Applied Science, Faculty of
Biomedical Engineering, School of
DSpace
UBCV
Dzikunu, Obed Korshie
Master of Applied Science - MASc
University of British Columbia
Detection and segmentation of lesions in metastatic prostate cancer patients via multimodal positron emission tomography (PET)/computed tomography (CT) imaging can have significant value in the clinical management spectrum, including staging, restaging, diagnosis, treatment planning, and prognosis. Conventional manual segmentation methods, such as tumor contouring, are labor intensive and time-consuming, driving interest in automated solutions using artificial intelligence, specifically deep neural networks. However, automated segmentation faces challenges due to tumor variability in size, shape, uptake intensity, and anatomical location, as well as significant class imbalance between lesion and background voxels, and variability in voxel classification difficulty. Loss function design is central to addressing these challenges. While Dice loss effectively handles class imbalance, it struggles with voxel-level classification difficulty. Nevertheless, distribution-based loss functions like cross-entropy and focal loss address classification difficulty but compromise Dice loss's balance of false positives and false negatives. Combining these loss functions into compound forms improves robustness but often trades off specificity for sensitivity, limiting clinical utility. To address these issues, we propose a novel loss function, L1-weighted Dice Focal Loss (L1DFL), which uses L1 norms of predicted probabilities and ground truth labels to prioritize underrepresented classes and hard-to-classify voxels. Performance was evaluated across multiple lesion scenarios, including images with single and multiple lesions, and varying molecular volumes. We also evaluated its robustness across various levels of spatial distribution of lesions, beyond the primary prostatic region. Subsequently, the analysis evaluated the loss function's clinical utility and its ability to reproduce clinical metrics. Experimental results demonstrate that L1DFL outperforms Dice Focal Loss and Dice Loss by 22% and 13%, respectively, on the Dice Similarity Coefficient, and achieves a superior sensitivity-specificity balance, improving F1 scores by 34% and 6%. L1DFL consistently performed well across lesion scenarios, surpassing other loss functions by at least 11%. It also quantified clinically relevant metrics such as the standardized uptake values (SUVmean, SUVmax), total lesion activity, and lesion count within acceptable margins. Coupled with its effectiveness across diverse neural network architectures (SegResNet, U-Net, Attention U-Net), the results highlight L1DFL's potential to enhance automated segmentation and support clinical decision-making in metastatic prostate cancer management.
Automated Detection andSegmentation of Metastatic ProstateCancer Lesions in PET/CT Images viaDeep Neural Networks TowardsAccurate and Routine Quantificationof Clinical MetricsbyObed Korshie DzikunuB.Sc., University of Ghana, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Biomedical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2024© Obed Korshie Dzikunu 2024The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Automated Detection and Segmentation of Metastatic Prostate Cancer Lesionsin PET/CT Images via Deep Neural Networks Towards Accurate and RoutineQuantification of Clinical Metricssubmitted by Obed Korshie Dzikunu in partial fulfillment of the require-ments for the degree of Master of Applied Sciencein Biomedical EngineeringExamining Committee:Arman Rahmim, Professor, Departments of Radiology and Physics & Astronomy,UBCSupervisorXiaoxiao Li, Assistant Professor, Department of Electrical and ComputerEngineering, UBCSupervisory Committee MemberCarlos Uribe, Clinical Assistant Professor, Department of Radiology, UBCSupervisory Committee MemberiiAbstractDetection and segmentation of lesions in metastatic prostate cancer patientsvia multimodal positron emission tomography (PET)/computed tomogra-phy (CT) imaging can have significant value in the clinical management spec-trum, including staging, restaging, diagnosis, treatment planning, and prog-nosis. Conventional manual segmentation methods, such as tumor contour-ing, are labor intensive and time-consuming, driving interest in automatedsolutions using artificial intelligence, specifically deep neural networks. How-ever, automated segmentation faces challenges due to tumor variability insize, shape, uptake intensity, and anatomical location, as well as significantclass imbalance between lesion and background voxels, and variability invoxel classification difficulty.Loss function design is central to addressing these challenges. WhileDice loss effectively handles class imbalance, it struggles with voxel-levelclassification difficulty. Nevertheless, distribution-based loss functions likecross-entropy and focal loss address classification difficulty but compromiseDice loss’s balance of false positives and false negatives. Combining theseloss functions into compound forms improves robustness but often trades offspecificity for sensitivity, limiting clinical utility.To address these issues, we propose a novel loss function, L1-weightedDice Focal Loss (L1DFL), which uses L1 norms of predicted probabilitiesand ground truth labels to prioritize underrepresented classes and hard-to-classify voxels. Performance was evaluated across multiple lesion scenarios,including images with single and multiple lesions, and varying molecularvolumes. We also evaluated its robustness across various levels of spatialdistribution of lesions, beyond the primary prostatic region. Subsequently,the analysis evaluated the loss function’s clinical utility and its ability toreproduce clinical metrics. Experimental results demonstrate that L1DFLoutperforms Dice Focal Loss and Dice Loss by 22% and 13%, respectively, onthe Dice Similarity Coefficient, and achieves a superior sensitivity-specificitybalance, improving F1 scores by 34% and 6%.L1DFL consistently performed well across lesion scenarios, surpassingother loss functions by at least 11%. It also quantified clinically relevantiiiAbstractmetrics such as the standardized uptake values (SUVmean, SUVmax), totallesion activity, and lesion count within acceptable margins. Coupled withits effectiveness across diverse neural network architectures (SegResNet, U-Net, Attention U-Net), the results highlight L1DFL’s potential to enhanceautomated segmentation and support clinical decision-making in metastaticprostate cancer management.ivLay SummaryAccurate lesion segmentation in medical images is essential for quantifyingclinical metrics that guide prostate cancer management, including staging,treatment planning, and prognosis. Automated segmentation methods relyon loss functions—mathematical formulas that train artificial intelligencemodels to accurately delineate tumors. However, due to the complex featuresof metastatic prostate cancer lesions, existing loss functions often struggle tobalance sensitivity and specificity, leading to missed lesions or false positives.This study introduces a novel loss function that adaptively adjusts themodel’s focus based on the complexity of segmenting a tumor or classifyingregions in an image. The loss function improved segmentation performanceby at least 11% compared to commonly used loss functions, achieving betteralignment with ground truth values. These advancements highlight its po-tential to enhance automated segmentation accuracy and support accuratequantification of clinical metrics for prostate cancer management.vPrefaceThis thesis focuses on automating the detection and segmentation ofmetastatic prostate cancer tumors from PET/CT scans to enable accuratequantification of clinical metrics critical for predicting patient outcomes andinforming treatment planning. A major contribution of this work is thedevelopment of a novel loss function that achieves a balanced focus, effec-tively minimizing false positive and negative rates. This loss function usesthe absolute differences between the model’s probability predictions and theground truth labels, L1 norms, to quantify the classification difficulty of spe-cific regions within the medical image. These differences are then used togenerate a weighting scheme that adaptively directs the model’s attention,enhancing voxel-level segmentation performance holistically.Additionally, this thesis evaluates the performance of other widely usedloss functions for medical image segmentation alongside various deep learn-ing neural networks, providing a comparative analysis. Beyond the conven-tional evaluation of segmentation performance using metrics like the DiceSimilarity Coefficient, this work emphasizes clinical relevance. It assesses theability of the proposed loss function and comparative methods to quantifycritical clinical metrics accurately, demonstrating their utility in real-worldapplications and advancing the field of automated medical image analysis.The datasets used in this thesis included PET/CT scans of patientsdiagnosed with biochemical recurrence metastatic prostate cancer as partof an ongoing clinical trial (ClinicalTrials.gov NCT02899312) with informedwritten consent obtained from the subjects. The study with number H16-01551 and title: 18F-DCFPyL Positron Emission Tomography / ComputedTomography (PET/CT) for Assessment of Recurrent Prostate Cancer, wasapproved by the University of British Columbia – BC Cancer Ethics Board.The material presented in this thesis has been disseminated through aconference presentation, and two peer-reviewed scientific papers are also be-ing submitted based on various chapters of this work. Below we outlinethese:O. Dzikunu, S. Ahamed, A. Toosi, S. Harsini, F. Benard, A.viPrefaceRahmim, C. Uribe. A 3D UNet for automated metastatic lesionsdetection and segmentation from PSMA-PET images of patientswith biochemical recurrence prostate cancer J. Nucl. Med., vol.65 (suppl. 2): 242504, 2024Parts of the material covered in Chapter 3 were presented at the Societyof Nuclear Medicine and Molecular Imaging (SNMMI) conference in 2024.In this work [1], I contributed as the primary author, developing the deeplearning model for the detection and segmentation of the lesions and wrotethe abstract. S. Ahamed and A. Toosi helped in data processing and offeredguidance in implementing the training framework. S. Harsini performed theground truth annotation of the lesions. The other co-authors helped in re-fining the abstract.O. Dzikunu, S. Ahamed, A. Toosi, X. Li, A. Rahmim. Whereto look: Adaptive L1-Weighted Loss for Automated Detectionand Segmentation of Metastatic Prostate Cancer Lesions in Mul-timodal PSMA PET/CT Images (Being Submitted).This paper [2] is based on the work presented in the chapter 3 of thisthesis. I conceptualized and formulated the proposed loss function, and im-plemented the segmentation pipeline. S. Ahamed and A. Toosi assisted indata preprocessing and offered expert guidance on pipeline implementation.X. Li and A. Rahmim provided expertise on implementation and clinical rel-evance. I wrote the manuscript and all co-authors assisted in proofreadingand refining the presented material.O. Dzikunu, A. Toosi, S. Ahamed, S. Harsini, X. Li, A. Rah-mim. Reproducibility Assessment of Clinical Metric Quantifica-tion Using L1-weighted Dice Focal Loss on PSMA PET/CT Scans(Being Submitted).This paper [3] is based on work presented in chapter 4 of this thesis andparts of chapter 3. I conceptualized and implemented the quantificationpipeline with help from A. Toosi and S. Ahamed. S. Harsini performed theground truth annotation of the datasets used for the model training andalso provided clinical advice. X. Li and A. Rahmim provided expertise onimplementation and clinical relevance. I wrote the manuscript and all co-authors assisted in proofreading and refining the presented material.In the course of this work, I was also involved in collaborative worksregarding the development of an open-source python-based reconstructionlibrary, PyTomography, to offer a GPU-accelerated platform for nuclearviiPrefacemedicine image reconstruction. This work has led to two journal publi-cations (see appendix A.3) outlined below. Relevant to these two papers,two conference works were also presented [4, 5].O. K. Dzikunu, M. Sabouri, S. Ahamed, C. Uribe, A. Rah-mim, L.A. Polson. SlicerSPECTRecon: A 3D Slicer Extension forSPECT Image Reconstruction. Journal of Open Source Software,9(104), 7399, 2024.This journal publication is based on the development of a 3D slicer ex-tension module for single photon emission computed tomography (SPECT)image reconstruction [6]. I contributed as the primary author, leading thedevelopment of the module. L. Polson suggested the idea and assisted invarious phases of development. M. Sabouri assisted in the development ofthe UI component. S. Ahamed, A. Rahmim and C. Uribe also contributedin various developmental stages of the module. This material is presentedin appendix A.3.L. Polson, R. Fedrigo, C. Li, M. Sabouri, O. Dzikunu, S. Ahamed,N. Karakatsanis, S. Kurkowska, P. Sh.Zadeh, P. Esquinas, A. Rah-mim, C. Uribe. PyTomography: A Python Library for MedicalImage Reconstruction (SoftwareX - accepted for publication).This work was led by L. Polson, and I participated in the development ofthe library focusing on implementing a collimator database with the aim ofleveraging source-detector distance for PSF modeling with a variable widthGaussian filter for SPECT image reconstruction. This work presents PyTo-mography and explores various use cases.viiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Medical Imaging and Nuclear Medicine in Prostate Cancer 11.1 Background on Medical Imaging . . . . . . . . . . . . . . . . 11.2 Medical Imaging Modalities . . . . . . . . . . . . . . . . . . . 21.2.1 X-Ray Imaging . . . . . . . . . . . . . . . . . . . . . 21.2.2 Computed Tomography Imaging . . . . . . . . . . . . 31.2.3 Magnetic Resonance Imaging . . . . . . . . . . . . . . 41.2.4 Ultrasound Imaging . . . . . . . . . . . . . . . . . . . 61.2.5 Introduction to Nuclear Medicine Imaging . . . . . . 71.3 Foundations of Nuclear Medicine Imaging . . . . . . . . . . . 81.3.1 Physics of Radiation in Nuclear Medicine Imaging . . 81.3.2 Radiopharmaceuticals in Nuclear Medicine Imaging . 101.4 Imaging Modalities in Nuclear Medicine . . . . . . . . . . . . 161.4.1 Planar Imaging or Scintigraphy . . . . . . . . . . . . 161.4.2 Single Photon Emission Computed Tomography . . . 171.4.3 Positron Emission Tomography . . . . . . . . . . . . 201.5 Prostate Cancer . . . . . . . . . . . . . . . . . . . . . . . . . 251.5.1 Overview and Pathophysiology . . . . . . . . . . . . . 25ixTable of Contents1.5.2 Role of PET/CT in Prostate Cancer Management . . 261.5.3 Prostate Cancer Segmentation . . . . . . . . . . . . . 271.6 Conclusion and Transition to Next Chapter . . . . . . . . . . 292 Artificial Intelligence in Medical Imaging . . . . . . . . . . 302.1 Artificial Intelligence For Medical Image Processing . . . . . 302.2 Artificial Intelligence in Medical Image Segmentation . . . . 322.3 Model Architectures in Image Segmentation . . . . . . . . . 332.3.1 Fully Connected Architectures . . . . . . . . . . . . . 352.3.2 Latent Representation Networks . . . . . . . . . . . . 372.3.3 Saliency Map Networks . . . . . . . . . . . . . . . . . 402.4 Loss Functions in Image Segmentation . . . . . . . . . . . . 422.4.1 Notation and Definitions . . . . . . . . . . . . . . . . 442.4.2 Region-Based Losses . . . . . . . . . . . . . . . . . . 452.4.3 Distribution-Based Losses . . . . . . . . . . . . . . . 462.4.4 Boundary-Based Losses . . . . . . . . . . . . . . . . . 462.4.5 Compound Losses . . . . . . . . . . . . . . . . . . . . 472.4.6 Performance Summary of Loss Functions . . . . . . . 472.5 Tumor Segmentation in PET/CT Images . . . . . . . . . . . 472.6 Conclusion and Transition to Next Chapter . . . . . . . . . . 493 Optimizing Lesion Segmentation with L1-Weighted Dice Fo-cal Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . 533.2.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2.2 Ground Truth Annotation . . . . . . . . . . . . . . . 543.2.3 Data Preprocessing and Augmentation . . . . . . . . 543.2.4 Architectures . . . . . . . . . . . . . . . . . . . . . . . 543.2.5 Model Training . . . . . . . . . . . . . . . . . . . . . 553.2.6 Loss Functions . . . . . . . . . . . . . . . . . . . . . . 553.2.7 Model Evaluation . . . . . . . . . . . . . . . . . . . . 583.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Segmentation performance across different networks . 613.3.2 Performance in single and multiple lesion scenarios . 643.3.3 Performance based on lesion metabolic volume . . . . 663.3.4 Performance based on lesion dissemination . . . . . . 683.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5 Conclusion and Transition to Next Chapter . . . . . . . . . . 73xTable of Contents4 Comparative Analysis of Loss Functions for Accurate Quan-tification of Clinical metrics . . . . . . . . . . . . . . . . . . . 744.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . 774.2.1 Model Training . . . . . . . . . . . . . . . . . . . . . 774.2.2 Model Evaluation . . . . . . . . . . . . . . . . . . . . 784.2.3 Assessment of Agreement with Ground Truth . . . . 784.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.1 Performance evaluation of loss functions across differ-ent networks . . . . . . . . . . . . . . . . . . . . . . . 814.3.2 Evaluation of concordance between predicted and groundtruth clinical metrics . . . . . . . . . . . . . . . . . . 814.3.3 Reproducibility of the ground truth metrics . . . . . 834.3.4 Evaluation of performance based on coverage proba-bility and total deviation index . . . . . . . . . . . . 854.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 Summary and Future Work . . . . . . . . . . . . . . . . . . . 915.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95AppendicesA Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137A.1 Mathematical Description of Clinical metrics . . . . . . . . . 137A.2 Validation and Test Set Segmentation Performance . . . . . 138A.3 Open-source SPECT Reconstruction Module . . . . . . . . . 139A.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 139A.3.2 Statement of need . . . . . . . . . . . . . . . . . . . . 139A.3.3 Overview of SlicerSPECTRecon . . . . . . . . . . . . 140xiList of Tables1.1 Various theranostic pairs in nuclear medicine imaging high-lighting their various half-life, decay modes, possible imagingmodality (positron emission tomography (PET), single pho-ton emission computed tomography (SPECT), Bremsstrahlungimaging (Brems.) and maximum particulate energy for beta-positive decay (Eβ+), beta-minus decay (Eβ−), alpha decay(Eα) and Auger electron (EAuger). EC = electron capture, IT= isomeric transition [7]. . . . . . . . . . . . . . . . . . . . . . 141.2 Properties of positron-emitting radionuclides relevant to nu-clear medicine imaging. The table shows the half-life (in min-utes), percentage of positron emission, maximum energy re-leased during decay (Emax) in MeV, and positron range inmillimeters [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Comparison of the loss functions on the test set based onmean and median patient-level DSC. The mean DSC, TP,FP, FN and F1 scores are presented with standard deviationsand the median DSC with interquartile ranges. A detection isconsidered a true positive if the predicted mask overlaps withthe voxel containing the SUVmax value in the ground truth.The P values reflect a one-tailed paired Wilcoxon signed-ranktest comparing the median difference in DSC between L1DFLand DL and DFL, with * indicating statistical significance atthe α = 0.05 level. . . . . . . . . . . . . . . . . . . . . . . . . 62xiiList of TablesA.1 Comparison of the loss functions (Dice Loss (DL), Dice Cross-Entropy Loss (DCE), Dice Focal Loss (DFL), L1-weightedDice Focal Loss (L1DFL)) based on a five-fold validation,and on the test set. We report performance based on meanand median patient-level DSC. The mean DSC, TP, FP, FNand F1 scores are presented with standard deviations andthe median DSC with interquartile ranges. A detection isconsidered a true positive if the predicted mask overlaps withthe voxel containing the SUVmax value in the ground truth. 139xiiiList of Figures1.1 The schematic diagram of an x-ray tube (top center) illus-trates the process of x-ray generation. Electrons are accel-erated from the cathode towards the anode, where they pro-duce x-rays. As these high-energy electrons collide with theanode atoms, they can generate either bremsstrahlung radia-tion (bottom left) or characteristic radiation (bottom right).In the case of characteristic radiation, the interaction mayresult in the emission of a characteristic x-ray or the releaseof an Auger electron [9]. . . . . . . . . . . . . . . . . . . . . 31.2 MRI scans compare T1- and T2-weighted imaging in a pa-tient with right-sided hemiparesis. (A) T1-weighted imagingshows a dark lesion in the left frontal-parietal region. (B) T2-weighted imaging highlights the lesion as bright with visiblesurrounding edema [10]. . . . . . . . . . . . . . . . . . . . . . 61.3 Comparison of [68Ga]Ga-PSMA PET and [18F]FDG PET imag-ing in a metastatic castration resistant prostate cancer pa-tient. High tumor activity is shown on the [68Ga]Ga-PSMAimage while minimal activity is evident on the [18F]FDG PETscan [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Growth and decay of 99mTc radioactivity (in millicurie units)during the decay of 99Mo, with equilibrium reached after ap-proximately 24 hours. The solid curve assumes that all disin-tegrations of 99Mo yield 99mTc. The dashed curve representsthe actual 99mTc radioactivity, accounting for the fact thatonly 92% of 99Mo disintegrations produce 99mTc [8]. . . . . . 16xivList of Figures1.5 Figure of a typical scintillation camera, highlighting its fourmain components: A (collimator), B (NaI crystal), C (lightpipe, photomultiplier tubes, and preamplifiers), and D (cath-ode ray tube for display). The detector head integrates com-ponents A, B, and C, while the console houses D along withpower supplies and related electronics. Historically, the cath-ode ray tube (CRT) in component D was used in analogsystems to display γ-ray interactions, where X and Y pulseswould deflect a light spot on the CRT or oscilloscope in pro-portion to the amplitude of the signals. In modern systems,however, a digital computer processes the X- and Y-signals,storing the data and the count at each location in memoryfor display and further analysis on a digital monitor [8]. . . . 191.6 Coronal and maximum intensity projection (MIP) images via[99mTc]Tc-PSMA-I&S SPECT/CT to assess metastatic castration-resistant prostate cancer (mCRPC), showing extensive tu-mor spread prior to therapy initiation in a patient with dis-ease progression under chemotherapy and a PSA level of 137ng/mL [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.7 Illustration of the various phenomena that lead to resolutiondegradation in PET imaging. When a γ-ray interacts with acrystal, it may be detected in a neighboring crystal due to twofactors: (i) the penetration effect, where the γ-ray entersthe crystal at an angle and extends into an adjacent crystal,or (ii) Compton scattering, where the γ-ray is scatteredinside the crystal and deflects to a different crystal. Thefigure demonstrates these effects with three adjacent crystals,showing how a photon entering the middle crystal may eitherpenetrate into an adjacent crystal or scatter to another crystalbefore detection [8]. . . . . . . . . . . . . . . . . . . . . . . . 221.8 This image illustrates the development of PET image recon-struction, starting with 3D filtered back projection (FBP),progressing to Poisson noise modeling (MLEM and OSEM),and advancing with point spread function (PSF) data mod-eling. More recently, regularization techniques have beenadopted to reduce noise and improve image quality [13]. . . 24xvList of Figures2.1 An example of a Deep Neural Network (DNN) architecture,specifically a Convolutional Neural Network (CNN), illustrat-ing its operations for extracting spatial features from inputimages. (1) The input image is processed using a series offilters (or kernels), which slide over the image in a systematicmanner to identify specific patterns. Initially, these filtersare randomly initialized but evolve during training to detectmeaningful features, such as edges or textures. (2) The out-put of the filtering process is a set of activation maps, thathighlight detected features (3) Nonlinear activation functions,such as ReLU or sigmoid, are applied to the activation mapspixel by pixel. This step introduces the ability to learn com-plex, nonlinear relationships in the data, allowing the networkto extract increasingly abstract features at deeper levels [14] 352.2 Illustration of fully connected (FC) layers commonly used atthe output stage of convolutional neural networks (CNNs).Activation maps from the preceding layers are flattened intoa one-dimensional vector and fed into the fully connectedlayer. Each element of the flattened vector is connected toevery node in the FC layer, with each connection assigned alearnable weight. The output of each node is computed as theweighted sum of all inputs, combined with a learnable bias,and passed through an activation function. This structureenables the FC layer to aggregate global information, makingit useful for classification tasks or as a final stage in hybridmodels [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 The U-Net architecture is a network structured as an encoder-decoder. The encoder (left side) performs downsampling throughmax pooling, capturing spatial features at progressively lowerresolutions. The decoder (right side) performs upsampling toreconstruct fine details. The structure inherently representslatency, as information passes through varying depths andprocessing stages, balancing feature extraction and resolutionrecovery [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38xviList of Figures2.4 Overview of a Vision Transformer (ViT) model, which servesas an example of transformer-based networks applied to imageanalysis. The input image is divided into fixed-size patches,which are linearly embedded into vectors. Positional embed-dings are added to retain spatial information, and the se-quence of embedded patches is fed into a Transformer en-coder. A learnable classification token is appended to the se-quence to facilitate tasks such as image classification. This ar-chitecture, by processing images as sequences, enables the useof attention mechanisms, which can produce saliency mapshighlighting the most relevant regions of an image for a giventask [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 An illustration of the dynamic weighting strategy in L1 weightedDice Focal Loss (L1DFL). (a) shows the ground truth (G),where 1 indicates lesion areas and 0 indicates background.(b) displays the predicted probability map, where green re-gions denote false positive predictions and red regions repre-sent ROIs. (c) shows the L1 norm values between the pre-dicted probabilities and ground truth, highlighting the errormagnitude for each voxel. (d) presents the density, D, whichreflects the frequency distribution of each L1 norm value, cap-turing the difficulty of correctly classifying each voxel. (e)depicts the final weights assigned to each voxel, where higherweights focus on rare, misclassified regions while down weight-ing well-classified background areas. This weighting strategyreduces the influence of abundant, correctly predicted vox-els and increases the focus on difficult, misclassified lesions,thereby enhancing the model’s focus on challenging samples. 583.2 Illustration for defining a true positive detection based on anoverlap with the voxel containing the maximum standardizeduptake value (SUVmax) in the ground truth lesion. G is theset of ground truth lesions and P is the set of predicted lesions. 60xviiList of Figures3.3 Segmentation performance of loss functions across two net-works: Attention U-Net (top row) and SegResNet (bottomrow). Each column represents the results for the loss func-tions: Dice loss (DL), Dice Focal loss (DFL), and L1-weightedDice Focal loss (L1DFL). The green-colored region denotesthe ground truth annotation of the lesion, while the red-colored regions represent the segmentations produced by themodels. The circled area highlights the lesion in the groundtruth image alongside the corresponding correct segmentationby the models. Red regions outside the circled area indicatefalse positive segmentations. . . . . . . . . . . . . . . . . . . 633.4 Box plots overlaid with swarm plots illustrating the distri-bution of Dice Similarity Coefficient (DSC) performance forthe three loss functions across two lesion scenarios: M (lesioncount > 2) and S (lesion count = 1). Each box plot’s top andbottom edges represent the interquartile range (IQR), whilethe horizontal lines within the boxes indicate the median DSCvalues. Whisker lengths are set to 1.5 times the IQR. (a) Per-formance on Attention U-Net architecture. (b) Performanceon SegResNet. The number of cases in the single lesion sce-nario was N = 34 and for the multiple lesion scenario, thenumber of cases was N = 23. . . . . . . . . . . . . . . . . . . 653.5 Patient and lesion-level median Dice score (DSC) performanceof the different loss functions as a function of various groundtruth tumor metabolic volume thresholds. Each column showsthe plots for the different lesion scenarios: single (S) andmultiple (M). At both the patient and lesion levels, the lineplots represent the trend of each loss function’s performanceacross different TMTV and MTV thresholds, respectively,with thresholds (t) based on the interquartile range from 0to the 85th percentile. Median DSC is calculated for all le-sions (l) where volume (vl > t), and each color indicates adifferent loss function. The first row of plots (a) shows theperformance of the loss functions with Attention U-Net. Thesecond row (b) shows the performance of the loss functionswith SegResNet. The third row (c) displays area plots ofmedian tumor lesion activity (TLA) values according to theTMTV and MTV thresholds, where median TLA values arecalculated for lesions with (vl > t). . . . . . . . . . . . . . . 