity [Bq/cm 3 ].5 Figure 3.9: Montage of dataset with uncorrelated noise. The sequence should be seen along rows starting from the upper left corner and ending in the lower right Concentration of activity [Bq/cm 3 ].5 Figure 3.: Montage of dataset with correlated noise. The sequence should be seen along rows starting from the upper left corner and ending in the lower right.

36 36 Characterization and Reduction of Noise in PET Data Using MVW-PCA The MSE was calculated on datasets reconstructed with; the first PC, the first two PCs, the first three PCs and the first four PCs. This was done on both the dataset with uncorrelated and correlated noise, and for all prenormalizations.

37 Materials and methods Clinical study In order to compare HOPC pre-normalization to other pre-normalization methods prior to MVW-PCA on clinical data, two VOIs were placed on regions with different kinetic behaviour. Dimension reduction was then used to reconstruct datasets from various numbers of early MVW-PCs. Two different estimates were used on the pre-normalized datasets in order to measure the performance of the pre-normalizations. In this way it was possible to compare the amount of signal accounted for by the MVW-PCs and also to compare the reduction of noise within each VOI. Dataset and VOIs A clinical dynamic PET dataset from a full body scan performed with the tracer FDG was used. The dataset had 4 frames and both FBP and OSEM reconstructions were available. The time protocol described in table 3.3 was used during the acquisition. Frame Length [min] Time, t [min] Table 3.3: Time protocol for a 4-frame full body scan (45 min) with the tracer FDG. Two VOIs were selected from the data reconstructed with OSEM. The first VOI was placed on a tumour on the liver and the second VOI on the stomach. Dimension reduction Both datasets reconstructed with FBP and OSEM were pre-normalized with the ROM, SV, BN and HOPC pre-normalization, creating eight new datasets. MVW-PCA was performed on all of these datasets and reconstructions were made with MVW-PC, with MVW-PC and MVW-PC 2, with MVW-PC, MVW-PC 2 and MVW-PC 3, and so on until all but the last MVW-PC (MVW-PC 4 ) was used in a reconstruction. In this way 4 datasets (2 4 3 = 4) were created.

38 38 Characterization and Reduction of Noise in PET Data Using MVW-PCA Estimates The most common way to measure the signal within a VOI or ROI is to calculate the arithmetic mean of the samples. Therefore it is important to compare the arithmetic mean in regions within the original dataset to the mean in the same regions in the dimension-reduced datasets. This is important since the difference in quantitative measurements should not deviate too much. In order to get a scalar estimate that could be used to compare the differences in arithmetic mean for all frames, the MSE of the mean was calculated with MSE mean = p p ( x[n] x[n]) 2 (3.6) n= where p is the number of frames, x[n] is the arithmetic mean within the VOI of the original data and x[n] is the mean within the VOI of the reconstructed data. Usually, when drawing a VOI within a structure, the volume is not completely homogeneous. There are variations in scale between the TACs but they all share a similar kinetic behaviour. Since TACs that only differs in scale can be described by one PC, MVW-PCA was performed on the VOI in order to find one MVW-PC than optimally represents the TACs in a mean square sense. This optimal component was reconstructed and referred to as ˆx[u, v, w, n]. To measure the deviation from ˆx[u, v, w, n] the MSE was used. The MSE between ˆx[u, v, w, n] and the reconstructed signal is MSEˆx = p N VOI p (ˆx[u, v, w, n] x[u, v, w, n]) 2. (3.7) n= VOI In other words, MSEˆx is the mean of the squared difference between the N VOI estimated optimal TACs and the corresponding reconstructed TACs. p is the number of frames.

