216 research outputs found

    Statistical modeling of bladder motion and deformation in prostate cancer radiotherapy

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    Prostate cancer is the most common cancer amongst the male population in most developed countries. It is the most common cancer amongst the male population in France (73.609 cases in 2014) and in Colombia (9564 cases in 2014). It is also the third most common cause of cancer deaths in males in both countries (9.3% and 7.1% in France and in Colombia in 2014, respectively). One of the standard treatment methods is external radiotherapy, which involves delivering ionizing radiation to a clinical target, namely the prostate and seminal vesicles. Due to the uncertain location of organs during treatment, which involves around forty (40) radiation fractions delivering a total dose ranging from 70 to 80Gy, safety margins are defined around the tumor target upon treatment planning. The radiation units are expressed in Grays, abbreviated as Gy, which represents 1 Jule/Kg. This leads to portions of healthy organs neighboring the prostate or organs at risk – the bladder and rectum – to be included in the target volume, potentially resulting in adverse events affecting patients' urinary (hematuria and cystitis, among others) or rectal (rectal bleeding, fecal incontinence, etc.) functions. Several studies have shown that increasing dose delivery to the prostate leads to improved local cancer control, up to approximately 80Gy. However, such dose increases are limited by their associated risks of treatment-related toxicity involving the organs at risk. The bladder is notorious for presenting the largest inter-fraction shape variations during treatment, caused by continuous changes in volume. These variations in shape introduce geometric uncertainties that render assessment of the actual dose delivered to the bladder during treatment difficult, thereby leading to dose uncertainties that limit the possibility of modeling dose-volume response for late genitourinary (GU) toxicity. The Quantitative Analysis of Normal Tissue Effects in the Clinic (QUANTEC) project has stated that a similar dose-response to that of late gastrointestinal (GI) toxicity is far from being established. The dosimetric variables obtained from the planning CT prove to be very poor surrogates for the real delivered dose. As a result, it appears crucial to quantify uncertainties produced by inter-fraction bladder variations in order to determine dosimetric factors that affect late GU complications. The aim of this thesis was thus to characterize and predict uncertainties produced by geometric variations of the bladder between fractions, using solely the planning CT as input information. In clinical practice, a single CT scan is only available for a typical patient during the treatment planning while on-treatment CTs/CBCTs are seldom available. In this thesis, we thereby used a population approach to obtain enough data to learn the most important directions of bladder motion and deformation using principal components analysis (PCA). As in groundwork, these directions were then used to develop population-based models in order to predict and quantify geometrical uncertainties of the bladder. However, we use a longitudinal analysis in order to properly characterize both patient-specific variance and modes from the population. We proposed to use mixed-effects (ME) models and hierarchical PCA to separate intra and inter-patient variability to control confounding cohort effects. Other than using PCA, bladder shapes were represented by using spherical harmonics (SPHARM) which additionally enables data compression without losing information. Subsequently, we presented PCA models as a tool to quantify dose uncertainties produced by bladder motion and deformation between fractions. We then estimated mean and variance of the dose delivered to the bladder using PCA-based models via Monte Carlo simulation and dose integration; and subsequently, we compared the estimated accumulated doses with the accumulated dose derived from non-rigid registration and patient's available images. We also calculated average voxel doses, local dose variability and dose-volume histogram uncertainties.Resumen: El cáncer de próstata es el cáncer más común entre la población masculina en muchos de los países desarrollados. Especificamente, es el cáncer más común de la población masculina tanto en Francia (73.609 casos en 2014) como en Colombia (9564 casos en 2014); y además, es también la tercera