Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Bevington, Connor William James
Doctor of Philosophy - PhD
University of British Columbia
Hybrid PET/MRI scanners are becoming more common in research and clinical settings, in particular for their ability to simultaneously acquire unique functional and structural information to probe the healthy and diseased brain. Data from both modalities need to be thoroughly processed to enhance signal-to-noise ratios (SNR), and the development and optimization of analysis algorithms is required to extract meaningful physiological quantities for medical research applications. This work provides significant contributions to these fields through the development, testing, and validation of four algorithms. PET data are intrinsically noisy due to the Poisson nature of positron emission events and the relatively low fraction of events detected by scanners, so we first improve upon an existing PET denoising algorithm to overcome limitations in quantitative accuracy in precision. This proves important in task-based PET imaging, where tasks presented to a subject during scanning result in a small temporal signal change that is difficult to distinguish from noise. The developed PET denoising algorithm is thus coupled with an advanced PET detection algorithm which provides a significant improvement in the detection sensitivity of task-induced signal changes. Motivated by the success of existing PET denoising algorithms, we then develop an MRI denoising algorithm for low SNR functional MRI data, which demonstrates significant improvements in quantitative accuracy and precision of downstream modeling metrics. Finally, we propose a pattern analysis extension of an existing joint PET/MRI analysis algorithm to identify disease-related alterations to simultaneously acquired multimodal data. We identify immediate applications of each algorithm in novel medical research studies, and proof-of-concept testing on human data is performed when available.
Development of advanced denoisingand analysis algorithms forapplications in hybrid PET/MRI brainimagingbyConnor William James BevingtonB.Sc., The University of Waterloo, 2017M.Sc., The University of British Columbia, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2023© Connor William James Bevington, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Development of advanced denoising and analysis algorithms for applications in hybridPET/MRI brain imagingsubmitted by Connor William James Bevington in partial fulfilment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Vesna Sossi, Professor, Physics, UBCSupervisorArman Rahmim, Professor, Physics, UBCSupervisory Committee MemberAlexander Rauscher, Associate Professor, Physics, UBCSupervisory Committee MemberJane Z. Wang, Professor, Department of Electrical and Computer Engineering, UBCUniversity ExaminerStefan Reinsberg, Associate Professor, Physics, UBCUniversity ExaminerRichard E. Carson, Professor, Biomedical Engineering & Radiology and Biomedical Imaging,Yale UniversityExternal ExaminerAdditional Supervisory Committee Members:Ludovic van Waerbeke, Professor, Physics, UBCSupervisory Committee MemberiiAbstractHybrid PET/MRI scanners are becoming more common in research and clinical settings, inparticular for their ability to simultaneously acquire unique functional and structural informa-tion to probe the healthy and diseased brain. Data from both modalities need to be thoroughlyprocessed to enhance signal-to-noise ratios (SNR), and the development and optimization ofanalysis algorithms is required to extract meaningful physiological quantities for medical re-search applications. This work provides significant contributions to these fields through thedevelopment, testing, and validation of four algorithms. PET data are intrinsically noisy dueto the Poisson nature of positron emission events and the relatively low fraction of events de-tected by scanners, so we first improve upon an existing PET denoising algorithm to overcomelimitations in quantitative accuracy in precision. This proves important in task-based PETimaging, where tasks presented to a subject during scanning result in a small temporal signalchange that is difficult to distinguish from noise. The developed PET denoising algorithm isthus coupled with an advanced PET detection algorithm which provides a significant improve-ment in the detection sensitivity of task-induced signal changes. Motivated by the success ofexisting PET denoising algorithms, we then develop an MRI denoising algorithm for low SNRfunctional MRI data, which demonstrates significant improvements in quantitative accuracy andprecision of downstream modeling metrics. Finally, we propose a pattern analysis extension ofan existing joint PET/MRI analysis algorithm to identify disease-related alterations to simul-taneously acquired multimodal data. We identify immediate applications of each algorithm innovel medical research studies, and proof-of-concept testing on human data is performed whenavailable.iiiLay SummaryHuman brain scanners allow the investigation of brain structure and function non-invasively—extremely important for diagnosing and characterizing diseases of the brain. The imaging dataacquired by these scanners can contain considerable noise, which makes understanding the datadifficult. Furthermore, information within the data that is useful to medical researchers oftenneeds to be extracted from the data through further modeling and analysis. Therefore, this workdevelops methods for removing noise and performing advanced analysis on brain imaging data,ultimately resulting in higher quality data that is more informative for medical research studies.The methods are first tested on simulated data where the truth is known to demonstrate theirimprovements over existing methods, then applied to real human data (if available) as a proofof concept.ivPrefaceA version of Chapter 2 has been published. Connor WJ Bevington, Ju-Chieh Cheng, andVesna Sossi. A 4-D iterative HYPR denoising operator improves PET image quality. IEEETrans. Radiat. Plasma Med. Sci., 6(6):641–655, 2022. I conceptualized the method, designedthe simulations, ran the analysis and wrote the manuscript. Dr. Ju-Chieh Cheng ran thesimulated and human image reconstructions and edited the manuscript. Professor Vesna Sossiedited the manuscript. I also presented part of the developed work at the 2020 IEEE MedicalImaging Conference (31 October–7 November 2020, virtual) and the 16th International Meetingon Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (19–23July, 2021).A version of Chapter 3 has been submitted for publication. I conceptualized the method,generated the simulations, and wrote the Introduction, Theory, and part of the Methods andResults sections. Jordan Hanania extended the method to account for heterogeneous binding inthe striatum, wrote part of the Methods and Results sections plus the Discussion. Analysis ofthe simulation data was split between myself and Jordan Hanania. Professor Vesna Sossi editedthe manuscript. I also presented part of the developed work at the 2021 IEEE Medical ImagingConference (16–23 October 2021, virtual), the XIII International Symposium of FunctionalNeuroreceptor Mapping of the Living Brain (14–16 December 2021, virtual), and the 31stInternational Symposium on Cerebral Blood Flow and Metabolism (12–15 June 2023, Brisbane,Australia).A version of Chapter 4 is being prepared as a manuscript. I conceptualized the method,generated and analyzed the simulations, analyzed all data, and wrote the manuscript. Dr.Rebecca Williams, Professor Bruce Pike, and Professor Vesna Sossi have provided an initialedit of the manuscript. I also presented part of the developed work at the 2022 IEEE MedicalImaging Conference (5–12 November 2022, Milan, Italy).I presented part of the developed work in Chapter 5 at the 31st International Symposiumon Cerebral Blood Flow and Metabolism (12–15 June 2023, Brisbane, Australia).Human data in Chapter 4 and Chapter 5 were obtained under the research studies H19-02892and H19-01839, approved by the UBC Clinical Research Ethics Board. I am a co-investigatoron both studies.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 PET imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 PET physics: positron emission and detection . . . . . . . . . . . . . . . 41.1.2 Challenges in detection: gamma ray interactions in matter . . . . . . . . 51.1.3 Resolution in PET imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7viTable of Contents1.3 Denoising PET data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Why are PET data so noisy? . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Post-processing denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 Denoising within image reconstruction . . . . . . . . . . . . . . . . . . . 171.4 Modeling PET data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.1 Compartmental models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.2 Reference tissue models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.3 Detecting neurotransmitter release using compartmental models . . . . . 261.4.4 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4.5 Graphical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.5 MRI: basic physics and utility in PET/MRI imaging . . . . . . . . . . . . . . . . 291.5.1 Nuclear magnetic resonance and its relationship to MRI . . . . . . . . . 291.5.2 Encoding spatial information . . . . . . . . . . . . . . . . . . . . . . . . . 311.5.3 MRI sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5.4 MRI sequences relevant to this work . . . . . . . . . . . . . . . . . . . . . 361.6 Application of MRI: quantification of cerebral metabolic rate of oxygen . . . . . 371.6.1 Physiological background . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.6.2 Quantifying CMRO2 in vivo . . . . . . . . . . . . . . . . . . . . . . . . . 381.6.3 Calibrating the BOLD signal . . . . . . . . . . . . . . . . . . . . . . . . . 381.6.4 Dual-calibrated BOLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.7 Pattern analysis of imaging data . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Development of an advanced PET denoising operator . . . . . . . . . . . . . 462.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.1.1 HYPR denoising: limitations and targeted improvements . . . . . . . . . 46viiTable of Contents2.1.2 Testing and validating IHYPR4D in a variety of scanning environments . 482.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.1 Phantom study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.2 Human simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.3 Human study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.1 Phantom Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.2 Human simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4.3 Human study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.5.1 Interpreting IHYPR4D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.5.2 An improvement over HYPR3D . . . . . . . . . . . . . . . . . . . . . . . 732.5.3 Coupling a denoising reconstruction with IHYPR4D . . . . . . . . . . . . 742.5.4 ROI generation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.5.6 Likely applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.5.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Development of an advanced dopamine release detection algorithm . . . . . 793.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.1.1 Physiological background . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.1.2 Transient release modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 813.1.3 Transient release detection . . . . . . . . . . . . . . . . . . . . . . . . . . 813.1.4 Disadvantages of model-based detection . . . . . . . . . . . . . . . . . . . 83viiiTable of Contents3.1.5 Isolating DA-induced perturbations via residuals analysis . . . . . . . . . 833.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.2.1 Transforming acquired data to a residuals space . . . . . . . . . . . . . . 853.2.2 Residual Space Detection (RSD) . . . . . . . . . . . . . . . . . . . . . . 883.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.3.1 Simulation design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.3.2 Implementation of RSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.3 Cluster size thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.4 Comparison to model-based F-test method . . . . . . . . . . . . . . . . . 933.3.5 Performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.4.1 Release in the presence of homogeneous binding . . . . . . . . . . . . . . 943.4.2 Release in the presence of heterogeneous binding . . . . . . . . . . . . . . 943.4.3 Effect of not accounting for heterogeneous binding in RSD . . . . . . . . 963.4.4 Comparing TAC modeling in lp-ntPET vs. RSD . . . . . . . . . . . . . . 963.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.5.1 Detection sensitivity improvements . . . . . . . . . . . . . . . . . . . . . 983.5.2 Handling heterogenous RAC binding . . . . . . . . . . . . . . . . . . . . 994 Development of an advanced MRI denoising algorithm . . . . . . . . . . . . . 1004.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.2 Denoising hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . 110ixTable of Contents4.3.3 OEF quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.4 Image quality metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.5 Human data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.4.2 Human data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.5.1 Denoising using cluster-based averaging . . . . . . . . . . . . . . . . . . . 1244.5.2 Relaxing the assumptions of clustering . . . . . . . . . . . . . . . . . . . 1254.5.3 Ability of the feature space to capture variation in OEF . . . . . . . . . 1264.5.4 Improvements in noise and error . . . . . . . . . . . . . . . . . . . . . . . 1264.5.5 Application to healthy human data . . . . . . . . . . . . . . . . . . . . . 1274.5.6 Reliable subject-level data? . . . . . . . . . . . . . . . . . . . . . . . . . 1274.5.7 Fine-tuning HYPR4D-FS hyperparameters . . . . . . . . . . . . . . . . . 1274.5.8 Combining OEF and CBF to estimate CMRO2 . . . . . . . . . . . . . . 1285 Joint analysis of metabolic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.1.1 Physiological background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.1.2 Previous work in joint analysis of CMRO2 and CMRglu . . . . . . . . . . 1305.1.3 Adapting and expanding joint analysis of CMRglu and CMRO2 . . . . . 1325.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.2.1 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.2.2 Data acquisition, image processing, and metric computation . . . . . . . 1335.2.3 Pattern analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134xTable of Contents5.2.4 Computation of OGI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3.1 OEF in healthy and PD subjects . . . . . . . . . . . . . . . . . . . . . . . 1365.3.2 CMRglu and CBF patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.3.3 rPWR and rCST patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.3.4 Absolute quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1 Motivations based on current research in PET/MRI image processing and anal-ysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1.1 PET denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1.2 MRI denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.1.3 PET/MRI image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.2 Improvements offered to PET/MRI image processing and analysis . . . . . . . . 1466.3 Strengths and limitations of the present work . . . . . . . . . . . . . . . . . . . . 1486.3.1 Development from first principles . . . . . . . . . . . . . . . . . . . . . . 1486.3.2 Accurate simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3.3 Quantitative improvements over previous algorithms . . . . . . . . . . . 1506.3.4 Computational speed of algorithms . . . . . . . . . . . . . . . . . . . . . 1506.3.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.4 Possible applications of the present work . . . . . . . . . . . . . . . . . . . . . . 1536.4.1 IHYPR4D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.4.2 RSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.4.3 HYPR4D-FS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.4.4 SSM-PCA applied to joint PET/MRI analysis metrics . . . . . . . . . . . 154xiTable of Contents6.5 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157AppendicesA Effects of inaccurate segmentation or PSF estimate on IHYPR4D . . . . . . 176A.1 Effect of inaccurate segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 176A.2 Effect of inaccurate PSF estimation . . . . . . . . . . . . . . . . . . . . . . . . . 178B Simulating the DA release timecourse . . . . . . . . . . . . . . . . . . . . . . . . 179C 2D Hampel filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182xiiList of Tables1.1 An example comparison of regional values in the healthy vs. diseased brain . . . 431.2 An example comparison of regional values in the healthy vs. diseased brain,demeaned across regions and then subjects . . . . . . . . . . . . . . . . . . . . . 432.1 Comparing %std in the cerebellum for human RAC data . . . . . . . . . . . . . . 703.1 Detection sensitivities for RSD and lp-ntPET + F-test . . . . . . . . . . . . . . . 954.1 Average image noise in the uniform simulation . . . . . . . . . . . . . . . . . . . 1174.2 Mean and standard deviation of regional OEF of the lesion . . . . . . . . . . . . 1225.1 Absolute group mean values of CMRglu, CMRO2, CBF, and OGI . . . . . . . . . 140xiiiList of Figures1.1 Various scenarios for gamma ray pairs resulting in a coincident detection . . . . . 61.2 Illustrations of compartmental models . . . . . . . . . . . . . . . . . . . . . . . . 241.3 The Simplified Reference Tissue Model . . . . . . . . . . . . . . . . . . . . . . . . 242.1 IHYPR4D k-means segmentation method . . . . . . . . . . . . . . . . . . . . . . 522.2 Contrast recovery versus noise in the phantom study . . . . . . . . . . . . . . . . 572.3 Contrast recovery versus voxel noise: effect of temporal kernel size . . . . . . . . 582.4 Concentration distributions in the phantom study . . . . . . . . . . . . . . . . . 602.5 Voxel noise as a function of frame counts in the phantom study . . . . . . . . . . 612.6 Contrast recovery versus voxel noise in the SIGNA phantom study . . . . . . . . 612.7 Bias and noise in the simulation study . . . . . . . . . . . . . . . . . . . . . . . . 622.8 Comparing HYPR3D and initial IHYPR4D initial composites in the simulationstudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.9 Axial slices of the simulation study images . . . . . . . . . . . . . . . . . . . . . . 662.10 Demonstrating noise-induced biases in fitted BPND in the simulation study . . . 672.11 Histograms of fitted BPND in the simulation study . . . . . . . . . . . . . . . . . 672.12 Bias and noise in BPND in the simulation study . . . . . . . . . . . . . . . . . . . 672.13 Line profiles of the simulated BPND gradient . . . . . . . . . . . . . . . . . . . . 682.14 Axial slices of human images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.15 Voxel TACs in the simulated and human data . . . . . . . . . . . . . . . . . . . . 71xivList of Figures2.16 Image-derived input functions for the first four minutes of the FDG scan of thehuman study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.17 Axial slices of parametric Ki images derived from human FDG data . . . . . . . 723.1 Simulated noisy voxel TACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.2 Demonstrating heterogeneous RAC binding in PD . . . . . . . . . . . . . . . . . 863.3 k-means results for noisy simulated data with an anterior-posterior/superior-inferior Bmax gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4 Summary of the Residual Space Detection method . . . . . . . . . . . . . . . . . 893.5 Testing the convergence of RSD’s iterative outputs . . . . . . . . . . . . . . . . . 903.6 Voxelwise prediction maps for release in the presence of homogeneous binding . . 943.7 Voxelwise prediction maps for release in the presence of heterogeneous binding . 953.8 Weights from the NNLS regression used to estimate voxelwise baseline TACs . . 963.9 The effect of not accounting for heterogeneous binding in RSD . . . . . . . . . . 973.10 Visualization of results in a negative (non-release) voxel from a single noisy re-alization for the homogeneous simulations . . . . . . . . . . . . . . . . . . . . . . 984.1 The effect of under/overestimations of various metrics on calculated OEF . . . . 1024.2 Design of the dual-calibrated BOLD protocol used at our centre . . . . . . . . . . 1074.3 High resolution atlas of tissue types . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4 OEF maps from denoised simulation data . . . . . . . . . . . . . . . . . . . . . . 1154.5 2D histograms of HYPR4D-FS feature space . . . . . . . . . . . . . . . . . . . . 1164.6 Histograms of OEF values for all voxels and noisy realizations of the uniformsimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.7 Voxel maps of %RMSE in the uniform simulation . . . . . . . . . . . . . . . . . . 1184.8 Voxel maps of absolute RMSE in the lesion simulation . . . . . . . . . . . . . . . 1194.9 Voxel maps of %RMSE in the spatially-varying simulation . . . . . . . . . . . . . 120xvList of Figures4.10 Median OEF images over all healthy subjects . . . . . . . . . . . . . . . . . . . . 1224.11 Individual OEF maps for two healthy subjects . . . . . . . . . . . . . . . . . . . 1235.1 Computation of relative power (rPWR) and relative aerobic cost (rCST) . . . . . 1345.2 Group median images of OEF for healthy and PD subjects . . . . . . . . . . . . 1375.3 PD-related CMRglu and CBF patterns . . . . . . . . . . . . . . . . . . . . . . . . 1385.4 PD-related rPWR and rCST patterns . . . . . . . . . . . . . . . . . . . . . . . . 1395.5 Oxygen-to-glucose ratio (OGI) as calculated by equation (5.1) in the posteriorputamen, for healthy and PD subjects . . . . . . . . . . . . . . . . . . . . . . . . 139A.1 The effect of inaccurate segmentation on IHYPR4D . . . . . . . . . . . . . . . . 177A.2 The effect of inaccurate PSF on IHYPR4D . . . . . . . . . . . . . . . . . . . . . 178B.1 Temporal DA release profile for the simulated motor/cognitive task . . . . . . . . 181xviList of SymbolsBelow is a list of symbols used in each chapter, in order of appearance in the text. Referencesto their definitions are provided when applicable.Chapter 1PET imagingSymbol Description Definition∆t timing offset of gamma ray arrival at detector equation (1.1)∆x offset in distance traveled by annihilation gamma raysc speed of lightθ Compton scattering angle equation (1.2)Rab random coincidence rate for detectors a and b equation (1.3)ra single event rate for detector arb single event rate for detector bτ coincidence timing windowxviiChapter 1Image reconstructionSymbol Description Definitiony vector of line-of-response counts equation (1.6)x vector of voxel counts equation (1.6)P system matrix mapping x to y equation (1.6)a vector of additive contamination counts for each line-of-responsei index over lines-of-responsej index over voxelsM number of lines-of-responseN number of voxelsn index over image reconstruction iteration numbers index over subset imagesDenoising PET dataSymbol Description DefinitionI imaging dataF filtering kernel∗ convolutionC HYPR composite data equation (1.22)IH HYPR-processed data equation (1.22)K kernel matrix for kernelized reconstruction equation (1.26)α kernel coefficients for kernelized reconstruction equation (1.26)κ(·) kernel function for kernelized reconstruction equation (1.28)l index over kernel matrix basis functionsL number of kernel matrix basis functionsxviiiChapter 1Modeling PET dataSymbol Description DefinitionCT (t) concentration in target region equation (1.34)CP (t) plasma input functionH(t) impulse response function equation (1.35)K1 influx rate constant from plasma to first compartmentk2 efflux rate constant from first compartment to plasmak3 influx rate constant from first to second compartmentk4 efflux rate constant from second to first compartmentCND(t) concentration of non-displaceable radiotracerCS(t) concentration of specifically-bound radiotracerVND distribution volume of non-displaceable radiotracer equation (1.37)VS distribution volume of specifically-bound radiotracer equation (1.38)VT total distribution volume equation (1.39)BP binding potential equation (1.40)Bmax density of receptorsBPND non-displaceable binding potential equation (1.41)CR(t) concentration in reference regionk2a apparent efflux rate constant from target region in SRTM equation (1.42)K ′1 influx rate constant from plasma to reference regionk′2 efflux rate constant from reference region to plasmaR1 radiotracer delivery to target region relative to referenceregionh(t) lp-ntPET neurotransmitter release basis functions equation (1.51)θ(t) Heaviside functiontD start time of neurotransmitter release equation (1.51)tP time of peak neurotransmitter release equation (1.51)α sharpness of neurotransmitter release profile equation (1.51)γ strength of neurotransmitter release equation (1.52)Ki net uptake into trapping compartment equation (1.56)CMRglu cerebral metabolic rate of glucose equation (1.57)xixChapter 1MRI: basic physics and utility in PET/MRI imagingSymbol Description DefinitionB0 external magnetic fieldµ± magnetic moment of aligned/anti-aligned spins equation (1.58)γ gyromagnetic ratioh Plank’s constantE± energy of aligned/anti-aligned spins in magnetic field equation (1.59)P± probability of spin being aligned/anti-aligned to magneticfieldequation (1.60)ω0 Larmor frequency equation (1.62)Mz(t) longitudinal component of magnetization equation (1.63)Mxy(t) transverse component of magnetization equation (1.64)T1 spin-lattice relaxation timeT2 spin-spin relaxation timeT∗2 effective spin-spin relaxation timeGx x-gradientGy y-gradientGz z-gradientω(x, t) spatiotemporal distribution of precession frequencies equation (1.68)φ(x, t) spatiotemporal distribution of phase accumulation equation (1.70)S(t) RF receiver coil signal equation (1.71)ρ(x) spatial distribution of proton densityk k-space vector equation (1.72)TE sequence echo timeTR sequence repetition timep blood perfusionCBF cerebral blood flow equation (1.77)λ brain/blood partition coefficientT1,blood spin-lattice relaxation time for bloodαL ASL labeling efficiency of bloodSPD proton-density weighted imageτ ASL labeling pulse durationPLD post-label delayxxChapter 1Application of MRI: quantification of cerebral metabolic rate of oxygenSymbol Description DefinitionCMRO2 cerebral metabolic rate of oxygen equation (1.78)CaO2 arterial oxygen concentration equation (1.90)OEF oxygen extraction fraction equation (1.92)BOLD blood oxygen level dependent signal equation (1.79)δS BOLD signal change relative to baseline equations (1.80, 1.88)R∗2 effective spin-spin relaxation rate equation (1.83)CBV cerebral blood volume[dHb] deoxyhemoglobin concentrationA deoxyhemoglobin concentration proportionality constantv blood volume fractionβ exponent relating [dHb] changes to δS changesM calibration constantα Grubb’s constantf cerebral blood flow relative to baselined deoxyhemoglobin concentration relative to baseline equation (1.89)ϕ carrying capacity of oxygen in blood[Hb] hemoglobin concentrationSaO2 arterial oxygen saturation equation (1.91)PaO2 partial pressure of arterial oxygenϵ solubility coefficient of oxygen in bloodPattern analysis of imaging dataSymbol Description DefinitionX imaging data matrixi index over spatial regionsj index over subjectsk index over PCA componentsZ SSM imaging data equations (1.99, 1.100)GMRj global mean rate for subject jGMPi group mean profile for spatial region iSS PCA subject score matrixC PCA component matrixxxiChapter 2: Development of an advanced PET denoising operatorChapter 2: Development of an advanced PET denoisingoperatorSymbol Description Definitioni index over voxelsj index over framesC(x) HYPR3D composite equation (2.1)IH(x, tj) HYPR3D-processed image for frame j equation (2.1)wj HYPR3D composite weighting for frame jF (x) spatial filtering kernelC(x, t) IHYPR4D initial composite equation (2.3)Ri set of voxels in the ROI containing voxel i| · | cardinality of a setP (x) point spread functionn index over IHYPR4D iteration numberIn=1H (x, t) IHYPR4D-processed data after iteration one equation (2.3)F (x, t) spatiotemporal filtering kernelIn+1H (x, t) IHYPR4D-processed data after iteration n+ 1 equation (2.4)%CRC percentage contrast recovery coefficient equation (2.5)CH concentration in hot regionCB concentration in background regionAH measured activity in hot regionAB measured activity in background region∆t frame durationλ decay constant of radiotracer%bias percentage bias equation (2.6)%std percentage standard deviation equation (2.7)nr index over noisy realizationsNR number of noisy realizationssnr voxel TAC or BPND value of noisy realization nrsGT ground truth voxel TAC or BPND values mean voxel TAC or BPND value over noisy realizationsxxiiChapter 3: Development of an advanced dopamine release detection algorithmChapter 3: Development of an advanced dopamine releasedetection algorithmSymbol Description Definitionk index over k-means clustersCi(t) time activity curve for striatal voxel iCˆi(t) hypothetical baseline time activity curve for striatal voxeliequation (3.4)CS,k(t) average time activity curve over voxels of cluster kCR(t) average reference region time activity curvetS start time of taskwS,k regression weight for cluster kwR regression weight for reference regionRi(t) residuals for striatal voxel i equation (3.5)E(t) event vectorH(t) PET response functionP (t) residuals predictor equation (3.6)tF end time of taskL broadness of response functionλ response function normalization constantβ general linear model regressor equation (3.7)ϵ error in general linear model fit equation (3.7)SA specific activity of radiotracerF (t) concentration of free radiotracerFDA(t) concentration of free dopamineB(t) concentration of bound radiotracerBDA(t) concentration of bound dopaminekon influx rate constant for specific radiotracer bindingkoff efflux rate constant for specific radiotracer bindingkDAon influx rate constant for dopamine bindingkDAoff efflux rate constant for dopamine bindingxxiiiChapter 4: Development of an advanced MRI denoising algorithmChapter 4: Development of an advanced MRI denoisingalgorithmSymbol Description Definitionφ(x) feature space position for voxel x equation (4.3)z(·) z-scoreδS(x) BOLD signal relative to baseline, averaged over timeδShc(x) BOLD signal relative to baseline, averaged over hyper-capnia periodδSho(x) BOLD signal relative to baseline, averaged over hyperoxiaperiodk index over k-means clustersSd(x, t) denoised BOLD or ASL data, reconstructed from basisfunctionsequation (4.9)Sk(x, t) basis function derived from cluster kwk(x) weight for cluster k in constructing denoised data forvoxel xequation (4.10)φk feature space centroid for cluster kh weighting kernel standard deviationSH(x, t) HYPR4D-FS-processed data equation (4.12)S0(x) signal scaling factor for voxel xPETO2 partial pressure of oxygen end-tidalsPETCO2 partial pressure of carbon dioxide end-tidalsMAD median absolute deviation equation (4.21)MED medianNR noisy realization%noise percentage noise level equation (4.22)RMSE root-mean-square error equation (4.23)%RMSE percentage root-mean-square error equation (4.24)xxivChapter 5: Joint analysis of metabolic dataChapter 5: Joint analysis of metabolic dataSymbol Description DefinitionOGI(x) spatial distribution of normalized oxygen-to-glucose in-dexequation (5.1)rPWR concurrence of relative oxygen and relative glucosemetabolismequation (5.4)rCST discordance between relative oxygen and relative glucosemetabolismequation (5.4)k index over SSM-PCA componentswk weight for SSM-PCA component kκ set of SSM-PCA components that separate PD andhealthy subjects with p < 0.2j index over subjectsCPD PD-related pattern equation (5.5)SSPD,j expression of PD-related pattern for subject j equation (5.5)xxvAcknowledgementsI would not be here without the support from Professor Vesna Sossi, my advisor. Her trust inme—especially in the early phases of my academic career—allowed me to take on some highrisk projects during my PhD. Thanks also goes to Professor Jon Stoessl, for bearing with themind of a physicist and helping keep the big picture in focus. I also owe thanks to Dr. Ju-ChiehCheng, for pointing out flaws in my methods and encouraging me to push my algorithms totheir limits.xxviDedicationTo my parents—for raising me in an environment that fostered intellectual exploration, thensupporting me in the meandering academic journey that followed.xxviiChapter 1IntroductionHybrid medical imaging scanners capable of simultaneously acquiring positron emission tomog-raphy (PET) and magnetic resonance imaging (MRI) data are becoming increasingly popularin research and clinical settings, since the imaging modalities can provide unique yet comple-mentary information for assessing and quantifying the structure and physiology of the healthyand diseased brain [1].PET is a nuclear medicine imaging modality that uses a radioactive tracer (radiotracer)to quantify molecular function; the molecule chosen will determine the molecular target orprocess of interest, then appropriate modeling of the imaging data will provide the correspondingquantitative assessment of relevance. MRI can provide both structural and functional imagesby including spatial encoding within nuclear magnetic resonance (NMR) techniques applied tohydrogen nuclei (protons) in the brain. Image contrast is determined by the MRI sequencechosen, which is a set of instructions for the radiofrequency excitation of protons and the localalteration of the external magnetic field that results in an image.With advances in scanner hardware plus novel PET radiotracers and MRI sequences, thequality and amount of physiological information embedded with the acquired data is increas-ing. At the same time, the hypotheses investigated in neuroimaging research require betterspatiotemporal resolution of these physiological processes. This often comes with the trade-offof noisier acquired data and hence decreased image quality.This provides a space for post-processing of the acquired data to rectify this loss of imagequality while maintaining high spatiotemporal resolution. The goal of post-processing is toimprove the precision of the data—i.e., perform denoising—without sacrificing the accuracy ofthe data. Especially on the PET side, denoising is essential before any meaningful informationcan be extracted from the data. Thus, a large body of research in the past few decades hasbeen dedicated to developing PET denoising algorithms.There is less of a need for denoising in MRI imaging, since the acquired data tend tohave much higher signal to noise ratio (SNR) than in PET. However, in certain applicationsof functional MRI—where tasks are presented to subjects in the scanner and a set of imagesis taken over time to track the neuronal response—the temporal task-induced signal changes1Chapter 1. Introductionare relatively small, meaning denoising is now essential for maximizing the SNR of these signalchanges—not just the signal itself—for reliable spatiotemporal localization and characterizationof neuronal response.More recently, the tracking of molecular changes in response to tasks using PET has beenmade theoretically possible. However, with PET already having poor SNR, the detectionof low magnitude, task-induced temporal signal changes is even more difficult than in MRI.Furthermore, the detection algorithms developed to date are fundamentally limited and havelow detection sensitivity.The discussion thus far has focused on PET and MRI separately, but in PET/MRI scannersthere is the valuable opportunity for joint analysis of the simultaneously-acquired metrics fromeach modality. Given the novelty of PET/MRI scanners in research, the number of PET/MRIdatasets in the literature is relatively small, and hence there has been little development of jointimage analysis algorithms to date.These observations motivate the development of advanced denoising and analysis for PETand/or MRI datasets. This work thus aims to contribute to these research fields by:1. Providing a significant advancement to current state-of-the art PET denoising algorithms(Chapter 2);2. Developing an advanced detection algorithm for PET that, in conjunction with the de-veloped PET denoising algorithm, provides a significant improvement in the detectionsensitivity of task-induced signal changes (Chapter 3);3. Developing an advanced MRI denoising algorithm to improve quantification from func-tional MRI datasets, inspired by the success of various methodologies in PET denoising(Chapter 4);4. Demonstrating the utility of a simple yet powerful joint analysis algorithm for simultane-ously acquired PET and MRI data (Chapter 5);5. Ensuring the developed algorithms are informed by physics, easy to implement, compu-tationally efficient, and applicable to a wide range of PET and/or MRI imaging datasets.This work is inherently multidisciplinary and draws upon knowledge from not only physics,but also biology, medicine, computer science, and engineering. However, the focus of this workis on the algorithmic development from physical first principles. Motivating information fromother fields is included when necessary, but is kept brief; footnotes and references are providedfor the interested reader.2Chapter 1. IntroductionAlgorithmic development starts with a motivation and description of the mathematical andcomputational operations performed by the algorithm. Testing proceeds usually on simulateddata first, such that the ground truth is known, thus allowing objective measures of accuracyand precision to be reported and compared to existing algorithms. Proof-of-concept applicationsof the developed algorithms on human datasets are provided in certain cases to validate theresults from the simulations. Any biological interpretations of these results are ancillary to themain goals of this work. Nevertheless, the development of these algorithms has immediatelyenabled progress in two neuroimaging research studies at our centre. These are:1. The effect of exercise on brain energetics and Parkinson’s disease. [18F]fluoro-deoxyglucose (FDG) PET imaging and dual-echo pCASL MRI imaging is performed onParkinson’s disease (PD) subjects. FDG data provides quantitative images on cerebralmetabolism of glucose (CMRglu), whereas the MRI data coupled with advanced denoisingand modeling provides quantitative images on cerebral metabolism of oxygen (CMRO2).Brain energetics can be quantified via the ratio CMRO2/CMRglu; mitochondrial dysfunc-tion can be inferred through a decrease in this ratio. There is evidence that mitochondrialdysfunction occurs in PD and that exercise may reduce this dysfunction [2], though thishas yet to be tested in humans. This research study thus obtains CMRO2 and CMRglumeasures from PD subjects, without a history of intense aerobic exercise, before and af-ter a six-month aerobic exercise intervention. Measures are also obtained from healthycontrols and PD subjects with a history of intense aerobic exercise for comparison withthe initial non-exercising PD subjects.2. Striatal Plasticity in Parkinson’s disease. Functions such as upper/lower limb move-ment, cognition, planning, and reward may result in neuronal activation and dopaminerelease in different spatial locations of the striatum, known as functional segregation. Itis hypothesized that there is a progressive loss of functional segregation in early PD [3],though this has yet to be studied systematically. This research study thus acquires simul-taneous fMRI and [11C]raclopride (RAC) PET imaging data in early PD subjects underfour conditions: (i) finger tapping; (ii) foot tapping; (iii) cognitive task; (iv) reward task.RAC data, processed and analyzed using advanced denoising and detection algorithms,localizes dopamine release, whereas fMRI data localizes neuronal activation.The remainder of this chapter provides a comprehensive physical background on PET andMRI. This includes the physics of the imaging process in each modality; PET and MRI imagereconstruction; the principals of PET denoising algorithms and important previous developmentin this area; the modeling of PET and MRI data to extract meaningful parameters from imagingdata; and descriptions of the image analysis algorithms used or built upon in this work.31.1. PET imaging1.1 PET imagingPET is an in vivo functional imaging modality that tracks a molecule of interest chelated to apositron-emitting radioisotope, collectively known as a radiotracer. The most commonly usedradiotracer is [18F]fluorodeoxyglucose (FDG), where one of the hydroxyl groups of glucose isreplaced by 18F. FDG images identify spatial variations in energy metabolism within the body,which may be normal [4], transiently varied in response to a task [5], or altered due to disease;in particular, the high energy demand of tumour cells allows cancer to be diagnosed and stagedvia PET imaging [6], and unique spatial patterns of hyper- and hypo-metabolism can be usedas a marker for neurodegenerative diseases, such as Parkinson’s disease [7] and Alzheimer’sdisease [8].Other radiotracers include the suite of H2[15O], C[15O], and [15O]2 for quantifying bloodflow, blood volume, and oxygen extraction fraction. A multitude of radiotracers for imagingprotein targets exist [9], some of which include radiotracers for β-amyloid and tau proteins(pathophysiological markers of Alzheimer’s disease [10]), as well as radiotracers for neurotrans-mitter transporters, terminals, and receptors.1.1.1 PET physics: positron emission and detectionRegardless of the radiotracer used, emission of a positron from the radiotracer leads to anannihilation event with a nearby electron, resulting in (nearly) collinear 511 keV gamma rayemissions in opposite directions1. Arrays of gamma cameras surround the subject to detect thegamma rays resulting from these annihilation events. The gamma rays from a single annihilationevent will arrive at two different detectors. The line in space connecting these detectors is calleda line of response (LOR) and is the closest localization of the path the gamma rays took. Wethus we presume the annihilation event occurred somewhere along the LOR.The brain is centred within the array of detectors. Gamma rays emitted from the centre ofthe brain will arrive at the same time, but gamma rays offset from centre will travel differentdistances before being detected. A coincidence timing window is used in an attempt to assignannihilation events to the correct LOR, which is limited by the temporal precision of the detectorelectronics. When a detector makes a detection, it will wait for the duration of the coincidencetiming window for a detection in another detector. If one occurs, this is termed a coincidenceevent and a count is assigned the LOR connecting the two detectors. If a second detection isnot made, then a single event is recorded by the detector. Given an offset in distance traveledof ∆x, the offset in arrival time will be:1If the annihilation occurs at rest, by conservation of momentum the two gamma rays must travel 180◦ fromeach other. However, the positron will often have some residual momentum at the time of annihilation and thusthe gamma ray momenta will not be perfectly collinear.41.1. PET imaging∆t =∆xc(1.1)where c is the speed of light. For even an extreme ∆x = 30 cm, the arrival time offset will onlybe one nanosecond; smaller ∆x will then have ∆t on the order of picoseconds. If ∆t can beprecisely measured by the detectors, then it can be exploited to localize the annihilation eventto a segment of the LOR. This is known as time-of-flight (TOF) imaging, but evidently requiresdetectors with sub-nanosecond timing resolution [11].1.1.2 Challenges in detection: gamma ray interactions in matterSeveral complications occur when attempting to detect gamma rays resulting from annihilationevents (Figure 1.1). The gamma rays are traveling through matter and thus have a probabilityof being scattered or attenuated before reaching the detector. Attenuated gamma rays evidentlyare not detected, but this loss can be corrected for during image reconstruction, which will bediscussed later. A scattered gamma ray will still reach a detector, but now it is impossible toassign the annihilation event to the correct LOR (Figure 1.1b).Photon interactions in matter to consider are Rayleigh scattering, photoelectric effect,Compton scattering, or pair production. For the 511 keV gamma rays involved in PET imaging,Compton scattering is usually only considered; relative interaction cross sections are negligiblefor Rayleigh scattering and the photoelectric effect, and the gamma ray energy is insufficientfor pair production.Compton scattering is inelastic and thus the scattered gamma ray loses some of its energy.The scattering angle is given by:θ = arccos[1−(511 keVE′)](1.2)where E′ is the energy of the scattered gamma ray. PET detectors have a modest amount ofenergy discrimination which allows some filtration of scattered events (as scattered gamma raysbelow the energy window are not detected), but from the above equation even a 20% energywindow would still accept photons scattered up to ∼ 40◦. Therefore, scatter is also correctedfor in image reconstruction, most often using a single-scatter simulation scaled to the intensityof the image [12], although more involved Monte Carlo simulations also exist [13].Random coincidences can also occur when two annihilation events occur nearly simulta-neously: in this case, the gamma rays detected first from each event are detected within thecoincidence window and thus are erroneously considered to be from the same annihilation event51.1. PET imaging(Figure 1.1c). The rate of random coincidences for a given LOR can be estimated and correctedfor during image reconstruction. Let the random coincidence rate for the LOR connecting de-tectors a and b be Rab, the single event rate for each detector be ra and rb, respectively, and thecoincidence timing window be τ . Then in one second, the total amount of time that detectora is “waiting” for a coincidence detection is 2τra. Therefore, the rate at which a detection indetector b occurs during this time is:Rab = 2τr1r2. (1.3)A final possibility is when more than two gamma rays are detected within the coincidencetiming window. In this case, it is impossible to determine which gamma rays came from a trueannihilation event, and so counts are not assigned to any LOR.Figure 1.1: Various scenarios for gamma ray pairs resulting in a coincident detection. Stars indicatethe location of the annihilation(s), solid lines trace the path of the gamma rays, and dashed lines indicatethe LOR the annihilation event would be assigned to. (a) a true coincidence where the photons travelunimpeded towards their respective detectors (note the LOR is still offset from the true gamma raypaths due to the finite detector size. (b) a scatter coincidence where one gamma ray has scattered,leading to the annihilation event being assigned to the wrong LOR. (c) a random coincidence where twoannihilation events occur nearly simultaneously and offset from the centre, such that the detection ofcloser photons from each annihilation event are mistakenly considered a coincidence event from a singleannihilation.1.1.3 Resolution in PET imagingPET resolution is fundamentally limited in localizing the radiotracer, since the positron emittedby the radiotracer will travel a short distance before annihilating, meaning even if one canperfectly localize the annihilation site, this will be radially offset from the true positron emissionsite. Different radionuclides used in PET will have different positron ranges, varying from < 1mm to several millimetres [14]. Resolution is further degraded by the aforementioned gamma61.2. Image reconstructionray acollinearity after annihilation, as well as detector blurring. The total effect of resolutiondegradation can be modeled by a point spread function (PSF), which is defined as the responseof an imaging system to a point source of radioactivity. Any radioactivity distribution can berepresented as a sum of point sources, so convolving the PSF with the true distribution g(x, y, z)yields the imaged distribution2, h(x, y, z), ignoring noise:h(x, y, z) = g(x, y, z) ∗ PSF(x, y, z). (1.4)The PSF itself is the convolution of the individual effects listed above and, in general, isspatially variant across the field of view of the scanner [15]. The PSF can be experimentallycharacterized by scanning point sources of radiation at various locations within the field ofview and computing the full-width at half maximum (FWHM) of the distribution resultingfrom these point sources. Hardware improvements over the decades have dramatically reducedthe amount of degradation due to detectors such that the FWHM of the PSF for the highestresolution imaging systems are < 3 mm near the centre of the field of view [16]. The remainingphysical effects can be partially compensated for by applying a PSF correction during imagereconstruction, but because the original data are bandlimited in the high frequency range oneseeks to restore, this will result in a ringing artifact near sharp intensity boundaries in thecorrected image [17].1.2 Image reconstructionThe set of LORs and the number of events detected along each LOR form a set of projec-tions through the object being imaged. An inverse problem must be solved to map the set ofprojections to an image. This task is ill-posed and is known as image reconstruction.The forward problem is the mapping of the unknown 3D image to projection space. LetM be the number of LORs and N be the number of voxels (3D pixels) in the correspondingimage. Also let y = [y1, y2, · · · , yM ] be a vector containing the number of counts detected ineach LOR and x = [x1, x2, · · · , xN ] be a vector containing the number of counts originatingfrom positron emissions in each voxel. Finally, let P be the system matrix with each elementPij being the probability that an emission in the jth voxel leads to a coincidence detectionalong the ith LOR. Then:y = Px. (1.5)2Here we are ignoring the fact that g is continuous whereas h is discretized into a grid of voxels. However,in practice the voxel size chosen is smaller than the resolution of the scanner such that voxel size does not limitimage resolution.71.2. Image reconstructionMost of the relevant physical details of the imaging operation can be encoded in P, in-cluding scanner geometry, attenuation of gamma rays, plus resolution degradation effects suchas detector blurring and positron range before annihilation [18]. The system matrix does notinclude the aforementioned contributions of scatter and random coincidences to the observedLOR counts, so these need to be added in a single vector a to have a full model for the physicsof imaging:y = Px+ a. (1.6)The number of LORs in a modern scanner can exceed 109 and the number of voxels inhigh resolution image reconstructions can approach 107. Therefore, storing the system matrixin computer memory is infeasible. Furthermore, inverting P to directly solve for x is alsoimpossible. Therefore, an iterative framework is required where an initial estimate of x isprovided, forward projected to yield an estimate of y that can be compared to the actualacquired data, then a correction factor is backprojected to image space to update the imageestimate.This framework is derived using the theory of Maximum Likelihood Expectation Maximiza-tion (MLEM) [19]. First, positron emission follows a Poisson distribution, and the observednumber of counts along each LOR is the sum of the emissions from voxels that contribute tothat LOR. Since the emissions from distinct voxels are independent, by the properties of thePoisson distribution this sum of independent Poisson-distributed emissions also is a Poissondistribution. The Poisson distribution states that the probability of observing k counts is:Pr(k) = e−λλkk!