67xviiiList of Figures3.6 Plots illustrating the Dice Similarity Coefficients (DSC) acrossfour interquartile ranges of Dmax. The first row (a) shows theperformance with Attention U-Net. The second row (b) showsthe performance with SegResNet. The groups G0 (0-9 cm),G1 (9-11cm), G2 (11-14cm), and G3 (14-60cm) correspond tothe interquartile ranges of 0− 25%, 25− 50%, 50− 75%, and75−100%, respectively. In the box plots, horizontal lines indi-cate the median, and whisker lengths are set to 1.5 times theinterquartile range (IQR), with outliers represented as circles.Accompanying bar plots (c) display the median total lesionactivity (TLA) values for cases within each Dmax group. Thedifferent loss functions are color-coded for clarity. The totalnumber of cases in each group was N = 6. . . . . . . . . . . 693.7 Plot of loss versus probability for samples with the groundtruth label as 1. The curves represent the behavior of eachloss function, highlighting how they respond to varying prob-abilities of correctly classifying examples. The plot illustratesthe differences in how each loss function handles both easy-to-classify and hard-to-classify cases based on their predictedprobabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.1 Radar plots showing the correlation between the predictedand ground truth metrics for the four loss functions, Dice loss(DL), Dice Cross-Entropy (DCE), Dice Focal loss (DFL), andL1-weighted Dice Focal Loss (L1DFL) assessed by the Lin’sConcordance Correlation Coefficient. The different radar plotsillustrate the performance of the three architectures - U-Net,Attention U-Net, and SegResNet. The radial lines representthe correlation coefficient values. . . . . . . . . . . . . . . . . 824.2 Scatter plots comparing predicted clinical metrics (y-axis) toground truth values (x-axis) for different architectures (columns:U-Net, SegResNet, and Attention U-Net) and clinical met-rics (rows: SUVmax, SUVmean, TMTV , TLA, L, and Dmax).Each data point represents a specific prediction, with colorsindicating different loss functions. The dashed line representsthe line of equality (y = x), and agreement between predictedand ground truth values is assessed using Lin’s concordancecorrelation coefficient. . . . . . . . . . . . . . . . . . . . . . . 83xixList of Figures4.3 Two one-sided forest plots illustrating the results of equiva-lence testing for predicted metrics versus ground truth acrossdifferent loss functions. Each row represents a specific met-ric, and the columns correspond to the different networks.The black vertical dashed lines represent the region of clinicalequivalence (±20% of the mean of each ground truth metric).The x-axis represents the mean difference between the pre-dicted and ground truth metrics. Green bars represent equiv-alence between predictions of a given loss function and groundtruth values at a significance level of α = 0.05, while red barsindicate non-equivalence. . . . . . . . . . . . . . . . . . . . . 844.4 Figure: Modified Bland-Altman plots illustrating the varia-tions (∆) between predicted and ground truth values (y-axis)against the ground truth metrics (x-axis) for different lossfunctions, represented by distinct colors. Each row corre-sponds to a specific clinical metric, and the columns repre-sent predictions from different network architectures (U-Net,SegResNet, and Attention U-Net). The dashed black hori-zontal lines indicate the limits of agreement (±20% of themean ground truth values), while the dotted line at ∆ = 0represents perfect agreement. Each data point reflects a sin-gle instance, highlighting the distribution of variations acrossground truth values. . . . . . . . . . . . . . . . . . . . . . . . 864.5 The left column displays plots of Coverage Probability (CP),representing the proportion of instances where the model’spredictions fell within ±20% of the corresponding clinicalmetric. Error bars indicate the 95% confidence intervals.The right column shows bar plots of the Total Deviation In-dex (TDI) for each metric, representing the 95th percentileof deviations between predictions and ground truth values.Rows correspond to specific metrics (SUVmax, SUVmean,Lesion Count (L), and TLA), while columns indicate differ-ent network architectures (U-Net, SegResNet, and AttentionU-Net). Distinct colors denote different loss functions: DL,DCE, DFL, and L1DFL. . . . . . . . . . . . . . . . . . . . . 88xxList of FiguresA.1 Left: user interface for the proposed extension. Right: re-constructed coronal slice from a patient receiving [177Lu]Lu-PSMA-617 radiopharmaceutical therapy (color) overlaid on acorresponding CT (greyscale). Raw SPECT data consisted oftwo bed positions that were automatically stitched togetherafter each was reconstructed; the raw data was acquired on aGE Discovery 670 camera. . . . . . . . . . . . . . . . . . . . 141xxiAcknowledgementsFirstly, I would like to thank my Lord Jesus Christ for granting me thegrace needed to carry me through this journey. Despite numerous obstacles,Your grace enabled me to maintain a positive outlook and overcome everychallenge.I would also like to express my profound thanks to my supervisor, ArmanRahmim, for his technical guidance and support throughout this journey.Your expectations were clear, and you were willing to provide any additionaltools that may be needed to help me in this work. You also showed muchinterest in my professional growth. I appreciate all the life lessons.I will also want to thank Shadab Ahamed who got me started on thisjourney. He offered me a systematic orientation into the field of lesion seg-mentation and provided me with tutorials to get started. Amirhossein Toosi,you also made this journey a memorable one. You were my go-to personwhen I was stuck. Shadab and Amir, thank you so much for allowing me toleverage your experiences and expertise to chart my own course.I am also very grateful to Luke Polson. Although a student, your men-torship is only next to an accomplished professor. You believed in my capa-bilities and led me along the lines of software development. Through yourmentorship, today I am proudly the lead developer of an open-source re-construction software module that will greatly impact the nuclear medicinecommunity.Shariar Siddique, thank you for being more than a colleague. I amexceptionally grateful to you for all you advise on course selection. Today,I look back and realize how much you have helped me in selecting coursesthat provide me with the requisite knowledge for my career development.Thank you so much Shariar.Xiaoxiao Li, our relationship started with me logging a complaint of notbeing able to register for your course on visual computing, but today I workunder your supervision as an intern with the prestigious Vector Institute.This opportunity has set me on the right path to achieving my academicgoals. Thank you Dr. Li.Carlos Uribe, you have always held a favorable estimation of my abil-xxiiAcknowledgementsities, and for that, I am deeply grateful. Thank you for believing in mypotential and for your invaluable supervision during the development of thereconstruction software.I would also like to thank all members of the Quantitative Radiomolec-ular Imaging and Therapy (Qurit) lab. You provided me with an enablingenvironment to support my academic growth. Narges, Maziar, Sara Harsini,Sara Kurkowska and Fereshteh, I greatly appreciate your help along the way.I am indebted to my family who have been my backbone; they are mygreatest cheerleaders. Rev. Raymond Anyang and Rev. Benjamin Amoah, Iam grateful for your mentorship throughout this whole period; you ensuredI remained focused on the goal, thank you. Finally, to you Adepa AkosuaAnyang, through whom this whole journey started. I am grateful to you forbelieving in my abilities and always urging me to push the boundaries. Youhave been an immense support to me motivating me to be the best in whatI do, and investing time and resources in ensuring that I have everything Ineed to make my journey a memorable one. Thank you so much for yourlove and support.xxiiiDedicationI dedicate this work to my mom, Emelia Dzikunu, whose immense supportfor my academic growth has been phenomenal. She set me on this path byencouraging me to read novels and present the key points to her. She alsospurred me on when the journey seemed bleak. Thanks mom, this is foryou!xxivChapter 1Medical Imaging and NuclearMedicine in Prostate Cancer1.1 Background on Medical ImagingMedical imaging is a cornerstone of modern healthcare, enabling the vi-sualization and assessment of internal structures and functions within thehuman body [16]. It encompasses a variety of technologies and methods de-signed to produce images that assist in diagnosing, monitoring, and treatingmedical conditions [17]. Medical imaging allows clinicians to detect abnor-malities, guide interventions, and assess treatment progress by providing anon-invasive way to observe bones, organs, muscles, blood vessels, and othertissues [18].The discipline can be broadly divided into two categories: diagnosticimaging and therapeutic imaging. Diagnostic imaging is used to detect andevaluate the severity of medical conditions [19]. It includes widely usedmodalities such as X-rays, computed tomography (CT), magnetic resonanceimaging (MRI), ultrasound, and nuclear medicine imaging [20]. These toolsprovide detailed insights into anatomical structures and physiological pro-cesses. By contrast, therapeutic imaging provides supportive functions fortreatments, such as surgery or radiation therapy, through real-time visu-alization using techniques like fluoroscopy, angiography, and interventionalradiology that enhance the precision of interventions [19].A general objective of medical imaging is to make a particular medicalcondition or disease visible by contrast, which refers to the difference inbrightness, color, or intensity regarding the surrounding tissues [17]. Thiscontrast allows the differentiation between normal and abnormal structures.For example, in X-ray imaging, bones appear bright on account of higherabsorption levels of X-rays by more dense tissues, while air-filled spacesappear black [16]. Similarly, MRI provides contrast based on the encodingof signals owing to the magnetic properties of protons in tissues, achievinghighly detailed images that reflect subtle differences in tissue characteristics11.2. Medical Imaging Modalities[17].The imaging process involves exposing the body to energy sources suchas X-rays, sound waves, or magnetic fields and capturing the resulting data[17]. This data encodes spatial and functional information, which is thenprocessed to generate interpretable images [17]. For example, X-rays rely onthe absorption of radiation as it passes through the body to highlight dif-ferences in tissue density [21]. MRI uses magnetic fields and radiofrequencypulses to create images based on the behavior of protons in the body’s tissues[22]. Each imaging modality uses a different physical mechanism to generatecontrast and thus offers complementary views of the internal structure andfunction of the body [17]. The widely used imaging modalities are discussedbelow, explicitly focusing on nuclear medicine imaging modalities.1.2 Medical Imaging Modalities1.2.1 X-Ray ImagingX-ray imaging is one of the oldest and most common imaging modalities.It depends on the features of electromagnetic radiation (X-rays) to produceimages for diagnosis [21]. X-rays are high-energy electromagnetic radiationthat can penetrate the body and thus give an image of structures withinit [23]. Primarily used to evaluate disorders and injuries to the skeletalsystem, X-rays detect fractures, structural disorders, and diseases such aspneumonia, pulmonary edema, and abdominal obstruction [19].In X-ray imaging, a high-voltage current is used to produce a streamof electrons aimed at a metal target in the X-ray tube [23]. The processproduces two types of radiation: bremsstrahlung and characteristic radia-tion. Bremsstrahlung occurs when electrons decelerate upon approachingan atomic nucleus [24], while characteristic radiation arises from electroninteractions that eject inner orbital electrons, releasing X-rays of specificenergies [25]. Additionally, during characteristic radiation, the ejection ofinner electrons can lead to the release of an Auger electron as the atom seeksto return to a stable state [9]. Figure 1.1 illustrates the process of X-raygeneration. The resulting X-ray beam is filtered and aimed at the body. Asthe X-rays penetrate tissues, they are differentially attenuated dependingon the tissue density. For example, bone is denser than soft tissue, so itabsorbs more X-rays and appears white on the resulting image, while softtissue appears gray and air appears black [16].Modern developments have changed X-ray imaging from film to digi-tal radiography, significantly improving image quality, the ability to archive21.2. Medical Imaging Modalitiesimages, and the ease of sharing images electronically [26]. Digital systemsemploy electronic detectors that capture and convert X-rays into electronicsignals [26], lowering radiation exposure and improving efficiency comparedto previous methods. Although X-rays are two-dimensional, they are rela-tively inexpensive, portable, and highly accessible, making them useful inemergencies and outpatient care [19].Figure 1.1: The schematic diagram of an x-ray tube (top center) illustratesthe process of x-ray generation. Electrons are accelerated from the cathodetowards the anode, where they produce x-rays. As these high-energy elec-trons collide with the anode atoms, they can generate either bremsstrahlungradiation (bottom left) or characteristic radiation (bottom right). In the caseof characteristic radiation, the interaction may result in the emission of acharacteristic x-ray or the release of an Auger electron [9].1.2.2 Computed Tomography ImagingComputed Tomography (CT) uses X-rays combined with computer algo-rithms to generate detailed cross-sectional and volumetric images of thebody [27]. It has revolutionized the practice of medicine by providingthree-dimensional (3D) reconstructions, enabling the visualization of com-plex anatomical structures and pathological conditions. CT’s foundationalprinciple lies in the differential absorption of X-ray beams as they traversetissues of varying densities. This differential absorption creates a series of at-31.2. Medical Imaging Modalitiestenuation profiles, which are then reconstructed mathematically into images[28].CT imaging starts with the acquisition of data, during which a rotatingX-ray source emits beams while detectors measure the attenuated signalsafter passing through the patient [29]. These detectors are usually arrangedin a gantry, and as the patient table moves, continuous or helical data ac-quisition occurs [29, 30]. Helical CT, typical of modern scanners, capturesvolumetric data in a spiral trajectory, thereby reducing scan time and mo-tion artifacts while improving image resolution [30].The mathematical reconstruction of images in CT is done by algorithmslike filtered back projection [31] or iterative reconstruction [32]. These tech-niques convert raw data into 2D slices or 3D volumes, with each voxel beingassigned a Hounsfield Unit (HU) representing tissue density [33]. DistinctHU values for air, fat, soft tissue, and bone provide inherent contrast fordifferentiating these anatomical structures [34]. The addition of contrast me-dia, like iodine-based solutions, enhances the visualization of vascular andsoft tissue structures for diagnosing pathologies, including tumors, vascularmalformations, and organ traumas [35].The development of CT technology has been characterized by succes-sive generations, with continuous improvement in image quality, acquisi-tion speed, and diagnostic capability. Modern scanners like multi-slice CT(MSCT) and cone-beam CT (CBCT) allow the performance of rapid acqui-sitions with submillimeter spatial resolution [36]. For example, MSCT hasseveral detector rows to cover larger areas in a much shorter time [37], thusbecoming indispensable in whole-body imaging, cardiac studies [38], andtrauma assessment [39]. CBCT finds primary applications in dental [40]and orthopedic uses [41], where high-resolution imaging of smaller anatom-ical regions is required.CT has several advantages over other modalities: its spatial and tem-poral resolution, especially in imaging dense tissues like bone and detectingsubtle abnormalities [17]. It also allows multiplanar reconstructions and ad-vanced visualizations such as maximum intensity projections (MIP) and 3Drenderings, useful in surgical planning and intervention [29, 42].1.2.3 Magnetic Resonance ImagingMagnetic resonance imaging (MRI) relies on magnetic fields and radiofre-quency waves to develop highly detailed anatomical and functional imagesof organs, tissues, and cells [43]. Unlike all methods of imaging that use ion-izing radiation, MRI uses the magnetic properties of atomic nuclei, mainly41.2. Medical Imaging Modalitieshydrogen within the human body which are either freely available as wateror complexly bound into organic molecules [22, 43]. The basic principles ofMRI are based on the physics of nuclear magnetic resonance, a process thataligns and perturbs hydrogen nuclei in a magnetic field [44].When a patient is placed inside the magnetic field, hydrogen nuclei inthe body align either parallel (low-energy state) or anti-parallel (high-energystate) to the field [43]. A slight excess of the nuclei aligning with the fieldresults in net magnetization. These aligned nuclei precess at a frequencyproportional to the field strength called the Larmor frequency [43]. Forimaging, a radio frequency (RF) pulse at the Larmor frequency is appliedto excite the nuclei, thereby tipping their net magnetization away from thelongitudinal axis into the transverse plane [44, 45]. Once the RF pulse isterminated, the excited nuclei return to equilibrium during processes referredto as relaxation [46]. Two major relaxation occur: T1 (longitudinal or spin-lattice relaxation) and T2 (transverse or spin-spin relaxation). T1 describesthe recovery of the longitudinal magnetization in which the net magneticmoments realign with the magnetic field, while T2 characterizes the lossof transverse magnetization caused by interactions of neighboring magneticmoments [46]. An example highlighting the variations in the images resultingfrom the different relaxations (T1 and T2) in visualizing a brain mass isshown in Figure 1.2.The spatial localization of signals in MRI relies on the application ofgradient magnetic fields along the axes x, y, and z [43]. These gradientscreate spatial variations in the magnetic field that enable the scanner toencode the position of the signals along specific directions [43, 47]. Thedata recorded populates k-space, a frequency domain representation of theimage, which is then transformed via the Fourier transform into the finalimage [43, 44].MRI is especially known for having superior soft-tissue contrast; imag-ing parameters such as repetition time and echo time can be changed tooutline certain features of tissues [48]. For example, T1-weighted imageshighlight anatomy, while T2-weighted images are very useful for the detec-tion of fluid-containing or pathological tissues, including edema and tumors[49]. Advanced techniques such as diffusion-weighted imaging (DWI) pro-vide information on the tissue cellularity and are highly helpful in cancerimaging [50, 51].MRI’s added advantages include its versatility in pulse sequences, eachserving a different diagnostic purpose. For example, spin-echo sequences arecommonly used for routine imaging [52], while gradient-echo sequences arepreferred for rapid acquisition or high temporal resolution applications [53],51.2. Medical Imaging Modalitiessuch as cardiac imaging. Diffusion-weighted sequences, as well as techniquesfor T1 and T2 mapping, allow quantitative assessment of tissue properties[50].Figure 1.2: MRI scans compare T1- and T2-weighted imaging in a patientwith right-sided hemiparesis. (A) T1-weighted imaging shows a dark lesionin the left frontal-parietal region. (B) T2-weighted imaging highlights thelesion as bright with visible surrounding edema [10].1.2.4 Ultrasound ImagingUltrasound imaging provides real-time, high-resolution images of the hu-man body without ionizing radiation. Its foundational principles rely onthe propagation of high-frequency sound waves through tissues, the inter-action of these waves with different structures, and the interpretation ofthe returning echoes [54]. Ultrasound waves are mechanical waves trans-mitted through a medium, in this case, biological tissue, via the vibrationof particles [55]. The physics of ultrasound revolves around parameters likewavelength, frequency, and amplitude [56].These waves have frequencies exceeding 20 kHz, beyond the range of hu-man hearing. They are generated and received by piezoelectric transducers[57]. These transducers rely on the piezoelectric effect, a property of certaincrystals, such as quartz, to produce electrical charges when subjected tomechanical stress [58]. The reverse piezoelectric effect allows these crystalsto convert electrical energy into mechanical vibrations [59]. In generatingthe ultrasound waves, the reverse piezoelectric effect occurs: conversion ofelectrical energy to mechanical vibrations. When echoes are received, the61.2. Medical Imaging Modalitieswaves induce mechanical stresses in the transducers, generating electricalsignals [59].As ultrasound waves travel through the body, they encounter interfacesbetween tissues with differing acoustic properties, such as muscle and fat.At these boundaries, the waves are reflected, refracted, or absorbed. Re-flected waves carry critical information about the internal structures andare captured by the transducer, causing piezoelectricity [57, 59]. The elec-tric signals are processed to generate images, providing insights into thecomposition and function of tissues.However, the ultrasound waves are pulsed. That is, the transducer sendsout intermittent pulses of sound waves and picks up reflected sound. Pul-sating the transmission of the sound waves enables the calculation of pulserepetition frequency, or the number of pulses transmitted per second, a crit-ical value in the depth measurement [60]. High-frequency ultrasound, forexample, 10-15 MHz, is preferred for imaging superficial structures becauseof its high resolution, whereas low-frequency ultrasound, for example, 2-5MHz, is used for deeper structures because of its greater tissue penetrationcapabilities [60].1.2.5 Introduction to Nuclear Medicine ImagingUnlike X-rays, CT, and MRI imaging, which primarily focus on visualizinganatomical structures, nuclear medicine imaging emphasizes physiologicaland metabolic activities. This functional imaging capability is achievedthrough radiopharmaceuticals, which combine a biologically active moleculewith a radionuclide capable of emitting radiation [61]. These radiopharma-ceuticals target specific tissues or cellular processes and emit radiation [62],typically gamma rays, captured by imaging systems such as gamma cam-eras [63]. This allows for the detection of molecular and biochemical changeswell before structural abnormalities become visible. For example, fluorine-18fluorodeoxyglucose (FDG) PET imaging outlines areas of increased glucosemetabolism [64], which is common in many cancers, thus offering early in-formation about tumor activity [65]. Likewise, nuclear medicine imagingcan assess organ-specific functions, such as iodine uptake in the thyroidgland [66] or myocardial blood flow [67], providing important informationin diagnosis and disease management. In so doing, nuclear medicine fillsan essential gap in medical imaging, complementing the structural focus ofconventional techniques.Moreover, by leveraging the different radioactive decay pathways of ra-dioisotopes, therapy and imaging can be performed simultaneously. This71.3. Foundations of Nuclear Medicine Imagingdual functionality is a hallmark of theranostic agents, which integrate treat-ment and visualization within a single radiopharmaceutical [68]. For exam-ple, iodine-131 (I-131) emits beta particles that deliver cytotoxic radiation totarget tissues, making it highly effective for treating thyroid cancer and hy-perthyroidism [69]. Simultaneously, its gamma emissions allow for imagingthe radiopharmaceutical distribution, enabling clinicians to assess iodine up-take and monitor therapeutic progress [70]. Similarly, lutetium-177 (Lu-177)emits beta particles for targeted cancer therapy, such as neuroendocrine tu-mors and prostate cancer [71]. Its gamma emissions can be detected throughSPECT imaging, providing real-time data on dose distribution and treat-ment response.1.3 Foundations of Nuclear Medicine Imaging1.3.1 Physics of Radiation in Nuclear Medicine ImagingAt the foundation of nuclear medicine imaging lies the science of radiationphysics, which governs the behavior of radionuclides and the mechanisms ofradioactive decay. Radionuclides, during decay, emit energy through pro-cesses such as gamma-ray emission or positron annihilation, and this energyis harnessed to generate images of functional activity [72]. Factors like half-life, energy emission, and decay pathways guide the choice of radionuclides[72]. The following section discusses the physics of radiation in nuclearmedicine imaging.Atomic Structure and the NucleusThe atom consists of a dense nucleus with a positive charge, composed ofprotons and neutrons, and is surrounded by electrons revolving in orbitalshells with a negative charge [73]. Two opposing forces hold the nucleustogether: the attractive, strong nuclear force between all nucleons and therepulsive Coulomb force between positively charged protons. The relativestrength of these forces determines whether the nucleus is stable [8]. Stablenuclei have certain optimal combinations of neutrons and protons; unstablenuclei, or radionuclides, are deficient in this respect and attempt to achievestability through radioactive decay [8].The behavior of radionuclides is determined by their atomic number, Z,and mass number, A [74], which further classify them into isotopes, isobars,or isotones [75]. For example, isotopes of an element have the same numberof protons but differ in neutrons. The stability and decay mechanisms of ra-81.3. Foundations of Nuclear Medicine Imagingdionuclides are based on nuclear binding energy, which is the energy neededto separate nucleons from the nucleus [8].Radioactive Decay ProcessesRadionuclides emit radiation to transition to more stable configurations.The primary modes of decay are:• Alpha Decay: Heavy radionuclides emit an alpha particle (a heliumnucleus consisting of two protons and two neutrons) [8]. This processreduces the mass number by four and the atomic number by two,resulting in a new element [8]. Alpha decay is common in radionuclideswith high atomic masses, such as radium-226.• Beta Decay: This occurs in two forms: Beta-minus (β−) decay,where a neutron converts into a proton, emitting an electron and anantineutrino [8]. Beta-plus (β+) decay, where a proton converts intoa neutron, emitting a positron and a neutrino [8]. The emitted positronquickly annihilates with an electron, producing two gamma photonsof 511 keV each, which are crucial for positron emission tomography(PET) [76]. Beta decay alters the atomic number while conserving themass number, facilitating transitions between elements.