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Linking radiomic PET features with metabolic tissue parameters using a hybrid mathematical model of tumor… Ahn, Hailey S. H. 2021-04

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Linking radiomic PET features with metabolic tissueparameters using a hybrid mathematical model of tumorgrowthbyHailey S.H. AhnA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFBachelor of Science in Honours BiophysicsinTHE FACULTY OF SCIENCE(Physics and Astronomy)The University of British Columbia(Vancouver)April 2021© Hailey S.H. Ahn, 2021AbstractTumor heterogeneity can be observed between and within tumors through med-ical imaging such as positron emission tomography (PET). Heterogeneity arisesdue to the genetic diversity in cancer cell populations and the dynamic microenvi-ronments. Understanding the relationship between tumor tissue microparametersand quantitative PET radiomic features can offer a better strategy for caner diag-nosis and treatment. Our goal was to develop a multiscale mathematical modelfor realistic tumor growth in vascularized tissue, and to generate synthetic PETimages from the simulated images to study this relationship. The hybrid mathe-matical model used in the simulation combines an agent grid and partial differ-ential equations to model the dynamic tumor microenvironments. The status ofthe cell and its behaviour is determined by the local concentration of oxygen andglucose which diffuse from the vessels to tissue. The simulated cell maps wereconverted to synthetic PET images by translating the spatial locations of the cellsto the corresponding pseudo-standardized tracer uptake values of the PET tracer18F-fluorodeoxyglucose, which are unique to each cell type. Using different com-binations of tissue microparameters, we were able to generate tumors with distinctphenotypic profiles that were visually distinguishable in the translated syntheticPET images. Four radiomic features were computed from the resulting images andthis demonstrated that unique tumor phenotypes can be linked to radiomic PET fea-tures. Moreover, the identified optimal radiomic features can be used as biomarkersfor tumor assessment.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . 12 Theory and Background . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Cancer Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Normal and Tumor Cell Metabolism . . . . . . . . . . . . 62.1.2 Tumor Vascular Network and Alterations . . . . . . . . . 82.1.3 Hypoxia . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.4 Apoptosis and Necrosis . . . . . . . . . . . . . . . . . . . 92.2 Positron Emission Tomography . . . . . . . . . . . . . . . . . . . 112.2.1 Radiotracer . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Signal Detection and Reconstruction . . . . . . . . . . . . 122.3 Radiomic Features . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Image Acquisition and Segmentation . . . . . . . . . . . 152.3.2 Radiomic Features Extraction . . . . . . . . . . . . . . . 152.3.3 Features Analysis . . . . . . . . . . . . . . . . . . . . . . 19iii2.4 Hybrid Mathematical Modeling . . . . . . . . . . . . . . . . . . 202.4.1 Agent Grid . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Partial Differential Equation Grid . . . . . . . . . . . . . 203 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Tumor Growth Model . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Types of Agents . . . . . . . . . . . . . . . . . . . . . . 233.1.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . 243.2 Translating Simulated Images to PET Images . . . . . . . . . . . 253.2.1 Advantages of Using Synthetic PET Images . . . . . . . . 273.3 Radiomics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 284 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1 Agent Grid + Oxygen PDE Grid . . . . . . . . . . . . . . . . . . 294.2 Agent Grid + Oxygen and Glucose PDE Grids . . . . . . . . . . . 305 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40ivList of TablesTable 2.1 Semantic and agnostic radiomic features . . . . . . . . . . . . 19Table 3.1 Model biological parameters . . . . . . . . . . . . . . . . . . 24Table 3.2 MATLAB functions for radiomics analysis. . . . . . . . . . . . 28Table 4.1 Radiomic feature values for 1cm tumors. . . . . . . . . . . . . 31vList of FiguresFigure 1.1 PET/CT image of a pancreatic cancer patient. Demonstra-tion of how the CT scan (A) and a PET scan (B) is integratedto provide a high resolution functional and anatomical image(C). The merged image allows localization of the FDG uptakevalues in the PET scan. Figures adapted from [5]. . . . . . . . 2Figure 1.2 Identification and validation of radiomic biomarkers forprecision medicine. There are two ways of developing imag-ing biomarkers: extraction of hand-crafted features and data-driven deep learning models. The identified features an be usedfor patient-specific decision making and prediction of clinicaloutcomes. Figure from [19] . . . . . . . . . . . . . . . . . . 4Figure 2.1 Tumor and normal cell metabolism. The three cellular metabolicpathways are illustrated. Tumor cells exhibit modified metabolismthat utilizes the aerobic glycolysis pathway, which is called theWarburg effect. Figure from [35] . . . . . . . . . . . . . . . . 7Figure 2.2 Aerobic, hypoxic, and necrotic regions with distance fromthe blood vessel. Cells that are in close proximity of the ves-sel are viable aerobic cells. However, with increasing distancefrom the vessel hypoxic and necrotic cells arise. Figure from [3]. 9Figure 2.3 Survival curve of normal tumor and hypoxic tumor cells.Hypoxic cells exhibit reduced radiation sensitivity and requiresa much higher radiation dose to achieve the equivalent effec-tiveness as in oxic cells. Figure from [27] . . . . . . . . . . . 10viFigure 2.4 Radioisotope decay and annihilation event. The radioactiveatom attached to the radioisotope goes through a beta decayand produces a positron (β+). The positron travels a few mil-limeters until it loses most of it kinetic energy and annihilateswith a nearby electron (β−). The annihilation event producesa pair of 511keV photons in opposite directions. If the twophotons are observed by the scintillation detectors within thecoincidence time window, the annihilation event is recorded.Figure from [29] . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.5 Positron emission tomography signal detection and imageconstruction. (a) The red circle represents one pixel of thePET image where coincident events are recorded by the detec-tors along the lines of response (LORs). The angle and dis-placement of the LORs are measured with respect to the redhorizontal line and the center of field of view. (b) A sinogramwhere the four LORs from panel (a) are added up over thescanning period. For example, LOR4 is along the center, sothe displacement is 0 and it makes a right angle with the hori-zontal line, so the angle is 90◦. The total counts along the fourLORs are stored in each position of the sinogram. (c) A sino-gram of all pixels in the image plane and darker spots indicategreater number of counts. Figure from [29] . . . . . . . . . . 14Figure 2.6 Radiomics analysis pipeline. (a) Image acquisition via com-puted tomography. (b) Image segmentation is performed onthe lesion to identify the ROI. (c) Radiomic features are ex-tracted from the ROI. The three feature categories are graylevel patterns, inter-voxel relationships, and shape. (d) Anal-ysis and classification of a subset of selected features. (e) Se-lected features are used as biomarkers in diagnosis and patientprognosis. Figure from [30] . . . . . . . . . . . . . . . . . . 16viiFigure 2.7 Gray level co-occurrence matrix computation. (a) Showshow adjacency is defined given a pixel of interest (red). Thefour possible angles are labeled and any choice of distance canbe used to compute the GLCM. (b) An example of how animage matrix is translated to a GLCM matrix. The parametersused are D=1 and θ = 0◦. Figure adapted from [25] and [4]. . 