(1.7)where λ > 0 is the expected number of counts, i.e. the mean number of counts observed overan infinite number of equally-timed observations. Ignoring scatter and random coincidencecontribution, for the ith LOR we thus have k = yi and λ =∑j Pijxj . By the multiplicationlaw of independent probabilities, the likelihood of all observations in the vector y is given by:L(y|x) =∏iPr(yi) =∏ie−∑j Pijxj(∑j Pijxj)yiyi!. (1.8)MLEM uses the log-likelihood as an objective function:81.2. Image reconstructionlogL(y|x) =∑i−∑jPijxj+ yi log∑jPijxj− log yi! (1.9)and thus is trying to solve:xˆ = argmaxxlogL(y|x). (1.10)Maximizing logL(y|x) expression with respect to xj yields the MLEM iterative updatealgorithm:xn+1j = xnj∑i Pijyi∑j Pijxnj∑i Pij= xnj xncorr,j . (1.11)xncorr,j is the correction factor to xnj , the current best estimate of the jth voxel of the image,which gives xn+1j , the next best estimate. The correction factor can be interpreted as follows.The current best estimate of the entire image is forward projected to provide an estimate ofthe counts along the ith LOR:yˆni =∑jPijxnj . (1.12)This is then compared, via division, to the actual observed counts along the ith LOR toprovide a correction factor yncorr,i in projection space:yncorr,i =yiyˆni=yi∑j Pijxnj. (1.13)Repeating this for all LORs, we then backproject these comparisons to image space andnormalize by the system matrix to yield the image space correction factor:xncorr,j =∑i Pijyncorr,i∑i Pij. (1.14)Another interpretation based on this last expression is that the image space correction factoris a weighted sum of the projection space correction factors, where the weights are system matrixLOR elements for the given voxel.Note that this derivation was performed without considering random or scatter coincidences91.2. Image reconstructionfor notational brevity. In practice, this is achieved by assuming λ =∑j Pijxj+ai in the Poissondistribution for the ith LOR such that the final MLEM algorithm is:xn+1j = xnj∑i Pijyi∑j Pijxnj +ai∑i Pij. (1.15)An initial estimate is required to start the algorithm, which is usually x = 1N , i.e. a vectorof ones of length N . After this initialization, MLEM is guaranteed to converge [19]. Therefore,with each iteration of the MLEM algorithm the image estimate becomes more accurate (lessbiased), but in the presence of noise in y, the estimate simultaneously becomes less precise(more noisy). Thus, it is traditional to terminate the algorithm when the decrement in bias isnot worth the corresponding increment in noise. In other words, the image remains somewhatbiased as a trade-off for lower noise.Convergence can be sped up to save computational time by partitioning the LOR data intoorthogonal angular subsets; this is the basis of the Ordered Subsets Expectation Maximization(OSEM) algorithm [20]. In OSEM, equation (1.15) is sequentially applied to each subset ofdata to yield an image estimate xn,sj based on the sth subset data, which then serves as theinitializing image for the next subset:xn,sj = xn,s−1j∑i Psijysi∑j Psijxn,s−1j +asi∑i Psij(1.16)where ys contains the counts for LORs within the sth subset, Ps is the system matrix indexingover all LORs of the sth subset and all voxels, and as contains the additive contamination countsto the LORs of the sth subset. A single iteration of OSEM (n → n + 1) involves updating allsubset images.PET images are normally reconstructed in 3D; the LOR counts can be summed up from theentire scan duration and reconstructed to yield a static image. Alternatively, the scan durationcan be partitioned into a set of consecutive time bins, or frames, during which the LOR countsare separately summed and reconstructed to yield a set of 3D images. This is known as a 4Ddynamic image and allows one to follow the temporal evolution of the radiotracer distribution—necessary for kinetic modeling, which is discussed in Section 1.4.101.3. Denoising PET data1.3 Denoising PET data1.3.1 Why are PET data so noisy?The goal of PET is to quantify the binding or metabolism of a target molecule across the brain,using a radiotracer. The reconstructed data are a spatial map of the estimated radiotraceractivity concentration. The number of counts emitted by each radiotracer molecule is random,following a Poisson distribution. Therefore, for repeated measurements the number of acquiredcounts will vary: if the mean number of expected counts within a fixed time interval is Nc, thenthe variance for any given measurement of the number of counts is also Nc. This inherentlylimits the SNR of the acquired data to Nc/√Nc =√Nc, meaning the data will be exceptionallynoisy unless a very large number of counts is acquired.As described in the above section, PET image reconstruction maps the acquired counts ontoa 3D grid of voxels, so the number of counts per voxel is of particular interest for SNR. This isfurther compounded by PET often being acquired dynamically in neuroimaging applications,meaning the total counts per voxel during the scan are binned to distinct temporal framesand the counts per voxel, per frame now determines the SNR of the data in a spatiotemporalsegment.Increasing counts per voxel per frame then can occur through any of the following: (i)increasing the injected dose for more global counts, (ii) increasing the voxel size to collect morecounts per voxel, and/or (iii) increasing the frame duration to collect more counts per frame.All these options improve the SNR of the data, but have negative consequences elsewhere:(i) is unethical and harmful to the subjects being imaged, who would have to experience ahigher radiation dose; (ii) reduces the spatial resolution of the data, making quantification lessaccurate—especially in small brain structures; and (iii) reduces the temporal resolution of thedata, meaning dynamic physiological processes cannot be captured as accurately. Therefore,there exists an intrinsic tension in PET data acquisition between the SNR and resolution;improving one negatively impacts the other.It also follows that voxels containing a higher concentration of radiotracer—due to higherbinding or metabolism of the radiotracer—will have more counts per voxel. Therefore, in generalthe SNR will vary spatially across the brain, as most physiological functions tracked by PETradiotracers also spatially vary.For precise quantification, further noise reduction, or denoising, is warranted—especially indynamic imaging. This can be incorporated within the reconstruction or as a post-processingstep after reconstruction. The aim of denoising is to reduce noise without simultaneouslyincreasing bias.111.3. Denoising PET data1.3.2 Post-processing denoisingSpatial filteringAt the resolution of PET images, the radiotracer distribution across the brain tends to consist offunctional regions with relatively homogeneous concentration, separated by sharp boundaries.As a result, within such regions the variance should be minimal, but in a reconstructed PETimage there will be considerable unwanted variance from image noise. To remove this, one canenforce correlations between voxels in these regions through spatial averaging. For a uniformset of N values corrupted by independent noise samples, a√N SNR gain is obtained fromaveraging3.In the simplest implementation, correlations can be enforced in a spatially invariant fashionthrough convolution of the 3D image, I(x), with a 3D filtering kernel, F (x), i.e. spatial filtering:IF (x) = I(x) ∗ F (x). (1.17)In most PET applications, the kernel used is an isotropic Gaussian. In dynamic PET,one can in theory extend filtering to the temporal dimension, but in practice it is common toperform spatial filtering separately to each frame.Since PET images are discretized objects, mathematically the convolution with a finite-sizedkernel has the effect of replacing the value of the jth voxel with a weighted average of the voxelsin its local neighbourhood, Nj :IF (xj , yj , zj) =∑k∈Nj wjk I(xk, yk, zk)∑k∈Nj wjk(1.18)where the weights are determined using the Gaussian kernel:wjk = exp[−((xj − xk)22σ2x+(yj − yk)22σ2y+(zj − zk)22σ2z)](1.19)and the σ describe the standard deviation of the kernel in each spatial dimension, i.e. howspatially broad the correlation introduced. While mathematically the convolution operationextends in all directions to infinity, the Gaussian kernel weights beyond a few standard devi-ations are negligible, so for computational efficiently the local neighbourhood Nj is typically3The limited resolution of PET means a given voxel will already be correlated with its nearest neighbours, solocal averaging has a less than√N improvement in SNR.121.3. Denoising PET datachosen as a 3D box around the central voxel, extending several standard deviations in eachdirection4. Computationally, spatial filtering is extremely quick because convolution can berepresented as a multiplication in Fourier space (Convolution Theorem):F{I ∗ F} = F{I} · F{F}. (1.20)Therefore, one can use the Fast Fourier Transform to compute F{I} and F{F}, then inverseFourier transform their product to yield the filtered image. The Fourier transform of a Gaussianis also a Gaussian, so viewed in Fourier space the multiplication of F{I} by a Gaussian preserveslow frequencies and attenuates the higher frequencies of the image, meaning it is a low-passfilter.Spatially invariant filtering is simple and effective at reducing noise, but implicit in itsdefinition is that local correlations should exist across the images. However, this is obviouslynot the case at sharp intensity boundaries where the tissue physiology tracked by the radiotracerrapidly changes. As a result, the correlations introduced will negatively bias the higher intensityside of the boundary and positively bias the lower intensity side of the boundary. This can alsobe understood in frequency space: noise is contained within the higher frequencies, but so aresharp boundaries in the image, so the attenuation of these higher frequencies has the effect ofsimultaneously removing noise and important image features.However, the imaging data can be transformed to a different frequency domain where theseparation of signal and noise may be easier. One popular method is wavelet denoising [21],where a wavelet transform is applied to the data. This decomposes the data into a weightedsum of wavelets, where the weights are called the wavelet coefficients. By thresholding thewavelet coefficients to cut off higher frequency wavelets and inverse transforming, one arrives ata denoised image with better—albeit still imperfect—high frequency feature preservation thanGaussian spatial filtering.In order to improve filtering-based denoising further, most sophisticated methods are re-quired where the structure of the kernel can adapt to the heterogeneous nature of PET image.These spatially variant filters, also known as adaptive filters, are discussed next.Adaptive filtersAn adaptive filter still leverages the fact that there are regions of relative homogeneity acrossthe brain and thus aims to introduce correlations through weighted spatial averaging, but now4For instance, MATLAB’s imgaussfilt3 function defaults to a kernel size of 2⌈2σ⌉+ 1, which guarantees atleast two standard deviations in each direction; beyond this, the weights are less than 5% of the peak centralweight.131.3. Denoising PET datathe weights within the kernel may vary spatially and/or iteratively.An early implementation of an adaptive filter in PET that found success was the Herholzfilter [22], where the difference between the time course of the central voxel to those of adjacentvoxels is computed to determine whether they should be included in the local averaging, thencomputing this local average and comparing its time course to the voxels of an expandedneighbourhood in an iterative fashion.While the filters discussed thus far focus on local similarity, the non-local means (NLM)filter [23] is also popular for exploiting the fact that physiologically similar tissues exist acrossthe brain. In NLM, theoretically each voxel is replaced by a weighted average of all other voxelsin the brain. The weights in this case are determined by comparing the similarity of their localspatiotemporal neighbourhoods within a Gaussian kernel. If ηj is the vector containing all nvoxel values in the spatiotemporal neighbourhood of the jth voxel, and ηk is that for the kthvoxel, then the weight between the two voxels is given by:wjk = exp(− 1n||ηj − ηk||2h2)(1.21)where h controls the amount of mixing between voxels; for a fixed similarity between neigh-bourhoods, higher h increases the weighting between voxels. Computationally, it is infeasibleto compute all wjk, so in practice when computing the weights for the jth voxel, one limits kto a 2D search window centred on the jth voxel.HighlY constrained backPRojection (HYPR)An adaptive filter for which significant development has been made in this work is the HighlYconstrained backPRojection (HYPR) operator [24]. HYPR aims to denoise data of arbitarydimension, I, without a loss of resolution. It achieves denoising through simple filtering, butreturns high frequency features lost during filtering with the aid of a higher SNR composite,C. The HYPR-denoised data are given as follows:IH = C · I ∗ FC ∗ F (1.22)where multiplication and division are performed in an elementwise fashion. The ratio C/(C ∗F )extracts the high frequency features from the composite. If dim(I) = N , then HYPR requires1 ≤ dim(C) ≤ N and dim(C) = dim(F ). For instance, the first application of HYPR indynamic PET data constructed a 3D composite image, C(x), as a weighted sum of all dynamicframes, and so HYPR is applied to all frames I(x, tj), j ∈ {1, 2, · · · , J}, separately:141.3. Denoising PET dataIH(x, tj) = C(x) · I(x, tj) ∗ F (x)C(x) ∗ F (x) . (1.23)In this work, we shall refer to this method as HYPR3D. In HYPR, increasing the size of thefiltering kernel F will introduce more features from C into IH . The noise level of IH is limitedby whichever is noisier, I ∗ F or C. Thus, once I ∗ F has a similar noise level to C, increasingF will not further decrease noise, only return more of the features of C to IH .Deep learning approaches to post-processingFinally, developments in deep learning have been transferred to the arena of PET image post-processing denoising. Denoising deep learning neural networks consist of an input layer, or thevector that stores a noisy input, which is mapped to a denoised output by passing it througha series of intermediate hidden layers, each of which may be partially- or fully-connected toadjacent layers. The elements of each layer are called nodes; a given node will receive inputsfrom any node it is connected to in the preceding layer, then produce a single output bycomputing a weighted sum of all the inputs. The weights are randomly initialized, but thenare iteratively updated to improve the accuracy and precision of the output, a process calledtraining.In the regime of supervised learning, networks need to be trained with a set of noisy inputsand “clean” target (i.e., less noisy or noiseless) pairs. An objective function, often called aloss function, is used to evaluate the performance of the network by comparing the denoisedoutput to the clean copy. With the random initialization of weights, the network initially hashigh error. Usually a version of gradient descent is used to update the weights, which involvescalculating the partial derivatives of the loss function with respect to the weights and changingthe weights by a value proportional to the gradient. The proportionality constant is called thelearning rate, one of many hyperparameters that can be tuned in training. Ocne the networkweights are optimized for the training set, the network is applied to an independent testing setof data. This ensures that the network performance is similar for both datasets; if not, it islikely that the network is overfitting to the training dataset.The architecture of the network—i.e., the number of layers, the number of nodes per layer,and the connections between layers—is usually empirically chosen, but a common choice inmedical imaging is the U-net architecture [25]. This is a class of deep learning networks calledconvolutional neural networks (CNN), which, as the name implies, contain convolutional layers.The structure of the input is often 2D or 3D and such that spatial information of the imagecan be encoded within the network. The convolution layers have learnable kernels, rather thansingle weights, which are convolved with the input of the previous layer to produce the output.151.3. Denoising PET dataThe various kernels may learn to preserve certain features, such as edges, while discarding otherfeatures—ideally noise.CNNs have been used to improve the image quality of low dose PET images to those offull dose images [26]. This was achieved by training the network with the full dose image asthe clean target, then downsampling the projection data to reconstruct a noisier version of thesame subject data as the noisy input. Once the network was optimized, it could restore theSNR of image features in the denoised low dose image to that of the full dose image.Another use of CNNs in PET is in the noise2noise framework [27], where two independentnoisy realizations of the same object are used: one passes through the network, then the L2loss between the output and the second noisy copy is minimized. This scheme is based on thefact that over many such independent noisy pairs, the output that minimizes the L2 loss is themean over the noisy realizations—a denoised version of any noisy input.Another deep learning architecture is the 4D denoising autoencoder. Our centre has appliedthis structure to post-process dynamic PET data [28]. The autoencoder consists of encodingand decoding steps. In the encoding step, the input noisy data (a 4D patch from the dynamicimage) is fed into a network structure of decreasing nodes in successive layers. Here the imagerepresentation is compressed in a trainable fashion to a latent space, such that the importantfeatures of the image are maintained and noise is removed. Then, from the latent space thenetwork architecture enters the decoding step where the number of nodes is increased untilit matches the input data size again, thus producing an output. The 4D autoencoder wasshown to be able to denoise both simulated and real data better than HYPR3D, but has thelimitations of relying on simulated data for training and needing to be trained separately fordifferent radiotracer distributions.At present, many deep learning-based denoising methods in medical imaging fundamentallylack generalizability. That is, separate networks often need to be trained for different radio-tracers, scanners, noise levels, and/or subjects. Supervised training of deep learning denoisingnetworks requires a large amount of data to properly discern signal from noise. This is anissue in PET imaging where the number of human imaging datasets within a centre is generallyinsufficient and the pooling of human datasets across centres is hampered by the ethical andtechnical aspects of data sharing. Preparing training datasets and optimizing hyperparametersfor training can also be time consuming. While some of the results in specific applications areimpressive, this work aims to provide advances to image denoising that are generalizable, sim-ple to implement, and constructed from physics-based first principles to have a fundamental,mechanistic understanding of their effectiveness. Therefore, we do not explore deep learningapproaches in this present work.161.3. Denoising PET data1.3.3 Denoising within image reconstructionRegularizing the reconstructionWithin reconstruction, denoising strategies include adding a penalty term to the log-likelihoodobjective function of equation (1.10) [29] such that the MLEM framework seeks to find:xˆ = argmaxxlogL(y|x)− βU(x) (1.24)where U(x) is a penalty function serving the role of regularizing the image estimate x, and βis a hyperparameter controlling the degree of regularization. U(x) may be spatially invariantto enforce local smoothness [30, 31]; this reflects the aforementioned fact that within manybrain regions the PET distribution is fairly homogeneous. However, a spatially invariant regu-larization cannot capture sharp boundaries between physiologically-distinct regions. Therefore,spatially variant versions of U(x) have also be suggested, such as a general form derived fromthe Gibbs distribution [32], i.e.:U(x) =N∑j=1∑k∈Njϕjk(xj − xk) (1.25)where Nj is the local neighbourhood of the jth voxel and ϕjk is a potential function whichaims to enforce local smoothness while maintaining sharp boundaries. Several choices of ϕjkhave been suggested [29], but by the argument xj − xk fed to ϕjk, all will be comparing theintensity of the central voxel to its neighbours and thus are locally adaptive to sharp transitionsin intensity—unlike a spatially invariant penalty function.The penalty function can also incorporate external information, such as a CT or MRIanatomical image. In this case, the sum over the potential functions can be weighted basedon these anatomical boundaries [33], or a more direct anatomical-based regularization can beperformed by enforcing some degree of similarity between the PET and anatomical gradientfields [34]. However, these methods have the disadvantage of requiring a very accurate co-registration of the two images to avoid introducing artifacts into the PET image. Additionally,there is an intrinsic trade-off between the strength of regularization and oversmoothing of PET-unique features; variance (not related to noise) in the PET image that is not present in theanatomical image will be erroneously reduced by the regularization.171.3. Denoising PET dataImage reconstruction using the kernel methodA newer class of reconstructions reparameterize the PET image as a weighted sum of basisfunctions, then iteratively estimate these weights (instead of the image values directly). Thisis known as the kernel method; the image is given by:x = Kα (1.26)where K is the kernel matrix containing the basis functions and α contains the weights foreach basis function. The MLEM framework is applied to estimate α, then the image can becomputed with the above equation.The elements Kjl of the kernel matrix, where l = {1, 2, · · · , L} for L basis functions, arecomputed using a specified kernel function, κ(fj , fl), where f is a feature space. Therefore, forthe jth voxel we have:xj =L∑l=1αlκ(fj , fl). (1.27)The original implementation of the kernel method in PET [35] divides the full 60-minutescan into three 20-minute frames and performs MLEM reconstruction of each frame to definef . Each voxel thus has three coordinates in the feature space. The kernel function operates inthis space and was chosen to be a Gaussian:κ(fj , fl) = exp(−||fj − fl||222σ2), (1.28)which essentially quantifies the similarity between the jth and lth voxel in feature space. Thekernel method was shown to outperform both traditional and regularized MLEM approaches,but has the obvious disadvantage in this formulation of requiring an initial reconstruction todefine the feature space. This is avoided if one uses external information, such as defining thefeature space vector for each voxel as the local neighbourhood of a high resolution anatomicalMRI image [36], but again has the disadvantages listed above for anatomically-guided regular-ization.More recently, it has been theoretically demonstrated that the kernel method can be repre-sented as a subject-specific trainable neural network model:181.3. Denoising PET datax = K(θ;Z)α (1.29)where Z are the data from the three 20-minute frames as described above and θ is a vectorof trainable parameters that computes Z 7→ f . Therefore, the neural network constructs thefeature space and thereafter the Gaussian kernel function can be used to compute the kernelmatrix. This method demonstrated improvements over the traditional kernel method, but stillrequires the initial reconstruction to compute Z and the neural network needs to be indepen-dently trained for each subject.In all cases above, the kernel matrix definition is temporally invariant, meaning only spatialcorrelations are being investigated to perform denoising. A temporal kernel combined with theoriginal spatial kernel has been suggested [37], but the temporal kernel extracts informationfrom averaging the entire projection space, meaning the effectiveness of this approach in a givenvoxel depends on the degree of similarity of the temporal evolution of the signal in this voxelto the average temporal signal across the brain.An adaptive kernelized reconstruction: HYPR4D-K-OSEMAt our centre, we have addressed these limitations with a spatiotemporal kernel matrix con-structed using a 4D version of the Highly constrained backPRojection (HYPR) operator [24]within an OSEM framework, which we term HYPR4D-K-OSEM [38]. By using OSEM, weare able to construct an iteratively-updated 4D HYPR composite by summing over the kernel-reconstructed subset images:Cn+14D =∑sKn4Dαn,s4D. (1.30)where, in this case, the kernel coefficients are estimated by OSEM:αn,s4D = αn,s−14D ·(Kn4D)T (Ps)T ys4DPsKn4Dαn,s−14D +as4D(Kn4D)T (Ps)T1M (1.31)where “T ” denotes the matrix transpose, 1M is a vector of ones of length M (the number ofLORs), and all multiplication and division operations are performed in an elementwise fashion.The kernel matrix itself then incorporates 4D HYPR:Kn4D = diag(hn)F4D (1.32)191.4. Modeling PET datawhere F4D is the spatiotemporally invariant Gaussian convolution and hn are the spatiotem-porally variant high frequency features extracted from Cn4D:hn =Cn4DCn4D ∗ F4D. (1.33)Therefore, the 4D basis functions are relatively smooth from spatiotemporal convolution,but retain the spatiotemporally variant features that are common across the subset images.By construction, the features retained are not only adaptive across space and time, but alsoacross iterations of the reconstruction; with each iteration a new composite is constructed, thusproviding a new set of features. The algorithm is initialized with one iteration of OSEM, whichis relatively smooth but biased. Therefore, iterating HYPR4D-K-OSEM improves the accuracyof the feature set extracted by HYPR4D while highly constraining the noise increment periteration.The 4D adaptive kernel in HYPR4D-K-OSEM was shown to outperform a non-adaptiveversion where spatial and temporal features are extracted separately (similar to [37]) in termsof noise reduction and feature preservation [38]. Furthermore, a version incorporating PSFcorrection, i.e. PSF-HYPR4D-K-OSEM, was shown to outperform regularized PSF-OSEMreconstructions available on our centre’s scanner in these same metrics [39]. Thus, in this workwe use HYPR4D-K-OSEM or PSF-HYPR4D-K-OSEM for reconstruction of real scanning data.1.4 Modeling PET dataOnce dynamic PET data are reconstructed and denoised, kinetic modeling ensues to extractphysiological parameters that may be of use for quantifying healthy or diseased brain functionat rest or in response to a task. An accurate kinetic model will capture the biophysical processof radiotracer exchange between blood plasma and tissue. Radiotracers have a specific target,so the primary endpoint of kinetic modeling is often a quantitative measure of the binding ofthe radiotracer to that target.Kinetic modeling is applied to a time activity curve (TAC), which is the temporal componentof a dynamic PET image. TACs may be computed from a single voxel, or averaged over aregion of interest. Dynamic PET models can be broadly separated into three methodologies:compartmental models, spectral analysis, and graphical analysis.201.4. Modeling PET data1.4.1 Compartmental modelsInjected radiotracer circulates through the body via blood plasma. When it reaches the brain,some radiotracer is exchanged into tissue where it may bind to receptors, enter a different chem-ical state as it passes through a physiological process, or be exchanged back into blood plasma.Therefore, within a given voxel radiotracer can be in a number of physical or chemical states,and exchange between different states can occur over time. In compartmental modeling, eachstate is represented by a compartment with a time-varying radiotracer concentration. Mathe-matically, this can be described by a set of coupled linear ordinary differential equations, whereeach compartment has a time-dependent concentration and a set of rate constants describingradiotracer influx to and efflux from plasma and/or other compartments (Figure 1.2a).Solutions for a model with N compartments can readily be obtained via Laplace transform[40]:CT (t) = CP (t) ∗H(t) (1.34)where CT (t) is the modeled TAC, CP (t) is the radiotracer concentration circulating in bloodplasma, and H(t) is the impulse response function:H(t) =N∑i=1ϕie−θit. (1.35)The coefficients ϕi and decay constants θi are a function of the rate constants of all com-partments.Since PET spatial resolution is on the order of millimetres, and temporal resolution is oftenon the order of minutes, there is a significant spatiotemporal averaging of the more complicatedcellular-level dynamics. As a result, the radiotracer kinetics observable to PET are remarkablywell-described by a small number of compartments [41].Commonly, a two tissue compartmental model is sufficient. One of the tissue compartmentsoften represents non-displaceable radiotracer, i.e. the radiotracer bound to non-specific bindingsites (receptors not specifically targeted by the radiotracer) plus the unbound radiotracer inextracellular space. The other compartment then represents the radiotracer bound to specificbinding sites. If these tissue compartments are placed in series (Figure 1.2b), then radiotracer isfirst exchanged between the non-displaceable compartment (ND), then further exchange occursinto the specific compartment (S). Thus, four rate constants are required to completely describethe system:211.4. Modeling PET data• K1 (units: mL plasma per minute per mL tissue): influx from plasma to ND• k2 (units: min−1): efflux from ND back into plasma• k3 (units: min−1): influx from ND into S• k4 (units: min−1): efflux from S back into ND.The units of K1 are different since it describes the transfer of radiotracer from the plasmato a tissue compartment, so a conversion from mL of plasma to mL of tissue is necessary; K1is capitalized to reflect this. The remaining rate constants all have units of inverse time, andthus reflect the fractional mass transfer of radiotracer per unit time between compartments(or, for k2, between a tissue compartment and plasma). It is typical to report rate constantsin min−1 to reflect the relatively slow kinetics of radiotracers. For instance, in [11C]raclopridetypical values may be K1 ≈ 0.15 mL min−1 mL−1, and k2 ≈ 0.1 min−1 in high binding regionsor k2 ≈ 0.35 min−1 in low binding regions [42].If we let CND be the concentration of non-displaceable radiotracer and CS be the concentra-tion of specifically-bound radiotracer, then the specific differential equations for the two tissuecompartmental model are:dCNDdt= K1CP − (k2 + k3)CND + k4CSdCSdt= k3CND − k4CS .(1.36)Note that although we model CND and CS separately, we can only observe their sum, i.e.CT = CND +CS . Useful parameters derived from these equations include the volume of distri-bution of the compartments, which is defined as the ratio of the compartment concentration toCP in equilibrium, i.e. whendCNDdt =dCSdt = 0. Equating the two equations above when this isthe case yields:K1CP − (k2 + k3)CND + k4CS = k3CND − k4CSK1CP = k2CND=⇒ VND := CNDCP=K1k2.(1.37)To get VS , we focus on the second differential equation whendCSdt = 0:221.4. Modeling PET data0 = k3CND − k4CSCSCND=k3k4=⇒ VS := CSCP=CNDCPCSCND= VNDCSCND=K1k3k2k4.(1.38)Therefore, the total volume of distribution is:VT = VND + VS =K1k2(1 +k3k4)(1.39)which tells us the ratio of how much radiotracer is being concentrated in the target tissueversus the plasma. The ratio k3/k4 is known as the non-displaceable binding potential (BPND).Binding potential itself is given by:BP =BmaxKD(1.40)where Bmax is the total density of target receptors and KD is the dissociation constant, whichis inversely proportional to the affinity of the radiotracer to the receptor [43]. BPND is then BPafter accounting for fND, the fraction of radiotracer that is free or non-specifically bound [44]:BPND = fNDBP =k3k4. (1.41)Therefore, with serial blood plasma samples to compute CP (t), one can use non-linear fittingof equation (1.34) to regional or voxel TACs to solve for the rate constants and hence VT orBPND. Parametric images of BPND are useful for visualizing the density of target receptorsacross the brain. This provides a quantitative measure of physiology in vivo, and thus can beused to investigate the effect of disease and intervention on neuroreceptor systems.1.4.2 Reference tissue modelsBlood sampling to obtain CP is invasive and requires a skilled nuclear medicine technician.However, if a region of the brain is devoid of specific binding, one can use a reference tissuemodel to eliminate the need for CP in solving for CT . A common reference tissue model is theSimplified Reference Tissue Model (SRTM) [41].SRTM is derived from first assuming a one tissue compartmental model can reasonablydescribe the radiotracer kinetics in both the target region (CT , with specific binding) and the231.4. Modeling PET data(a) (b)Figure 1.2: (a) An illustration of a compartmental model with N tissue compartments; CP is theradiotracer concentration circulating in blood plasma, and CT is the sum of the radiotracer concentrationin all tissue compartments. From [40]. (b) The two tissue compartmental model in series.Figure 1.3: The The Simplified Reference Tissue Model (SRTM). In (a) we show the kinetics of thetwo tissue compartmental model for the target region, which are subsequently replaced by a single tissuecompartmental model in (b).241.4. Modeling PET datareference region (CR, without specific binding). In CR, this assumption is reasonable becausewithout any receptors for specific binding, a single ND compartment will accurately describethe kinetics. However, where specific binding does exist, to replace the aforementioned twotissue compartmental model (having ND and S compartments) with a single compartmentrequires the assumption that radiotracer kinetics between ND and S are rapid, that is k3 →∞and k4 → ∞. This then allows us to define an apparent efflux rate out of the single tissuecompartment, which is denoted k2a (Figure 1.3). In this case, the volume of distribution for thesingle tissue compartment is equivalent to the total volume of distribution for the two tissuecompartmental model [equation (1.39)]:K1k2a=K1k2(1 + BPND) =⇒ k2a = k21 + BPND(1.42)which gives us a method of estimating BPND. The differential equations for this system are asfollows:dCTdt= K1CP − k2aCTdCRdt= K ′1CP − k′2CR.(1.43)These can be combined to eliminate CP :dCTdt=K1K ′1dCRdt+K1K ′1k′2CR − k2aCT . (1.44)We can then define the meaningful parameter R1 = K1/K′1, which characterizes the relativedelivery of radiotracer to the target compared to the reference region. SRTM further assumesthe distribution volume for non-displaceable radiotracer is identical across the brain, i.e. inboth the target and the reference region:K1k2=K ′1k′2(1.45)which yields the following simplified expression:dCTdt= R1dCRdt+ k2CR − k2aCT . (1.46)The solution can then be found via Laplace transform:251.4. Modeling PET dataCT (t) = R1CR(t) + (k2 −R1k2a)CR(t) ∗ e−k2at (1.47)and can be fit to the dynamic data either using nonlinear fitting techniques, or pre-specifyingbasis functions B(t; k2a) = CR(t)∗ e−k2at with physiologically-plausible values of k2a and fittinglinearly [45]. Alternatively, integrating both sides of equation (1.46) gives:CT (t) = R1CR(t) + k2∫ t0CR(u) du− k2a∫ t0CT (u) du (1.48)which for time samples taken at {t1, t2, · · · , tN} is multilinear in:X =CR(t1)∫ t10 CR(u) du∫ t10 CT (u) duCR(t2)∫ t20 CR(u) du∫ t20 CT (u) du.........CR(tN )∫ tN0 CR(u) du∫ tN0 CT (u) du (1.49)meaning θ = [R1 k2 k2a] can be determined by solving:CT = Xθ (1.50)using linear regression. This is known as the Multilinear Reference Tissue Model (MRTM) [46].1.4.3 Detecting neurotransmitter release using compartmental modelsMost compartmental models characterize the dynamics of the radiotracer with the assumptionthat the physiology of the subject is in a steady state. However, many neuroimaging studieswish to investigate physiology in response to tasks presented to the subject during the scan.For instance, one may wish to study the temporal dynamics of neurotransmitter release inresponse to a task. If one has an appropriate radiotracer that competes for binding with theneurotransmitter of interest, then one can model the dynamics of this competition using thelinear parametric neurotransmitter PET (lp-ntPET) model [47].Radiotracer and unlabeled neurotransmitter are competing for binding with the same setof receptors, so when neurotransmitter concentration increases in response to a task, this willdisplace some of the bound radiotracer molecules, thus reducing radiotracer binding and ul-timately leading to a decrease in the temporal PET signal. To account for these observabletransient changes in the data, lp-ntPET modifies k2a, the effective efflux rate constant for bound261.4. Modeling PET dataradiotracer in MRTM, to be time-varying in response to changes in endogenous neurotransmit-ter concentrations. The neurotransmitter release timecourse is modeled by a family of gammavariate basis functions:h(t) = θ(t− tD)(t− tDtP − tD)αexp[−α(t− tPtP − tD)](1.51)where α controls the “sharpness” of the shape, tD is the onset of the release, tP is the timeof peak release, and θ(t) is the Heaviside function. This effectively adds an additional term toMRTM [equation (1.48)], and so the full model is given by:CT (t) = R1CR(t) + k2∫ t0CR(u)du− k2a∫ t0CT (u)du− γ∫ t0CT (u)h(u)du (1.52)where γ controls the strength of release. Detection of release is traditionally achieved througha model comparison between lp-ntPET and MRTM using an F-test. If the extra term in lp-ntPET—the contribution of transient neurotransmitter release to the PET signal—provides asignificantly better fit to a TAC, this suggests release is present in the signal [48]. In such regions,parametric “movies” of γh(t) can be constructed to visualize the contribution of endogenousneurotransmitter release to the observed PET signal over time [49].1.4.4 Spectral analysisCompartmental models require one to specify the number of compartments in advance. How-ever, from the general solution of equation (1.34) one can decompose CT into a discretizedkinetic spectrum, where each spectral element is a particular decaying exponential (e−θit) andits weight is the corresponding ϕi [50]. Thus, the spectrum will inform on both the number ofcompartments and the quantitative aspects of the kinetics without pre-specifying a compart-mental model.1.4.5 Graphical analysisThe final approach for kinetic modeling considered in this work, graphical analysis, appliesa linearizing transformation to the data. This is particularly useful for irreversibly boundradiotracers, i.e. where the radiotracer becomes trapped. For instance, the Patlak model [51]—which is featured in this work—takes the differential form of the two tissue compartmental modelwith a single trapping compartment, i.e. where there is no efflux out of the compartment with271.4. Modeling PET dataspecifically-bound radiotracer (i.e., k4 = 0 in Figure 1.2b):dCNDdt= K1CP − (k2 + k3)CNDdCSdt= k3CND(1.53)and applies the following transformation:dCSdt=K1k3k2 + k3CP − k3k2 + k3dCNDdtCS =K1k3k2 + k3∫ t0CP du− k3k2 + k3CND=⇒ CT = CND + CS = K1k3k2 + k3∫ t0CP du+k2k2 + k3CNDCTCP=K1k3k2 + k3∫ t0 CP duCP+k2k2 + k3CNDCP(1.54)The ratio CND/CP reaches a steady-state value some time post-injection. After this time,the second term above becomes constant and thus the transformed data CT /CP are linear in∫ t0 CP du/CP :CTCP=K1k3k2 + k3∫ t0 CP duCP+k2k2 + k3VND = KiX +B. (1.55)whereKi =K1k3k2 + k3(1.56)is the rate of net uptake into the trapping compartment.The kinetics of the radiotracer FDG—used extensively in visualizing and quantifying glucosemetabolism—is well-described by a two tissue compartmental model with a single trappingcompartment. FDG cannot pass through phosphorylation like ordinary glucose molecules andbecomes trapped at this stage of metabolism as a result. Once the transformed data becomelinear, regression after this point yields an estimate for Ki, which can subsequently be used toquantify the cerebral metabolic rate of glucose, CMRglu:281.5. MRI: basic physics and utility in PET/MRI imagingCMRglu =CgluLCKi (1.57)where LC is a constant accounting for different rates of transport and phosphorylation betweenFDG and glucose and Cglu is a measurement of the subject’s plasma glucose concentration.1.5 MRI: basic physics and utility in PET/MRI imagingThe contrast in PET images is often much different than the anatomical structure of the brain.Therefore, to map the values in a PET image to the anatomical location within the brain,a separate 3D structural image is required. MRI offers high soft tissue contrast and is thussuitable for this task, but also can provide images with contrast sensitive to multitude of otheraspects of tissue physiology. Computed tomography (CT) also offers 3D structural images byrotating an X-ray camera around the subject. CT thus obtains several planar projections ofthe brain, which can be reconstructed into a 3D image using similar methods as PET imagereconstruction. Many clinical hybrid scanners are PET/CT, since CT is much cheaper thanMRI and some patients cannot tolerate MRI due to claustrophobia or medical implants thatare unsafe in the MRI environment. However, CT lacks the imaging versatility of MRI and hasthe disadvantage of providing an additional radiation dose to the subject. Therefore, hybridPET/MRI scanners may be preferred in a research setting, since MRI will provide importantphysiological information beyond structure and will not deliver an additional radiation dose toresearch volunteers.From a hardware perspective, an MRI scanner consists of a superconducting magnet, ra-diofrequency (RF) transmitter and receiver coils, and a series of gradient coils which can locallymodulate the magnetic field. For neuroimaging MRIs, the superconducting magnet is a set ofmetal rings encased on liquid helium, which generates an external magnetic field, B0, that isoriented axially (i.e., B0 = B0zˆ) with respect to a cylindrical bore. A scanner bed moves thesubject into the bore such that their head is at the isocentre. The B0 field strength is usuallyat least 3T in research scanners, but may be as low 1.5T in clinical scanners.1.5.1 Nuclear magnetic resonance and its relationship to MRIMRI imaging uses the physics of nuclear magnetic resonance (NMR) applied to the spins ofhydrogen nuclei, i.e. protons, which are ubiquitous across the brain. In the presence of B0, theproton spins will be either aligned (+) or anti-aligned (−) to B0, yielding magnetic momentsof:291.5. MRI: basic physics and utility in PET/MRI imagingµ± = ±12γhzˆ (1.58)where γ is the gyromagnetic ratio of the proton and h is Plank’s constant. The correspondingenergies are:E± = −µ± ·B0 = ∓12γhB0. (1.59)From the Boltzmann distribution, the relative probability of aligned to anti-aligned statesis:P+P−=exp(−E+/kT )exp(−E−/kT ) = exp(γhB0kT)(1.60)where k is the Boltzmann constant and T is the temperature. Scanning occurs at room tem-perature, so at 3T:P+P−= exp((4.2577× 107) · (6.626× 10−34) · 3(1.38× 10−23) · 293)≈ 1.00002. (1.61)Therefore, there exists a (small) preference for the aligned state, leading to a net magneticmoment in the z-direction. The spins will precess about B0 at the Larmor frequency:ω0 = γ|B0|. (1.62)It is conventional in MRI to view the system in a reference frame rotating at ω0, such thatthe magnetization vector is stationary along the z-axis. More generally, in a rotating framethere is an “apparent” magnetic field along the z-axis, B′z = Bz−ω′/γ, where ω′ is the rotationfrequency of the reference frame, such that if ω′ = ω0 and Bz = B0 we have B′z = 0.An NMR signal is obtained by perturbing the spins with an RF pulse applied at or close toω0, which applies a much weaker oscillating secondary magnetic field B1 orthogonal to B0. Inthe rotating reference frame, the magnetization vector is tipped off axis towards the transverseplane by the RF pulse. After the pulse, the excited protons will undergo relaxation effects.There are two types of relaxation effects. Spin-lattice relaxation causes spins to return to thethermal equilibrium state of the ensemble, i.e. the time for the longitudinal component of themagnetization, Mz to return to its equilibrium value M0:301.5. MRI: basic physics and utility in PET/MRI imagingMz(t) =M0(1− e−t/T1) (1.63)where the characteristic time T1 varies based on the type of tissue, which allows for contrastin the final image. When spins are tipped off the longitudinal axis, spin-spin interactionswill cause a dephasing about the transverse plane, meaning the transverse component of themagnetization will also relax:Mxy(t) =Mxy(0)e−t/T2 (1.64)where again the characteristic time T2 varies based on the type of tissue. In practice, the decaytime observed is actually T ∗2 < T2 since the external magnetic field cannot be made perfectlyhomogeneous, and thus local homogeneities lead to a more rapid dephasing.To quantify T1, T2, or T∗2 , a specific pulse sequence is required. For instance, in a Carr-Purcell pulse sequence [52] a 90◦ RF pulse is first applied to tip the magnetization vector intothe transverse plane, where the spins will begin to dephase. Following this with a 180◦ pulseafter a short time delay of TE/2 will flip the spins in the transverse plane, thus reversing thedirection of the dephasing. This causes the spins to refocus in a what is known as a spin echo.The peak of the echo will occur at a time TE after the initial pulse, known as the echo time.During the echo, a sinusoidal voltage will be induced in the RF receiver coil from the precessionof the spins. After the echo peak, the sinusoidal voltage is enveloped by a e−t/T ∗2 decay as thespins dephase again under spin-spin relaxation. This signal is known as free induction decay.1.5.2 Encoding spatial informationThus far, we have discussed NMR where the signal obtained comes from the net magnetizationof the entire spin ensemble. The difference between an NMR setup and an MRI scanner isthe gradient coils, which adds a spatially-varying magnetic field on top of B0 to encode spatialinformation in the signal. The gradient fields are usually separated into their x-, y-, and z-components to understand their contribution to the imaging process. Therefore, the set of RFpulses plus the timing and magnitude of x-, y-, and z-gradients forms an MRI pulse sequence.Slice selectionMRI images are usually acquired in a slice-by-slice fashion. For slices acquired in an axialorientation, the z-gradient, Gz, is used for slice selection. If a linear gradient in the magneticfield is established along the z-axis during the RF pulse, the Larmor frequency will vary along311.5. MRI: basic physics and utility in PET/MRI imagingthe z-axis:ω(z) = γ(B0 + zGz). (1.65)Therefore, only spins located where ω(z) closely matches the frequency of the RF pulse willbe excited, meaning no contribution to the signal will come from outside a small axial slice. Theslice thickness can be varied by changing the steepness of Gz or the bandwidth of frequenciestransmitted in the RF pulse.Approximating the set of protons within this axial slice as a continuous distribution de-scribed by an effective proton density5 ρ(x), then the observed signal in the RF receiver coil isan integral over all precessing spin contributions within the selected slice:S(t) =∫xdx ρ(x)e2πiω0t. (1.66)With only a slice-selective Gz, all spins within the slice are precessing at ω0. The goalthen is to modulate this precession based the x- and y-position of the spins, such that spatialinformation is encoded within the observed signal. Often, the techniques of frequency and phaseencoding are used to accomplish this.Frequency encodingIn frequency encoding, a gradient is set up along either the x- or y-axis to alter the localLarmor frequency within the slice. We shall choose the x-axis without loss of generality, soupon applying the x-gradient the spins will precess at different frequencies depending on thex-coordinate of their position within the slice. The observed signal now becomes:S(t) =∫xdx ρ(x)e2πiω(x,t) (1.67)where in the rotating reference frame the precession frequency is given byω(x, t) = γx∫ t0Gx(u) du. (1.68)5The signal from a region of space depends not only on the proton density, but also the relaxation rates T1and T2.321.5. MRI: basic physics and utility in PET/MRI imagingPhase encodingTo localize the y-coordinate, we can encode spatial information in the phase of the spin pre-cession as well; this is the basis of phase encoding. A phase encoding gradient is applied inbetween the slice selection and frequency encoding gradient during readout. When the phaseencoding gradient is turned on, the precession rate will be spatially varied along the y-axis,meaning some spins will precess faster and others will precess slower. Once it is turned off, thespins return to precessing at ω0, but now have a phase shift depending on their position alongthe y-axis; spins that precessed faster will lead spins that precessed slower. The observed signalis now:S(t) =∫xdx ρ(x)e2πiω(x,t)+2πiφ(y,t) (1.69)where φ(y, t) is the phase accumulated, which is determined by the y-gradient, Gy, and they-position:φ(y, t) = γy∫ t0Gy(u) du. (1.70)In summary, every point in space within the slice now has a unique frequency and phasecombination that is related to its spatial position. The observed signal can be concisely writtenas:S(t) =∫xdx ρ(x)e2πik·x (1.71)wherek = γ∫ t0G(u) du (1.72)with G = [Gx Gy].k-space formalism and disentangling spatial informationAfter frequency and phase encoding, the equation for S(t) above looks suspiciously like the 2DFourier transform of ρ(x). This implies we need to repeat the entire procedure of slice selection,then phase encoding, then frequency encoding multiple times—each time varying the intensity331.5. MRI: basic physics and utility in PET/MRI imagingof the phase encoding gradient—then structure the data in 2D with each row containing thesampled S(t) with a unique phase encoding gradient. Each column is then a measurement of aunique S(t) at a fixed time point.In this 2D space, taking the inverse Fourier transform along the x-axis disentangles thefrequency information from the signal, whereas the inverse Fourier transform along the y-axisdisentangles the phase-encoded information from a given time point across all measured S(t).Finally, taking two orthogonal inverse Fourier transforms is the same as taking the 2D inverseFourier transform, so taking the inverse 2D Fourier transform of this space gives us a 2D grid ofvalues where each pixel contains a signal from a spatial location. Remarkably, we have arrivedat an image.This 2D space is termed k-space [53, 54]. The position in k-space, given by equation (1.72),depends on the frequency and phase encoding gradients applied; each time we apply the fre-quency encoding gradient, Gx(t), we are moving across kx to store sampled values of the signal,and similarly the phase encoding gradient, Gy(t), selects the row in k-space that is being filled,i.e. the value of ky. Therefore:dkxdt∝ Gx(t)dkydt∝ Gy(t)(1.73)In other words, the gradients control the velocity in k-space, and thus a pulse sequencespecifies a trajectory through k-space.In practice, k-space is discretely sampled for each slice and stored in a matrix. Repeatingthis for each slice allows us to reconstruct the entire image using the inverse