• Gamma Decay: Excited nuclei release excess energy in the form ofgamma rays, which are electromagnetic waves with high energy [8].This process occurs without changing the atomic or mass number andis often a secondary transition following other decay modes.Radioactive KineticsThe rate of radioactive decay is governed by the decay constant (λ), whichdefines the probability of decay per unit time [8]. The activity (A) of aradionuclide sample, representing the number of decays per second, followsan exponential law:A(t) = A0e−λt (1.1)where A0 is the initial activity. The time required for half the nuclei in asample to decay is known as the half-life (t1/2), a critical parameter in select-ing radionuclides for medical imaging. For example, fluorine-18, commonlyused in PET, has a half-life of approximately 109 minutes [77], balancingeffective imaging duration and minimal patient radiation exposure.91.3. Foundations of Nuclear Medicine ImagingRadiopharmaceuticals are designed to target specific tissues or processesand use the predictable decay properties of the radionuclides to emit signalsthat can be detected and reconstructed into images. Techniques such asSPECT and PET use the unique characteristics of gamma rays and positronannihilation events, respectively [8, 78], to provide detailed functional andmetabolic information not offered by purely anatomical imaging modalities.1.3.2 Radiopharmaceuticals in Nuclear Medicine ImagingRadiopharmaceuticals are the fundamental compounds used in nuclearmedicine for diagnosis and therapeutics. They have two essential compo-nents: a radionuclide that emits radiation and a biochemical carrier thatselectively targets a particular tissue or cellular process. Radiopharmaceu-ticals’ unique ability to selectively bind to biological targets enables clini-cians to study physiological processes with high specificity while minimizingsystemic exposure [79].The targeting mechanism of radiopharmaceuticals is mediated by conju-gating radionuclides to molecules such as monoclonal antibodies, peptides,or small organic compounds. For example, prostate-specific membrane anti-gen (PSMA) ligands labeled with gallium-68 (68Ga) [80] or lutetium-177(177Lu) target PSMA-expressing prostate cancer cells [81], enabling bothprecise imaging and targeted therapy . This mechanism exploits molecularoverexpression in diseased tissues, ensuring high sensitivity and specificityfor disease detection and management. Somatostatin analogs labeled withradionuclides such as [68 Ga]Ga-DOTATATE for imaging and [177Lu]Lu-DOTATATE for therapy are well known in the treatment of neuroendocrinetumors [82, 83], showing the duality of radiopharmaceuticals in diagnosisand treatment.In nuclear medicine imaging, gamma emitters such as technetium-99m(99mTc) find widespread applications due to their ideal photon energy andhalf-life [84], yielding high-resolution images of organ function with mini-mal radiation exposure [85]. The prevalent positron emitters for PET aregallium-68 (68Ga), fluorine-18 (18F), with 18F-fluorodeoxyglucose (FDG) be-ing the most commonly used tracer in assessing tumor glucose metabolism[86].In contrast, therapeutic applications depend on the use of radionuclidesthat emit radiation of a particular type and energy. Beta emitters, suchas yttrium-90 (Y-90) and Lu-177, are effective in treating large tumor vol-umes because of their crossfire effects [87, 88]. Radiation destroys tumorcells in areas where the radiopharmaceutical is not directly in contact. Al-101.3. Foundations of Nuclear Medicine Imagingpha emitters like actinium-225 (Ac-225) deliver densely ionizing radiationwith extremely high linear energy transfer [89], making it ideal for targetingmicrometastatic disease while sparing surrounding healthy tissue.Targeting PSMA Protein for PET ImagingPSMA represents a pivotal target in PET imaging for prostate cancer due toits high expression in prostate adenocarcinomas and increases with tumoraggressiveness, progression, and resistance to therapies, which makes it agood and reliable indicator for advanced disease stages [90]. The uniqueextracellular domain of this transmembrane glycoprotein enables the bindingof imaging agents, while its internalization property increases the retentionof radiopharmaceuticals within the cancer cells [91, 92]. These attributesenable precise imaging with minimal off-target effects.The development of radiopharmaceuticals targeting this protein, suchas [68Ga]Ga-PSMA-11 and [18F]PSMA-1007, allows for better sensitivityand specificity [80, 93] than traditional methods such as CT or MRI. Theirpotential to find micrometastases and early lesions even at low PSA levels[94] addresses some of the critical limitations of conventional imaging. Thisprecision significantly aids in the staging, treatment planning, and follow-up of metastatic and recurrent PCa [93, 95]. Additionally, PSMA imaginghas a better false positive detection rate than choline or amino acid-basedtracers, which lack prostate cancer specificity [96, 97].Compared to amino acid tracers like [11C]methionine or hypoxia trac-ers such as Fluorine-18 fluoromisonidazole ([18F]FMISO) [98], which lackprostate cancer specificity, targeting PSMA provides greater diagnostic pre-cision. Also, [18F]FDG, which is commonly used to detect a wide arrayof cancer types like lung cancer, lymphoma, and head and neck cancer,finds limited application in prostate cancer detection due to its typicallylow glycolysis rate [99–101]. Figure 1.3 below shows PSMA PET images ofa patient with metastatic castration resistant prostate cancer showing high[68Ga]Ga-PSMA activity whiles minimal activity is shown on the [18F]FDGscan. PSMA imaging also addresses the limitations of receptor-based imag-ing and bone metabolism tracers like Technetium-99m methylene diphospho-nate ([99mTc]Tc-MDP) or Fluorine-18 sodium fluoride ([18F]NaF), which areprone to false positives [102]. This precision significantly reduces diagnosticuncertainty and helps detect micrometastases, assess disease progression,and manage advanced prostate cancer.111.3. Foundations of Nuclear Medicine ImagingFigure 1.3: Comparison of [68Ga]Ga-PSMA PET and [18F]FDG PET imag-ing in a metastatic castration resistant prostate cancer patient. High tumoractivity is shown on the [68Ga]Ga-PSMA image while minimal activity isevident on the [18F]FDG PET scan [11].Significance of [18F]DCFPyL in PET ImagingPiflufolastat Fluorine-18 ([18F]DCFPyL) is a second-generation fluorine-18-labeled radiotracer that represents a significant advancement in PSMA-targeted PET imaging [103]. Compared with [68Ga]Ga-PSMA, [18F]DCFPyLhas a longer half-life, and superior positron emission characteristics [104–106]. These features improve tumor-to-background contrast and lesion de-tectability, thus being especially effective for the detection of small-volumedisease [102].The extended half-life of [18F]DCFPyL supports centralized productionand distribution, overcoming logistical barriers in facilities without on-sitecyclotron capabilities [103]. Its ability to facilitate flexible imaging protocolsand delayed imaging further optimizes diagnostic workflows [103]. Clinicalstudies have shown that [18F]DCFPyL frequently identifies lesions missed byconventional imaging or other tracers, leading to changes in clinical manage-ment [102]. In theranostics, [18F]DCFPyL provides critical information onPSMA expression, enabling patient selection for radioligand therapies like[177Lu]Lu-PSMA [107] even though it is not a direct molecular pair with thetherapeutic agent.121.3. Foundations of Nuclear Medicine ImagingTheranosticsTheranostics is a major frontier in the field of nuclear medicine where ra-diopharmaceutical imaging is combined with radiopharmaceutical therapy.Theranostics enables personalized precision medicine by using either (1) thesame radiopharmaceutical for both visualization and treatment of patients,or (ii) different radiopharmaceuticals that target the same molecule such asPSMA. A well-known example for (i) is the use of iodine-131 (131I) for diag-nostic and therapeutic purposes in thyroid cancer [69]. For (ii), modern ad-vancements have introduced theranostic pairs such as [68Ga]Ga-DOTATATEand [177Lu]Lu-DOTATATE, targeting somatostatin receptors in neuroen-docrine tumors [108]. Similarly, prostate-specific membrane antigen-basedtheranostics for prostate cancer use [68Ga]Ga-PSMA-11 or [18F]DCFPyLfor imaging and [177Lu]Lu-PSMA-617 or [225Ac]Ac-PSMA-617 for targetedtherapy [109], treating disease with minimal off-target effects. Table 1.1 be-low provides a summary of common theranostic pairs in nuclear medicineimaging and therapy along with decay information:131.3. Foundations of Nuclear Medicine ImagingTable 1.1: Various theranostic pairs in nuclear medicine imaging highlight-ing their various half-life, decay modes, possible imaging modality (positronemission tomography (PET), single photon emission computed tomography(SPECT), Bremsstrahlung imaging (Brems.) and maximum particulate en-ergy for beta-positive decay (Eβ+), beta-minus decay (Eβ−), alpha decay(Eα) and Auger electron (EAuger). EC = electron capture, IT = isomerictransition [7].Interaction of Radiation with MatterThe interaction of radiation with matter forms the basis of the mechanismof action of radiopharmaceuticals. These interactions are dominated bythree main processes: the photoelectric effect, Compton scattering, and pair141.3. Foundations of Nuclear Medicine Imagingproduction.The photoelectric effect dominates at low photon energies and is mostsignificant in tissues with higher atomic numbers, such as bones. This pro-cess involves the complete transfer of photon energy to an inner-shell elec-tron, resulting in its ejection and the subsequent emission of characteristicX-rays [8]. The photoelectric effect increases contrast in imaging but de-creases efficiency at higher photon energies [8].At intermediate photon energies, Compton scattering predominates. Inthis process, the photon imparts some of its energy to a loosely bound outer-shell electron, ejecting it while the photon continues at a deflected angle [8].While Compton scattering is essential to imaging deep tissues, it adds noiseand diminishes contrast, requiring correction algorithms to enhance imagequality.Pair production is a process that starts when the energy of a photonexceeds 1.022 MeV and essentially consists of the transformation of theenergy of a photon into an electron-positron pair [8]. The annihilation ofthe positron produces two gamma photons, each with 511 keV of energy,which are responsible for PET imaging [8]. Coincident detection of thesephotons enables the exact co-localization of the radiotracer activity in thebody and makes PET an important modality for functional imaging [110].Production of RadionuclidesThe production of the radionuclides used in radiopharmaceuticals is an im-portant aspect of nuclear medicine. Processes in nuclear reactors, cyclotrons,or radionuclide generators produce radionuclides. Nuclear reactors usuallymanufacture radionuclides by neutron capture [111], for example, the con-version of molybdenum-98 (98Mo) to molybdenum-99 (99Mo), which decaysto produce 99mTc [112], a very commonly used isotope in nuclear medicineimaging.Also, Cyclotrons, by accelerating charged particles like protons [113],induce nuclear reactions that create positron-emitting radionuclides suchas 18F [8], essential for PET imaging. Radionuclide generators, such asthe 99Mo/99mTc system essential for SPECT imaging [114] or germanium-68/gallium-68 generator for PET imaging [115], provides a practical solutionfor obtaining short-lived isotopes ensuring accessibility even in facilities farfrom production centers. Figure 1.4 below shows the decay of 99Mo andsubsequent growth of 99mTc typically occurring in 99Mo/99mTc radionuclidegenerator system.151.4. Imaging Modalities in Nuclear MedicineFigure 1.4: Growth and decay of 99mTc radioactivity (in millicurie units)during the decay of 99Mo, with equilibrium reached after approximately 24hours. The solid curve assumes that all disintegrations of 99Mo yield 99mTc.The dashed curve represents the actual 99mTc radioactivity, accounting forthe fact that only 92% of 99Mo disintegrations produce 99mTc [8].1.4 Imaging Modalities in Nuclear Medicine1.4.1 Planar Imaging or ScintigraphyPlanar allows for the two-dimensional visualization of the distribution of aradiopharmaceutical within the body; it involves the use of gamma cam-eras to detect gamma radiations emitted from radiopharmaceuticals [116].While, for most diagnostic applications, this modality has been replaced byother more advanced modalities, such as SPECT and PET, because of itssimplicity, low cost, and speed of acquisition, planar imaging remains helpfulin various clinical situations.The gamma camera remains central to planar imaging. It consists of alarge, thallium-activated sodium iodide (NaI(Tl)) scintillation crystal thatgenerates light upon interaction with gamma photons [8, 117]. This light161.4. Imaging Modalities in Nuclear Medicineis then detected by an array of photomultiplier tubes (PMTs) that convertthe light into electrical signals [117]. By analyzing the relative intensitiesof the signals coming from the PMTs, the camera’s electronics determinethe location of the gamma photon interaction, hence producing spatiallyaccurate images [8]. Early issues of non-uniformity and spatial distortionhave been overcome with advances in electronics and real-time correctionalgorithms, making planar imaging much more accurate [116]. Among themain challenges of planar imaging is deterioration in image quality due toscattered photons, which reduce contrast and resolution [116].Moreover, the instrumentation of planar imaging serves as the basis forSPECT imaging. In planar imaging, the gamma camera is stationary andcaptures 2D projections of radiotracer distribution in the body from a singleangle or a limited number of angles [116]. In SPECT imaging, the gammacamera is also used, however, the camera rotates around the patient toacquire projections from multiple angles, enabling 3D reconstruction of theradiotracer distribution [118]. The Anger gamma camera has remained thedominating technology in planar imaging despite searches for alternativedesigns such as multi-crystal arrays and solid-state detectors [116]. Thisimaging modality however lacks depth and precision compared to newerthree-dimensional modalities like SPECT and PET [119].1.4.2 Single Photon Emission Computed TomographyPrinciples of SPECT ImagingSPECT imaging initiates with administering radiopharmaceuticals that tar-get specific tissues or physiological processes. These compounds decay, emit-ting gamma photons, and a gamma camera detects the emitted photons.The gamma camera consists of a scintillation crystal, typically sodium io-dide (NaI), doped with thallium, which interacts with incoming photons toproduce flashes of light [8]. These light signals are subsequently convertedinto electrical signals through the action of PMTs, further amplifying andlocalizing the photon interactions [117]. For 3-D imaging, the gamma cam-era rotates around the patient in order to acquire projection data at manydifferent angles [118].Each projection yields a 2D representation of the radiotracer distribu-tion. A collimator, a lead or tungsten structure that filters the photonsdepending on their trajectory, controls the spatial localization of the de-tected photons [8]. That way, only photons moving along a specific pathcan be detected, and the spatial resolution increases. After this, reconstruc-171.4. Imaging Modalities in Nuclear Medicinetion algorithms like filtered back projection (FBP) or iterative algorithms,such as the Ordered Subset Expectation Maximization (OSEM) algorithm,convert the projection data into cross-sectional or volumetric images [8].To promote the design and deployment of various tomographic recon-struction algorithms, we developed an open-source python library, PyTo-mography [120], which leverages GPU-acceleration for SPECT/PET imagereconstruction. In appendix A.3, I present a user-friendly extension module,which I led the development of, built on PyTomography and the 3D SlicerAPI [121] for SPECT reconstruction.Radiation Detection Components and Photon InteractionsThe gamma camera is, however, at the heart of SPECT imaging, incorporat-ing a combination of components: the scintillation crystal, photomultipliertubes (PMTs), and collimators [8]. In selecting a collimator, applicationneeds and energy of the employed radionuclide remain factors. For instance,the low-energy high-resolution (LEHR) collimator is suited to cardiac imag-ing due to finer spatial resolution, while applications like iodine-131 (131I)involve medium-energy or high-energy collimators [8]. The main componentsof the gamma camera is highlighted in Figure 1.5 below:The interaction of photons with matter significantly affects image qual-ity. The gamma photons can be subjected to Compton scattering, a pro-cess where photons lose energy and change direction after interacting withloosely bound electrons, hence adding noise and artifacts[122]. The pho-tons can also be absorbed through the photoelectric effect, contributing tothe image signal [122]. Attenuation, caused by absorption by interveningtissues, is another challenge in SPECT imaging [123]. These effects arereduced through scatter correction algorithms and attenuation correctiontechniques, often made possible using hybrid SPECT/CT systems employ-ing CT-derived attenuation maps to enhance image clarity and quantitativeaccuracy [123].Applications and Advancements in SPECT ImagingIn cardiology, myocardial perfusion studies evaluate blood flow to the heartmuscle, which helps diagnose coronary artery disease and assess ischemicregions [124]. In neurology, SPECT can monitor brain perfusion and func-tion, supporting the diagnosis of conditions such as Alzheimer’s disease andepilepsy [125, 126]. Oncological applications include detecting and stag-ing tumors by visualizing metabolic activity or receptor binding [127]. Also,181.4. Imaging Modalities in Nuclear MedicineFigure 1.5: Figure of a typical scintillation camera, highlighting its fourmain components: A (collimator), B (NaI crystal), C (light pipe, photomul-tiplier tubes, and preamplifiers), and D (cathode ray tube for display). Thedetector head integrates components A, B, and C, while the console housesD along with power supplies and related electronics. Historically, the cath-ode ray tube (CRT) in component D was used in analog systems to displayγ-ray interactions, where X and Y pulses would deflect a light spot on theCRT or oscilloscope in proportion to the amplitude of the signals. In mod-ern systems, however, a digital computer processes the X- and Y-signals,storing the data and the count at each location in memory for display andfurther analysis on a digital monitor [8].SPECT imaging provides essential information regarding the efficacy of can-cer treatments. Changes in tumor uptake before and after treatment maysuggest response, progression, or resistance. For instance, in thyroid cancer,123I or 131I imaging is used post-thyroidectomy for the assessment of residualdisease or metastatic spread and to guide radioactive iodine therapy [128].Hybrid imaging systems, such as SPECT/CT, combine the functionalcapabilities of SPECT with the anatomical detail of CT, improving diag-nostic precision and localization. For instance, SPECT/CT can localizeareas of abnormal tracer uptake, thus allowing for much more precise dif-ferentiation between pathological and physiological activity [123]. Figure1.6 below shows an example application of SPECT/CT imaging for tumorlocalization based on differences in tracer uptake. Recent detector technol-ogy developments, including semiconductor-based systems such as cadmium191.4. Imaging Modalities in Nuclear Medicinezinc telluride detectors, have further improved resolution and sensitivity inSPECT imaging [8].Figure 1.6: Coronal and maximum intensity projection (MIP) images via[99mTc]Tc-PSMA-I&S SPECT/CT to assess metastatic castration-resistantprostate cancer (mCRPC), showing extensive tumor spread prior to therapyinitiation in a patient with disease progression under chemotherapy and aPSA level of 137 ng/mL [12].1.4.3 Positron Emission TomographyPrinciples of PET ImagingPET images are formed based on the detection of gamma photons generatedin a process called positron annihilation [129]. Radiotracers labeled withpositron-emitting radionuclides, such as fluorine-18 (18F), are administeredto the patient. By undergoing β+ decay, these radionuclides emit positronsthat travel a short distance in the surrounding tissue losing energy throughinteractions with electrons [129]. In this case, when a positron combineswith an electron, there is a resulting annihilation event where two gammaphotons are produced with energies of 511 keV each, nearly 180° apart [8].These photons are detected by a ring of detectors surrounding the patient,defining a ”line of response” (LOR) between the two detection points [8].201.4. Imaging Modalities in Nuclear MedicineCoincidence detection for PET identifies photon pairs that come from thesame annihilation event, thus avoiding the need for physical collimators, asin SPECT, and increasing sensitivity [8, 130]. Table 1.2 below highlightsthe characteristics of various positron-emitting radionuclides of interest inPET imaging.Table 1.2: Properties of positron-emitting radionuclides relevant to nuclearmedicine imaging. The table shows the half-life (in minutes), percentage ofpositron emission, maximum energy released during decay (Emax) in MeV,and positron range in millimeters [8].Instrumentation and Data AcquisitionThe PET scanner consists of several detector blocks arranged in a circle ora polygon. The detectors consist of scintillation crystals such as lutetiumoxyorthosilicate (LSO) or bismuth germanate (BGO), which interact withthe gamma photons [8]. An interaction of the photon with the crystalsproduces a flash of light proportional to the energy of the photon. Theselights are further amplified and digitized by the photomultiplier tubes orsilicon photomultipliers (SiPMs) [8].Modern PET systems often incorporate Time-of-Flight (TOF) technol-ogy, which measures the slight difference in arrival times of the two photons[131]. The TOF improves the localization of the annihilation event alongthe LOR, enhancing image quality and reducing noise [8]. This capabil-ity is particularly useful in improving image resolution and the accuracy ofquantitative data.Challenges in PET ImagingSeveral physical phenomena affect the spatial resolution and image quality inPET. The positron range, or the distance a positron travels before annihilat-211.4. Imaging Modalities in Nuclear Medicineing, introduces blurring in reconstructed images [132]. Other limitations in-clude photon acollinearity: the residual momentum of the positron-electronsystem means that the gamma photons are not quite 180° apart, whichcauses resolution degradation, especially for larger detector ring diameters[8].Furthermore, scatter, whereby photons interact with tissue before reach-ing a detector, adds noise and leads to a reduction in contrast [8]. The illus-tration below (Figure 1.7) summarizes the various phenomena that limit im-age resolution in PET, highlighting instances where a γ-ray enters a crystalat an angle and penetrates other crystals (penetration effect), and Comptonscattering leading to a deflection into other crystals and photon acollinearity.Figure 1.7: Illustration of the various phenomena that lead to resolutiondegradation in PET imaging. When a γ-ray interacts with a crystal, it maybe detected in a neighboring crystal due to two factors: (i) the penetrationeffect, where the γ-ray enters the crystal at an angle and extends into anadjacent crystal, or (ii) Compton scattering, where the γ-ray is scatteredinside the crystal and deflects to a different crystal. The figure demonstratesthese effects with three adjacent crystals, showing how a photon entering themiddle crystal may either penetrate into an adjacent crystal or scatter toanother crystal before detection [8].Image Reconstruction and Correction MethodsReconstructing PET images involves transforming detected LORs into cross-sectional or three dimensional (3D) representations of radiotracer distribu-tion [8]. Iterative statistical reconstruction algorithms, such as Maximum221.4. Imaging Modalities in Nuclear MedicineLikelihood Expectation Maximization (MLEM) [133] and Ordered SubsetExpectation Maximization (OSEM) [134], model the physics of photon inter-actions, the detector response, and the Poisson noise in detection to producehigh-quality quantitatively reliable images.In PET imaging, attenuation correction is crucial because photons em-anating from deeper tissues have greater chances of being absorbed or scat-tered, and their incomplete distribution may cause inaccuracies in imagereconstruction [8]. The attenuation map provided by CT data is usuallyapplied in PET/CT systems to account for these effects, enabling correc-tions that account for photon loss and restoring accurate tracer distributionmeasurements [135].Normalization or uniformity correction compensates for variations in thesensitivity of detector pairs [8]. Differences in individual detector efficiency,photon incidence angles, and geometric factors can result in non-uniformresponses even for a uniform source [8]. To address this, normalizationfactors are derived from calibration scans, often using a uniform rod source[136]. These factors adjust for discrepancies in detection sensitivity, ensuringconsistent and accurate measurements. Modern methods use component-based normalization to reduce calibration time while maintaining precisionby separately modeling the contributions of each detector component [8,137].Corrections for scatter and random coincidences are crucial to improvePET image accuracy. Scatter occurs when photons interact with tissues,distorting their trajectory and leading to an incorrect line of response detec-tion [138]. Scatter correction typically uses estimation techniques to modeland subtract the distribution of scattered photons during the reconstructionprocess [8]. This approach estimates scatter iteratively by first performing areconstruction without correction, then refining it by incorporating scatterestimates from the initial image and attenuation maps, using models likeKlein-Nishina [139] or Monte Carlo methods [140].Random coincidences arise from unrelated photons detected simultane-ously, causing false events. These previously were addressed through de-layed coincidence timing windows [8, 141]. In more recent methods, theseare incorporated in the denominator of iterative reconstruction methods likeOrdered Subset Expectation Maximization (OSEM), or the detection the ofsingles are used to estimate the random coincidences. Figure 1.8 belowshows the improvement of PET image quality over the years as a result ofa progressive modeling of imaging system parameters.231.4. Imaging Modalities in Nuclear MedicineFigure 1.8: This image illustrates the development of PET image reconstruc-tion, starting with 3D filtered back projection (FBP), progressing to Poissonnoise modeling (MLEM and OSEM), and advancing with point spread func-tion (PSF) data modeling. More recently, regularization techniques havebeen adopted to reduce noise and improve image quality [13].Clinical RelevanceThe ability of PET to detect molecular changes before structural abnor-malities make it an important modality in personalized medicine. In oncol-ogy, fluorine-18 fluorodeoxyglucose ([18F]FDG) is widely used for detectinghypermetabolic activity in tumors, thus providing helpful information forstaging, treatment planning, and response monitoring in cancer [64]. Spe-cialized tracers, such as 68Ga-DOTATATE for neuroendocrine tumors orPSMA-labeled prostate cancer agents, further enhance PET’s diagnosticspecificity [82].In cardiology, the assessment of myocardial viability and perfusion usingPET helps guide appropriate interventions in ischemic heart disease [142].In neurology, PET tracers such as 18F-florbetapir are used for studyingamyloid plaques in Alzheimer’s disease [143]. At the same time, [18F]FDGhelps in the mapping of epileptic foci, thus aiding in the understanding ofneurodegenerative disorders [144].Hybrid imaging systems, such as PET/CT and PET/MRI, combine241.5. Prostate CancerPET’s functional capabilities with the anatomical precision of CT or MRI,thus enabling better disease localization and characterization [129]. Thesedevelopments have significantly expanded PET’s applications, with increaseddiagnostic accuracy and improved therapeutic decisions.1.5 Prostate Cancer1.5.1 Overview and PathophysiologyProstate cancer (PCa) is the most common cancer currently diagnosed inmen from all over the world and thus makes a significant contribution tocancer-related morbidity and mortality [145]. Generally, it is a disease ofolder men; there is a steep rise in incidence after the age of 50 years [146].The prostate gland is one of the most susceptible organs to malignancybecause of its critical dependency on androgen signaling and its unique tissueenvironment [147].Pathogenesis and Genetic InfluencesProstate cancer arises primarily as a carcinoma of the epithelial cells liningthe prostatic acini [148]. Genetic mutations, epigenetic modifications, andhormonal regulation may contribute to transforming benign epithelial cellsinto malignant phenotypes [149]. Key genetic alterations include mutationsin tumor suppressor genes such as PTEN and BRCA1/2 and chromosomalrearrangements involving the ETS family of genes, especially TMPRSS2-ERG fusions [150]. These genetic alterations usually result in the dysregula-tion of cellular growth, apoptosis, and metabolism [151]. Hereditary factorsalso come into play, as family history can increase the risk of the disease[152].The development and progression of prostate cancer depend on androgenreceptor (AR) signaling [153]. Testosterone and dihydrotestosterone activatethe AR, which in turn activates the transcription of target genes responsiblefor cell proliferation and survival [154]. However, long-term androgen abla-tion therapy, a conventional treatment for advanced prostate cancer, oftenleads to the development of castration-resistant prostate cancer [155]. Inthis condition, the tumor becomes more aggressive due to AR-independentpathways [155].251.5. Prostate CancerDiagnosis and BiomarkersThe diagnosis of prostate cancer includes several approaches, such as prostate-specific antigen (PSA) testing, digital rectal examination (DRE), PET imag-ing, and confirmatory biopsies [156, 157]. PSA is a widely used biomarker,though it lacks specificity because it can also be elevated in cases of benignprostatic hyperplasia (BPH) or prostatitis [158]. Advances in molecular di-agnostics have included the introduction of biomarkers such as PCA3 andgenomic assays that improve specificity and aid in risk stratification [159].Metabolic Adaptations in Prostate CancerMetabolic reprogramming is one of the hallmarks of prostate cancer, whichis driven by both AR signaling and tumor microenvironmental changes. Un-like in many other cancers, where tumors are largely dependent on glycolysis,prostate cancer cells have both aerobic glycolysis and oxidative phosphory-lation [160]. The tumor uses this dual metabolic strategy for rapid prolifera-tion and survival in variable conditions [160]. Pathways such as the pentosephosphate pathway and lipid biosynthesis are upregulated to meet increasedbiosynthetic and redox demands in advanced prostate cancer [161].Treatment ApproachesTreatment options in prostate cancer range from active surveillance in low-risk disease to aggressive interventions like prostatectomy and radiation ther-apy [162]. Androgen deprivation remains the primary therapy of advanceddisease management. However, this frequently leads to the development ofresistance [155], and alternative strategies involve AR-targeted therapies,chemotherapy, and immunotherapy [163].1.5.2 Role of PET/CT in Prostate Cancer ManagementPET/CT integrates the molecular imaging capability of PET with high-resolution anatomic imaging provided by CT [129]. This allows for accu-rate prostate cancer detection, staging, follow-up, and treatment planning[164]. The development of prostate-specific radiotracers has widened theuse of PET/CT in managing prostate cancer. Although [18F]FDG PET haslimited use because of the low glucose metabolism of prostate tumors, thesensitivity of PET/CT is enhanced by agents, such as gallium-68 (68Ga)-PSMA and fluorine-18 (18F)-PSMA, targeting prostate-specific antigen thatis overexpressed on most PCa cells [80, 93, 99]. These tracers allow for261.5. Prostate Cancerhigh tumor-to-background contrast, thus enabling precise lesion detection.They can also be labeled with therapeutic isotopes like 177Lu or 225Ac fortheranostic applications [81, 165].One of the most critical applications of PET/CT in the management ofPCa involves primary staging and restaging following biochemical recurrence[166]. With biochemical recurrence, lesions are often small and have lowlevels of PSA. However, 68Ga-PSMA PET/CT, for example, demonstratedan excellent ability to detect recurrent disease even in patients with PSA aslow as 0.2 ng/mL, surpassing the ability of conventional imaging modalitiessuch as MRI and CT [167].Moreover, PET/CT helps in guiding therapeutic decisions in PCa. Ac-curately delineating the extent of disease helps oncologists differentiate be-tween localized and systemic disease, influencing choices such as curativesurgery or systemic therapies. The integration of PET/CT in radiotherapyplanning has also enhanced precision. For instance, it identifies metaboli-cally active lymph nodes that might otherwise be missed, ensuring targetedradiation [168]. Furthermore, functional imaging using PET tracers allowsclinicians to monitor treatment response and disease progression. Cliniciansassess the efficacy of systemic therapies, such as androgen deprivation orradioligand therapy, by evaluating tumor metabolism and viability, therebyguiding adaptation in treatment strategies [169].1.5.3 Prostate Cancer SegmentationImportance of Tumor SegmentationSegmentation of prostate cancer tumors in PET/CT scans involves outlin-ing the tumor region of interest in the images. It provides data on tumorbehavior, aggressiveness, and progression and can be helpful for treatmentplanning, dosimetry, and