18Figure 2.8 Hybrid mathematical modelling with agent and PDE grids.The illustration shows one agent grid holding an agent (yellowdot) and two PDEs for oxygen and glucose. All grids can in-terface easily and the agent can retrieve the information fromthe corresponding pixels in the PDEs (yellow border). . . . . 21Figure 3.1 Tumor growth simulation flowchart. The flowchart showsthe step sequence of how the simulation is initialized, howagents determine their state, and how the PDE grids function(diffusion and consumption of molecules) . . . . . . . . . . . 26Figure 3.2 Moore neighbourhood. Represents the eight locations sur-rounding the center pixel, in this case an agent, that border anedge or a corner of the cell. The arrows indicate the possiblelocations an agent can divide into in a Moore neighbourhood.Figure adapted from [9] . . . . . . . . . . . . . . . . . . . . 27Figure 4.1 Simulated tumors with distinct tumor phenotypes. The modelwith different combinations of tumor tissue microenvironmentswas able to produce three tumor types with distinct tumor phe-notypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 4.2 Translation of the Type C tumor longitudinal growth simu-lation to PET images. Simulated images were translated intosynthetic PET images by converting the cell map into corre-sponding pSUV values. . . . . . . . . . . . . . . . . . . . . . 31Figure 4.3 Biological time for tumor growth. The cell count was plottedagainst the biological time to compare the growth rate of eachtumor type. The tumors were grown up to ∼ 1cm in diameter. 32viiiFigure 4.4 Radiomics analysis on synthetic PET images. Four PET ra-diomic features were plotted against simulated biological timefor tumor growth. The mean and standard deviations of thedata points were calculated using three separate tumor growthsimulation rounds. The error bar represents one standard devi-ation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 4.5 Two distinct tumor phenotypes simulated with oxygen andglucose PDE grids. The tumor progression is shown at threedifferent time points (biological hours) until it reached 1cm indiameter. The bottom panels represent the magnifications ofthe regions indicated with the white box. . . . . . . . . . . . . 34ixAcknowledgmentsI would like to express my sincere gratitude towards my supervisors Dr. IvanKlyuzhin and Dr. Arman Rahmim from the Quantitative Radiomolecular Imaging& Therapy (QURIT) Lab at the BC Cancer Research Institute for the opportunityto work on this project. Their support and advice were what made this project pos-sible and engaging despite of the remote nature of the work. I am also immenselygrateful for my parents and my brother for their wholehearted support throughoutmy undergraduate degree. Many thanks to my dear friends as well, for shaping methe way I am today.xChapter 1Introduction and MotivationCancer refers to a class of diseases characterized by cells that proliferate at un-controlled rates and spread throughout the body resulting in destruction of normaltissue. In fact, cancer accounted for about 9.6 million deaths in 2018, making itthe second leading cause of death globally [7]. Benign tumors turn into malignanttumors as they acquire hallmarks for cancer progression, which includes resistanceto programmed cell death (apoptosis), invasion of adjacent tissue, and induction ofnew blood vessel growth (angiogenesis). These hallmarks along with the dynamictumor tissue microenvironments contribute to various phenotypes that introducecomplications for cancer treatment.Early detection and treatment for cancer have shown to effectively prevent pro-gression of tumor which is crucial for patient survival. The most widely usedmethod for cancer diagnosis is medical imaging and there are various modalitiesthat allow physicians to detect abnormal properties of cancerous tissue such as in-creased metabolic activity or appearance without a surgical incision. The imagingmethods are classified into three different categories: anatomic, functional, andmolecular imaging. The typical resolution for medical images varies between eachmode: 1mm for MRI and CT; 2-5mm for PET. Oftentimes anatomical and func-tional imaging methods are used together to acquire certain information that cannotbe obtained via one imaging modality.• Anatomical Imaging provides detailed structural information about the anatomythat yields high resolution clinical images. Magnetic resonance imaging1(MRI) and computed tomography (CT) imaging are examples of anatomi-cal imaging modalities.• Functional Imaging highlights physiological information including metabolismand local chemical composition. These imaging modalities include positronemission tomography (PET) and single photon emission computed tomogra-phy (SPECT) which provide information on tumor biological functions andmicroenvironment at a limited spatial resolution. Oftentimes, functional andanatomical images are used side by side for an integrated visualization oftumor biology with high resolution imaging (Figure 1.1).• Molecular Imaging is used to obtain detailed pictures of human body ata molecular/cellular level. It provides functional information without theneeds for more invasive procedures such as biopsy or surgery.Figure 1.1: PET/CT image of a pancreatic cancer patient. Demonstrationof how the CT scan (A) and a PET scan (B) is integrated to provide ahigh resolution functional and anatomical image (C). The merged imageallows localization of the FDG uptake values in the PET scan. Figuresadapted from [5].2There are four conventional types of cancer treatment: chemotherapy, radia-tion therapy, immunotherapy, and surgical removal of tumors. Surgical removal isonly applicable when the tumor is accessible and has not metastasized. However,metastasis is prevalent in most cancers and requires other treatments involving highdose radiation that also kills healthy cells. The current strategy applied to cancertreatment is ”one-size-fits-all”, meaning that it has been standardized to be applica-ble to most people. However, this treatment approach is only effective in a subsetof people due to the complexity in tumor intrinsic properties and progressions pat-terns that should be targeted differently [23]. In fact, a study performed in 2017estimated that any particular category of standardized cancer drugs is ineffectivein 75% of the patients [1]. The inherent variability in cancer led to the growingfield of precision medicine which includes cancer genomics studies and radiomicsanalysis on medical images. Careful extraction and selection of meaningful infor-mation from these patient-specific approaches enable healthcare professionals todeliver customized treatment for increased efficiency and survival rates.One of the emerging fields of precision medicine is radiomics, which is amethod of converting medical images into high dimensional data with numerousquantifiable features. Analysis of these features can elucidate the underlying patho-physiology captured by medical images, revealing tumor characteristics that canassist cancer diagnosis, prognosis, and the development prediction models. Quan-titative features based on size, shape, texture, and intensity can provide uniqueinformation about tumor heterogeneity and its microenvironments that cannot beobtained through other laboratory techniques [15]. There are hundreds of met-rics that have been extracted from the images with the potential to reveal tumorcharacteristics that cannot be identified with human eyes [2]. The features can becarefully selected and used as biomarkers for patient-specific treatment planningand prognosis (Figure 1.2).Tumor heterogeneity refers to observable variations in tumor phenotype and itcan be observed through medical imaging or histopathological studies. Examplesof heterogeneity are distinct morphologies, gene expression patterns, and sensitiv-ity to treatment. Many research have shown that more heterogeneous tumors tendto exhibit more aggressive growth and resilience to treatment, leading to poor pa-tient outcomes [18]. Additionally, tumors with similar characteristics have shown3Figure 1.2: Identification and validation of radiomic biomarkers for pre-cision medicine. There are two ways of developing imaging