Fast Fouriertransform. For the Fast Fourier Transform algorithm, it is helpful to use an N by N matrixwhere N = 2n, n ∈ Z+; typical choices are N = 64, 128, or 256.1.5.3 MRI sequencesDifferent sequences will yield unique image contrasts that are sensitive to particular tissuestructures and/or physiological processes [55]. In the simple spin echo sequence of the abovesection, the spins are selectively excited within the prescribed slice by the RF pulse and Gz att = 0. As soon as the spins are tipped off the longitudinal axis into the transverse plane, T2effects cause the transverse component to decay. Signal acquisition occurs at t = TE (duringthe spin echo), so the choice of TE will influence the how T2 effects influence the final imagecontrast. Recall that multiple repetitions of the sequence are required to fully sample k-space,but after the initial RF excitation T1 effects are also causing the longitudinal magnetization341.5. MRI: basic physics and utility in PET/MRI imagingcomponent to regrow. Therefore, at the next excitation the amount of magnetization availablewill depend on the repetition time of the sequence, TR, and the local value of T1. Finally, thedensity of protons within a voxel, ρ, will also be directly proportional to the intensity of thesignal. To summarize these effects:S ∝ ρe−TE/T2(1− e−TR/T1). (1.74)From this, choices of TE and TR can be made such that the contrast in the resulting imageis weighted towards the local to proton density, T1, or T2:• Proton density weighted: Long TR (relative to T1) to minimize T1 effects and shortTE (relative to T2) to minimize T2 effects• T1 weighted: TR on the order of T1 for tissue-of-interest to maximize T1 differentiationand short TE (relative to T2) to minimize T2 effects• T2 weighted: Long TR (relative to T1) to minimize T1 effects and TE on the order ofT2 for tissue-of-interest to maximize T2 differentiation.Using equation (1.74), one can obtain specific values for TR or TE such that the T1 or T2contrast between two tissue types are maximized. For instance, imagine we wish to maximizeT1 contrast between grey and white matter. TE → 0 minimizes T2 contrast, and assumingnegligible grey–white matter differences in proton density (for simplicity), image contrast isthen given by:C = (1− e−TR/T(wm)1 )− (1− e−TR/T(gm)1 ) = e−TR/T(gm)1 − e−TR/T(wm)1 . (1.75)Taking the first derivative with respect to TR, equating to zero, and solving yields:TR = log(T(gm)1T(wm)1)(T(wm)1 T(gm)1T(gm)1 − T(wm)1). (1.76)For healthy tissue at 3T, T(gm)1 ≈ 1600 ms and T(wm)1 ≈ 800 ms [56], resulting in TR ≈ 1100ms to maximize grey–white matter T1 contrast. At this time, the ratio of white matter signalto that of grey matter is about 1.5. Because such a T1-weighted image provides grey–whitematter contrast, it is useful as an anatomical reference in PET; segmentation algorithms canbe applied to identify anatomical structures to use as regions of interest in PET images.351.5. MRI: basic physics and utility in PET/MRI imagingNumerous MRI sequences that differ from the simple spin echo sequence considered herehave been developed, producing images sensitive to effects beyond those listed above. Theseinclude Arterial Spin Labeling (ASL) [57], which produces images sensitive to blood perfusion;the Blood Oxygen Level Dependent (BOLD) sequence [58], which captures changes in bloodoxygenation in response to neuronal activity; Diffusion Tensor Imaging (DTI) [59], which tracksthe motion of water molecules through white matter fibres; and susceptibility-weighted imaging(SWI) [60], which can be used to visualize or quantify local magnetic susceptibility (a potentialproxy for iron deposition).1.5.4 MRI sequences relevant to this workIn Chapter 4 we develop an MRI denoising operator for ASL and BOLD datasets, which togetherare used to quantify the cerebral metabolic rate of oxygen (CMRO2). The computation ofCMRO2 is described in the section below; in this section we first outline the physics andacquisition of these two sequences.To quantify CMRO2, ASL and BOLD images need to be repeatedly acquired in a sequentialfashion using a dual-echo pseudocontinuous Arterial Spin Labeling (pCASL) sequence. Dual-echo in this case means echoes corresponding to two unique contrasts are present per repetitionof the sequence, thus providing two images per repetition; in our image the first echo providesthe ASL image and the second echo the BOLD image.As mentioned, ASL images are sensitive to blood perfusion, p. The ASL sequence acquires aset of alternating “label” and control images. The control image is acquired before the labelingprocess and is mostly sensitive to proton density. The label image is acquired shortly after a180◦ RF pulse alongside a z-gradient to invert spins only within a slab below the brain; thismagnetically labels blood water spins entering the brain. The labeled spins enter the imagingslice and have a different magnetization than existing spins within the slice, so an exchangeof magnetization occurs between the two sets of spins. The label image will have a differentcontrast than the control image as a result of this magnetization exchange, so subtracting thelabel image from the control image6 theoretically yields a separate image that is sensitive toblood perfusion.A consensus model7 [61] exists that transforms qualitative perfusion images into quantitativecerebral blood flow (CBF) images. The model accounts for imperfect labeling efficiency, spin6The label image has lower intensity because the labeled spins have an opposite polarity magnetization frominversion.7The ISMRM Perfusion Study Group and the European ASL in Dementia consortium met in 2012 and cameto consensus on specific formulae and parameters for the modeling of CBF from ASL data. Since the formulaeand parameters chosen evidently affect the absolute values of CBF obtained, by identifying a consensus modelthere is now a means for standardizing MRI-derived CBF in the literature.361.6. Application of MRI: quantification of cerebral metabolic rate of oxygenrelaxation effects based on scanner acquisition parameters, local proton density, and differencesin concentrations of labeled blood in the perfused tissue versus arterial blood:CBF =6000 · λ · p · ePLDT1,blood2 · αL · T1,blood · SPD ·(1− e−τT1,blood) (1.77)where λ is the brain/blood partition coefficient, T1,blood is the spin-lattice relaxation time ofblood, αL is the efficiency of labeling the blood, SPD is a proton weighted image (which canbe approximated by the control image), τ is the duration of the labeling pulse, and PLD is thepost-label delay.For the second echo, BOLD-weighted images rely on differences in magnetic susceptibilityin oxygenated versus deoxygenated blood. Oxyhemoglobin (Hb) is diamagnetic, meaning zeromagnetic moment, whereas deoxyhemoglobin (dHb) is paramagnetic and thus will locally distortthe magnetic field, ultimately reducing the acquired MRI signal. Neurons require a supply ofoxygenated blood to function; increases in neuronal activity need to be supplemented by anincrease in oxygenated blood delivery, which is achieved by increasing the flow and volume ofoxygenated blood locally. This has the effect of reducing the concentration of dHb, resulting ina transient increase in the BOLD signal [58].1.6 Application of MRI: quantification of cerebral metabolicrate of oxygen1.6.1 Physiological backgroundCerebral metabolic rate of oxygen (CMRO2) is a physiological quantity that reflects the combi-nation of arterial oxygenation, blood flow, and oxygen extraction in the brain. As such, CMRO2,or its constituents, can be useful in vivo markers of metabolic dysfunction in diseases such asstroke [62–65], tumour [66–68], or neurodegeneration [69–71], as well as a tool for studyingmetabolic changes during healthy aging [72, 73]. CMRO2 coupled with information of glucosemetabolism has also been used to demonstrate regional aerobic glycolysis in the healthy brain[74], and may sensitive to mitochondrial dysfunction which is hypothesized to be present insome neurodegenerative diseases, such as Alzheimer’s [75], Parkinson’s [76], and Huntington’sdisease [77].Oxygenated blood is delivered to the brain through the arteries before it branches throughthe capillary bed to diffuse to cells for metabolism. Thus, metabolism of oxygen is regulated bythe initial oxygenation of the blood entering the brain (CaO2), the fraction of oxygen extracted371.6. Application of MRI: quantification of cerebral metabolic rate of oxygenfrom the blood (OEF), and the local rate of delivery of blood, or cerebral blood flow (CBF). IfCaO2 mL of oxygen per mL of blood is delivered at a rate CBF mL of blood per minute per100g of tissue and the oxygen extraction fraction is OEF (unitless; 0 indicates no extraction ofoxygen, 1 indicates all oxygen is extracted), then CMRO2 can be quantified in units of mL ofoxygen per minute per 100g of tissue:CMRO2 = CaO2 · CBF ·OEF. (1.78)Normal ranges are ∼ 0.2 mL oxygen/mL blood for CaO2, ∼ 30–60 mL blood/min/100g forCBF and 0.4 for OEF (uniform in the healthy brain), meaning reasonable CMRO2 estimatesrange from 2–5 mL/min/100g, with primary regional variation due to white matter vs. greymatter differences in CBF [78–80].1.6.2 Quantifying CMRO2 in vivoFor human quantification of voxelwise CMRO2, the gold standard method usually requires threeseparate 15O-PET scans [80]. This protocol is challenging8 and administers a radioactive doseto subjects. Therefore, in the past decade there has been a push towards MRI-based methodsto achieve reliable voxelwise OEF and CMRO2 quantification [81, 82]. These methods applybiophysical models to isolate the OEF or CMRO2 signal from either BOLD images [79, 83, 84]velocity-selective spin labeling images [85, 86], or quantitative susceptibility mapping (QSM)images [87–89]. Alternatively, one can substitute 1H-MRI for 17O-MRI and dynamically imageinhaled 17O to obtain CMRO2 using compartmental modelling [90, 91], though these techniquesare limited by the high cost of 17O and significantly lower signal in 17O-MRI than 1H-MRI.1.6.3 Calibrating the BOLD signalThe earliest attempts to map OEF and CMRO2 used the method of calibrating the BOLD signalwith respiratory manipulations, namely hypercapnia (increased CO2) [83] and/or hyperoxia(increased O2) [79, 92]. To formalize, the BOLD signal is sensitive to the transverse relaxationrate of spins, R∗2 = 1/T∗2:BOLD ∝ e−TE·R∗2 (1.79)where TE is the echo time chosen for the sequence.8With a 2.05-minute half life, 15O is limited to centres with a fast connection between the cyclotron generatingthe radiotracer and the scanner.381.6. Application of MRI: quantification of cerebral metabolic rate of oxygenHypercapnia or hyperoxia will cause a change in the BOLD signal, which we can normalizeby the baseline BOLD signal to define the relative BOLD signal change from the stimulus, δS:δS =∆BOLDBOLD0=BOLD −BOLD0BOLD0=BOLDBOLD0− 1 (1.80)Substituting from above:δS =e−TE·R∗2e−TE·R∗2|0− 1 = e−TE·∆R∗2 − 1. (1.81)Generally, TE ·∆R∗2 << 1, so we can expand and keep only the first order term:δS ≈ −TE ·∆R∗2. (1.82)R∗2 for venous dHb varies as follows [93]:R∗2 = A · CBV · [dHb]β (1.83)where A is constant governed by field strength and differences in volume susceptibility betweentissue and blood; CBV is the cerebral blood volume, and the exponent β depends on vessel sizeand geometry. Thus:δS = TE ·A(CBV0 · [dHb]β0 − CBV · [dHb]β)(1.84)= TE ·A · CBV0 · [dHb]β0(1−(CBVCBV0)([dHb][dHb]0)β)(1.85)δS =M(1−(CBVCBV0)([dHb][dHb]0)β). (1.86)The constant M lumps together several physiological and scanner-dependent constants andis estimated using the hypercapnia or hyperoxia experiment. Therefore, it is referred to as thecalibration constant. δS = M when all deoxyhemoglobin has been removed from blood, i.e.[dHb] = 0, and thus M can also be viewed as the maximum possible BOLD response. Fromthe Grubb relationship [94]:391.6. Application of MRI: quantification of cerebral metabolic rate of oxygenCBVCBV0=(CBFCBF0)α(1.87)we can recast the ratio of blood volumes with a ratio of blood flows, the latter of which we canmeasure using ASL imaging. To simpify notation, we indicate baseline-normalized physiologicalquantities by a lowercase variable, i.e. f = CBF/CBF0 and d = [dHb]/[dHb]0 such that:δS =M(1− fαdβ). (1.88)This is known as the extended Davis model [83], and demonstrates that under an iso-CMRO2 assumption [95], the BOLD signal can be modulated by changes in blood flow and/ordHb concentrations. Based on measurements in humans, α = 0.2 is recommended [96] andan appropriate value of β should be chosen depending on field strength9. d is governed bya separate biophysical model derived from Fick’s principle of mass conservation as oxygen isexchanged in the capillary bed [79]:d =1f− 1[dHb]0{1ϕ(CaO2 − CaO2|0f)+ [Hb](1f− 1)}(1.89)where ϕ = 1.34 mL O2/g Hb is the carrying capacity of oxygen in Hb. f is measured from arte-rial spin labeling (ASL) imaging followed by CBF quantification as described in Section 1.5.4,[Hb] can be measured from a blood sample or assumed from population averages, and CaO2can be measured or assumed:CaO2 = ϕ · [Hb] · SaO2 + ϵ · PaO2 (1.90)where SaO2 is the arterial oxygen saturation, ϵ = 0.000031 mL O2/mL blood/mmHg, and PaO2is the partial pressure of oxygen in arterial blood, measured in mmHg. SaO2 can be measuredfrom pulse oximetry or derived from assuming measured end-tidal partial pressures of oxygenare equal to arterial partial pressures of oxygen, PaO2, and invoking the Severinghaus equation[100]:SaO2 =11 + 23400(PaO2)3+150·PaO2. (1.91)Finally, OEF can be expressed as:9β ∈ [0.9, 1.5] is often selected for 3T [97–99].401.6. Application of MRI: quantification of cerebral metabolic rate of oxygenOEF = 1− ϕCaO2|0 ([Hb]− [dHb]0) . (1.92)Thus, with the suite of equations above the two unknowns in estimating OEF are thecalibration constant, M , and [dHb]0. The dual-calibrated approach first uses a hypercapniaexperiment to isolate M , then uses this estimate within a separate hyperoxia experiment toisolate [dHb]0. The estimate of [dHb]0 is used within equation (1.92) to obtain OEF, which thencan be combined with the estimates of CaO2|0 and CBF0 to calculate CMRO2 in equation (1.78).The section below details these computations.1.6.4 Dual-calibrated BOLDHypercapnia calibrationCO2 is a vasodilator, meaning inducing hypercapnia will result in f > 1. This increased flow ofoxygenated blood means hypercapnia also results in d < 1; the net effect is an increase in theBOLD signal from baseline, as seen in equation (1.88). If oxygen end-tidals are held constantduring hypercapnia, then CaO2 = CaO2|0 during hypercapnia. This simplifies equation (1.89)to:d =1f−1− 1f[dHb]0(CaO2|0ϕ− [Hb]). (1.93)For CaO2|0 = ϕ · [Hb] · SaO2|0 + ϵ ·PaO2|0, typical values are [Hb] = 0.15 g Hb / mL blood[101], SaO2|0 = 0.98 [102], and PaO2|0 = 100 mmHg [103], meaning the first term (O2 boundto Hb) dominates the second term (dissolved O2). Thus, only keeping the first term we have:d =1f−1− 1f[dHb]0([Hb] · SaO2 − [Hb])=1f−(1− 1f)[Hb][dHb]0(SaO2 − 1) . (1.94)Mild hypercapnia, induced for the purposes of calibration experiments, will typically resultin f ∼ 1.6 at most. Along with normal OEF ∼ 0.4 [80] this leads to [Hb]/[dHb]0 ∼ 2.5. Thismeans the 1/f term is roughly 30x larger than the second term above. Thus, if we ignore thissecond term, i.e. d ≈ 1/f , and substitute back into equation (1.88), we have:411.6. Application of MRI: quantification of cerebral metabolic rate of oxygenδShc =M(1− fα−βhc)(1.95)where the subscript “hc” indicates “determined or measured from hypercapnia”. With thisequation, one can obtain the calibration constant M by measurements of the BOLD and CBFsignal during baseline and hypercapnia:Mhc =δShc1− fα−βhc. (1.96)Hyperoxia calibrationIf one begins breathing air greatly enhanced with oxygen, they will enter hyperoxia and thepercentage of saturated hemoglobin molecules will approach 100%, i.e. SaO2 → 1. As a result,the BOLD signal will increase as dHb is diluted.Mathematically, an increase in SaO2 leads to a corresponding increase in CaO2. Changesin CaO2 from CaO2|0 during hyperoxia lead to the second term in equation (1.89) being non-negligible (unlike in hypercapnia). Therefore, [dHb]0 can be isolated as:[dHb]0 =1ϕ(CaO2|ho − CaO2|0fho)+ [Hb](1fho− 1)1fho− d . (1.97)d is the only unknown in this equation. With M determined from the hypercapnia experi-ment, we solve for d using equation (1.88):d =(1− δShoMhcfαho)1/β. (1.98)Therefore, hyperoxia measurements of (δSho, fho,CaO2|ho),Mhc determined from hypercap-nia, plus a baseline measurement of CaO2|0 gives us the necessary information to determine[dHb]0 and hence OEF. In practice, blood flow changes during hyperoxia are small and mea-sured voxelwise fho are very noisy. Therefore, fho is often assumed to be a global constantto avoid propagation of errors in fho through the highly nonlinear equation (1.97). A globalvalue of fho can be assumed based on the intensity of hyperoxia [99] or a global average of thevoxelwise estimates can be used instead. We opt for the latter strategy in this work, using themedian of voxelwise fho as the global constant.421.7. Pattern analysis of imaging data1.7 Pattern analysis of imaging dataIn imaging datasets, images from each subject will contain millions of voxels—or in the caseof regional analysis, often up to 100 regions—each containing a value that could potentiallydiffer between healthy and diseased subjects. Direct statistical comparisons are thus hamperedby the multiple comparisons problem, but more importantly are ill-equipped to capture spatialcovariance that is related to disease.This latter point can be illustrated by an example. Let there be three regions where we arecomparing an image-derived metric in the healthy versus diseased brain. In Table 1.1, we seethat region-by-region statistical comparison tells us only regions 1 and 3 are different in thediseased versus healthy brain. In reality, all regions contribute to a pattern of disease, since therelative distribution has been altered.This can be seen by first demeaning across regions for each subject, then demeaning acrosssubjects for each region, shown in Table 1.2. In region 1 healthy and diseased subjects havevalues equal to their global average and so the relative value here is identical. However, inregion 2 diseased subjects have relatively elevated values compared to healthy subjects, whereasin region 3 diseased subjects have relatively lowered values.Region 1 Region 2 Region 3 MeanValue in healthy brain 10± 1 12± 1 8± 1 10± 0.6Value in diseased brain 8± 1 12± 1 4± 1 8± 0.6Statistically different? yes no yes yesTable 1.1: An example comparison of regional values in the healthy vs. diseased brain.Region 1 Region 2 Region 3 MeanValue in healthy brain 0± 1 −1± 1 +1± 1 0Value in diseased brain 0± 1 +1± 1 −1± 1 0Statistically different? no yes yes n/aTable 1.2: An example comparison of regional values in the healthy vs. diseased brain, demeanedacross regions and then subjects.The above description characterizes a pattern of disease. Identified patterns can be extractedfrom data or prospectively projected onto new data to calculate the expression of the patternin individual subjects, offering a single statistical value for testing rather than a set of region-by-region statistical tests. This thus illustrates the power of studying disease through patterns,which delve into the effects of disease on spatial covariance, rather than implicitly assumingthe effect of disease independently affects individual regions. This characterization of diseaseis more concordant with fact that the brain is an interconnected system.The data preparation method described above that demeans across regions and then subjects431.7. Pattern analysis of imaging datais part of an analysis framework known as Scaled Subprofile Modeling Principal ComponentAnalysis (SSM-PCA) [104]. SSM is the data centring step. If we prepare a data matrix X withelements Xij being the logarithm of metabolism data10, where i indexes over voxels or regionsand j indexes over subjects, then we transform the data as follows:Zij = Xij −GMRj −GMPi (1.99)where GMRj is the global mean rate (average across brain) for subject j and GMPi is thegroup mean profile in region i (average of Xij − GMRj across subjects). Note that roughlyequal numbers of diseased and healthy subjects should be included in the matrix to best capturethe variance that discriminates between the healthy and diseased brain.PCA is then applied to Z, which decomposes the SSM data into a linear combinationof k spatial components contained in a matrix C. Each component is a spatial covariancepattern, rank-ordered by the percentage of variance in the dataset explained by the pattern.Mathemetically, each component consists of a weight in each spatial region, which representsthe contribution of that region/voxel towards the pattern, so the elements of C are Cik.To reconstruct the SSM data of a given subject using a linear combination of the components,there needs to be a coefficient for each component. Since these coefficients are different for eachsubject (in general) they are termed subject scores. The subject score quantifies the expressionof a given component for that subject. The subject scores for each subject, for each component,i.e. SSjk, are stored in a separate matrix SS. The PCA reconstruction of the SSM data Z isthus succinctly written as:Z = SS ·CT . (1.100)Since changing the sign of both the subject scores for a given component and componentweights themselves leaves Z unaffected, by convention we change the sign such that, on average,diseased subjects have a positive subject score. This allows one to view the component as adisease-related pattern.Pattern analysis has been used extensively in the analysis of metabolic data to identify avariety of disease-related patterns [105], such as in PD, Huntington’s, and Alzheimer’s. No-tably, the PD-related CMRglu pattern [106] has been robustly identified and reproduced acrossseveral populations. It is characterized by relative hypermetabolism in cerebellum, pons, andposterior putamen, as well as hypometabolism in occipitoparietal regions. Furthermore, pat-10The logarithm is thus taken to transform these imaging data, which often have long right-tails, to a semi-normal distribution.441.7. Pattern analysis of imaging datatern expression correlates with disease severity, illustrating how disease-related patterns maybe used as imaging biomarkers for the progression of disease.45Chapter 2Development of an advanced PETdenoising operator2.1 MotivationThis work first begins with the development of a denoising operator specific to dynamic PETapplications. As mentioned in the Introduction, the SNR of PET images is typically muchlower than that of MRI except in a few select applications1. Therefore, we aim to design a PETdenoising operator that may be used for any radiotracer and can be easily integrated within aprocessing and analysis pipeline for various dynamic PET applications.2.1.1 HYPR denoising: limitations and targeted improvementsThe development centres around the HighlY constrained backPRojection (HYPR) post-process-ing operator (introduced in Section 1.3), since it is favoured in dynamic PET imaging forits simplicity and similar computational time to post-filtering [24]. However, in its initialdeveloped formulation, HYPR has two specific disadvantages (detailed below) that are mitigatedby modifications to its formulation in this chapter.As a brief reminder, HYPR uses filtering to denoise, which removes high frequency noiseas desired, but also has the undesired consequence of removing high frequency image featuresas well—effectively reducing the resolution of the image and thus increasing bias (i.e., quan-tification error in the true activity within a given point in space). HYPR reintroduces highfrequency features through a high SNR composite image to restore image resolution, tradingreduced bias for a partial increase in noise. This increase generally depends on the noise levelin the composite. In traditional HYPR applications, the composite is a weighted sum of thetemporal image frames2, and the HYPR operator is applied in 3D separately to each frame:1In Chapter 4 we develop an MRI denoising operator that is targeted towards one such MRI applicationwhere low SNR hinders the utility of the method, hence motivating the development of a MRI-specific denoisingalgorithm.2As a reminder, frames in PET imaging are 3D images reconstructed from a temporal subset of data, meaningthe activity reflected in the 3D image is the average activity during the frame duration,462.1. MotivationC(x) =∑jwj · I(x, tj)IH(x, tj) = C(x) · I(x, tj) ∗ F (x)C(x) ∗ F (x)(2.1)where the weights usually account for uneven frame duration of the scan. We shall refer tothis method as HYPR3D, which was first developed in [24]. There are two major limitationsof HYPR3D:1. Assumption of similar contrast between each frame and the composite image.For radiotracers where the spatial distribution changes over time, this is not true. Theimage contrast may change drastically throughout the scan, whereas the composite imageis a weighted sum of all frames, meaning this temporally varying information is lost. Asa result, the image features extracted from the composite image will not match the imagefeatures of some frames, thus introducing bias into these frames.2. Noise in the composite. The method of temporal averaging to construct a higher SNRcomposite is limited, since the noise level is limited to the number of counts per voxelsummed across all frames. Therefore, in scans with high spatial resolution and/or lowerdose, the counts per voxel can remain relatively low, leading to relatively high compositenoise levels. The high frequency features extracted by HYPR include noise, meaning fora relatively noisy composite an unacceptable level of noise may be reintroduced into theimage after the filtering step.With regards to (1), all radiotracers have an uptake phase where the distribution mostlyreflects blood flow, which is primarily contrast between grey matter and white matter. Afteruptake, the radiotracer distribution transitions to an equilibrium phase, which may be com-pletely different. Then, based on the rate of clearance of the radiotracer with respect to thescan duration, there may be a washout phase where the distribution changes yet again. TheHYPR3D composite will be a mixture of all phases; for a given frame and region where thecontrast is overestimated by the composite, a positive bias will be introduced by HYPR3D, andvice versa for underestimated contrast.In this chapter we introduce an iterative application of a 4D version of the HYPR op-erator (IHYPR4D) to address these limitations of HYPR3D. A 4D composite is generatedby spatial averaging over regions of interest (ROI) for each frame, thus preserving temporalinformation in the composite. The 4D nature of the composite addresses limitation (1) ofHYPR3D. Additionally, averaging over ROIs with 100s to 1000s of voxels in generating theIHYPR4D composite—rather than averaging over 10s of frames in HYPR3D—leads to a less472.1. Motivationnoisy composite, addressing limitation (2) of HYPR3D. However, implicit in this constructionis an assumption of spatial similarity within each ROI, which may not be true. Therefore, weintroduce an iterative step to update the composite and undo any bias introduced by the initialcomposite. Intuitively, IHYPR4D resembles a noise-suppressed trajectory of iterative recon-struction in bias-noise space: we start with a smooth yet biased 4D composite, then iterate the4D HYPR operator to reduce bias while maintaining low noise.We demonstrate through simulation, phantom, and human studies that processing withIHYPR4D yields images with greater accuracy and precision as compared to HYPR3D. Thispresent work aims to improve upon current implementations of HYPR3D, which have beenshown to outperform other advanced denoising commonly used in PET imaging. Thus, we donot offer a comprehensive comparison to other advanced, non-HYPR-based methods. For this,we recommend readers cross-reference our results to works where comparisons of HYPR3D withother advanced methods are made [23, 28].2.1.2 Testing and validating IHYPR4D in a variety of scanningenvironmentsThis chapter is organized as follows. We begin with describing the theoretical development ofIHYPR4D, then proceed to testing of the method with phantom3 and simulation studies, andfinish with validation on human data.We use a 18F phantom study framed to match the temporal count distribution of a rep-resentative human [11C]raclopride (RAC) scan; the short half-life of 11C in general presents agreater challenge for a given denoising operator. This is repeated in two scanning environments:1. The Siemens High Resolution Research Tomograph [16] (HRRT), which has in-trinsically high resolution, resulting in a low number of acquired counts per voxel andthus high noise.2. A commercially available, higher sensitivity scanner with time-of-flight (TOF) capability,the GE SIGNA PET/MR [107] (hereafter SIGNA). The higher sensitivity is exploitedto decrease frame duration, which again leads to a low number of counts per voxel.The HRRT is a dedicated brain scanner developed in the late 1990s/early 2000s, yet remainsone of the highest resolution human PET scanners (2.3 × 2.3 × 2.5 mm FWHM) due to its small3A phantom is a low-attenuation container with several inserts that are filled with known concentrations ofaqueous radioactive solutions. These are placed into the scanner and thus provide a motion-free dataset to assessthe quantification of reconstructed and/or post-processed data.482.2. Theorydetector crystal size and use of depth of interaction4 [16]. The SIGNA was developed in the2010s and is the first total-body PET/MR scanner based on silicon photomultipliers with TOFcapability, but because it is not a dedicated brain scanner the crystal size is larger and hencethe resolution is poorer than the HRRT (3.48 × 3.43 × 4.67 mm FWHM) [107]. In terms ofnoise characteristics, the SIGNA has roughly 2.5x better detection sensitivity [16, 107], which,coupled with the TOF capability, contributes to less image noise. A comparison of the imagequality between these scanners using image point spread function (PSF) reconstructions andpost-filtering demonstrated the lower noise levels of SIGNA images, though image quantificationwas comparable between scanners [108].The phantom studies provide an initial investigation to the convergence properties of IHYPR-4D and its relative performance to HYPR3D. From here, we move to a simulated RAC studyon the HRRT5 where the ground truth is known, to study trade-offs between accuracy and pre-cision. RAC is a radiotracer with a distribution that changes significantly throughout the scan.Finally, we compare results from the simulated data to those obtained from human RAC scans,and a human FDG scan to study the performance on a more diffuse radiotracer distribution.2.2 TheoryAs mentioned, in HYPR3D the composite is a weighted sum of the temporal frames; the biasinduced is low if the contrast in the composite image closely matches the contrast in eachtarget frame, i.e. time-independent contrast. For time-dependent contrast, it has been shownthat using multiple composites—each a weighted sum of a subset of the temporal frames—reduces HYPR3D-induced bias, but at the cost of increased noise [109]. Furthermore, while thecomposite has higher SNR than the target, it may have considerable spatial noise even whenall frames are summed—especially in high resolution PET. Thus, if the filtered target has lowernoise than the composite, the composite limits noise reduction.In IHYPR4D, we reduce these limitations by using a 4D composite image, C(x, t), that isvery smooth in the spatial and temporal dimensions; the temporal dimension allows preservationof the potentially time-dependent contrast in the target, and its smoothness makes it less likelyto limit noise reduction. The composite is constructed using spatial averages over regions ofinterest (ROIs): the time activity curve (TAC) from each voxel within an ROI is replaced withthe spatial average over all voxel TACs of that ROI.In theory, any set of ROIs may be used to compute the initial composite. In this work, we4Depth of interaction in scintillation PET detectors refers to better localization of where a gamma ray im-pinges within a detector crystal, thus providing more precise LORs for annihilation events and ultimately higherresolution.5Presently, we have no means of accurately simulating SIGNA TOF data.492.2. Theorytest two ROI-generation approaches: using k-means clustering on the PET data and anatom-ical ROIs from an MRI segmentation. Such methods have shown promise in other denoisingapplications [110, 111]. The target has resolution limitations that generally exceed a singlevoxel, whereas ROI delineations in the composite form sharp boundaries. To avoid artificialintroduction of such boundaries into the denoised images, the composite image is filtered witha spatial kernel, P (x), representing the system/reconstruction PSFC ′(xi, tj) =1|Ri|∑r∈RiI(xr, tj)C(x, t) = C ′(x, tj) ∗ P (x)(2.2)where i indexes over space, j indexes over time, I(x, t) is the target image, Ri is the set of allvoxels in the ROI containing voxel xi, and | · | denotes the cardinality of the set. As a resultof this composite design, if the radiotracer distribution is inhomogeneous within an ROI, i.e.the ROI has subregional features, the voxelwise bias in the composite will be high. To recoverthese features, we apply the 4D HYPR operator iteratively. The initial iteration is given byIn=1H (x, t) = C(x, t) ·I(x, t) ∗ F (x, t)C(x, t) ∗ F (x, t) (2.3)where F (x, t) is a 4D Gaussian filtering kernel. The ratio of the composite to the filtered com-posite extracts high frequency spatiotemporal features from the composite. Since the compositehas low noise, these features will be ROI boundaries and regional kinetics, rather than high fre-quency noise. To recover subregional features, iteration can proceed by replacing the compositewith the denoised image of the previous iterationIn+1H = InH(x, t) ·I(x, t) ∗ F (x, t)InH(x, t ∗ F (x, t). (2.4)With each iteration, the composite is noisier, but less biased. The initial composite is lessnoisy than the filtered target, so the noise level in equation (2.3) is limited by the filtered target,and thus will remain relatively low for several iterations, depending on the filtering kernel size.A larger kernel size will replace more high frequency features in the target with those fromthe iteratively updated composite. Thus, we predict a larger kernel will have a smaller noiseincrement but will require more iterations to recover subregional features. Any subregionalfeatures will initially be present in a reduced resolution form; successive iterations serve toasymptotically return to the initial resolution.502.3. Methods2.3 Methods2.3.1 Phantom studyWe explore the convergence properties of IHYPR4D in terms of a contrast recovery versus voxelnoise trade-off in the controlled setting of a phantom study. Three conditions were tested:1. High spatial resolution setting, standard reconstruction. A modified Esser phan-tom containing 16, 12, 8, 6, and 4 mm hot inserts each filled with 18F in a 4:1 hot-to-background ratio was scanned on the HRRT for 60 minutes. The data were binned tomatch the temporal count distribution of a 4 x 1, 3 x 2, 8 x 5, 1 x 10 min RAC scan.Such a framing sequence is common in brain PET studies where the relative change in ra-dioactivity concentration is small after the initial uptake phase. Ordinary Poisson OSEM(OP-OSEM, 16 subsets, six iterations) [112] was used for reconstruction.2. High spatial resolution setting, denoising reconstruction. The same Esser datasetabove was reconstructed using the denoising reconstruction HYPR4D-K-OSEM [38] (16subsets, 10 iterations, 5 mm/four frame kernel size) to study IHYPR4D with substantiallydenoised input data. See Section 1.3.3 for details on the theory of HYPR4D-K-OSEM.3. High spatiotemporal resolution setting. We scanned a phantom with 10 and 17 mmhot inserts filled with 18F in a 4:1 hot-to-background ratio on the SIGNA (for technicalreasons the Esser phantom could not be scanned on the SIGNA). The framing was set tomatch the temporal count distribution of a high temporal resolution human RAC scanwith 16 x 15, 6 x 30, 4 x 45, 50 x 60 s frames. The manufacturer’s PSF + TOF OSEMreconstruction (PSF-TOFOSEM, 28 subsets, four iterations) [113] was used. This settingis useful when the effects of a rapid, externally induced change on radiotracer bindingneed to be evaluated [5, 49, 114–116].For each setting, IHYPR4D was applied to the reconstructed data. Three methods ofgenerating the ROIs for the initial composite were evaluated:1. No segmentation. A single hand-drawn ROI encompassing the entire phantom was usedas the “segmentation”; the hot inserts thus serve as subregional features not included inthe initial composite.2. Hand-drawn segmentation. ROIs for each insert were hand-drawn to simulate anMRI-based segmentation in the scenario where the hot inserts are anatomical features; inthis case the hot inserts are spatial features included in the initial composite.512.3. Methods3. k-means segmentation (Figure 2.1). This was performed as a four-step process: (i)the mean image over all frames was computed; (ii) this mean image was passed to thedeterministic imsegkmeans3 MATLAB function to yield 30 clusters6; due to high spatialnoise, the cluster members are mostly incongruous; (iii) an initial IHYPR4D compos-ite was formed using the ROIs from step (ii) and the mean over all composite frameswas computed; (iv) this new mean image was passed to imsegkmeans3 to yield 30 morecongruous clusters, which served as final IHYPR4D ROIs.Figure 2.1: Demonstrating the four-step k-means segmentation method for obtaining ROIs on theOP-OSEM phantom data. In (i) and (iii) colour indicates activity levels; in (ii) and (iv) each uniquecolour indicates an ROI.In this introductory evaluation of IHYPR4D, we have elected to test ROI generation meth-ods that are automated, simple to implement, and are considerate of different spatiotemporalradiotracer distributions. Using an MRI segmentation is useful when PET functional bound-aries mostly coincide with a subset of anatomical boundaries, whereas the PET-derived k-means630 was empirically chosen to avoid under-segmenting the image while ensuring region sizes were not toosmall.522.3. Methodsapproach may be preferred when distinct functional features exist within anatomical boundariesor functional and anatomical boundaries largely do not intersect.IHYPR4D was applied for up to 50 iterations with a spatial kernel size of either 1x, 1.5x,or 2x the PSF7 and a temporal kernel size of two, four, or eight frames. For comparison,the images were also denoised using HYPR3D with matching spatial kernel sizes to IHYPR4Dand a frame-duration-weighted sum of all temporal frames as the composite. Percent contrastrecovery (%CRC) was analyzed for all hot inserts%CRC = 100% · CH/CB − 1AH/AB − 1 (2.5)where CH and CB are the regional estimated hot and background concentrations from theimages, and AH/AB is the independently measured hot-to-background activity ratio (4:1).%CRC is compared before and after denoising to study the effect of post-processing on %CRC;an ideal method reduces noise without affecting %CRC. To study denoising efficacy, voxel noiseis computed using coefficient of variation in a large background region of the phantom (∼ 700cm3) at each frame. For each kernel size in IHYPR4D, %CRC and voxel noise are averaged overall frames and plotted for increasing iterations to study the trade-off between contrast recoveryand image noise. The target %CRC is defined as the mean %CRC over all frames obtainedwithout any denoising.2.3.2 Human simulation studyThe simulated human RAC study allows us to test IHYPR4D with a time-varying radiotracerdistribution where the ground truth is known. Realistic noiseless simulated dynamic imageswere generated based on data from an 8 mCi RAC scan of a healthy volunteer on the HRRT. Ananatomical (T1-weighted) MRI image of the volunteer was segmented using Freesurfer [117] intoa set of anatomical ROIs, then co-registered to the PET data. A regional TAC was computedfor each ROI by spatially averaging the PET data over these anatomical regions at each frame,then fitted with SRTM2 [42] to obtain a set of smooth ROI TACs. An initial dynamic image wascreated by filling the voxels of each ROI with these smooth TACs, resampled to 4 x 1, 3 x 2, 8 x5, 1 x 10 min frames. Finally, the noiseless ground truth image was computed by filtering thisimage with a 2.5 mm isotropic Gaussian to simulate the PSF of the HRRT. This noiseless imagewas then forward projected while accounting for attenuation, normalization, radioactive decay,and frame duration. Poisson noise was added to produce 20 noisy simulated projection datasets,which were reconstructed using OP-OSEM and HYPR4D-K-OSEM as described above.To test recovery of subregional features, we placed two spherical hot spots of different size7(2.5 mm)3 for the HRRT [16] and 1.8 x 1.8 x 3.6 mm for the SIGNA [107].532.3. Methodsrelated to an elevated non-displaceable binding potential (BPND) in the left putamen (large hotspot: ∼ 7.5 mm diameter, i.e. 3x PSF or ∼ 100 voxels; small hot spot: ∼ 5 mm diameter, i.e.2x PSF or ∼ 30 voxels) and a disease-induced gradient [118] in the right putamen. IHYPR4Dwith either an MRI-based or k-means segmentation and HYPR3D was then applied as in thephantom study8. Weighted least squares fitting to was applied to each denoised dataset withSRTM2 to extract parametric BPND images9. Using several noisy realizations and the groundtruth concentration and BPND, the percentage bias and standard deviation were computedacross all noisy realizations:%bias = 100%× 1NNR∑nr=1snr − sGTsNR(2.6)%std = 100%1s√√√√ 1NR− 1NR∑nr=1(snr − s)2 (2.7)where N = 20 is the number of noisy realizations, snr is the voxel TAC or BPND value of noisyrealization nr, sGT is the corresponding ground truth value, and s is the corresponding meanvalue over all realizations.For voxel TACs, the metrics are computed at each frame and are thus time-varying. Thevoxelwise metrics were averaged separately over three ROIs: one around the large hot spot (∼0.2 cm3), one around the small hot spot (∼ 0.06 cm3), and one encompassing the entire caudate(∼ 6.5 cm3). Results from the two hot spot ROIs indicate the recovery of high contrast featuresnot included in the anatomical segmentation and thus offer a comparison to the hot insertresults of the phantom study. Results from the caudate ROI indicate the recovery of a uniformrelatively low contrast feature that is included in the anatomical segmentation. By computingover all realizations and averaging over many voxels, %bias largely captures systematic biasesinduced by post-processing rather than noise-induced bias.The reconstructed images fed to the denoising algorithms tested have small biases from theground truth to start. Image reconstruction starts with a uniform image estimate and iterativelycompares forward-projected image estimates with the ground truth data in an attempt torecover image features. However, as mentioned in Section 1.2, the image reconstruction is8Only post-uptake frames (15 minutes onward) were used to compute a mean image for k-means clusteringto better capture RAC contrast.9A diagonal weighting matrix was used, with the diagonal elements being the inverse of the estimated varianceat each time point assuming a Poisson distribution (accounting for frame duration and decay correction), i.e.σ2(t) = C(t)∆t eλtwhere C(t) is the denoised TAC, ∆t is the frame duration, and λ is the decay constant of 11C.While IHYPR4D involves temporal filtering and thus violates independence of the temporal data, meaning a non-diagonal weighting matrix would be required, the reintroduction of high frequency features in each applicationof HYPR returns anticorrelation and thus we retain the use of a diagonal matrix for simplicity.542.3. Methodsusually terminated before all features are perfectly recovered, because each iteration introducesmore noise into the image estimate. Furthermore, because the simulated data have time varyingcontrast, the amount of bias in the reconstructed image varies from frame-to-frame, leading toa temporal bias pattern, i.e. %bias(t). Thus, %bias(t) was computed before and after denoisingto determine any additional bias introduced by denoising.To analyze trade-offs between bias and standard deviation (i.e. accuracy versus precision),the absolute value of the bias for a given ROI at each frame was taken and then averaged overall frames, i.e. mean |%bias(t)|, then plotted against the mean %std(t) over all frames in thecerebellum—a uniform low-binding region in RAC scans—yielding bias-noise trajectories forIHYPR4D. For BPND, the trade-off between accuracy and precision was similarly studied byplotting %bias versus %std for each ROI. Finally, line profiles along the simulated diseased-induced gradient in the right putamen were compared in the denoised and ground truth images,to test the ability of each method to recover a subregional spatial gradient.2.3.3 Human studyWe applied IHYPR4D to three human datasets: one RAC scan on the HRRT (matching oursimulation); one RAC scan on the SIGNA (same radiotracer, different scanner/reconstructionfrom our simulation); and one [18F]fluorodeoxyglucose (FDG) scan on the SIGNA (differentradiotracer, different scanner/reconstruction). All scans were 60 minutes in duration. Fram-ing for the HRRT RAC scan was identical to the HRRT phantom and simulation studies.Framing for the SIGNA scans was of higher temporal resolution, matching that of the SIGNAphantom study. List mode data were reconstructed using OP-OSEM for the HRRT scan andPSF-TOFOSEM for the SIGNA scans. MRI images used for segmentation in IHYPR4D wereacquired on the Philips Achieva 3T for the HRRT data and on the SIGNA for the two PET/MRscans. MRI images were co-registered to the PET data with SPM12 (Wellcome Centre for Hu-man Neuroimaging, https://www.fil.ion.ucl.ac.uk/spm/software/spm12) and automati-cally segmented using Freesurfer [117].The FDG dataset was chosen to evaluate IHYPR4D for an irreversibly binding radiotracerwhere the radiotracer distribution does not change as rapidly as in the RAC dataset. Weevaluate the effect of IHYPR4D on parametric images derived from compartmental modellingin the RAC simulation study; in this human FDG dataset we evaluate the effect of IHYPR4Don parametric images obtained with the Patlak graphical analysis. An individually calibratedinput function based on the administered activity and initial distribution volume was modelled[119] and regression was performed over data from the last 30 minutes of the scan to extractparametric Ki images.552.4. Results2.4 Results2.4.1 Phantom StudyComparing ROI generation methods (Figure 2.2)No segmentation. At early IHYPR4D iterations %CRC is negatively biased from the target%CRC (i.e. the %CRC without any denoising, shown as solid black lines in Figure 2.2). Withfurther iterations, the %CRC increases until it generally matches the target %CRC. Thus,subregional features are recovered by IHYPR4D with sufficient iteration.Hand-drawn segmentation. When the inserts are parcellated as regional features, the%CRC closely matches the target %CRC after just a single iteration. Small discrepanciesfrom the target %CRC are generally reduced after a few iterations.k-means segmentation. The sharp activity boundaries that exist in the phantom are de-graded in the images by limited PET resolution, and thus the k-means clustering tends toextend the boundary of the insert beyond that of the true boundary—resulting in an initialreduction from the target %CRC. Nevertheless, with iteration the %CRC vs. noise trajectoriesare similar or slightly worse than those from the hand-drawn segmentation method and betterthan those from when no segmentation is used.Effect of spatial kernel sizeA larger kernel size borrows more features from the composite. In the no segmentation method,these features are not present in the initial composite, and thus the first iteration in the tra-jectories of Figure 2.2 yields lower %CRC with a larger kernel size. However, since the initialcomposite does not limit noise, the noise is also initially lower with a larger kernel. With in-creasing iterations, these effects generally offset each other as the trajectories are largely similarfor the kernel sizes studied. However, small differences arise as the feature size approaches thespatial kernel size: Figure 2.2d shows 1x PSF IHYPR4D recovering the 4 mm (∼ 1.6x PSF)target %CRC after three iterations at 16.2% voxel noise, whereas Figure 2.2f shows 2x PSFIHYPR4D recovering the target %CRC after 25 iterations at 17.2% voxel noise. Additionally,larger kernel sizes produce both smaller %CRC and noise increments, meaning more iterationsare required to reach a given point on the trajectories.In contrast, when the hot inserts are segmented and thus included in the initial composite,using a larger kernel is beneficial since these features are returned after a few or even a single562.4. ResultsFigure 2.2: %CRC versus voxel noise trajectories in the three smallest hot inserts for various kernelsizes Different ROI generation methods are compared for OP-OSEM + IHYPR4D (a-c) or HYPR4D-K-OSEM + IHYPR4D (d-f). Both contrast recovery and voxel noise have been averaged over all frames.Each IHYPR4D marker represents an iteration, with iterations increasing left-to-right. NoniterativeHYPR3D %CRC and voxel noise is shown as a single marker. Solid black lines indicate the target%CRC (i.e. %CRC without denoising).572.4. Resultsiteration at lower noise compared to using smaller kernels. For instance, Figure 2.2d shows1x PSF IHYPR4D recovering the 8 mm target %CRC after a single iteration at 12.2% voxelnoise, whereas Figure 2.2f shows 2x PSF IHYPR4D recovering the target %CRC after a singleiteration at 8.7% voxel noise.Effect of temporal kernel sizeUsing a larger IHYPR4D temporal kernel size lowers voxel noise through the initial filtering(Figure 2.3). Since the hot insert TACs and the background region TACs are scalar multiplesof each other, when the hot inserts are not segmented the initial IHYPR4D composite has thesame temporal shape as the insert and background TACs, meaning the larger kernel does notintroduce any further negative bias from the target %CRC.However, unsegmented features may have TACs that vary substantially from the TACs ofthe surrounding tissue. For instance, the carotid arteries deliver radiotracer to the brain andthus have sharply-peaked concentration shortly after injection, whereas the surrounding tissuehas little radiotracer uptake at this point. In the application of using the carotid arteries tocompute an image-derived input function (IDIF), the temporal kernel size needs to be carefullyselected to fully recover this feature. This is discussed in Section 2.4.3.Figure 2.3: Esser phantom %CRC versus voxel noise trajectories for IHYPR4D (1.5x PSF kernel size,no segmentation of inserts) with different temporal kernel sizes. Shown are the three smallest inserts ofthe phantom on the HRRT for OP-OSEM and HYPR4D-K-OSEM reconstructions. Both %CRC andvoxel noise have been averaged over all frames. Solid black lines indicate the target %CRC (i.e. %CRCwithout denoising).Effect of feature sizeIn the no segmentation method, contrast in the larger inserts is recovered with fewer IHYPR4Diterations. Though we only show the 4, 6, and 8 mm analyses here, this trend continues for 12and 16 mm.582.4. ResultsWhen the inserts are segmented, IHYPR4D %CRC is similar to the target %CRC after asingle iteration in the 6 and 8 mm inserts; small differences are due to (i) inaccurate positioningof the hand-drawn or k-means-generated insert ROIs and/or (ii) imprecise delineation of theinsert due to voxel discretization. Nonetheless, the %CRC is an improvement over the %CRCwith no