39 Chapter 4 Results 4. Characterization of noise 4.. Frames To visualize the difference in arithmetic mean and sample standard deviation for the ROIs in different frames, the arithmetic mean of the measurements in all slices were calculated. The result is shown in figure 4.. (a) (b) (c) Concentration of activity [Bq/cm 3 ] (d) (e) (f) Figure 4.: Arithmetic mean (a) (c) and sample standard deviation (d) (f) for FBP (dashed lines) and OSEM (solid lines) over time in the three ROIs: High activity ROI (a) and (d), low activity ROI (b) and (e), and no activity ROI (c) and (f). OSEM had a slightly higher expectation value in the high and low activity ROIs compared to FBP, whereas the expectation value in the no activity ROI is a lot lower for OSEM compared to FBP. OSEM also showed a heavy decrease in standard deviation in the low and no activity ROIs compared to FBP. Apart from the differences between FBP and OSEM one can see that the arithmetic

40 4 Characterization and Reduction of Noise in PET Data Using MVW-PCA mean is constant, whereas the standard deviation shows a positive trend over time. Correlation between different frames should be close to zero and the estimated correlation coefficient for the closest neighbouring frames, shown in figure 4.2, confirms this. Correlation coefficient.5 (a) (b) Frame.5 (c) Figure 4.2: Correlation coefficient for FBP (dashed lines) and OSEM (solid lines) for the closest neighbouring frames in the three ROIs: High activity ROI (a), low activity ROI (b), and no activity ROI (c) Slices To visualize the difference in arithmetic mean and sample standard deviation for the ROIs in different slices, the arithmetic mean of the measurements in all frames were calculated. The result is shown in figure 4.3. Concentration of activity [Bq/cm (a) 2 2 (d) (b) 2 2 (e) (c) (f) 2 2 Axial offset [mm] Figure 4.3: Arithmetic mean (a) (c) and sample standard deviation (d) (f) for FBP (dashed lines) and OSEM (solid lines) for different axial offsets in the three ROIs: High activity ROI (a) and (d), low activity ROI (b) and (e), and no activity ROI (c) and (f). The same differences in expectation value and standard deviation between OSEM and FBP seen in figure 4. where visible in 4.3. Other general observations were that the outer slices close to the opening of the gantry had lower expectation value than the ones near the centre and that the standard deviation

41 Results 4 had a characteristic shape with high values near the edges and two peeks at slice 25 and 37. The estimated expectation value in the no activity ROI also had this shape. One can also see that odd slices tend to have a slightly higher standard deviation compared to the even neighbouring slices, resulting in a jagged appearance. An inspection of the slice correlation matrix diagonals indicated that there were correlations to the closest neighbouring slice. The estimated correlation coefficient to the closest and second closest neighbouring slice is shown in figure 4.4. (a) (b) (c) Correlation coefficient 2 2 (d) (e) (f) Slice Figure 4.4: Correlation coefficient for the closest (a) (c), and the second closest (d) (f) neighbouring slices for FBP (dashed lines) and OSEM (solid lines) in the three ROIs: High activity ROI (a), low activity ROI (b), and no activity ROI (c). The only visible difference in correlation between slices when comparing FBP and OSEM is in the no activity region where FBP has a lower correlation coefficient compared to OSEM. In general, the correlation coefficient between neighbouring slices is approximately.4 near the centre of the gantry, compared to.2 closer to the edges. One can also distinguish a pattern in the correlation with an almost flat peak between the 26th and 36th slice pair. The correlation to the second nearest neighbouring slice is close to zero Correlation between samples within the same slice The sample correlation at the centre coordinate within the three regions is shown in figure 4.5. The first column shows sum images with a marked dot and small rectangle. The dot indicates the centre coordinate, and the rectangle shows an pixel region of neighbouring samples to which the correlation has been calculated. The second column shows the correlation in FBP data, and the third column show the correlation in OSEM data. The correlation in data reconstructed with FBP and OSEM has very different properties. In regions where both FBP and OSEM has isotropic correlation, as can be seen in the upper row, the width of the correlation in FBP is smaller than