causa de muerte por cancer en los hombres para ambos países (9.3% y 7.1% en Francia y en Colombia en 2014, respectivamente). Uno de los métodos de tratamiento más común es la radioterapia externa, el cual consiste en enviar una radiación ionizante a un objetivo clínico, en este caso la próstata y las vesículas seminales. Debido a incertidumbres producidas por variaciones anat\'omicas de los órganos durante el tratamiento, el cual consiste en 40 fracciones para un total de dosis entre 70 y 80 Gy, márgenes de seguridad son definidos alrededor del tumor durante la planeación del tratamiento. Lo anterior, conlleva a que partes de los órganos sanos, también llamados órganos en riesgo, que son cercanos a la próstata y vesículas seminales -como la vejiga y el recto- también sean irradiados, potencialmente resultando en eventos adversos que afectan las funciones urinarias (hematuria e infección urinaria) o rectal (sangrado rectal, incontinencia fecal) del paciente. Algunos estudios han demostrado que el incrementando la dosis a la próstata permite un mejor control local del cáncer (por encima de los 80Gy aproximadamente). Sin embargo, tales incrementos de dosis son limitados por sus riesgos asociados de toxicidad para los órganos en riesgo. La vejiga es particular por presentar las variaciones de forma más grandes entre las fracciones del tratamiento, las cuales son causadas por continuos cambios de volumen. Estas variaciones de forma de la vejiga introducen incertidumbres geométricas que hacen dif\'icil la determinacion de la verdadera dosis entregada a la vejiga durante el tratamiento. Estas incertidumbres limitan la posibilidad de modelar una relación dosis-volumen para la toxicidad Genitourinario tardia (GU). El proyecto Quantitative Analysis of Normal Tissue Effects in the Clinic (QUANTEC) ha establecido que una respuesta dosis-volumen similar a la que se tiene para la toxicidad Gastrointestinal tardia (GI) está lejos de ser establecida. Las variables dosimétricas que se obtienen de la tomografía computarizada de planeación pueden ser débilmente represenativas de la verdadera dosis suministrada. Por lo tanto, es crucial identificar las incertidumbres producidas por el movimiento y deformación entre fracciones de la vejiga con el fin de determinar los factores dosimétricos que afectan las complicaciones GU tardías. El propósito de esta tesis fue entonces caracterizar y predecir las incertidumbres geométricas producidas por las variaciones geométricas de la vejiga entre fracciones, usando solamente el CT de planeación como información de entrada. En la pr\'actica clínica, un sola tomografía computarizada esta disponible en la fase de planeación del tratamiento para un paciente típico, mientras que imágenes suplementarias durante el tratamiento estan raramente disponibles. Por lo tanto, en esta tesis fue usado un enfoque poblacional para obtener suficientes datos para aprender las direcciones más importantes de movimiento y deformación de la vejiga usando análisis de componentes principales (ACP). Tal como en trabajos anteriores, estas direcciones fueron también usadas para desarrollar modelos poblacionales para predecir y cuantificar incertidumbres geométricas de la vejiga. Sin embargo, en esta tesis se propuso un análisis longitudinal con el fin de adecuadamente caracterizar la varianza y los modos del paciente de la población. Básicamente, se propuso usar modelos de efectos-mixtos (ME) y ACP jerárquico para separar la variabilidad intra e inter-paciente para controlar efectos confusos de la población. Adicional a ACP, la superficie de la vejiga también fue representada usando esféricos harmónicos (SPHARM), lo cual nos permitio adicionalmente comprimir los datos sin perder información. Finalmente, se presentan los modelos basado en ACP como una herramienta para cuantificar las incertidumbres de la dosis producidas por el movimiento y deformación de la vejiga entre fracciones. El promedio y la varianza de la dosis entregada a la vejiga fueron estimadas utilizando modelos ACP via simulación de Monte Carlo e integración de la dosis; y posteriormente, se comparo las dosis acumuladas estimadas con la dosis acumulada obtenida a partir de registro no-rigido y las imagenes disponibles del paciente. Igualmente, se calcularon los valores promedio de la dosis por voxel, la variabilidad local de la dosis y las incertidumbres de los histogramas dosis-volumen.Résumé: Le cancer de la prostate est le cancer le plus fréquent chez les hommes dans la plupart des pays développés. C’est le cancer le plus fréquent