segmentation, and the effect of inaccurate segmentation is mitigated after a few iter-ations. For the 4 mm insert, the initial IHYPR4D %CRC is noticeably lower than the target%CRC even with segmentation—especially when OP-OSEM is used to generate the target.This is discussed below.Effect of reconstructionData obtained with OP-OSEM without any denoising have over 6x higher voxel noise thanthose obtained with the denoising reconstruction HYPR4D-K-OSEM. Consequently, HYPR4D-K-OSEM + IHYPR4D trajectories reach the target %CRC at much lower voxel noise thanOP-OSEM + IHYPR4D. For all inserts except 4 mm, the target %CRC is also higher forHYPR4D-K-OSEM versus OP-OSEM, likely due to less zero trapping and being less biasedtoward the last subset of the data [38].OP-OSEM + IHYPR4D with k-means or hand-drawn segmentation reaches the target%CRC at similar voxel noise to HYPR4D-K-OSEM without any post-processing denoising.However, as a post-processing method IHYPR4D cannot undo the biases introduced by OP-OSEM, so HYPR4D-K-OSEM without post-processing still has higher %CRC than OP-OSEM+ IHYPR4D. Nevertheless, coupling HYPR4D-K-OSEM + IHYPR4D reduces noise furtherwhile maintaining higher %CRC than OP-OSEM + IHYPR4D.For the 4 mm insert the target %CRC is ∼ 10% for HYPR4D-K-OSEM compared to ∼ 15%for OP-OSEM. This is a noise-induced difference: the ground truth insert activity is uniform,but high noise in the OP-OSEM image leads to a broad distribution of voxel concentrationestimates, with positive noise spikes outweighing zero-trapped voxels in the 4 mm insert (Fig-ure 2.4), leading to an artificially higher regional %CRC. Both HYPR4D-K-OSEM reconstruc-tion and IHYPR4D post-processing reduce zero-magnitude voxels and positive noise spikes,consequently lowering %CRC in this insert. In other inserts, the OP-OSEM zero-trapping out-weighs the positive noise spikes, leading to reduced %CRC as compared to HYPR4D-K-OSEM.Noise suppression effects in reconstruction (HYPR4D-K-OSEM) or post-processing (IHYPR4D)reduce positive noise spikes, thus lowering %CRC. Thus, the OP-OSEM + IHYPR4D 4 mminsert trajectories may be misleading: IHYPR4D needs to overcome any initial resolution lossor insert segmentation error and produce the broad concentration distribution that arises withhigher noise.592.4. ResultsFigure 2.4: Concentration distributions with and without IHYPR4D denoising (1.5x PSF, four frames,10 iterations) in the 4 mm insert of the Esser phantom on the HRRT when (a) OP-OSEM and (b)HYPR4D-K-OSEM is used as the input reconstruction.IHYPR4D compared to HYPR3DResults demonstrate that IHYPR4D can fully recover segmented and unsegmented features,i.e. recover the target %CRC of the inserts, provided enough iterations are run. NoniterativeHYPR3D also matches the target %CRC; increasing kernel size increases noise reduction with-out changing %CRC. However, in this phantom study the contrast is time-independent. Thus,the contrast in the HYPR3D composite is perfectly matched to each frame. In a more realisticscenario with time-dependent contrast, any contrast mismatch between the composite and aparticular dynamic frame will be amplified by a larger kernel (see section B).Even in this time-dependent scenario, OP-OSEM + IHYPR4D substantially outperformsOP-OSEM + HYPR3D in terms of voxel noise, since the noisy OP-OSEM images result in anoisy HYPR3D composite. When using the less noisy HYPR4D-K-OSEM images, the HYPR3Dcomposite is also less noisy, so HYPR3D voxel noise is close to that of IHYPR4D after feweriterations.Voxel noise in Figure 2.2 is an average over all frames, which have count levels spanning∼ 2–12 x 107. Figure 2.5 demonstrates that IHYPR4D has lower voxel noise on average thanHYPR3D at equivalent %CRC but has higher noise for the first frame of the dataset (one of thelow count frames). Because in our implementation the temporal kernel is padded with duplicateframes when the kernel extends beyond the first or last frame, the noise adds coherently amongthe duplicates, resulting in decreased noise reduction near the edge frames. The highest countframe corresponds to the last frame, where a similar effect is observed.SIGNA phantom studySimilar IHYPR4D trajectories were observed for the 10 mm and 17 mm inserts of the SIGNAphantom scan (Figure 2.6), with IHYPR4D matching the target %CRC at lower voxel noisethan HYPR3D. The noise levels of the PSF-TOFOSEM SIGNA target images were lower than602.4. ResultsFigure 2.5: Voxel noise in a background region of the Esser phantom as a function of frame countsin the dynamic scan. IHYPR4D voxel noise is shown with a 1x PSF kernel after three iterations suchthat %CRC is approximately matched to HYPR3D. Voxel noise is very similar between IHYPR4Dsegmentation methods and thus the data mostly overlap.Figure 2.6: IHYPR4D %CRC versus voxel noise trajectories for the 10 and 17 mm inserts of a phantomscanned on the SIGNA with a (a) 1x PSF and (b) 2x PSF kernel size. Both %CRC and voxel noise havebeen averaged over all frames. Each marker represents an IHYPR4D iteration, increasing left-to-right.Noniterative HYPR3D %CRC and voxel noise is shown as a single marker. Solid black lines indicatethe target %CRC (i.e. %CRC without denoising).those of the OP-OSEM HRRT images (mostly due to the higher sensitivity of the SIGNA).Thus, IHYPR4D appears to behave consistently across a wide range of noise levels.2.4.2 Human simulation studyBias and reproducibility in concentrationTemporal bias patterns (Figure 2.7a-d). In the simulation study, there is a time-dependentcontrast due to the regionally variant RAC kinetics. In the early frames during the radiotraceruptake phase (0–4 minutes), there is little contrast between regions. As the radiotracer distri-bution transitions to equilibrium in the middle frames (4–40 minutes), there is notable contrastbetween high binding regions of the striatum and the rest of the brain which continues to in-crease during radiotracer washout in the later frames (40–60 minutes). The HYPR3D compositehas contrast best matching that of the middle frames. As a result, positive bias is introduced612.4. ResultsFigure 2.7: Temporal bias patterns in the small hot spot with different kernel sizes for IHYPR4Dand HYPR3D when using OP-OSEM (a-b) or HYPR4D-K-OSEM (c-d) to generate the target (threeIHYPR4D iterations are used for 1x PSF kernel and 15 iterations for 2x PSF kernel); bias-noise trade-offfor IHYPR4D and HYPR3D in all three regions studied when using a 1x PSF kernel (e-g) or 2x PSFkernel (h-j). For the bias-noise trade-off, solid and dashed black lines indicate the target mean |%bias(t)|without any denoising for OP-OSEM and HYPR4D-K-OSEM, respectively. Black arrows indicate howthe k-means segmentation for IHYPR4D recovers the temporal bias pattern of the small hot spot betterthan using the MRI segmentation when the feature size is less than the kernel size used, since thek-means segmentation can parcellate the hot spots from the surrounding putamen tissue (unlike theMRI segmentation). The pink arrow in (a) demonstrates how HYPR3D-induced positive bias partiallycancels with OP-OSEM-induced negative bias in early frames, leading to lower mean |%bias(t)| than thetarget, as identified by the pink arrow in (g). HYPR4D-K-OSEM has %bias(t) close to zero, thus theHYPR3D-induced positive bias in this case leads to higher mean |%bias(t)| than the target.622.4. Resultsinto the striatal regions during uptake and negative bias during washout, as shown for thesmall hot spot in Figure 2.7a-d (similar results are seen in the caudate and small hot spot).This HYPR3D-induced bias is amplified with increasing kernel size (compare Figure 2.7a withFigure 2.7b and Figure 2.7c with Figure 2.7d). In comparison, IHYPR4D, with a better matchbetween the contrast in the composite and in the individual frames, reproduces the %bias(t) ofthe target images—independent of kernel size and reconstruction used to generate the targetimages—provided sufficient iterations are run.Bias-noise trade-off (Figure 2.7e-j). Using MRI segmentation, the mean |%bias(t)| inthe segmented caudate closely matches the target mean |%bias(t)| after a single IHYPR4Diteration, whereas mean |%bias(t)| in the unsegmented hot spots is initially high but reducesto the target mean |%bias(t)| after sufficient iterations, with more iterations required for largerkernels (similar to the unsegmented hot spots in the phantom study, c.f. section IV.A.1).Effect of reconstruction. As in the phantom study, OP-OSEM + IHYPR4D consistentlyreaches a stabilized, minimum mean |%bias(t)| similar to that of the target at lower noisethan HYPR3D and HYPR4D-K-OSEM + IHYPR4D offers a significant reduction in noisecompared to OP-OSEM + IHYPR4D. HYPR4D-K-OSEM + IHYPR4D reaches minimizedmean |%bias(t)| at lower noise than HYPR4D-K-OSEM + HYPR3D with a 1x PSF kernel (tominimize the HYPR3D-induced bias) or has reduced mean |%bias(t)| at equivalent noise toHYPR3D with a larger kernel.Comparing ROI generation methods. The MRI segmentation does not parcellate thefunctional hot spots since they are not present in the MRI data, while the k-means segmen-tation identifies the hot spots from the PET data as separate clusters from the surroundingputamen tissue (see the initial composites for each method in Figure 2.8), leading to bias-noisetrajectories lying below those from the MRI segmentation method (Figure 2.7f,g and Fig-ure 2.7i,j). Conversely, the MRI-segmented caudate ROI better delineates the true functionalcaudate boundary than the k-means segmentation, yielding slightly worse bias-noise trajectoriesfor the k-means segmentation method in the caudate (Figure 2.7e and Figure 2.7h).Visualizing the bias-noise trade-off (Figure 2.9: 2x PSF spatial kernel sizes forHYPR3D and IHYPR4D). After a single iteration, the IHYPR4D images are visuallysmoother than HYPR3D. After 18 iterations, OP-OSEM + IHYPR4D is still less noisy thanOP-OSEM + HYPR3D and HYPR4D-K-OSEM + IHYPR4D has comparable noise levels toHYPR4D-K-OSEM + HYPR3D. However, by 18 iterations structural boundaries are degradedby this increased noise. In general, 3 iterations of IHYPR4D with k-means segmentation or632.4. ResultsFigure 2.8: The HYPR3D and initial IHYPR4D composites for the simulation study. Summing thenoisy OP-OSEM images lead to a noisy HYPR3D composite, whereas spatially averaging the sameimages yields much less noisy IHYPR4D composites. The denoised HYPR4D-K-OSEM images lead tosimilar composite noise levels for HYPR3D and IHYPR4D. Pink arrows demonstrate how the k-meanssegmentation parcellates the hot spots, while the MRI segmentation does not.642.4. Results6 iterations with MRI segmentation mostly recovers the functional hot spot features whilemaintaining low noise.Bias and standard deviation in BPNDParametric images, generated by fitting to voxel TACs, will have biases sensitive to the temporalbiases in the concentration estimates, i.e. %bias(t), and the kinetic modeling method employed.OP-OSEM images on the HRRT will have TACs with high temporal noise due to the low countsper voxel. In TACs where large, positive noise spikes exist in the last few frames, the fits ofkinetic models will be skewed positively as well (Figure 2.10a), producing a ∼ 20% positivelybiased fitted BPND on average, in all regions. Since (i) there is a positivity constraint in theOP-OSEM images; (ii) mathematically BPND ∈ [−1,∞); and (iii) the majority of ground truthBPND in the region of analysis (i.e. striatum) fall between ∼ 2 and ∼ 7, on average BPND willbe more likely to be positively biased, leading to a right-skewed distribution of fitted BPNDin OP-OSEM images (Figure 2.11a). With the positive noise spikes largely removed by thespatiotemporal denoising of IHYPR4D (Figure 2.10b), the distribution of fitted BPND moreclosely resembles the ground truth distribution (Figure 2.11b).From the concentration analysis, ∼ 3 iterations of 1x PSF IHYPR4D or ∼ 15 iterations of 2xPSF IHYPR4D produces %bias(t) matching the %bias(t) of the target. At these points alongthe BPND bias-noise trajectories (Figure 2.12), denoising produces concentration images withless severe temporal noise spikes. This reduces the right-skewedness of the BPND distributionwhile mostly preserving the negatively biased peak, thus leaving a small negative %bias inBPND estimates. Since a smaller kernel size is less effective at suppressing noise, with furtheriteration the %bias in BPND continues to trend upward for OP-OSEM + 1x PSF IHYPR4Das the right-skewedness returns to the BPND distribution. This effect is not present with a 2xPSF kernel.HYPR3D, having systematically different %bias(t) than the target, produces more nega-tively biased BPND estimates at higher %std than IHYPR4D after the above-mentioned num-ber of iterations. Increasing HYPR3D kernel size amplifies this bias, with a small reduction in%std.For HYPR4D-K-OSEM data, the denoising during reconstruction similarly reduces theright-skewedness, leading to a much smaller (∼ 2%) positive bias in BPND in the hot spots.The remaining right-skewedness is removed by IHYPR4D after sufficient iterations, leaving es-sentially no bias in BPND in the hot spots. HYPR4D-K-OSEM yields negatively-biased BPNDestimates in the caudate which is preserved by IHYPR4D after an appropriate number of iter-ations—a fault of the reconstruction rather than IHYPR4D.652.4. ResultsFigure 2.9: Axial slices (zoomed in on the striatum) of a single iteration of the HRRT RAC simulationfor (a) an equilibrium frame and (b) the mean of the uptake period. For HYPR3D and IHYPR4D a2x PSF spatial kernel size is used. Upper and lower panes are for OP-OSEM and HYPR4D-K-OSEMdata, respectively. IHYPR4D is shown for various representative iteration numbers and for both MRIand k-means segmentation methods. Pink arrows in (a) indicate the better initial recovery of the hotspots using k-means versus MRI segmentation; however, after many iterations the hot spots are similarlyrecovered. Orange arrows in (a) indicate better recovery of the true caudate boundaries when using MRIversus k-means segmentation at early iterations, but again these differences are less apparent by lateriterations. Black arrows in Fig. (b) indicate the positive bias introduced by HYPR3D during uptake inthe caudate and especially the caudate, whereas the uptake contrast is better estimated by IHYPR4D.662.4. ResultsFigure 2.10: (a) an OP-OSEM voxel TAC where noise has induced a 432% bias in the fitted BPND.(b) the same voxel TAC after IHYPR4D denoising results in a -7% bias.Figure 2.11: The estimated BPND distribution for all caudate voxels in a representative noisy real-ization when OP-OSEM is used as the input reconstruction (a) without any denoising, and (b) withIHYPR4D applied.Figure 2.12: Bias-noise plots for BPND estimates in the three regions studied for HYPR3D andIHYPR4D when using a 1x PSF (a-c) or 2x PSF (d-f) kernel. Solid and dashed lines represent the%bias in BPND without any denoising for OP-OSEM and HYPR4D-K-OSEM, respectively.672.4. ResultsFigure 2.13: Average line profiles (solid lines) ± one standard deviation (dashed lines) over all noisyrealizations of the simulated BPND gradient in the putamen.Recovery of a spatially variant BPND patternIHYPR4D recovers the disease-induced gradient in BPND in the right putamen with high accu-racy and higher precision than HYPR3D (Fig. 2.13). Systematic biases from the ground truthprofile that remain after denoising are likely from early termination of reconstruction sincethey are consistent among the post-processing methods. However, small differences betweenthe k-means and MRI segmentation methods in IHYPR4D exist because the k-means segmen-tation parcellates the putamen gradient into multiple ROIs—instead of a single ROI in theMRI segmentation—allowing a slightly more accurate recovery of the gradient for the k-meanssegmentation at equivalent number of iterations.2.4.3 Human studyImage qualityQualitatively, the IHYPR4D images from the human study are less noisy while maintainingcomparable regional contrast as the HYPR3D images (Figure 2.14). These observations areconsistent across scanners and tracers. This lower noise level in IHYPR4D increases the sharp-ness of features such as the boundaries of the striatum in the RAC images and cortical greymatter in the FDG images.Quantitatively, we compute %std in the cerebellum of the RAC scans—a low-binding, rel-atively uniform region for RAC—to assess noise levels (Table 2.1). In the HRRT RAC scan,without any denoising %std is 174.1%; this is reduced to 16.2% by IHYPR4D with MRI segmen-tation after a single iteration and increases to 40.1% after 18 iterations. However, this is stilllower than 54.2% for HYPR3D. Similar reductions are seen when using k-means segmentationin IHYPR4D and similar trends are observed in the SIGNA RAC scan.682.4. ResultsFigure 2.14: Axial slices from equilibrium frames of the (a) HRRT RAC (b) SIGNA RAC, and (c)SIGNA RAC human studies. For HYPR3D and IHYPR4D a 2x PSF spatial kernel size is used. Upperand lower panes are for OP-OSEM and HYPR4D-K-OSEM data, respectively. IHYPR4D is shown forvarious representative iteration numbers and for both MRI and k-means segmentation methods.692.4. ResultsMethod HRRT SIGNANo denoising 174.1% 108.1%HYPR3D 54.2% 50.1%IHYPR4D1 iter 16.2% 16.3%MRI 3 iter 23.5% 24.5%segmentation 6 iter 29.2% 29.0%18 iter 40.1% 34.7%IHYPR4D1 iter 21.8% 18.8%k-means 3 iter 26.7% 25.2%segmentation 6 iter 31.3% 29.2%18 iter 41.1% 34.7%Table 2.1: Comparing %std in the cerebellum for human RAC data.Representative voxel-level TACs in Figure 2.15a-d showcase the reduced temporal noiseobtained with IHYPR4D. These results, as well as the denoising performance of HYPR3D,are similar between the simulated and real HRRT RAC data (compare Figure 2.15a with Fig-ure 2.15b). Additionally, in the case of the HRRT, for HYPR3D one may be summing overTACs with large noise spikes, such as in Figs. 2.13a and 2.13b, leading to incorrect correctionfactors for such voxels (the ratio of the composite to filtered composite in the HYPR operator),thus incorrectly scaling the denoised TACs. IHYPR4D reduces this possibility by summingover many more voxels in space instead of the same voxel in time, reducing the possibility ofthis incorrect scaling. For the FDG scan, we also qualitatively show that a HYPR3D-processedimage obtained from the average of the last 20 minutes of data acquisition, when the radio-tracer distribution is mostly constant, has comparable noise to a single one-minute frame imagedenoised with IHYPR4D (Figure 2.15e).Effect of temporal kernel size on an image-derived input functionIn our human FDG scan, the current 76-frame protocol can capture an image-derived inputfunction (IDIF) with a 15-second temporal resolution for the first four minutes (Figure 2.16),giving the IDIF a FWHM of ∼ 2 frames. If the arteries are not segmented from the backgroundtissue within the initial IHYPR4D composite, the 15-frame temporal kernel size of IHYPR4Dis too large to recover an accurate IDIF from the filtered data. What is needed then is atemporally variant temporal kernel size; this is effectively achieved by resampling the data toshorter frame durations (i.e. interpolating) such that the IDIF is not oversmoothed initially.In Figure 2.16, we plot the IDIF calculated from the non-denoised images using a manuallydefined ROI of the carotids and compare to the IDIF after denoising. Without resampling, theIDIF is not recovered by IHYPR4D, but after resampling the first two minutes to two secondframes (such that the IDIF peak has a FWHM ∼ 14 resampled frames), the IHYPR4D IDIF702.4. ResultsFigure 2.15: A representative voxel TACs for (a) a caudate voxel in the HRRT RAC simulation; (b) acaudate voxel in the HRRT RAC human study; (c) a caudate voxel in the SIGNA RAC human study;(d) a grey matter voxel in the SIGNA FDG human study. (e) An axial slice of the SIGNA FDG humanstudy over the last 20 minutes without denoising and with HYPR3D applied, which is juxtaposed withone-minute frame during this time period after IHYPR4D (k-means segmentation method) is applied. Inall cases, HYPR3D and IHYPR4D use a 2x PSF kernel size and IHYPR4D is shown after six iterations.largely resembles the non-denoised IDIF. Thus, the IHYPR4D temporal kernel size should beselected to match the FWHM of the shortest temporal feature one wishes to detect in the(potentially resampled) data. Conversely, HYPR3D has a biased IDIF since the high contrastof the carotids only exists for a few frames and thus is heavily diminished in the HYPR3Dcomposite.Figure 2.16: Image-derived input functions for the first four minutes of the FDG scan of the humanstudy, with and without denoising.Parametric analysisWith the human SIGNA FDG Patlak analysis, the Ki images are not as impacted by noise asthe simulated RAC HRRT SRTM2 analysis, and IHYPR4D only offers a modest improvement712.5. Discussioncompared to no denoising (Figure 2.17a). This difference exists because (i) the SIGNA hashigher sensitivity and lower spatial resolution than the HRRT, leading to less noisy input datato IHYPR4D; and (ii) the HRRT protocol has a smaller number of frames (16 versus 76), solarge noise spikes more easily skewed BPND estimates.With HYPR3D, an additional disadvantage was discovered through analysis of this humandataset. The HYPR3D composite—a weighted sum of all frames—is sensitive to imperfectmotion correction (which likely exists even after motion correction algorithms are applied),thus appearing blurred when compared to the IHYPR4D MRI segmentation-based compositein Figure 2.17b. As a result, HYPR3D cannot recover the resolution of the target images fromthis degraded composite, leading to parametric Ki images similar to those generated from a 3mm Gaussian post-filtered dataset (Figure 2.17a). The IHYPR4D k-means segmentation-basedcomposite in Figure 2.17b also suffers from motion induced blurring, but unlike HYPR3D theIHYPR4D composite is iteratively updated to partially undo this effect—leading to similar Kiimages as from IHYPR4D using MRI segmentation.Figure 2.17: (a) Axial slices of parametric Ki images derived from human FDG data for variousdenoising methods, alongside a zoomed in cortical section. The subject’s co-registered T1 MRI image isshown as a structural reference. (b) The 3D composite for HYPR3D and a single frame of similar contrastfrom the IHYPR4D composites, demonstrating the how imperfections in motion correction result in ablurring effect that reduces the resolution of the composite in HYPR3D and IHYPR4D with k-meanssegmentation. IHYPR4D iterative recovers this resolution loss, while HYPR3D does not, leading to Kiimages for HYPR3D that also appear blurred (compare to the Ki images derived from 3 mm post-filtereddata in (a)).2.5 Discussion2.5.1 Interpreting IHYPR4DIn PET imaging, the (unknown) ground truth image is generally characterized by regions of rela-tively homogeneous tissue concentration with sharp boundaries commonly shared with anatom-ical boundaries. However, acquired PET images may have low SNR, and most of the variabilityin the concentration estimates may arise from noise rather than concentration inhomogeneities.722.5. DiscussionA spatially invariant filtering operation on the acquired image will enforce local similarity acrossthe entire image space, penalizing regions with sharp changes in radiotracer distribution. Toavoid this, a spatially variant operator that enforces local similarity only within appropriateboundaries is required.IHYPR4D is a step towards this ideal operator. It can be thought of as a compressionalgorithm that relaxes with each iteration; the similarity of concentration within anatomicalregions is exploited by initially describing the regional concentration distributions as uniformin the initial IHYPR4D composite. This harsh compression would produce an image closelymatching the ground truth image if the concentration were indeed homogeneous within eachregion. However, given that true inhomogeneities may exist, the initial composite image iscompared to the acquired data through the HYPR operator. This produces a denoised imagewith similar SNR to a filtering operation with the same kernel used in the HYPR operator,albeit with regional boundaries intact. By replacing the composite with the denoised image,the HYPR operator can be iterated to further relax the “compression” level: allowing inhomo-geneities within regions to be fully recovered at the cost of reintroducing some noise-inducedconcentration variability.2.5.2 An improvement over HYPR3DThe phantom, simulation, and human studies have collectively demonstrated that IHYPR4Dprovides an improvement over HYPR3D in terms of noise and bias reduction. HYPR3D uses a3D composite image formed via a weighted sum of all temporal frames to extract high frequencyspatial features and return them to a filtered version of the individual frames. By moving to4D, IHYPR4D mitigates some of the limitations of HYPR3D, namely the implicit assumptionthat the contrast in the composite and individual frames is matched, as well as the noise levelsin the HYPR3D images being limited by the noise level in the composite. HYPR3D uses aspatial filter in its formulation, thus limiting denoising and feature extraction to the spatialdomain only. The IHYPR4D formalism employs a spatiotemporal filter, allowing for improvednoise reduction as well as spatiotemporal feature extraction.By iteratively updating the 4D composite image in IHYPR4D, contrast mismatch betweenthe composite and input data is minimized, thus reinstating the contrast lost from filtering,as shown in the phantom study (Figure 2.2). The contrast mismatch of HYPR3D ultimatelyalters the temporal shape of the TACs, potentially leading to biases in parametric analyses aswell (Figure 2.7).IHYPR4D computes the initial composite by averaging over the voxels of ROIs, instead ofthe same voxel over temporal frames as in HYPR3D. Noise will mostly add decoherently andthe SNR in the composite will roughly increase as the square root of the number of data points.732.5. DiscussionIn this work we investigated using anatomical ROIs or a k-means segmentation algorithmwith 30 regions for IHYPR4D. In high resolution PET, both strategies yield regions rangingfrom hundreds to several thousand voxels, as compared to tens of temporal frames typicallyused in dynamic PET. With iteration, the IHYPR4D composite noise level will increase ina suppressed manner, so that one can terminate at an iteration where bias is appropriatelyminimized. Such a termination point could be experimentally selected for a given scanner,framing, and reconstruction protocol.The aim of this work was to present an improved HYPR operator and thus we did not com-pare HYPR to other denoising methods. However, it is also briefly worth mentioning anotherproposed modification to HYPR3D: employing the HYPR3D operator with multiple manuallychosen composites across different frames was shown to reduce the implications of contrastmismatch, but at the cost of increased noise over the traditional HYPR3D implementation[109]. IHYPR4D largely removes the contrast mismatch while simultaneously reducing noiseover HYPR3D, without the need to manually select multiple composites.2.5.3 Coupling a denoising reconstruction with IHYPR4DHYPR3D performance is improved when using a denoising reconstruction such as HYPR4D-K-OSEM [38] to generate the target images. However, the denoising reconstruction also ben-efits IHYPR4D so that both methods have similar noise levels, though the issue of increasedbias from contrast mismatch in HYPR3D remains—regardless of reconstruction. Thus, of thereconstruction/post-processing combinations studied, HYPR4D-K-OSEM + IHYPR4D yieldsimages with the highest accuracy and precision (as characterized by low bias and noise).While both IHYPR4D and HYPR4D-K-OSEM use a 4D HYPR operator to extract fea-tures, they use entirely different methods to construct a low noise composite from which thesefeatures are extracted. HYPR4D-K-OSEM exploits the inconsistent noise patterns across thesubset images of the OSEM reconstruction, summing over these subset images to compute thecomposite after each iteration. Thus, HYPR4D-K-OSEM (i) still does not exploit possible voxelsimilarity within ROIs by incorporating spatial averaging and (ii) requires an initial iterationof OP-OSEM for initialization. (ii) implies the starting point for HYPR4D-K-OSEM has lowcontrast and low noise, thus requiring several iterations to improve the contrast while increasingnoise—albeit at a slower rate than OP-OSEM. If one were to modify HYPR4D-K-OSEM toincorporate spatial averaging in the composite definition after each iteration, the assumptionof uniformity within regions would improve noise reduction, but it may be offset by an inferiorability to recover difficult-to-parcellate low contrast features from early updates (as well asslowdown of convergence rate in image features), thus undermining the benefit of low bias andnoise that HYPR4D-K-OSEM practically offers.742.5. DiscussionBy contrast, separately running HYPR4D-K-OSEM + IHYPR4D is more beneficial be-cause HYPR4D-K-OSEM first effectively minimizes bias without any regional assumptions(and reduces noise, as compared to OP-OSEM), then IHYPR4D can employ spatial averag-ing to reduce noise further with a minimal effect on bias. We do not believe an alteration toHYPR4D-K-OSEM as described above would arrive at the same location in bias-noise spaceas HYPR4D-K-OSEM + IHYPR4D. However, this has yet to be investigated and could be anarea of future research.2.5.4 ROI generation methodsAny set of ROIs may be used to generate the initial IHYPR4D composite. The choice of ROIgeneration method should ideally group together voxels with identical kinetics to minimizebias in the initial composite. In this work, we tested two relatively simple methods: MRIsegmentation of a co-registered T1 image to the PET data and a four-step k-means approachwhere k-means segmentation is run on a mean PET image over frames with relevant radiotracercontrast.The ROI generation method selected should be reflective of the expected radiotracer dis-tribution. For tracers where functional boundaries are consistent with a subset of anatomicalboundaries, such as FDG and RAC, an MRI segmentation is useful as the ROI boundaries aresharper in anatomical MRI than the PET image itself. Conversely, if nonuniform radiotracerdistribution is expected within anatomical boundaries, such as disease-induced radiotracer bind-ing gradients within a structure or in task-based FDG/RAC studies, then it would be preferableto extract the ROIs from the PET data themselves to avoid a large bias in the initial composite,despite the functional boundaries being degraded by PVE. PET-derived segmentation methodsinclude the k-means approach discussed in this work, affinity propagation discussed in [111],or kinetic clustering [120]. Selection of an ROI generation method should also be considerateof image noise. For instance, high resolution and/or low dose images with high temporal noiseare likely not amenable to kinetic clustering or a single k-means segmentation.These recommendations are supported by our simulated results: when functional featuresexist within an anatomical ROI, i.e. the hot spots or BPND gradient in our simulated RACdataset, then IHYPR4D images will be biased at early iterations. However, depending on thesize and contrast of the feature, the majority of functional feature recovery occurs within threeiterations for a 1x PSF kernel and the first 10 iterations for a 2x PSF kernel, at which pointthe noise and bias are both lower than HYPR3D in general (see the bias-noise plots of the hotspots in Figure 2.7).Conversely, these features can be parcellated initially using a k-means approach, thoughfeature boundaries are more difficult to delineate in PET images, resulting in imperfect parcel-752.5. Discussionlation. This ultimately means a single IHYPR4D iteration will often still bias these features,albeit less so than unparcellated features in the MRI segmentation. Nonetheless, after only twoiterations for a 1x PSF kernel or five iterations for a 2x PSF kernel most of the bias introducedhas been removed. A disadvantage of the k-means method is that the number of clusters needsto be pre-specified. In this work we empirically chose 30 clusters as a balance between segment-ing small features and maintaining reasonably sized ROIs. Optimizing the number of clustersis beyond the scope of this work and likely is dependent on the level of inhomogeneity of theradiotracer distribution and input image noise level.2.5.5 ConvergenceIHYPR4D is an iterative algorithm and thus a choice when to terminate the algorithm mustbe made. Akin to iterative reconstruction, the noise level of IHYPR4D increases with eachiteration, so it is advised to terminate the algorithm before convergence (if any), when theincrement in contrast recovery or decrement in bias is no longer worth the increase in noise. Wehave suggested using a phantom and/or simulation study reproducing specific human imagingfor this purpose. We have noted that IHYPR4D can fully recover features not parcellated inthe segmentation, though the closer in size a feature is to the kernel size, the more iterationsrequired for recovery. Thus, a hot insert/simulated hot spot comparable in size to the smallestfeature one wishes to recover should be present in the phantom/simulation. In our investigationwe found that using a 2x PSF spatial kernel balances fast recovery of small features with a smallenough noise increment for fine-tuning.2.5.6 Likely applicationsIHYPR4D outperforms HYPR3D especially in cases where the voxel counts are low and/orwhen the radiotracer distribution changes significantly over time. This includes high spatialand/or temporal resolution dynamic scans, as well as low dose scans, which is of increasedinterest for the reduced radiation doses to participants and lower scan costs [121–123].High temporal resolution PET has various applications such as computing image-derivedinput functions and detecting short-duration, low magnitude concentration changes in task-based neurotransmitter [49, 114] or FDG studies [5, 115, 116]. In Chapter 3, we develop anovel dopamine release detection algorithm that, coupled with IHYPR4D, enables greatly im-proved detection sensitivity of low magnitude dopamine release—demonstrating the immediateapplicability of IHYPR4D.We have indirectly shown that IHYPR4D helps improve a PET-based segmentation: in ourk-means segmentation method, the initial mean image over several frames used for an initial k-762.5. Discussionmeans segmentation is too noisy to provide reliable feature parcellation, but generating a smoothIHYPR4D initial composite with this suboptimal segmentation followed by a second k-meanssegmentation results in improved feature parcellation. A direct investigation whereby IHYPR4Dand PET segmentation algorithm(s) are alternated to benefit each other would be useful, similarto the approach of [124] Additionally, more complex PET segmentation algorithms, such askinetic clustering [120], may be useful in this context, but were not investigated in this worksince they were not amenable to the high temporal noise of HRRT images.Finally, HYPR3D has been used in conjunction with partial volume correction (PVC) al-gorithms: PVC will increase spatial resolution but also noise, so applying HYPR3D beforeand after PVC has been shown to minimize this noise increased while maintainin the increasedresolution from PVC [125]. Given IHYPR4D shows improvements over HYPR3D, it is likelythat applying IHYPR4D before and after PVC will offer further improvements, especially inhigh spatiotemporal resolution applications that would benefit from both denoising and PVC,such as extracting image-derived input functions with high temporal resolution from the carotidarteries.2.5.7 LimitationsSegmentation requirementIHYPR4D requires an accurate segmentation to produce the initial composite. Inaccuraciesin the segmentation will introduce bias at early IHYPR4D iterations, though these biases areminimized after appropriate iteration and have a minor effect on image quality (see quantitativeresults in Appendix A.1).PSF knowledge requirementThe resolution of the system/reconstruction needs to be known to filter the initial IHYPR4Dcomposite image obtained by segmentation, provided the resolution of the segmentation ishigher than that of the PET image. This ensures features of size below the PET imagingresolution are not artificially introduced to the images. The PSF of the scanner is usuallyestimated from scanner characterization studies, but may not be known exactly. In this workwe modelled the PSF as a spatially invariant Gaussian.Additionally, factors such as uncorrected in-frame subject motion which degrade resolutionbeyond scanner and reconstruction effects are not accounted for in the PSF filtering, meaningthe target PET data and initial IHYPR4D concept may have slightly mismatched resolution.See Appendix A.2 for the effect of over- and under-estimations of the PSF, which again have a772.5. Discussionsmall effect on IHYPR4D image quality.Additional computational time over HYPR3DHYPR3D processing with a 2x PSF kernel of a 16-frame HRRT dataset on a Lenovo workstationusing an Intel Xeon E-2146G 3.5 GHz Processor with 64 GB of RAM took ∼ 0.5 seconds perframe. In comparison, the first IHYPR4D iteration takes ∼ 4 seconds per frame (with a 2x PSF,four frame kernel on the same computing platform) and subsequent iterations ∼ 2 seconds perframe; the input reconstructed image and the composite are filtered in the first iteration, butonly the updated composite is filtered in subsequent iterations. The time to generate an MRIsegmentation for the IHYPR4D initial composite was generally several hours using Freesurfersoftware, though MRI segmentations are often required for subsequent image analysis pipelines.By contrast, the k-means segmentation method generates the initial composite in under 10minutes.78Chapter 3Development of an advanceddopamine release detectionalgorithm3.1 MotivationUntil recently, task-based neuroimaging—the detection of brain changes in response to a taskpresented to the subject during scanning—has been mostly confined to functional MRI (fMRI),which indirectly measures neuronal activation through a transient increase in blood oxygena-tion [58]. However, improvements in SNR of PET data and newer modeling approaches haveenabled the detection of transient molecular changes in response to tasks, such task-based neu-rotransmitter release [49, 126, 127] or synthesis [128], as well as changes in glucose metabolismas certain brain regions increase their energy consumption during a task [5, 115, 116].In this chapter we focus on the challenging problem of detecting relatively localized andlow amplitude dopamine (DA) release using PET. To date, PET-based transient DA releasestudies have focused on tasks with relatively large PET signal changes, such as those inducedby cigarette smoking [49, 126]. Developing a detection algorithm that can reliably detect loweramplitude changes will enable investigations into motor, cognitive, and reward-based tasks thatalso induce DA release. Furthermore, comparisons of the location and magnitude of DA releasebetween healthy and diseased subjects may provide insights into the pathophysiology of disease.For this to be possible and—more importantly—to have the best chance at detecting subtledifferences in the localization and amplitude of DA release between groups, (i) the PET dataneed to be thoroughly denoised, and (ii) the analysis framework used to identify release needsto have high detection sensitivity.These two requirements are intertwined: sufficiently denoised data will improve the detectionsensitivity of any analysis framework, as DA-induced changes to the temporal PET signal will beeasier to distinguish from noise-induced signal changes. For requirement (i), we have detaileda denoising algorithm in the previous chapter, IHYPR4D, that greatly reduces spatial and793.1. Motivationtemporal noise. This chapter thus aims to satisfy (ii) by developing and testing an improvedDA detection framework on simulated data, denoised with IHYPR4D. Simulated data is usedto quantify the improvements in detection sensitivity over previous methods.3.1.1 Physiological backgroundTo date, the most commonly employed mechanism for detection of neurotransmitter releaseusing PET relies on competition for binding between endogenous neurotransmitter and the ra-diotracer [129]; during release, a localized time-varying increase in endogenous neurotransmitterconcentration results in a decrease in radiotracer concentration in that region—ultimately caus-ing a decrease in the PET TAC.In theory, any neurotransmitter can be investigated in this manner, provided a competingradiotracer exists. However, the majority of research thus far has focused on DA, due to theimportant role DA plays in healthy brain functioning and to the favourable binding propertiesof the relatively commonly available radiotracer [11C]raclopride (RAC), a D2/D3 receptor an-tagonist. RAC DA detection is thus mostly sensitive to changes in synaptic DA concentration.Detection frameworks mostly confine their search to the D2-rich regions of the striatum wherethe SNR of RAC images is adequate for reliable detection. DA is involved in motor function,executive function, and reward [130–132], making it a useful research target for neurodegen-erative diseases, neuropsychiatric disorders, and neuropsychological tasks in the healthy anddiseased brain.In its earliest form, this competition mechanism was exploited in a double scan protocol(double RAC), where one scan occurred with the subject in the baseline condition and a secondscan occurred with a sustained stimulus evoking DA release [133]. The change in bindingpotential estimate from the baseline to stimulus scan is a proxy for the evaluation of themagnitude of DA release.However, such a protocol has several disadvantages: scan-to-scan variability in the bindingpotential confounds stimulus-evoked changes; increased commitment and radiation dose is re-quired for study participants; and the release magnitude is assumed to be constant over timefor analysis purposes, which may not be true. More advanced double RAC analysis methodsleveraged the residual behaviour between the baseline and activated scans by use of singularvalue decomposition in order to accurately determine DA release at varied task timings [134],however this method is practically limited since N + 1 PET scans are required to analyze Ntasks [135].803.1. Motivation3.1.2 Transient release modelingMore recently, several competition models have been suggested to quantify the transient kineticsof neurotransmitter release [47, 136, 137]. With transient displacement modeling, scan protocolscan be reduced to a single scan: the subject is scanned in baseline condition during a first partof the scan that includes radiotracer uptake, then the stimulus is presented later in the scan toobserve and quantify any transient responses.Such models use compartmental modeling as discussed in Section 1.4.3, where separatebiological compartments model endogenous neurotransmitter and ligand concentrations. Fromthe rate constants describing the radiotracer influx and efflux of these compartments, a time-varying profile of the release magnitude can be estimated. This is usually expressed as apercentage change in ligand binding potential rather than neurotransmitter concentration.Stimulus-evoked endogenous DA changes often occur over second or sub-second intervals[138]. However, the clearance rate of RAC is on the order of minutes, meaning the RAC responseto a change in DA is a temporally broadened version of the release response. Mathematically,this is essentially a convolution of the release response with a diffuse kernel, i.e., the highfrequency endogenous changes are mapped to low frequency RAC changes. This is similar—conceptually—to the hemodynamic response in fMRI: neuronal activation occurs on the order ofmilliseconds, but the measured hemodynamic response via the Blood Oxygen Level Dependent(BOLD) signal evolves on the order of seconds [139, 140].3.1.3 Transient release detectionWith a transient model fit to TACs from a single mid-scan-stimulus acquisition, detection nowaims to determine whether the estimated contribution from endogenous DA change in the modelis statistically significant or not. For instance, the competition model of Section 1.4.3, the lin-ear parametric neurotransmitter PET (lp-ntPET) model [47], has a single term responsiblefor the contribution of endogenous DA, so one can test whether the coefficient of this term issignificantly different from zero. If so, this is indicative of DA release. Similarly (and morecommonly), the competition model is statistically compared to a model incapable of incorpo-rating DA competition with RAC, such as the Multilinear Reference Tissue Model (MRTM)[141] of Section 1.4.3, using an F-test [48, 49].If the location of DA release can be hypothesized, then TACs can be calculated by averagingover all voxels of a particular region of interest (ROI). However, such an approach is likely toaverage over voxels corresponding to regions where release does not occur and/or omit voxelsoutside of the defined ROI where true release may occur. Therefore, it is beneficial to adopta voxelwise approach, where significance testing over all voxels is performed alongside multiple813.1. Motivationcomparisons correction to yield data-defined clusters of DA release. The main disadvantage ofthis approach is the high level of statistical noise often present in voxel-level TACs, which—without advanced denoising—often exceeds the magnitude of stimulus-evoked responses in thePET signal.An initial voxelwise model comparison framework [48, 49] employed a voxelwise F-test be-tween MRTM and lp-ntPET, which are shown below for reference:MRTM: CT (t) = R1CR(t) + k2∫ t0CR(u)du− k2a∫ t0CT (u)du (3.1)lp-ntPET: CT (t) = R1CR(t) + k2∫ t0CR(u)du− k2a∫ t0CT (u)du− γ∫ t0CT (u)h(u)du(3.2)where CT (t) is the target region tissue concentration; CR(t) is the reference region tissue con-centration; h(t) is the pre-specified shape of the TAC response to DA release, characterized bythe onset of release, tD, peak of release, tP , and sharpness of release, α:h(t) = θ(t− tD)(t− tDtP − tD)αexp[−α(t− tPtP − tD)](3.3)where θ(t) is the Heaviside function. The coefficients of each term of MRTM/lp-ntPET aredetermined through linear regression. HYPR3D denoising [24] (discussed in Chapter 2) wasapplied to the data, which has been shown to improve detection sensitivity by decreasingtemporal noise levels [48]. Applying this methodology to cigarette smoking, spatiotemporaldifferences in the release profiles of men and women have been detected [126]. A separate studywas able to map out consistent spatiotemporal patterns of release in response to cannabissmoking [127].More recently, direct reconstruction with lp-ntPET [114], a machine learning denoisingapproach using a feed-forward perceptron neural network [142], a machine learning detectionmethod that trains personalized neural networks on simulated lp-ntPET data [28], a MonteCarlo extension to the F-test approach [143], and a Bayesian framework for lp-ntPET parameterestimation [144] have been developed in an effort to increase the accuracy and reliability ofdetection. At the core, though, all of these methods rely on the accuracy of a model—namelylp-ntPET—and a reliable fit of this model to the data.823.1. Motivation3.1.4 Disadvantages of model-based detectionThe lp-ntPETmodel incorporates a family of basis functions by discretely modulating {tD, tP , α}in h(t), which describe physiologically-plausible release timecourses. Thus, the number of de-grees of freedom used to parametrize the F-distribution in significance testing needs to bemodified based on the number of basis functions chosen; failing to do so will alter the expectedfalse positive rate [145]. Additionally, the F-test itself assumes that the data are normallydistributed, but image noise in PET is best described by a gamma distribution rather than anormal distribution [146]. Furthermore, detection of release is greatly improved by significantpost-processing [48], which presumably further violates the normality assumption of the F-test.While the model comparison approach has produced high detection sensitivity in simula-tions of relatively high levels of DA release expected in pharmaceutical challenges, such ascigarette smoking [48, 49], model comparison in simulations of lower DA release levels—whichtypically are present in motor, cognitive, or neuropsychological tasks [128, 138, 147]—provedless successful [143]. The major goals of this work are thus twofold:1. Avoid the use of direct kinetic modeling for the detection of DA release;2. Improve detection sensitivity for lower levels of DA release.Kinetic models like lp-ntPET were initially developed to more accurately quantify release[47] and not specifically to be embedded within a detection framework. Thus, we hypothesizedecoupling kinetic modeling from detection will increase the reliability and localization of DArelease, after which lp-ntPET or a similar model may be used to accurately quantify the releaseprofiles at these locations. As a consequence of this decoupling, a lack of kinetic modelingavoids the complications of parameterizing multiple tasks within an experiment. Finally, if ourhypothesis that the decoupling improves detection sensitivity—especially for lower levels of DArelease—PET has the potential to become a meaningful tool for the investigation of short-livedfluctuations of DA in response to motor, cognitive, or neuropsychological tasks in the healthyand diseased brain.3.1.5 Isolating DA-induced perturbations via residuals analysisWhile we have noted a conceptual similarity between fMRI and RAC-PET in the mapping ofphysiological processes to observable temporal signals, major differences lie in the nature of thesignal and how it is used for detection and analysis. In fMRI, the baseline hemodynamic statecan be estimated with one or more BOLD images acquired while the subject lies at rest in thescanner. Then, upon commencing a task, the effect of the task can be directly measured by833.1. Motivationtaking the percentage difference in the BOLD signal between the rest and task periods (ac-counting for hemodynamic lag), and voxelwise activation can be detected by comparing simpletask-specific predictors with the time-varying BOLD signal in a general linear model (GLM).This is possible because the hemodynamic signal reaches steady-state during rest periods.In PET, however, the radiotracer concentration is never in steady-state during a bolusinjection protocol1 due to plasma clearance of the radiotracer. Attempts to reach steady-stateconcentrations where plasma clearance is balanced with a constant infusion of radiotracer afterbolus (i.e., bolus plus constant infusion protocols) are technically difficult and often rely onpre-specifying the infusion rate to maintain radiotracer concentration steady-state based onpopulation averages—introducing the possibility of not actually being in steady-state for agiven individual subject [148]. Thus, often the changes to the RAC signal due to DA releaseare confounded by the baseline pharmacokinetics of the radiotracer. As a result, DA releasedetection in PET must either follow the aforementioned double RAC protocol—one scan forbaseline binding potential estimation and a second for binding potential estimation during thetask—or the transient release protocol, where the task period is placed late in the scan suchthat modeling approaches can (ideally) accurately and reliably estimate baseline binding andtransient changes in response to the task from the entire TAC.As an initial effort to decouple kinetic modeling from detection, we propose to shift detectionto a residuals space where DA-induced perturbations to the TAC are isolated. The observedTAC is subtracted from a hypothetical baseline TAC to generate voxelwise residuals. Beforethe