42 42 Characterization and Reduction of Noise in PET Data Using MVW-PCA the width of the correlation in OSEM. But, this is not true for all coordinates since the amount of correlation is highly dependent on the orientation of the correlation. In this dataset the correlation in the FBP data was oriented towards the strongest region of activity while the correlation in OSEM was orthogonally oriented towards the closest region of activity. Overall, the correlation in OSEM has a larger extent in the non-orientation direction and is more isotropic in regions with activity compared to the correlation in FBP. Coordinate & Region FBP OSEM Concentration of activity.5.5 Correlation coefficient.5.5 Correlation coefficient Concentration of activity.5.5 Correlation coefficient.5.5 Correlation coefficient Concentration of activity.5.5 Correlation coefficient.5.5 Correlation coefficient Figure 4.5: Sample correlation coefficient for data reconstructed with FBP and OSEM for the centre coordinate within each ROI Correlation between samples within the same slice from one realization The result of the ACF estimate for WSS regions is shown in figure 4.6. In the first column two rectangles are drawn on the slice used when calculating the ACF estimate. The largest rectangle corresponds to the region A where the signal should be WSS and the smallest region has the size of the pixel region B for which the correlation is calculated. Since the region A is assumed to be WSS the whole region will have the same estimated ACF, with a size equal to that of B. The correlation for FBP and OSEM data is shown in column two and three. There are similarities between this estimated correlation and the sample correlation, for example in the high intensity region (first row) the correlation in the FBP data has a small extent compared to that of OSEM, and the correlation in the FBP data has the same orientation as the sample correlation. In this slice

43 Results 43 realization the orientation of the correlation in OSEM was differed from the sample correlation. For example within the high intensity region the correlation was not as isotropic as the sample correlation and the orientation in the low and no activity regions is not as easy to distinguish as it is with the sample correlation. Another observation is that this estimate much more often yields negative correlation compared to the sample correlation. Regions FBP OSEM Concentration of activity.5.5 Correlation coefficient.5.5 Correlation coefficient Concentration of activity.5.5 Correlation coefficient.5.5 Correlation coefficient Concentration of activity.5.5 Correlation coefficient.5.5 Correlation coefficient Figure 4.6: Correlation coefficient for each ROI in data reconstructed with FBP and OSEM, estimated with the ACF estimate for WSS regions.

44 44 Characterization and Reduction of Noise in PET Data Using MVW-PCA 4.2 Reduction of noise 4.2. Higher-order principal component pre-normalization The scaled standard deviation of data with one to three low-order MVW-PCs removed and the estimated standard deviation of the background noise is shown in figure 4.7. Scaled standard deviation [Bq/cm 3 ] (a) Frame (b) Figure 4.7: Scaled plots of the standard deviation used by the HOPC prenormalization, from a full body study using the tracer FDG (a) and a brain study using the tracer PIB (b). HOPC ( ), HOPC 2 ( ) and HOPC 3 ( ) are compared to the background noise ( ). The HOPC index corresponds to the number of low-order MVW-PCs that where removed in the pre-normalization step. To measure the deviation from the standard deviation of the reconstructions compared to the background noise, the MSE was calculated for reconstructions with up to 3 low-order MVW-PCs removed. The result is shown in (a) 7 (b) MSE [(Bq/cm 3 ) 2 ] Low-order MVW-PCs removed Figure 4.8: MSE of the scaled standard deviation, used by HOPC prenormalization, with respect to the standard deviation of the background noise. (a) shows the result from the full body study and (b) shows the result from the brain study. Figure 4.8 shows that reconstruction without the first or the two first MVW- PCs, in the HOPC pre-normalization, results in scaling coefficients that are close to the BN pre-normalization in a mean square sense.