extracting radiomic features for predictive model-ing [170–172]. Additionally, biomarkers such as standardized uptake valuessuch as maximum uptake (SUVmax), mean uptake (SUVmean), peak up-take (SUVpeak), molecular tumor volume (MTV), and total lesion activity(TLA) are directly derived from these segmented regions, providing predic-tive information on treatment response and survival outcomes [170].Conventional Segmentation MethodsManual segmentation methods, such as manual contouring, have been thegold standard for tumor boundary delineation in medical images. In manualcontouring, expert annotators visually inspect the imaging data and draw271.5. Prostate Cancerthe tumor ROI based on their interpretation of the imaging features. Al-though manual methods are the basis for tumor segmentation, they haveshortcomings.Manual contouring is highly labor-intensive, requiring extensive experi-ence and attention to minute details: identifying complex tumor structures,differentiating tumor boundaries from normal tissues, and accounting forvarious image artifacts. Another commonly used technique is thresholding.This involves selecting a range of intensity or SUV values that distinguishtumor tissue from surrounding normal tissue [173]. Both methods aim toprovide accurate delineation essential for extracting relevant biomarkers andfurther analysis. Thresholding may be sensitive to changes in tumor uptake,noise, and inconsistencies in the imaging protocol, leading to suboptimaldelineations [174]. Consequently, these methods (manual contouring andthresholding) usually fail to provide the reproducibility and scalability thatmodern clinical and research work demands.Challenges of Manual Tumor SegmentationManual segmentation depends on the expertise and interpretation of the in-dividual annotators, which can result in variability in the delineation of theregions of interest [175, 176]. This inconsistency between annotators (inter-observer variability) or for a specific annotator segmenting same volumeat different times (intra-observer variability), respectively, results in differ-ent interpretations of the same medical images, affecting the reliability ofthe biomarkers extracted and influencing subsequent analyses [177], such asthe evaluation of treatment response or survival prediction. Besides, whenapplied to large cohorts of datasets or multiple centers, the variations inannotator expertise and institutional protocols can increase inconsistencies,reducing reliability of the results.Another major challenge is the labor-intensive and time-consuming na-ture of manual segmentation. Accurately delineating tumors or complexanatomical structures requires careful effort, often demanding hours of workper case. This demand can lead to bottlenecks in clinical settings with highpatient volumes, delaying critical tasks such as treatment planning or ra-diomics analysis. Moreover, a limited number of trained experts can performsuch detailed work, further compounding the problem and making manualsegmentation challenging for high-volume applications.Prostate cancer tumors themselves pose additional challenges due totheir biological complexity. These lesions are heterogenous in size, shape,and radiotracer uptake, and in the case of metastases, they can vary in281.6. Conclusion and Transition to Next Chapteranatomical location. Such characteristics can make it extremely difficult foreven experienced annotators to accurately define tumor boundaries, leadingto higher levels of uncertainty and potential errors in segmentation. Thesedifficulties are further highlighted by imaging artifacts and variations inimaging quality.1.6 Conclusion and Transition to Next ChapterPET/CT imaging is used in the different clinical stages for the managementof prostate cancer. This cancer type is associated with an overexpressionof the prostate-specific membrane antigen, and with the use of radiophar-maceuticals that target this antigen, the tumors are accurately localized.After detection, lesions may be segmented to obtain quantitative metricslike tumor size, activity, or other radiomic features. The segmentation pro-cess involves manually delineating the boundaries of the tumors, which isa labor-intensive task that is also prone to errors. Thus, there has beenan increased interest in automating the lesion segmentation process usingartificial intelligence (AI)-based techniques. In the next chapter, we discussthe potential of AI methods for oncologic image segmentation in generaland prostate cancer specifically and consider different techniques employedto achieve this task.29Chapter 2Artificial Intelligence inMedical Imaging2.1 Artificial Intelligence For Medical ImageProcessingThe research and development of artificial intelligence (AI) tools for medicalapplications has risen significantly in the past decade. It has been appliedin various capacities for diagnosis, treatment planning, and patient monitor-ing [178, 179]. By taking advantage of recent advances in machine learningand deep learning, AI systems can process and interpret massive amounts ofdata in imaging with unique efficiency and precision. These advancementshave significantly enhanced the capabilities of radiology and other imaging-based specialties, reshaping the landscape of contemporary healthcare [180].Across imaging modalities—including CT, MRI, PET, ultrasound, and X-ray—AI applications span from fundamental image enhancement to sophis-ticated tasks such as predictive modeling and diagnostic report generation,offering unparalleled benefits to clinicians and patients alike [181, 182].The success stories of AI in medical imaging are numerous, but proba-bly one of the most notable examples is in the image enhancement domain.Over the past few years, deep learning (DL) has enhanced the quality andutility of imaging data [183]. For example, it is possible to denoise low-doseCT scans with DL techniques such as convolutional neural networks (CNNs)and generative models like generative adversarial networks (GANs) and dif-fusion models or to reconstruct high-resolution MRI images from noisy orincomplete data [184–187]. These are novel solutions to age-old problemsin radiology, such as the trade-off between image quality and radiation ex-posure or the prolongation of patient scanning times [182]. By enhancingresolution and reducing artifacts, AI-based image reconstruction methodsmight drive medical imaging towards reaching greater potentials with mini-mal risks and discomfort for the patient. The role of AI in nuclear medicine,particularly in PET imaging, further highlights its transformative impact.302.1. Artificial Intelligence For Medical Image ProcessingConventional limitations of PET have included poor spatial resolution andnoise [188]. Yet, these algorithms have significantly improved the imagequality and quantitative accuracy through denoising and artifact correction[189, 190]. Additionally, based on insights obtained from medical images,AI is increasingly being explored to redefine precision medicine and pre-dictive modeling by allowing clinicians to personalize treatments for eachpatient. Radiomics is a subfield of AI that extracts quantitative featuresfrom imaging data, uncovering subtle patterns in correlation with diseaseprogression, treatment response, and patient outcomes [191]. For instance,AI models have been used in oncology to identify imaging biomarkers thatpredict the efficacy of chemotherapy, immunotherapy, and radiation ther-apy [192, 193]. These models integrate imaging data with clinical variables,enabling personalized treatment plans that improve patient prognosis whileminimizing unnecessary interventions [194]. It also has applications in car-diovascular imaging, where AI algorithms analyze serial scans for monitor-ing plaque progression [195] and predicting events such as heart attacks orstrokes [196, 197].Another scope of clinical application where AI methods are being de-veloped is generating medical reports for the automated interpretation ofimaging data into structured, clinically meaningful descriptions [198, 199].This application combines computer vision for image analysis with naturallanguage processing for generating textual reports. For example, in chest ra-diograph interpretation, AI models trained on large datasets such as MIMICchest radiograph (MIMIC-CXR) have demonstrated the ability to generatepreliminary diagnostic reports that align with the findings of expert radiol-ogists [200]. These systems integrate attention mechanisms that allow themto focus on clinically relevant regions of the image, ensuring the reportsgenerated are accurate and detailed [201, 202]. Such systems can be in-valuable tools in resource-limited settings with fewer radiologists, enablingfaster triage and diagnosis [203]. The applications of AI methods continue toexpand to find use as tools for automated tumor detection and characteriza-tion, a task previously considered to be beyond the reach of automation; thesubsequent sections of this chapter will delve into the specific contributionsof AI to achieve automation of image segmentation.312.2. Artificial Intelligence in Medical Image Segmentation2.2 Artificial Intelligence in Medical ImageSegmentationSegmentation has become one of the most impactful applications of AI inmedical imaging, forming the basis for a range of clinical tasks, includingthe extraction of radiomic features [204], the quantification of clinically rel-evant tumor metrics such as tumor volume [205], organ delineation [206],and preoperative surgical planning [207]. Traditional segmentation meth-ods, such as thresholding, region growing, and edge detection [208], are in-creasingly being substituted with AI-driven approaches. While these tradi-tional methods are effective in some controlled scenarios [209], image-relatedchallenges like complex anatomical structures, image variability, and low-contrast regions make the traditional methods insufficient for dealing withsuch data, necessitating the adoption of more advanced, data-driven tech-niques [210, 211]. Before deep learning, AI-related techniques, mainly ma-chine learning algorithms such as support vector machines, regression trees,and other clustering algorithms, yielded marginal performance [209]. How-ever, deep learning marked a turning point in medical image segmentation,enabling automated systems to achieve competitive precision and reliabil-ity. Encoder-decoder architectures like U-Net and its variants have becomefoundational tools. These models leverage a combination of convolutionaloperations, downsampling, and upsampling to extract features at multipleresolutions [212]. U-Net preserves spatial details while learning hierarchi-cal features by incorporating skip connections between encoder and decoderstages, making it highly effective for delineating fine boundaries in challeng-ing datasets [212]. For example, in brain tumor segmentation in MRI scans,U-Net-based models have demonstrated exceptional performance, achiev-ing high Dice similarity coefficients and significantly improving consistency[213, 214].AI has also been transformative in other domains of medical imaging.In hepatic imaging, segmentation algorithms are crucial for outlining pri-mary liver tumors and metastases on computed tomography and MRI stud-ies [215]. Automation not only speeds up tumor delineation [216] but alsoincreases the diagnostic accuracy of the process. For example, AI modelshave been applied to the segmentation of liver tumors with sensitivity valuesgreater than 90% [217]. Similarly, in breast imaging, segmentation modelshelp identify masses and calcifications in mammograms and breast MRIs, en-hancing BI-RADS (breast imaging reporting and data system) classificationand reducing false positives [218]. These advances are especially valuable in322.3. Model Architectures in Image Segmentationdense breast tissue cases, where lesion detectability is more challenging, yetAI systems have achieved sensitivity rates that exceed 90% [219].Beyond tumor segmentation, AI-powered tools have revolutionized vas-cular imaging, where accurate segmentation of blood vessels is crucial forsurgical planning and risk assessment. CNN-based models, including U-Netand its attention-enhanced variants, have achieved outstanding performancein several tasks, such as segmentation of retinal blood vessels with accu-racy values higher than 90% on datasets like DRIVE (digital retinal imagesfor vessel extraction) and STARE (structured analysis of the retina) [220].Similarly, in cardiovascular applications, 3D U-Nets have performed well inidentifying coronary arteries in CT angiography, enabling early diagnosis ofvascular diseases, and helping cardiologists plan treatment [221].The integration of transformer-based architectures into segmentationmodels has advanced the field by addressing limitations in capturing globalcontext and subtle pathologies [222]. Hybrid models, such as TransUNet,combine the local feature extraction capabilities of CNNs with the globalcontextual awareness provided by transformers, allowing accurate segmen-tation of complex structures such as small lesions or vasculatures [223]. Thiscapability is particularly impactful in oncology, where delineating complextumor boundaries is crucial for treatment planning.AI-driven segmentation has extended its influence in modalities andapplications, including PET/CT imaging, where volumetric segmentationmodels are crucial to assessing whole-body tumor burden [224]. These cap-ture interslice dependencies and reconstruct 3D structures to evaluate andmonitor the disease comprehensively [225]. Automation in segmentation hasreduced the time-consuming tasks associated with manual delineation andenhanced reproducibility, reducing intra- and inter-observer variability.2.3 Model Architectures in Image SegmentationArtificial Neural Networks (ANN) form the backbone of contemporary ma-chine learning systems, drawing inspiration from biological neural networks[226]. The traditional variants of ANNs, such as MLPs, have proven to solvesimple tasks; however, these models are shallow, generally with one or twohidden layers, and thus cannot capture the rich sets of complex hierarchicalpatterns present in the data [227]. This limitation gave rise to deep neuralnetworks, which are simply extensions of the ANN architecture by addingmultiple hidden layers, hence having a much greater representational power[228].332.3. Model Architectures in Image SegmentationDeep Neural Networks (DNNs) excel by leveraging their depth to learnhierarchical features from raw data. In lower-level vision tasks such as imagesegmentation, early layers of DNNs detect simple edges and textures, whiledeeper layers capture higher-order structures like shapes or objects [229].This aspect of abstraction at every layer makes DNNs excel at handling andunderstanding complex patterns, which is crucial for applications involvingaccuracy and context awareness.It follows that DNN architectures are described with depth that defineshow often a network can process input in the sequence of transformations[228]. A wide variety of DNN models have a common structure in everyone of their layers: a combination of weighted inputs and nonlinear acti-vation functions produces the output. Choosing an appropriate activationfunction - the rectified linear units function (ReLU), sigmoid, or tanh - isimportant in guaranteeing the network can model non-linear relationshipsbetween features and the output value [230]. ReLU is a standard in deeplearning due to its computational efficiency and performance in avoiding thevanishing gradient problem, which had been a problem when trying to traindeep networks [231]. An example of a DNN is shown in Figure 2.1 below.Training DNNs minimizes the difference between the model’s predictedand actual output by optimizing millions or billions of parameters (weightsand biases). This optimization relies heavily on algorithms like stochasticgradient descent and its variants, such as Adam, which iteratively adjustparameters based on the gradient of the loss function [232]. Regularizationtechniques like dropout and batch normalization ensure that the networkgeneralizes well to unseen data by avoiding overfitting and stabilizing thetraining process [232].Advances in computational resources and data availability have fueledthe rise of DNNs. The pervasive use of graphics processing units (GPUs)and specialized hardware such as tensor processing units (TPUs) has drasti-cally reduced the training time for deep models [233]. Moreover, large-scaledatasets, such as ImageNet, have provided the volume and diversity of datarequired to harness the potential of DNNs [234]. These developments havemade deep learning accessible and scalable for real-world applications.Deep neural networks have revolutionized image segmentation, assign-ing a label to each pixel in an image. The following sections discuss somecommonly used DNNs for medical image segmentation. The networks arebroadly grouped into the following categories:• Fully connected architectures• Networks with latent representations342.3. Model Architectures in Image Segmentation• Networks that leverage saliency mapsFigure 2.1: An example of a Deep Neural Network (DNN) architecture,specifically a Convolutional Neural Network (CNN), illustrating its opera-tions for extracting spatial features from input images. (1) The input imageis processed using a series of filters (or kernels), which slide over the imagein a systematic manner to identify specific patterns. Initially, these filtersare randomly initialized but evolve during training to detect meaningful fea-tures, such as edges or textures. (2) The output of the filtering process isa set of activation maps, that highlight detected features (3) Nonlinear ac-tivation functions, such as ReLU or sigmoid, are applied to the activationmaps pixel by pixel. This step introduces the ability to learn complex, non-linear relationships in the data, allowing the network to extract increasinglyabstract features at deeper levels [14]2.3.1 Fully Connected ArchitecturesFully connected architectures are those which have densely connected layers;each neuron is connected to every neuron in the next layer [235]. Althoughthese architectures are less used directly for segmentation tasks becausethey have no spatial awareness, they play a supportive role in hybrid mod-els. Fully connected networks aggregate global information, making themvaluable in scenarios where overall context is as important as localized fea-tures [236]. For instance, in medical segmentation tasks like the analysisof brain lesions, fully connected layers can be used at the output stage toclassify lesion presence or severity based on features extracted from earlier352.3. Model Architectures in Image Segmentationlayers [237].These layers are usually combined in the heads of classification for mul-titask network architectures, where one model is likely to perform bothsegmentation and diagnosis. An example is shown in Figure 2.2 where thefully connected layers are used at the output stage of a convolutional neuralnetwork. However, these kinds of architectures cannot preserve the spa-tial structure of input data and are unsuitable as standalone solutions forpixel-wise segmentation [238]. Thus, their application in medical image seg-mentation is usually limited to augmenting more spatially aware networks.Figure 2.2: Illustration of fully connected (FC) layers commonly used at theoutput stage of convolutional neural networks (CNNs). Activation mapsfrom the preceding layers are flattened into a one-dimensional vector andfed into the fully connected layer. Each element of the flattened vector isconnected to every node in the FC layer, with each connection assigned alearnable weight. The output of each node is computed as the weightedsum of all inputs, combined with a learnable bias, and passed through anactivation function. This structure enables the FC layer to aggregate globalinformation, making it useful for classification tasks or as a final stage inhybrid models [14].362.3. Model Architectures in Image Segmentation2.3.2 Latent Representation NetworksLatent representation networks have become a fundamental approach inmedical image segmentation, allowing for effective encoding of input imagesinto compact, high-level feature spaces. These networks extract essentialfeatures while discarding irrelevant details, thus offering accurate and effi-cient segmentation [238]. At the core of this category are encoder-decoderarchitectures, which balance global contextual understanding with precisespatial localization [239]. Of these, U-Net, ResNet, and their numerousadaptations have set the paradigm for exploiting latent representations inmedical imaging. The strengths of latent representation networks lie in theirversatility, computational efficiency, and the feature hierarchies they learn.Latent representation networks provide excellent performance in a vari-ety of medical imaging tasks. For instance, in brain imaging, variations ofU-Net have found widespread application in segmenting gliomas, accuratelycapturing tumor boundaries, necrotic cores, and surrounding edema [213].In hepatic imaging, these models help in the segmentation of primary livertumors and metastases [240], thus carrying out the quantification of tumorvolumes with accuracy [241], which becomes very important for treatmentplanning. In PET/CT imaging, the latent representation networks form thebackbone in detecting and segmenting cancerous lesions that show highlyvariable tracer uptakes. By leveraging anatomical and functional data frommultimodal inputs of CT or MRI with PET scans, they have been reportedto yield robust segmentations [242].U-Net and Its VariantsThe U-Net architecture of [212] has proven itself to be a keystone in theapplications of medical image segmentation. It adopts the architecture ofan encoder-decoder structure: while the encoding path decreases spatial res-olution progressively along with the extraction of semantic and contextualfeatures via convolutional and max-pooling layers, the decoding path returnsthe resolution using upsampling operations, while enhancing the resolutionof captured features in order to derive accurate segmentation output [212].A distinctive feature of U-Net is its skip connections, bridging correspond-ing layers of the encoder and decoder, merging high-resolution features withupsampled outputs [212, 243]. This integrates fine-grained spatial detailswhile maintaining robust feature reconstruction [243], making U-Net par-ticularly effective in tasks requiring accurate boundary delineation, such asliver segmentation. Figure 2.3 below shows the architecture of the U-Net372.3. Model Architectures in Image Segmentationhighlighting the encoder-decoder structure as proposed by [212].Figure 2.3: The U-Net architecture is a network structured as an encoder-decoder. The encoder (left side) performs downsampling through max pool-ing, capturing spatial features at progressively lower resolutions. The de-coder (right side) performs upsampling to reconstruct fine details. Thestructure inherently represents latency, as information passes through vary-ing depths and processing stages, balancing feature extraction and resolutionrecovery [14].Despite its success, the base U-Net architecture has limitations in seg-menting highly complex or low-contrast regions. For this reason, severalvariants have been developed [244]. The 3D U-Net for volumetric data suchas 3D PET or CT scans uses 3D convolutions to capture interslice dependen-cies, which allows for accurate segmentation of lesions with irregular shapesspanning multiple slices [243]. Other variants, like Residual U-Net, employresidual connections to help with gradient flow, allowing for the learningof much deeper, hierarchical representations [244, 245]. These skip connec-tions resolve issues such as vanishing gradients and enable the network tosegment subtle or diffuse abnormalities [246]. Similarly, Dense U-Net usesdense connectivity within layers, encouraging feature reuse and capturingcomplex spatial relationships that are crucial for tasks such as whole-bodylesion segmentation [247].Flexibility of U-Net allows for hybrid designs. For example, transformer-augmented U-Nets combine the latent representation framework with the382.3. Model Architectures in Image Segmentationattention mechanisms of transformers, thus capturing both local and globalcontextual information [248]. These hybrids are especially effective in sce-narios where long-range dependencies are crucial, such as segmenting scat-tered or poorly defined lesions in oncology. In PET/CT imaging, U-Netadaptations integrate anatomical information from CT with functional in-formation from PET [249], leading to robust segmentation results and sup-porting downstream tasks such as radiomics-based analysis.ResNetAnother prominent architecture of latent representation networks is theResidual Neural Network, ResNet, proposed by [246]. Its main noveltyrelies on using residual blocks with skip connections across a few layers,allowing earlier inputs to directly connect further down in outputs [246]. Inthis design, vanishing gradient problems are minimized to ensure very deepnetworks effectively learn. ResNet’s residual units facilitate the extractionof high-resolution features, making it particularly suited for tasks requir-ing multi-scale feature integration, such as lesion segmentation in PET/CTimaging [250].The extensions of ResNet, like 3D Residual U-Net, combine residualblock concepts with volumetric data processing, allowing the segmenta-tion of structures across slices with interslice continuity [251]. Attention-augmented ResNet models have gone a step further in enhancing their capa-bilities by emphasizing regions of interest while filtering out irrelevant noise[252], which is particularly useful in complex modalities such as PET/CT,where functional and anatomical data have to be harmonized.SegResNet is a ResNet variant specially adapted for medical image seg-mentation. Combining residual learning blocks into an encoder-decoderframework, SegResNet ensures high performance in the precise tasks oftumor delineation or bone segmentation [253]. This network incorporatesmultiscale feature extraction and spatial reconstruction into a frameworksupported by residual blocks for local and global contextual informationbalance [253]. Recent adaptations, such as context-aware SegResNet, en-hance performance by introducing attention mechanisms that model spatialdependencies [254], further improving segmentation accuracy in challengingdatasets, such as spinal imaging or oncological PET/CT scans.392.3. Model Architectures in Image Segmentation2.3.3 Saliency Map NetworksNetworks using saliency maps leverage mechanisms like attention and self-attention to dynamically focus on the most relevant regions of an image,improving segmentation accuracy and interpretability [255]. These archi-tectures emphasize the importance of identifying and weighting salient fea-tures in input data, making them particularly effective for tasks with small,subtle, or scattered regions of interest. This integration of attention mech-anisms has enabled segmentation networks to capture not only local detailbut also global context with much greater accuracy [15].Transformer NetworksTransformer networks, originally proposed for natural language processingby [256], introduced a revolutionary change in the field of deep learning:the self-attention mechanism. Unlike CNNs using local receptive fields andtranslation equivariance, transformers model global dependencies across theentire input. This capability makes them a very good fit for medical imagesegmentation, where capturing not only fine details but broader anatomicalcontext is vital.Vision Transformers (ViTs) adapt the transformer architecture to imagedata by treating image patches as tokens [15] (Figure 2.4). Each image issegmented into patches of fixed size, flattened, and then linearly embeddedbefore being fed into the transformer encoder [15]. This architecture excelsat learning global inter-patch relationships, hence finding applications in abroad range of tasks in radiology and other imaging modalities. However,their scalability and excellent performance at state-of-the-art comes at thehigh price of requiring large datasets with annotated data for their pre-training, which for medical imaging is often sparse [257].Building upon the model from ViTs, the Swin Transformers introducehierarchy through shifted windows [258]. By dividing the input into non-overlapping local windows for attention computation and reducing computa-tional complexity, this architecture effectively captures both local and globaldependencies. The mechanism of shifting the window ensures better model-ing of inter-window relations and enhances the segmentation accuracy of vol-umetric data [258]. For example, Swin Transformers have outperformed tra-ditional CNN-based models in tasks such as segmentation of the brain [259]and for delineation of the clinical target volume and lymph node drainagearea for postoperative prostate cancer radiotherapy [260].402.3. Model Architectures in Image SegmentationFigure 2.4: Overview of a Vision Transformer (ViT) model, which serves asan example of transformer-based networks applied to image analysis. Theinput image is divided into fixed-size patches, which are linearly embeddedinto vectors. Positional embeddings are added to retain spatial information,and the sequence of embedded patches is fed into a Transformer encoder.A learnable classification token is appended to the sequence to facilitatetasks such as image classification. This architecture, by processing imagesas sequences, enables the use of attention mechanisms, which can producesaliency maps highlighting the most relevant regions of an image for a giventask [15].Hybrid Transformer ModelsHybrid transformer models merge the strengths of CNNs in local featureextraction with the capability of transformers in global contextual modeling.Architectures such as TransUNet embody this hybrid approach by using aCNN encoder for preliminary feature extraction, followed by transformerblocks for global refinement [223]. This is highly effective in multi-organsegmentation, where both the local anatomical details and global spatialrelationships are of importance.Attention-Enhanced Residual NetworksAttention-enhanced networks extend the capability of traditional residual ar-chitectures, such as ResNet, by incorporating attention mechanisms. Resid-ual Attention Networks (RANs) merge residual blocks with attention gates412.4. Loss Functions in Image Segmentationthat allow the model to selectively focus on relevant features while sup-pressing background noise [261]. These architectures are particularly usefulin segmenting small or complex lesions, such