task, the subject is in baseline condition, so in theory the residuals during this time shouldbe solely due to signal noise, i.e. unstructured with zero mean. Any change to the observedTAC during the task will then appear as structured deviations in residuals space. This mappingto the residuals space thus removes the confound of baseline pharmacokinetics and provides aconstant signal during baseline, meaning model comparison is no longer required and simplerdetection methods—similar to those used in fMRI—may be applied. We hypothesize that thisisolation of DA-induced perturbations will increase detection sensitivity through the removalof this confound and the aforementioned disadvantages of the model comparison approach.Due to high levels of noise in voxel-level TACs it is necessary to apply advanced denoising tothe data before applying subsequent analyses on voxel residuals. To best characterize the voxelresiduals we rely on the novel denoising approach detailed in Chapter 2, IHYPR4D [149], whichprovides improvements in joint noise and bias reduction and inherently bypasses limitationsimposed by HYPR3D—the denoising approach used in current frameworks [48, 49].This chapter proceeds as follows. We first outline how to calculate the residuals from1A bolus injection protocol injects the entire dose of radiotracer into the subject over a short period of time,usually a minute or less. Bolus plus contast infusion protocols instead inject a fraction of the entire radiotracerdose at the beginning of the scan, then inject the remaining fraction continuously throughout the scan.843.2. TheoryFigure 3.1: Simulated noisy voxel TACs with different reconstruction and denoising approaches. Gaus-sian noise is added to simulated two-task release ground truth voxel TACs, with noise levels matchingreal noise measured on the GE SIGNA PET/MR for each reconstruction protocol. Denoising is per-formed on noisy TACs simulated with PSF-HYPR4D-K-TOFOSEM [39]. Results in this chapter usePSF-HYPR4D-K-TOFOSEM+IHYPR4D [39, 149] (right panel).acquired data in the context of a DA release experiment. We then propose a detection methodin residual space inspired by fMRI: using a simpler GLM rather than kinetic modeling in imagespace. Collectively, we refer to this process as Residual Space Detection (RSD). We thentest and optimize RSD on simulated single-task DA release experiments under the conditionsof homogeneous and heterogeneous RAC binding, comparing the results with the traditionallp-ntPET model comparison method.3.2 Theory3.2.1 Transforming acquired data to a residuals spaceThe first step of this work is to define a hypothetical baseline TAC for each voxel that is expectedto be observed in the absence of DA release. DA release in response to non-pharmaceuticalinterventions is likely to be localized, low amplitude, and short-lived, meaning that a significantfraction of striatal voxel TACs are unlikely to contain DA-induced perturbations. Therefore,it is reasonable to assume that the mean striatal TAC appears mostly devoid of DA-inducedperturbations and can approximate baseline conditions.Under the assumption that the mean striatal and measured voxel TACs behave very similarlyprior to a DA release-inducing task, a first attempt to define the voxel-level baseline TAC is touse the mean striatal TAC and fit it to a given voxel TAC for timepoints prior to any tasks;to attain a full scan-length voxel-level baseline TAC, the weight determined from this pre-task853.2. Theoryfit can be applied to the entire mean striatal TAC, hence extrapolating baseline behaviour tolater timepoints. Taking the difference between this extrapolated fit and the measured voxelTAC yields structured residuals in voxels where DA release is present.Two immediate issues arise when attempting this protocol: (i) TACs of voxels near theboundaries of the striatum have poor fits to the mean striatum TAC due to partial volumeeffects2, and (ii) in the presence of heterogeneous radiotracer binding within the striatum,defining a single mean striatal TAC as a regressor will result in biased fits and extrapolationsfor voxels that have higher or lower non-displaceable binding potentials (BPND) than average.The first issue is remedied by extracting the mean TAC from a reference region devoid ofspecific binding, in the case of raclopride using the cerebellum, and adding this as a regressorto the pre-task fit. The second issue may not arise in general as the density of D2 receptorsin the healthy striatum is relatively uniform [150], but this is not always true—particularly forconditions affecting the striatum such as Parkinson’s disease (PD, see Figure 3.2). To resolvethis, we propose applying k-means clustering to pre-task voxel TACs, normalized to their peakvalue, with k = 3 to produce three spatial regions representing relatively low, mid, and highBPND. Mean TACs for each cluster can then be defined as regressors in a non-negative leastsquares (NNLS) fitting framework for each voxel TAC (Figure 3.3), assuming a sufficiently largenumber of voxels are found in each k-region.Figure 3.2: Activity concentration of a single RAC frame (25 minutes post-injection) for a healthy con-trol (HC) and Parkinson’s disease (PD) subject. These images were reconstructed with PSF-HYPR4D-K-TOFOSEM [39].2The partial volume effect is an apparent loss of activity in relatively small structures due to limited scannerresolution. In our scenario of high activity within striatal tissue surrounded by tissue with lower activity, voxelsnear the edge of the striatum will have lower measured activity than central striatal voxels (assuming the trueactivity is uniform throughout the striatum), since the lower activity of the surrounding tissue appears to “spillinginto” the striatum during the imaging process.863.2. TheoryFigure 3.3: k-means results for noisy simulated data with an anterior-posterior/superior-inferior Bmaxgradient. (a) Concentration of a simulated noise-free PET frame (frame 10, t = 25 min) displaying theeffect of a simulated Bmax gradient in the striatum. (b) Spatial k-regions found by running k-meanson pre-task normalized TACs from a noisy simulation. Normalized mean TACs of each k-region aredisplayed on the plot, demonstrating spatially-varied binding characteristics. These TACs, along withthe mean cerebellum TAC, are used to define the voxelwise set of baseline TACs. Voxel and TAC colorsare consistent.Before the commencement of a DA release-inducing task, a striatal voxel TAC, Ci(t), is thusapproximated asCi(t) ≈ Cˆi(t) =3∑k=1wS,kCS,k(t) + wRCR(t); t < tS , wS,k, wR > 0 (3.4)i.e., Cˆi(t) is the hypothetical baseline TAC, determined by performing NNLS fitting to the ithmeasured voxel TAC prior to the task start time tS . The NNLS regressors are {CS,k(t) : k =1, 2, 3}, the set of mean TACs over the three k-regions, and CR(t), the mean reference regionTAC, here used to account for spill-in of non-target-related activity from surrounding tissue.This assumes that the baseline behaviour of each voxel TAC is adequately described by a linearcombination of the mean k-region TACs and the reference region.The corresponding regressor weights are wS,k and wR. These weights, obtained from apply-ing NNLS over t < tS , are applied to their regressors defined over the entire scan duration toextrapolate Cˆi(t) ∀ t. The residuals are then defined asRi(t) = 100%×(1− Ci(t)Cˆi(t)); ∀ t. (3.5)Note, by this construction, a negative DA-induced perturbation in the voxel TAC post-intervention, Ci(t > tS) manifests as positively structured residuals (Figure 3.4a).873.2. Theory3.2.2 Residual Space Detection (RSD)Transforming the PET TACs to residual space allows for analysis and subsequent detectionof voxel TACs that deviate from the extrapolated mean baseline behavior as a consequenceof task-evoked response. The basis of RSD is to compare residuals to a predictor, P (t), thatapproximates an expected residual response to the task (Figure 3.4b) We initially define P (t)based on the assumed shape of DA release in response to a block task design; however, to avoidpotential bias in results introduced by an incorrect initial assumption, we iteratively re-defineP (t) using both the data and the results from the first iteration.Inspired by fMRI detection approaches—where task events/blocks are convolved with ahemodynamic response function to yield predictors—we first define the predictor as the convo-lution between an event vector, E(t), and a gamma variate response function3 H(t)P (t) = λE(t) ∗H(t)= λ [θ(t− tS)− θ(t− tF )] ∗[tLexp(− tL)]; maxλ(P ) = 1(3.6)where θ(t) is the Heaviside function, tS is again the delay time indicating the start of thetask relative to the scan start time, tF is the task end time (tS < tF ), λ is a normalizationconstant, and L is a parameter governing the broadness of the gamma variate response function.Empirically, we found that a gamma variate function with L = 10 min reasonably matched theshape of residuals derived from noiseless simulations.We are now in an analysis paradigm similar to that of fMRI: in fMRI we have normalizedBOLD signal changes and predicted hemodynamic responses, which are compared in a voxelwisefashion using GLM. Therefore, we adopt a similar analysis strategy in RSD by comparingvoxelwise residuals to predictors using GLM:Ri(t) = βiP (t) + ϵi(t) (3.7)where Ri (1 x time) are the residuals for voxel i, P (1 x time) is the predictor [initially con-structed with equation (3.6)], βi is a weight from regression which indicates the maximumpercentage signal change observed (enforced by the residuals definition and normalization con-dition on P (t)), and ϵi contains error in the fit, ideally satisfying typical GLM requirements ofmean(ϵi) = 0, var(ϵi) = σ2.3While conventional PET modeling could potentially be used to estimate the response function, we opt forthe strategy of a simple a priori estimate followed by a data-driven, iterative refinement of the predictor (seethe paragraphs below).883.2. TheoryFigure 3.4: Summary of the Residual Space Detection (RSD) method. (a) Voxelwise residuals areextracted by taking the percent difference between voxel TACs and their predicted baseline TAC [equa-tions (3.4) and (3.5)]. (b) A predicted residual response is defined [equation (3.6)] and GLM is computed,assigning beta weights to each voxel. (c) A single iterative update is performed in which beta valuesfrom the first iteration define new baseline TAC regressors and residual predictors. After iterating, betamaps are thresholded to achieve a desired false positive rate. (d) In-plane edge voxels are removed frombinary maps due to partial volume effects, and a cluster size threshold is applied.893.3. MethodsFigure 3.5: Testing the convergence of RSD’s iterative outputs. Mean beta values across ground truthclusters (described in the main text) are shown vs iteration number, in which residuals and predictorsare updated as a function of the previous iteration’s beta values. Convergence is reached after just oneiteration.Finally, to increase accuracy in our results, we shift the method to being more data-driven byiteratively (i) redefining predictors to reduce bias introduced by an incorrectly assumed responsefunction, and (ii) redefining baseline TACs to remove contaminant release signals (Figure 3.4c).Baseline TACs, and thus residuals, are redefined following an initial determination of β: voxelsin the top 20% of beta values are temporarily removed from k-regions and the mean TACs arecomputed, resulting in new baseline TACs C′s,k(t) that contain less DA-release related signal,helping to amplify detection of low-magnitude signals. Data-driven predictors are then definedusing β values: voxel residuals are normalized to a peak value of 1, weighted by their βi value,and averaged. The result is then renormalized to a peak of 1, providing a new predictor P ′(t).From simulations we observed that only one iteration of this protocol is required to achieveconvergent predictions of β values in simulated regions of release and no release (Figure 3.5).3.3 Methods3.3.1 Simulation designTo test our method in comparison to the traditional F-test methods, we simulate 70-minutesingle-task and 79-minute two-task RAC scans on the GE SIGNA PET/MR system (hereafterSIGNA; see Section 2.1.2 for a previous discussion of this scanner). A T1 MRI from a healthyhuman subject scanned on the SIGNA was segmented using Freesurfer [117] to generate ananatomical reference for simulating regions with pre-determined RAC kinetics, which was thenresampled to the PET voxel dimensions (1.39 x 1.39 x 2.78 mm) using nearest-neighbours903.3. Methodsinterpolation. The striatum was defined using bilateral ROIs from the putamen, caudate, andnucleus accumbens.Noiseless TACs are simulated following the complete ntPET model, which accounts for con-centration exchange between the free and bound compartments for both RAC and endogenousDA ([137]):CT (t) = SA [F (t) +B(t)] (3.8)dFdt= K1Cp(t)− k2F (t)− kon[Bmax −B(t)−BDA(t)]F (t) (3.9)dBdt= kon[Bmax −B(t)−BDA(t)]F (t)− koffB(t) (3.10)where F (t) and B(t) are the free and bound molar concentrations of RAC, solved by numericallyintegrating their differential equations above, and SA is the specific activity that converts themolar RAC concentrations to radioactivity concentration units (kBq/mL). The arterial inputfunction, Cp(t), is modeled as in [119] to match concentration levels observed in 15 mCi RACinjections on the GE SIGNA PET/MR scanner at our centre. BDA(t) is the bound DA, whichis found by numerically integrating the following differential equation:dBDAdt= kDAon[Bmax −B(t)−BDA(t)]FDA(t)− kDAoffBDA(t) (3.11)where FDA(t) is the transient endogenous DA molar concentration. For non-release voxels,FDA(t) is a constant 100 pmol/mL to reflect basal DA concentration. For release voxels,FDA(t) starts at this basal concentration, then at t = tS a pre-specified release timecourseis added as a result of the task (see Appendix B for details). Realistic values for the kineticparameters {K1, k2, kon, koff , kDAon , kDAoff} are taken from [151] to allow a numerical solution ofthe above differential equations4.DA release clusters were placed in the left caudate (cluster size: 118 voxels), in the rightputamen (160 voxels), and in the right caudate (104 voxels) to simulate a possible response toa single, 10-minute functional task block starting at 32 minutes, involving motor and cognitiveprocessing (Figure 3.4b). The peak simulated DA release magnitude was set to 120% increasefrom basal DA concentration; this is relatively low compared to previous simulations whichsimulated peak DA increases of 200% or more relative to baseline [48, 49].To ensure our method applies more generally, we then performed identical simulations in4BDA(ti → ti+1) is first solved since the exchange rate of DA is much faster than RAC, followed by F (ti →ti+1) then B(ti → ti+1).913.3. Methodsthe presence of spatially heterogeneous RAC binding to assess the performance of the k-meansmethodology. Motivated by observed binding heterogeneities in subjects with PD (Figure 3.2)we simulate a gradient in the binding potential (BPND), accomplished by modulating Bmax ∝BPND (see equations 1.40 and 1.41). The gradient is designed to approximate the direction andmagnitude of observed concentration gradients in human data. The simulated Bmax rangedfrom 70 pmol/mL in the superior and posterior regions of the striatum, to 30 pmol/mL in theanterior and inferior regions, with a slope of -30◦ in the anterior direction. In the homogeneousbinding simulation, a constant striatal value of Bmax = 44 pmol/ml is used.The TACs were framed 4 x 1, 3 x 2, 4 x 5 minutes for estimation of the baseline por-tion of the scan, then 20 x 2 minutes for higher temporal sampling during DA release, andfinally 1 x 4, and 2 x 5 minutes at the end of the scan for better counting statistics. Thedynamic images were then spatially filtered with a 1.8 x 1.8 x 3.6 mm FWHM Gaussian tosimulate the SIGNA PSF reconstruction resolution [107]. We generated 100 noisy realizations(NRs) using scaled Gaussian noise [49], before post-processing with IHYPR4D (7.2 mm, twoframes spatiotemporal kernel, one iteration); the scaling coefficient was set so that the noiselevel after applying IHYPR4D matched human RAC scan noise levels reconstructed with PSF-HYPR4D-K-TOFOSEM [39] (5.6 mm, four frames spatiotemporal kernel, 10 iterations) andpost-processed with IHYPR4D.3.3.2 Implementation of RSDRSD was performed as described in Section 3.2.2, and beta maps for all noisy realizations werevariably thresholded at a common value across NRs to obtain cluster-level receiver operatingcharacteristic (ROC) curves. From these curves, a critical beta value corresponding to a 5% falsepositive rate (FPR) was identified. Binary classification maps were generated by thresholdingat this critical beta value, i.e. replacing all voxels with beta exceeding the critical beta valuewith 1, then replacing all other voxels with 0. In-plane edge voxels were then removed frombinary maps to minimize false positives caused by partial volume effects.3.3.3 Cluster size thresholdingTo control for the multiple comparisons problem, clusters of predicted DA release voxels in thebinary maps are subject to a cluster size threshold (CST), where clusters smaller than the CSTare rejected as DA release clusters [48]. In our simulations the left putamen was kept devoidof release (Figure 3.4b), therefore we used this region to define the CST. The CST was setfor each simulated dataset and methodology independently and defined so that 90% of noisyrealizations resulted in the left putamen being completely free of false positives.923.3. Methods3.3.4 Comparison to model-based F-test methodWe compare our proposed RSD method to the existing model-based F-test method as describedin [48]. Both lp-ntPET and MRTM [equation (3.2)] were fit to each IHYPR4D-processed voxelTAC in the striatum. Parameters for lp-ntPET basis functions [equation (3.3)] were discretizedas follows: (1) tD ∈ [tS − 5min, tS + 5min] in one-minute intervals; (2) tP ∈ [tD + 1min, tF ] inone-minute intervals; (3) α ∈ {0.25, 1, 4}.Weighted residual sum of squares (WRSS) was calculated for lp-ntPET and MRTM fits,which permitted the calculation of the F-statistic and thus the generation of parametric F-maps. Like in RSD, thresholding the F-maps from all noisy realizations at a varying commonF-value yields ROC curves that can be compared to those of RSD. From here, a critical F-valuecorresponding to 5% FPR was determined and used to generate binary classification maps,which then went through the same in-plane edge voxel removal and CST processing as RSD.3.3.5 Performance metricsTo determine the performance of our method in comparison with the model-based F-test methodwe used two sensitivity measures as defined in [48]:1. Voxelwise sensitivity (VWS): for a given true positive voxel, this is the percentage ofnoisy realizations in which the voxel was correctly classified as a release voxel after CSTwas applied. Mean voxelwise sensitivity is defined as the mean of this quantity across allvoxels within regions where DA release was simulated. This measure can be interpretedas the mean true positive rate, and spatial maps of voxelwise sensitivity were used tocompare the ability of a given method to determine the accurate shape and extent ofrelease within a region for a single scan.2. Binary sensitivity (BS): for a given true positive cluster, this is the percentage of noisyrealizations in which at least a subset of the true release voxels in the cluster survivedthe CST. This is an important metric for scanning studies; if release is present, we wouldwant binary sensitivity as close to 100% as possible such that if release is present, we havea very high likelihood of detecting it, though the spatial extent of the release may not beprecisely characterized.933.4. Results3.4 Results3.4.1 Release in the presence of homogeneous bindingFigure 3.6 displays voxelwise prediction maps aggregated from binary classification maps (thresh-olded at 5% FPR) across 100 NRs for the lp-ntPET + F-test method vs. the proposed RSDmethod, in addition to ROC curves for the bilateral caudate clusters and right putamen cluster.CST was determined for each method from these binary maps and was subsequently applied tothe individual NR binary maps; the resulting voxelwise sensitivity values after CST applicationare given in Table 3.1.As seen in the ROC curves, RSD drastically outperforms lp-ntPET across all FPRs forthis low-magnitude single-task release (∼ 120 pmol/ml peak increase). Binary sensitivity (BS)for lp-ntPET was 79% and 80% bilateral caudate and putamen clusters, respectively, afterapplication of CST; RSD had a BS of 99% for all clusters after CST.Figure 3.6: Left: Simulated ground truth clusters are placed bilaterally in the caudate and in theright putamen. Center: Voxelwise prediction maps for the lpnt-PET + F-test method and RSD methodapplied to single-task simulations of DA release in the presence of homogeneous binding. Color barrepresents percentage of noisy realizations in which the voxel was predicted as release, determined using100 noisy realizations described in Methods. Right: ROC curves are shown for each type of cluster forthe lp-ntPET + F-test method and RSD, prior to CST application.3.4.2 Release in the presence of heterogeneous bindingResults for the heterogeneous binding simulations as measured by the ROC curve are verysimilar to those observed in the homogeneous binding case (Figure 3.7). Table 3.1 summarizes943.4. Resultsthe heterogeneous binding simulation results post-CST application. Sensitivities of RSD aresimilar between simulations, indicating that RSD is effective at accounting for the simulatedgradient in Bmax by use of its k-means step without altering detection performance.Shown in Figure 3.8 are the weights from NNLS regression of pre-task baseline TACs to eachstriatal voxel TAC for a single noisy realization of heterogeneous single task release data. Theweights behave as expected, with maximal amplitude about the nearest k-region. The weightaccounting for spill-in, wR, modelled using the cerebellum TAC, is relatively low and specificto edge voxels, as expected.Figure 3.7: Left: Simulated ground truth clusters are placed bilaterally in the caudate and in theright putamen. Center: Voxelwise prediction maps for the lpnt-PET + F-test method and RSD methodapplied to single-task simulations of DA release in the presence of heterogenous binding. The colour barrepresents percentage of noisy realizations in which the voxel was predicted as release, determined using100 noisy realizations described in the Methods. Right: ROC curves are shown for each type of clusterfor the lp-ntPET + F-test method and RSD, prior to CST application.Simulationlp-ntPET + F-test RSDCSTBilateral RightCSTBilateral Rightcaudate putamen caudate putamenMeanBSMeanBSMeanBSMeanBSVWS VWS VWS VWSHomogenous 30 24.4% 79% 37.8% 80% 30 51.0% 99% 77.0% 99%Heterogeneous 49 22.8% 67% 39.5% 72% 32 54.4% 99% 78.6% 99%Table 3.1: Comparison between RSD and lp-ntPET + F-test, with sensitivities and CST for bothsimulation sets. Binary maps were produced from beta (RSD) and F-statistic (lp-ntPET) maps bythresholding at 5% FPR over 100 noisy realizations to determine the CST. Mean VWS was then deter-mined after application of the CST. CST: cluster size threshold, VWS: voxelwise sensitivity, FPR: falsepositive rate.953.4. ResultsFigure 3.8: Weights from NNLS regression of pre-task basis functions (3 mean k-region TACs and 1cerebellum TAC) to each striatal voxel TAC [from equation (3.4)] for a single noisy realization. K-regionlocations are shown for reference.3.4.3 Effect of not accounting for heterogeneous binding in RSDIn the presence of heterogeneous binding, by simply using the mean striatum TAC and meancerebellum TAC as regressors when defining voxel residuals, one encounters severely biasedpredictions of DA release. In particular, lower-than-average binding regions without releasewill have post-task activity that is less than the mean striatal TAC, leading to positively biasedresiduals. As a result, the residuals are structured despite no release being present, leadingto false positive detection. This phenomenon is displayed in Figure 3.9; in our simulatedheterogeneous binding pattern the anterior striatum contains lower binding, so we observeelevated DA release prediction as we increase along this axis—regardless if release is presentor not. This demonstrates the need to model heterogeneities in the data using a method likek-means clustering.3.4.4 Comparing TAC modeling in lp-ntPET vs. RSDWe display a single baseline voxel from a noisy realization in Figure 3.10. Due to the aggressivedenoising of IHYPR4D, the denoised voxel TACs are relatively smooth which proves to begenerally beneficial to the performance of both lp-ntPET and RSD. However, lp-ntPET isprone to noise-induced bias present in voxel TAC shapes, as seen in Figure 3.10; a baselinevoxel is falsely classified as release because lp-ntPET is capable of better modeling the noise-induced shape alteration compared to MRTM as quantified by an F-test. Because RSD makes963.5. DiscussionFigure 3.9: Voxelwise sensitivity maps for the RSD in the absence of gradient modeling vs using k-means clustering. Sensitivities were determined using 100 noisy realizations described in Methods andMaterial. Orientation is in radiological convention.use of the higher SNR k-region TACs in defining each voxelwise baseline signal, the voxel isaccurately modeled pre-task and therefore classified correctly by GLM.3.5 DiscussionThis work presents a novel residual space detection framework (RSD) for PET-based DA re-lease detection, most advantageous for spatially localized and low-magnitude responses whichare inherently difficult to detect. Such release may be expected in motor, cognitive, and/orneuropsychological reward-based tasks rather than pharmaceutical interventions [128, 138, 147],where the spatial extent and magnitude of release is much greater. Moreover, the scan pro-tocol only requires a single bolus injection of RAC, providing a simpler means to study DArelease patterns. Given the increasing incidence of addiction and neurodegenerative diseases,which alter dopaminergic function, the ability to detect dopamine release in response to non-pharmacological stimuli is becoming increasingly more important.Our detection framework has two major differences compared to existing methods whichprimarily incorporate significance testing between models to detect release:1. Moving detection to a residual space. We leverage the condition of spatially lo-calized, low-magnitude release in behavioral tasks to construct voxelwise hypothetical973.5. DiscussionFigure 3.10: Visualization of results in a negative (non-release) voxel from a single noisy realization forthe homogeneous simulations. The plot shows a simulated noisy striatal voxel along with its denoisedTAC and fits from RSD (left) and lp-ntPET/MRTM (right).baseline TACs from regional averages of the striatum. Regressing these against actualvoxel TACs and computing residuals moves the analysis to a space where structuredtemporal signals are most likely indicative of DA release.2. Decoupling kinetic modeling from detection. The transformation to residual spaceremoves the effect of baseline kinetics; in TAC space both release and non-release TACshave structured signals, whereas only DA-release TACs should have structure in residualspace, so modeling the residuals with a kinetic model is unnecessary for detection. Rather,using GLM to identify similarities between task-informed predictors and the residuals issufficient.3.5.1 Detection sensitivity improvementsIn this work we simulated two separate single-task dopamine release datasets: one with ho-mogeneous RAC binding and another with heterogeneous RAC binding. In both simulations,RSD outperformed existing model comparison methods employing MRTM/lp-ntPET fittingalongside an F-test. RSD provides higher binary and voxelwise sensitivities for clusters of lowamplitude DA release when compared to lp-ntPET + F-test at equivalent false positive rates,exhibiting improved robustness to noise and a greater ability to distinguish release from non-release. Importantly, the binary sensitivity—a reflection of the likelihood of detecting a least aportion of true release for a given noisy realization—was 99% for RSD in all simulation types,compared to ∼ 75% for lp-ntPET + F-test, suggesting that in human studies the latter methodwould introduce a confound of “missed” release in certain subjects, whereas RSD would detecttrue release in virtually all subjects (if present).983.5. DiscussionIn these simulation studies, RSD has been applied to data denoised with IHYPR4D [149]to greatly reduce temporal noise in voxel TACs (see Figure 3.10). While we did not test thesemethods on non-denoised data, we can confidently claim that the denoising algorithms employedsignificantly aided DA release detection, as has been shown in previous work with HYPR-baseddenoising applied to voxelwise DA release detection [48].In the presence of positive noise spikes prior to task initiation and/or negative noise spikesduring the task, lp-ntPET is especially prone to predict release when it does not exist. In thesescenarios lp-ntPET overfits to the noise-altered TAC by tweaking any of its four free parameters,{R1, k2, k2a, γ} and three discrete parameters, {tD, tP , α}, simultaneously to achieve the best fitto a given voxel (Figure 3.10). Although RSD can still overfit to noise, it performs its analysisin two steps which acts to decouple several of its degrees of freedom. First, a voxel baseline TACis predicted from the k-means region TACs alongside the cerebellum TAC, dictated by four freeparameters, {ws,1, ws,2, ws,3, wR}, though for a given voxel only one of these parameters typicallycontribute to its baseline prediction (Figure 3.8). Lastly, GLM is performed on voxel residuals,governed initially by two fixed parameters defining predictors, {tD, L}, that are iterativelyreplaced by data-driven predictors, and one free parameter, β. In total, RSD effectively usestwo free parameters, {ws,k, β}, for a given voxel in region k, in a sequential manner, makingit less prone to overfitting. Bias is indeed observed when the baseline fit is erroneous (e.g.when drastic changes in binding occur within a k-region), however these instances are readilyidentified and can be accounted for by increasing the number of k-regions from k = 3, or bycomparison to the lp-ntPET result.3.5.2 Handling heterogenous RAC bindingWhen observing scans of patients with PD, it was clear that the binding homogeneity as-sumption was not always satisfied (Figure 3.2), and thus using a single mean striatal TAC asregressor in RSD was insufficient; when the mean striatal TAC is fit (pre-task) to a relativelylow-BP voxel TAC, the resulting baseline prediction will consistently overshoot the measuredvoxel TAC, resulting in large positive residuals post-task that are consistently predicted asDA release (Figure 3.9). To remedy this, we introduced a k-means step that supplies furtherbaseline regressors in a data-driven manner, which effectively resolved this issue. Results fromRSD for single- task simulations with heterogeneous binding showed high detection sensitivi-ties of simulated clusters with minimal bias introduced by the k-means processing method (seeTable 3.1).99Chapter 4Development of an advanced MRIdenoising algorithm4.1 MotivationThus far, we have focused on developments on denoising and analysis algorithms specifically forPET applications. As mentioned, the SNR of PET data is typically much poorer than MRI,leading to a greater demand for effective PET denoising operators in studies acquiring datafrom both modalities. As detailed in the previous chapter, PET can identify molecular changesin response a task presented to subjects during the scan, so in simultaneous PET/MRI studiesa natural MRI pairing would be fMRI, such that task-induced hemodynamic and molecularresponses can be concurrently investigated [152–155].In fMRI, the SNR is relatively low since whole-brain Blood Oxygen Level Dependent(BOLD) images need to be rapidly acquired to adequately sample the hemodynamic responseto neuronal activation—which occurs on the order of several seconds [139, 140]. Conversely, instructural MRI there is no physiological temporal constraint, so the acquisition time is usuallyseveral minutes and thus the SNR is generally much higher. More importantly, in fMRI thetemporal signal change in response to a task or stimulus can be quite low, so combining this withintrinsically high temporal noise means denoising is now necessary to obtain robust detection ofneuronal activation. In most task-based fMRI studies, spatial filtering alone can increase SNRsufficiently. Therefore, in this chapter we will focus on developing an MRI denoising operatorfor a particularly challenging application of fMRI: the dual-calibrated BOLD protocol detailedin Section 1.6.4.In theory, this protocol provides a method to estimate voxelwise cerebral metabolic rateof oxygen (CMRO2) from dual-echo pCASL data (alternating acquisition of ASL images, toquantify cerebral blood flow (CBF), then BOLD images) through a set a biophysical modelsquantifying the BOLD and CBF response to hypercapnia (breathing in air enriched with CO2)and hyperoxia (breathing in air enriched with O2) stimuli. However, in practice the SNR of theBOLD/ASL images is relatively poor, leading to unreliable voxelwise estimates after modeling1004.1. Motivation[79]. As a reminder, CMRO2 is quantified by:CMRO2 = CaO2|0 · CBF0 ·OEF (4.1)where CBF0 is CBF during baseline breathing of room air, OEF is the fraction of oxygenextracted during metabolism, derived from the BOLD/ASL response during the respiratorystimuli, and CaO2|0 is the arterial oxygen concentration estimated from respiratory measure-ments at baseline. CBF estimates during baseline, hypercapnia, and hyperoxia are especiallynoisy due to the pairwise subtraction of ASL label and control images, which acts as a highpass filter and ultimately amplifies noise. Nevertheless, these estimates on their own are robust1and are not directly to blame for unreliable CMRO2 data; rather, the propagation of noise inthese estimates through the equations calculating OEF leads to nonphysical values (OEF < 0or OEF > 1), which in turn yield nonphysical values of CMRO2.Combining equations 1.92, 1.97, 1.98, and 1.96 reveals the highly nonlinear dependence ofOEF on δS and f as measured during the hypercapnia and hyperoxia experiments:OEF = 1− ϕCaO2|0[Hb]−1ϕ(CaO2|ho − CaO2|0fho)+ [Hb](1fho− 1)1fho−(1fαho− δSho(1−fα−βhc)δShc·fαho)1/β . (4.2)In practice, fho is set be to constant across the brain. This means noise-induced pertur-bations in OEF estimates are primarily through noise in fhc, δShc, and δSho estimates. Usingthe realistic physiological values of fhc = 1.4, δShc = 0.03, and δSho = 0.01, we plot OEF asa function of each of these metrics, spanning a 50% underestimation to 50% overestimation,in Figure 4.1. The results demonstrate that noise-induced negative bias in fhc can lead to adrastic overestimate of OEF. Therefore, to enable reliable OEF and hence CMRO2 estimates,the SNR of the underlying BOLD and ASL data needs to be greatly improved.The SNR of the reconstructed data from the scanner is limited by a short acquisitiontime—required to capture temporal resolution of the seconds-long hemodynamic response—andscanner hardware, so further SNR increases must occur from post-processing denoising of thedata. BOLD and ASL data used for calculating CMRO2 is typically processed with high-passtemporal filtering (to remove low-frequency components unrelated to the calibration tasks, suchas scanner drift [158]) as well as low-pass spatial filtering—typically Gaussian smoothing—in1MRI-based CBF estimates are physiologically plausible and generally correlate with the gold standard PET-based CBF measurements using H2[15O], though absolute quantification in MRI is often biased from PET[156, 157].1014.2. TheoryFigure 4.1: The effect of under/overestimations of various metrics on calculated OEF.an attempt to increase SNR.Gaussian smoothing is quite limiting as it only performs a local weighted average of thedataset; in hyperoxia and hypercapnia experiments, the changes across the brain may be verysimilar for certain extended regions, such as across sections of cortical grey matter, deep greymatter, and white matter [79, 99]. Therefore, improved denoising may exist by leveragingthis observation and performing non-local weighted averaging between voxels having similartissue characteristics, similar to the non-local means algorithm employed in PET denoising(Section 1.3.2).This chapter thus focuses on developing an advanced MRI denoising operator to enablereliable voxelwise OEF quantification. This in turn would enable reliable, non-invasive quan-tification of CMRO2 using MRI rather than PET. As mentioned in Section 1.6.1, OEF and/orCMRO2 may be altered in stroke [62–65], tumour [66–68], neurodegeneration [69–71], or inhealthy aging [72, 73], making them potentially useful biomarkers of disease.The dataset for dual-calibrated BOLD is 4D; based on the success of extending the HYPRoperator to 4D in Chapter 2 for PET data, we similarly devise a 4D HYPR operator forprocessing dual-echo pCASL data used for calculating OEF. The theoretical construction ofthis operator can be easily adapted to other applications, though this is beyond the scope ofthis present work.4.2 TheoryInspired by recent advances in cluster-based averaging in CMRO2 quantification through QSM[89], as well as the approach of non-local means filtering algorithm [23], we propose a method1024.2. Theorythat employs non-local averaging of signals across the brain that have similar tissue propertiesrelevant to dual-calibrated BOLD quantification, namely M and OEF.We begin by mapping the dual-calibrated BOLD signal at each voxel, δS(x, t), into a 2Dnormalized feature space:δS(x, t) 7→ φ(x) =[z(δS(x)), z(δShc(x)δSho(x))](4.3)where z(·) implies taking the z-score across all voxels. The first dimension averages the relativeBOLD signal over all time, meaning the resulting value will be sensitive primarily to variation inM and flow changes during hypercapnia across the brain. In the healthy brain, this will separatewhite and grey matter. The second dimension divides the time-averaged relative BOLD signalin hypercapnia by that in hyperoxia. The division removes the effect of MδShc(x)δSho(x)=M(1− fα−βhc )M(1− fαhodβ)=1− fα−βhc1− fαhodβ(4.4)making this measure more sensitive to OEF. This can be seen by ignoring the relatively smallchange (if any) in flow during hyperoxiaδShc(x)δSho(x)≈ 1− fα−βhc1− dβ (4.5)and then because d ∼ 1 in hyperoxia2, we can Taylor expand to first orderδShc(x)δSho(x)≈ 1− fα−βhc−β(d− 1) . (4.6)Still assuming fho ≈ 1, we haved− 1 ≈ −CaO2|ho − CaO2|0ϕ · [dHb]0 (4.7)which demonstrates the second dimension is sensitive to [dHb]0 and hence OEF2Using equations (1.89) and (1.90) and assuming no flow changes during hyperoxia, if SaO2 is saturated to100% from a typical baseline of 98%, then this yields d ≈ 0.90–0.96 for reasonable values of OEF.1034.2. TheoryδShc(x)δSho(x)≈ [dHb]0 · ϕ(1− fα−βhc)β (CaO2|ho − CaO2|0) (4.8)This division of noisy signals can lead to a number of outlying values. To mitigate these, wefirst apply an initial spatial filtering of 6 mm to the BOLD and ASL images before computingthe non-z-scored feature space metrics. Then, local outliers are identified and replaced in eachdimension with a 2D Hampel filter [159] (see Appendix C for details). After these two filteringoperations, the z-score is taken over each dimension of the feature space.Within this feature space, we can perform k-means clustering to group voxels with similarMand OEF. The centroids of such clusters can then be used as “nodes” in the feature space fromwhich each voxel vector, φ(x) =[z(δS(x)), z(δShc(x)δSho(x))], is separated by a certain distance.Each voxel BOLD/ASL signal can then be increased in SNR by performing a weightedaverage of voxels with similar M and OEF, where the weights are guided by the separationbetween the voxel vector in feature space and the feature space centroids. Thus each denoisedvoxel signal Sd(x, t) is defined asSd(x, t) =1∑Nk=1wk(x)N∑k=1wk(x) · Sk(x, t) (4.9)where Sk(x, t) =1|xk|∑x∈xk S(x, t) is the signal averaged over the set of voxels belonging to thekth cluster, xk, and wk(x) are the corresponding weights defined by a Gaussian kernelwk(x) = exp(−||φ(x)− φk||22h2)(4.10)with φ(x) being the voxel vector in feature space, φk being the kth centroid in the featurespace, and h the standard deviation of the kernel. Thus, each weight falls between zero andone. Weights close to zero indicate the separation between the voxel vector and the centroidis large, whereas weights close to one indicate the separation is small; h controls how quicklythe weight approaches zero with increasing separation, with a larger h causing the weight toapproach zero more slowly.The set of Sk(x, t) can be thought of as basis functions which ideally span the range oftrue physiological signals observed in the subject. However, to ensure each basis function isphysiologically plausible, we compute OEF for each cluster using BOLDk(x, t) and ASLk(x, t)and remove any k-clusters from consideration that produce OEFk /∈ (0, 1).1044.2. TheoryConceptually, this framework of mapping the data a feature space and using distances in thefeature space to define basis functions shares some similarity to the kernel method of PET imagereconstruction (Section 1.3.3); the feature space is a lower dimensional, denoised representationof the original data, and thus can be used to effectively guide the construction of basis functions.Implicit in this formulation is that (unknown) true voxel signal is well-described by aweighted sum of the basis functions. Since this may not be true, we relax this assumptionusing 4D HYPR:SH(x, t) = C(x, t) · S(x, t) ∗ F (x, t)C(x, t) ∗ F (x, t) (4.11)where as always C(x, t) is the 4D composite and F (x, t) is a 4D Gaussian filter. In our case, wemay use C(x, t) = Sd(x, t), since Sd is a high SNR version of the original dataset, but may belocally biased since it has assumed each voxel signal can be replaced by a linear combination ofthe cluster-averaged regional signals, which may not always be true:SH(x, t) = Sd(x, t) · S(x, t) ∗ F (x, t)Sd(x, t) ∗ F (x, t) (4.12)Just like in the 4D implementation of HYPR in Chapter 2, i.e. IHYPR4D, in this applicationHYPR serves as a “relaxation” of the assumptions in Sd, allowing some local variation backinto the signal if Sd(x, t) ∗ F (x, t) is sufficiently different than S(x, t) ∗ F (x, t). The filteringof each signal before comparison (through division) means high-frequency noise and signal areremoved; thus, HYPR identifies any discrepancies in low frequency features, i.e. contrast,between Sd(x, t) and S(x, t) and issues a correction factor to Sd(x, t). On the other hand, ifSd(x, t) ∗ F (x, t) and S(x, t) ∗ F (x, t) are sufficiently similar,S(x, t) ∗ F (x, t)Sd(x, t) ∗ F (x, t) ≈ 1 =⇒ SH(x, t) ≈ Sd(x, t). (4.13)Collectively, we refer to this denoising method of 4D HYPR with a composite derived usinga feature space as HYPR4D-FS. Unlike in IHYPR4D, we do not iterate in HYPR4D-FS for tworeasons:1. While iteration may reduce bias further if there are discrepancies between the compositeand (unknown) ground truth data, iteration increases noise. As shown in Figure 4.1 thecomputation of OEF is highly sensitive to noise, so we prefer to maximize denoising ratherthan minimize bias.2. The construction of the HYPR4D-FS composite by combining basis functions means the1054.3. Methodsinitial composite is no longer relying on a uniformity assumption within spatial ROIs (likein IHYPR4D). As a result, the composite in HYPR4D-FS is more adaptive to nonunifor-mities within the data than the initial composite of IHYPR4D.4.3 Methods4.3.1 SimulationsWe assess the feasibility of the proposed denoising method first in simulations where the groundtruth is known. Three simulations with distinct spatial OEF distributions are used to test theability of the denoising method to preserve these distributions in the process of removing noise.The simulations are:1. Uniform OEF: Spatially uniform OEF = 0.4 is used to simulate a healthy brain.2. Uniform OEF with lesion: Spatially uniform OEF = 0.4 except for a 30 mm lesionwith OEF = 0.1. This simulates a stroke-type lesion where oxygen extraction has beenrestricted.3. Spatially-varying OEF: A spatial disease pattern [106] is scaled to create a 20% varia-tion in OEF centred at OEF = 0.4. This simulates a hypothetical disease-induced patternwhere certain brain regions have locally different OEF as a result of metabolic dysfunction.In each simulation, we simulate our centre’s dual-calibrated BOLD protocol of two two-minute +8 mmHg hypercapnia challenges and two four-minute +230 mmHg hyperoxia chal-lenges, each separated by baseline breathing periods for a total sequence duration of 20:56(Figure 4.2). The BOLD and ASL datasets are simulated based on an EPI dual-echo pseudo-continuous arterial spin labeling (pCASL) sequence where BOLD and ASL echoes are acquiredwithin a TR of 4.187 s, yielding 300 volumes for each dataset.BOLD and ASL datasets are generated for each simulation type using these above OEFdistributions as follows.Simulation tissue type segmentationThe 1 mm MNI152 template tissue probability maps of grey matter, white matter, and CSFwere used to generate a segmentation of the brain into grey matter, white matter, CSF, anddraining veins (Figure 4.3). Voxel tissue probabilities for grey (white) matter exceeding 0.51064.3. MethodsFigure 4.2: Design of the dual-calibrated BOLD protocol used at our centre, consisting of two hyper-capnia and hyperoxia challenges each.were assigned as grey (white) matter, whereas CSF tissue weights exceeding 0.6 were assignedas CSF. Voxels not assigned a tissue type based on these above classifications were then assignedto draining veins. In the case of the lesion simulation, a 30 mm region of interest was insertedin the right frontal area and was assigned to the lesion.Each unique tissue segmentation type was then uniformly filled with physiologically-plausibleASL/BOLD signals corresponding to that tissue type.Figure 4.3: High resolution atlas of tissue types used to place ASL/BOLD signals for each simulation.Note the lesion tissue type is only used for the lesion simulation.Simulating the BOLD signalThe Davis equation [83] was used to simulate the BOLD signal for voxel, i.e.BOLD(x, t)S0(x)= δS(x, t) + 1 =M(x) ·(1− f(x, t)α · d(x, t)β)+ 1. (4.14)S0(x) is the arbitrary signal scaling factor that is irrelevant for quantifying OEF, since δSis used instead of the absolute BOLD signal. To calculate the voxelwise BOLD signal, we then1074.3. Methodsneed to know the calibration constant M(x), relative CBF changes f(x, t), and relative [dHb]0changes d(x, t).M(x) will be spatially varying because of differences in blood volume fraction, v, in thedifferent tissue types, as well as any spatial variation in OEF and hence [dHb]0(x). The followingvalues were used for v: 0.035 for grey matter; 0.017 for white matter; 1 for draining veins; 0.0153for the lesion.Real human respiratory PETO2 data were used to calculate CaO2(t), which in combinationof our above OEF(x) maps allows one to compute [dHb]0(x)[dHb]0(x) = [Hb]− (1−OEF(x)) · CaO2|0ϕ(4.15)where we set [Hb] = 0.15 g Hb / mL blood. For OEF = 0.4, this yields [dHb]0 = 0.06. If we thenuse a typically-observed grey matter value of MGM := M(v = 0.035, [dHb]0 = 0.06) = 0.08,using the definition of MM ∝ v · [dHb]β0 (4.16)we can then obtainM for a given voxel with arbitrary v and [dHb]0 with the appropriate scalingM(v, [dHb]0) = 0.08 ·( v0.035)( [dHb]00.06)β(4.17)For f(x, t), the same real PETO2 data as well as PETCO2 were scaled to achieve a meanof f = 1 during baseline, f = 1.4 hypercapnia, and f = 0.95 during hyperoxia. These signalswere made to be spatially uniform.Finally, for d(x, t) we use equation (1.89) with the appropriate [dHb]0(x) for each voxel,plus the calculated f(t), CaO2(t), and [Hb] = 0.15 g Hb/mL blood described above.This provides all necessary components for computing relative BOLD signal changes usingequation (4.14), which is then scaled by 10000 in grey matter and the draining veins and 8000in white matter and the lesion to yield absolute BOLD images with similar magnitudes on ourscanner. The CSF signal is then computed as the average of the grey and white matter signals.1084.3. MethodsSimulating the ASL signalThe ASL signal is reverse-calculated from the CBF signal, which can be simulated using theconsensus CBF model from [61] in conjunction with a pre-defined baseline CBF, i.e. CBF0,and the previously-defined relative CBF changes, f(t).This is performed in three steps. First, the consensus CBF model is rearranged to isolatethe perfusion term, p(t)p(t) = CBF0 · f(t) · 2 · αL · T1,blood · SPD · (1− e− τT1,blood )6000 · λ · ePLDT1,blood(4.18)where αL = 0.85 is the labeling efficiency, T1,blood = 1650 ms at 3T, SPD is the proton-density weighted signal intensity which is approximated by the mean BOLD signal over baselinevolumes, λ = 0.9 mL/g is the blood-brain partition coefficient, τ = 1600 ms is the label duration,and PLD = 1724 ms is the post-label delay. All parameters are taken from [61] except thefinal two, which are the scanner-specific parameters employed on our scanner. CBF0 is set to30 and 60 mL/min/100g in white and grey matter, respectively. In the lesion simulation, thewhite matter CBF0 is scaled using the Grubb relationship to obtain CBF0 in the lesion, i.e.CBF0|lesion = CBF0|WM(vlesionvWM)1/α.Since p(t) is the sequential subtraction of control and label images, it has half the temporalresolution of f(t). Thus, in the next step p(t) calculated above is resampled to half the temporalresolution to yield p(t′).The final step then uses p(t′) to define the ASL signal. In our pCASL sequence the ASLsignal is acquired during the first echo and thus has a greater absolute magnitude, so we scalethe BOLD signal by 1.5 and use this as every odd ASL volume. The even volumes are thencalculated using the p(t′), i.e.ASL(2t) = 1.5 · BOLD(2t− 1) + p(t′). (4.19)where for our 300-volume acquisition t/TR ∈ {1, 300} and t′/(2 · TR) ∈ {1, 150}.Adding noiseThe 1 mm-resolution ASL and BOLD images were resampled to 3.5 mm x 3.5 mm x 6 mmto match the voxel size of the dual-calibrated BOLD protocol on our scanner. Gaussian noiseis then added to the noiseless BOLD and ASL voxel signals defined above, using a standard1094.3. Methodsdeviation such that we achieve an SNR of 100 in grey matter, approximately matching theSNR observed in human scans at our centre with this protocol. (The lower signal in whitematter consequently results in a lower SNR after noise is added.) Twenty noisy realizations aregenerated for each simulation type.4.3.2 Denoising hyperparametersHYPR4D-FS has three hyperparameters:1. k: the number of clusters to identify in feature space;2. h: the smoothing parameter used in defining the HYPR composite;3. the spatiotemporal kernel size of the HYPR operator itself.Choosing k too small means we are underclustering the feature space and thus the basisfunctions generated from averaging over clusters does not properly span the true underlyingvariance present in feature space. Choosing k too large means we are overclustering: in thiscase some clusters will be redundant and their separation in feature space will be due to noiserather than true underlying variance, ultimately degrading the denoising efficacy. For instance,in the uniform simulation the ideal number of clusters is one, since all voxels have the sameOEF—any additional clusters will be separated purely on the basis of noise—whereas in thelesion simulation ideally there will be two clusters: one for the lesion and one for the rest of thebrain.However, in practice many more clusters are needed to avoid “missing” features like alesion, and in