45 Results Synthetic images PCs and eigenvectors PCA was performed on the dataset without any added noise. The data was normalized with the default removal of mean pre-normalization. The PCs, eigenvectors as well as the corresponding eigenvalues are shown in figure 4.9. PC PC 2 PC 3 PC 4 PC 5 e e 2 e 3 e 4 e λ 6. λ 2.2 λ λ λ 5 Figure 4.9: The first five PCs, corresponding eigenvectors and eigenvalues for synthetic data without noise. Samples with red colour are positive while samples in blue colour are negative. PCA on the dataset without any added noise gave four principal components according to the fifth eigenvalue that is zero, indicating that the eigenvector has zero length. According to the first eigenvector, PC is similar to a mean image but has a slight emphasis on early pre-normalized frames. One can also see that all PCs contain both positive and negative samples. The PCA on pre-normalized datasets with added uncorrelated and correlated Gaussian noise is shown in figure 4. and 4.. The used pre-normalizations were Removal of Mean (ROM), SV, BN and HOPC pre-normalization with MVW-PC removed. Removal of Mean (ROM) pre-normalization gave one PC with low noise while the other three methods gave two PCs with very similar appearance. This can also be seen in the corresponding eigenvectors since late eigenvectors tend to have a Dirac, indicating that the PC describes the noise in that frame very well. The BN and HOPC pre-normalization resulted in early eigenvectors with very similar appearance. No significant differences were seen between the datasets with correlated and uncorrelated noise.

46 46 Characterization and Reduction of Noise in PET Data Using MVW-PCA PC PC 2 PC 3 PC 4 ROM SV BN HOPC e e 2 e 3 e ROM SV BN HOPC Figure 4.: PCs and eigen vectors for synthetic data without correlation that were normalized with ROM, SV, BN and HOPC pre-normalization.

47 Results 47 PC PC 2 PC 3 PC 4 ROM SV BN HOPC e e 2 e 3 e ROM SV BN HOPC Figure 4.: PCs and eigenvectors for synthetic data with correlation that were normalized with ROM, SV, BN and HOPC pre-normalization.

48 48 Characterization and Reduction of Noise in PET Data Using MVW-PCA Noise variance in pre-normalized datasets The expectation value in each point, defined by the signal dataset, was used in order to calculate the standard deviation in every slice, seen in figure 4.2. Concentration of activity [Bq/cm 3 ] RoM SV BN HOPC Concentration of activity [Bq/cm 3 ] RoM SV BN HOPC (a) (b) Figure 4.2: Standard deviation in pre-normalized data with uncorrelated noise (a) and correlated noise (b). The noise standard deviation in the BN pre-normalized dataset is close to constant, while the standard deviation of the noise in the datasets pre-normalized with the other three pre-normalizations show a negative trend. Mean squared error of reconstructions The calculated MSE for datasets with uncorrelated and correlated noise are shown in figure ROM SV BN HOPC.7.6 ROM SV BN HOPC.5.5 MSE [Bq 2 /cm 6 ].4.3 MSE [Bq 2 /cm 6 ] Number of early PCs used in reconstruction Number of early PCs used in reconstruction (a) (b) Figure 4.3: MSE of reconstructions with early PCs for datasets with uncorrelated noise (a) and correlated noise(b). The best reconstruction is obtained by pre-normalizing with BN and reconstructing the dataset with MVW-PC and MVW-PC 2. The next best result is obtained by using HOPC or SV pre-normalization.