as scattered metastases or low-contrast structures.Attention U-Net proposes the use of attention gates in the expansivepath, refining segmentation by weighing feature maps in terms of their rel-evance to the task at hand [262]. This type of selective processing is evenmore helpful when anatomical structures overlap each other, or pathologi-cal regions are subtle; this boosts the accuracy of segmentations made fromchallenging datasets.Advanced Saliency Based Map NetworksSwin UNETR unites the advantages of a hierarchical attention mechanisminherent in Swin Transformers and spatial reconstruction capabilities of U-Net [259]. Swin UNETR was designed for 3D medical image segmentation.The Swin Transformer encoder captures features hierarchically, capturingboth global and local contexts, while the CNN-based decoder reconstructshigh-resolution segmentation maps. The shifted window attention mecha-nism ensures computational efficiency while maintaining long-range depen-dency modeling [259].In the same vein, UNETR replaces the usual CNN-based encoder in U-Net with a Vision Transformer [263]. This architecture uses self-attentionmechanisms to model long-range dependencies directly from 3D volumetricdata, which makes it effective for fine-grained segmentation tasks. UNETRhas shown outstanding performance multi-organ delineation by effectivelycombining global context with detailed spatial localization through skip con-nections [264].2.4 Loss Functions in Image SegmentationThe loss function is a mathematical function that assigns a real numberto some variable values or outcomes; this number is the ”cost” or ”error”of the outcome, which the optimization process tries to minimize [265]. Inmost applications involving the process of mathematical optimization, theminimization of the loss function is the ultimate objective [266], since itreflects the performance of the model or system being optimized [267]. Onthe other hand, depending on the problem at hand, the aim may be tomaximize a related function [266], commonly called an objective function.In machine learning, loss functions are essential since they measure how far422.4. Loss Functions in Image Segmentationapart the predicted values are from the actual values; thus, models learnthrough iteration to optimize this function.The definition of a loss function is a critical step in supervised learning.Loss functions provide a scalar value that represents the model’s predictiveaccuracy or error and serves to guide optimization. Their design is relatedto the specific problem at hand, such as classification, regression, or imagesegmentation, and to the nature of the data and the structure of the learn-ing algorithm [268]. Minimizing a loss function not only improves modelperformance but also helps in achieving the desired task-specific objectives.Therefore, the development and choice of effective loss functions play animportant role in AI-based tasks.Generally, to have an optimized machine learning model, one minimizesa defined loss function within a structured framework which can be mathe-matically formalized as structural risk minimization [269], written as:minf1NN∑i=1Lθ(f(xi)) + λR(f), (2.1)where N represents the number of training examples, Lθ(·) is the loss func-tion parameterized by θ, f(xi) denotes the model’s prediction for input xi,R(f) is a regularization term that discourages overly complex models, andλ is a positive scalar that balances the empirical risk and the regularizationterm [270].In this equation, the first term is called empirical risk. It represents theaverage loss across the training data and hence reflects how well the modelfits the observed data [270]. Then, the second term serves to penalize over-fitting by adjusting the model complexity [270]. Choosing an appropriateloss function is central to any optimization process since, apart from direct-ing the model toward meaningful patterns and convergence to optimality, apoorly chosen loss may lead to suboptimal performance.Given their fundamental role, loss functions are an indispensable part inthe training of deep neural networks, especially for complex tasks such asmedical image segmentation. The choice of loss function directly impactsthe model’s ability to accurately delineate structures of interest, particularlyin challenging scenarios like imbalanced datasets, fine boundary delineation,or multi-class segmentation.Loss functions can be broadly categorized into region-based, distribution-based, boundary-based, and compound losses [271], each with specific formu-lations and applications tailored to the unique challenges of segmentationtasks. This section introduces the mathematical foundations of these loss432.4. Loss Functions in Image Segmentationfunctions, followed by a discussion of their performance across various seg-mentation scenarios.2.4.1 Notation and DefinitionsIn the equations describing the loss functions, the following symbols areused:• yi: The ground truth label for pixel i.• pi: The predicted probability for pixel i.• N : The total number of pixels in the segmentation map.• ϵ: A small positive constant used to avoid division by zero.• C: The number of classes in a multi-class segmentation task.• wc: The weight assigned to class c in Generalized Dice Loss, definedas wc =1(∑Ni=1 yci )2to balance contributions across classes.• α, β: Hyperparameters in Tversky Loss that control the relative em-phasis on false positives and false negatives.• αyi : Hyperparameter in Focal loss that balances the contribution ofeach class in imbalance scenarios.• γ: A focusing parameter in Focal Loss that adjusts the weight givento hard-to-classify examples.• ∂Y and ∂P : The boundaries of the ground truth and predicted seg-mentations, respectively, in boundary-based losses.• x: A spatial coordinate on the predicted boundary ∂P .• dpi : Distance transform map for the predicted segmentation. It com-putes the shortest distance of pixel i from the boundary of the pre-dicted region.• dyi : Distance transform map for the ground truth segmentation. Itcomputes the shortest distance of pixel i from the boundary of theground truth region.• λ: A weighting factor used in compound losses to balance the contri-butions of different loss components.442.4. Loss Functions in Image SegmentationThese symbols will be used consistently throughout the formulations ofthe loss functions to describe their mathematical properties and applications.2.4.2 Region-Based LossesRegion-based losses directly measure the overlap between ground-truth andpredicted segmentations [271], hence especially effective when handling classimbalance by giving importance to the accuracy between regions.Dice Loss [272] is one of the most used region-based losses, derived fromthe Dice Similarity Coefficient [273]. It is particularly suitable for tasks suchas tumor segmentation in MRI and PET/CT imaging when the foregroundregions are small compared to the background. However, the sole relianceon measures of overlap may sometimes result in less optimal smoothing atthe boundaries [274], reducing accuracy for fine structures. Mathematically,LDice = 1− 2∑Ni=1 yipi + ϵ∑Ni=1 yi +∑Ni=1 pi + ϵ(2.2)There other variant of the Dice Loss includes a denominator with squaredterms:LDice = 1− 2∑Ni=1 yipi + ϵ∑Ni=1 y2i +∑Ni=1 p2i + ϵ(2.3)Generalized Dice Loss (GDL) [275] extends Dice Loss by introducingclass-specific weights, making it effective in multi-class segmentation tasks.Mathematically,LGDL = 1− 2∑Cc=1wc∑Ni=1 yci pci∑Cc=1wc(∑Ni=1 yci +∑Ni=1 pci) (2.4)Other region-based losses, including the Tversky Loss [276] and the Jac-card Loss [277], offer increased flexibility and specificity. Tversky Loss pro-vides tunable emphasis on false positives and false negatives [276] and isparticularly suited to applications such as liver tumor segmentation, wherea low rate of false negatives is important [278]. Jaccard Loss, or Intersec-tion over Union (IoU) Loss, has also shown particular efficacy in improvingsensitivity in retinal vessel segmentation [279].452.4. Loss Functions in Image Segmentation2.4.3 Distribution-Based LossesDistribution-based losses focus on minimizing the dissimilarity between pre-dicted and true label distributions [271]. The most common example isCross Entropy (CE) Loss [271], which is effective for balanced datasets butstruggles in scenarios with class imbalance [280]. Mathematically,LCE = − 1NN∑i=1(yi log(pi) + (1− yi) log(1− pi)) (2.5)Focal Loss [281] builds on CE by emphasizing hard-to-classify examples,thus improving sensitivity in detecting small lesions or rare pathologies, suchas lung nodules in CT scans [282] or breast lesions in mammograms [283].Mathematically,LFocal = −∑iαyi(1− pi(yi))γ log(pi(yi)) (2.6)2.4.4 Boundary-Based LossesBoundary-based losses prioritize the accurate delineation of object bound-aries, making them essential for segmenting fine structures or irregularshapes [284].Boundary Loss [284] minimizes the distance between predicted and trueboundaries [271], enhancing edge accuracy. It has proven to be effectivein capturing fine details at edges of anatomical structures of tumors andreducing over-smoothing [285]. Mathematically,LBoundary =∫∂Yminx∈∂P∥x− y∥2dx (2.7)Hausdorff Distance Loss [286] focuses on the largest boundary distancebetween the segmentation map and the ground truth [287]. Its applicationhas yielded high accuracy in tasks such as brain tumor segmentation in MRIscans [288]. Mathematically,LHD =1NC∑c=1N∑i=1[(pci − yci ) ◦ (d2yi + d2yi)](2.8)462.5. Tumor Segmentation in PET/CT Images2.4.5 Compound LossesCompound losses combine complementary loss functions to leverage theirstrengths. For example, Dice Cross-Entropy (DCE) Loss merges Dice Loss’soverlap optimization with CE Loss’s distribution focus [289], offering bal-anced performance in multi-organ segmentation tasks. Mathematically,LDCE = λLDice + (1− λ)LCE (2.9)Similarly, Dice Focal Loss (DFL) combines the Dice Loss with the FocalLoss [290] addressing both class imbalance and boundary accuracy, provingeffective in PET/CT lesion segmentation where sensitivity and specificityare critical. Mathematically,LDFL = λLDice + (1− λ)Lfocal (2.10)2.4.6 Performance Summary of Loss FunctionsBased on the segmentation task, the different loss functions have variedperformances. In general, Region-based losses handle class imbalance anddetection of small structures [291], like tumors or lesions, effectively, whiledistribution-based losses are effective in balanced datasets or hard-to-classifyregions [281]. Boundary-based losses give superior boundary accuracy withfine or irregular structures [284, 285], and compound losses offer a versa-tile approach as they address diverse segmentation challenges by combiningmultiple strengths [287, 289]. The varying strengths of these loss functionsemphasize the importance of selecting task-specific loss functions in segmen-tation workflows.2.5 Tumor Segmentation in PET/CT ImagesAI approaches have created significant interest in enhancing lesion segmenta-tion in PET/CT imaging, especially for the detection of metastatic prostatecancer. Their translation into clinical routine requires robustly validatedand approved systems. For instance, aPROMISE [292] is an FDA-clearedand CE-marked software since 2021 that supports standardized analysis andreporting of PSMA PET/CT. It uses a U-Net architecture for the segmen-tation of organs in low-dose CT scans and uses a classical blob detectionwithin defined organ masks for the identification of hotspots [293]. Highsensitivity, ranging from 87% for bone lesions to 92% for lymph nodes, isreported in retrospective studies [293]. These results are, however, partly472.5. Tumor Segmentation in PET/CT Imagesaffected by the high rates of false-positives, averaging eight per patient forbone lesions and as many as 90 in lymph node lesions [293], thus underliningthe continuing struggle to balance sensitivity of detection with specificity.Notably, aPROMISE-assisted readings have shown great inter-reader consis-tency, as reflected in Cohen kappa values of 0.77 to 0.90 across the differentstages [292].Other models also aimed to improve PET/CT lesion segmentation. [294]proposed a triple-combining 2.5D U-Net pipeline, trained on [68Ga]Ga-PSMA-11 PET/CT scans from 193 patients across three centers. This model per-formed well for bone lesions (99% sensitivity) and lymph nodes (90%), butunfortunately underperformed for prostatic lesions with a sensitivity of only61% [294]. These results underline the variability of model performanceacross lesion type and anatomical location because of the inherent hetero-geneity of the appearance and metabolic activity of lesions.[295] increased the scope to segment primary tumors, metastases, andwhole-body tumor burden by using 660 [18F]PSMA-1007 PET/CT. Themodel had sensitivities of 79% for primary tumors and lymph nodes and62% for bone metastases, slightly better than the clinicians’ sensitivity of59% [295]. These results again highlight how the delineation of lesions re-mains challenging in cases where their contours are indistinct or the traceruptake is poor, as often happens with PET/CT imaging. [242] investigatedthe use of off-the-shelf segmentation frameworks, such as nnU-Net [296], indetecting PET/CT lesions. The results, assessed with the Dice SimilarityCoefficient, were suboptimal although it yielded better segmentations thanthe manual contouring method (mean DSC of 0.49 versus 0.33 obtained byhuman observers). Such a relatively low DSC further highlights the difficultyin automating the detection and segmentation of prostate cancer lesions, es-pecially when there is a high variability of lesions or complex anatomicalbackgrounds. Another approach for lesion segmentation involves hybridmethods, such as qPSMA, developed by [297]. This software combines ma-chine learning and SUV thresholding to create bone and organ uptake masksthat enhance detection accuracy. However, reliance on liver uptake thresh-olds presents a limitation, with the sensitivity of lesions compromised incases of extensive hepatic metastases [298].A consistent theme throughout these studies is the challenge in achiev-ing a balance between sensitivity and precision. While models such as thoseby [294] and [295] are good at the detection performance for certain lesiontypes, this often comes at the cost of high false-positive rates or reduced per-formance in other regions. This highlights the need for improved techniques,including advanced loss functions and novel training strategies, to address482.6. Conclusion and Transition to Next Chapternuances in PET/CT imaging, such as lesion heterogeneity, varying tracers,and noisy backgrounds. Despite significant advancements in AI-based lesionsegmentation methods, the low DSC values and high percentages of false pos-itives—largely resulting from non-diverse, proprietary datasets—highlightlimitations in robustness and generalizability. These findings encourage thedevelopment of more robust algorithms, training strategies, and loss func-tions that can handle these particular weaknesses. The issues will be furtherexplored in Chapter 3, which will introduce a new loss function developedto improve precision-recall balance and enhance segmentation accuracy inPET/CT imaging.2.6 Conclusion and Transition to Next ChapterArtificial intelligence has made significant advancements in medical imaging,contributing to areas such as image enhancement, computer-aided diagnosis,and predictive modeling. These developments are closely tied to the evo-lution of deep neural networks and the design of loss functions that guidethese networks toward achieving optimal results. The choice of loss func-tion is typically determined by the specific challenges of the task at hand.In tumor segmentation, region-based loss functions, such as Dice Loss, arecommonly used, however, these functions can struggle with the varying com-plexities of classifying regions of interest (ROIs), leading to issues like missedlesions or increased false positives. To address these limitations, other lossesare constantly being explored such as the combination of region-based lossfunctions with distribution-based ones to improve segmentation accuracy.In the next chapter, we explore the comparative performance of variousloss functions in detecting and segmenting foreground voxels, with a focuson minimizing false positive segmentations. The analysis considers diverselesion scenarios, categorized by the number of lesions in PET/CT volumetricimages, as well as variations in lesion size and distribution throughout thebody.49Chapter 3Optimizing LesionSegmentation withL1-Weighted Dice Focal Loss3.1 IntroductionSemantic segmentation involves mapping every pixel of an image to a class,whether an object or a background [299]. Semantic segmentation operatesat a pixel level, unlike standard image classification, where a single label ispredicted for an entire image [300]. It requires both a local and contextualunderstanding of the image [301], making it particularly difficult since neigh-boring pixels are highly related; thus, information on the spatial context andthe boundary’s smoothness is essential in a segmentation model [302]. Thistype of segmentation finds many applications ranging from natural scene un-derstanding to medical image analysis [303]. Particularly in medical imageanalysis, higher accuracy levels are required for lesion or tumor segmen-tation since false positives may lead to unwarranted medical interventions(such as unnecessary surgical procedures or excess radiation doses), and falsenegatives may lead to incomplete surgical margins [304]. However, medicalimages are fraught with an imbalance in the proportion of voxels belongingto the regions of interest (ROIs) compared to the background making a bal-ance in precision and sensitivity in automated segmentation a challengingtask.Whole-body positron emission tomography (PET) scans of cancer pa-tients are an example of such medical images where voxels corresponding tothe foreground class are overwhelmingly outnumbered by background voxels.These images are characterized by high variability in lesion size, shape, andintensity, as well as significant heterogeneity in tracer uptake [305]. Specifi-cally, in the case of metastatic prostate cancer images, lesions can range fromsmall, low-contrast regions barely distinguishable from surrounding tissuesto large, high-uptake areas with well-defined boundaries, and may occupy503.1. Introductiondifferent anatomical sites [306]. Additionally, background noise, physiolog-ical uptake in non-cancerous tissues and artifacts from adjacent structuresfurther complicate segmentation [307]. But these complexities arise from thevery definition of how lesions are localized in PET scans. PET imaging mea-sures changes in metabolic activities in tissues and organs by leveraging thedistribution and absorption of radioactive substances known as radiotracers[129].These radiotracers consist of radioisotopes that are chemically boundto pharmacologically relevant molecules like choline, acetate, or prostate-specific membrane antigen (PSMA)-targeted ligands [308–310]. For exam-ple, PSMA-targeted radiotracers such as 68Ga-PSMA or 18F-PSMA are de-signed to bind to PSMA, a cell surface protein that is overexpressed inprostate cancer cells [309]. However, PSMA expression is not exclusive toprostate cancer lesions; it is also found in other tissues, including salivaryglands, kidneys, and the small intestine, as well as in certain non-prostaticmalignancies and inflammatory processes [311]. This broad expression canlead to radiotracer uptake in non-cancerous or non-prostatic regions, com-plicating the interpretation of PET scans and challenging the specificity ofcomputer aided detection (CAD) algorithms. Thus, the challenge in artifi-cial intelligence (AI) based lesion detection and segmentation in PET scansof metastatic prostate cancer lesions is avoiding false-positive segmentationof anatomical structures and other artifacts while maintaining accurate le-sion detection.With most of the automated approaches, improved sensitivity in the de-tection of lesions often comes at the cost of specificity, yet both extremespose potential risks. One commonly used strategy for limiting false posi-tive rate is standardised uptake value (SUV) thresholding, which involvesdefining all voxels with SUV greater than some value, typically consideringthe background uptake of the liver, to be foreground and all other voxels asbackground [174]. This method may be included as part of the automatedsegmentation process or as a post processing step, and has been shown toreduce false positive counts [312]. Yet, its major drawback is the potentialof having a reduced sensitivity for lesions that exhibit lower avidity for theradiotracer, or in patients with extensive liver metastases [298].Loss function design is another strategy that has received much attentionand has led to an improvement in the performance of AI based segmenta-tion methods. The widely used loss function is the Dice loss. It is based onthe Dice Similarity Coefficient (DSC) and it penalizes pixel-wise discrepan-cies between the predicted maps and their corresponding ground truth foreach class [291]. A model trained using the Dice loss objective maximizes513.1. IntroductionDSC inherently incorporating a balance in sensitivity and specificity [272].Nevertheless, the Dice loss is not well-suited for handling another type ofimbalance caused by a difference in the difficulty in classifying pixels [313].Within a class, be it the minority foreground class or majority backgroundclass, some pixels might be easier to classify, with models achieving greaterthan 50% probability of correctly classifying such pixels [313]. However,others may be more challenging to predict, yielding lower probabilities [313]and are therefore designated as hard-to-classify. Without a focusing strat-egy implemented, the Dice loss can be easily overwhelmed by the easy-to-classify samples and may perform poorly on the harder ones [314]. Besides,it exhibits an inconsistent performance when segmenting multiple lesions ofvarying sizes in an image [315]. These limitations have prompted the devel-opment of alternative loss functions. Distribution-based loss functions likeFocal and Cross-Entropy losses are frequently used alternatives due to theirdynamic focusing ability. Yet, functions based on overlap measurementshave been identified to be more robust for medical segmentation tasks [275],thus, these distribution-based loss functions are often combined with theDice loss and its variants to form hybrid or compound losses [291]. With anadditional ability of controlling the model’s focus on detecting ROIs, thesecombo losses generalize well [291, 316]. The challenge, however, remainson the balance of sensitivity and specificity as these compound losses tendto lose the Dice loss component, potentially leading to more false positivesegmentation [317]. Thus, various regularizations and additional constraintsare required to improve model accuracy for a medical lesion segmentationtask [318].This work proposes a new loss function, L1-weighted Dice Focal loss(L1DFL), with a dynamic weighting strategy based on L1 norms of pre-dicted probabilities and ground truth labels. Using L1 norms, the loss func-tion considers the difficulty of each sample and the prevalence of similarsamples in the dataset. In medical image segmentation tasks, datasets areoften imbalanced (disproportionately higher number of background voxelsthan ROIs) and dominated by easy samples (voxels that are easily classifiedcorrectly); the L1 norms of these easy samples tend to approach zero. Ourtechnique focuses on down-weighting both easy samples and those with highL1 norm counts, which are likely to represent the background. We demon-strate the efficiency of L1DFL by experiments on [18F]DCFPyL PET/CTscans of patients with metastatic prostate cancer, for which lesions havevaried sizes, primarily small, with low uptake values.The main contributions of this work are as follows: (i) we introduce anovel loss function, L1-weighted Dice Focal loss (L1DFL), incorporating a523.2. Materials and Methodsdynamic weighting strategy; (ii) we evaluate the performance of L1DFL,Dice loss, and Dice Focal loss functions in handling false positives and falsenegative rates; (iii) we compare the loss function segmentation performancebased on different lesion scenarios, single-lesion, and multiple lesions. Thesescenarios are based on the number of lesions in an image; (iv) we evaluate theperformance of these loss functions on different clinical metrics like moleculartumor volumes and the extent of lesion distribution in the body; and (v) weassess performance using two network architectures, Attention U-Net [262]and SegResNet [253].3.2 Materials and Methods3.2.1 DatasetWe analyzed 380 prostate-specific membrane antigen (PSMA) [18F]DCFPyLPET/CT scans of patients diagnosed with biochemical recurrence metastaticprostate cancer as part of an ongoing clinical trial (NCT02899312) with in-formed written consent obtained from the subjects. The image data col-lection was approved by the University of British Columbia – BC CancerEthics Board. The inclusion criteria for the patients were: (1) histologi-cally proven prostate cancer with biochemical recurrence after initial cura-tive therapy with radical prostatectomy, with a PSA > 0.4 ng/mL and anadditional PSA measurement confirming an increase; and (2) histologicallyproven prostate cancer with biochemical recurrence after initial curative ra-diotherapy, with a PSA > 2 ng/mL after therapy, with five or fewer lesionsidentified on [18F]DCFPyL PET/CT [319].Each patient received an average dose of 350 MBq [18F]DCFPyL af-ter fasting for four hours, with dose adjustments based on body weight.Imaging was conducted 120 minutes after injection using a GE Discovery600 or 690 scanner (GE Healthcare, USA). A non-contrast-enhanced CTscan was acquired for localization and attenuation correction (120 kV, auto-matic mA adjustment from 30–200 mA, and a noise index of 20). FollowingCT acquisition, a whole-body PET scan was conducted at 2–4 minutes perbed position, depending on the participant’s size, with reconstruction per-formed using an ordered subset expectation maximization algorithm andpoint-spread function modeling. Each transaxial PET image had a matrixresolution of 192 × 192 pixels, with a physical pixel size of 3.64mm2. Anexpert nuclear medicine physician performed manual segmentation on allactive lesions [319].There were a total of 684 prostate cancer (PCa) lesions across the entire533.2. Materials and Methodsdataset with a mean active lesion volume of 6.68 ± 10.20 ml. The lesionsvaried in uptake with the average maximum standardized uptake value (SU-Vmax) and mean standardized uptake value (SUVmean) being 12.65±14.46and 4.62 ± 3.88, respectively. The CT image had an average long-axis di-ameter of 12.96 ± 10.11 mm.3.2.2 Ground Truth AnnotationThe ground truth segmentations were performed by a board-certified nu-clear medicine physician from the BC Cancer Research Institute. All lesionswere annotated using the PET Edge tool, a semi-automated gradient-basedsegmentation tool, and contours refined using the 3D Brush tool. Both toolsare available in the MIM workstation (MIM software, Ohio, USA). Thesesegmentations served as the ground truth labels for training the models.3.2.3 Data Preprocessing and AugmentationAfter acquiring the images, PET activity values were converted to decay-corrected standardized uptake values (SUV), and the CT intensity values (inHaunsfield units) were clipped to the range [−1000, 3000] before normalizingto a uniform range of [0, 1]. The PET images however were left in their abso-lute SUV values. We resampled all images, including PET, CT and groundtruth (GT) masks to an isotropic voxel spacing of [2 mm, 2 mm, 2 mm] us-ing bilinear interpolation for the CT and PET images and nearest-neighborinterpolation for the GT masks. For augmentation, we applied random crop-ping and affine transformations for the training images, including transla-tions in (-10,10) voxels in every spatial dimension, rotations of up to π15 , andscaling by a factor of up to 1.1. We obtained cubic patches of dimensions128 × 128 × 128 voxels with an 80% probability of being centered arounda foreground voxel. The augmented CT and PET patches were channel-concatenated and fed as input to the networks.3.2.4 ArchitecturesIn this work, we implemented and studied two convolutional neural networkarchitectures: Attention U-Net [262] and SegResNet [253]. These networks,built on top of a 3D-CNN backbone, comprise encoder and decoder blocks.The SegResNet uses residual blocks with skip connections for feature propa-gation and maintaining gradient stability during training [253]. The encoderpath contained blocks with the following number of layers: 1, 2, 2, and 4, al-lowing for increasing levels of feature extraction at different resolutions. The543.2. Materials and Methodsinitial filter size was set to 16 channels, then doubled through each downsam-pling step to capture progressively higher-level features. Each convolutionallayer employs a ReLU activation function, and Group Normalization [320].The decoder path, containing one layer in each block, gradually reconstructsthe segmented output while leveraging skip connections to preserve spatialdetails from the encoder. The input and output channels were set to 2,accommodating multi-channel input images and dual-class segmented out-puts.On the other hand, Attention U-Net extends the U-Net architecture toinclude attention gates along the encoder-decoder path, allowing the networkto selectively pay attention to essential features [262]. The encoder pathconsisted of five layers, with each layer increasing in channel depth from16, 32, 64, 128, to 256 channels, and each layer downsampled by a strideof 2. These attention gates guide the network in emphasizing regions thatcontribute significantly to the segmentation task. Each convolutional layeremploys a ReLU activation function, and Batch Normalization as in theoriginal implementation [262]. The output layer matched the input withtwo channels for multi-modal inputs and dual-class output, representingbackground and foreground classes.3.2.5 Model TrainingWe divided the dataset into training, validation, and testing sets containing258, 65, and 57 samples, respectively. The training objective was to minimizethe loss functions in the training set. We used AdamW optimizer with weightdecay of 10−5 to optimize the loss functions. We adopted a cosine annealingscheduler to decay the learning rate from 2× 10−4 to zero for 1000 epochs.The loss was first computed for each batch within an epoch and the overallloss for an epoch was calculated by taking a mean over all the batch losses.The model with the highest mean DSC in the