cases where gradients in OEF exist, such as the spatially-varying simulation,then a sufficient number of clusters is required for the basis functions to capture the extremesof the gradients and avoid a too coarse interpolation of the gradients themselves. Since theunderlying variance in OEF cannot be known a priori, it is safer to err towards overclustering,meaning clusters will subdivide regions of identical physiological signal due to noise-inducedvariance. Empirically we determined k = 30 consistently separated the lesion in feature spaceand captured the spatial pattern of the spatially-varying simulation and thus use this value forall simulated and human data presented.For a given voxel, a larger h allows greater weighting of regional signals from clustersfurther away from the voxel values in feature space, thus making the composite smoother. Alarger spatiotemporal HYPR kernel size borrows more from the composite when calculatingthe HYPR-denoised image. Therefore, the two hyperparameters work together to modulate thesmoothness (and presumably the error) of the final image.1104.3. MethodsWe test h2 ∈ [0.5, 1, 2] and HYPR spatial kernel sizes of 10, 15, and 20 mm FWHM eachwith a fixed temporal kernel of 60 s FWHM. The temporal kernel size is maximized to half theduration of the shortest respiratory challenge such that the temporal contrast of the challengeis not removed from the initial temporal filtering in the HYPR operator.For comparison, we also denoise the images with a standard Gaussian filter matched to thespatial kernel size of the HYPR operator.4.3.3 OEF quantificationCBF(t) was computed from the denoised ASL data using the pCASL consensus model [61]. Themean of all baseline ASL control images were used to approximate the proton density signalintensity for the model. The computed CBF(t) was then resampled to the temporal resolutionof BOLD(t).O2 and CO2 end-tidals from the real human data used to generate the simulations wereshifted temporally to maximize the cross-correlation between the mean of BOLD(t) over allgrey matter voxels, then resampled to the temporal resolution of BOLD(t). This then allowsone to define baseline volumes as when the resampled, time-shifted end-tidals are within ±2mmHg of the targeted baseline PETCO2 and within±10 mmHg of the targeted baseline PETO23.These baseline volumes are used in the averaging of baseline ASL control images above, plus inthe computation of δS(t) and f(t).Hypercapnia volumes are defined as when the resampled, time-shifted end-tidals are within±2 mmHg of the targeted hypercapnic PETCO2 and within ±10 mmHg of the targeted baselinePETO2. Similarly, hyperoxia volumes are defined when the end-tidals are within ±2 mmHgof the targeted baseline PETCO2 and within ±10 mmHg of the targeted hyperoxic PETO2.Collectively, these allow one to compute δShc, δSho, fhc, and fho.OEF maps were calculated as described in Section 1.6.3 using the computed metrics above.4.3.4 Image quality metricsWith any denoising algorithm, the goal is to reduce the noise level of the processed imagewithout introducing any additional deviations from the ground truth, i.e. achieve both lownoise and error. Therefore, below we outline the metrics used to assess noise and error presentin OEF images after denoising that can be used to provide a quantitative comparison betweenstandard3This is the reported targeting precision of the RespirAct RA-MR system (Thornhill Medical, Toronto,Canada), our respiratory manipulation hardware.1114.3. MethodsSpatial noiseWithin a single OEF image, noise introduces spatial variation on top of any existing spatialvariation from the true underlying signal, thus making it difficult to interpret the noisy image.However, in the uniform OEF simulation we know the underlying signal to have no spatialvariation. Hence, all spatial variation in a single noisy realization of the uniform simulation isdue to noise. After denoising, the degree of remaining variation is thus reflective of the efficacyof the denoising algorithm.OEF maps calculated using standard denoising (i.e., spatial filtering alone) have a numberof outlying values, specially voxels with OEF > 1 due to the nonlinear propagation of noiseinto OEF from noise fhc, as shown in Figure 4.1. Therefore, even when the ground truth OEFis uniform, the distribution of estimated OEF from the noisy data will be asymmetric with along right tail. Typically, spatial noise is assessed by taking the standard deviation of a uniformregion of an image to assess the amount of variation due to noise:σ(s) =√√√√ 1N − 1N∑i=1(si − s)2 (4.20)where si are the voxel values in the region and s is the mean of all voxel values in the region.This is only a meaningful metric for noise level when the distribution of the noisy imagingmetric is normal or close to normal; in our case where we have an asymmetric distribution witha right tail, the outlying OEF > 1 values would cause s to be right-skewed from the peak ofthe distribution, and thus (si − s)2 would also be skewed, making the result of σ(s) difficult tointerpret.A more appropriate metric to assess variation in this case is the median absolute deviation(MAD):MAD(s) =MEDi(|si −MEDi(s)|) (4.21)where MEDi indicates taking the median over all voxels in the region. MEDi(s) will be closerto the peak of the distribution of OEF, and thus taking the median of the absolute value of thedeviation between each si and MEDi(s) will be less sensitive to outlying values. Therefore, toassess noise in OEF calculated a given noisy realization (NR) of BOLD/ASL data, we computeMAD over all voxels (x) in the OEF image, normalized by the median value over all voxels:%noise(NR) = 100%× MADx[OEF (x,NR)]MEDx[OEF (x,NR)]. (4.22)1124.3. MethodsThe mean of %noise(NR) taken over all noisy realizations then specifies, on average, theremaining spatial noise present after denoising.Voxel errorIn each simulation we have a noiseless ground truth image, OEFGT (x), and thus the errorpresent in the denoised image can be quantified. Error from the ground truth can come fromresidual noise in the denoised image, as well as any bias introduced by the denoising processitself. For instance, spatial filtering will introduce correlations between neighbouring voxels, soany sharp spatial boundaries will be biased by the filtering operation.We leverage multiple noisy realizations to compute the root-mean-square error (RMSE) ateach voxel across noisy realizations:RMSE(x) =√1N∑NR(OEF (x,NR)−OEFGT (x))2 (4.23)The result is an image, which can pinpoint regions with relatively higher or lower RMSE.As well, RMSE(x) can be averaged over a set of voxels or the entire brain to yield regional orglobal error. It is often more informative to normalize RMSE in OEF by the ground truth OEFto yield a percentage RMSE:%RMSE(x) = 100%×√√√√ 1N∑NR(OEF (x,NR)−OEFGT (x)OEFGT (x))2. (4.24)However, because the ground truth OEF is in the denominator, if this is close to zero theneven small absolute errors in OEF can lead to high %RMSE, making this metric less meaningful.This is only the case in the lesion simulation where OEF = 0.1 within the lesion, so in thiscase we present results using RMSE. Otherwise, in the uniform and spatially-varying OEFsimulations, we present results using %RMSE.Regional estimatesSince OEF images typically are quite noisy even after standard denoising, voxelwise estimatesare often averaged over region(s) for a given subject, such as subregions of cortical grey matter[79, 99, 160], to reduce the effect of noise on quantitative results. While any given voxel estimatemay not be robust due to high spatial noise, the median over the region may be close to theground truth—provided the modeling of OEF is accurate and the denoising method used does1134.3. Methodsnot introduce any additional deviation from the ground truth. Therefore, while RMSE definedabove will assess both noise-induced error and additional error introduced by the denoisingmethod, a median regional estimate minimizes the amount of noise-induced error and thus ismore sensitive to any error introduced by the denoising method itself.Therefore, we also compute regional median estimates of OEF over (i) the entire brain in theuniform simulation and (ii) an eroded region-of-interest for the lesion in the lesion simulation(this avoids edge voxels of the lesion which are most affected by the partial volume effect of therelatively large voxel size). We then take the average and standard deviation of these regionalestimates across all noisy realizations. One can then observe if these regional estimates areconsistent with the ground truth; if so, then this suggests any bias introduced by the denoisingmethod is negligible from the point of view of regional quantification.4.3.5 Human dataAcquisitionNine healthy controls (four female, age 60±14 years) were scanned on the SIGNA GE PET/MRusing the same dual-echo pCASL sequence featured in the simulations, with TR/TE1/TE2 =4187/20/32 ms, 1600 ms label duration, 900 ms post-label delay (first slice), bottom-up ac-quisition, tag location 23 mm below inferior-most slice, 3.5 x 3.5 mm in-plane voxel size and5 mm slice thickness (+ 1 mm gap), 20 slices, and 300 volumes. The subjects underwent thesame simultaneous gas challenege protocol described above in developing the simulations: twotwo-minute +8 mmHg hypercapnic challenges and two four-minute +230 mmHg hyperoxic chal-lenges separated by baseline breathing (Figure 4.2). Respiratory modulation was performed bythe RespirAct RA-MR system (Thornhill Medical, Toronto, Canada). MPRAGE T1 imageswere also acquired for each subject.Image processingBOLD and ASL datasets were realigned and rigidly co-registered to the T1 image using theAFNI toolbox. The data were passed through a high-pass filter with a cutoff of 480 s (i.e., twicethe length of the longest gas challenge) before the proposed denoising was applied as describedin Section 4.2. Like in the simulations, the realigned/co-registered datasets were also separatelydenoised using spatial filtering as a comparison.1144.4. ResultsOEF quantificationQuantification of OEF from the denoised images was performed as described in Section 4.3.3.A warp field for the T1 image of each subject to the 65 year-old 2 mm T1 template of [161]was defined using the FNIRT function of the FSL toolbox. This warp field was then applied toeach computed OEF image, thus moving the image of each subject to a common space wheregroup-level averaging can be performed.4.4 Results4.4.1 SimulationsQualitative image observationsFigure 4.4: Slices of OEF maps derived from denoised data for each simulation type, shown alongsidethe ground truth OEF maps. For each denoising method, a representative noisy realization is shownas well as the median over all 20 noisy realizations. The single noisy realization images visualize imagequality for data from a single subject, whereas the median images are indicative of group-averaged imagequality.Figure 4.4 provides slices of OEF maps derived from denoised data from a representativenoisy realization as well as the median over all noisy realizations for each simulation. Whenusing 10 mm spatial filtering, the voxelwise estimates in a single noisy realization containfrequent outliers, i.e. nonphysical OEF values. This indicates that subject-level data withthis level of denoising is unreliable. When averaging over all noisy realizations, the spatialdistribution is more coherent and visually approximates the ground truth with some residual1154.4. Resultsnoise still present. This suggests that group-level data may be interpretable. However, in thecase of studying subjects with lesions where individual localized differences may be expected,group averaging cannot be employed.Increasing to 20 mm spatial filtering reduces the frequency of voxels with outlying valuessuch that subject-level data is now somewhat interpretable for the uniform simulation. How-ever, the noise level remaining still obscures the underlying pattern in the spatially-varyingsimulation, and such a large spatial kernel greatly reduces the contrast between the lesion andsurrounding healthy tissue in the lesion simulation. Averaging over all noisy realizations yieldsa median image with better correspondence to the ground truth (except for the positive biasin the lesion).In contrast, the HYPR4D-FS images show good correspondence to the ground truth, evenin single a noisy realization, for all simulations. 2D histograms of the feature space and clustercentroids are shown in Figure 4.5, demonstrating the ability of k-means clustering to isolatedraining veins (grey arrow), separate the lesion from surrounding tissue in the lesion simulation(green arrow), and properly characterize the spatial pattern of OEF in the spatially-varyingsimulation (gold arrow). In addition, there is minimal difference between the single noisyrealization images and the median images, which demonstrates HYPR4D-FS denoising producesOEF images that are consistent across noisy realizations.Figure 4.5: 2D histograms of HYPR4D-FS feature space indicating the location frequency of voxelvalues in each simulation type. Red stars overlaid indicate the k-means clustering centroids identified,demonstrating the proper sampling of feature space to identify features such as draining veins (greyarrow), the lesion in the lesion simulation (green arrow), and the range of OEF values in the spatially-varying simulation (gold arrow).Histograms of voxelwise OEF in the uniform simulation in Figure 4.6 are symmetricallycentred near the ground truth OEF = 0.4 for all kernel sizes of HYPR4D-FS, whereas a right-skewed distribution of OEF estimates is present for spatial filtering—especially at smaller kernelsize—leading to a number of outlying, overestimated values. As mentioned, this stems fromnoise in fhc being nonlinearly propagated into OEF (Figure 4.1). However, with HYPR4D-FSthe noise is sufficiently reduced in fhc such that OEF is approximately linear across the rangeof denoised fhc values, yielding more symmetric distributions.1164.4. ResultsFigure 4.6: Histograms of OEF values for all voxels and noisy realizations of the uniform simulation,for both spatial filtering and HYPR4D-FS at different kernel sizes.Noise reductionImage noise is significantly reduced by HYPR4D-FS as compared to spatial filtering (Table 4.1),yielding %noise ∼ 3% for any combination of the tested hyperparameters. %noise for spatialfiltering is over 6x higher when using a 10 mm kernel and over 2.5x higher even when using a20 mm kernel.The noise level in HYPR4D-FS-processed images depends on the noise level in the filteredimage and the noise level in the composite image. Since increasing the spatial kernel sizefor HYPR4D-FS has a relatively small effect on image noise, this indicates that the compositeimage is limiting the amount of denoising. The smoothness of the composite can be increased byincreasing the hyperparameter h, which increases the amount of mixing between basis functionsin constructing the composite. However, from Table 4.1 increasing h also has little effect on theresulting noise level. This tells us that the OEF signal from each basis function is sufficientlyclose to begin with, such that increasing the degree of mixing between basis functions has onlya minimal effect on reducing the image noise.Filtering HYPR denoisingh2 = 0.5 h2 = 1 h2 = 210 mm 19.2 3.1 2.9 2.915 mm 11.3 2.9 2.7 2.620 mm 8.0 2.7 2.5 2.4Table 4.1: Average image noise in the uniform simulation, as measured by the average %noise(NR)across all noisy realizations.Voxel errorHYPR4D-FS also reduces voxel error significantly in each simulation compared to spatial filter-ing, as visualized using RMSE images in Figures 4.7–4.9 (uniform, lesion, and spatially-varyingsimulations respectively). Spatial filtering with a 10 mm kernel insufficiently reduces noise such1174.4. ResultsFigure 4.7: Voxel maps of %RMSE in the uniform simulation for all denoising methods. The whole-brain median %RMSE is listed below each map.1184.4. ResultsFigure 4.8: Voxel maps of absolute RMSE in the lesion simulation for all denoising methods. Thewhole-brain median RMSE is listed below each map.1194.4. ResultsFigure 4.9: Voxel maps of %RMSE in the spatially-varying simulation for all denoising methods. Thewhole-brain median %RMSE is listed below each map.1204.4. Resultsthat the resulting OEF images contain considerable noise-induced error, ultimately yieldingglobal RMSE in excess of 45%. A 15 mm or 20 mm kernel significantly reduces global RMSE,but it is still more than 2x higher than that of HYPR4D-FS.In the uniform and spatially-varying simulations, HYPR4D-FS has RMSE of ∼ 6%, com-pared to ∼ 14% for spatial filtering with a 20 mm kernel. Furthermore, there is little spatialvariation in voxel error for HYPR4D-FS in these simulations, whereas spatial filtering tends toproduce higher error in white matter (where the SNR is lower) as well as in voxels close to theventricles or containing draining veins (where partial volume effects from spatial filtering aremost pronounced).However, in the lesion simulation all methods have elevated RMSE near and/or inside thelesion. For spatial filtering, this is because the filter is introducing correlations between thelow OEF voxels within the lesion and the higher OEF voxels adjacent to the lesion. ForHYPR4D-FS, the voxel error is only relatively elevated on the boundary of the lesion, whichsuggests variations in the delineation of the lesion in the composite image (imposed through theseparation of low and higher OEF voxels in feature space) across different noisy realizations.Nevertheless, the absolute RMSE is still 4-16x higher for the lesion when using spatial filteringcompared to HYPR4D-FS.Regional estimation accuracyWhole-brain median estimates of OEF in the uniform simulation are 0.42± 0.02 for all denois-ing methods—consistent with the ground truth of 0.4. This suggests there is no global biasintroduced by filtering or HYPR4D-FS when OEF is spatially uniform, i.e. as expected inhealthy controls. When using spatial filtering, voxelwise estimates of OEF can be nonphysical,but this result tells us that median global estimates are robust and reproducible on a subjectlevel. However, this is of little utility in a diseased population where spatial variations in OEFmay exist.In the lesion simulation, median regional estimates of OEF in the lesion are fairly consistentwith the ground truth of 0.1 for HYPR4D-FS (Table 4.2), though increasing h and the HYPRkernel size leads to a small underestimation. This would suggest small biases in the basis func-tion(s) used to construct the composite image voxels signal within the lesion, which would beamplified by increasing the kernel size and/or h. However, all regional estimates are consistentwithin their standard deviations, so this effect remains minimal.For spatial filtering in the lesion simulation, the 10 mm kernel leads to an overestimationdue to the correlation of lesion voxels (OEF = 0.1) with adjacent normal tissue (OEF = 0.4)voxels imposed by the spatially-invariant filter. Increasing the kernel size evidently results in1214.4. Resultsa greater overestimate since the larger kernels are approaching the size of the 30 mm-diameterlesion.Filtering HYPR denoisingh2 = 0.5 h2 = 1 h2 = 210 mm 0.17± 0.03 0.09± 0.04 0.09± 0.03 0.09± 0.0315 mm 0.29± 0.07 0.08± 0.03 0.07± 0.03 0.07± 0.0320 mm 0.34± 0.06 0.08± 0.03 0.07± 0.03 0.07± 0.03Table 4.2: Mean and standard deviation of regional OEF of the lesion, taken over all noisy realizations.For reference, OEF = 0.1 is the ground truth.4.4.2 Human dataThe median OEF maps for the healthy human data are shown in Figure 4.10, for both HYPR4D-FS (h2 = 1, 20 mm spatial kernel size) and spatial filtering with a 20 mm kernel. No distinctspatial patterns in OEF are observed, as is expected for healthy subjects. The image qualityof the median maps for HYPR4D-FS are similar to those of the uniform simulation. Average%noise taken over all subjects data in their native space is 2.1% for HYPR4D-FS, which, likethe simulations, is significantly lower than the 16.7% observed for 20 mm spatial filtering.Figure 4.10: Median OEF images over all subjects (N=10). OEF was computed from denoised data ineach subject’s native space, then warped to a common template before the voxelwise median was taken.Due to limited axial coverage of the brain and inconsistencies in the warping operation, some voxels inthe common space do not have a value for each subject; here we have only kept voxels where at least6/10 subjects contribute to the median.Data from the noisiest and least noisiest subject in their native space (as measured via%noise) are also shown in Figure 4.11. Like in the simulations, the global median values aresimilar for HYPR4D-FS and spatial filtering, but spatial filtering will still occasionally yieldoutlying values, especially in lower SNR white matter regions or near structural boundariessuch as the ventricles as seen in subject 2.1224.4. ResultsFigure 4.11: Individual OEF maps for two subjects, comparing HYPR4D-FS to spatial filtering. Asmeasured by %noise for spatial filtering, subject 1 has the lowest noise and subject 2 has the highestnoise.1234.5. Discussion4.5 DiscussionDual-calibrated BOLD has the potential to provide maps of OEF and CMRO2 noninvasivelyusing MRI and respiratory manipulations. However, the traditional approach of preprocessingusing spatial filtering insufficiently denoises the ASL/BOLD signal such that voxelwise estima-tions of OEF and hence CMRO2 are unreliable at the subject level. Nevertheless, as shown in[79, 99, 160] this method still produces useful global grey matter estimates or physiologically-plausible voxelwise estimates when averaged over several subjects. However, this is only of usein the case of healthy populations where OEF is expected to be spatially uniform.In cases of disease such as stroke where affected regions are expected to have different OEFfrom the rest of the brain, evidently subject-specific delineation and quantification of lesions isnecessary. In other disease populations such as neurodegeneration where CMRO2 is hypothe-sized to be altered, robust OEF maps are necessary to determine whether any CMRO2 changesare due to CBF changes, OEF changes, or both. The progressive nature of neurodegenera-tion may lead to the magnitude of alteration in CMRO2 changing as a function of diseaseprogression, which again means subject-level quantification is important.There has been extensive research into optimizing the models and specific protocols used fordual-calibrated BOLD [79, 95–99], but less on attempting to improve the SNR of the data itselfbefore quantifying OEF. The SNR of dual-calibrated BOLD data is modulated by the scannerand scanning protocols chosen to a certain degree, but there is also a fundamental limit onthe magnitude of respiratory manipulations that can be tolerated by patient populations andhence the magnitude of the BOLD response4. Therefore, denoising remains the final option forimproving SNR and ultimately improving the robustness of OEF estimation.4.5.1 Denoising using cluster-based averagingAlternatives to dual-calibrated BOLD which aim to quantify OEF using entirely different MRI-based approached have been proposed [84–91]. In particular, the quantitative BOLD and QSM-based approach (QQ) of [89] has shown superior spatial resolution and SNR in OEF estimatesby employing cluster averaging of voxels with similar time evolution. However, QQ relies ona heavily-regularized minimization problem, since the biophysical models used are extremelysensitive to initial parameter guesses. As a result of this regularization, outputted OEF mapsof healthy subjects will tend to agree with global grey matter estimates from dual-calibratedBOLD, but often have discretized distributions of OEF stemming from the initial guesses ofeach cluster (see the Supplementary Information of [160]), giving the misleading impression4From an initial technical development study using the RespirAct in conjunction with our scanner, we foundhealthy subjects reasonably tolerated +8 mmHg from baseline PETCO2, which usually induces ∆BOLD < 5%.At +10 mmHg, subjects reported discomfort and/or a feeling of shortness of breath.1244.5. Discussionthat there actually are “clusters” within the brain with unique OEF even in healthy subjects.Nevertheless, we were inspired by this cluster-based approach and adapted the idea todual-calibrated BOLD datasets while attempting to minimize the effect of the initial clustering.Instead of clustering in the data space, we devised a low-dimensional feature space that attemptsto isolate the major sources of underlying variation in the theoretical BOLD signal, namely thecalibration constantM (driven primarily by blood volume fraction differences across the brain)and [dHb]0, which in turn is related to OEF.Within the feature space we use k-means clustering to identify voxels with similar physiol-ogy. Since the mapping from data space to feature space removes any geometrical information,averaging over these k-means clusters is a form of non-local averaging. Non-local averaginginformed by physiology can be thought of as a spatially-variant filtering operation; this canhandle sudden changes in tissue physiology across space by minimizing the correlation intro-duced between physiologically-distinct regions, whereas a spatially-invariant filter will introducecorrelations within a fixed spatial neighbourhood.4.5.2 Relaxing the assumptions of clusteringOur k-means clustering in feature space identifies the voxels that can be viewed as physiologi-cally similar. However, if one were to simply replace each voxel within a given cluster with thecluster average, then this is assuming that each voxel within the cluster has identical physiology.This is evidently a bad assumption, especially for voxels in feature space roughly equidistantfrom two or more cluster centroids.A better assumption is that if sufficient clusters are identified, then the cluster regional sig-nals can be thought of as basis functions which span the underlying variation of the dataset. Onecan then construct a denoised representation of the voxelwise data from these basis functions,i.e. each voxel signal can be mathematically recast as a weighted sum of the basis functions.In effect, one is“interpolating” in feature space between the basis functions.While this is a better approach, we relax the assumptions of clustering further by comparingthe low frequency features (i.e., contrast) between our denoised representation of the data andthe original data using a 4D HYPR operator. As shown in equation (4.12), HYPR will updatethe denoised representation if there are low frequency discrepancies from the original data, thusforming the final HYPR4D-FS denoised product.1254.5. Discussion4.5.3 Ability of the feature space to capture variation in OEFIn Figure 4.5, we demonstrate that the feature space is able to capture variations in OEF forthree different simulations: a uniform OEF simulation, a simulation with uniform OEF exceptfor a 30 mm lesion, and a simulation with a 20% spatially-varying OEF pattern.The feature space is self-normalizing along each dimension, so with a fixed k chosen fork-means clustering the clusters identified will map to regions of essentially identical OEF, ifthere is no underlying variation in OEF (i.e., in the uniform simulation), or sample from thedistribution of underlying OEF variation (i.e., in the lesion and spatially-varying simulations).In the lesion simulation, voxels within the lesion having lower OEF are captured by distinctclusters that are well-separated from the remaining clusters, meaning the denoised lesion voxelsignals essentially have no contribution from voxels outside the lesion.Along the other dimension, sensitive to M , we also discovered the feature space is ableto identify voxels with drastically elevated blood volume fraction, namely voxels containingdraining veins which often lead to outlying OEF values in the surrounding tissue when usingstandard denoising.4.5.4 Improvements in noise and errorAverage voxel noise in estimated OEF was several times higher for spatial filtering as comparedto HYPR4D-FS, and as a result greater consistency was visually observed in the denoised imagesacross noisy realizations for HYPR4D-FS. The denoising efficacy of spatial filtering is limitedby the size of the kernel and hence the number of voxels included in local averaging. The non-local averaging in HYPR4D-FS allows for many more voxels to be included in the computationof each basis function. The basis functions are thus smoother than voxel signals after spatialfiltering, and consequently the HYPR4D-FS BOLD/ASL voxel signals are smoother. Withless noise in BOLD/ASL measures, less noise propagation occurs in the computation of OEFimages.The reduction in spatial noise by HYPR4D-FS also reduces noise-induced voxel error (asmeasured by RMSE). More importantly, the voxel error is relatively uniform across the brainfor HYPR4D-FS, even in simulations of spatially-varying OEF, indicating both accuracy andprecision in OEF quantification is improved. However, in the lesion simulation we observedan inconsistent delineation of the boundary of the lesion across noisy realizations when usingHYPR4D-FS, leading to slightly elevated RMSE here. Nevertheless, an even higher error is in-duced by spatial filtering both on the boundary and within the lesion, due to spatial correlationsintroduced from tissue outside the lesion.1264.5. Discussion4.5.5 Application to healthy human dataObservations from the simulations are mostly consistent across real data from nine healthysubjects. Though the ground truth is not known, the individual and median OEF maps forHYPR4D-FS are physiologically plausible with OEF ∼ 0.45. The maps are also visibly uni-form and quantitatively have similar average voxel noise levels as the uniform simulation. Incomparison, the noise level for spatial filtering in the real data is over 2x greater than that inthe simulations (8%). This may stem from subject-to-subject variation in SNR, causing thelower SNR subjects to have more outlying values and consequently much higher %noise thanaverage.4.5.6 Reliable subject-level data?Traditional spatial filtering of dual-calibrated BOLD data is usually with a 6–10 mm kerneland produces unreliable voxelwise estimates; we decided to test even larger kernels in this workto see if this sufficiently improved the robustness of the OEF maps.In Figure 4.4, we demonstrate how the number of outlying voxels present with 10 mm filter-ing in subject-level OEF maps prevent reliable quantification of any underlying OEF variation.With 20 mm filtering, there are less outlying values, but separating noise-induced variationfrom any underlying variation is difficult, meaning it is hard to distinguish a single realizationof the uniform simulation from the spatially-varying simulation. Furthermore, the bias intro-duced by this large a kernel obscures the contrast of the lesion in the lesion simulation, makingdelineation and quantification of the lesion difficult.By contrast, HYPR4D-FS images are only slightly more noisy in appearance than themedian images over all noisy realizations, indicative of the much improved subject-level noiseand reproducibility. The lesion contrast is relatively preserved in the lesion simulation andnoise-induced variation is sufficiently low that a single realizations of the uniform simulationcan be distinguished from the spatially-varying simulation. These observations, combined withthe above mentioned similarities between the results of the simulations and real human data,demonstrate the ability of HYPR4D-FS to provide robust single-subject OEF maps.4.5.7 Fine-tuning HYPR4D-FS hyperparametersHYPR4D-FS has three hyperparameters: the number of clusters identified in feature space,k; the smoothing parameter used in defining the weighting of basis functions in the HYPRcomposite, h; the kernel size of HYPR operator itself. Though any combination of the hyper-parameters tested lead to significant improvements in image quality over spatial filtering, it is1274.5. Discussionstill important to consider fine-tuning the hyperparameters.We empirically chose k = 30 to err on the side of overclustering vs. underclustering. Whileoverclustering may produce redundant clusters, we found that underclustering sometimes leadsto the lesion being missed in feature space, and that there is little change in noise and re-producibility for k = 15 vs. k = 30 (results not shown). Visually, k = 30 also demonstratesa comprehensive sampling of voxel values in feature space (Figure 4.5). In theory, one mayemploy an X-means or similar data-driven approach to choosing the number of clusters, butimplementing this was beyond the scope of this present work.Together, h and the HYPR4D-FS kernel size modulate the smoothness of the final denoisedimage. Global error is generally reduced after increasing h and/or the HYPR4D-FS kernel size,but this is primarily through the reduction of noise-induced error. If there is disagreement inhigh frequency features between the composite and true high frequency features in the data(i.e., not noise), then error will be increased by increasing the kernel size; this can be seen in theoccipitoparietal region for the h2 = 2 case in Figure 4.9. Relatedly, if h is increased too largethen some voxels may be receiving too large a contribution from basis functions with dissimilarphysiology. This can be seen for the draining vein voxels of in Figure 4.7: as h is increased, thevoxel error increases in these regions. As a result, we would recommend using a modest h2 = 1to avoid oversmoothing the composite with a HYPR kernel size of 10–20 mm.4.5.8 Combining OEF and CBF to estimate CMRO2The purpose of this work was to denoise ASL/BOLD data to improve the robustness of OEFestimates. However, in many applications the parameter of interest is CMRO2 = CaO2 ·CBF0 · OEF. Since OEF is most sensitive to noise of the metrics used to compute CMRO2,by demonstrating the ability of HYPR4D-FS to produce robust OEF estimates in this work, itfollows that CMRO2 estimates will become more robust as well.HYPR4D-FS denoises ASL data in the course of computing OEF, which means denoisedestimates of CBF0 are also available. These can be combined with a CaO2|0 estimate from therespiratory data to compute CMRO2, or, alternatively, an estimate of CBF0 can be obtainedfrom an additional, higher SNR ASL sequence. The latter option may be desirable, since theCBF0 estimates from HYPR4D-FS tend to be biased; the feature space in HYPR4D-FS hasbeen constructed primarily to identify spatial variation in OEF rather than CBF0, so theretends to be a mixing of voxels with the same OEF but different CBF0.128Chapter 5Joint analysis of metabolic data5.1 MotivationThe previous chapters have developed denoising and analysis algorithms which are separatelydesigned for PET and MRI datasets. This is by necessity: for denoising, we have demonstratedthat the source of noise and noise characteristics vary between the two modalities; for analysis,there is unique information to be extracted from each modality. However, in certain cases thedata from each modality can be complementary, or knowledge of the discordance between PET-and MRI-based measures can be informative. In this case, there is a need for a joint analysisalgorithm to identify concurrences and/or discordance between two imaging metrics.Previous sophisticated methods1 of analyzing multimodality PET/MRI (or multi-radiotracer)data include correlation analyses between structural information (from MRI), functional con-nectivity networks (from fMRI data), metabolic information (from PET), and/or metabolic con-nectivity networks (from PET) to assess the relationship between neuronal structure/activityand molecular function [152, 162–166]; Canonical Correlation Analysis to identify common andunique information from two datasets, such as between multiple PET tracers [167] or betweenPET and MRI data [168–171]; and data fusion of PET/MRI data to aid in disease diagnosisand prognosis [172–175].In this chapter we develop a joint analysis algorithm that can be applied to any two imagingmeasures. However, we frame the development around the application of brain energetics,quantified by metabolic rates of oxygen (CMRO2) and glucose (CMRglu), due to an ongoingresearch study at our centre investigating brain energetics in healthy and Parkinson’s disease(PD) subjects. These measures can be quantified simultaneously in a PET/MRI scanner:CMRO2 from dual-calibrated BOLD fMRI (as discussed extensively in the previous chapter)and CMRglu from [18F]FDG PET (Section 1.4.5).1These are methods that go beyond voxelwise or regional parametric testing between the quantitative imagingdata from each modality.1295.1. Motivation5.1.1 Physiological backgroundBrain energetics refers to the production and usage of energy in the brain. Energy is gener-ated through the production of adenosine triphosphate (ATP), which can occur through twopathways:1. Krebs cycle: glucose is metabolized in the mitochondria of cells using oxygen as acatalyst, producing a relatively large number of ATP molecules, but at a relatively slowrate. The presence of oxygen indicates this is aerobic energy production.2. Glycolysis: glucose is metabolized without the presence of oxygen to produce relativelyfew ATP molecules, but at a relatively fast rate. The lack of oxygen indicates anaerobicenergy production.When oxygen is unavailable, evidently the only energy production pathway is glycolysis.However, even in the presence of oxygen glycolysis may still occur—a process thus termedaerobic glycolysis. If the energy production is fully aerobic, then by the molar ratio of oxygento glucose in cellular respiration one would observe the oxygen-to-glucose index as OGI =CMRO2/CMRglu = 6. Therefore, one can infer some degree of aerobic glycolysis if OGI < 6.Aerobic glycolysis was first observed in cancer cells [176], but more recently variations inaerobic glycolysis have been observed in the healthy brain [74]. The mechanism(s) for aerobicglycolysis is not fully understood [177], but a possible pathophysiological mechanism may bemitchondrial dysfunction, which is hypothesized to be present in PD [178]. With mitochondrialdysfunction, the mitochondria are unable to complete the Krebs cycle despite oxygen beingpresent, leading to an increase in glycolysis in an attempt to meet local energy demands.5.1.2 Previous work in joint analysis of CMRO2 and CMRgluOGI is traditionally determined from two PET scanning sessions. As mentioned, an [18F]FDGscan quantifies CMRglu and another scanning session with15O-labeled water, carbon monoxide,and oxygen gas quantifies CMRO2 [43]. A two-compartment model of15O-labeled oxygen gasallows one to calculate OEF, provided separate measures of CBF and cerebral blood volume(CBV) are obtained. A separate 15O-labeled water scan will yield CBF, while CBV comes froma separate 15O-labeled carbon monoxide scan. With OEF and CBF, a measure of CaO2 froman arterial blood sample provides the necessary metrics to compute CMRO2.Modeling assumptions in the calculation of CMRO2 and CMRglu limit the accuracy ofabsolute quantification. In turn, this limits the accuracy of directly-computed OGI. In CMRO2,the blood volumes within capillary and venous subcompartments—required the calculation of1305.1. MotivationOEF—are assumed from population averages, so subject-level variation in these values wouldnot be captured. Since OEF is uniform in healthy controls, errors from assuming population-averaged parameters would have a global scaling effect. In CMRglu, one assumes a globalscaling factor called the “lumped constant” [equation (1.57)] to account for phosphorylationdifferences between FDG and glucose, though as long as the same value is used across subjectsthe error introduced by this assumption is identical. Furthermore, the voxelwise metrics are stillrelatively noisy—even after denoising—meaning voxelwise division will amplify noise further.Directly computing OGI from PET-based measures of CMRglu and CMRO2 in early PD andhealthy controls failed to find any differences between the groups [76]. However, the difficultieslisted above may have confounded any small underlying differences. As well, diurnal variationsin CMRglu and CMRO2 can exceed 20% [74], meaning the results may be further confounded bynot acquiring CMRglu and CMRO2 simultaneously. Finally, variation in quantitative measuresmay exceed the variation between groups, making the detection of group differences difficultunless a large number of subjects are imaged.To reduce the limitations of absolute quantification, the mapping of OGI in [74] was in-stead achieved by dividing CMRglu and CMRO2 by their whole-brain means before computingCMRO2/CMRglu, then scaling by a whole-brain mean OGI literature value of 5.323, i.e.OGI(x) = 5.323 · CMRO2(x)/CMRO2(x)CMRglu(x)/CMRglu(x). (5.1)The division by the whole-brain mean removes the global effects of the lumped constant inCMRglu) and population average assumptions in CMRO2. However, this strategy normalizesall subjects to a common whole-brain mean and consequently any absolute differences betweensubjects are lost. Nevertheless, this still allows one to quantify regional or voxelwise patternsof relative values that are consistent across subjects, thus identifying regions with more orless aerobic glycolysis relative to the whole-brain average. This pattern of aerobic glycolysisis presumably similar across healthy subjects, and indeed in [74] the variance across subjectsin regional OGI estimates, as calculated above, is relatively small (coefficients of variation2ranging from 2–8%).Because OGI is inversely related to the amount of aerobic glycolysis, a simpler-to-interpretmetric termed the glycolytic index (GI) was also introduced in [74]. GI is computed by regressingthe whole-brain-normalized CMRglu onto the whole-brain-normalized CMRO2 and scaling theresulting residuals of this regression by 1000. The line of regression assumes a perfect linearrelationship between the two normalized metrics, i.e.2The standard deviation of regional values taken across subjects normalized by the mean of regional valuestaken across subjects.1315.1. MotivationCMRglu(x)CMRglu(x)= a · CMRO2(x)CMRO2(x). (5.2)So for a given voxel, the line of regression predicts a value for CMRglu(x)/CMRglu(x) basedon the measured value of CMRO2(x)/CMRO2(x). If the actual measured CMRglu(x)/CMRglu(x)is higher than the predicted value, GI will be positive and this suggests glycolysis in excess thanwhat is predicted, i.e. more aerobic glycolysis within this voxel.A similar strategy for jointly investigating brain energy utilization—as measured by CMRglu—and, in this case, neuronal demand—as measured through the logarithm of the local functionalconnectivity density, log(lFCD), a resting state fMRI metric—was employed in [163]. The abovemetrics were z-scored normalized instead of normalized by their whole-brain mean, then plottedon orthogonal axes. Instead of regression, a change of basis via a 45◦ rotation was applied toidentify two new orthogonal metrics, the relative power (rPWR) and relative cost (rCST):[rPWRrCST]=[cos π4 sinπ4− sin π4 cos π4][z(log(lFCD))z(CMRglu)]. (5.3)The z-scoring ensures the variance contributed by each metric is equal. rPWR thus capturesthe variance in the data explained by concurrent energy metabolism and neuronal demand,whereas rCST captures variance explained by discordance between energy metabolism andneuronal demand, with positive rCST indicating metabolism exceeds demand. This methodcan easily be adapted to the study of brain energetics by replacing log(lFCD) with CMRO2:[rPWRrCST]=[cos π4 sinπ4− sin π4 cos π4][z(CMRO2)z(CMRglu)]. (5.4)Now, rPWR indicates the concurrent oxygen and glucose metabolism, which can be thoughtof as a proxy for relative energy production across the brain, whereas rCST indicates the level ofaerobic glycolysis, since positive rCST implies glucose metabolism exceeds oxygen metabolism(both relative to their whole-brain means).5.1.3 Adapting and expanding joint analysis of CMRglu and CMRO2Either GI or rPWR and rCST as defined in equation (5.4) could be used to investigate brainenergetics in healthy and PD subjects. For the specific interest in aerobic glycolysis, GI hasthe disadvantage of relying on regression to identify deviation from predicted glycolysis in agiven region, whereas rCST provides a guaranteed orthogonal measure of discordance between1325.2. Methodsglucose and oxygen metabolism. Furthermore, the z-scoring in the preparation of rPWR andrCST is helpful in our application where CMRO2 and CMRglu are obtained from differentimaging modalities with different levels of noise-induced variance, as the z-score ensures equalcontribution of each metric to rPWR and rCST.Therefore, in this chapter we explore rPWR and rCST in healthy and PD subjects as a novelparametric approach to jointly analyzing CMRO2 and CMRglu. As discussed in Section 1.7,there are inherent limitations in region-by-region statistical comparison of metrics betweenhealthy and PD subjects. Therefore, the second novelty introduced in this chapter is to applySSM-PCA pattern analysis to rPWR and rCST in an attempt to identify PD-related alterationsto brain energy production (i.e., using rPWR) and aerobic glycolysis (i.e., in rCST).5.2 Methods5.2.1 SubjectsRecruitment was ongoing at the time of preparing this thesis; 10 healthy controls (4 male, 6female, age 59.5 ± 12.4 years) and 14 PD subjects (8 male, 6 female, age 65.4 ± 8.2 years,disease duration 6.3± 3.7 years, UPDRS-III3 21.3± 5.2) have been scanned and their CMRO2and CMRglu measures computed.5.2.2 Data acquisition, image processing, and metric computationA 21-minute dual-echo pCASL sequence on the GE SIGNA PET/MR scanner acquired 300sequential ASL and BOLD images while subjects underwent a respiratory modulation protocolinvolving hypercapnia and hyperoxia (see Section 4.3.5 for details). HYPR4D-FS (h2 = 1, kernelsize of 10 mm and 60 s) was used to denoise the dual-echo pCASL data before computing OEFas detailed in Section 4.3.3, which was then combined with the temporal average of CBF imagestaken during baseline breathing conditions (i.e., CBF0) to compute CMRO2.Immediately following the dual-echo pCASL scan, subjects underwent a 60-minute, 5 mCiFDG scan. Data were histogrammed into 4 x 1, 3 x 2, 8 x 5, 1 x 10 minute frames andreconstructed using PSF-HYPR4D-K-TOFOSEM [39]. Serial arterialized venous blood sampleswere taken during the scan to help compute an image-derived input function developed at ourcentre [39]. Briefly, a sagittal maximum-intensity projection image was computed from thefirst minute post-injection to identify candidate voxels of the carotid arteries4, then a temporal3Unified Parkinson’s Disease Rating Scale [179]. Part III assesses motor symptom severity, with a higherscore indicating more severe symptoms.4A maximum-intensity projection image (MIP) is constructed by taking parallel rays through the 3D image1335.2. MethodsFigure 5.1: Computation of relative power (rPWR) and relative aerobic cost (rCST), adapted from[163]. Regional measures of z-scored CMRglu and CMRO2 are plotted on orthogonal axes. A 45◦ rotationyields new axes of rPWR, i.e. relative energy production through oxygen and glucose metabolism, andrCST, i.e. the degree of discordance between CMRglu and CMRO2. Positive rCST indicates relativeglucose metabolism in excess of relative oxygen metabolism, which indicates increased aerobic glycolysis.constraint is imposed such that only voxels that have values and temporal patterns that matchthree venous blood samples taken during the last 30 minutes of the scan (when venous andarterial blood radioactivity is sufficiently close). CMRglu data were computed using the Patlakmodel in equation (1.57) using LC = 0.65 and the average Cglu from two blood samples (thefirst just before injection, the second at the end of the scan).Voxelwise CMRO2 and CMRglu measures were averaged over 92 regions from a cortical andsubcortical Freesurfer segmentation [117] of a T1 MPRAGE image acquired simultaneouslyduring the FDG scan for each subject. Because there is often a gradient in the PD-relatedalterations along the anterior-posterior axis of the putamen, the Freesurfer putamen ROI wassubdivided into anterior, middle, and posterior putamen ROIs for all subjects. For PD subjects,bilateral regions were flipped such that the more affect side corresponded to the left side.rPWR and rCST were computed as in equation (5.4) using regional values for each subject (seeFigure 5.1 for a visualization).5.2.3 Pattern analysisWe perform an initial validation of our data by running SSM-PCA (see Section 1.7 for details)on the regional CMRglu data to reproduce the well-known PD-related glucose pattern [106]; re-gional analysis was used to have improved statistics for the relatively low number of subjects. Ato the projection plane and mapping the maximal value along each ray to the projection plane. This is usefulfor producing a 2D visualization of objects of high intensity embedded within the 3D volume; in this case, thecarotid arteries.1345.2. Methodssearch for a PD-related pattern proceeded as in [180]. Briefly, all components identified by PCAthat (i) explain at least 5% of the total variance in the data, and (ii) whose subject scores dis-criminated healthy subjects from PD subjects by p < 0.2, were entered into a logistic regressionframework, where components are linearly combined such that the Akaike information criterion(AIC) of the logistic model discriminating healthy subjects from PD subjects is reduced. Upondetermining the weights wk for k ∈ κ, κ being the set of indices for the components that satisfy(i) and (ii) above, the combined PD-related component, CPD and corresponding subject scorefor subject j, SSPD,j , are computed asCPD =∑k∈κwkCkSSPD,j =∑k∈κwkSSjk.(5.5)Note that components that do not reduce AIC when entered into the logistic regressionmodel are assigned a weight of zero.SSM-PCA has yet to be applied to CMRO2 directly. However, a simpler direct statisticalcomparison of relative values identified spatially similar patterns of relative changes in CMRO2,CBF, and CMRglu [181]. Relatedly for CBF, a component of CMRO2, attempts to identify aPD-related blood flow pattern using PCA have had minimal success [182], but projecting thePD-related glucose pattern onto CBF and CMRglu data of the same subjects yields highlycorrelated subject scores [183]. This suggests that similar alterations occur to CMRO2, CBF,and CMRglu. We thus apply SSM-PCA to CMRO2 to investigate whether our novel analysisframework yields similar conclusions.We then proceed with pattern analysis of rPWR and rCST. The rPWR and rCST mea-sures are already centred for each subject, so taking the logarithm of these values in the SSMtransformation is not possible. The raw rPWR and rCST values are thus used and SSM hasthe effect of further centring the data across subjects. The rPWR pattern, obtained whencomparing PD and HC data, will identify PD-related alterations to overall energy production,and correspondingly the rCST pattern will identify PD-related alterations to aerobic glycolysis.Because rPWR and rCST are orthogonal, it is possible to have any combination of thetwo. For instance, a region with positive values in the PD-related rPWR and PD-related rCSTpatterns would imply that, compared to healthy subjects, PD subjects have higher overallenergy production in this region to relative to their brain average, but also more aerobic gly-colysis relative to their brain average. Therefore, it is imperative to interpret these measuressimultaneously to fully understand brain energetics alterations.1355.3. Results5.2.4 Computation of OGIPattern analysis of rCST will identify regions where OGI is likely to be different between PDand healthy subjects; since a healthy pattern of discordance between CMRO2 and CMRgluexists [74], centring rCST data across PD and healthy subjects will reveal consistent alterationsto this pattern in PD subjects. Therefore, in these regions we also perform between groupcomparisons of OGI as computed in equation (5.1).5.3 Results5.3.1 OEF in healthy and PD subjectsComputing OEF using HYPR4D-FS revealed a spatially uniform distribution for both healthyand PD subjects; this was first determined visually by observing group median maps of OEF(Figure 5.2), then applying SSM-PCA to OEF data itself yielded no spatial pattern that sepa-rated healthy and PC subjects. Global medians of OEF for each group were also not significantlydifferent, with 0.47± 0.08 for healthy subjects and 0.42± 0.09 for PD subjects.For the purposes of CMRO2 pattern analysis—which seeks to identify spatial covariancepatterns related to disease—a uniform OEF does not contribute any spatial variance to theCMRO2 data matrix. This is due to the demeaning step in the preparation of the data [equa-tion (1.99)]: if OEF is constant across regions, thenXij = log(CMRO2)ij= log [CBFij ·OEFj · (CaO2)j ]Xij = logCBFij + logOEFj + log(CaO2)j(5.6)such that in the SSM demeaning step across regions for each subject, the spatially uniformcontributions from OEF and CaO2 cancelXij −GMRj = Xij − 1NN∑i=1Xij= log(CMRO2)ij − 1NN∑i=1log(CMRO2)ijXij −GMRj = logCBFij − 1NN∑i=1log CBFij .