chez les hommes en France (73.609 cas en 2014) et en Colombie (9564 cas en 2014). En outre, c’est la troisième cause de décès par cancer chez les hommes dans les deux pays (9,3 % en France et 7,1 % en Colombie en 2014). L’une des techniques de traitement est la radiothérapie externe, qui consiste à délivrer un rayonnement ionisant à une cible clinique, à savoir la prostate et les vésicules séminales. En raison des variations anatomiques au cours du traitement, qui consiste en environ 40 fractions de rayonnement délivrant une dose totale allant de 70 à 80Gy, des marges de sécurité sont définies autour de la cible tumorale lors de la planification du traitement. Ceci entraîne des portions d’organes sains voisins de la prostate - la vessie et le rectum - à être inclus dans le volume cible, pouvant conduire à des événements indésirables affectant les fonctions urinaires (hématurie et cystite, entre autres) ou rectale (saignement rectal, incontinence fécale, Etc.). Plusieurs études ont montré que l’augmentation de la dose administrée à la prostate conduit à une amélioration du contrôle local du cancer (jusqu’à environ 80Gy). Cependant, de telles augmentations de dose sont limitées par les risques associés de toxicité pour les organes à risque. La vessie présente les plus grandes variations de forme entre fractions de traitement, provoquées par des changements continus de volume. Ces variations de forme introduisent des incertitudes géométriques qui rendent difficile l’évaluation de la dose réellement délivrée à la vessie pendant le traitement. Ces incertitudes limitent la possibilité de modéliser une relation dose-volume pour la toxicité génito-urinaire tardive (GU). Le projet QUANTEC (Quantitative Analysis of Normal Tissue Effects in the Clinic) a déclaré que la relation dose-réponse pour la toxicité gastro-intestinale tardive (GI) était loin d’être établie. Les variables dosimétriques obtenues à partir de la tomodensitométrie de planification peuvent être faiblement représentative de la dose effectivement administrée. En conséquence, il est crucial de quantifier les incertitudes produites par les variations inter-fraction de la vessie afin de déterminer les facteurs dosimétriques qui affectent les complications GU tardives. Le but de cette thèse était donc de caractériser et de prédire les incertitudes produites par les variations géométriques de la vessie entre les fractions de traitement, en utilisant uniquement la tomodensitométrie de planification comme information d’entrée. En pratique clinique, une seule tomodensitométrie est disponible au moment de la planification du traitement pour un patient typique, alors que des images supplémentaires peuvent être acquises en cours de traitement. Dans cette thèse une approche population a été utilisée pour obtenir suffisamment de données pour apprendre les directions les plus importantes du mouvement et de la déformation de la vessie en utilisant l’analyse en composante principales (ACP). Comme dans les travaux de référence, ces directions ont ensuite été utilisées pour développer des modèles basés population pour prédire et quantifier les incertitudes géométriques de la vessie. Cependant, nous avons utilisé une analyse longitudinale afin de caractériser correctement la variance du patient et les modes spécifiques du patient à partir de la population. Nous avons proposé d’utiliser un modèle à effets mixtes (ME) et une ACP hiérarchique pour séparer la variabilité intra et inter-patients afin de contrôler les effets de cohorte confondus. Outre l’ACP, la forme de la vessie a été représentée par l’utilisation d’harmoniques sphériques (SPHARM), ce qui a permis la compression des données sans perte d’information. Par la suite, nous avons cherché à quantifier les incertitudes de dose produites par le mouvement de la vessie et la déformation entre les fractions. Finalement, nous avons présenté des modèles sur l’APC comme un outil pour quantifier des incertitudes de la dose produit par le mouvement et déformation de la vessie entre fractions. Nous avons ensuite estimé la moyenne et la variance de la dose délivrée à la vessie en utilisant des modèles basés sur l’APC par la méthode de simulation de Monte-Carlo et l’intégrale de la dose; et par la suite, nous avons comparé les doses cumulées estimées avec la dose accumulée obtenue en utilisant un recalage d’image non-rigide et les images du patient. Également, nous avons calculé la moyenne de la dose par voxel, la variabilité de la dose locale et des incertitudes d’histogramme de volume de dose.Doctorad