validation phase was chosenfor further evaluation of the test set.3.2.6 Loss FunctionsThe L1-weighted Dice Focal loss (L1DFL) applies a weighting strategy to theDice loss (DL) based on the L1 norm between the predicted probabilities andthe ground truth labels and combines it with the Focal loss (FL). The Diceloss generally has two variants, one with squared terms in the denominator(Eq. 2.3) [321] and one without (Eq. 2.2) [272]. In L1DFL, we employthe Dice loss with the squared denominator which was preferred over the553.2. Materials and Methodsnon-squared version following its performance in [322]. We describe themathematical formulation of the weighting strategy next.First, we compute the L1 norm, ∆, between the predicted probabilitiesp and the ground truth labels g,∆i = ∥gi − pi∥1, for i = 1, 2, ...N (3.1)where N is the total number of samples. Next, we partition the range of L1norm values ∆ into bins of a consistent nominal bin width of Γ. Each binB is defined by its center Bk:Bk = Γ · k, for k = 0, 1, 2, ...n (3.2)where n = ⌈ 1Γ + 1⌉ is the total number of bins. For instance, with a binwidth of Γ = 0.1 and a total of n = 11 bins, Bk takes values at regularintervals in the range [0,1], such that for k = 0, 1, 2, . . . , 10, bin centers (Bk)= [0, 0.1, 0.2, . . . , 1.0].We then calculate the count of ∆ values that fall within each bin:C(Bk) =N∑i=1δκ(Bk,∆i) (3.3)where δκ(Bk,∆i) is an indicator function defined as:δκ(Bk,∆i) ={1, if |Bk −∆i| ≤ Γ20, otherwise(3.4)To account for potential truncation near the boundaries of [0,1], we calculatethe effective bin width for each bin as:λκ(Bk) = min{Bk +Γ2, 1}−max{Bk − Γ2, 0}(3.5)Using λκ(Bk), the norm density, D(B), for each bin is defined as:D(Bk) = C(Bk)λκ(Bk)(3.6)The density D(Bk) represents the concentration of samples around the bincenter Bk. For bins with centers near 0, the L1 norm values (∆i) are alsoclose to 0, indicating easy samples where the predictions align closely withthe ground truth. In contrast, bins with centers near 1 represent L1 normscloser to 1, indicating hard-to-classify samples where predictions deviate563.2. Materials and Methodssignificantly from the ground the truth. Thus, based on the norm density,we can evaluate how common or rare a particular prediction difficulty is,allowing the weighting to be adjusted accordingly. This adjustment ensuresthat less frequent and difficult samples, which often correspond to foregroundsamples in an imbalanced scenario receive higher weights than the easy-to-classify samples. The weight for each bin is then calculated as:w(Bk) =ND(Bk) (3.7)These weights are applied to the numerator and the denominator of theDice Loss to account for the imbalances caused by the variations in ∆LwDice = 1− 2∑Ni=1wiyipi + ϵ∑Ni=1wi(y2i + p2i ) + ϵ(3.8)where wi = w(Bk) if ∆i belongs to Bin Bk. If the L1 norms of the examplesare uniformly distributed, the density D(Bk) will be the same for all bins,resulting in equal weights w(Bk) for all bins. In this case, the weighted DiceLoss ((LwDice) will reduce to the standard Dice Loss.Finally, we combine LwDice with the Focal loss. The full expression ofL1DFL thus is:L1DFL = LwDice + LFocal (3.9)We illustrate the dynamic weighting strategy of L1DFL below. For agiven 4x4 matrix, Figure 3.1 highlights how L1DFL downweights contri-butions from the background class and focuses on regions of classificationdifficulty. Specifically, the figure shows that higher weights are applied forvoxels with L1 norm (∆) close to 1, but for voxels with L1 norms close to 0,representing that those voxels are easily classified, lower weights are applied.Thus, regions corresponding to false positives and negatives receive muchattention from the model.In our experiments, we empirically selected a constant bin width, Γ of 0.1.We compared the performance of L1DFL with using only Dice loss (Eq. 2.2)or only Dice Focal loss (Eq. 2.10). For the Dice Focal loss formulation andthe Focal loss component of L1DFL, we set γ = 2 and α = 1 in all ourexperiments similar to the implementation reported in [323]. Also, for theDice Focal Loss we maintained an equal weighting between the Dice Lossand Focal Loss components.573.2. Materials and MethodsFigure 3.1: An illustration of the dynamic weighting strategy in L1 weightedDice Focal Loss (L1DFL). (a) shows the ground truth (G), where 1 indicateslesion areas and 0 indicates background. (b) displays the predicted probabil-ity map, where green regions denote false positive predictions and red regionsrepresent ROIs. (c) shows the L1 norm values between the predicted prob-abilities and ground truth, highlighting the error magnitude for each voxel.(d) presents the density, D, which reflects the frequency distribution of eachL1 norm value, capturing the difficulty of correctly classifying each voxel.(e) depicts the final weights assigned to each voxel, where higher weights fo-cus on rare, misclassified regions while down weighting well-classified back-ground areas. This weighting strategy reduces the influence of abundant,correctly predicted voxels and increases the focus on difficult, misclassifiedlesions, thereby enhancing the model’s focus on challenging samples.3.2.7 Model EvaluationWe assessed the performance of the loss functions on the overall test data.We made predictions on 2-channel PET and CT whole-body images usingthe sliding-window technique [324] with a window of dimension (128, 128,128) for all networks. The test set predictions were resampled to the coordi-nates of the original ground truth masks to compute the evaluation metrics.In addition to assessing performance on the overall test set, we performedevaluations based on two different lesion scenarios, single and multiple le-sion scenarios. First, we categorized the set of images (W ) into two subsets:images with a single-lesion (S) and images with multiple-lesion (M). For animage x with n(Gl) ground truth lesions, we defined these groups as follows:S = {x ∈W | n(Gl) = 1}M = {x ∈W | n(Gl) ≥ 2}Then, for each scenario, as well as on the overall test set, we performedpatient-level and lesion-level assessments based on the methodologies below.583.2. Materials and MethodsPatient-Level AnalysisSegmentation metrics: We evaluated the segmentation performance ofthe loss functions using the DSC. We report the mean DSC together withthe standard deviation as well as the median DSC with the inter-quartileranges. Let G and P represent the ground truth and predicted masks of animage at the patient level. The patient-level DSC is defined as:DSC =2|G ∩ P ||G|+ |P | (3.10)Detection metrics: We defined true positive (TP), false positive (FP),and false negative (FN) detections at the patient level, reporting their meanvalues across the test set and computed F1 scores per patient. A detec-tion is considered as TP if there is a matched pair of Gl and Pl such thatGl,SUVmax ∩ Pl ̸= 0, where Gl,SUVmax is the voxel in Gl with the maximumstandardized uptake value (SUVmax) [170]. For a prediction Pl, for whichthere is no corresponding overlap with a ground truth lesion Gl, the detec-tion is designated as an FP. Similarly, a false negative (FN) occurs whenGl,SUVmax ∩ Pl = 0. We provide a visualize illustration of the definition ofthe detection metrics in Figure 3.2. The F1 score is thus computed as:F1 =TPTP + 12(FP + FN)(3.11)Clinical metrics: For each scenario, we analyzed the performance of theloss functions across different groupings of molecular tumor volume. Specifi-cally, we evaluated the performance of the loss functions on DSC on differentthresholds of total molecular tumor volume (TMTV). We computed thresh-olds (t) based on the inter-quartile range from 0 to the 85th percentile ofthe ground truth TMTV values and calculated median DSC for all lesions(l) where volume (vl > t). Similarly, in the multiple-lesion scenario, weassessed the loss function performance based on the spatial extent of lesiondistribution, Dmax, which is measured as the maximum distance betweenany pair of foreground voxels in the image.For a given ground truth mask, let vi denote foreground voxels, wherei = 1, 2, . . . , N and N is the total number of lesion voxels. We calculated theEuclidean distance between every pair of lesion voxels, vi and vj , accountingfor voxel spacing in each dimension. For voxel coordinates (xi, yi, zi) and593.2. Materials and MethodsFigure 3.2: Illustration for defining a true positive detection based on anoverlap with the voxel containing the maximum standardized uptake value(SUVmax) in the ground truth lesion. G is the set of ground truth lesionsand P is the set of predicted lesions.(xj , yj , zj) with spacing (sx, sy, sz), the distance dij is given by equation(3.12) below:dij =√(sx(xi − xj))2 + (sy(yi − yj))2 + (sz(zi − zj))2 (3.12)The lesion dissemination, Dmax, is calculated as the maximum distanceamong all calculated distances; Dmax = maxi,j dij . We categorized thecalculated distances into inter-quartile ranges indicating the first, second,third and fourth IQR and analyzed the performance of the loss functions oneach group.Lesion-Level AnalysisSegmentation metrics: For lesion-level analysis, we defined the DSCbased on the overlap between each predicted lesion mask and its corre-sponding ground truth mask. We identified which predicted lesion voxelscorresponded to the ground truth voxels, especially in the scenario of mul-tiple lesions in an image, based on a voxel-wise matching strategy. Specif-ically, lesions were segmented as individual connected components in boththe ground truth and predicted masks using 18-connectivity to ensure spa-tial continuity within each identified lesion. This produced unique integerlabels for each lesion. Each connected component in the predicted mask603.3. Resultswas then assessed for spatial overlap with each connected component in theground truth mask.For each pair of ground truth and predicted lesions, a match was de-fined if there was any voxel overlap between the components (i.e., if anyvoxels in a predicted lesion occupied the same spatial locations as those ina ground truth lesion). This was computed by checking if any intersectingvoxels existed between the two labeled components. When an overlap wasidentified, the corresponding pair of ground truth and predicted lesion labelswas recorded as a matched lesion pair. These matches were used to computelesion-wise metrics, including lesion-level DSC, by comparing the voxel dis-tributions in each matched pair of lesions. We defined the lesion-wise DSCas:DSCl =2|Gl ∩ Pl||Gl|+ |Pl| (3.13)where Gl and Pl denote the ground truth and predicted masks of a specificlesion. For a given ground truth lesion with no match, |Gl∩Pl| = 0 yieldinga DSC of 0.Clinical metrics: At the lesion level, analysis was performed only on themolecular tumor volumes of the lesions. Similar to the threshold analysisperformed at the patient level, we calculated thresholds (t) based on theinter-quartile range from 0 to the 85th percentile of the ground truth MTVvalues (metabolic volume of individual lesions) and calculated median DSCfor all lesions (l) where volume (vl > t). For each scenario, we analyzed theperformance of the loss functions across different groupings of MTV.3.3 Results3.3.1 Segmentation performance across different networksIn this section, we present results for the segmentation performance of thethree loss functions, DL, DFL, and L1DFL using the Attention U-Net andSegResNet architectures (Table 3.1). We report the average and medianDSCs at the patient level, with mean values of true positive, false positive,and false negative counts on the test set and the F1 score performance.A one-tailed paired Wilcoxon signed-rank test was performed to evaluatethe significance of the performance of L1DFL from those of DL and DFLseparately, with corresponding p-values reported. Importantly, significant613.3. Resultsdifferences were found in the performance metrics, providing insights intothe comparative performance of the loss functions.Network Loss FunctionDSC Mean RatesF1 scores P valuesMean Median TP FP FNAttention U-NetDL 0.54 ± 0.24 0.58 [0.44, 0.72] 0.81 ± 0.35 2.06 ± 1.89 0.19 ± 0.35 0.50 ± 0.28 < 0.01*DFL 0.51 ± 0.26 0.54 [0.33, 0.73] 0.77 ± 0.36 2.65 ± 2.60 0.23 ± 0.36 0.44 ± 0.27 < 0.01*L1DFL 0.58 ± 0.27 0.66 [0.51, 0.77] 0.77 ± 0.37 0.42 ± 0.65 0.23 ± 0.37 0.69 ± 0.34SegResNetDL 0.52 ± 0.30 0.60 [0.29, 0.76] 0.73 ± 0.39 0.73 ± 1.10 0.27 ± 0.39 0.62 ± 0.35 0.24DFL 0.52 ± 0.25 0.59 [0.41, 0.71] 0.81 ± 0.34 2.05 ± 2.00 0.19 ± 0.34 0.49 ± 0.27 0.015*L1DFL 0.57 ± 0.29 0.68 [0.47, 0.78] 0.76 ± 0.37 0.52 ± 0.76 0.24 ± 0.37 0.66 ± 0.34Table 3.1: Comparison of the loss functions on the test set based on meanand median patient-level DSC. The mean DSC, TP, FP, FN and F1 scoresare presented with standard deviations and the median DSC with interquar-tile ranges. A detection is considered a true positive if the predicted maskoverlaps with the voxel containing the SUVmax value in the ground truth.The P values reflect a one-tailed paired Wilcoxon signed-rank test compar-ing the median difference in DSC between L1DFL and DL and DFL, with* indicating statistical significance at the α = 0.05 level.The median DSCs are higher than the average DSCs, with SegResNetslightly outperforming Attention U-Net in DSC. However, Attention U-Netgenerally got better true positive rates except in the case of DFL, whereSegResNet performed better with a 5% increment. In both architectures,Dice loss always performed better than Dice Focal loss in minimizing falsepositives. However, both loss functions showed varied performances on thedifferent networks regarding true positive rates. Specifically, while the Diceloss surpassed the other loss functions with a true positive rate of 0.81 ±0.35 on the Attention U-Net, Dice Focal loss also achieved a performancebut on the SegResNet architecture. However, this higher sensitivity of Diceloss on Attention U-Net came at the cost of more false positives. Comparedto a false positive rate of 0.73 ± 1.10 on SegResNet, the Dice loss had arate of 2.06 ± 1.89 on Attention U-Net, about three times the value on theformer network. This imbalance in true and false positives on AttentionU-Net led to a lower F1 score for Dice loss despite its higher sensitivity.Conversely, L1DFL achieved the best balance of true positive detectionsand minimum false positives. For example, on Attention U-Net, where Diceloss and Dice Focal loss had mean false positive rates above 2.0, the L1DFLmaintained better specificity while outperforming DFL in the true positivecounts. Although it had a slightly lower specificity on SegResNet, its falsepositives count was still low compared to DL and DFL. Its superior meanand median DSC performance reflect its balance between true and false de-tections. Specifically, on Attention U-Net, L1DFL reached a median DSC623.3. Resultsof 0.66, outperforming DFL and DL by 22% and 13%, respectively. More-over, the high interquartile range of L1DFL indicates that its DSC valuesfell within a narrow range of high values.Figure 3.3: Segmentation performance of loss functions across two networks:Attention U-Net (top row) and SegResNet (bottom row). Each columnrepresents the results for the loss functions: Dice loss (DL), Dice Focalloss (DFL), and L1-weighted Dice Focal loss (L1DFL). The green-coloredregion denotes the ground truth annotation of the lesion, while the red-colored regions represent the segmentations produced by the models. Thecircled area highlights the lesion in the ground truth image alongside thecorresponding correct segmentation by the models. Red regions outside thecircled area indicate false positive segmentations.Figure 3.3 shows a sample segmentation performance of the loss functionson a test PET/CT image. The first-row segmentations are from AttentionU-Net, while the second row is from SegResNet, with columns having differ-ent loss functions. The circled region in the figure depicts correct segmen-tation according to the ground truth. The figure highlights the differencesin segmentation performance between the loss functions and the networkarchitectures. In this example, Attention U-Net yielded better segmenta-tion of the lesion with higher Dice scores than SegResNet. Yet, for bothnetworks, the Dice and the Dice Focal loss functions produced instances633.3. Resultsof false positives further impacting the overall DSC. L1DFL, however, re-mained robust, providing a consistentperformance. Although the one-tailedWilcoxon signed-rank test comparing DL and L1DFL did not provide enoughevidence to reject the null hypothesis of no significant difference between thetwo on the SegResNet architecture, statistically significant differences wereobtained on the Attention U-Net. Also, comparing DFL versus L1DFL re-sulted in statistically significant differences in favor of L1DFL across bothnetworks.3.3.2 Performance in single and multiple lesion scenariosWe evaluated the performance of the loss functions in two different lesionscenarios: multiple lesions in one PET/CT volumetric image and only onelesion in the volume constituting the single lesion scenario. Figure 3.4 showsthat the loss functions performed differently under these scenarios; a widerspread of Dice score values is observed in the single lesion scenario than inthe multiple scenario for both networks. Also, the Dice scores beyond the75th percentile for each loss function in the single lesion scenario were higherthan those in the multiple lesion scenario, with more cases achieving DSCsgreater than 0.8. Despite this observation, each loss function with AttentionU-Net had higher median Dice scores in the multiple-lesion scenario than inthe single-lesion scenario (Figure 3.4a). Besides, in the single lesion case, allthe loss functions have a more significant number of lower outliers, whichare instances where the network’s performance is significantly worse thanthe average, reflecting occasional underperformance. On the other hand, forall the loss functions, there are fewer outliers in the multiple-lesion scenario.While similar observations are made on the SegResNet architecture, themedian DSC of L1DFL in the single lesion scenario remained slightly higherthan the median DSC in the multiple lesion case (Figure 3.4b).Dice loss generally had higher median DSC for both architectures whencompared with DFL, although there were variations in performance betweenlesion scenarios and architectures. In a single lesion scenario using the At-tention U-Net architecture, for example, DL demonstrated more consistentperformance as evidenced by an IQR of 0.33 compared to DFL with an IQRof 0.45. However, on SegResNet, DL’s performance became highly incoher-ent, with a significantly wider IQR of 0.72 compared to the DFL’s narrowerIQR of 0.37. In the case of multiple lesions, while both loss functions haveshown similar IQRs of 0.26 in SegResNet, on Attention U-Net, the DSCvalues from DL are more spread out with lower values in the lower quartilerange compared to DFL and L1DFL. Thus, Dice loss may not be consistent643.3. ResultsFigure 3.4: Box plots overlaid with swarm plots illustrating the distributionof Dice Similarity Coefficient (DSC) performance for the three loss functionsacross two lesion scenarios: M (lesion count > 2) and S (lesion count = 1).Each box plot’s top and bottom edges represent the interquartile range(IQR), while the horizontal lines within the boxes indicate the median DSCvalues. Whisker lengths are set to 1.5 times the IQR. (a) Performance onAttention U-Net architecture. (b) Performance on SegResNet. The numberof cases in the single lesion scenario was N = 34 and for the multiple lesionscenario, the number of cases was N = 23.in some scenarios and architectures, particularly where the cases are morecomplex.In contrast, L1DFL outperformed DL and DFL in both lesion scenarioson the Attention U-Net; the median Dice scores of the multiple and singlelesions were 0.67 and 0.65, respectively. These values are at least 11% higherthan the median Dice scores of DL in both cases. Moreover, on SegResNetin the single lesion case, L1DFL achieved a median DSC of 0.69, whileDL and DFL had 0.56 and 0.55, respectively. However, in the multiplelesion scenario, Dice loss slightly outperformed L1DFL by 4%, althoughthey both had similar IQR. Nevertheless, in all lesion scenarios and on botharchitectures, L1DFL outperformed the Dice Focal loss function.653.3. Results3.3.3 Performance based on lesion metabolic volumeFor each lesion scenario, we evaluated the performance of the loss func-tions across different TMTV thresholds at both the patient and lesion levels(Figure 3.5). For the single lesion case, higher DSC values were gener-ally observed when assessing the performance at the lesion level. However,patient-level DSC values tended to be slightly lower due to false-positivesegmentations. Notably, Dice Loss and Dice Focal Loss were more suscep-tible to false positives, resulting in a reduction of median DSC by at least10% in the patient-level assessment. This trend was also observed in themultiple-lesion scenario.Again, in this scenario, SegResNet displayed distinctive behavior. Withgenerally fewer false positives, SegResNet showed an increase in DSC asa function of TMTV, indicating better segmentation of larger lesions thansmaller ones. For lesions with TMTV below 5 ml, however, higher DSC val-ues were observed at the patient level than at the lesion level, highlighting alimitation of the Dice Similarity Coefficient. When larger lesions are present,a missed small lesion may have minimal impact on the image’s overall DSCcompared to a larger lesion.The results further highlight the difference in the loss functions’ perfor-mance in this scenario. For instance, Dice loss struggled to accurately seg-ment lesions as TMTV increased, leading to a decline in DSC. This trendwas evident in single and multiple lesion cases and the Attention U-Net ar-chitecture; for tumor volumes above 5 ml, where TLA also increased, DiceLoss showed a negative trend in DSC. Generally, brighter and larger lesionsare easier to segment [307]; however, as tumor volume increases, the SU-Vmax voxels occupy a smaller proportion of the lesion, leading to reducedsegmentation performance. In contrast, for smaller lesions, the SUVmaxvoxel represents a larger fraction of the lesion volume, resulting in higherDice scores when segmentation focuses on this voxel. The results suggestthat Dice Loss segments regions with the highest uptake values. As lesionvolume increases and the relative proportion of high-uptake voxels decreases,segmentation performance declines.Moreover, SegResNet demonstrated that segmentation performance isinfluenced by more than just lesion size and uptake. For example, at thepatient level and in the 0-10 ml threshold range, higher TLA values wereobserved in the multiple lesion scenario compared to the single lesion casedue to the summed TLA values of individual lesions. However, higher DSCvalues were achieved in the single lesion scenario, indicating that beyondTLA, lesion count also affects performance. This points to the influence of663.3. ResultsFigure 3.5: Patient and lesion-level median Dice score (DSC) performanceof the different loss functions as a function of various ground truth tumormetabolic volume thresholds. Each column shows the plots for the differentlesion scenarios: single (S) and multiple (M). At both the patient and lesionlevels, the line plots represent the trend of each loss function’s performanceacross different TMTV and MTV thresholds, respectively, with thresholds(t) based on the interquartile range from 0 to the 85th percentile. MedianDSC is calculated for all lesions (l) where volume (vl > t), and each colorindicates a different loss function. The first row of plots (a) shows theperformance of the loss functions with Attention U-Net. The second row(b) shows the performance of the loss functions with SegResNet. The thirdrow (c) displays area plots of median tumor lesion activity (TLA) valuesaccording to the TMTV and MTV thresholds, where median TLA valuesare calculated for lesions with (vl > t).673.3. Resultsother factors, such as lesion anatomical location, proximity to organs likethe bladder, lesion count, and spread impacting segmentation performance.Regarding overall performance, Dice Loss performed better on SegRes-Net than on Attention U-Net, outperforming Dice Focal Loss at the patientlevel in both scenarios due to Dice Focal Loss’ higher false positive rate. DiceLoss and Dice Focal Loss showed similar trends at the lesion level. How-ever, with Attention U-Net, Dice Focal Loss performed better than DiceLoss. L1DFL, on the other hand, maintained stable performance acrossboth architectures and lesion scenarios. This consistency suggests the ef-fectiveness of L1DFL’s dynamic weighting strategy, making it more robustto false positives at the patient level and achieving better overlap with theground truth at the lesion level.3.3.4 Performance based on lesion disseminationTo quantify the spatial distribution of lesions, we calculated Dmax, the max-imum Euclidean distance between any two lesions in the image.For the performance assessment of the loss functions based on the Dmax,we categorized the distances into the following groups: 0-9 cm (G0), 9-11cm (G1), 11-14 cm (G2), and 14-60 cm (G3), representing the first, second,third and fourth quartile ranges, respectively. Patient-level analysis of DSCshowed consistent improvement across these ranges (Figure 3.6). The fig-ure also shows differences in performance across the different architectures.Generally, the Dice loss performed better on SegResNet than on AttentionU-Net. The Dice Focal loss, on the other hand, performed better on At-tention U-Net than on SegResNet. L1DFL achieved similar performancein median DSC but had a wider IQR on SegResNet than on Attention U-Net. On Attention U-Net initially, median DSC values for all loss functionsranged from 0.4 to 0.6, increasing in the third quartile to 0.6-0.8. In the lastquartile, G3, representing a more expansive lesion spread in the image, theAttention U-Net had variability in the segmentation performance.The results with SegResNet also show a similar trend. Specifically, thereis a general increase in DSC values up to the third quartile. For both net-works, all loss functions showed a drop in their performance metrics withinthe highest range, G3. However, the results suggest differences in how theloss functions performed in this group. On Attention U-Net, DFL performedthe worst in this group, and L1DFL and Dice loss presented similar perfor-mances, with a median of 0.62. Dice Focal loss and L1DFL had smallerinterquartile ranges in G3, reflecting less spread across Dice scores in thatcategory, though the median DSC values were slightly lower. The median683.3. ResultsDSC of L1DFL dropped from 0.76 to 0.62 by 18.4%, and the DFL droppedfrom 0.65 to 0.58 by 10.3%. Dice loss also slightly declined in performancewith a broader spread in IQR of its Dice scores. On SegResNet, the Diceloss had the worst performance in this group.Figure 3.6: Plots illustrating the Dice Similarity Coefficients (DSC) acrossfour interquartile ranges of Dmax. The first row (a) shows the performancewith Attention U-Net. The second row (b) shows the performance withSegResNet. The groups G0 (0-9 cm), G1 (9-11cm), G2 (11-14cm), and G3(14-60cm) correspond to the interquartile ranges of 0 − 25%, 25 − 50%,50 − 75%, and 75 − 100%, respectively. In the box plots, horizontal linesindicate the median, and whisker lengths are set to 1.5 times the interquartilerange (IQR), with outliers represented as circles. Accompanying bar plots(c) display the median total lesion activity (TLA) values for cases withineach Dmax group. The different loss functions are color-coded for clarity.The total number of cases in each group was N = 6.Also, narrower IQRs were obtained on Attention U-Net in this groupthan with SegResNet. The differences in the network performance are alsohighlighted in the performance in the lower groups. Overall, there werehigher median Dice scores with SegResNet than with Attention U-Net in693.4. DiscussionG0, G1, and G2. Despite performing better than the Dice loss on the At-tention U-Net architecture, the Dice Focal loss had lower performance onSegResNet. DFL had the worst performance in G1 and G2. The Dice loss,in contrast, yielded better segmentation in the Dmax ranges which charac-terize less lesion spread in the body on SegResNet than on Attention U-Net.Moreover, as lesion spread increased, the results show that L1DFL is morerobust, outperforming DL and DFL.Figure 3.6 also depicts a higher median total lesion activity (TLA) acrossthe different groups, indicating an increased lesion size or activity withgreater lesion spread. Although model performance generally improves withhigher lesion activity, the slight decrease in median DSC in the highest quar-tile of Dmax, despite high TLA, suggests that other factors might contributeto the segmentation performance such as lesion site and count.3.4 DiscussionIn this work, we evaluated the Dice loss function in segmenting metastaticprostate cancer lesions in images with one lesion and images with multiplelesions. We compared its performance with Dice Focal loss and our proposedloss function, L1 weighted Dice Focal loss. Moreover, we report how the sizeof a lesion and its spatial distribution influence the segmentation result.The volume and standardized uptake values (SUV) were relatively low inthe dataset we worked with thus introducing unique challenges for correctsegmentations.Our findings showed that the performance of the loss functions is notuniform across different architectures. For instance, we found that Dice lossgenerated more false positives on Attention U-Net compared to SegResNet,despite showing higher sensitivity on the former. Similarly, the Dice Focalloss exhibited lower specificity on the Attention U-Net architecture due toincreased false positives.Segmentation performance across the single-lesion and multiple-lesionscenarios showed subtle differences. While the median Dice scores wereslightly higher in the multiple-lesion scenario, the single-lesion scenario ex-hibited a wider IQR with instances of both under- and over-performance.Additionally, the results highlighted differences in segmentation performancebased on varying lesion sizes. Specifically, while the TMTV increased withan associated increase in total lesion activity, Dice loss tended to performbetter in the single lesion case and poorer in the multiple lesion. At thelesion level, however, considering a volume threshold range of 0 - 7.5ml, the703.4. DiscussionDice loss produced better segmentation with increasing volumes, but witha decrease in Dice scores beyond this range. The L1DFL and DFL in con-trast maintained consistent performances across the different lesion scenar-ios, generally demonstrating an upward trend