(5.7)1365.3. ResultsAs a result, patterns in CMRO2 will be identical to patterns in CBF, which is supportedby the literature [181]. Similarly, for the z-score normalization of CMRO2 in computing rPWRand rCST, we havez [(CMRO2)ij ] =(CMRO2)ij − 1N∑Ni=1(CMRO2)ijσi [(CMRO2)ij ]=OEFj · (CaO2)j(CBFij − 1N∑Ni=1CBFij)OEFj · (CaO2)j · σi [CBFij ]=CBFij − 1N∑Ni=1CBFijσi [CBFij ]z [(CMRO2)ij ] = z [CBFij ](5.8)where σi[·] indicates taking the standard deviation over regions. Finally, in computing OGI,the effect of uniform OEF will also cancel out in the numerator of equation (5.1) such thatCMRO2/CMRO2 = CBF/CBF. Therefore, we entered CBF data rather than CMRO2 datainto these measures to avoid unnecessary noise propagation into the already noisy CBF data—especially given the small number of subjects to date.Figure 5.2: Group median images of OEF for healthy and PD subjects, demonstrating a spatiallyuniform distribution of OEF for both groups.5.3.2 CMRglu and CBF patternsA regional PD-related CMRglu pattern was identified that separates healthy and PD subjectsat p < 0.00005, characterized by relative hypermetabolism in bilateral cerebellum and the moreaffected posterior putamen, plus relative hypometabolism in bilateral caudate and occipitopari-etal regions, consistent with the well-known previously identified pattern [106]. Additionally, wealso identify a PD-related CBF pattern that separates healthy and PD subjects at p < 0.01 andshares similar regional weights with the CMRglu pattern in cerebellar grey matter and lateraloccipital cortex—regions with high absolute weights, meaning they strongly contribute to both1375.3. Resultspatterns (Figure 5.3)—though the pattern regional weights do not correlate overall (ρ = 0.02).Both patterns describe ∼ 27% of the total variance.Figure 5.3: (a) z-scored regional weights that characterize the identified PD-related CMRglu and CBFpatterns. Only regions with |z| > 1.5 for either pattern are shown for clarity; these are the regionsthat contribute most to the patterns. Note that “ventral DC” is a Freesurfer region consisting of thehypothalamus, mammillary body, subthalamic nuclei, substantia nigra, red nucleus, lateral geniculatenucleus, and medial geniculate nucleus. Also note that PD data have been flipped as necessary suchthat the more-affected side is on the left. Positive weights indicate relative hypermetabolism whereasnegative weights indicate relative hypometabolism. The projection of these weights onto the data ofeach subject yields subject scores; (b) shows subject scores for the CMRglu pattern, which separate PDsfrom HCs at p < 0.00005, and (c) shows subject scores for the CBF pattern, which separate PDs fromHCs at p < 0.01. VAF = variance accounted for in the data by the pattern.5.3.3 rPWR and rCST patternsPD-related patterns were also identified for rPWR and rCST (Figure 5.4). rPWR is elevatedin PDs compared to healthy subjects in bilateral cerebellum and lowered in occipitoparietalregions; rCST is elevated in PDs compared to HCs in posterior putamen (more affected side),suggesting more aerobic glycolysis relative to the brain average here, respectively. In termsof directly computed OGI, a significant decrease in PD was observed in the more affectedside of the posterior putamen (p < 0.0001, Figure 5.5). Additionally, the PD-related rCSTpatterns shows lower rCST in cerebellum and brain stem. For the cerebellum, higher rPWRand lower rCST suggests elevated energy production (relative to the brain average) in PD, butthis elevation does not occur through glycolysis.1385.3. ResultsFigure 5.4: (a) z-scored regional weights that characterize the identified PD-related rPWR and rCSTpatterns. (b) Subject scores for the rPWR pattern, which separate PDs from HCs at p < 0.001. (c)Subject scores for the rCST pattern, which separate PDs from HCs at p < 0.005.Figure 5.5: Oxygen-to-glucose ratio (OGI) in the posterior putamen of the more affected side in PDs(left side in HCs), where OGI is significantly reduced in PDs (p < 0.0001). OGI = 6 means all energy isproduced aerobically; OGI < 6 means some aerobic glycolysis is present.1395.3. Results5.3.4 Absolute quantificationAs mentioned, in pattern analysis we are looking at the distribution of metabolism acrossthe brain relative to the brain average and comparing these relative values between groups.Therefore, for a given amount of total oxygen or glucose delivered to the brain, the patternsabove identify regions where relatively more or relatively less of this oxygen or glucose is beingmetabolized. We have previously discussed the advantages of this approach, but one disadvan-tage is that biological interpretation of such patterns is difficult without any absolute valuesof metabolism. Therefore, we supplement the above results with group comparisons of whole-brain, grey/white matter, and featured regional averages of CMRglu, CMRO2, and CBF, plusraw OGI (i.e., absolute CMRO2 divided by absolute CMRglu) and normalized OGI [i.e., calcu-lated using equation (5.1)], in Table 5.1.Whole-brain Grey Matter White Matter Post. put. Cerebellum Brain StemCMRgluHC 17.8± 2.1 20.5± 2.3 14.1± 1.8 25.6± 3.3 16.1± 1.5 12.4± 1.5PD 16.9± 3.0 19.6± 3.5 13.3± 2.4 27.9± 5.7 16.8± 2.9 11.8± 2.3CMRO2HC 120± 38 128± 39 110± 37 123± 46 100± 32 72± 24PD 106± 38 115± 42 96± 35 97± 37 98± 55 63± 22CBFHC 44.5± 9.2 48.7± 9.7 39.1± 8.5 47.6± 7.2 40.7± 9.3 35.4± 7.6PD 38.7± 6.1 42.8± 6.7 33.6± 5.6 38.2± 6.3 42.9± 10.4 35.1± 8.2OGI (raw)HC 6.83± 2.37 6.35± 2.18 7.92± 2.83 4.98± 2.15 6.25± 2.12 5.84± 2.19PD 6.41± 2.32 5.99± 2.18 7.40± 2.70 3.61± 1.55 5.94± 3.30 5.58± 2.34OGI (norm)HC 5.46± 0.05 5.43± 0.05 6.23± 0.64 4.31± 0.54 5.75± 0.69 6.57± 0.94PD 5.48± 0.06 5.43± 0.06 6.57± 0.84 3.44± 0.36 6.32± 0.91 7.49± 1.56Table 5.1: Absolute group mean values of CMRglu, CMRO2, and CBF, plus raw OGI values (i.e., ab-solute CMRO2 divided by absolute CMRglu) and normalized OGI [i.e., calculated using equation (5.1)].Units of CMRglu and CMRO2 are µmol/100 g/min. Units of CBF are mL/100 g/min.Variation within groups for absolute values is fairly high, meaning with the low number ofsubjects in the study to date there are no significant group differences in these values, exceptfor the aforementioned normalized OGI in the more affected side of the posterior putamen.However, a few observations can still be made from trends in the data:• Whole-brain (plus grey and white matter) CMRO2 and CMRglu are lower in PD, whichsuggests overall lower energy production.• The more affected posterior putamen in PD has higher CMRglu than healthy subjects—despite the whole-brain average being lower. By contrast, CMRO2 is lower in this regionfor PD. This is consistent an increase in aerobic glycolysis, as was suggested by the rCSTpattern.• CMRO2 and CMRglu are similar between PD and healthy subjects in the cerebellum. Thissuggests energy production being preserved in PD in this region, consistent with a lack ofpathology in this region for PD [184]. Given the whole-brain averages of both measures1405.4. Discussionare lower, this results in the relative energy production being increased, confirming theelevated rPWR in this region for PD.5.4 DiscussionRegional pattern analysis performed on a relatively small number of PD and healthy subjectsidentified the well-known PD-related CMRglu pattern [106], thus satisfying an initial validationof our dataset. We also performed pattern analysis on the same subjects for CBF (as a proxy forCMRO2), which revealed alterations similar to those observed in CMRglu in the cerebral greymatter and the lateral occipital cortex. This is consistent with previous observations [181], butthe fact that the alterations are not identical across all regions suggests there may be changesto brain energetics in PD.To investigate this further, we jointly analyzed normalized CBF and CMRglu data by de-composing the joint dataset for each subject into two orthogonal, normalized measures: relativepower (rPWR) and relative cost (rCST), adapted from [163]. rPWR tracks overall relative en-ergy alterations as a result of both CMRO2 and CMRglu being altered relative to the brainaverage, whereas rCST is a proxy for the level of aerobic glycolysis relative to the brain av-erage, with positive rCST indicating relative glucose metabolism exceeding relative oxygenmetabolism.Since aerobic glycolysis has been previously found to vary across the healthy brain [74],performing pattern analysis on these measures allowed us to identify PD-related alterationsnot only to overall energy production (rPWR), but also a metric that can be interpreted tobeing related to aerobic glycolysis (rCST). From the PD-related rPWR and rCST patterns, weidentified higher rPWR and higher rCST in the more affected posterior putamen. This meansCMRglu is relatively increased without a corresponding relative increase in CMRO2, likelyindicating increased aerobic glycolysis which may suggest mitochondrial dysfunction. This isalso supported by trends in absolute metabolism values, with absolute CMRO2 decreased andCMRglu increased in PD compared to healthy controls. Furthermore, a direct computationof the oxygen-to-glucose ratio in this region confirmed a significant decrease in OGI for PDsubjects. Conversely, in bilateral cerebellum and brain stem, the PD-related patterns indicatedincreases in rPWR but decreases in rCST, meaning relative increases in CMRO2 exceed relativeincreases in CMRglu. This suggests elevated energy production (relative to the brain average)without relying on aerobic glycolysis; see [74] for possible biological interpretations, thoughthese cannot be confirmed or denied with the current methodology.While further investigation is needed to fully understand the biological interpretation andclinical significance of these findings, for the purposes of analysis algorithm development we have1415.4. Discussiondemonstrated a simple yet powerful method to jointly analyze CBF and CMRglu, consistingof SSM-PCA pattern analysis applied to the orthogonal measures of rPWR and rCST derivedfrom the joint dataset. Evidently, this analysis method could be applied to any populationof disease to investigate alterations from healthy patterns of energy production and aerobicglycolysis across the brain. Additionally, the method could be adapted for any two metricswhere one is interested in group-related alterations to patterns of concurrence and discordancebetween these parameters.142Chapter 6ConclusionsOver the past few decades, PET/MRI scanners are becoming more common in research and—tosome degree—in clinical settings. Despite their high cost, the functional information providedby PET [9] and the variety of structural and functional contrasts that can be jointly providedby MRI [55, 57–60] provides an unparalleled tool for studying the healthy and diseased brainin vivo [1].As research studies and clinical evaluations ask for more quantitative and specific informa-tion from PET/MRI scanners, there is a need for higher quality acquired data, advanced imagereconstruction and post-processing algorithms to further improve signal-to-noise ratio (SNR),and more sophisticated analysis algorithms to extract relevant physiological information fromthis higher quality data. The quality of the acquired data is improved through innovations inscanner hardware, whereas improved image reconstruction, post-processing, and analysis comefrom algorithm development. This work has focused on the latter aspect.6.1 Motivations based on current research in PET/MRIimage processing and analysis6.1.1 PET denoisingDenoising is a necessary post-processing step for PET studies, since the data are intrinsicallynoisy. PET produces quantitative images of radioactivity from an injected radiotracer, meaningthe spatial SNR in images depends on the number of detected radioactive emissions per voxel(3D pixel). These emissions follow a Poisson distribution, often leading to high voxelwisevariability. As a result, there is an extensive literature of denoising algorithm development, buttrade-offs often exist between noise reduction, bias, and broad applicability (see Section 1.3).We thus identified a need for improvements upon these existing algorithms.1436.1. Motivations based on current research in PET/MRI image processing and analysis6.1.2 MRI denoisingThe MRI signal, on the other hand, relies on the physics of nuclear magnetic resonance (NMR).In NMR, proton spins are weakly polarized by an external magnetic field and the resulting netmagnetization is modulated by RF pulses, which results in the emission of a corresponding RFsignal that is measured. Unlike NMR, MRI also applies magnetic gradient fields on top of theexternal field to locally alter spin precession, such that the signal contains spatial informationthat can be reconstructed into an image. The development of denoising algorithms is less presentin the MRI literature, where SNR is relatively high for most image acquisition sequences (theset of RF pulses transmitted and magnetic field gradients applied to produce the measured RFsignal that is transformed to an image). However, in particular applications such as functionalMRI (fMRI)—where whole-brain images are acquired within a few seconds to capture transientchanges in blood oxygenation in response to a task or stimulus [58]—the fast acquisition and lowbiological signal change leads to lower SNR. In this case post-processing denoising is applied, butusually through spatially-invariant filtering, which leads to a loss of resolution by introducinglocal correlations, and does not exploit non-local similarities in tissue characteristics across thebrain.A recent application of fMRI is dual-calibrated BOLD [79, 99], where respiratory challengesare used to modulate cerebral blood flow and oxygenation in an attempt to quantify cerebralmetabolic rate of oxygen (CMRO2), which may be a useful marker of metabolic dysfunction instroke [62–65], tumour [66–68], or neurodegeneration [69–71]. The biophysical equations usedto quantify CMRO2 are complex and highly nonlinear, leading to significant noise propagationfrom the already noisy input fMRI data (see Section 4.1). We thus recognized an opportunityfor improved denoising in this area, by leveraging the ideas and methods from PET denoisingalgorithm development and adapting these to the MRI field.6.1.3 PET/MRI image analysisReconstructed and processed PET/MRI datasets on their own often do not provide sufficientinformation to address the hypotheses of researchers in fields such as neuroscience, neurology,psychology, and psychiatry who help design imaging studies. Rather, the relevant informationis often encoded within the image or dynamic signal, meaning image analysis is needed toextract this information. For instance, dynamic PET data follow the temporal evolution of theradiotracer distribution post-injection: the initial uptake as radiotracer is exchanged from bloodplasma into tissue, followed by a possible dynamic equilibrium as the radiotracer selectivelybinds to its target. By modeling these radiotracer kinetics, one can extract physiologically-relevant parameters, such as the non-displaceable binding potential (a proxy for target receptordensity). See Section 1.4 for more details.1446.1. Motivations based on current research in PET/MRI image processing and analysisIf the subject is being scanned without any external stimulus, then the signal acquired byPET and/or MRI is presumably reflective of baseline physiology. However, it is common topresent a task or stimulus to the subject during the scan, at which point aspects of physiologymay be altered in response. As mentioned, this is the basis of tasked-based fMRI: the taskor stimulus results in a subset of neurons being activated, which requires a local increase inoxygenated blood. The magnetic properties of oxygenated versus deoxygenated blood differ, sothis local change in oxygenation induces a local change in magnetization that can be detectedby MRI [93]. Albeit a less developed field, task-based PET imaging is also possible: glucosemetabolism [5, 115, 116], or aspects of neurotransmitter synthesis [128] or release [49, 126, 127],may be transiently altered by a task, resulting in perturbations to the PET kinetics of theradiotracers that image these molecular processes.In both task-based MRI and PET imaging, a detection algorithm is needed to localize theregions of the brain where a physiological response occurs. The task-induced signal change canbe relatively small—especially for neuropsychological tasks1 [147]—so especially in PET wherethe SNR is already low, detecting these changes can be challenging: for instance, in PET-baseddopamine release detection, current detection algorithms have less than a 40% true positiverate of detection based on simulations of release in response to motor and cognitive tasks (seeTable 3.1). Therefore, significant improvements—in both denoising of the data and the methodof detection—were needed to enable reliable detection and applicability of the method in humanstudies.Finally, the advent of PET/MRI permits simultaneous acquisition of potentially comple-mentary metrics from each modality, where the identification of concurrence or discordancebetween the two metrics is thus important. For instance, an ongoing research study at the timeof writing this thesis from our centre seeks to investigate changes in brain energetics in Parkin-son’s disease (PD), by simultaneously quantifying CMRO2 and CMRglu (cerebral metabolicrate of glucose) in PD subjects and age-matched healthy controls. From the existing literature,a simple joint analysis algorithm has been developed for quantifying concurrence and discor-dance between CMRglu and neuronal demand [163], but we identified that this could easily beadapted to jointly analyze CMRglu and CMRO2. Direct statistical comparison of the resultingconcurrence/discordance values across the brain to identify PD-related alterations is limiting,since this strategy implicitly treats different regions of the brain as being statistically indepen-dent; the brain has a high degree of functional and molecular connectivity. Therefore, exploringthe spatial covariance of imaging data—using pattern analysis—is more keeping in line withthis connectivity. As a result, pattern analysis of CMRglu data have yielded a variety of robustdisease-related patterns [105]. We thus identified an opportunity to include pattern analysiswithin a joint analysis framework.1This is referring to anything not induced by a pharmaceutical intervention, where signal changes may actuallybe large.1456.2. Improvements offered to PET/MRI image processing and analysis6.2 Improvements offered to PET/MRI image processing andanalysisThis work has succeeded in its goal to develop physics-based, broadly applicable denoisingand analysis algorithms for data acquired from PET, MRI, or PET/MRI scanners. Thesealgorithms address the current limitations and areas of improvement within the research fieldsof PET/MRI image processing and analysis, as outlined above. In total, four algorithms havebeen developed:1. IHYPR4D (Chapter 2): a post-processing denoising algorithm for dynamic PET datathat demonstrated marked improvements in noise reduction while maintaining the lowbias (error) of the input reconstructed data. IHYPR4D is an extension of HYPR3D [24].Both IHYPR4D and HYPR3D denoise through filtering, then reintroduce high frequencyfeatures lost by filtering using a guiding composite image; HYPR3D uses a 3D compositeimage for every temporal frame in the data, whereas IHYPR4D uses a 4D composite anditeratively improves the composite. These two alterations result in:• Significant improvements in noise reduction compared to HYPR3D in high spa-tiotemporal resolution datasets where voxel noise is very high;• Minimal bias introduced compared to HYPR3D by better matching the contrast inthe composite image and the input data.IHYPR4D was applied to data from multiple scanning environments and radiotracers.In all cases, significant improvements in joint noise and bias reduction as compared toHYPR3D were observed, demonstrating its broad applicability.2. Residuals space detection (RSD, Chapter 3): an analysis algorithm for detectingtask-based dopamine (DA) release from PET data. Existing detection frameworks usea model-based detection strategy, where models capable and incapable of modeling DArelease are fit to voxel TACs; if an F-test indicates the model incorporating DA release fitssignificantly better, then this is indicative of DA release being present. This frameworksuffers from low detection sensitivity in simulations of low amplitude of release reflectiveof neuropsychological tasks [143]. The small signal change coupled with low SNR in thedata makes separation of true positives from true negatives challenging, and the methodrequires a model to accurately characterize the baseline radiotracer kinetics plus the DA-induced perturbations—without overfitting to noise. We mitigated these limitations inRSD by applying IHYPR4D to greatly improve the SNR of the data, then isolated theDA-induced signal perturbations by transforming the data to a residuals space. In thisspace, structured residuals are indicative of DA release and unstructured residuals (i.e., amean close to zero and a variance reflective of the temporal noise in the signal) suggest no1466.2. Improvements offered to PET/MRI image processing and analysisrelease. Therefore, the baseline kinetics have been removed and a simpler general linearmodel can be applied for significance detection in a fashion similar to fMRI analysis. Fromsimulations of low amplitude release, we demonstrated this novel detection frameworkgreatly outperforms the traditional model-based detection framework, with:• Over 2x higher true positive rate at equivalent false positive rate, indicating morereliable detection and conformal localization of release;• Virtually a 100% chance of detecting a least a portion of the voxels within simulatedclusters of release, compared to a ∼ 75% chance with model-based detection.While we demonstrated the feasibility of RSD for detecting DA release, in theory thismethod could be used for any PET-based detection of neurotransmitter release, provideda radiotracer that competes for binding with the neurotransmitter exists.3. HYPR4D-FS (Chapter 4): an MRI denoising algorithm for dual-calibrated BOLDdata, used for absolute quantification of CMRO2 = CaO2 ·CBF0 ·OEF. With traditionaldenoising of fMRI data using spatial filtering, voxelwise OEF estimates are still often unre-liable, with a subset of voxels containing outlying values beyond the range of physiologicalplausibility (i.e., OEF < 0 or OEF > 1), which, in turn, results in unreliable CMRO2images [79]. Spatial filtering only performs local, spatially-invariant averaging, whereasthe tissue characteristics relevant to the fMRI signal can be similar in non-local regions[79, 99]. Therefore, inspired by the success of the non-local means algorithm [23] andthe kernel method [35] in PET denoising, we developed a denoising algorithm that mapsthe 4D fMRI data to a low-dimensional feature space, where the dimensions are sensitiveto physiological parameters that drive contrast in the fMRI images. Within the featurespace, k-means clustering is performed to define basis functions which span the variabilityin the data. Then, these basis functions are used to construct a 4D composite image ofthe original data, which is finally compared to the original data and updated based on lowfrequency mismatches between the original data and composite (using HYPR) to formthe final denoised product. Through simulations, we demonstrate HYPR4D-FS producesOEF images with:• Significantly reduced noise and error, compared to traditional fMRI denoising ofspatial filtering;• No outlying values;• Reliable characterization of simulated lesions resulting in sharp changes in OEF inthe brain, as well as spatially-varying distributions of OEF.Applying HYPR4D-FS to healthy human data produced physiologically plausible OEFimages with noise levels consistent with those observed in simulations, whereas data pro-cessed with spatial filtering had ∼ 8x higher noise levels and many outlying voxel val-1476.3. Strengths and limitations of the present workues. Although the development of HYPR4D-FS centred around the application of dual-calibrated BOLD, the theory of HYPR4D-FS—mapping to a feature space to identifybasis functions that can reconstruct the original data with the guidance of HYPR—canbe easily adapted to other denoising applications.4. SSM-PCA applied to joint PET/MRI analysis metrics (Chapter 5): we inte-grated the successes of SSM-PCA pattern analysis (see Section 1.7) in identifying disease-related spatial covariance patterns in metabolic imaging data with a simple joint analysismetric of two imaging metrics [163] for the specific application of jointly analyzing CMRO2and CMRglu data in PD subjects and age-matched healthy controls. The joint analysismetric decomposes the z-scored normalized metabolism metrics into orthogonal metricsof relative power (rPWR), sensitive to relative differences in energy production across thebrain, and relative cost (rCST), sensitive to the degree of aerobic glycolysis relative tothe brain average, i.e. how much relative CMRglu exceeds relative CMRO2. By applyingSSM-PCA to the original metabolism metrics as well as the joint analysis metrics fromexisting human data, we identified regional PD-related patterns in:• CMRglu, which agreed with existing patterns in the literature [106];• CBF, which contained similar spatial weights in PD-affected regions, consistent with[181, 183];• rPWR, which revealed altered energy production in PD-affected regions;• rCST, which, supplemented with absolute quantification, indicated increased aerobicglycolysis in the more affected side of posterior putamen (a region involved in thepathology of PD).These results serve as a proof of concept for identifying alterations between groups inconcurrence and discordance of two imaging metrics; the specific application here wassearching for PD-related alterations to brain energy production and the imaging met-rics were CMRO2 and CMRglu data, but the developed method is generalizable to jointanalysis of any two imaging metrics.6.3 Strengths and limitations of the present work6.3.1 Development from first principlesThe above algorithms were mostly developed from first principles. For the denoising algorithms,observations on the physical sources of noise and the effects of noise on image quantificationwere identified, then further observations on the underlying physiology producing contrast inthe images were leveraged to hypothesize targeted strategies for denoising. For the analysis1486.3. Strengths and limitations of the present workalgorithms, limitations in current analysis algorithms were identified and/or methodologiesfrom other domains were adapted for relevant PET/MRI applications. This serves as a strength,since the mechanisms by which the algorithms operate can be clearly stated, and thus observedimprovements over previous algorithms can be soundly understood.6.3.2 Accurate simulation studiesIHYPR4D, RSD, and HYPR4D-FS were thoroughly tested in controlled yet realistic simulationstudies, which allowed us to quantitatively compare these proposed algorithms to existing algo-rithms. The simulations were designed based on real human data and incorporated as many ofthe physical effects of imaging as possible. In particular, the IHYPR4D simulation study used aforward projection algorithm that incorporates the imaging physics of the HRRT scanner [16],and Poisson noise was added to the projection data to accurately model the noise characteris-tics of acquired PET data. For the DA release simulations for RSD, unlike similar simulationsin the literature [48, 49] we simulated the effect of the scanner point spread function (PSF),which is important as the limited resolution of the PET scanner degrades the observed DArelease magnitude. Not accurately simulating the PSF thus leads to overestimated detectionsensitivity for a given true release magnitude; for a ∼ 100% increase in DA concentration in oursimulations we found standard detection methods had ∼ 38% true positive rate, whereas similarsimulations in [49] without PSF modeling found 80.9% true positive rate2. Finally, for test-ing HYPR4D-FS we generated the first known digital brain phantom of dual-calibrated fMRIdata, by modeling the unique tissue characteristics across the brain and accurately simulatingspatially-varying SNR of real human data (lower SNR in white matter than grey matter).Accurately simulated data is imperative for the translation of simulated results to resultsfrom human studies, where the ground truth no longer exists; if simulated images and realimages are sufficiently close, then the observed improvements in simulations should be similar inhuman data. In cases where human data was available, qualitative and quantitative assessmentsof the developed algorithms were made and compared to similar assessments made in thesimulation studies. Overall, these results were consistent, suggesting our simulations servedas a good approximation of real imaging data. For instance, qualitatively the HRRT RACsimulations in Chapter 2 have similar temporal noise levels of real data (see Figure 2.14), andquantitatively the average noise levels in our simulated and human OEF images of Chapter 4are nearly identical (2.5% and 2.1%, respectively).2Our ∼ 38% true positive rate at ∼ 100% increase in DA concentration is more consistent with their simulatedresults at a 50% increase (35.8% true positive rate). This especially demonstrates the importance of accuratemodeling at relatively low levels of release, where detection sensitivity decreases significantly.1496.3. Strengths and limitations of the present work6.3.3 Quantitative improvements over previous algorithmsThe denoising algorithms IHYPR4D and HYPR4D-FS both showed marked improvements inboth noise reduction and error reduction compared to existing algorithms—in simulated andreal human data. The reduction in noise for IHYPR4D is broadly beneficial to quantitativeprecision in dynamic PET. However, for HYPR4D-FS the reduction in noise enables reliablevoxelwise estimates of MRI-derived CMRO2 from dual-calibrated BOLD data, which has notbeen demonstrated to date.Similarly, the DA release detection framework of RSD offers a 2x improvement in detectionsensitivity over existing detection frameworks based on our simulations of low amplitude release,but more importantly RSD improves the likelihood of detecting at least a portion of true releasevoxels from ∼ 75% to 99%. Therefore, in human studies the confound of undetected releaseacross a set of subjects is essentially removed at these levels of release.To our knowledge, SSM-PCA applied to the rPWR and rCST joint analysis metrics is thefirst application of pattern analysis to imaging metrics derived from a multimodal dataset, sono quantitative comparison between this proposed analysis framework and existing frameworkscould be made. Nonetheless, the resulting PD-related alterations identified by this frameworkare physiologically plausible based on previous knowledge of metabolic alterations in PD [106,181, 184]. Furthermore, interpretations drawn from the pattern analysis results were consistentwith trends present in absolute quantification results. Therefore, this analysis framework mayenable a more comprehensive understanding of brain energetics deficits in PD as more subjectdata becomes available.6.3.4 Computational speed of algorithmsSpatially-invariant filtering remains a popular choice for denoising PET/MRI imaging despiteits inherent limitations, in part because of its computational speed: despite spatial filteringrepresenting a voxel-by-voxel averaging of its local neighbourhood, in practice one invokes theConvolution Theorem to simplify this operation to a multiplication of the Fourier-transformedimage and filtering kernel [equation (1.20)], which is computationally more efficient and thusallows spatial filtering of high resolution PET images3 in seconds or less. Advanced denoisingalgorithms that rely on spatially-variant filtering can no longer use the Convolution Theorem,since the kernel varies across space, and thus the computational time can increase dramatically.For instance, the non-local means algorithm in PET [23] theoretically defines a unique kernelfor each voxel by comparing their local spatiotemporal neighbourhood to that of every othervoxel in the image. However, in order to keep computational times to the order of minutes,3High resolution PET images have 100s of voxels in each dimension, often yielding > 107 total voxels.1506.3. Strengths and limitations of the present workin practice one restricts non-local means by only comparing neighbourhoods between voxelswithin a local 2D patch.In contrast, both of the denoising algorithms developed in this work rely on HYPR, whichhas a similar computational speed to spatial filtering. Instead of performing voxel-by-voxel non-local averaging, HYPR filters the initial image and then defines a voxel-specific update factor toreintroduce features lost by filtering. The update factor itself is simply the elementwise divisionof a composite image by the filtered composite image, so computationally HYPR performs twospatially-invariant filtering operations and a single instance each of elementwise multiplicationand division. Multiplication and division—even in high resolution PET—still only takes afraction of a second, so as a result HYPR only takes a few seconds to implement. In IHYPR4Dwe iteratively apply HYPR, but based on simulated results one only needs ∼ 10 iterations tominimize any bias introduced through denoising, meaning the entire denoising process takes afew minutes.In HYPR4D-FS, non-local averaging is performed, but the strategy of defining the kernelweights for non-local voxels is different than non-local means in PET such that the compu-tational speed is drastically improved. HYPR4D-FS maps to a low-dimensional feature spaceand uses k-means clustering to average voxels with similar positions in feature space—whichcorresponds to non-local averaging in image space—rather than doing a voxelwise comparisonof neighbourhoods, as is done in the non-local means algorithm in PET. k-means clusteringin this low-dimensional feature space takes only a few seconds, meaning this process plus theapplication of HYPR results in rapid computation of the denoised data.6.3.5 LimitationsAs discussed in Section 2.5.7, the primary limitation of IHYPR4D is its reliance on a segmenta-tion of the imaging data to perform regional averaging in defining the initial 4D composite. Thisis necessary to exploit spatiotemporal denoising and feature extraction from a 4D composite,but has the drawback of potentially introducing uniformity within a region where the underly-ing radiotracer distribution is heterogeneous. However, by iteratively updating the compositewe were able to minimize the bias introduced by such erroneous uniformity assumptions. Inthe development of IHYPR4D we tested two simple segmentation approaches: (i) an externalMRI segmentation to define anatomical regions that are averaged over in the PET image, and(ii) a two-step k-means segmentation of the time-averaged PET image. We found that bothstrategies converged to similar locations in bias-noise space (and thus their improvements inthese metrics over HYPR3D were similar), though their initial starting points were different:compared to MRI-based segmentation, k-means segmentation produced higher initial noise butlower bias after one IHYPR4D iteration.1516.3. Strengths and limitations of the present workFor RSD, the method of transforming the data to a residuals space relies on estimatingthe hypothetical baseline behaviour (i.e., the temporal evolution of the signal if no task werepresented) at each voxel. This is achieved by running k-means clustering over all striatalvoxels TACs (pre-task) to identify a set of cluster-averaged regional TACs that approximatethe baseline behaviour across the striatum. Implicit in this construction is the assumptionthat the release within the striatum is localized and low magnitude, such that the cluster-averaged regional TACs have very little contamination from release voxels. This means RSDwill not work well if the release distribution is widespread across the striatum and/or highmagnitude. However, the motivation for developing RSD was precisely for detecting localized,low amplitude release DA release, since widespread, high amplitude release—more relevant forpharmaceutical interventions rather than the more diverse set of neuropsychological tasks—canbe detected with existing methods [49, 126].For HYPR4D-FS, the feature space developed was designed to detect variance related mostlyto differences in OEF across the brain, since it is the most noisy component of computingCMRO2 = CaO2 ·CBF0 ·OEF. While simulations showed that HYPR4D-FS achieves superiornoise and error reduction in OEF images compared to traditional spatial filtering, the method isnot optimized to produce baseline cerebral blood flow (CBF0) images with minimal error (sincevoxels with different CBF0 can have similar OEF, and thus are correlated by HYPR4D-FS).Therefore, a CBF0 image computed from ASL data without HYPR4D-FS processing is neededto finish the calculation of CMRO2 with minimal error.Finally, SSM-PCA applied to rPWR and rCST carries the existing limitations of patternanalysis. By design, pattern analysis removes any absolute differences between groups and thusthe interpretations of the patterns are limited to relative changes in metabolism. These patternsof relative values are powerful since they are derived from covariance, meaning the brain is beinganalyzed more like a system rather than independent parts, and thus can characterize disease-related alterations better than repeated statistical comparison of absolute metabolism valuesacross regions. However, to fully interpret the biological meaning of the patterns, one stillneeds absolute metabolism data. For instance, the well-known PD-related CMRglu is partlycharacterized by relative hypermetabolism in the cerebellum. At first glance this may suggestpathology in this region, but ex vivo research demonstrates the cerebellum is relative spared[184]. Instead, relative hypermetabolism appears here because whole-brain absolute CMRglu islowered in PD, yet conserved in the cerebellum.1526.4. Possible applications of the present work6.4 Possible applications of the present work6.4.1 IHYPR4DThe PET denoising algorithm developed in this work, IHYPR4D, demonstrated improvementsin noise reduction compared to HYPR3D across multiple scanning environments and radiotrac-ers, indicating its use is appropriate across a range of dynamic PET datasets. The segmentationmethod used to define the initial IHYPR4D composite can be chosen based on the general char-acteristics of the radiotracer distribution within the data. For instance, in [18F]FDG image con-trast post-uptake shares some similarity with anatomical MRI images and thus the MRI-basedsegmentation may be most appropriate. However, in [11C]raclopride the contrast post-uptakeis primarily between the striatum and the rest of the brain, so a simpler segmentation of thesetwo regions may be more appropriate. Beyond this initial segmentation choice, there are noassumptions within IHYPR4D that would limit its application to a given dynamic PET dataset.Furthermore, one may elect to use a data-driven segmentation approach—such as the k-meansclustering of a time-averaged subset of the data featured in this work—in which case the initialuniformity assumptions come from the data itself.IHYPR4D was shown to offer the greatest relative improvement over HYPR3D in terms ofnoise reduction in datasets with high spatial noise, i.e. in high spatial resolution and/or lowdose scanning applications where the counts per voxel are low. This is because the HYPR3Dcomposite is the sum over tens of temporal frames, so its noise level is essentially driven bythe number of counts per voxel across the entire scan duration. By contrast, the IHYPR4Dcomposite is the average over spatial regions which contain hundreds or thousands of voxels, andthus the composite noise level remains low even in low counts per voxel applications. Therefore,IHYPR4D is best suited in these applications, though we demonstrated improvements overHYPR3D in lower noise datasets (such as the relatively high sensitivity GE SIGNA PET/MRIscanner) and when IHYPR4D is coupled with a denoising reconstruction, such that the inputto IHYPR4D already is considerably denoised.An immediate application of IHYPR4D was demonstrated within Chapter 3 of this work:using IHYPR4D to denoise simulated [11C]raclopride data with transient DA release greatlyreduced temporal noise (see Figure 3.1), such that integrating IHYPR4D within an improveddetection framework demonstrated the feasibility of reliably detecting low amplitude DA release.6.4.2 RSDAs mentioned in the previous section, RSD is designed to detect localized, low amplitudeneurotransmitter release, and the simulation studies of Chapter 3 validate its ability to do1536.4. Possible applications of the present workso. These simulations involved DA release, but in theory RSD can be used to detect anyneurotransmitter that competes for binding with a radiotracer.An ongoing research study at our centre is investigating DA release in PD subjects andhealthy controls in response to motor, cognitive, and reward-based tasks. RSD is currently beingused to analyze these data as they become available—demonstrating an immediate applicationof the method. Given DA is heavily involved in the motor system [130], alterations to DArelease in movement disorders other than PD may be investigated using this method. As well,DA release related to cognitive and reward-based tasks is likely altered by addiction [185–187],providing another target for PET-based DA release characterization by RSD.6.4.3 HYPR4D-FSThe development of HYPR4D-FS was centred around the specific application of dual-calibratedBOLD to quantify OEF and hence CMRO2. In Chapter 5, HYPR4D-FS was immediately ap-plied to data from PD subjects and healthy controls. This helped identify PD-related alterationsto CMRO2 and, coupled with CMRglu data, brain energetics as a whole. Additionally, group-level analysis of preliminary data indicated no apparent spatial alterations to OEF in PD,meaning alterations to CMRO2 may be driven solely by alterations to CBF. As demonstratedin Section 4.4.2, without HYPR4D-FS subject level OEF images have a considerable numberof outlying values, and thus obtaining the results of Chapter 5 would not have been possiblewithout HYPR4D-FS.Other applications of HYPR4D-FS applied to dual-calibrated BOLD data could be to char-acterize subject-specific metabolic disruptions due to stroke or tumour, or to test for group-leveldifferences in absolute metabolism in neurodegeneration or healthy aging.6.4.4 SSM-PCA applied to joint PET/MRI analysis metricsThe joint analysis framework developed in Chapter 5 was immediately applied to CMRO2and CMRglu data in PD and healthy controls. This framework was successful in identifyingPD-related alterations of brain energetics from a limited number of subjects; both CMRO2 andCMRglu data were computed from denoised data—CMRO2 with HYPR4D-FS and CMRglu withPSF-HYPR4D-K-OSEM (see Section 1.3.3)—highlighting how coupling advanced denoising andanalysis algorithms can provide meaningful results from relatively few subjects.As mentioned, the joint analysis framework can be adapted to any two imaging metricswhere some degree of concurrence and discordance can be expected. This is especially relevantfor physiological processes that rely on multiple inputs: for energy production this is CMRO21546.5. Future directionsand CMRglu, but other applications may be the balance between neuronal demand (measuredby resting-state fMRI metrics) and metabolism (measured by CMRglu)—as was previouslyinvestigated without pattern analysis [163]—or the balance between structural connectivity(measured by diffusion tensor imaging metrics) and functional connectivity (measured by restingstate fMRI metrics).6.5 Future directionsWith more PET/MRI scanners appearing in research settings, the number of multimodaldatasets will increase in the coming years. This will spawn the development of new algo-rithms to analyze the influx of larger and more comprehensive datasets. Optimistically, futurepushes for data sharing across centres will also allow more robust characterizations of variancein imaging datasets due to disease and interventions. Code sharing across centres will allowintegration and adaptation of new and existing methodologies to construct more powerful anal-ysis frameworks, similar to Chapter 5 of this work where the previously-published rPWR/rCSTmetrics were adapted to a new application and combined with the existing SSM-PCA patternanalysis algorithm.While PET/MRI hardware improvements increase the SNR of acquired data, especially inPET there will always be a need for better denoising algorithms to further improve the SNRof reconstructed and processed data; the Poisson noise characteristics leads to a diminishingsquare-root improvement in SNR as the number of counts acquired is increased. Furthermore,to reduce concerns about radiation exposure and the cost of PET imaging, there is an ongoinginterest in reducing the injected dose and hence number of acquired counts [121–123], meaningadvanced denoising will be necessary to maintain image quality at lower dose.With regards to the specific algorithms developed in this work, future research will bedirected at combining the HYPR-based denoising methods of Chapters 2 and 4 for PET imagedenoising. The noise level in IHYPR4D is in part related to the number of iterations run,since noise monotonically increases with iteration number. Iterating is necessary to undo anybiases introduced by the initial composite. However, if the contrast in the initial composite issufficiently close to that of the (unknown) ground truth—while still having much lower noisethan the input data—then the number of iterations and hence final noise level required will belower. HYPR4D-FS instead constructs the composite via a weighted sum of basis functions,which are derived from k-means clustering in a low dimensional feature space based on theoriginal fMRI data. Thus, unlike IHYPR4D, there is no assumption of uniformity withinthe initial composite regions. A similar strategy could be adopted for dynamic PET data toinvestigate whether this helps reduce the amount of iteration necessary in IHYPR4D: the featurespace could be defined by temporal averages of different subsets of the data (corresponding to1556.5. Future directionssignificant changes in the radiotracer distribution). For instance, separate averages for uptake,equilibrium, and washout could define a 3D feature space where voxels with similar temporalevolution could be identified, then used to form the basis functions that construct the initialcomposite.The RSD algorithm was developed and tested on single-task DA release data, but in theorythe method could be extended to an arbitrary number of tasks. This may be useful in a studydesign where the task performed by the subject requires both motor and cognitive function,such as pressing buttons within a cognitive task. In this case one may wish to disentangle themotor and cognitive contributions to the spatiotemporal release pattern by performing an initialmotor control task, followed by the desired motor plus cognitive task. A two-task detectionframework to enable the above example protocol is currently in development at our centre atthe time of writing this thesis.RSD is also not designed to provide quantitative information of the magnitude and timingof the release; the present investigation focused on the spatial localization of release. However,our centre is also investigating a means to follow this spatial localization with a quantitativetemporal characterization of release, which may be important for identifying disease-relatedalterations to the timing and magnitude of task-induced DA release.Finally, the joint analysis algorithm of Chapter 5 presently analyzes two imaging metrics,but the same principal of decomposing two metrics into concurrence and discordance could beexpanded to a higher number of metrics, though the complexity of interpreting these metricswould increase. Nevertheless, this may enable an expanded understanding of neuronal functionby jointly analyzing neuronal demand (from resting-state fMRI metrics) and the correspondinglocal energetics in response (quantified by CMRO2 and CMRglu).156Bibliography[1] Sikkandhar Musafargani, Krishna Kanta Ghosh, Sachin Mishra, Pachaiyappan Mahalak-shmi, Parasuraman Padmanabhan, and Bala´zs Gulya´s. PET/MRI: a frontier in era ofcomplementary hybrid imaging. Eur. J. Hybrid Imaging, 2(1):12, June 2018.[2] Nguyen Thanh Nhu, Yu-Jung Cheng, and Shin-Da Lee. Effects of treadmill exerciseon neural mitochondrial functions in parkinson’s disease: A systematic review of animalstudies. Biomedicines, 9(8):1011, August 2021.[3] Antonio P Strafella, Ji Hyun Ko, Joshua Grant, Maria Fraraccio, and Oury Monchi.Corticostriatal functional interactions in Parkinson’s disease: a rTMS/[11C]raclopridePET study. Eur. J. Neurosci., 22(11):2946–2952, December 2005.[4] Marc C Huisman, Larissa W van Golen, Nikie J Hoetjes, Henri N Greuter, PatrickSchober, Richard G Ijzerman, Michaela Diamant, and Adriaan A Lammertsma. Cerebralblood flow and glucose metabolism in healthy volunteers measured using a high-resolutionPET scanner. EJNMMI Res., 2(1):63, November 2012.[5] Lucas Rischka, Gregor Gryglewski, Sarah Pfaff, Thomas Vanicek, Marius Hienert, Man-fred Klo¨bl, Markus Hartenbach, Alexander Haug, Wolfgang Wadsak, Markus Mitter-hauser, Marcus Hacker, Siegfried Kasper, Rupert Lanzenberger, and Andreas Hahn. Re-duced task durations in functional PET imaging with [18F]FDG approaching that offunctional MRI. Neuroimage, 181:323–330, November 2018.[6] Juliane Becker, Sarah M Schwarzenbo¨ck, and Bernd J Krause. FDG PET hybrid imaging.Recent Results Cancer Res., 216:625–667, 2020.[7] Si-Chun Gu, Qing Ye, and Can-Xing Yuan. Metabolic pattern analysis of 18F-FDGPET as a marker for Parkinson’s disease: a systematic review and meta-analysis. Rev.Neurosci., 30(7):743–756, October 2019.[8] Louise Rice and Sotirios Bisdas. The diagnostic value of FDG and amyloid PET inAlzheimer’s disease-A systematic review. Eur. J. Radiol., 94:16–24, September 2017.[9] Roger N Gunn, Mark Slifstein, Graham E Searle, and Julie C Price. Quantitative imagingof protein targets in the human brain with PET. Phys. Med. Biol., 60(22):R363–411,November 2015.157Bibliography[10] Avinash Chandra, Polytimi-Eleni Valkimadi, Gennaro Pagano, Oliver Cousins, GeorgeDervenoulas, Marios Politis, and Alzheimer’s Disease Neuroimaging Initiative. Applica-tions of amyloid, tau, and neuroinflammation PET imaging to Alzheimer’s disease andmild cognitive impairment. Hum. Brain Mapp., 40(18):5424–5442, December 2019.