biomark-ers: extraction of hand-crafted features and data-driven deep learningmodels. The identified features an be used for patient-specific decisionmaking and prediction of clinical outcomes. Figure from [19]similar progression patterns regardless of the location of the tumor [14, 17, 28].Thus, understanding how tumor tissue microparameters are linked to observablefeatures in medical imaging is of great interest. Unfortunately, we currently don’thave a clear understanding of how specific tumor phenotypes are reflected in clin-ical images. Thus, our goal is to develop a multiscale mathematical model of re-alistic tumor growth in vascularized tissue, and use the simulated images to pro-duce synthetic PET images showing metabolic profiles of the tumor. Performingradiomics analysis on these synthetic PET images allows us to establish a link be-tween specific tumor phenotypes and tissue microparameters. More accurate anddetailed characterization of the tumors using this approach enables optimized treat-ment based on individual tumor characteristics.4Chapter 2Theory and Background2.1 Cancer BiologyCancer cells proliferate in uncontrolled manners, unlike normal cells with sophis-ticated balance between proliferation and differentiation until programmed celldeath. Advances in cancer research revealed that cancer cells have mutations intheir cell genome that disrupt this balance resulting in uncontrolled growth. Whencancer is diagnosed in a patient, it has typically developed into solid tumor, whichis a collection of of many components such as cancer cells, healthy cells, bloodvessels, and fibroblasts. The current speculation of a solid tumor is that it origi-nates from a single tumor cell that has accumulated genetic mutations [10]. Thiscell proliferates forming a benign lesion, which progressively develops into malig-nant tumors with interactions with environmental factors such as carcinogens. Amalignant tumor is capable of growing into normal tissue until it breaks throughthe basal membrane, which causes the cancer cells to spread to distant locationsvia a process called metastasis [39].Despite of the uncontrolled proliferation patterns of tumor cells, their behaviouris dependent on the interactions with the tumor tissue microenvironments. It relieson external factors like signals from surrounding cells and the nutrient availability.This motivates the evaluation of solid tumors and their surrounding microenviron-ment as a whole, which allows us to observe the phenotypic variations in complextumor tissue. Progression of solid tumors leads to drastic changes in the tumor5environment; for example, growing tumors can alter vascular networks by enhanc-ing or inactivating vessels, invade into normal tissue, and become resistant to thehost’s immune response. The interactions between genetic diversity in tumor cellsand modified tumor microenvironments give rise to both intra- and intertumoralheterogeneity. Previous studies have shown that tumors with similar phenotypicprofiles show similar progression patterns and patient outcomes even if the tumorsare located in different parts of the body [14, 17, 28]. There is also evidence thatsupports the claim that more heterogeneous tumors are often more aggressive andresistant to cancer treatments [18]. Hence, understanding how tumor heterogeneityis related to tumor prognosis and image properties is crucial for patient diagnosisand treatment planning. The following sections will explain the causes and mech-anism of how heterogeneity arise in solid tumors.2.1.1 Normal and Tumor Cell MetabolismTumor cells proliferate at a much higher rate in comparison to normal cells duemutations in the genes that regulate normal regulation of cell division. Becausea cell division requires a lot of energy for biosynthesis of various proteins, cancercells typically consume glucose and oxygen at a much higher rate relative to normalcells. Blood is the source of oxygen and glucose which diffuses through the tissuefor cells to consume. Nutrient rich blood travels from the heart to different parts ofthe body through bulk motion and diffusion happens mostly at the capillary level.Capillaries are the ideal place for nutrients to diffuse out as the walls are made upof only a single layer of cells and the increased surface area with a typical diameterof ∼ 5−12µm facilitates the diffusion process [36].To understand cancer cell metabolism and how the different nutrient consump-tion rates arise, one needs to know the different metabolic pathways that cells taketo generate adenosine triphosphate (ATP). The three pathways are oxidative phos-phorylation (OXPHOS), anaerobic glycolysis, and aerobic glycolysis (Figure 2.1).A normal cell under oxygen saturated condition mainly utilizes the most efficientOXPHOS pathway for cellular metabolism, yielding 36ATP/glucose. Tumor cells,however, have shown to take the aerobic glycolysis pathway regardless of the pres-ence of oxygen in the microenvironment, which only yields 4ATP/glucose; this is6only about one-ninth of what can be harvested through OXPHOS [35]. One of themost common speculation for this phenomenon is that glycolysis is up to ∼ 100folds faster than that of OXPHOS, allowing cancer cells to produce much moreATP in a given time [20, 24]. However, this comes at a cost of increased glucoseconsumption due to the low efficiency of this metabolic pathway. This is called theWarburg Effect and it leads to the variations in glucose uptake rates that are mostconveniently observed through PET imaging.Figure 2.1: Tumor and normal cell metabolism. The three cellularmetabolic pathways are illustrated. Tumor cells exhibit modifiedmetabolism that utilizes the aerobic glycolysis pathway, which is calledthe Warburg effect. Figure from [35]If a cell were to go through both OXPHOS and aerobic glycolysis, the com-bined net chemical equation for aerobic respiration becomesG+5O2+29ADP+29P→ 29ATPwith glucose (G), oxygen (O2), adenosine diphosphate (ADP), and phosphate (P)7[31].2.1.2 Tumor Vascular Network and AlterationsThe capillary network in the human body widely varies between tissue types. Forinstance, the skin has a capillary density of∼ 20/mm2 [33] and the skeletal musclehas ∼ 100− 500/mm2 [26]. Since tumor metabolism and rapid proliferation areclosely associated with the supply of nutrients from the capillaries, where the tumoris located significantly influences the tumor progression patterns and the exhibitedphenotype. The available oxygen and nutrients decrease with distance from thecapillary. Hypoxia typically occurs at about 100µm from the vessel and necrosisoccurs at 150µm (Figure 2.2) [3]. Moreover, tumor progression results in a numberof vascular network alterations that give rise to observable heterogeneity in tumortissue. This includes angiogenesis and dilated capillaries up to 200µm in diameter[11]. On the other hand, with the rapid growth of solid tumors, host vessels areremodeled and pushed away, or even obstructed. Prolonged vessel occlusion causesnecrosis to occur, typically starting at the center of the tumor that spreads outwardsas more active vessels are removed by increasing cancer cell density [38]. Smallervessels are first to be closed, followed by occlusion of larger vessels in later stageswhich leads to complete necrosis of the tumor.2.1.3 HypoxiaHypoxia is one of the major contributors to tumor heterogeneity. It refers to a con-dition where a cell is deprived of oxygen supply due to increased oxygen consump-tion, inadequate oxygen diffusion from nearby blood vessels, or a combinationof both. This phenomenon arises in solid tumors either in a diffusion-dependent(chronic) manner or as acute hypoxia if the vessel in the local environment be-comes temporarily inactive. Thus, hypoxia itself is also a spatial and temporalheterogeneity and the dynamics are specific to the local neighborhood. It is alsoknown to induce adaptive processes such as angiogenesis, which is a process wheretumor tissue grows its own blood vessels to draw extra nutrients required for rapidproliferation. Moreover, hypoxia reduces the effectiveness of the treatment by re-duction of the oxygen effect. The