with L1DFL outperformingDFL. Despite these patterns, the SegResNet architecture which generallyyielded lower false positive rates had higher median dice scores in the mul-tiple lesion scenario at the patient level than at the lesion level for TMTVvalues less than 5ml.In addition, considering the performance at the lesion level for this lesionscenario, the loss functions generally performed better on the larger lesionsthan on the smaller lesions. When DSC is calculated at the patient level,it compares the overlap of the entire predicted mask with the entire groundtruth mask. Large lesions contribute a significant number of voxels to thisoverall overlap calculation, thus having a proportionally higher impact onthe patient-level DSC. However, calculating DSC at the lesion level assessesoverlap at the scale of each lesion. Small lesions contribute fewer voxels, soany segmentation error in these smaller lesions reduces the lesion-level DSC.Thus, if larger lesions are segmented accurately, the patient-level DSC couldbe high even if small lesions are missed or poorly segmented. That is, smalllesions contribute only a small fraction of the total voxel count. If thesesmall lesions are inaccurately segmented or missed altogether, the impacton the patient-level DSC remains minimal but will be evident in the lesion-level assessment of DSC. Thus, the results showing a lower performanceon smaller lesion, but better segmentations of larger lesions at the lesionlevel yet with higher DSC at the patient level in the multiple lesion scenariohighlights this limitation of the Dice Similarity Coefficient.Nevertheless, the results show that the performance of the Dice losstends to decline beyond an MTV greater than 5ml. This could suggest thatDice loss is highly sensitive to the voxels of high uptake values. Once thelesion volume increases, the proportionality of such high-uptake voxels to thelesion volume diminishes, reducing the segmentation accuracy. Dice loss isinherently biased toward smaller regions [291], generally performing poorlyon larger lesions with more variable SUV distribution across the volume,thus affecting the precision of segmentation. Consequently, the Dice lossfunction resulted in DSC values for the lower levels of tumor volume thanthe Dice Focal loss, which on the other hand performed consistently fordifferent lesion size cases but at the cost of specificity.Lesion spread, or dissemination, further complicates segmentation, es-pecially in metastatic cases. In the multiple-lesion case, the segmentationperformance increased with higher lesion dissemination, dmax, measured as713.4. Discussionthe maximum distance between lesions. The increase in dmax was accom-panied by slight increases in median TLA values. However, we found thatmedian Dice scores decreased across all loss functions in cases within thetop quartile of dmax (14–60 cm); of these, Dice Focal loss was affected moston Attention U-Net and the Dice loss on SegResNet. The decreased perfor-mance at larger values of dmax suggests that while increased lesion spread canenhance segmentation by emphasizing the clear regions of lesions, larger dis-tances between lesions may introduce new challenges for the model. Besides,the anatomical location of lesions also affects detection and segmentation.Figure 3.7: Plot of loss versus probability for samples with the ground truthlabel as 1. The curves represent the behavior of each loss function, high-lighting how they respond to varying probabilities of correctly classifyingexamples. The plot illustrates the differences in how each loss function han-dles both easy-to-classify and hard-to-classify cases based on their predictedprobabilities.The L1 Dice-Focal loss comprises a dynamic weighting strategy thatadapts the model’s focus based on sample difficulty. For easier-to-classifyexamples (characterized by predicted probabilities P > 0.5), L1DFL ap-plies a reduced penalty while penalizing more challenging examples (withP < 0.5) more heavily (Figure 3.7). By contrast, Dice loss applies a nearlyuniform weighting, balancing sensitivity and specificity. This balance maybe beneficial in scenarios where a balanced performance is of interest, suchas in medical settings. Loss functions based on overlap measurements, likethe Dice loss, are more robust in scenarios with significant data imbalance,as in lesion segmentation tasks [275], therefore, distribution-based losses are723.5. Conclusion and Transition to Next Chapteroften combined with the Dice loss or its variants to improve their robustness.However, without adapting its focus to classification difficulty, the Dice lossfails to adjust dynamically toward heterogeneous lesion characteristics.Nevertheless, the Dice Focal loss increases the penalty on samples witha lower probability of correctly classifying the ground truth label, P(G),encouraging the model to focus more on ”hard” samples, but it applies amuch steeper penalty. Thus, it tends to lose the Dice loss component, wherepositive and negative examples remain equally weighted, potentially leadingto more false positive rates due to misclassification of background areas aslesions [317]. On the other hand, the weighting strategy of L1DFL allowsfor a more balanced approach to penalizing misclassification, consequentlyreducing false positive rates. Our results indicate that L1DFL yielded thehighest F1 and Dice scores in this study, underlining its more robust per-formance across various lesion scenarios and levels of lesion dissemination.This adaptive weighting strategy shows promise in improving segmentationperformance in complex metastatic imaging cases.3.5 Conclusion and Transition to Next ChapterWe evaluated the performance of a custom loss function, L1-weighted DiceFocal loss (L1DFL), in the detection and segmentation of metastatic prostatecancer lesions in PSMA PET/CT scans across single and multiple lesion sce-narios, characterized by lesion count. L1DFL leverages a dynamic weightingscheme that is based on the L1 norms of the predicted probabilities andground truth labels to modulate the Dice loss and then combined with theFocal loss to emphasize challenging pixel classifications.We compared L1DFL’s performance against Dice loss and Dice Focal lossusing two 3D CNN architectures, Attention U-Net, and SegResNet. WhileDice loss demonstrated sensitivity for small lesions, it usually suffered fromlow precision. Dice Focal loss showed higher false positive rates and variableperformance across the different networks. On the other hand, L1DFL con-sistently achieved a balanced segmentation performance across the lesionsscenarios and architectures, achieving superior F1 scores and Dice Similar-ity Coefficients. In the following chapter, we evaluate the performance ofL1DFL in quantifying clinical metrics which have been identified as relevantfor prognostication, risk stratification, and treatment planning.73Chapter 4Comparative Analysis ofLoss Functions for AccurateQuantification of Clinicalmetrics4.1 IntroductionProstate cancer arises from abnormal proliferation of cells in the prostategland [325]. In the early stages, these cells depend on testosterone forgrowth, and often androgen deprivation may control the disease [326]. Mostpatients are treated with curative intent using radical prostatectomy (RP)or radiotherapy (RT) [327]. Despite these interventions, about 20 − 40%of patients develop biochemical recurrence (BCR), which typically involvesa rise in prostate-specific antigen (PSA) [328, 329]. However, dependingon the type of primary treatment, specific definitions of BCR may be em-ployed. For instance, after RP, BCR is indicated by two consecutive PSAvalues ≥ 0.2 ng/mL [330]. Although BCR does not invariably indicate clin-ical progression, it is associated with an increased risk of metastasis andmortality in certain subsets of patients [331]. The course of BCR is quiteheterogeneous and depends on several factors, including PSA kinetics (suchas doubling time, velocity), Gleason score, and pathologic features related tosurgical margins and seminal vesicle involvement [332, 333]. For example, aPSA doubling time of less than three months is associated with unfavorableoutcomes, while the interval to recurrence often stratifies disease aggressive-ness; early recurrence typically indicates a more aggressive cancer phenotype[334].In managing BCR, imaging modalities like prostate-specific membraneantigen (PSMA) positron emission tomography (PET) may be used and havesubstantially increased the detection of early metastatic disease [335]. Iden-tified lesions may be manually segmented by physicians to extract various744.1. Introductionclinical metrics such as total molecular tumor volume (TMTV), mean andmaximum standardized uptake value (SUVmean and SUVmax), and totallesion activity (TLA) [170, 336]. These metrics are increasingly being iden-tified as important in the prediction of patient outcomes and in treatmentdecisions. The manual process of lesion segmentation and quantification ofthese metrics, however, is time-consuming, leading to an interest in artificialintelligence (AI)-based methods to automate the process.Despite their promise, automated segmentation introduces its own uniqueset of challenges. Typically, lesions that resurface after treatment are small,heterogeneous in shape, location, and uptake activity, and occupy a smallportion compared to the background volume[305]. Consequently, most meth-ods struggle to balance lesion sensitivity and specificity, often presentinghigh rates of false positives that may pose clinical risks due to unnecessaryinterventions (such as excess radiation or surgery). Several techniques havethus been explored to address these challenges, including developing mod-els for specific anatomical sites [337, 338], using standardized uptake value(SUV) thresholding methods [174], and using healthy organs as anatomypriors to refine model attention [298]. More recently, loss functions, suchas focal loss and cross-entropy loss, that are tailored to segmentation taskshave also been proposed [291, 313, 317].While these various methods have led to some progress, many still tendto fail in balancing precision with recall leading to an overall lower DiceSimilarity Coefficient (DSC) value. For instance, SUV thresholding maydecrease false positive rate yet affect sensitivity to lesions with lower ra-diotracer avidity. Furthermore, previous studies (e.g. [339] and [307]) havedemonstrated improvements in segmentation using diffusion models, andcustom foreground voxel weighting loss functions, respectively. While [307]achieved a mean lesion sensitivity of 0.64, the overall mean DSC remainedlow at 0.43 due to high false positive rates. Similarly, diffusion models out-performed architectures like U-Net, Flexible U-Net, and transformer-basedmodels like U-NetR but were limited to a DSC of 0.53 [339].Furthermore, automated segmentation studies often emphasize overlap-based metrics like the DSC to assess performance. While these metricspresent a useful understanding of the quality of segmentations, they do notalways correlate with the clinical value of the results, especially when quan-tifying disease-relevant metrics [340]. Therefore, in addition to reportingDSC, it is vital to evaluate the ability of an algorithm to reproduce clini-cally meaningful outcomes [170]. Accurate quantification of clinical metricsis dependent on the accuracy of segmentation methods. Thus, to achieveclinically relevant results, training automated methods to optimize a balance754.1. Introductionin sensitivity and specificity is important. Loss functions are one way thefocus of models is tuned to improve segmentation performance. Specifically,different groups of loss functions have been explored, including region-basedloss functions like the Dice loss, distribution-based loss functions like cross-entropy loss, boundary-based loss functions like the Hausdorff distance loss,and compound loss functions like Dice cross-entropy loss [271]. Each grouphas a specific focus; for instance, the region-based loss functions are com-monly used in medical segmentation tasks due to the significant imbalancein the distribution of voxels between the background class and regions ofinterest [291].Moreover, they lack robustness regarding handling differences in voxelclassification difficulties [313]. That is, within the background and fore-ground classes, some voxels are easily correctly classified. In contrast, oth-ers are more difficult to classify, causing the models to achieve suboptimalperformances on those voxels. Distribution-based loss functions have beenfound to achieve better results regarding this additional type of imbalance[281], so they are combined with the region-based loss functions to formcompound losses with improved robustness and generalizability [287]. Yet,by combining these loss functions, a trade-off is often required because thebalance in sensitivity and specificity preserved by the region-based loss func-tions is often lost, and specificity is often traded for sensitivity [317]. Toaddress this challenge, we proposed a novel loss function, L1-weighted DiceFocal loss (chapter 3), which leverages the L1 norms of predicted probabili-ties and ground truth labels to adjust the model’s focus. It yielded superiorperformance over the Dice loss and Dice Focal loss, achieving higher DSCand F1 scores.In this chapter, we evaluate the performance of the L1-weighted DiceFocal loss (L1DFL; Eq. 3.9) in quantifying six important clinical metrics fordisease prognosis, namely SUVmean, SUVmax, TLA, TMTV, number of le-sions (L), and lesion dissemination (Dmax). We compared its performanceagainst three commonly used loss functions: Dice loss, Dice Cross-Entropyloss, and Dice Focal loss, and assess its generalizability across different con-volutional neural network (CNN)-based architectures: U-Net, SegResNet,and Attention U-Net.764.2. Materials and Methods4.2 Materials and Methods4.2.1 Model TrainingIn addition to the networks, Attention U-Net and SegResNet, evaluated in3, we also assess the performance of L1DFL (defined in Chapter 3, Eq. 3.9)on the U-Net architecture. The model comprised a symmetric encoder-decoder path. The encoder pathway embedded the PET/CT input volumeinto a lower-dimensional feature space while progressively downsampling inthe spatial dimensions. Each encoder block consisted two residual units,each made up of a 3D convolutional layer, batch normalization, and ReLUactivation. It achieved downsampling through convolutional layers with 2×2 × 2 strides, effectively reducing the spatial dimensions by half at eachlevel, increasing feature channels from 16 to 512. The decoder path followedthe modular structure of the encoder, each block comprising two residualunits. The feature maps were progressively upsampled using transposedconvolutions with the same stride as in the encoder path.Also, the models were trained similar to what was described in 3.2.5with the difference being the data split. Specifically, we divided the datasetinto training, and testing sets with a split ratio of 323:57, respectively. Thetraining set was further split into training and validation subsets for a 5-fold cross-validation; models were trained on four folds and validated onthe remaining fold. We trained each network with four loss functions: Diceloss (Eq. 2.2), Dice Focal loss (Eq. 2.10), Dice Cross Entropy (Eq. 2.9) andthe L1-weighted Dice Focal loss (Eq. 3.9). For both Dice Focal Loss andDice Cross Entropy Loss, an equal weighting was maintained between thedifferent loss components in the compound loss.The best-performing models from each fold, based on their DSC scoreson the validation set, were used to predict the test set. A majority votingstrategy was applied at the voxel level, where a voxel was classified as alesion if for a network, at least three out of the five fold models predicted itas such. Mathematically, the ensemble prediction E(v) for each voxel v wasdefined as:E(v) ={1 if∑5i=1 Pi(v) ≥ 3,0 otherwise,(4.1)where Pi(v) represents the binary prediction of the i-th fold model for voxelv, with Pi(v) = 1 indicating a lesion and Pi(v) = 0 indicating non-lesion.774.2. Materials and Methods4.2.2 Model EvaluationTo evaluate segmentation performance at the patient level, we calculatedseveral clinical metrics: mean standardized uptake value (SUVmean), max-imum standardized uptake value (SUVmax), total molecular tumor volume(TMTV), total lesion activity (TLA), lesion dissemination (Dmax), and le-sion count. These metrics provide insights into various lesion characteristicsand are shown to correlate with biochemical progression. For example, SU-Vmean and SUVmax, which measure the average and maximum uptake in-tensity, respectively, serve as indicators of tumor activity and are relevant forrisk stratification and prognosis in metastatic castration-resistant prostatecancer [341, 342]. TMTV, which represents the total tumor volume, is shownto be predictive of overall patient survival [343]. TLA, calculated as the sumof the product of SUVmean and the molecular tumor volume, reflects thetotal tumor burden and is also linked to patient outcomes and progression[344]. Additionally, Dmax, which measures the maximum spread of lesions,can influence staging, prognosis, and treatment decisions, such as whethersystemic or localized therapy. The number and anatomical distribution oflesions have also been shown to significantly correlate with biochemical pro-gression [319]. The detailed mathematical description of these metrics areprovided in the appendix (A.1).4.2.3 Assessment of Agreement with Ground TruthWe evaluated the agreement between the predicted and ground truth clini-cal metrics using Concordance Correlation Coefficient (CCC), EquivalenceTesting, Bland-Altman Analysis, Coverage Probability (CP) and Total De-viation Index (TDI). For each clinical metric, we performed a paired eval-uation at the patient level, comparing the predicted value for each patientwith its corresponding ground truth value in the test set with the aim of as-sessing the overall agreement between predictions and ground truth for eachloss function and architecture in reproducing clinical metrics. The detailsfor each method are provided below:Concordance Correlation Coefficient (CCC)The CCC was used to quantify the agreement between predicted values (yˆi)and ground truth values (yi) for each clinical metric. It measures bothprecision and accuracy, providing a robust assessment of agreement [345].784.2. Materials and MethodsThe CCC is defined as:ρccc =2ρσyˆiσyσ2yˆi + σ2y + (µyˆi − µy)2, (4.2)where ρ is the Pearson correlation coefficient between yˆi and y, σ2yˆiand σ2yare the variances, and µyˆi and µy are the means of yˆi and y, respectively.CCC takes on values in the range [-1, 1] where a value of 1 implies perfectagreement and a value of -1 implies perfect disagreement. For values be-tween 0 and 1, we employ the interpretation suggested by McBride(2005)where CCC > 0.99 is interpreted as an almost perfect agreement, CCCvalues between 0.95 and 0.99 are considered a substantial agreement, valuesbetween 0.90 and 0.95 are considered moderate agreement and values lessthan 0.90 are interpreted as poor.Equivalence TestingWe performed equivalence testing using the two one-sided t-tests procedure[346]. Following consultation with physicians, we set a predefined margin of±20%. Based on this margin, we computed the upper and lower equivalencebounds for each metric as ∆ = 0.2 × y¯, where y¯ is the mean of the groundtruth values for the metric over all patients in the test set. The differencebetween predicted values (yˆ) and the ground truth values y for each patientwas calculated as ∆yi = yˆi − yi, for an ith patient. For the TOST proce-dure, we defined the null as H0 : ∆µ ≤ −∆ or ∆µ ≥ ∆ and alternatehypotheses as: H1 : −∆ < ∆µ < ∆, with ∆ expressed in the absolute unitsof the clinical metric.Also, we computed two one-sided p-values, pl = P (∆µ > −∆) and pu =P (∆µ < ∆), such that the null hypothesis of non-equivalence is rejectedwhen pl < α and pu < α. We also constructed a 90% confidence interval(CI) for ∆µ such that equivalence was established if and only if the entireCI fell within the equivalence bounds [−∆,∆] and both pl and pu are lessthan the significance level α, which we set to 0.05.Bland-Altman AnalysisWe used the Bland-Altman analysis to visualize and quantify the agreementbetween predictions and ground truth. We used a modified version of theBland-Altman plots, where the values on the y-axis are the computed meandifference between the predictions and the ground truth of a given clinicalmetric and the x-axis are the ground truth values of the metric. Points794.2. Materials and Methodsoutside the limits of agreement indicate poor agreement and values aboveand below a mean difference of zero indicate instances of over and under-prediction, respectively.Coverage Probability (CP)The coverage probability assesses the proportion of instances where the con-fidence interval encompasses the true value in a repeated run [347]. Adopt-ing the clinical equivalence region of ±20%, CP thus could be defined asthe proportion of instances where the difference in predicted and groundtruth values lie within the clinical range. This provides a complementaryassessment to TOST where the consistency of the model’s performance atthe patient level is evaluated. We compute CP as:CP =1NN∑i=1I (|yˆi − yi| ≤ δ · yi) (4.3)where N is the total number of patients in the test set, I is the indicatorfunction, δ is the acceptable clinical margin (0.2). Higher CP values reflectbetter precision and clinical agreement between predictions and the groundtruth.Total Deviation Index (TDI)The Total Deviation Index (TDI) was used to quantify the deviation ofpredictions from the ground truth at a specified percentile (τ) [347][348].This method reverses the Coverage Probability calculation, such that, givena proportion of instances (τ), TDI evaluates the margin within which thedifferences fall. Based on the calculated margin, a decision was made onwhether the margin is narrow enough for the model to be considered as goodas the ground truth method (which in this case is manual segmentation).For this study, (τ) was set as the 95th percentile. We calculate the TDI as:TDIτ = Qτ({|yˆi − yi|}Ni=1) (4.4)where Qτ is the quantile function which returns the (τ -th) quantile of theabsolute deviations, and {|yˆi − yi|}Ni=1 is the set of absolute deviations be-tween the predictions (yˆi) and the ground truth (yi) values for all the Npatients in the test set. Lower TDI values indicate tighter agreement be-tween predictions and ground truth values.804.3. Results4.3 Results4.3.1 Performance evaluation of loss functions acrossdifferent networksThe segmentation performance of the loss functions on the different networksas assessed by the DSC and the detection metrics (true positive rate, falsepositive rate, false negative rate, and F1 score) are provided in the appendix(A.2) We report the average DSC obtained from five-fold cross-validationand performance on the held-out test set. Briefly, the loss functions had theirhigher mean and median DSC on the SegResNet architecture but showedvariable performance on the U-Net and Attention U-Net. Also, L1DFLoutperformed the loss functions achieving better false positive rates and F1scores.4.3.2 Evaluation of concordance between predicted andground truth clinical metricsIn this section, we assess the agreement between the predicted values of theclinical metrics and the ground truth values using the Lin’s concordancecorrelation coefficient. The results suggest variable performances by thedifferent architectures and loss functions. Figure 4.1 show that the differ-ent loss functions achieved higher correlation coefficients on Attention U-Netand had the least values on SegResNet. Also, although not consistent acrossthe different networks, the loss function exhibited better agreements withthe ground truth on SUVmax, SUVmean and TLA. Comparing the perfor-mance of the loss functions highlights their different levels of agreements withthe ground truth values on the different networks. For example, althoughoverall, Dice Focal loss exhibited the weakest correlation on SegResNet, itslightly outperformed DCE and Dice loss on the U-Net architecture.Similarly, the DCE had a weak correlation on U-Net but maintained astrong measure of agreement with the ground truth measurement on At-tention U-Net. Moreover, L1DFL obtained higher values of the correlationcoefficient, sometimes at a moderate or substantial level of performance,that is, between 0.90-0.99. Especially on the Attention U-Net model, it of-ten outperformed others with an overall strong correlation in metrics suchas TLA and SUVmax. Meanwhile, DCE demonstrated a largely moder-ate performance, with weaker correlations for metrics like Dmax and LesionCount. However, it outperformed L1DFL, achieving a stronger correlationfor SUVmean and SUVmax with SegResNet. While the overall performance814.3. ResultsFigure 4.1: Radar plots showing the correlation between the predicted andground truth metrics for the four loss functions, Dice loss (DL), Dice Cross-Entropy (DCE), Dice Focal loss (DFL), and L1-weighted Dice Focal Loss(L1DFL) assessed by the Lin’s Concordance Correlation Coefficient. Thedifferent radar plots illustrate the performance of the three architectures- U-Net, Attention U-Net, and SegResNet. The radial lines represent thecorrelation coefficient values.for both the Dice and Dice Focal losses was generally poor, as most of thevalues of correlation fell below 0.90, showing weaker alignment with theclinical ground truth across metrics and architectures, they achieved bettercorrelation coefficients with U-Net and Attention U-Net than with SegRes-Net.Figure 4.2 further highlights the differences in the performance of themodels in predicting the ground truth metrics. The scatter plots showinstances of overprediction and underprediction, highlighting metrics withpoor alignment to the ground truth. Similarly, there are occasional outlierpredictions for example, as shown in the performance of the Dice and DiceFocal losses in predicting the ground truth SUVmax values on SegResNet.However, L1DFL consistently showed better alignment with ground truthvalues, particularly for TMTV and TLA, where its predictions remainedclose to the equality line even for higher values. Dice Loss (DL), in contrast,showed noticeable underperformance for higher SUVmean and TLA values,often underestimating their magnitude, especially with U-Net. As shown inthe radar plots, Attention U-Net and SegResNet tended to produce tighterclustering than U-Net for all metrics, although variability increased withlarger lesion counts (L) and lesion spread (Dmax), where SegResNet showedlarger deviations. The lesion count metric (L) demonstrated greater spread824.3. ResultsFigure 4.2: Scatter plots comparing predicted clinical metrics (y-axis) toground truth values (x-axis) for different architectures (columns: U-Net,SegResNet, and Attention U-Net) and clinical metrics (rows: SUVmax,SUVmean, TMTV , TLA, L, and Dmax). Each data point represents a spe-cific prediction, with colors indicating different loss functions. The dashedline represents the line of equality (y = x), and agreement between predictedand ground truth values is assessed using Lin’s concordance correlation co-efficient.in predictions across all architectures and loss functions, indicating this re-mains a challenging metric to predict. However, L1DFL reduced variabilityand showed better performance overall, particularly for cases with higherlesion counts and larger lesion distributions.4.3.3 Reproducibility of the ground truth metricsWe evaluate the reproducibility of the ground truth metrics using two one-sided t-tests (TOST), assessing equivalence at a significance level of α = 0.05and an equivalence bound of ∆ = 20%. Reproducibility was achieved foronly four metrics, SUVmax, SUVmean, Lesion Count (L), TLA as depicted in834.3. ResultsFigure 4.3, but performance was loss function and architecture dependent.In the U-Net architecture, all the loss functions were able to reproduce theSUVmean and SUVmax metrics, however, on SegResNet, only DCE was ableto reproduce both metrics. For Attention U-Net, while all loss functionsreproduced the SUVmean, only DCE, and L1DFL achieved reproducibilityfor the SUVmax metric. Lesion Count and TLA were harder to reproduce bythe models, yet it was achieved by the L1DFL loss function on the AttentionU-Net and SegResNet for Lesion Count and only Attention U-Net for TLA.The other metrics for which reproducibility was not achieved had largerconfidence intervals, reflecting greater uncertainties of their estimates.Figure 4.3: Two one-sided forest plots illustrating the results of equivalencetesting for predicted metrics versus ground truth across different loss func-tions. Each row represents a specific metric, and the columns correspond tothe different networks. The black vertical dashed lines represent the region ofclinical equivalence (±20% of the mean of each ground truth metric). Thex-axis represents the mean difference between the predicted and groundtruth metrics. Green bars represent equivalence between predictions of agiven loss function and ground truth values at a significance level of α = 0.05,while red bars indicate non-equivalence.The Bland-Altman analysis across U-Net, SegResNet, and Attention844.3. ResultsU-Net architectures demonstrates that L1DFL consistently agrees betterwith ground truth values for all evaluated clinical metrics. For SUVmaxand SUVmean, differences remained minimal across all ground truth values.However, Dice loss and DCE exhibited greater variability at higher values,while L1DFL maintained tighter limits of agreement. Similar trends wereobserved for TMTV, where L1DFL demonstrated reduced variability andminimal bias, contrasting with Dice Loss, which tended to overestimate val-ues at larger TMTVg. All the loss functions generally agreed better with theground truth values for TLA except for a few under predictions at highervalues. Greater variability is shown for the lesion count metric. Specifically,the dataset included images of up to five lesions. For each count, there arescenarios of false positive estimation leading to instances of overprediction,as shown in the figure. For instance, for a lesion count of 1, Dice loss, DiceFocal loss, and DCE are shown to deviate from the ground truth by an ex-cess of at least 4 false positive predictions. At higher counts of 5 lesions, theDice loss, Dice Focal loss, and L1DFL are shown to underestimate the countdue to instances of false negative prediction. However, the loss functions areshown to estimate the lesion count better on SegResNet than on AttentionU-Net and U-Net.Similarly, for Dmax, SegResNet yielded better estimations than U-Netand