    Development and evaluation of low-dose rate radioactive gold nanoparticles for application in nanobrachytherapy

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    Depuis les dix dernières années, l’innovation des traitements d’oncologie a fait une utilisation croissante de la nanotechnologie. De nouveaux traitements à base de nanoparticules (NPs) sont notamment rendus au stade de l’essai clinique. Possédant des caractéristiques physico-chimiques particulières, les NPs peuvent être utilisées afin de bonifier l’effet thérapeutique des traitements actuels. Par exemple, l’amélioration de la curiethérapie (c.-à-d. radiothérapie interne) nécessite le développement de nouvelles procédures permettant de diminuer la taille des implants, et ce, tout en augmentant l’homogénéité de la dose déposée dans les tumeurs. Des études théoriques et expérimentales ont démontré que l’injection de NPs d’or à proximité des implants traditionnels de curiethérapie de faible débit de dose (par ex. 125I, 103Pd) permettrait d’augmenter significativement leur efficacité thérapeutique. L'interaction entre l’or et les photons émis par les implants de curiethérapie (c.-à-d. l’effet de radiosensibilisation) génère des rayonnements divers (photoélectrons, électrons Auger, rayons X caractéristiques) qui augmentent significativement la dose administrée. Dans le cadre de cette thèse, l’approche proposée était de développer des NPs d’or radioactives comme nouveau traitement de curiethérapie contre le cancer de la prostate. L’aspect novateur et unique était de synthétiser une particule coeurcoquille (Pd@Au) en utilisant l’isotope actuellement employé en curiethérapie de la prostate: le palladium-103 (103Pd, 20 keV). Dans ce cas-ci, la présence d’atomes d’or permet de produire l’effet de radiosensibilisation et d’augmenter la dose déposée. La preuve de concept a été démontrée par la synthèse et la caractérisation des NPs 103Pd@Au-PEG NPs. Ensuite, une étude longitudinale in vivo impliquant l’injection des NPs dans un modèle xénogreffe de tumeurs de la prostate chez la souris a été effectuée. L’efficacité thérapeutique induite par les NPs a été démontrée par le retard de la croissance tumorale des souris injectées par rapport aux souris non injectées (contrôles). Enfin, une étude de cartographie de la dose générée par les NPs à l’échelle cellulaire et tumorale a permis de comprendre davantage les mécanismes thérapeutiques liés aux NPs radioactives. En résumé, l’ensemble des travaux présentés dans cette thèse font office de précurseurs relativement au domaine de la nanocuriethérapie, et pourraient ouvrir la voie à une nouvelle génération de NPs pour la radiothérapie.The last decade saw the emergence of new innovative oncology treatments based on nanotechnology. New treatments using nanoparticles (NPs) are now translated to clinical trials. NPs possess unique physical and chemical properties that can be advantageously used to improve the therapeutic effect of current treatments. For instance, therapeutic efficiency enhancement related to internal radiotherapy (i.e., brachytherapy), requires the development of new procedures leading to a decrease of the implant size, while increasing the dose homogeneity and distribution in tumors. Several theoretical and experimental studies based on low-dose brachytherapy seeds (e.g., 125I and 103Pd) combined with gold nanoparticles (Au NPs) showed very promising results in terms of dose enhancement. Gold is a radiosensitizer that enhances the efficiency of radiotherapy by increasing the energy deposition in the surrounding tissues. Dose enhancement is caused by the photoelectric products (photoelectrons, Auger electrons, characteristic X-rays) that are generated after the irradiation of Au NPs. In this thesis, the proposed approach was to develop radioactive Au NPs as a new brachytherapy treatment for prostate cancer. The unique and innovative aspect of this strategy was to synthesize core-shell NPs based on the radioisotope palladium-103 (103Pd, 20 keV), which is currently used in low-dose rate prostate cancer brachytherapy. In this concept, the administrated dose is increased via the radiosensitization effect that is generated through the interactions of low-energy photons with the gold atoms. The proof-ofconcept of this approach was first demonstrated by the synthesis and characterization of the core-shell NPs (103Pd@Au-PEG NPs). Then, a longitudinal in vivo study following the injection of NPs in a prostate cancer xenograft murine model was performed. The therapeutic efficiency was confirmed by the tumor growth delay of the treated group as compared to the control group (untreated). Finally, a mapping study of the dose distribution generated by the NPs at the cellular and tumor levels provided new insights about the therapeutic mechanisms related to radioactive NPs. In summary, the studies presented in this thesis are precursors works in the field of nanobrachytherapy, and could pave the way for a new generation of NPs for radiotherapy

    Analysis of contrast-enhanced medical images.