current treatment for cancer is to target cancer8Figure 2.2: Aerobic, hypoxic, and necrotic regions with distance from theblood vessel. Cells that are in close proximity of the vessel are viableaerobic cells. However, with increasing distance from the vessel hy-poxic and necrotic cells arise. Figure from [3].cell DNA with high-energy radiation to cause damage and induce cell death. Here,oxygen acts as a radiosensitizer in radiotherapy by participating in chemical reac-tions that induce DNA damage. In fact, the direct effect of radiation to eliminatecancer cells only accounts for about 35%, and the other 65% is due to the oxygeneffect [27]. Thus, hypoxic tumors require 2-3 times higher radiation dose relativeto what oxygenated tumor tissue would require to be effective (Figure 2.3) [32].2.1.4 Apoptosis and NecrosisThere are two mechanisms for cell death, which are apoptosis and necrosis. Apop-tosis refers to programmed cell death where cells that are damaged beyond repaircommit to death at a certain point during their cell cycle. Cellular stimuli such asdepletion of growth factors and hormones can also induce apoptosis. Apoptosisis a delicate ATP dependent process which any amount of disruption in its regula-tion can lead to devastating conditions like altered immune system functions andcancer. During the cell cycle, there are multiple checkpoints that ensure the cellis viable and is able to go through another round of division. A cell cycle arrest9Figure 2.3: Survival curve of normal tumor and hypoxic tumor cells. Hy-poxic cells exhibit reduced radiation sensitivity and requires a muchhigher radiation dose to achieve the equivalent effectiveness as in oxiccells. Figure from [27]at one of the checkpoint initiates apoptosis which involves condensing of chro-matin and organelles [12]. These collapsed cells are engulfed by macrophages inthe body that digest the apoptotic cell and remove the waste products. The secondform of cell death is necrosis. Unlike apoptosis, necrosis is caused by externalstress, such as nutrient deficiency, and the process is independent of ATP availabil-ity. In general, there are two modes that trigger necrosis: cell death due to extremehypoxia or ATP production below a threshold level. Because apoptosis involvesmultiple ATP-dependent steps, necrosis is the predominant pathway for cell deathin a condition lacking ATP [12, 34]. A lump of necrotic cells called necrotic coreis a phenomena frequently observed in solid tumors. This is caused by the rapidgrowth of solid tumors which makes it increasingly difficult for sufficient amountof nutrients to reach its core. Necrosis is not programmed like apoptosis and theprocess is triggered when one of the two conditions mentioned earlier is sustainedfor a prolonged period of time, ranging from hours to days [12]. Along with hy-poxia, necrosis is one of the solid tumor characteristics that we modelled through10the tumor growth simulation.2.2 Positron Emission TomographyThis section will go over the underlying physics of positron emission tomography,a medical imaging method that falls under the functional imaging category. Most ofthe information comes from The Oxford Handbook of Functional Brain Imagingin Neuropsychology and Cognitive Neurosciences - Positron Emission Tomogra-phy: Blood Flow and Metabolic Imaging [29] and Nuclear Medicine Physics: TheBasics [8].2.2.1 RadiotracerRadiotracer refers to a molecule that is used up by organs during normal physiolog-ical processes with an additional radioactive atom attached. A small amount of thisradioactive substance is injected into a patient’s body to examine the metabolic ac-tivity of specific organs and tissue. A commonly used radiotracer for PET imagingis 18F-fluorodeoxyglucose (18F-FDG) which is an analog of 2-deoxy-D-glucosethat is taken up by cells for metabolism. The attached radionuclide is fluorine-18which undergoes β+, emitting a positron. Although fluorine is not a physiologi-cally relevant molecule, it is powerful for PET imaging purposes due to its half-lifeof ∼ 2 hours and the ease of substituting the hydroxyl group on glucose moleculeswithout disturbing its biological properties. The positron emitted from the transientradioactive decay travels for a few millimeters and annihilates with an electron inthe tissue, resulting in two photons that are emitted in opposite directions (Figure2.4).Cell proliferation costs large amounts of energy as it requires increased synthe-sis of macromolecules for a successful cell division. The increased energy demandand consequently increased glucose consumption leads to higher 18F-FDG uptakein tumors. A PET scan captures the change in metabolism, revealing spatial andtemporal information of the tumor tissue.11Figure 2.4: Radioisotope decay and annihilation event. The radioactiveatom attached to the radioisotope goes through a beta decay and pro-duces a positron (β+). The positron travels a few millimeters until itloses most of it kinetic energy and annihilates with a nearby electron(β−). The annihilation event produces a pair of 511keV photons in op-posite directions. If the two photons are observed by the scintillationdetectors within the coincidence time window, the annihilation event isrecorded. Figure from [29]2.2.2 Signal Detection and ReconstructionAn illustration of a PET scanner with a patient situated inside the ring of photon de-tectors is shown in Figure 2.5. Soon after a small amount of radiotracer is injected12into the patient’s body, the unstable proton inside the nucleus of a radioisotope gothrough a β+ decay, releasing a positron. It travels for a few millimeters untilit loses most of its kinetic energy and annihilates with an electron in tissue. Theannihilation event produces two 511keV photons traveling in opposite directions.When these two gamma photons are detected by the PET scanner within a 6-12nswindow, they are recorded as coincident events and the line of response (LOR)is defined. The LOR passes through the point of annihilation which eliminatesthe need for collimators, which is required in other imaging methods like single-photon emission computed tomography (SPECT) to prevent exposure from outsideof the region of interest.Inside each gamma camera, there are scintillators containing crystals (Lu2SiO5 :Ce or Bi4Ge3O12) that allows radiation detection. These crystals produce lightwhen high energy radiation is absorbed which is used to count the photons. Unlessthe pair of photons is emitted from the midpoint of the LOR, there exists a time de-lay between the detection of the two coincident photons. This is referred to as timeof light (TOF) and it enables localization of the annihilation event. The signals arecollected in the form of a sinogram (Figure 2.5 b,c) that needs to be corrected forsources of errors such as random coincidence events, scatter, and attenuation fora more accurate image reconstruction. The processed signal can be reconstructedthrough iterative or analytical methods.In PET imaging, the radiotracer uptake is assessed through a semi-quantitativemetric called standard uptake value (SUV). SUV is given by the equation:SUV =Tracer activity in tissueTotal injected dose per patient weight.The minimum, maximum, and mean SUV values can be computed from the regionof interest based on the individual values stored in each pixel of the image. Thecomputed values are used to asses the abnormal characteristics of tumor tissue,where malignant tumors typically have SUV values over 2.5-3 [21].13Figure 2.5: Positron emission tomography signal detection and imageconstruction. (a) The red circle represents one pixel of the PET imagewhere coincident events are recorded by the detectors along the lines ofresponse (LORs). The angle and displacement of the LORs are mea-sured with respect to the red horizontal line and the center of field ofview. (b) A sinogram where the four LORs from panel (a) are addedup over the scanning period. For example, LOR4 is along the center, sothe displacement is 0 and it makes a right angle with the horizontal line,so the angle is 90◦. The total counts along the four LORs are stored ineach position of the sinogram. (c) A