[11] Dennis R Schaart. Physics and technology of time-of-flight PET detectors. Phys. Med.Biol., 66(9):09TR01, April 2021.[12] CC Watson. New, faster, image-based scatter correction for 3D PET. IEEE Trans. Nucl.Sci., 47(4):1587–1594, 2000.[13] CS Levin, Y-C Tai, EJ Hoffman, M Dahlbom, and TH Farquhar. Removal of the effectof Compton scattering in 3-D whole body positron emission tomography by Monte Carlo.In 1995 IEEE Nuclear Science Symposium and Medical Imaging Conference Record, vol-ume 2, pages 1050–1054, 1995.[14] LM Carter, Adam Leon Kesner, EC Pratt, VA Sanders, AVF Massicano, CS Cutler,SE Lapi, and Jason S Lewis. The impact of positron range on PET resolution, eval-uated with phantoms and PHITS Monte Carlo simulations for conventional and non-conventional radionuclides. Mol. Imaging Biol., 22(1):73–84, February 2020.[15] B Karuta and R Lecomte. Effect of detector weighting functions on the point spreadfunction of high-resolution PET tomographs: a simulation study. IEEE Trans. Med.Imaging, 11(3):379–385, 1992.[16] Hugo WAM de Jong, Floris HP van Velden, Reina W Kloet, Fred L Buijs, Ronald Boel-laard, and Adriaan A Lammertsma. Performance evaluation of the ECAT HRRT: anLSO-LYSO double layer high resolution, high sensitivity scanner. Phys. Med. Biol., 52(5):1505–1526, March 2007.[17] Simon Stute and Claude Comtat. Practical considerations for image-based PSF and blobsreconstruction in PET. Phys. Med. Biol., 58(11):3849–3870, June 2013.[18] Arman Rahmim, Jinyi Qi, and Vesna Sossi. Resolution modeling in PET imaging: theory,practice, benefits, and pitfalls. Med. Phys., 40(6):064301, June 2013.[19] K Lange and R Carson. EM reconstruction algorithms for emission and transmissiontomography. J. Comput. Assist. Tomogr., 8(2):306–316, April 1984.[20] HM Hudson and RS Larkin. Accelerated image reconstruction using ordered subsets ofprojection data. IEEE Trans. Med. Imaging, 13(4):601–609, 1994.[21] David L Donoho and Iain M Johnstone. Ideal spatial adaptation by wavelet shrinkage.Biometrika, 81(3):425–455, 09 1994.158Bibliography[22] K Herholz. Non-stationary spatial filtering and accelerated curve fitting for parametricimaging with dynamic PET. Eur. J. Nucl. Med., 14(9-10):477–484, 1988.[23] Joyita Dutta, Richard M Leahy, and Quanzheng Li. Non-local means denoising of dynamicPET images. PLoS One, 8(12):e81390, December 2013.[24] Bradley T Christian, Nicholas T Vandehey, John M Floberg, and Charles A Mistretta.Dynamic PET denoising with HYPR processing. J. Nucl. Med., 51(7):1147–1154, July2010.[25] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional networksfor biomedical image segmentation. In Nassir Navab, Joachim Hornegger, William MWells, and Alejandro F Frangi, editors, Medical Image Computing and Computer-AssistedIntervention–MICCAI 2015, pages 234–241, Cham, 2015. Springer International Publish-ing.[26] Yan Xing, Wenli Qiao, Taisong Wang, Ying Wang, Chenwei Li, Yang Lv, Chen Xi, ShuLiao, Zheng Qian, and Jinhua Zhao. Deep learning-assisted PET imaging achieves fastscan/low-dose examination. EJNMMI Phys., 9(1):7, February 2022.[27] Si Young Yie, Seung Kwan Kang, Donghwi Hwang, and Jae Sung Lee. Self-supervisedPET denoising. Nucl. Med. Mol. Imaging, 54(6):299–304, December 2020.[28] Ivan S Klyuzhin, Ju-Chieh Cheng, Connor Bevington, and Vesna Sossi. Use of a tracer-specific deep artificial neural net to denoise dynamic PET images. IEEE Trans. Med.Imaging, 39(2):366–376, February 2020.[29] Jinyi Qi and Richard M Leahy. Iterative reconstruction techniques in emission computedtomography. Phys. Med. Biol., 51(15):R541–78, August 2006.[30] E Levitan and GT Herman. A maximum a posteriori probability expectation maxi-mization algorithm for image reconstruction in emission tomography. IEEE Trans. Med.Imaging, 6(3):185–192, 1987.[31] K Lange, M Bahn, and R Little. A theoretical study of some maximum likelihood al-gorithms for emission and transmission tomography. IEEE Trans. Med. Imaging, 6(2):106–114, 1987.[32] Stuart Ganan and D McClure. Bayesian image analysis: An application to single photonemission tomography. Amer. Statist. Assoc, pages 12–18, 1985.[33] JE Bowsher, VE Johnson, TG Turkington, RJ Jaszczak, CR Floyd, and RE Coleman.Bayesian reconstruction and use of anatomical a priori information for emission tomog-raphy. IEEE Trans. Med. Imaging, 15(5):673–686, 1996.159Bibliography[34] Matthias J Ehrhardt, Pawel Markiewicz, Maria Liljeroth, Anna Barnes, VilleKolehmainen, John S Duncan, Luis Pizarro, David Atkinson, Brian F Hutton, SebastienOurselin, Kris Thielemans, and Simon R Arridge. PET reconstruction with an anatom-ical MRI prior using parallel level sets. IEEE Trans. Med. Imaging, 35(9):2189–2199,September 2016.[35] Guobao Wang and Jinyi Qi. PET image reconstruction using kernel method. IEEE Trans.Med. Imaging, 34(1):61–71, January 2015.[36] Will Hutchcroft, Guobao Wang, Kevin T Chen, Ciprian Catana, and Jinyi Qi.Anatomically-aided PET reconstruction using the kernel method. Phys. Med. Biol., 61(18):6668–6683, September 2016.[37] Guobao Wang. High temporal-resolution dynamic PET image reconstruction using a newspatiotemporal kernel method. IEEE Trans. Med. Imaging, 38(3):664–674, March 2019.[38] Ju-Chieh Kevin Cheng, Connor Bevington, Arman Rahmim, Ivan Klyuzhin, JulianMatthews, Ronald Boellaard, and Vesna Sossi. Dynamic PET image reconstruction uti-lizing intrinsic data-driven HYPR4D denoising kernel. Med. Phys., 48(5):2230–2244, May2021.[39] Ju-Chieh Kevin Cheng, Connor WJ Bevington, and Vesna Sossi. HYPR4D kernel methodon TOF PET data with validations including image-derived input function. EJNMMIPhys., 9(1):78, November 2022.[40] RN Gunn, SR Gunn, and VJ Cunningham. Positron emission tomography compartmentalmodels. J. Cereb. Blood Flow Metab., 21(6):635–652, June 2001.[41] AA Lammertsma and SP Hume. Simplified reference tissue model for PET receptorstudies. Neuroimage, 4(3):153–158, December 1996.[42] Yanjun Wu and Richard E Carson. Noise reduction in the simplified reference tissuemodel for neuroreceptor functional imaging. J. Cereb. Blood Flow Metab., 22(12):1440–1452, December 2002.[43] MA Mintun, ME Raichle, MR Kilbourn, GF Wooten, and MJ Welch. A quantitativemodel for the in vivo assessment of drug binding sites with positron emission tomography.Ann. Neurol., 15(3):217–227, March 1984.[44] Robert B Innis, Vincent J Cunningham, Jacques Delforge, Masahiro Fujita, AlbertGjedde, Roger N Gunn, James Holden, Sylvain Houle, Sung-Cheng Huang, MasanoriIchise, Hidehiro Iida, Hiroshi Ito, Yuichi Kimura, Robert A Koeppe, Gitte M Knud-sen, Juhani Knuuti, Adriaan A Lammertsma, Marc Laruelle, Jean Logan, Ralph PaulMaguire, Mark A Mintun, Evan D Morris, Ramin Parsey, Julie C Price, Mark Slifstein,160BibliographyVesna Sossi, Tetsuya Suhara, John R Votaw, Dean F Wong, and Richard E Carson. Con-sensus nomenclature for in vivo imaging of reversibly binding radioligands. J. Cereb.Blood Flow Metab., 27(9):1533–1539, May 2007.[45] RN Gunn, AA Lammertsma, SP Hume, and VJ Cunningham. Parametric imaging ofligand-receptor binding in PET using a simplified reference region model. Neuroimage, 6(4):279–287, November 1997.[46] Masanori Ichise, Jeih-San Liow, Jian-Qiang Lu, Akihiro Takano, Kendra Model, HiroshiToyama, Tetsuya Suhara, Kazutoshi Suzuki, Robert B Innis, and Richard E Carson. Lin-earized reference tissue parametric imaging methods: application to [11C]DASB positronemission tomography studies of the serotonin transporter in human brain. J. Cereb. BloodFlow Metab., 23(9):1096–1112, September 2003.[47] Marc D Normandin, Wynne K Schiffer, and Evan D Morris. A linear model for estimationof neurotransmitter response profiles from dynamic PET data. Neuroimage, 59(3):2689–2699, February 2012.[48] Shuo Wang, Sujin Kim, Kelly P Cosgrove, and Evan D Morris. A framework for design-ing dynamic lp-ntPET studies to maximize the sensitivity to transient neurotransmitterresponses to drugs: Application to dopamine and smoking. Neuroimage, 146:701–714,February 2017.[49] Su Jin Kim, Jenna M Sullivan, Shuo Wang, Kelly P Cosgrove, and Evan D Morris. Vox-elwise lp-ntPET for detecting localized, transient dopamine release of unknown timing:Sensitivity analysis and application to cigarette smoking in the PET scanner. Hum. BrainMapp., 35(9):4876–4891, 2014.[50] Mattia Veronese, Gaia Rizzo, Alessandra Bertoldo, and Federico E Turkheimer. Spectralanalysis of dynamic PET studies: A review of 20 years of method developments andapplications. Comput. Math. Methods Med., 2016:7187541, December 2016.[51] CS Patlak, RG Blasberg, and JD Fenstermacher. Graphical evaluation of blood-to-braintransfer constants from multiple-time uptake data. J. Cereb. Blood Flow Metab., 3(1):1–7, March 1983.[52] Herman Y Carr and Edward M Purcell. Effects of diffusion on free precession in nuclearmagnetic resonance experiments. Phys. Rev., 94(3):630, 1954.[53] Stig Ljunggren. Imaging methods. J. Magn. Reson., 54:338–343, 1983.[54] DB Twieg. The k-trajectory formulation of the NMR imaging process with applications inanalysis and synthesis of imaging methods. Med. Phys., 10(5):610–621, September 1983.161Bibliography[55] M Ethan MacDonald and G Bruce Pike. MRI of healthy brain aging: A review. NMRBiomed., 34(9):e4564, September 2021.[56] PJ Wright, OE Mougin, JJ Totman, AM Peters, MJ Brookes, R Coxon, PE Morris,M Clemence, ST Francis, RW Bowtell, and PA Gowland. Water proton T1 measurementsin brain tissue at 7, 3, and 1.5 T using IR-EPI, IR-TSE, and MPRAGE: results andoptimization. MAGMA, 21(1-2):121–130, March 2008.[57] JA Detre, JS Leigh, DS Williams, and AP Koretsky. Perfusion imaging. Magn. Reson.Med., 23(1):37–45, January 1992.[58] S Ogawa, TM Lee, AR Kay, and DW Tank. Brain magnetic resonance imaging withcontrast dependent on blood oxygenation. Proc. Natl. Acad. Sci. U. S. A., 87(24):9868–9872, December 1990.[59] PJ Basser, J Mattiello, and D LeBihan. MR diffusion tensor spectroscopy and imaging.Biophys. J., 66(1):259–267, January 1994.[60] JR Reichenbach, R Venkatesan, DJ Schillinger, DK Kido, and EM Haacke. Small vesselsin the human brain: MR venography with deoxyhemoglobin as an intrinsic contrast agent.Radiology, 204(1):272–277, July 1997.[61] David C Alsop, John A Detre, Xavier Golay, Matthias Gu¨nther, Jeroen Hendrikse, LuisHernandez-Garcia, Hanzhang Lu, Bradley J MacIntosh, Laura M Parkes, Marion Smits,Matthias JP van Osch, Danny JJ Wang, Eric C Wong, and Greg Zaharchuk. Recom-mended implementation of arterial spin-labeled perfusion MRI for clinical applications:A consensus of the ISMRM perfusion study group and the European consortium for ASLin dementia. Magn. Reson. Med., 73(1):102–116, January 2015.[62] Colin P Derdeyn, Tom O Videen, Kent D Yundt, Susanne M Fritsch, David A Carpenter,Robert L Grubb, and William J Powers. Variability of cerebral blood volume and oxygenextraction: stages of cerebral haemodynamic impairment revisited. Brain, 125(3):595–607, March 2002.[63] Masanobu Ibaraki, Eku Shimosegawa, Shuichi Miura, Kazuhiro Takahashi, Hiroshi Ito,Iwao Kanno, and Jun Hatazawa. PET measurements of CBF, OEF, and CMRO2 withoutarterial sampling in hyperacute ischemic stroke: method and error analysis. Ann. Nucl.Med., 18(1):35–44, February 2004.[64] Takashi Temma, Yuji Kuge, Kohei Sano, Junko Kamihashi, Naoyuki Obokata, HidekazuKawashima, Yasuhiro Magata, and Hideo Saji. PET O-15 cerebral blood flow andmetabolism after acute stroke in spontaneously hypertensive rats. Brain Res., 1212:18–24, May 2008.162Bibliography[65] Shun Zhang, Junghun Cho, Thanh D Nguyen, Pascal Spincemaille, Ajay Gupta, WenzhenZhu, and Yi Wang. Initial experience of challenge-free MRI-based oxygen extractionfraction mapping of ischemic stroke at various stages: Comparison with perfusion anddiffusion mapping. Front. Neurosci., 14:535441, September 2020.[66] Andrei G Vlassenko, Jonathan McConathy, Lars E Couture, Yi Su, Parinaz Mas-soumzadeh, Hayden S Leeds, Michael R Chicoine, David D Tran, Jiayi Huang, SonikaDahiya, Daniel S Marcus, Sarah Jost Fouke, Keith M Rich, Marcus E Raichle, and Tam-mie LS Benzinger. Aerobic glycolysis as a marker of tumor aggressiveness: Preliminarydata in high grade human brain tumors. Dis. Markers, 2015:874904, September 2015.[67] Daniel Paech, Armin M Nagel, Miriam N Schultheiss, Reiner Umathum, Sebastian Regn-ery, Moritz Scherer, Antje Wick, Tanja Platt, Wolfgang Wick, Martin Bendszus, AndreasUnterberg, Heinz-Peter Schlemmer, Mark E Ladd, and Sebastian C Niesporek. Quanti-tative dynamic oxygen 17 MRI at 7.0 T for the cerebral oxygen metabolism in glioma.Radiology, 295(1):181–189, April 2020.[68] Patrick LY Tang, Alejandra Me´ndez Romero, Jaap PM Jaspers, and Esther AH Warnert.The potential of advanced MR techniques for precision radiotherapy of glioblastoma.MAGMA, 35(1):127–143, February 2022.[69] Yulin Ge, Zhongwei Zhang, Hanzhang Lu, Lin Tang, Hina Jaggi, Joseph Herbert, James SBabb, Henry Rusinek, and Robert I Grossman. Characterizing brain oxygen metabolismin patients with multiple sclerosis with T2-relaxation-under-spin-tagging MRI. J. Cereb.Blood Flow Metab., 32(3):403–412, March 2012.[70] Peter Klinkmueller, Martin Kronenbuerger, Xinyuan Miao, Jee Bang, Kia E Ultz, AdrianPaez, Xiaoyu Zhang, Wenzhen Duan, Russell L Margolis, Peter Cm van Zijl, Christo-pher A Ross, and Jun Hua. Impaired response of cerebral oxygen metabolism to visualstimulation in Huntington’s disease. J. Cereb. Blood Flow Metab., 41(5):1119–1130, May2021.[71] W Hudson Robb, Omair A Khan, Humza A Ahmed, Judy Li, Elizabeth E Moore, Fran-cis E Cambronero, Kimberly R Pechman, Dandan Liu, Katherine A Gifford, Bennett ALandman, Manus J Donahue, Timothy J Hohman, and Angela L Jefferson. Lower cere-bral oxygen utilization is associated with Alzheimer’s disease-related neurodegenerationand poorer cognitive performance among apolipoprotein E ϵ4 carriers. J. Cereb. BloodFlow Metab., 42(4):642–655, April 2022.[72] Joel Aanerud, Per Borghammer, M Mallar Chakravarty, Kim Vang, Anders B Rodell,Kristjana Y Jo´nsdottir, Arne Møller, Mahmoud Ashkanian, Manouchehr S Vafaee, PeterIversen, Peter Johannsen, and Albert Gjedde. Brain energy metabolism and blood flowdifferences in healthy aging. J. Cereb. Blood Flow Metab., 32(7):1177–1187, July 2012.163Bibliography[73] Sarah J Catchlove, Helen Macpherson, Matthew E Hughes, Yufen Chen, Todd B Parrish,and Andrew Pipingas. An investigation of cerebral oxygen utilization, blood flow andcognition in healthy aging. PLoS One, 13(5):e0197055, May 2018.[74] S Neil Vaishnavi, Andrei G Vlassenko, Melissa M Rundle, Abraham Z Snyder, Mark AMintun, and Marcus E Raichle. Regional aerobic glycolysis in the human brain. Proc.Natl. Acad. Sci. U. S. A., 107(41):17757–17762, October 2010.[75] Andrei G Vlassenko, S Neil Vaishnavi, Lars Couture, Dana Sacco, Benjamin J Shannon,Robert H Mach, John C Morris, Marcus E Raichle, and Mark A Mintun. Spatial correla-tion between brain aerobic glycolysis and amyloid-β (Aβ) deposition. Proc. Natl. Acad.Sci. U. S. A., 107(41):17763–17767, October 2010.[76] William J Powers, Tom O Videen, Joanne Markham, Kevin J Black, Nima Golchin, andJoel S Perlmutter. Cerebral mitochondrial metabolism in early Parkinson’s disease. J.Cereb. Blood Flow Metab., 28(10):1754–1760, October 2008.[77] William J Powers, Tom O Videen, Joanne Markham, Lori McGee-Minnich, Joann VAntenor-Dorsey, Tamara Hershey, and Joel S Perlmutter. Selective defect of in vivoglycolysis in early Huntington’s disease striatum. Proc. Natl. Acad. Sci. U. S. A., 104(8):2945–2949, February 2007.[78] H An, W Lin, A Celik, and Y Z Lee. Quantitative measurements of cerebral metabolicrate of oxygen utilization using MRI: a volunteer study. NMR Biomed., 14(7-8):441–447,November 2001.[79] Richard GWise, Ashley D Harris, Alan J Stone, and Kevin Murphy. Measurement of OEFand absolute CMRO2: MRI-based methods using interleaved and combined hypercapniaand hyperoxia. Neuroimage, 83:135–147, December 2013.[80] Audrey P Fan, Hongyu An, Farshad Moradi, Jarrett Rosenberg, Yosuke Ishii, TadashiNariai, Hidehiko Okazawa, and Greg Zaharchuk. Quantification of brain oxygen extrac-tion and metabolism with [15O]-gas PET: A technical review in the era of PET/MRI.Neuroimage, 220(117136):117136, October 2020.[81] Zachary B Rodgers, John A Detre, and Felix W Wehrli. MRI-based methods for quan-tification of the cerebral metabolic rate of oxygen. J. Cereb. Blood Flow Metab., 36(7):1165–1185, July 2016.[82] Dengrong Jiang and Hanzhang Lu. Cerebral oxygen extraction fraction MRI: Techniquesand applications. Magn. Reson. Med., 88(2):575–600, August 2022.[83] TL Davis, KK Kwong, RMWeisskoff, and BR Rosen. Calibrated functional MRI: mappingthe dynamics of oxidative metabolism. Proc. Natl. Acad. Sci. U. S. A., 95(4):1834–1839,February 1998.164Bibliography[84] Xiang He and Dmitriy A Yablonskiy. Quantitative BOLD: mapping of human cerebraldeoxygenated blood volume and oxygen extraction fraction: default state. Magn. Reson.Med., 57(1):115–126, January 2007.[85] DS Bolar, BR Rosen, AG Sorensen, and E Adalsteinsson. QUantitative Imaging of eX-traction of oxygen and TIssue consumption (QUIXOTIC) using venular-targeted velocity-selective spin labeling. Magn. Reson. Med., 66(6):1550–1562, December 2011.[86] Jia Guo and Eric CWong. Venous oxygenation mapping using velocity-selective excitationand arterial nulling. Magn. Reson. Med., 68(5):1458–1471, November 2012.[87] Jingwei Zhang, Tian Liu, Ajay Gupta, Pascal Spincemaille, Thanh D Nguyen, andYi Wang. Quantitative mapping of cerebral metabolic rate of oxygen (CMRO2) usingquantitative susceptibility mapping (QSM). Magn. Reson. Med., 74(4):945–952, October2015.[88] Junghun Cho, Youngwook Kee, Pascal Spincemaille, Thanh D Nguyen, Jingwei Zhang,Ajay Gupta, Shun Zhang, and Yi Wang. Cerebral metabolic rate of oxygen (CMRO2)mapping by combining quantitative susceptibility mapping (QSM) and quantitative bloodoxygenation level-dependent imaging (qBOLD). Magn. Reson. Med., 80(4):1595–1604,October 2018.[89] Junghun Cho, Shun Zhang, Youngwook Kee, Pascal Spincemaille, Thanh D Nguyen, Si-mon Hubertus, Ajay Gupta, and Yi Wang. Cluster analysis of time evolution (CAT)for quantitative susceptibility mapping (QSM) and quantitative blood oxygen level-dependent magnitude (qBOLD)-based oxygen extraction fraction (OEF) and cerebralmetabolic rate of oxygen (CMRO2) mapping. Magn. Reson. Med., 83(3):844–857, March2020.[90] Dmitry Kurzhunov, Robert Borowiak, Marco Reisert, Ali Caglar O¨zen, and Michael Bock.Direct estimation of 17 O MR images (DIESIS) for quantification of oxygen metabolismin the human brain with partial volume correction. Magn. Reson. Med., 80(6):2717–2725,December 2018.[91] Sebastian C Niesporek, Reiner Umathum, Jonathan M Lommen, Nicolas GR Behl, DanielPaech, Peter Bachert, Mark E Ladd, and Armin M Nagel. Reproducibility of CMRO2determination using dynamic 17 O MRI. Magn. Reson. Med., 79(6):2923–2934, June2018.[92] Peter A Chiarelli, Daniel P Bulte, Richard Wise, Daniel Gallichan, and Peter Jezzard. Acalibration method for quantitative BOLD fMRI based on hyperoxia. Neuroimage, 37(3):808–820, September 2007.165Bibliography[93] JL Boxerman, LM Hamberg, BR Rosen, and RM Weisskoff. MR contrast due to intravas-cular magnetic susceptibility perturbations. Magn. Reson. Med., 34(4):555–566, October1995.[94] Robert L Grubb, Jr, Marcus E Raichle, John O Eichling, and Michel M Ter-Pogossian.The effects of changes in PaCO2: cerebral blood volume, blood flow, and vascular meantransit time. Stroke, 5(5):630–639, September 1974.[95] J Jean Chen and G Bruce Pike. Global cerebral oxidative metabolism during hypercapniaand hypocapnia in humans: implications for BOLD fMRI. J. Cereb. Blood Flow Metab.,30(6):1094–1099, June 2010.[96] J Jean Chen and G Bruce Pike. BOLD-specific cerebral blood volume and blood flowchanges during neuronal activation in humans. NMR Biomed., 22(10):1054–1062, Decem-ber 2009.[97] Valerie EM Griffeth and Richard B Buxton. A theoretical framework for estimatingcerebral oxygen metabolism changes using the calibrated-BOLD method: modeling theeffects of blood volume distribution, hematocrit, oxygen extraction fraction, and tissuesignal properties on the BOLD signal. Neuroimage, 58(1):198–212, September 2011.[98] DP Bulte, M Kelly, M Germuska, J Xie, MA Chappell, TW Okell, MG Bright, andP Jezzard. Quantitative measurement of cerebral physiology using respiratory-calibratedMRI. Neuroimage, 60(1):582–591, March 2012.[99] CJ Gauthier and RD Hoge. Magnetic resonance imaging of resting OEF and CMRO2using a generalized calibration model for hypercapnia and hyperoxia. Neuroimage, 60(2):1212–1225, April 2012.[100] JW Severinghaus. Simple, accurate equations for human blood O2 dissociation computa-tions. J. Appl. Physiol: Respirat. Environ. Exercise Physiol., 46(3):599–602, 1979.[101] S Gustard, EJ Williams, LD Hall, JD Pickard, and TA Carpenter. Influence of baselinehematocrit on between-subject BOLD signal change using gradient echo and asymmetricspin echo EPI. Magn. Reson. Imaging, 21(6):599–607, July 2003.[102] Elise Klæstrup, Torleif Trydal, Jan F Pedersen, Jacob M Larsen, Søren Lundbye-Christensen, and Søren Risom Kristensen. Reference intervals and age and gender depen-dency for arterial blood gases and electrolytes in adults. Clin. Chem. Lab. Med., 49(9):1495–1500, September 2011.[103] CC Chernecky and Berger BJ. Blood gases, arterial (ABG) - blood. In: Chernecky CC,Berger BJ, eds. Laboratory Tests and Diagnostic Procedures. Elsevier Saunders, St. Louis,MO, 6 edition, 2013.166Bibliography[104] JR Moeller and SC Strother. A regional covariance approach to the analysis of functionalpatterns in positron emission tomographic data. J. Cereb. Blood Flow Metab., 11(2):A121–35, March 1991.[105] David Eidelberg. Metabolic brain networks in neurodegenerative disorders: a functionalimaging approach. Trends Neurosci., 32(10):548–557, October 2009.[106] Yilong Ma, Chengke Tang, Phoebe G Spetsieris, Vijay Dhawan, and David Eidelberg.Abnormal metabolic network activity in Parkinson’s disease: test–retest reproducibility.J. Cereb. Blood Flow Metab., 27(3):597–605, March 2007.[107] Alexander M Grant, Timothy W Deller, Mohammad Mehdi Khalighi, Sri Harsha Maram-raju, Gaspar Delso, and Craig S Levin. NEMA NU 2-2012 performance studies for theSiPM-based ToF-PET component of the GE SIGNA PET/MR system. Med. Phys., 43(5):2334, May 2016.[108] Julia G Mannheim, Ju-Chieh Kevin Cheng, Nasim Vafai, Elham Shahinfard, Carolyn En-glish, Jessamyn McKenzie, Jing Zhang, Laura Barlow, and Vesna Sossi. Cross-validationstudy between the HRRT and the PET component of the SIGNA PET/MRI system withfocus on neuroimaging. EJNMMI Phys., 8(1):20, February 2021.[109] John M Floberg, Charles A Mistretta, Jamey P Weichert, Lance T Hall, James E Holden,and Bradley T Christian. Improved kinetic analysis of dynamic PET data with optimizedHYPR-LR. Med. Phys., 39(6):3319–3331, June 2012.[110] James Bland, Abolfazl Mehranian, Martin A Belzunce, Sam Ellis, Casper da Costa-Luis,Colm J McGinnity, Alexander Hammers, and Andrew J Reader. Intercomparison of MR-informed PET image reconstruction methods. Med. Phys., 46(11):5055–5074, November2019.[111] Ziyue Xu, Mingchen Gao, Georgios Z Papadakis, Brian Luna, Sanjay Jain, Daniel JMollura, and Ulas Bagci. Joint solution for PET image segmentation, denoising, andpartial volume correction. Med. Image Anal., 46:229–243, May 2018.[112] C Comtat, F Bataille, C Michel, JP Jones, M Sibomana, L Janeiro, and R Trebossen.OSEM-3D reconstruction strategies for the ECAT HRRT. In IEEE Symposium Confer-ence Record Nuclear Science 2004, volume 6, pages 3492–3496, 2004.[113] Clinical implementation of VUE point FX. White Paper PET-0228-09.09-EN-US, GEHealthcare, Waukesha, WI, September 2009.[114] Georgios I Angelis, John E Gillam, William J Ryder, Roger R Fulton, and Steven RMeikle. Direct estimation of voxel-wise neurotransmitter response maps from dynamicPET data. IEEE Trans. Med. Imaging, 38(6):1371–1383, June 2019.167Bibliography[115] Marjorie Villien, Hsiao-Ying Wey, Joseph B Mandeville, Ciprian Catana, Jonathan RPolimeni, Christin Y Sander, Nicole R Zu¨rcher, Daniel B Chonde, Joanna S Fowler,Bruce R Rosen, and Jacob M Hooker. Dynamic functional imaging of brain glucoseutilization using fPET-FDG. Neuroimage, 100:192–199, October 2014.[116] Sharna D Jamadar, Phillip Gd Ward, Shenpeng Li, Francesco Sforazzini, Jakub Baran,Zhaolin Chen, and Gary F Egan. Simultaneous task-based BOLD-fMRI and [18-F] FDGfunctional PET for measurement of neuronal metabolism in the human visual cortex.Neuroimage, 189:258–266, April 2019.[117] Bruce Fischl. FreeSurfer. Neuroimage, 62(2):774–781, August 2012.[118] Ivan S Klyuzhin, Jessie F Fu, Nikolay Shenkov, Arman Rahmim, and Vesna Sossi. Use ofgenerative disease models for analysis and selection of radiomic features in PET. IEEETrans. Radiat. Plasma Med. Sci., 3(2):178–191, 2019.[119] Dennis Vriens, Lioe-Fee de Geus-Oei, Wim JG Oyen, and Eric P Visser. A curve-fittingapproach to estimate the arterial plasma input function for the assessment of glucosemetabolic rate and response to treatment. J. Nucl. Med., 50(12):1933–1939, December2009.[120] S Mouysset, H Zbib, S Stute, JM Girault, J Charara, J Noailles, S Chalon, I Buvat, andC Tauber. Segmentation of dynamic PET images with kinetic spectral clustering. Phys.Med. Biol., 58(19):6931–6944, October 2013.[121] Spencer C Behr, Emma Bahroos, Randall A Hawkins, Lorenzo Nardo, Vahid Ravanfar,Emily V Capbarat, and Youngho Seo. Quantitative and visual assessments toward po-tential sub-mSv or ultrafast FDG PET using high-sensitivity TOF PET in PET/MRI.Mol. Imaging Biol., 20(3):492–500, June 2018.[122] Pietro Zucchetta, Marco Branchini, Alessandra Zorz, Valentina Bodanza, Diego Cecchin,Marta Paiusco, and Franco Bui. Quantitative analysis of image metrics for reduced andstandard dose pediatric 18F-FDG PET/MRI examinations. Br. J. Radiol., 92(1095):20180438, March 2019.[123] Jeffrey P Schmall, Suleman Surti, Hansel J Otero, Sabah Servaes, Joel S Karp, andLisa J States. Investigating low-dose image quality in whole-body pediatric 18F-FDGscans using time-of-flight PET/MRI. J. Nucl. Med., 62(1):123–130, January 2021.[124] Ziyue Xu, Ulas Bagci, Jurgen Seidel, David Thomasson, Jeff Solomon, and Daniel JMollura. Segmentation based denoising of PET images: an iterative approach via regionalmeans and affinity propagation. Med. Image Comput. Comput. Assist. Interv., 17(1):698–705, 2014.168Bibliography[125] Sandeep SV Golla, Mark Lubberink, Bart NM van Berckel, Adriaan A Lammertsma,and Ronald Boellaard. Partial volume correction of brain PET studies using iterativedeconvolution in combination with HYPR denoising. EJNMMI Res., 7(1):36, December2017.[126] Kelly P Cosgrove, Shuo Wang, Su-Jin Kim, Erin McGovern, Nabeel Nabulsi, Hong Gao,David Labaree, Hemant D Tagare, Jenna M Sullivan, and Evan D Morris. Sex differencesin the brain’s dopamine signature of cigarette smoking. J. Neurosci., 34(50):16851–16855,December 2014.[127] Katina C Calakos, Heather Liu, Yihuan Lu, Jon Mikael Anderson, David Matuskey,Nabeel Nabulsi, Yunpeng Ye, Patrick D Skosnik, Deepak Cyril D’Souza, Evan D Morris,Kelly P Cosgrove, and Ansel T Hillmer. Assessment of transient dopamine responses tosmoked cannabis. Drug Alcohol Depend., 227:108920, October 2021.[128] Andreas Hahn, Murray B Reed, Verena Pichler, Paul Michenthaler, Lucas Rischka, God-ber M Godbersen, Wolfgang Wadsak, Marcus Hacker, and Rupert Lanzenberger. Func-tional dynamics of dopamine synthesis during monetary reward and punishment process-ing. J. Cereb. Blood Flow Metab., 41(11):2973–2985, November 2021.[129] M Laruelle. Imaging synaptic neurotransmission with in vivo binding competition tech-niques: a critical review. J. Cereb. Blood Flow Metab., 20(3):423–451, March 2000.[130] Ann M Graybiel, Toshihiko Aosaki, Alice W Flaherty, and Minoru Kimura. The basalganglia and adaptive motor control. Science, 265(5180):1826–1831, September 1994.[131] Jeffery R Wickens, John NJ Reynolds, and Brian I Hyland. Neural mechanisms of reward-related motor learning. Curr. Opin. Neurobiol., 13(6):685–690, December 2003.[132] Ethan S Bromberg-Martin, Masayuki Matsumoto, and Okihide Hikosaka. Dopamine inmotivational control: Rewarding, aversive, and alerting. Neuron, 68(5):815–834, Decem-ber 2010.[133] R de la Fuente-Fernandez, V Sossi, A Jon Stoessl, J Lu, D Calne, Z Huang, and S Fur-tado. Levodopa-induced changes in synaptic dopamine levels increase with progression ofParkinson’s disease: implications for dyskinesias. Brain, 127(12):2747–2754, November2004.[134] Cristian C Constantinescu, Charles Bouman, and Evan D Morris. Nonparametric ex-traction of transient changes in neurotransmitter concentration from dynamic PET data.IEEE Trans. Med. Imaging, 26(3):359–373, March 2007.[135] ED Morris, CC Constantinescu, JM Sullivan, MD Normandin, and LA Christopher. Non-invasive visualization of human dopamine dynamics from PET images. NeuroImage, 51(1):135–144, May 2010.169Bibliography[136] Nathaniel M Alpert, Rajendra D Badgaiyan, Elijahu Livni, and Alan J Fischman. A novelmethod for noninvasive detection of neuromodulatory changes in specific neurotransmittersystems. NeuroImage, 19(3):1049–1060, 2003. ISSN 1053-8119.[137] Evan D Morris, Karmen K Yoder, Chunzhi Wang, Marc D Normandin, Qi-Huang Zheng,Bruce Mock, Raymond F Muzic, Jr, and Janice C Froehlich. ntPET: a new applicationof PET imaging for characterizing the kinetics of endogenous neurotransmitter release.Mol. Imaging, 4(4):473–489, October 2005.[138] Rachel N Lippert, Anna Lena Cremer, Sharmili Edwin Thanarajah, Clio Korn, ThomasJahans-Price, Lauren M Burgeno, Marc Tittgemeyer, Jens C Bru¨ning, Mark E Walton,and Heiko Backes. Time-dependent assessment of stimulus-evoked regional dopaminerelease. Nat. Commun., 10(1):336, January 2019.[139] TS Woodward, TA Cairo, CC Ruff, Y Takane, MA Hunter, and ETC Ngan. Functionalconnectivity reveals load dependent neural systems underlying encoding and maintenancein verbal working memory. Neuroscience, 139(1):317–325, April 2006.[140] Martin Havlicek and Kaˆmil Uludag˘. A dynamical model of the laminar BOLD response.Neuroimage, 204(116209):116209, January 2020.[141] Masanori Ichise, James R Ballinger, Haim Golan, Douglass Vines, Angela Luong, ScottTsai, and Hank F Kung. Noninvasive quantification of dopamine D2 receptors withIodine-123-IBF SPECT. J. Nucl. Med., 37(3):513–520, 1996.[142] GI Angelis, OK Fuller, JE Gillam, and SR Meikle. Denoising non-steady state dynamicPET data using a feed-forward neural network. Phys. Med. Biol., 66(3):034001, January2021.[143] Connor WJ Bevington, Ju-Chieh (Kevin) Cheng, Ivan S Klyuzhin, Mariya V Cherkasova,Catharine A Winstanley, and Vesna Sossi. A Monte Carlo approach for improving tran-sient dopamine release detection sensitivity. J. Cereb. Blood Flow Metab., 41(1):116–131,2021.[144] Zacharie Irace, Ine´s Me´rida, Je´roˆme Redoute´, Clara Fonteneau, Marie-Franc¸oise Suaud-Chagny, Je´roˆme Brunelin, Benjamin Vidal, Luc Zimmer, Anthonin Reilhac, and NicolasCostes. Bayesian estimation of the ntPET model in single-scan competition PET studies.Front. Physiol., 11:498, May 2020.[145] Heather Liu and Evan D Morris. Model comparison metrics require adaptive correctionif parameters are discretized: Proof-of-concept applied to transient signals in dynamicPET. IEEE Trans. Med. Imaging, 39(7):2451–2460, July 2020.170Bibliography[146] A Teymurazyan, T Riauka, H-S Jans, and D Robinson. Properties of noise in positronemission tomography images reconstructed with filtered-backprojection and row-actionmaximum likelihood algorithm. J. Digit. Imaging, 26(3):447–456, June 2013.[147] Alice Egerton, Mitul A Mehta, Andrew J Montgomery, Julia M Lappin, Oliver D Howes,Suzanne J Reeves, Vincent J Cunningham, and Paul M Grasby. The dopaminergic basisof human behaviors: A review of molecular imaging studies. Neurosci. Biobehav. Rev.,33(7):1109–1132, July 2009.[148] RE Carson. PET physiological measurements using constant infusion. Nucl. Med. Biol.,27(7):657–660, October 2000.[149] Connor WJ Bevington, Ju-Chieh Cheng, and Vesna Sossi. A 4-D iterative HYPR denois-ing operator improves PET image quality. IEEE Trans. Radiat. Plasma Med. Sci., 6(6):641–655, 2022.[150] H˚akan Hall, Go¨ran Sedvall, Olle Magnusson, Jutta Kopp, Christer Halldin, and LarsFarde. Distribution of D1- and D2-dopamine receptors, and dopamine and its metabolitesin the human brain. Neuropsychopharmacology, 11(4):245–256, December 1994.[151] S Pappata, S Dehaene, JB Poline, MC Gregoire, A Jobert, J Delforge, V Frouin, M Bot-tlaender, F Dolle, L Di Giamberardino, and A Syrota. In vivo detection of striataldopamine release during reward: A PET study with [11C]raclopride and a single dy-namic scan approach. Neuroimage, 16(4):1015–1027, August 2002.[152] Diego Cecchin, Alessandro Palombit, Marco Castellaro, Erica Silvestri, Franco Bui, Hen-ryk Barthel, Osama Sabri, Maurizio Corbetta, and Alessandra Bertoldo. Brain PET andfunctional MRI: why simultaneously using hybrid PET/MR systems? Q. J. Nucl. Med.Mol. Imaging, 61(4):345–359, December 2017.[153] Giacomo Tondo, Marcello Esposito, George Dervenoulas, Heather Wilson, Marios Politis,and Gennaro Pagano. Hybrid PET-MRI applications in movement disorders. Int. Rev.Neurobiol., 144:211–257, 2019.[154] Nicole R Zu¨rcher, Erin C Walsh, Rachel D Phillips, Paul M Cernasov, Chieh-En J Tseng,Ayarah Dharanikota, Eric Smith, Zibo Li, Jessica L Kinard, Joshua C Bizzell, Rachel KGreene, Daniel Dillon, Diego A Pizzagalli, David Izquierdo-Garcia, Kinh Truong, DavidLalush, Jacob M Hooker, and Gabriel S Dichter. A simultaneous [11C]raclopride positronemission tomography and functional magnetic resonance imaging investigation of striataldopamine binding in autism. Transl. Psychiatry, 11(1):33, January 2021.[155] Filip Grill, Jarkko Johansson, Jan Axelsson, Patrik Brynolfsson, Lars Nyberg, and AnnaRieckmann. Dissecting motor and cognitive component processes of a finger-tapping task171Bibliographywith hybrid dopamine positron emission tomography and functional Magnetic ResonanceImaging. Front. Hum. Neurosci., 15:733091, November 2021.[156] Ke Zhang, Hans Herzog, Jo¨rg Mauler, Christian Filss, Thomas W Okell, Elena Rota Kops,Lutz Tellmann, Thomas Fischer, Burkhard Brocke, Walter Sturm, Heinz H Coenen, andN Jon Shah. Comparison of cerebral blood flow acquired by simultaneous [15O]waterpositron emission tomography and arterial spin labeling magnetic resonance imaging. J.Cereb. Blood Flow Metab., 34(8):1373–1380, August 2014.[157] Markus Fahlstro¨m, Lieuwe Appel, Eva Kumlien, Torsten Danfors, Mathias Engstro¨m,Johan Wikstro¨m, Gunnar Antoni, Elna-Marie Larsson, and Mark Lubberink. Evaluationof arterial spin labeling MRI-comparison with 15O-water PET on an integrated PET/MRscanner. Diagnostics (Basel), 11(5), May 2021.[158] AM Smith, BK Lewis, UE Ruttimann, FQ Ye, TM Sinnwell, Y Yang, JH Duyn, andJA Frank. Investigation of low frequency drift in fMRI signal. Neuroimage, 9(5):526–533,May 1999.[159] J Astola and P Kuosmanen. Fundamentals of nonlinear digital filtering. CRC Press, BocaRaton, FL, 1997.[160] Junghun Cho, John Lee, Hongyu An, Manu S Goyal, Yi Su, and Yi Wang. Cerebral oxygenextraction fraction (OEF): Comparison of challenge-free gradient echo QSM+qBOLD(QQ) with 15O PET in healthy adults. J. Cereb. Blood Flow Metab., 41(7):1658–1668,July 2021.[161] Paul T Fillmore, Michelle C Phillips-Meek, and John E Richards. Age-specific MRI brainand head templates for healthy adults from 20 through 89 years of age. Front. AgingNeurosci., 7:44, April 2015.[162] J Annen, L Heine, E Ziegler, G Frasso, M Bahri, C Di Perri, J Stender, C Martial,S Wannez, K D’ostilio, E Amico, G Antonopoulos, C Bernard, F Tshibanda, R Hustinx,and S Laureys. Function-structure connectivity in patients with severe brain injury asmeasured by MRI-DWI and FDG-PET. Hum. Brain Mapp., 37(11):3707–3720, November2016.[163] Ehsan Shokri-Kojori, Dardo Tomasi, Babak Alipanahi, Corinde E Wiers, Gene-JackWang, and Nora D Volkow. Correspondence between cerebral glucose metabolism andBOLD reveals relative power and cost in human brain. Nat. Commun., 10(1):690, Febru-ary 2019.[164] Shaozhen Yan, Chaojie Zheng, Bixiao Cui, Zhigang Qi, Zhilian Zhao, Yanhong An, LiyanQiao, Ying Han, Yun Zhou, and Jie Lu. Multiparametric imaging hippocampal neu-172Bibliographyrodegeneration and functional connectivity with simultaneous PET/MRI in alzheimer’sdisease. Eur. J. Nucl. Med. Mol. Imaging, 47(10):2440–2452, September 2020.[165] Sharna D Jamadar, Phillip G D Ward, Emma X Liang, Edwina R Orchard, ZhaolinChen, and Gary F Egan. Metabolic and hemodynamic resting-state connectivity of thehuman brain: A high-temporal resolution simultaneous BOLD-fMRI and FDG-fPETmultimodality study. Cereb. Cortex, 31(6):2855–2867, May 2021.[166] Gayane Aghakhanyan, M Rullmann, J Rumpf, M L Schroeter, C Scherlach, M Patt,M Brendel, N Koglin, A W Stephens, J Classen, K T Hoffmann, O Sabri, and H Barthel.Interplay of tau and functional network connectivity in progressive supranuclear palsy: a[18F]PI-2620 PET/MRI study. Eur. J. Nucl. Med. Mol. Imaging, 50(1):103–114, Decem-ber 2022.[167] Jessie Fanglu Fu, Ivan Klyuzhin, Jessamyn McKenzie, Nicole Neilson, Elham Shahinfard,Katie Dinelle, Martin J McKeown, A Jon Stoessl, and Vesna Sossi. Joint pattern analysisapplied to PET DAT and VMAT2 imaging reveals new insights into parkinson’s diseaseinduced presynaptic alterations. NeuroImage Clin., 23(101856):101856, May 2019.[168] K J Worsley, J B Poline, K J Friston, and A C Evans. Characterizing the response of PETand fMRI data using multivariate linear models. Neuroimage, 6(4):305–319, November1997.[169] Xiaofeng Zhu, Heung-Il Suk, and Dinggang Shen. Multi-modality canonical feature se-lection for alzheimer’s disease diagnosis. Med. Image Comput. Comput. Assist. Interv.,17(Pt 2):162–169, 2014.[170] Xiaofeng Zhu, Heung-Il Suk, Seong-Whan Lee, and Dinggang Shen. Canonical featureselection for joint regression and multi-class identification in alzheimer’s disease diagnosis.Brain Imaging Behav., 10(3):818–828, September 2016.[171] Daniel M Stout, Monte S Buchsbaum, Andrea D Spadoni, Victoria B Risbrough, Irina AStrigo, Scott C Matthews, and Alan N Simmons. Multimodal canonical correlation re-veals converging neural circuitry across trauma-related disorders of affect and cognition.Neurobiol. Stress, 9:241–250, November 2018.[172] Hyunjin Park, David Wood, Hero Hussain, Charles R Meyer, Rajal B Shah, Timothy DJohnson, Thomas Chenevert, and Morand Piert. Introducing parametric fusion PET/MRIof primary prostate cancer. J. Nucl. Med., 53(4):546–551, April 2012.[173] Aaron F Struck, Lance T Hall, Joanna E Kusmirek, Catherine L Gallagher, John MFloberg, Christine J Jaskowiak, and Scott B Perlman. (18)F-DOPA PET with and with-out MRI fusion, a receiver operator characteristics comparison. Am. J. Nucl. Med. Mol.Imaging, 2(4):475–482, October 2012.173Bibliography[174] Yingbing Wang, Ruth Lim, Ranish Deedar Ali Khawaja, and Michael A Blake. Improvingconcordance of MRI and PET/CT interpretations with retrospectively coregistered MRIand PET/CT data sets. Curr. Probl. Diagn. Radiol., 44(3):232–236, May 2015.[175] Bicao Li, Jenq-Neng Hwang, Zhoufeng Liu, Chunlei Li, and Zongmin Wang. PET andMRI image fusion based on a dense convolutional network with dual attention. Comput.Biol. Med., 151(Pt B):106339, December 2022.[176] Otto Warburg. The Metabolism of Carcinoma Cells. J. Cancer Res., 9(1):148–163, 031925.[177] Maria V Liberti and Jason W Locasale. The Warburg Effect: How does it benefit cancercells? Trends Biochem. Sci., 41(3):211–218, March 2016.[178] Sudhakar Raja Subramaniam and Marie-Francoise Chesselet. Mitochondrial dysfunctionand oxidative stress in Parkinson’s disease. Prog. Neurobiol., 106-107:17–32, July 2013.[179] Christopher G Goetz, Barbara C Tilley, Stephanie R Shaftman, Glenn T Stebbins, Stan-ley Fahn, Pablo Martinez-Martin, Werner Poewe, Cristina Sampaio, Matthew B Stern,Richard Dodel, Bruno Dubois, Robert Holloway, Joseph Jankovic, Jaime Kulisevsky, An-thony E Lang, Andrew Lees, Sue Leurgans, Peter A LeWitt, David Nyenhuis, C WarrenOlanow, Olivier Rascol, Anette Schrag, Jeanne A Teresi, Jacobus J van Hilten, NancyLaPelle, and Movement Disorder Society UPDRS Revision Task Force. Movement dis-order society-sponsored revision of the unified parkinson’s disease rating scale (MDS-UPDRS): scale presentation and clinimetric testing results. Mov. Disord., 23(15):2129–2170, November 2008.[180] Phoebe Spetsieris, Yilong Ma, Shichun Peng, Ji Hyun Ko, Vijay Dhawan, Chris C Tang,and David Eidelberg. Identification of disease-related spatial covariance patterns usingneuroimaging data. J. Vis. Exp., (76), June 2013.[181] Per Borghammer, Paul Cumming, Karen Østergaard, Albert Gjedde, Anders Rodell,Christopher J Bailey, and Manoucher S Vafaee. Cerebral oxygen metabolism in patientswith early Parkinson’s disease. J. Neurol. Sci., 313(1-2):123–128, February 2012.[182] Swati Rane, Natalie Koh, John Oakley, Christina Caso, Cyrus P Zabetian, Brenna Choler-ton, Thomas J Montine, and Thomas Grabowski. Arterial spin labeling detects perfusionpatterns related to motor symptoms in Parkinson’s disease. Parkinsonism Relat. Disord.,76:21–28, July 2020.[183] Yilong Ma, Chaorui Huang, Jonathan P Dyke, Hong Pan, David Alsop, Andrew Fei-gin, and David Eidelberg. Parkinson’s disease spatial covariance pattern: noninvasivequantification with perfusion MRI. J. Cereb. Blood Flow Metab., 30(3):505–509, March2010.174[184] Heiko Braak, Kelly Del Tredici, Udo Ru¨b, Rob AI de Vos, Ernst NH Jansen Steur, andEva Braak. Staging of brain pathology related to sporadic Parkinson’s disease. Neurobiol.Aging, 24(2):197–211, March 2003.[185] Nora D Volkow, Gene-Jack Wang, Joanna S Fowler, Dardo Tomasi, and Frank Telang.Addiction: beyond dopamine reward circuitry. Proc. Natl. Acad. Sci. U. S. A., 108(37):15037–15042, 2011.[186] David J Nutt, Anne Lingford-Hughes, David Erritzoe, and Paul RA Stokes. The dopaminetheory of addiction: 40 years of highs and lows. Nat. Rev. Neurosci., 16(5):305–312, May2015.[187] Roy A Wise and Mykel A Robble. Dopamine and addiction. Annu. Rev. Psychol., 71(1):79–106, January 2020.175Appendix AEffects of inaccurate segmentationor PSF estimate on IHYPR4DA.1 Effect of inaccurate segmentationImplicit in the IHYPR4D formalism is that the PET images and the segmented image are per-fectly co-registered. This is not an issue if the PET data is used to determine segmentation,but if one elects to use another modality to perform segmentation—such as the MRI segmen-tation method investigated in this work—then there will potentially be some non-negligibleerrors in co-registration. Furthermore, the segmentation of the MRI image may be imperfectin its delineation of anatomical boundaries. In our simulation study, an MRI segmentation wasused to define the anatomical boundaries of our simulation brain phantom, before PET-uniquefeatures were added. The same MRI segmentation was used in the application of IHYPR4D,so we were investigating the method with (i) perfect co-registration and (ii) perfect estimationof the ground truth anatomical boundaries.To investigate how the method performs when these situations are not met, we co-registeredthe MRI image used to generate the ground truth MRI segmentation separately to all 20 noisyOP-OSEM realizations of the PET simulation (mean of the last 30 minutes), using SPM12.Since the images are perfectly co-registered to start, any translations or rotations introducedby SPM12 are due to noise and are considered a co-registration error. We identified the noisyrealization with the highest co-registration error, which is computed asEcoreg =√T 2x + T2y + T2z + r2(θ2x + θ2y + θ2z)(A.1)where the T ’s are the translations (in mm) and the θ’s are the rotations (in radians) applied bySPM12, and r = 50 mm is used to compute the error introduced by a rotation at a radius of 50mm1. We re-segmented the co-registered MRI image with the highest error and then reapplied1Angular registration errors evidently have a greater effect at greater distances from the centre of the brain,so we empirically chose r = 50 mm as a compromise between assessing the effect on central subcortical regionsand more distant cortical regions.176A.1. Effect of inaccurate segmentationIHYPR4D with using this inaccurate segmentation. For simplicity, we studied the effects inthe caudate—a region likely used for analysis in a RAC study and also a region more likelyto be affected by misregistration and segmentation errors because of its smaller size and lowerbinding in our simulation than the putamen.In MRI image with the highest co-registration error, the segmentation of the caudate was3498 voxels in size, compared to 3607 voxels in the ground truth case. Between these segmenta-tions, 2845 voxels intersected. Therefore, 762 voxels were missing in the inaccurate segmenta-tion, and 653 voxels were wrongly assigned to the caudate. The mean absolute bias across timeas a function of IHYPR4D iteration was investigated for the caudate voxels missing from thebad segmentation (Figure A.1a), the voxels wrongly assigned to the caudate (Figure A.1b), andthe entire ground truth caudate region (Figure A.1c). Unsurprisingly, the missing and wrongvoxels see greatly increased bias initially, since the regions they are assigned to are of muchlower and higher concentration than the caudate, respectively. However, after iteration the biasis reduced to levels slightly higher than if the correct segmentation is used. Overall, the biasover the entire caudate is minimally affected after the optimal number of iterations suggestedin this work (∼ 10).Figure A.1: Mean |%bias(t)| in the caudate of the simulation study as a function of IHYPR4D iteration,when using a correct or incorrect segmentation to construct the initial IHYPR4D composite. IHYPR4Dis applied with a 1.5x PSF, four frame kernel. Missing caudate voxels are those belonging to the groundtruth caudate, but not assigned to the caudate in the incorrect segmentation. Wrong caudate voxelsare those not belonging to the ground truth caudate but are assigned to the caudate in the incorrectsegmentation.177A.2. Effect of inaccurate PSF estimationA.2 Effect of inaccurate PSF estimationThe initial IHYPR4D composite is filtered with the system/reconstruction PSF such that sharpboundaries present in the segmented image are not artificially transferred to the target image.As the composite is used to reintroduce high frequency features to the filtered target in theHYPR operator, if one underestimates the PSF, then higher resolution features will still beartificially introduced to IHYPR4D images. If it is overestimated, then the IHYPR4D imageswill incur a loss of resolution compared to the target. To test this, we compared the meandifference in concentration in the caudate of our simulation study (across all voxels and framesfor a representative realization) between IHYPR4D (MRI segmentation) processed with anincorrectly- and correctly-estimated PSF.10% over- and under-estimations of the PSF led to mean concentration differences < 1%even after a single iteration and < 0.5% after 10 iterations (Figure A.2). As expected, anunderestimated PSF yields a positive difference and an overestimated PSF yields a negativedifference, though these differences are minimal.Figure A.2: (a) Mean difference in concentration in the caudate of the simulation study (over all voxelsand frames) between the IHYPR4D images (1.5x PSF, four frames) with an incorrectly- and correctly-estimated PSF, as a function of IHYPR4D iteration; (b) the difference in concentration and a functionof time (i.e. when averaging over caudate voxels only) after two IHYPR4D iterations.178Appendix BSimulating the DA releasetimecourseThe simulations aim to provide a realistic spatiotemporal release pattern in response to amotor/cognition control task. In the simulation, motor-based DA release, including that relatedto the cognitive aspect of the motor task, is assumed to be spatiotemporally identical; motor-based release is confined to a single side of the putamen since the subject usually only requires asingle hand to operate a mouse or control pad, and bilateral release is simulated in the caudatefor cognition-based release.The following task protocol is used to guide the simulation of DA release timecourse and isbased on the real task presented to the human subject of this study. After task start, subjectstapped their fingers for 10 minutes, in blocks of 2 minutes with 15 seconds of rest betweenblocks. The task involved serial tapping of the index finger, middle finger, ring finger, and littlefinger, then reversed (‘1-2-3-4, 4-3-2-1’. . . ), at the subject’s own pace.The spatiotemporal progression of extracellular DA concentration has been modeled asfollows [138]:De(r, t) =∑i,j(t≥tij)ℓsVsCsα (4πD(t− tij))3/2exp[− (r − ri)24D(t− tij) − k(t− tij)](B.1)where i indexes over synapses, j indexes over release times, Vs is the volume of the synapse, Csthe DA concentration within the synapse, D the apparent diffusion coefficient, α the fractionof extracellular volume, ℓs the fraction of synaptic leakage or spillover, and k is the effectiveremoval rate constant. The average extracellular DA at any point in time within a PET voxelis then the integration of De over the volume of the voxel, divided by the volume of the voxel.For simplicity, we assume a uniform distribution of synapses within the voxel that all fire atthe same time, and model a PET voxel as a spherical volume with radius a. LettingAj(t) =ℓsVsCsα (4πD(t− tij))3/2, (B.2)179Appendix B. Simulating the DA release timecoursewe have:⟨De⟩a = 143πa3∫ a0De(r, t)4πr′2dr′ (B.3)=143πa3∑jAj(t)∫ a0exp[− r′24D(t− tj)]dr′ (B.4)⟨De⟩a = 3Da3∑jAj(t) · (t− tj)[√4πD(t− tj)erf(a√4D(t− tj))− 2a exp(− a24D(t− tj))].(B.5)From [138], typical striatal values of D are on the order of 10−6 cm2/s, and for most PETscanners a is on the order of 0.1 cm. Therefore for t on the order of tj , erf(a√4D(t−tj))→ 1and exp(− a24D(t−tj))→ 0. As a result:⟨De⟩a ≈ 3a3∑jℓsVsCs4παexp [−k(t− tj)] (B.6)Returning to our series of task events described above, we thus model the time-varying DAconcentration fed to the ntPET model [137] asFDA(t) = b+∑jgjθ(t− tj) exp(− t− tjτj)(B.7)where b is the basal DA concentration of 100 pmol/mL, j indexes over each task event response,gj scales the magnitude of the jth response, and τj controls the decay rate of the jth response.Cognition-based response is assumed to occur every 20 seconds during each 10-minute taskblock, i.e. when the cards are first presented. Motor-based response occurs every 20 secondsduring the 10-minute task blocks, two seconds after the cards are presented. g and τ are keptconstant for each response within a class of response; empirically we set gmotor = gcognitive = 34pmol/mL and τmotor = τcognitive = 60 s to yield ∼ 100% increase from basal for the motor andcognitive clusters, respectively. See Figure B.1 for the temporal release profile.180Appendix B. Simulating the DA release timecourseFigure B.1: Temporal DA release profile, FDA(t), for the simulated motor/cognitive task performance.The numerical curve is fed to the ntPET model to generate noiseless DA release TACs. Due to the PETtemporal resolution on the order of minutes, the sawtooth pattern present in FDA(t) does not appear inthe noiseless TACs.181Appendix C2D Hampel filterLet f(x) be a 2D image and ηi be the N -by-N neighbourhood of pixels centred on the ithpixel. The 2D Hampel filter uses a N -by-N pixel sliding window to compute the local median,medx∈ηi[f(x)], and median absolute deviation, MADx∈ηi[f(x)], then replaces the central pixel value,f(xi), with the local median if|f(xi)−medx∈ηi[f(x)]| > T ·MADx∈ηi[f(x)]. (C.1)In this work we emperically chose N = 7 and T = 3 to identify outliers in feature space.182
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Sossi, Vesna
Development of advanced denoising and analysis algorithms for applications in hybrid PET/MRI brain imaging
Text