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    Early detection of human organ diseases is of great importance for the accurate diagnosis and institution of appropriate therapies. This can potentially prevent progression to end-stage disease by detecting precursors that evaluate organ functionality. In addition, it also assists the clinicians for therapy evaluation, tracking diseases progression, and surgery operations. Advances in functional and contrast-enhanced (CE) medical images enabled accurate noninvasive evaluation of organ functionality due to their ability to provide superior anatomical and functional information about the tissue-of-interest. The main objective of this dissertation is to develop a computer-aided diagnostic (CAD) system for analyzing complex data from CE magnetic resonance imaging (MRI). The developed CAD system has been tested in three case studies: (i) early detection of acute renal transplant rejection, (ii) evaluation of myocardial perfusion in patients with ischemic heart disease after heart attack; and (iii), early detection of prostate cancer. However, developing a noninvasive CAD system for the analysis of CE medical images is subject to multiple challenges, including, but are not limited to, image noise and inhomogeneity, nonlinear signal intensity changes of the images over the time course of data acquisition, appearances and shape changes (deformations) of the organ-of-interest during data acquisition, determination of the best features (indexes) that describe the perfusion of a contrast agent (CA) into the tissue. To address these challenges, this dissertation focuses on building new mathematical models and learning techniques that facilitate accurate analysis of CAs perfusion in living organs and include: (i) accurate mathematical models for the segmentation of the object-of-interest, which integrate object shape and appearance features in terms of pixel/voxel-wise image intensities and their spatial interactions; (ii) motion correction techniques that combine both global and local models, which exploit geometric features, rather than image intensities to avoid problems associated with nonlinear intensity variations of the CE images; (iii) fusion of multiple features using the genetic algorithm. The proposed techniques have been integrated into CAD systems that have been tested in, but not limited to, three clinical studies. First, a noninvasive CAD system is proposed for the early and accurate diagnosis of acute renal transplant rejection using dynamic contrast-enhanced MRI (DCE-MRI). Acute rejection–the immunological response of the human immune system to a foreign kidney–is the most sever cause of renal dysfunction among other diagnostic possibilities, including acute tubular necrosis and immune drug toxicity. In the U.S., approximately 17,736 renal transplants are performed annually, and given the limited number of donors, transplanted kidney salvage is an important medical concern. Thus far, biopsy remains the gold standard for the assessment of renal transplant dysfunction, but only as the last resort because of its invasive nature, high cost, and potential morbidity rates. The diagnostic results of the proposed CAD system, based on the analysis of 50 independent in-vivo cases were 96% with a 95% confidence interval. These results clearly demonstrate the promise of the proposed image-based diagnostic CAD system as a supplement to the current technologies, such as nuclear imaging and ultrasonography, to determine the type of kidney dysfunction. Second, a comprehensive CAD system is developed for the characterization of myocardial perfusion and clinical status in heart failure and novel myoregeneration therapy using cardiac first-pass MRI (FP-MRI). Heart failure is considered the most important cause of morbidity and mortality in cardiovascular disease, which affects approximately 6 million U.S. patients annually. Ischemic heart disease is considered the most common underlying cause of heart failure. Therefore, the detection of the heart failure in its earliest forms is essential to prevent its relentless progression to premature death. While current medical studies focus on detecting pathological tissue and assessing contractile function of the diseased heart, this dissertation address the key issue of the effects of the myoregeneration therapy on the associated blood nutrient supply. Quantitative and qualitative assessment in a cohort of 24 perfusion data sets demonstrated the ability of the proposed framework to reveal regional perfusion improvements with therapy, and transmural perfusion differences across the myocardial wall; thus, it can aid in follow-up on treatment for patients undergoing the myoregeneration therapy. Finally, an image-based CAD system for early detection of prostate cancer using DCE-MRI is introduced. Prostate cancer is the most frequently diagnosed malignancy among men and remains the second leading cause of cancer-related death in the USA with more than 238,000 new cases and a mortality rate of about 30,000 in 2013. Therefore, early diagnosis of prostate cancer can improve the effectiveness of treatment and increase the patient’s chance of survival. Currently, needle biopsy is the gold standard for the diagnosis of prostate cancer. However, it is an invasive procedure with high costs and potential morbidity rates. Additionally, it has a higher possibility of producing false positive diagnosis due to relatively small needle biopsy samples. Application of the proposed CAD yield promising results in a cohort of 30 patients that would, in the near future, represent a supplement of the current technologies to determine prostate cancer type. The developed techniques have been compared to the state-of-the-art methods and demonstrated higher accuracy as shown in this dissertation. The proposed models (higher-order spatial interaction models, shape models, motion correction models, and perfusion analysis models) can be used in many of today’s CAD applications for early detection of a variety of diseases and medical conditions, and are expected to notably amplify the accuracy of CAD decisions based on the automated analysis of CE images

    Imaging and radiotherapy in prostate cancer: advances in biomarkers and treatment

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    A novel NMF-based DWI CAD framework for prostate cancer.