Attention U-Net. Notwithstanding, Lesion count and Dmax had greatervariability than the other metrics. For all the other metrics, the modelstrained with the different loss functions either had instances of over or under-predictions at lower or higher ground truth values. However, for these twometrics, variability occurs across the entire range of values, with Dice loss,Dice Focal loss, and DCE being the most affected, particularly on Dmax.For SUVmax, SUVmean, and TLA, Attention U-Net and U-Net generallyoutperformed SegResNet, showing reduced bias and variability.4.3.4 Evaluation of performance based on coverageprobability and total deviation indexWe considered the metrics that were reproduced by the models based onthe two one-sided t-tests, and we evaluated the proportion of predictionsthat align with the ground truth metrics using coverage probability (CP)and total deviation index (TDI). In general, higher values of CP and lowervalues of TDI are associated with more accurate and precise predictions.Compared to SegResNet, U-Net, and Attention U-Net yielded generallylower CP values. However, when combined with L1DFL, Attention U-Netproduced relatively higher CP values. The values obtained were all con-854.3. ResultsFigure 4.4: Figure: Modified Bland-Altman plots illustrating the variations(∆) between predicted and ground truth values (y-axis) against the groundtruth metrics (x-axis) for different loss functions, represented by distinctcolors. Each row corresponds to a specific clinical metric, and the columnsrepresent predictions from different network architectures (U-Net, SegRes-Net, and Attention U-Net). The dashed black horizontal lines indicate thelimits of agreement (±20% of the mean ground truth values), while the dot-ted line at ∆ = 0 represents perfect agreement. Each data point reflectsa single instance, highlighting the distribution of variations across groundtruth values.sistently higher for SUVmax, and all the loss functions gave CP greaterthan 0.6, implying more than 60% of the predictions fell into the clinicalequivalence margin, ∆ = ±20%. Notably, SegResNet gave CP values over0.8 with all loss functions for SUVmax, showing a high portion of clini-cally acceptable predictions. Dice Focal Loss showed the best performanceamong different loss functions on U-Net, reaching a CP of 0.78 for SUVmax,whereas L1DFL achieved this value on Attention U-Net. For lesion count,Dice Focal Loss again performed better on U-Net, reaching a CP of 0.47,equal to DCE on SegResNet, but both were outperformed by L1DFL with aCP of 0.54 on Attention U-Net. Lower proportions were obtained on TLA,864.4. Discussioneven for L1DFL which reproduced the metric as assessed by TOST. Thus,although the average deviation in the predicted TLA values and the groundtruth fall within the clinical margin, the lower CP value suggests that theperformance of L1DFL on this metric was not consistent in the differentpatient cases. U-Net with Dice loss had the lowest proportion of clinicallyequivalent predictions across the metrics.TDI analysis showed a similar pattern where the Dice loss has the high-est deviation on at least three metrics, SUVmax, SUVmean, and Lesioncount, on all architectures. The Dice Focal loss also did not yield precisepredictions, with its deviation on SUVmax on SegResNet being almost threetimes the value for L1DFL. In fact, on this architecture, the Dice Focal lossstruggled to minimize inconsistent performance on all metrics. It still main-tained high deviations on Attention U-Net and U-Net on SUVmean, L, andTLA. Contrary to the relative superior performance of DCE on the CP as-sessment, its overall performance on the test set highlights instances of overand under-predictions resulting in higher deviations. It had the worse de-viation on Lesion count and SUVmean. L1DFL on the other achieved thelowest deviations on all metrics, although with different architectures. Theconsistently higher CP values and lower TDI values for L1DFL points to itspotential to its effectiveness in maintaining predictions close to the groundtruth metrics.4.4 DiscussionIn this work, we evaluated the performance of a novel loss function, L1DFL,in reproducing relevant clinical metrics, including SUVmean, SUVmax, le-sion count, Dmax, TLA, and TMTV. We compared its performance withother commonly used loss functions, Dice loss, Dice Cross Entropy loss, andDice Focal loss on three different 3D CNN architectures, U-Net, AttentionU-Net, and SegResNet. The results highlight differences in the performanceof the loss functions and the various architectures based on true positivedetections and their balance with false positive detection rate.The balance between both detection rates is crucial in ensuring patientcare in medical applications. The results further suggest that better perfor-mance on lesion segmentation, as assessed by the Dice Similarity Coefficient,does not imply superior performance in reproducing clinical metrics. Forexample, the loss functions achieved higher Dice scores on the SegResNetarchitecture than on U-Net and Attention U-Net. However, Attention U-Net had better concordance correlation coefficients on the clinical metrics.874.4. DiscussionFigure 4.5: The left column displays plots of Coverage Probability (CP),representing the proportion of instances where the model’s predictions fellwithin ±20% of the corresponding clinical metric. Error bars indicate the95% confidence intervals. The right column shows bar plots of the TotalDeviation Index (TDI) for each metric, representing the 95th percentile ofdeviations between predictions and ground truth values. Rows correspond tospecific metrics (SUVmax, SUVmean, Lesion Count (L), and TLA), whilecolumns indicate different network architectures (U-Net, SegResNet, andAttention U-Net). Distinct colors denote different loss functions: DL, DCE,DFL, and L1DFL.The SegResNet had the worst correlation. Correlation also varied across thedifferent clinical metrics.Moderate to strong correlation (CCC > 0.9) was obtained on SUVmean,884.4. DiscussionSUVmax, and TLA, but a weak correlation was achieved on lesion count,Dmax, and TMTV. Poor correlation between the predictions on Dmax andlesion count and their ground truth values could be attributed to the oc-currence of false positive segmentation and, in some cases, false negativedetections. As shown in the Bland Altman plots 4.4, there are occurrencesof overprediction on Dmax, highlighting increased distances between lesionpairs compared to ground truth due to false positive detections. On thismetric, Attention U-Net had the most increased variability in performance,with the Dice loss and Dice Focal loss yielding the most over predictionresults. This observation is supported by their higher mean false positiverates shown on this architecture in appendix A.2. SegResNet and U-Net alsohad instances of overprediction on Dmax, but they had a narrower range ofvariation between the predicted results and the ground truth values.Similarly, a wider deviation spread was noted in the performance on le-sion count. The loss functions had instances of overprediction for a lesioncount of one owing to false positive segmentations. However, there wereunderpredictions for a lesion count of five due to false negatives. L1DFLhad the most robust performance for both metrics, generally yielding moreconsistent results, followed by the Dice Cross-Entropy loss function. All theloss functions, however, had instances of underprediction on higher valuesof TMTV suggesting an increase in false negative rates. This result is sup-ported by the observation of underestimation of lesion count, especially fora lesion count of five. Thus, the loss functions generally struggle in segment-ing all lesions as the number of lesions present increases. This is expected aslesions may occupy different anatomical locations with lesion metastasis andvary in size, increasing the detection complexity. With weaker correlationsobtained for TMTV, Dmax, and lesion count, the loss functions failed toreproduce these metrics. Nevertheless, L1DFL reproduced lesion count onAttention U-Net and SegResNet.For SUVmean and SUVmax, the predictions from the four loss func-tions were statistically equivalent to the ground truth values, although thisperformance was architecture-dependent. For instance, for SUVmean, onlyL1DFL and DCE yielded equivalence to the ground truth on Attention U-Net. Similarly, only DCE reproduced SUVmax on SegResNet. Meanwhile,the proportion of predictions that fall within the clinically acceptable marginof ∆ = ±20% varies even amongst the metrics for which reproducibility wasachieved. Specifically, up to 90% of the predictions of SUVmax were withinthe acceptable clinical margin, but this value decreased to between 30% and80% for SUVmean and less than 60% on lesion count. Overall, L1DFL hada higher proportion of predictions in the clinical margin followed by DCE,894.5. Conclusionwith the least being the Dice loss function. Besides, the Dice loss and theDice Focal loss yielded higher deviations from the ground truth metrics,suggesting lower overall precision between predictions obtained on the testset and their actual values.4.5 ConclusionAmong the six evaluated metrics, only four—TLA, SUVmean, SUVmax,and Lesion Count— were successfully reproduced and had deviations fallingwithin the clinically acceptable range. The L1-weighted Dice Focal Loss(L1DFL) demonstrated superior performance by accurately reproducing allfour metrics with at least one architecture, whereas other loss functionswere limited to successfully reproducing only the SUVmean and SUVmaxmetrics. Furthermore, L1DFL achieved a better balance between precisionand recall, as reflected in its stronger correlation with ground truth valuesand reduced tendencies for overestimation or underestimation of the metrics.90Chapter 5Summary and Future Work5.1 SummaryImaging is repeatedly used throughout the patient care cycle due to the po-tential for biochemical recurrence, where timely identification of metastaticlesions is paramount. Accurate segmentation allows for precise quantifica-tion of lesion characteristics, which may inform clinical decisions. However,lesion segmentation related to prostate cancer presents unique challenges,in particular in biochemical recurrence where lesions are more often small,with generally lower uptake intensities, and highly heterogeneous.Manual segmentation is inherently time-consuming, prone to interob-server variability, and has limited scalability. Meanwhile, automated seg-mentation methods, though promising, face significant challenges due to theinherent heterogeneity in lesion characteristics, such as size, shape, and up-take intensities. In recurrent disease, suboptimal performances are common,with lesions often showing subtle metabolic activity that complicates theirdetection and delineation. These limitations outline the necessity for moreadvanced methodologies that, while overcoming these challenges, achievea balance between sensitivity and specificity critical to avoid unnecessaryinterventions and missed detections.Thus, this thesis helps address these challenges by proposing a new lossfunction to apply in the automatic segmentation of lesions in prostate cancerin PET/CT scans. The approach aims to perform adaptive voxel weighting-that is, dynamic self-adjustment of the model’s focus based on the difficultyof classifying voxels. This novel loss function assigns weights depending onL1 norms of predicted probabilities and ground truth labels, prioritizing theweighting of hard-to-classify voxels. This methodology enhances the model’sability to classify both high and low-uptake lesion voxels and backgroundvoxels better, addressing the heterogeneity challenge in lesion segmentation.The proposed loss function, L1-weighted Dice Focal Loss, yielded a goodbalance between sensitivity and specificity, ensuring clinically relevant seg-mentations.This work also quantifies clinical metrics integral to patient prognosis915.2. Future Workand treatment planning. Metrics such as SUVmean and SUVmax, repre-senting average and maximum uptake intensities, respectively, are criticaltumor activity indicators that help in the risk stratification and prognosisof metastatic castration-resistant prostate cancer. TMTV, an indicator ofoverall tumor burden, has been reported to correlate with overall survival.Moreover, the combination of SUVmean with tumor volume, TLA repre-sents the total molecular tumor burden and is relevant in predicting patientoutcomes and disease progression. Besides, Dmax quantifies the maximumlesion spread and can provide information for staging, prognosis, and ther-apeutic decisions, such as the choice between systemic or localized therapy.Lesions’ count and anatomical distribution of tumors further correlate withbiochemical progression, underlining their prognostic significance.For the quantification of clinical metrics, the proposed loss functiondemonstrated equivalence to the ground truth method on TLA, lesion count,SUVmean, and SUVmax. Additionally, it yielded minimum overestimationand underestimation errors, ensuring consistency for various lesion scenariosand architectures. This work addresses a critical gap in lesion segmentation,providing automated segmentation outputs that align closely with groundtruth quantification, contributing to improved clinical decision-making andpatient outcomes. The ability of the proposed method to balance sensitivityand specificity ensures that prostate cancer lesions are accurately identifiedand quantified for downstream tasks like risk stratification and prognostica-tion.5.2 Future WorkDespite the contributions of this work to the field of prostate cancer lesionsegmentation, it has a few limitations. One significant limitation was thatthe dataset included up to six lesions per image. This narrow focus did notallow for an assessment of how well the proposed L1-weighted Dice FocalLoss (L1DFL) and other loss functions generalize to higher tumor burden asthe segmentation task could be more difficult with the lesions overlappingor being located closely. How the loss functions perform in such scenariosremains open. Also, this work focused solely on multimodal PET/CT imag-ing acquired from prostate cancer patients. Consequently, the results mightnot generalize well to other types of cancers or imaging modalities. More-over, the current study did not explore different hyperparameter settings,such as the number of bins and bin width, to evaluate which may affect therobustness of the loss functions.925.2. Future WorkFuture work should address these limitations by extending the scopeof research. The performance of L1DFL in datasets with a higher tumorburden, including cases with more lesions, would be of utmost importanceto understand the scalability and robustness of the proposed loss function.Additionally, datasets from other cancer types, such as lymphoma, couldbe included, along with different radiotracers, to test the generalizabilityof the approach across diverse pathological and radiopharmaceutical sce-narios. Such a mixed-pathology dataset would further challenge the modelto differentiate between distinct tumor types and provide a more thoroughassessment of its robustness.Another promising direction could be exploring the applicability of L1DFLbeyond PET/CT imaging. Investigation into its performance on other imag-ing modalities, such as PSMA SPECT [167], MRI and ultrasound, wouldexhibit its versatility and probable extension into broader clinical usage. Ex-tending this work to include advanced network architectures, such as trans-formers, could indicate whether these models can leverage the adaptive voxelweighting strategy in L1DFL to improve segmentation performance further.Also, additional work could investigate segmentation-free outcome pre-diction using L1DFL. Instead of relying on voxel-level delineation of le-sions, models could directly use imaging data to predict clinically relevantoutcomes such as tumor burden, progression risk, or response to therapy[349, 350]. This approach could streamline clinical workflows and minimizeerrors associated with segmentation while maintaining clinical relevance. Byfocusing on high-level predictions, this methodology would complement thesegmentation-focused application of L1DFL, further expanding its clinicalutility.In addition to segmentation-free methodologies, training frameworks likemulti-angle maximum intensity projections (MA-MIPs) offer another promis-ing avenue. The MA-MIPs approach involves creating multiple 2D maxi-mum intensity projection views of a 3D PET/CT scan from various angles,which are then used for model training. This technique reduces the com-putational complexity of handling 3D data while retaining critical lesioninformation. Our group has shown improved performance in tumor detec-tion and segmentation tasks from PSMA PET images due to the simplifiedyet informative representations of lesions across different views [225, 339].Integrating L1DFL with the MA-MIPs framework could enhance both thedetection and segmentation of tumors by leveraging the adaptive focusingcapabilities of the loss function alongside the efficient training strategy ofMA-MIPs.Another direction for future work involves incorporating region-specific935.2. Future Workassessments of segmentation performance. Evaluating how well the lossfunction and models perform across specific anatomical regions, such asthe bones, visceral organs, and lymph nodes, would provide insights intocontext-specific strengths and limitations. This type of assessment couldidentify areas where targeted improvements are necessary, enhancing theoverall reliability of the method. Similarly, the inclusion of anatomical pri-ors, such as segmentations of healthy organs, as additional model inputscould provide further context during training [298], potentially improvingthe accuracy of tumor segmentation by incorporating structural informa-tion about the surrounding anatomy.Lesion proximity to the bladder is another factor that may influencesegmentation performance. Some studies have reported that lesions locatedfurther away from the bladder are more challenging to detect. However, oncethey are detected, the model is better at contouring these than the lesionsadjacent the bladder [307]. Future studies could systematically evaluatethe robustness of L1DFL in detecting and segmenting lesions based on theirdistance from the bladder. Such an analysis could inform further refinementsto the loss function, optimizing its performance in anatomically challengingscenarios.Beyond imaging data, incorporating clinical and pathological factors,such as Gleason scores, PSA levels, PSA doubling times, prior treatments,time to recurrence, and hormone sensitivity, into segmentation and predic-tion models could improve their clinical utility. Combining these factorswith imaging data would create a multimodal framework capable of provid-ing more comprehensive insights into disease progression, treatment plan-ning, and prognosis. Similarly, integrating molecular data, such as geneticmarkers, could further enhance model predictions by aligning imaging-basedinsights with biological characteristics of the disease.Lastly, further research should systematically explore different hyperpa-rameter settings that may affect the model’s performance. 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Physics in Medicine andBiology, 49(19):4543–4561, October 2004.136Appendix AAppendicesA.1 Mathematical Description of Clinical metricsBelow are the mathematical descriptions of the various clinical metrics thatwere quantified.SUVmean: SUVmean refers to the mean standardized uptake value withina segmented lesion and is representative of the activity of the lesion. It isconsidered an indicator of aggressiveness and response to therapy. We cal-culated the patient-level SUVmean of a segmented lesion as follows:SUVmean =∑i Pi ·Mimax (∑iMi, 1)(A.1)where Pi is the standardized uptake value at voxel i from the 3D PET image,Mi is the binary mask value at voxel i with a value of 1 if the voxel is partof a lesion, and 0 otherwise.SUVmax: SUVmax is the maximum standardized uptake value in a seg-mented lesion and corresponds to the most active part of the tumor. It allowsfor the identification of hotspots and may help guide medical interventions.The patient-level maximum SUV (SUVmax) is given by:SUVmax = max (Pi ·Mi) (A.2)where Pi is the standardized uptake value at voxel i from the 3D PET image,Mi is the binary mask value at voxel i with a value of 1 if the voxel is partof a lesion, and 0 otherwise.TMTV: TMTV is the sum of the molecular tumor volumes across all le-sions in an image. It is indicative of the whole tumor burden; poor prognosisis generally associated with higher TMTV. TMTV in cubic centimeters isgiven by:TMTV =(∏3d=1 spacingd1000)×∑iMi (A.3)137A.2. Validation and Test Set Segmentation Performancewhere∏3d=1 spacingd is the product of the voxel spacings along the three axes(x, y, z) in millimeters (mm), giving the voxel volume in cubic millimetersand converted to cubic centimeters by the division term to obtain,Vvoxel,∑iMi is the total number of lesion voxels.TLA: TLA corresponds to the sum of the product of molecular tumorvolume and SUVmean for all lesions and represents the total lesion activity ofthe tumor burden. TLA incorporates tumor volume and molecular activityinformation, providing an overall quantitative measure of the disease andtherapeutic response. The total lesion activity (TLA) is given by:TLA =n∑i=1SUVmean,i × Vi (A.4)where SUVmean,i is the mean SUV of lesion i, Vi = Vvoxel×Nvoxels,i, Nvoxels,iis the number of voxels in lesion i and Vvoxel is the voxel volume in cubiccentimeters.Dmax: Dmax describes the maximum distance between two lesion voxelsand indicates the spatial extent of the tumor spread. Its value may af-fect staging, prognosis, and therapy choice, whether systemic or localized.Considering the voxel spacing, it is calculated as the maximum Euclideandistance between any two voxels in the lesion masks. Mathematically, thiswas expressed as:Dmax =maxi,j ∥(v⃗i − v⃗j) · s⃗∥10, (A.5)where v⃗i and v⃗j denote the coordinates of voxels i and j, s⃗ represents thevoxel spacing in millimeters, and the factor of 10 converts millimeters tocentimeters.Lesion Count: The number of lesions was determined using connectedcomponent analysis on the binary lesion mask. Each connected region in themask was identified as a separate lesion, and the total number of connectedcomponents was reported as the lesion count.A.2 Validation and Test Set SegmentationPerformanceThe table below reports the segmentation performance of the different lossfunctions and model architectures on the validation and the test sets. We138A.3. Open-source SPECT Reconstruction Modulereport the mean and median DSC scores as well as the true positive, falsepositive, and false negative rates as well as the F1 scores.Validation Set Test SetModel Mean Dice Score Mean Dice Score Median Dice Score Mean TP Mean FP Mean FN Mean F1 ScoreU-NetDL 0.48 ± 0.02 0.54 ± 0.29 0.62 [0.36, 0.78] 0.76 ± 0.35 0.73 ± 0.89 0.24 ± 0.35 0.63 ± 0.31DCE 0.49 ± 0.02 0.57 ± 0.29 0.67 [0.44, 0.80] 0.77 ± 0.36 0.87 ± 1.24 0.23 ± 0.36 0.63 ± 0.33DFL 0.49 ± 0.03 0.55 ± 0.29 0.62 [0.38, 0.79] 0.75 ± 0.38 0.70 ± 1.02 0.25 ± 0.38 0.63 ± 0.35L1DFL 0.49 ± 0.03 0.56 ± 0.29 0.66 [0.48, 0.77] 0.73 ± 0.39 0.47 ± 0.68 0.27 ± 0.39 0.65 ± 0.34SegResNetDL 0.49 ± 0.03 0.61 ± 0.25 0.68 [0.49, 0.80] 0.77 ± 0.36 0.54 ± 1.00 0.23 ± 0.36 0.70 ± 0.34DCE 0.50 ± 0.02 0.60 ± 0.27 0.68 [0.49, 0.80] 0.80 ± 0.33 0.56 ± 0.79 0.20 ± 0.33 0.71 ± 0.30DFL 0.50 ± 0.02 0.60 ± 0.26 0.65 [0.51, 0.79] 0.79 ± 0.34 0.70 ± 0.98 0.21 ± 0.34 0.68 ± 0.33L1DFL 0.52 ± 0.03 0.61 ± 0.28 0.71 [0.52, 0.81] 0.79 ± 0.36 0.31 ± 0.48 0.21 ± 0.36 0.72 ± 0.33Attention U-NetDL 0.48 ± 0.01 0.59 ± 0.26 0.66 [0.47, 0.80] 0.82 ± 0.34 1.09 ± 1.30 0.19 ± 0.34 0.62 ± 0.31DCE 0.50 ± 0.03 0.58 ± 0.27 0.65 [0.44, 0.78] 0.77 ± 0.36 0.69 ± 0.93 0.23 ± 0.36 0.65 ± 0.33DFL 0.49 ± 0.02 0.59 ± 0.27 0.64 [0.50, 0.81] 0.79 ± 0.34 0.84 ± 1.07 0.21 ± 0.34 0.66 ± 0.30L1DFL 0.52 ± 0.03 0.60 ± 0.28 0.67 [0.54, 0.79] 0.78 ± 0.37 0.30 ± 0.62 0.22 ± 0.37 0.73 ± 0.35Table A.1: Comparison of the loss functions (Dice Loss (DL), Dice Cross-Entropy Loss (DCE), Dice Focal Loss (DFL), L1-weighted Dice Focal Loss(L1DFL)) based on a five-fold validation, and on the test set. We reportperformance based on mean and median patient-level DSC. The mean DSC,TP, FP, FN and F1 scores are presented with standard deviations and themedian DSC with interquartile ranges. A detection is considered a true pos-itive if the predicted mask overlaps with the voxel containing the SUVmaxvalue in the ground truth.A.3 Open-source SPECT Reconstruction ModuleA.3.1 SummarySlicerSPECTRecon is a 3D Slicer [351] extension designed for Single PhotonEmission Computed Tomography (SPECT) image reconstruction. It offersa range of popular reconstruction algorithms and requires raw projectiondata from clinical or Monte Carlo simulated scanners. Built with the Py-Tomography Python library [120], it features GPU-accelerated functionalityfor fast reconstruction. The extension includes a graphical user interface forthe selection of reconstruction parameters, and reconstructed images can bepost-processed using all available 3D Slicer functionalities.A.3.2 Statement of needSPECT imaging is used to measure the 3D distribution of a radioactivemolecule within a patient; it requires (i) acquisition of 2D projection im-139A.3. Open-source SPECT Reconstruction Moduleages at different angles using a gamma camera followed by (ii) the use ofa tomographic image reconstruction algorithm to obtain a 3D radioactiv-ity distribution consistent with the acquired data. While there exist manycommercial products for image reconstruction, as well as multiple low-levelsoftware packages [120, 352, 353], there is currently no open source graphicuser interface (GUI) that provides access to the latest reconstruction algo-rithms. Due to continuing research interest in the implications of differentreconstruction techniques on various clinical tasks [354, 355], there is a needfor a open-source GUI for SPECT reconstruction.A.3.3 Overview of SlicerSPECTReconTypical reconstruction algorithms in nuclear medicine attempt to maximizea likelihood function viaxˆ = argmaxx;AL(y|x,H, s) (A.6)where xˆ is the 3D image estimate, A is the reconstruction algorithm, L is astatistical likelihood function that describes the acquired data y,H is a linearoperator that models the imaging system, s is an additive term that cor-rects for scatter, and argmaxx;A signifies maximization of the likelihood withrespect to x using algorithm A. Based on this formalism, SlicerSPECTRe-con is partitioned into four main components: input data, system modeling,likelihoods, and reconstruction algorithms. There are additional modulesfor input data conversion as well as post reconstruction applications.Once the raw data is loaded into the 3D slicer interface, the user canselect the projection data y to reconstruct in the Input Data tab. Data frommultiple bed positions can be selected at this stage; resulting reconstructionswill contain a single 3D image where all the bed positions are stitched. Uponselection of the projection data, necessary information about the acquiredenergy windows becomes stored internally. Users then select which energywindow they want to reconstruct via the Photopeak option; it is possible toto reconstruct data from multiple photopeaks simultaneously provided therelative weighting between calibration factors is provided for each peak.The System Modeling component considers the system matrix H, as wellas additional corrections such as scatter s. Selecting Attenuation Correctionenables attenuation correction during reconstruction; users must specify acorresponding CT image used to generate a mu-map. Selecting CollimatorDetector Response Modeling enables modeling of the collimator and detectorspatial resolution in image reconstruction. Users must specify the collima-tor code, defined on the PyTomography data page, as well as the intrinsic140A.3. Open-source SPECT Reconstruction Modulespatial resolution of the scintillator crystals. Selecting Scatter Correctionenables scatter correction during reconstruction; users must select the scat-ter correction method as well as supporting data required for the method.Figure A.1: Left: user interface for the proposed extension. Right: recon-structed coronal slice from a patient receiving [177Lu]Lu-PSMA-617 radio-pharmaceutical therapy (color) overlaid on a corresponding CT (greyscale).Raw SPECT data consisted of two bed positions that were automaticallystitched together after each was reconstructed; the raw data was acquiredon a GE Discovery 670 camera.The third component, likelihoods, considers the likelihood function L.Currently, the extension only supports the PoissonLogLikelihood likelihood,which correctly describes the data acquired in SPECT imaging. It may bedesirable in the future to test alternative likelihood functions, so this is leftas a separate module.The fourth component, algorithms, considers the reconstruction algo-rithm A. Currently, the extension supports the ordered subset expectationmaximum (OSEM, [356]), block sequential regularized expectation maxi-141A.3. Open-source SPECT Reconstruction Modulemum (BSREM, [357]) and ordered subset maximum a posteriori one steplate (OSMAPOSL, [358]) algorithms. Regularized algorithms can use thequadratic, log-cosh, and relative difference penalty [359] priors; these priorscan also utilize a provided anatomical image to modify the weighting by onlyincluding contributions from the top N neighbours. Additional algorithmsmay be added based on community request.The Data Converters component permits the conversion of raw SPECTdata from various file formats, such as SIMIND [360] and GATE [361], intoDICOM format so it can be loaded using the native data readers of 3DSlicer. While the extension currently has support for the conversion of SI-MIND data, more data converters may be added in the future depending oncommunity request. The Post-Reconstruction component contains function-ality that is applicable for use on reconstructed images. One such exampleis the computation of uncertainty on total counts within segmented regionsof interest. Figure A.1 contains a screenshot of the extension along with asample reconstructed image in the 3D Slicer viewer.142
Thesis/Dissertation
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Biomedical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Rahmim, Arman
Automated detection and segmentation of metastatic prostate cancer lesions in PET/CT images via deep neural networks towards accurate and routine quantification of clinical metrics
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