sinogram of all pixels in the imageplane and darker spots indicate greater number of counts. Figure from[29]2.3 Radiomic FeaturesRadiomics is a powerful tool used to extract quantitative information from clini-cal images. Radiomics analysis converts digital images to minable data, offeringinsight into the relationship between image properties and cancer phenotypic pro-files. Due to the significant potential for elucidating unique tumor characteristics,it is an emerging field of research in medical physics with a main focus on de-veloping novel biomarkers for cancer diagnosis and prognosis. The developmentof machine-learning models and processes for high-throughput extraction of im-14age features pushed the studies in the direction of developing prediction modelsrather than finding biological meaning from the analysis. There have been effortsto reconnect the prediction models and the biological significance through differentapproaches such as genomics and microscopic image texture (histological) stud-ies. The four key steps of radiomics analysis are image acquisition, segmentation,quantitative image features extraction, and analysis of the features as shown inFigure Image Acquisition and SegmentationImages for radiomics analysis can be acquired through various modern imagingtechniques including CT, MRI, and PET. Numerical data can be extracted fromthese images, however, there must be a standardized image reconstruction protocolto remove variations in the intrinsic image properties that are not due to biolog-ical effects; an alternative method would be to introduce error bars to take thesemeasures into account. Identifying the volume of interest through image segmen-tation is fundamental in radiation oncology. The identified volume is called theregion of interest (ROI). There may be one or more tumor sites detected from oneimage and it is important to identify all suspected lesions as the volume of inter-est. This step is crucial since all subsequent processes depend on the segmentedvolume. Currently, images can be segmented manually by radiologists or by au-tomated computer-aided segmentation. The general consensus is that automatedsegmentation yields higher reproducibility due to the individual variability in man-ual segmentation.2.3.2 Radiomic Features ExtractionRadiomic features extraction from the segmented images is the essence of thispipeline. The features can be separated into two broad categories which are se-mantic and agnostic. A subset of features in each category are shown in Table 2.1.Semantic features are used to describe the regions of interest by visual assessmentof radiologists. Agnostic features on the other hand quantitatively capture tumorcharacteristics based on mathematical and statistical analysis. The latter can befurther divided into four classes described below:15Figure 2.6: Radiomics analysis pipeline. (a) Image acquisition via com-puted tomography. (b) Image segmentation is performed on the lesionto identify the ROI. (c) Radiomic features are extracted from the ROI.The three feature categories are gray level patterns, inter-voxel relation-ships, and shape. (d) Analysis and classification of a subset of selectedfeatures. (e) Selected features are used as biomarkers in diagnosis andpatient prognosis. Figure from [30]16• morphological features are used to define the shape of the defined ROI. Size,shape, sphericity, and compactness are a subset of features that fall underthis category. For instance, compactness of a 2D segmented image can becomputed using the perimeter to surface ratio of the ROI, given byperimeter to surface ratio=PA,where P is the perimeter and A is the area of ROI. A smaller ratio indicatesa more compact (circular) shape.• First-order statistical features describe the individual voxel without consid-ering the spatial relationship to the surroundings. A subset of first-order fea-tures includes histogram-based relationships that are summarized into singlevalues such as mean, maximum, minimum, asymmetry, and flatness. Forexample, the mean pSUV value of the region of interest can be calculated bythe following equation:mean=1NpNp∑i=1X(i),where Np is the number of voxels in the ROI and X is the set of pSUV valuesof the individual voxels in the ROI.• Second-order texture features characterize the texture of the ROI, or the spa-tial and statistical relationships between pixels. Texture analysis was firstintroduced by Haralick in 1973, which significantly enhanced the study ofintratumoral texture heterogeneity [16]. Gray level co-occurrence matrices(GLCMs) was introduced by Haralick as one of the earliest methods for tex-ture features extraction. This matrix (P) is an (N x N) square matrix withN possible number of gray levels from an image. The (i,j)th entry of P tellsus the number of times a pixel with intensity i is adjacent to that of j. Notethat adjacency is defined in four directions (right-horizontal, up-vertical, leftand right upper diagonal) as shown in Figure 2.7 (a). Several matrices canbe computed using different combinations of the distance (D) and angle (θ )parameters. An example of how the GLCM matrix is computed is shown17in Figure 2.7 (b), and the entries indicate how often a specific offset valueoccur between pixel pairs in the image.Figure 2.7: Gray level co-occurrence matrix computation. (a) Shows howadjacency is defined given a pixel of interest (red). The four possibleangles are labeled and any choice of distance can be used to computethe GLCM. (b) An example of how an image matrix is translated toa GLCM matrix. The parameters used are D=1 and θ = 0◦. Figureadapted from [25] and [4].Contrast is one of the Haralick features:contrast=Ng∑i=1Ng∑j=1(i− j)2p(i, j).The term p(i, j) refers to the normalized co-occurrence matrix given byP(i, j)∑P(i, j).In general terms, contrast indicates variations in local intensity, where highcontrast correlates to greater differences in intensity.Homogeneity is another example of an Haralick feature:homogeneity=Ng∑i=1Ng∑j=1p(i, j)1+ |i− j| .This value measures local homogeneity and a larger value indicates more18Table 2.1: Semantic and agnostic radiomic featuresSemantic features Agnostic features (Quantitative))Size Skewness (asymmetry)Shape Haralick texturesLocation WaveletsVascularity Laws texturesNecrosis Fractal dimensionsuniformity.• Higher-order statistical features elicit repetitive and non-repetitive patternsimposed on images with filter grids. Filter transform of images refers tomultiplying an image by a specific filter matrix. For instance, wavelet trans-form filters multiply the image by a matrix of complex waves to decomposethe image into details, and Laplacian transform filters extract coarse texturepatterns from the images.2.3.3 Features AnalysisThe main approach for radiomic feature analysis is through machine learning algo-rithms. The program learns from a given data set and becomes capable of identify-ing patterns with minimal human intervention when new sets of data are given. Theautomation of large quantity segmentation and extraction of image features enableda fast and efficient data-driven analysis of medical images. Over 1000 radiomicfeatures have been identified from numerous ROIs; however, the reproducibilityand significance of each feature have been achieved in only a small subset of thefeatures. Such a large complex data set may lead to over-fitting of the data; thus,identifying select features that are robust to noise and produce quality data is animportant step in radiomic analysis. Identification of ideal radiomic features en-ables important diagnostic, prognostic, and predictive information to be capturedfrom the images. Many suggest that will eventually become routine practice inclinics for a more precise diagnosis and patient-specific decision-making [15].192.4 Hybrid Mathematical ModelingThe tumor growth simulation model uses mathematical oncology which is a methodto study cancer using mathematics and simulation. The hybrid part of this model isthe use of both agent-based models and partial differential equation models (PDEs)to simulate realistic tumor microenvironments. PDEs store information, such asconcentration