49 Results Clinical study Dimension reduction Figure 4.4 shows the arithmetic mean in the original data and reconstructions with low-order MVW-PCs measured in the tumour-voi and the stomach-voi. This shows the differences between the tumour and stomach TAC, and also that the regions require a different amount of MVW-PCs in order to be reconstructed with an adequately small error. The number of MVW-PCs depends on both region and pre-normalization method. Concentration of activity [Bq/cm 3 ] x 4 (a) x (e) 2 4 x 4 (b) x (f) 2 4 x 4 (c) x (g) 2 4 x (h) Original ROM SV BG HOPC 2 4 Figure 4.4: Arithmetic mean within the tumour-voi (a) (c) and the stomach- VOI (d) (g), measured for different reconstructions. (a) and (b) is reconstruction with only the first MVW-PC, (c) and (d) is with the first two, (e) and (f) is with the first three and (g) is with the first four MVW-PCs. Measurements The MSE of the arithmetic mean in the reconstructed datasets is shown in figure 4.5. It should be observed that the vertical axes have a logarithmic scale. The error decreases for every added MVW-PC in the reconstruction. MVW-PCA with BN and HOPC pre-normalizations describe the arithmetic mean for the tumour better than MVW-PCA with the ROM and SV pre-normalizations in early MVW-PCs. The differences between the pre-normalizations are significant in the FBP data, not because BN and HOPC performs better on FBP data than on OSEM data, but because ROM and SV performs worse with FBP than on the OSEM data. This difference can be seen in figure 4.5 (a) (b) with reconstruction [3, 4, 5, 6] and in (c) (d) with reconstruction [4, 5, 6]. To get an understanding of what a certain MSE corresponds to in deviation from the correct mean, the MSE, in figure 4.5 (b) and (d) can be compared to the mean curves in figure 4.4. For example the sudden reduction in MSE for both the BN and HOPC pre-normalizations when reconstructing the stomach with the first three and the first four MVW-PCs (reconstruction 3 and 4) in figure 4.5 (d) is visible in figure 4.4 (f) and (g).

50 5 Characterization and Reduction of Noise in PET Data Using MVW-PCA ROM SV Background HOPC HOPC 2 9 (a) 8 (b) 8 MSEmean [(Bq/cm 3 ) 2 ] (c) (d) Reconstruction Figure 4.5: MSE of the arithmetic mean in different reconstructions for different pre-normalizations. The MSE of the data compared to on the optimal reconstruction of the VOI (described in section 3.2.6) is shown in figure 4.6. These bar graphs show that the optimal reconstruction of the tumour is retrieved from reconstructing the first three MVW-PCs in the HOPC pre-normalized dataset, for both FBP and OSEM. The optimal reconstruction of the stomach with FBP data is retrieved from reconstructing the first five MVW-PCs with BN pre-normalization, and the optimal reconstruction with OSEM data is retrieved from the first six MVW- PCs with SV pre-normalization. The ROM and SV pre-normalizations did not perform well on FBP reconstructed data, and BN pre-normalization did not perform well on OSEM reconstructed data. HOPC pre-normalization performed fairly well on both the FBP and OSEM reconstructed dataset.

51 Results 5 ROM SV Background HOPC HOPC 2 9 (a) (b) 8 9 MSEˆx [(Bq/cm 3 ) 2 ] (c) (d) Reconstruction Figure 4.6: Mean squared error in different reconstructions for different prenormalizations, compared to the optimal signal ˆx[u, v, w, n].

52 52 Characterization and Reduction of Noise in PET Data Using MVW-PCA Qualitative examples Slices from the three lowest-order MVW-PCs from datasets reconstructed with FBP and OSEM is illustrated in 4.7. HOPC pre-normalization was used when performing the MVW-PCA. (a) (b) (c) (d) (e) (f) 2 2 Figure 4.7: Slices from the three firt MVW-PCs from datasets reconstructed with FBP in (a)-(c) and OSEM in (d)-(f). Most of the adrenal tumour and the general tracer behaviour is described by MVW-PC in (a) and (d). MVW-PC 2 in (b) and (e) describes the early tracer accumulation in the kidneys, while the tracer concentration in the stomach can be separated with MVW-PC 3 shown in (c) and (f). An example of a slice from a frame in a dimension-reduced dataset is shown in figure 4.8. This dataset was reconstructed with OSEM and pre-normalized with the HOPC pre-normalization. It was then dimension-reduced to the five lowest-order MVW-PCs. The dimension reduced dataset have a more homogeneous appearance and the flickering of pixels, when viewing several frames in a sequence, have been reduced. (a) (b) x Concentration of activity [Bq/cm 3 ] Figure 4.8: One of the slices in the original (a) and dimension reduced (b) datasets showing the tumour.