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    In this thesis, a computer aided diagnostic (CAD) framework for detecting prostate cancer in DWI data is proposed. The proposed CAD method consists of two frameworks that use nonnegative matrix factorization (NMF) to learn meaningful features from sets of high-dimensional data. The first technique, is a three dimensional (3D) level-set DWI prostate segmentation algorithm guided by a novel probabilistic speed function. This speed function is driven by the features learned by NMF from 3D appearance, shape, and spatial data. The second technique, is a probabilistic classifier that seeks to label a prostate segmented from DWI data as either alignat, contain cancer, or benign, containing no cancer. This approach uses a NMF-based feature fusion to create a feature space where data classes are clustered. In addition, using DWI data acquired at a wide range of b-values (i.e. magnetic field strengths) is investigated. Experimental analysis indicates that for both of these frameworks, using NMF producing more accurate segmentation and classification results, respectively, and that combining the information from DWI data at several b-values can assist in detecting prostate cancer

    Radiobiological Optimization of Lung and Prostate Radiotherapy Treatments - A Macroscopic Approach

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    Radiobiological modelling is applied to different treatment individualization & optimization strategies and the resulting improvements in treatment outcomes quantified for lung and prostate cancer treatments; this is compared to current approaches based on delivering identical dose in fixed (size and number of) fractions to a given tumour type. In the first investigation, dose-based escalation (level 0) is compared to Iso-toxic (i.e. Iso-NTCP) fixed fraction number prescriptions (level 1) and Iso-toxic prescription dose & fraction number optimization (level 2). NTCP, dose-scaling factor and number of fractions are the parameters used to optimize TCP; multiple dose-limiting OAR endpoints are accounted for in the above analysis. It is shown that Iso-toxic (at 8.6% NTCP) dose fractionation optimization improves the average TCP of the lung cohort by ~19% compared to standard dose fractionation (55 Gy in 20 fractions yielding 8.6% average NTCP). Similarly, for prostate cancer treatment, it is demonstrated that level-2 optimization is superior to standard treatment 60 Gy in 20 fractions): a population TCP increase of 12.6% (α/β=10 Gy) and 9.7% (α/β=1.5 Gy) is observed at 10% Iso-toxic NTCP. The entire analysis is performed with the software ‘RadOpt’ which was written as part of this PhD work and is described in detail in chapter three. ‘RadOpt’ will be made available for other researchers to perform similar cohort comparison analyses (to compare changes in parameters, regimens, etcetera as demonstrated in this thesis). In chapter 4, the effect of patient-specific radiosensitivity information on treatment optimization & strategy selection is explored. For lung-tumour treatments it is shown that if patients are stratified into 3 or 5 subgroups of tumour radiosensitivity, the average TCP of the cohort would increase by about ~7.5% at 15 fractions (for most patients) after implementing level-2 optimization, compared to the current scenario where patient specific radiosensitivity information is not available. For the prostate cancer cohort, level-2 TCP is improved marginally by ~1-2% at a reduced NTCP (1.7% lower NTCP compared to optimization where such information is unavailable). Further, it is reported that patients at the extreme ends of normally distributed tumour radiosensitivity would benefit significantly from changes in treatment strategies if patient-specific tumour radiosensitivity information is accounted for in treatment optimization. In chapter 5, the superiority of radiobiological-parameter-driven VMAT inverse treatment plans (in terms of TCP, NTCP and standard dose-volume metrics) compared to dose-volume parameter based VMAT treatment plans for 4 patients (2 with lung & 2 with prostate cancer) is demonstrated. The analysis also shows that heterogeneous dose-distribution based planning can yield improved TCP and dose sparing of OARs compared to standard planning approach that aims for a fixed and homogeneous tumour dose deposition. Further, it is observed that employing radiobiological model-based objectives/constraints reduces the risk of cold spots in the tumour and improves planning efficiency as additional ‘dummy structures’ for sparing an OAR (e.g. rectum) would not be required. Chapter 6 introduces a novel method of iso-NTCP conversion of normal-tissue dose-volume metrics (Vxx) from one regimen to the other. The analysis is carried out for two (each) lung and prostate treatment endpoints. We introduce two methods to perform this analysis (graphical & mathematical). The graphical method allows a clinician to find the equivalent Vxx for the new regimen such that the NTCPs of the OAR for the two regimens are equal
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