on each grid location, and use mathematical relationships to trackthe changes in values due to diffusion and consumption of molecules. Agents,namely autonomous decision-making entities, have access to the information fromthe PDEs which is used to determine the current state and the following actionin each iteration step. The two components are completely decoupled and areexecuted independently while allowing the exchange of information between thegrids. This allows any combination of agent grids and PDE grids for a flexiblehybrid modeling (Figure 2.8). The particular strength of the hybrid model is thatit provides insights into mechanistic feedback between the tumor cells and tumortissue environment.2.4.1 Agent GridThe agent grid in this simulation is used as a container for the agents, in this case,the different types of tumor cells. The specific grid used is a 2D on-lattice gridwhere the agents are bound to lattice locations rather than a continuous plane. Theagent occupies a pixel in the grid space and keeps track of its location as well as itscurrent state. The agents can sample the neighbouring locations to determine thelocal population and decide if it can proceed with cell division.2.4.2 Partial Differential Equation GridTumor tissue is dynamic and the tumor local microenvironments are constantlychanging. In our tumor growth model, we use 2D continuous partial differentialequations to simulate the concentration of various molecules in the biological sys-tem. The PDE grids can model the complex internal dynamics while updating theconcentration fields in each grid location to mimic realistic tumor tissue microen-vironments at a microscopic level. The dynamics include diffusion of nutrients andconsumption by cells on the agent-based model. Diffusion is run by adding deltas20Figure 2.8: Hybrid mathematical modelling with agent and PDE grids.The illustration shows one agent grid holding an agent (yellow dot) andtwo PDEs for oxygen and glucose. All grids can interface easily and theagent can retrieve the information from the corresponding pixels in thePDEs (yellow border).to the current field with a wrap-around (periodic) boundary condition. This meansthat the simulation grid is surrounded by identical translated copies of itself. Peri-odic boundary conditions are often used to simulate a small part of a larger tissueto minimize edge effects.The consumption of molecules follows the Michaelis-Menten (MM) kineticssince the consumption of molecules depends on the enzymatic turnover rate toharvest ATP. The chemical formula for this reaction isE+S⇀↽ ES−→ E+Pwhere E stands for an enzyme, S for substrate, and ES for the enzyme-substratecomplex. The consumption rate equation is given byf =−Vmax ∗ [S]KM+[S] ,21where f is the rate of reaction, Vmax is the maximum rate of the system, and KM isthe concentration of substrate at half the maximum rate. Glucose consumption isdependent on the ATP needs of the cell. Tumor cells have a greater energy demandto maintain their uncontrolled growth, requiring a larger Vmax. Thus, the tumorcells in the model haveVmax which is 10-20 times higher compared to normal cells.An assumption was made in the model that the oxygen consumption rate foreach cell type is constant. This is because KM for oxygen in tissue is significantlylower than the oxygen concentration in tissue (KM¡¡ S), simplifying the model tof =−Vmax ∗ [S]KM+[S] =−Vmax.The diffusion of molecules was modeled by Fick’s diffusion law, together with theconsumption of molecules. The equation is given by∂C∂ t= D∇2C+ fiThe parameters are concentration (C), time (t), diffusion coefficient (D), and con-sumption rate of molecules ( fi).Finally, knowing the consumption rates of oxygen and glucose, we can calcu-late the ATP production rate which will be used to partially determine the necro-sis behavior of the cells in this model. The total aerobic respiration yields ∼ 27ATP per glucose molecule as seen in section 2.1.1. Cancer cells that shift theirmetabolic pathway to glycolysis produce ∼ 2 ATP per glucose molecule. Thus theoverall ATP production rate is given byfATP =−(2 fG+ 27 fO5 ).22Chapter 3Methods3.1 Tumor Growth ModelWe model realistic tumor growth in vascularized tissue to study the relationshipbetween measurable PET radiomic features and distinct tumor phenotypes. Thesimulation consists of one agent grid and two partial differential equation grids.The agent grid holds the different cell types and cross-sectional capillaries. ThePDE grids have two main roles: tracking the concentration of nutrients at each gridlocation, and adding differentials (diffusion and consumption) of the moleculesuntil a steady state is reached. The two PDE grids in the model track the oxygen andglucose concentrations at each pixel. The full list of realistic biological parametersused in this model can be found in Table 3.1. All three grids in the model areexecuted independently while the agents have access to the PDE grid data at alltimes. This is especially important since the type of the agent and its behaviordepends on the concentration of nutrients, just like real cells in a biological system.The simulated grid size was 1000x1000 cells (2x2cm).3.1.1 Types of AgentsThe four types of agents in the model are normal tumor, hypoxic tumor, necrotictumor cells, and capillaries. The agent type is determined by the local concentra-tion of nutrients that can be tracked using the PDE grids. Due to the altered cellularmetabolism in cancer cells, each type of agent has its unique consumption rates for23Table 3.1: Model biological parametersBiological constant ValuesAverage cell (unit) size [µm] 20Average time for aerobic cell division [hours] 24Capillary density range [#/mm2] 20-100Oxygen diffusion coefficient in tissue [cm2/s] 1.65×10−5Glucose diffusion coefficient in tissue [cm2/s] 2.7×10−6Oxygen flux from capillary [mol/min/unit] 2.81×10−12Glucose concentration in capillary [mol/L] 5×10−3Oxygen consumption rate - normal tumor [mol/min/cell] 4×10−15Oxygen consumption rate - hypoxic tumor [mol/min/cell] 2×10−15Oxygen consumption rate - non-cancer cell [mol/min/cell] 2.5×10−18Glucose Vmax - normal tumor [mol/min/cell] 5×10−14Glucose Vmax - hypoxic tumor [mol/min/cell] 1.02×10−13Glucose Vmax - non-cancer cell [mol/min/cell] 5×10−15Glucose KM [mol/min/cell] 2×10−14Death ATP production rate (DPR) [mol/min/cell] 2.57×10−14nutrients; the oxygen consumption rate is constant for each cell type, whereas theglucose consumption rate depends on the local concentration of glucose. Capillar-ies were modeled as cross-sectional points and initialized with uniform density orcompletely randomized placements across the agent grid. In between each diffu-sion step, the nutrient concentrations in the capillaries were reset to the basal levelsin the blood, as the source of glucose and oxygen that diffuse through the tissue.3.1.2 Simulation DetailsAgents are able to determine its current state and the behaviour in the followinground of iteration by accessing the information on the PDE grid that tracks theoxygen and glucose concentrations. The decision-making process of an agent isshown as a flowchart in Figure 3.1. First, the simulation model is initialized byplacing blood vessels and diffusing them until a steady state. Once a steady state isreached, a single normal tumor cell is implanted at the center of the grid. This cellgoes through a series of conditional questions where the cell determines whetherit will remain a normal tumor cell, or turn into a hypoxic or necrotic cell. The24cell turns hypoxic when the oxygen concentration at that cell location is below athreshold level (values listed in Table 3.1). Cell necrosis happens when the ATPproduction rate falls below the death rate or if the oxygen level is below the deathconcentration. This state determination mechanism is based on the realistic be-haviour of a cancer cell in vivo as explained in section 2.1.Agents can also divide to produce a daughter cell of the same type. Onlynormal tumor cells and hypoxic cells can go through division and the probabilitydepends on the concentration of oxygen and glucose to generate ATP. When the cellmeets the condition to divide, it samples the surrounding Moore neighbourhood(Figure 3.2) to look for an available location. Available locations are non-cancercell or vessel locations.The parameters that were adjusted to simulate various tumor phenotypes were1) blood vessel density, 2) spatial arrangement of vessels (random or uniform den-sity), 3) probability of vessel removal, and 4) probability of death by insufficientnutrients. The probability of vessel removal by a tumor cell was used to simulatethe extent of vessel obstruction in real solid tumors. The death probability wasimplemented because the cells do not die immediately after depletion of nutrients;instead, prolonged shortage of oxygen or glucose is required for necrosis. For allsimulations, the death probability due to shortage of oxygen wasP= 1− local concentrationnecrotic concentration,and that of ATP was either the same as oxygen orP= 1− ( local concentrationnecrotic concentration)2.3.2 Translating Simulated Images to PET ImagesThe tumor growth simulation state is exported every 10 hours of biological time inthe TIFF file format. The time-sequenced simulated images are then translated intosynthetic PET images for further analysis using radiomics. The translation stepinvolves two key steps: mapping the agent locations and associating them withradiotracer uptake values, and matching the resolution of the simulated images25Figure 3.1: Tumor growth simulation flowchart. The flowchart shows thestep sequence of how the simulation is initialized, how agents determinetheir state, and how the PDE grids function (diffusion and consumptionof molecules)to PET images. The time sequences of simulated cell maps were converted intopseudo standardized tracer uptake value (pSUV) for the 18F-FDG tracer, which isdependent on the type of individual cells. The relative tracer uptake values were1, 8, 12, 0 for normal cells, tumor cells, hypoxic tumor cells, and necrotic cells,respectively. Next, we used a Gaussian filter with a smoothing kernel to reducethe simulated image resolution of 20µm to a typical PET resolution of ∼ 3mm.Lastly, realistic intensity Gaussian noise was also added to make it more clinicallyrelevant. An example of time-sequenced translated images are shown in Figure4.2.26Figure 3.2: Moore neighbourhood. Represents the eight locations surround-ing the center pixel, in this case an agent, that border an edge or a cornerof the cell. The arrows indicate the possible locations an agent can di-vide into in a Moore neighbourhood. Figure adapted from [9]3.2.1 Advantages of Using Synthetic PET ImagesThere are two clear reasons to why we chose to use synthetic PET images fromthe tumor growth simulation for radiomics analysis. First, the model begins from asingle tumor cell allowing us to obtain images throughout the progression of tumorgrowth to study the evolution of image features with biological time. Such infor-mation is impossible to obtain from human subjects since PET imaging involvesionizing radiation that limits the maximum radiation dose that can be administeredto a patient. The second advantage is that we can test with any combination of bio-logical parameters in the simulation within the biologically feasible range to studythe affects on tumor progressions and image properties. The hybrid model is alsoextremely flexible and allows us to add layers of PDE grids to model with differentmolecules of interest.273.3 Radiomics AnalysisThe translated synthetic PET images were analyzed using MATLAB functions forradiomics analysis which is used widely in radiomics research. We used the typ-ical 40% SUVmax to determine the ROI in the synthetic PET images. To assesthe plausibility of our model, we computed four conventional PET radiomic fea-tures that have been validated for its reproducibility and significance [22]. Thefour features are mean pSUV, shape-compactness, texture-contrast, and texture-homogeneity. The simulations produced time sequenced images of tumor growthat every 10 biological hours, so the features were also extracted at the same inter-val. For the extraction process, all ROI’s were converted into gray-scale to gener-ate gray-level co-occurrence matrices with all four offsets defined as described byHaralick. The list of MATLAB functions and user-defined functions used for theanalysis is shown in Table 3.2.Table 3.2: MATLAB functions for radiomics analysis.28Chapter 4Results4.1 Agent Grid + Oxygen PDE GridWe were able to generate three types of tumors with distinct phenotypes usingdifferent combinations of microparameters in the hybrid mathematical model fortumor growth (Figure 4.1). The adjusted parameters were blood vessel density(Types A,C: 20/mm2, Type B: 100/mm2), spatial arrangement of vessels (A,C: Ran-dom placement, B: Uniform density placement), and tumor vascular network alter-ations. The third parameter specifically refers to the obstruction of blood vesselsdue to tumor growth; this was modelled by introducing a probability of vessel re-moval (A,B: 1, C: 0,05) by a cancer cell when the cell attempts to divide into avessel location. The vessels that are first to be obstructed represent smaller vesselsand the ones that remain until later stages of tumor growth depict larger vessels insolid tumors.The resulting simulated tumor images were translated into synthetic PET im-ages with each pixel reflecting the corresponding pSUV values. Figure 4.2 shows aseries of translated Type C tumor images with its longitudinal growth. This demon-strates that we can study the progression of the PET images properties with tumorgrowth, which is one of the advantages of using the simulation as discussed in theprevious sections.The synthetic PET images of the three different types of tumor were visuallydistinguishable in the corresponding mean pseudo standard uptake value (pSUV)29Figure 4.1: Simulated tumors with distinct tumor phenotypes. The modelwith different combinations of tumor tissue microenvironments wasable to produce three tumor types with distinct tumor phenotypes.PET images, reflecting the distinct phenotypic profiles. The comparison of thegrowth rates for each type demonstrated that the model was able to produce diverseand realistic growth rates of about 80-310 days to grow a 1cm diameter tumor (Fig-ure 4.3). Using MATLAB radiomics analysis library, four radiomic features werecomputed from the generated images: mean pSUV, shape-compactness, Haralicktexture-contrast, and Haralick texture-homogeneity. As shown in Figure 4.4, theseradiomic feature values were plotted as a function of tumor progression in biologi-cal hours. Numerical values of the four features at their fully grown size are listedin Table 4.1. With distinguishable feature values between tumor types, the combi-nation of the four features enables unique association of the tumor phenotype withits specific tissue parameters.4.2 Agent Grid + Oxygen and Glucose PDE GridsFollowed by the simulation with the oxygen PDE grid, the model was further de-veloped by adding a glucose PDE grid to track the ATP production rate based30Figure 4.2: Translation of the Type C tumor longitudinal growth simula-tion to PET images. Simulated images were translated into syntheticPET images by converting the cell map into corresponding pSUV val-ues.Table 4.1: Radiomic feature values for 1cm tumors.Feature Type A Type B Type CpSUV 4.58 ± 0.03 7.63 ± 0.72 4.71 ± 0.04Compactness 0.91 ± 0.01 0.95 ± 0.01 0.94 ± 0.005Contrast 1.67 ± 0.08 1.58 ± 0.04 1.36 ± 0.1Homogeneity 0.71 ± 0.01 0.80 ± 0.006 0.79 ± 0.01on the nutrient concentrations. By changing the death probability from ATP de-ficiency (Type D: 1- [ATP]localDPR , Type E: 1-([ATP]localDPR )2), vessel density (D: 20/mm2,E:100/mm2) , and vessel removal probability (D: 0.05, E: 1), we were able to gen-erate two tumors with distinct phenotypic profiles. The preliminary results fromthis simulation model are shown in Figure 4.5. Tumor type D showed significantlyincreased tumor tissue dynamics throughout its growth compared to tumor type C,which was simulated with the same blood vessel density and vessel removal prob-31Figure 4.3: Biological time for tumor growth. The cell count was plottedagainst the biological time to compare the growth rate of each tumortype. The tumors were grown up to ∼ 1cm in diameter.ability. The images also portray more realistic tumor tissue characteristics, whichclosely resemble a molecular or histological image of a tumor. By simulating tu-mor Type E, we were able to demonstrate that a necrotic core can be modelledusing this tumor growth simulation. With maximum vessel removal probabilityand increased death probability from ATP deficiency, a pronounced necrotic re-gion was generated with viable cells surrounding only the grid locations in closeproximity of active vessels.32Figure 4.4: Radiomics analysis on synthetic PET images. Four PET ra-diomic features were plotted against simulated biological time for tu-mor growth. The mean and standard deviations of the data points werecalculated using three separate tumor growth simulation rounds. Theerror bar represents one standard deviation.33Figure 4.5: Two distinct tumor phenotypes simulated with oxygen andglucose PDE grids. The tumor progression is shown at three differenttime points (biological hours) until it reached 1cm in diameter. Thebottom panels represent the magnifications of the regions indicated withthe white box. 34Chapter 5DiscussionTumor population is highly heterogeneous and under constant evolution towardsmore malignant phenotypes which challenges effective cancer diagnosis and treat-ment planning. Heterogeneous tumor cell populations have different sensitivityto treatment. However, molecular classification of tumors are difficult to achievewith the small region sampling due to intratumoral heterogeneity. In contrast, med-ical imaging allows for visualization of heterogeneity in the entire tumor region.This motivated us to develop the tumor growth model for studying the microscopicchanges and characteristics of tumor tissues reflected in PET imaging. We wereable to successfully generate five distinct tumor profiles using different combina-tions of biological parameters.The first three tumor types simulated with the oxygen PDE grid (Types A-C) were analysed using quantitative PET radiomic features which showed distinctprogression patterns with tumor growth. This suggests that realistic tumor growthcan be achieved via simulating tumor tissue and its parameters, and the resultingmicroscopic tumor phenotypes are measurable through PET imaging. This allowsus to establish the 1:1 correspondence between tumor phenotypic parameters andPET radiomic features, which can be used as image-based biomarkers for cancerdiagnosis, prognosis, and tracking treatment outcomes.The improved version of the simulation with the addition of glucose PDE wasable to produce two visually distinguishable tumor phenotypes. The addition ofglucose PDE grid significantly enhanced the ability to show the dynamics of the35tumor tissues throughout their development. The resulting images were also dis-playing more realistic features of solid tumors. In tumor Type D, the gradual re-moval of smaller vessels resulted in viable tissue surrounding the larger vesselsthat have not been removed; beyond the diffusion-limited range from the vesselswas a mixture of hypoxia and necrosis with some accumulation of necrotic cells.The simulation of the Type E tumor features a definite necrotic core. With vesselremoval probability of 1, simulating the vessel removal including the larger ones,we could observe that the center of the tumor became completely necrotic. Theseresults are consistent with what was seen in real solid tumor progressions studiedby other groups [11, 13, 38].Medical imaging is known for its macroscopic visualization of tumor hetero-geneity due to limited resolution of the imaging techniques. In this study, wedemonstrated that microscopic changes in tumor tissue can be observed and trackedthrough macroscopic PET images via a novel method of using a tumor growth sim-ulation model. This opens up the potential for radiomics studies on microscopictumor properties that could replace or complement histopathological approaches toassociate the predictive radiomic models with biological significance. Microscopicinformation from the radiomic analysis would reduce the needs of invasive sam-pling and molecular assays that are costly and burdening to patients. Additionally,the current methods of biopsies on a small region of tumor cannot fully capture thestate due to the spatial and temporal heterogeneity. Contrarily, medical imaging isa much less invasive method that can capture the entire region of interest; it pro-vides a wealth of information that can be used to monitor tumor progression andresponse to treatment. This also provides a significant potential to largely advanceprecision medicine that have been hindered by the limitation to fully capture spatialand temporal tumor heterogeneity through conventional sampling methods [2].There are a few limitations to the simulation and its applications for back track-ing the biological parameters from lower resolution PET images. Our method ofdown-sampling the simulated tumor images to PET image resolution to see how themicroscopic changes are reflected in PET scans was a convenient process relativeto the reverse process. However, backtracking from the macroscopic PET imagesto microscopic tumor properties would be much more difficult. This suggests that,the mechanism for extrapolating the biological parameters and tumor phenotypes36given a real PET image still remains a challenge. Also, the model has an upper limitto the dimensionless diffusion coefficient of molecules and any value over the limitwill disturb the partial differential equations making it unstable. This requires timeincrement of the diffusion and consumption steps to be extremely small (∼0.03sin this model) relative to the cellular (agent) time step of 1 hour between everyiteration. This required thousands of diffusion steps between cell steps to achievesteady state, which demands for a progressively greater computational power.37Chapter 6Future DirectionsThis study demonstrated the possibility of using tumor growth simulations to studythe relationship between tumor tissue microparameters and the resulting heteroge-neous tumor phenotypic profiles captured in PET images. The identified 1:1 linkbetween PET image features and tumor tissue characteristics can be used as image-based biomarkers for personalized cancer diagnosis and treatment planning.The unique strength of the hybrid model allows for further expansion anddevelopment of the model with additional PDE grids. For instance, the alteredmetabolism increases acid production, which affects the pH of the microenviron-ments [6]. Tumor cells are better adapted to acidic conditions than normal cellswhich promotes their aggressive growth and invasion into normal tissue. Anotherexample is polypeptide growth factors (GF) associated with cancer progression;these molecules are involved in resistance to therapy and colonization of distanttissues [37]. All of these could be incorporated into the hybrid model by addingPDE grids for protons (H+) and GF for a more realistic simulation. The increasedbiological relevance could offer more insight into the interlink between PET im-age properties and tumor tissue characteristics. Once the model becomes completeenough to accurately simulate a real biological tissue, it can be used as a referencefor simulating the effectiveness of cancer drugs and radiation therapy.Another improvement can be made in the translation process of simulated im-ages into synthetic PET images. The current method was to translate the relativetracer uptake values of the cell types into pSUV values and to apply the Gaussian38filter and noise across the entire image. However, this does not account for themany sources of errors in a realistic PET scan, such as scatter and random coin-cident events. A more realistic approach for the translation would be to performa Monte Carlo simulation of the photon emission (while the two emitted photonsmake a 180◦) with annihilation rates that depend on the individual cell’s glucoseconsumption rates. 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