The mathematical models in this book are concerned with a variety of approaches to the manner in which the clinical radiologic treatment of human neoplasms can be improved. These improvements comprise ways of delivering radiation to the malignan cies so as to create considerable damage to tumor cells while sparing neighboring normal tissues. There is no unique way of dealing with these improvements. Accord ingly, in this book a number of different presentations are given. Each presentation has as its goal some aspect of the improvement, or optimization, of radiotherapy. This book is a collection of current ideas concerned with the optimization of human cancer radiotherapy. It is hoped that readers will build on this collection and develop superior approaches for the understanding of the ways to improve therapy. The author owes a special debt of thanks to Kathy Prindle who breezed through the typing of this book with considerable dexterity. TABLE OF CONTENTS Chapter GENERAL INTRODUCTION 1. 1 Introduction 1 1. 2 History of Cancer and its Treatment by Radiotherapy 8 1. 3 Some Mathematical Models of Tumor Growth 12 1. 4 Spatial Distribution of the Radiation Dose 20 Chapter 2 SURVIVAL CURVES FROM STATISTICAL MODELS 24 2. 1 Introduction 24 2. 2 The Target Model 26 2. 3 Single-hit-to-kill Model 27 2. 4 Multitarget, Single-hit Survival 29 2. 5 Multitarget, Multihit Survival 31 2. 6 Single-target, Multihit Survival 31 2.

Biomathematical Problems in Optimization of Cancer Radiotherapy provides insight into the role of cell population heterogeneity in the optimal control of fractionated irradiation of tumors. The book emphasizes the mathematical modeling aspect of the problem and presents the state of the art in the stochastic description of irradiated cell survival. Some of the results are of general theoretical interest and can be applied to other areas of optimal control methodology. Detailed explanations of all mathematical statements are provided throughout the text. The book is excellent for biomathematicians, radiotherapists, oncologists, health physicists, and other researchers and students interested in the topic.

Radiation therapy has developed and advanced dramatically in the last few decades. However, very little has been published or done in the area of biologically optimized treatment planning. Development of Biologically Optimized Radiation Therapy aims to fill and close an important gap in the literature with a well-focused and in-depth content.The book covers the biological, physical and clinical background of advanced biologically based radiation therapy optimization with focus on modern radiation therapy modalities such as electron, photon and light ion therapy. Highly recommended for its strong interdisciplinary profile, the book contains a meritorious compilation of previously unpublished materials in many areas of modern science. Undergraduates, researchers and practitioners such as oncologists, medical physicists and radiation biologists alike should find the book immensely informative and comprehensively thorough.

The aim of this book is to provide a uniquely comprehensive source of information on the entire field of radiation therapy physics. The very significant advances in imaging, computational, and accelerator technologies receive full consideration, as do such topics as the dosimetry of radiolabeled antibodies and dose calculation models. The scope of the book and the expertise of the authors make it essential reading for interested physicians and physicists and for radiation dosimetrists.

Radiation therapy is a complex process where a given target volume receives a given dose of radiation divided over one or multiple treatments. Every step in this process can introduce some types of uncertainties into the problem which may compromise the quality of the treatment. Typically, a volume larger than the actual tumor is irradiated to make the treatment more robust against these uncertainties. This comes at the cost of normal tissue irradiation and an increased risk of toxicity. In this dissertation, we investigate approaches to managing uncertainties in radiation therapy treatments for lung cancer patients. In the first part of the dissertation, we focus on the process of designing a treatment plan which involves selecting appropriate beam angles and deciding the right amount of radiation dose to the tumor cells, while sparing the normal tissue surrounding the tumor. Selecting the optimal set of treatment beam angles, called beam angle optimization (BAO), involves a very large-scale combinatorial optimization problem with many local minima. In order to identify an efficient approach to obtain high quality beam angles, we first examine the strengths and weaknesses of some existing BAO optimization methods including both global and local search algorithms. We then propose a hybrid framework to overcome some of the weaknesses observed in these methods. Next, we perform an in-depth study into the impact of interplay effect, which results from relative motion of the tumor and proton beam, on the dose distribution in the patient with lung cancer. The dynamic dose distribution, that provides an estimation of delivered dose under the influence of interplay effect, is calculated by simulating the machine delivery processes on the moving patient described by 4D computed tomography (4DCT) during the dose delivery process by linking timestamps of each on/off switch of proton spots, spills, energies, and fields to patient respiratory cycles. We introduce a clinically applicable metric for clinicians to use for determining the magnitude of the uncertainties caused by interplay effects. We then explore the techniques of fractionation and iso-layered re-scanning for mitigating these interplay effects. In the last part of the dissertation, we develop a robust adaptive optimization framework for intensity modulated radiation therapy (IMRT) for lung cancer, where temporal variation of tumor volume and its associated uncertainties throughout the course of the treatment are accounted for to re-optimize the treatment plan for the following sessions. This framework gives an insight into the trade-off between sparing the healthy tissues and ensuring that the tumor receives a sufficient dose. With this trade-off in mind, we demonstrate that our robust adaptive solution outperforms a non-adaptive solution and a nominal (no uncertainty) solution on a clinical case.

External beam radiotherapy is delivered from outside the body aimed at cancer cells to damage their DNA making them unable to divide and reproduce. The beams travel through the body and may damage nearby healthy tissues unless carefullyplanned. Therefore, the goal of treatment plan optimization is to find the best system configuration to deliver sufficient dose to target structures while avoiding damage to healthy tissues. This thesis investigates optimization approaches for two external beam radiation therapy techniques: Intensity-Modulated Radiation Therapy (IMRT) and Volumetric-Modulated Arc Therapy (VMAT). We develop an automated treatment planning technology for IMRT which generates several high-quality treatment plans satisfying the provided requirements in a single invocation and without human guidance. Our approach is based on an existing linear programming-based fluence map optimization model that approximates dose-volume requirements using conditional value-at-risk (C-VaR) constraints. We show how the parameters of the C-VaR constraints can be used to control various metrics of treatment plan quality. A novel bi-criteria scoring based beam selection algorithm is developed which finds the best beam configuration at least ten times faster for real-life brain, prostate, and head and neck cases as compared to an exact mixed integer programming model. Patient anatomy changes due to breathing during the treatment of lung cancer need to be considered in treatment planning. To date, a single phase of the breathing cycle is typically selected for treatment and radiation is shut-off in other phases. We investigate optimization technology that finds optimal fluence maps for each phase of the breathing cycle by considering the overall dose delivered to a patient using image registration algorithms to track target structures and organs at risk. Because the optimization exploits the opportunities provided in each phase, better treatment plans are obtained. The improvements are shown on a real-life lung case. VMAT is a recent radiation treatment technology which has the potential to provide treatments in less time compared to other delivery techniques. This enhances patient comfort and allows for the treatment of more patients. We build a large-scale mixed-integer programming model for VMAT treatment plan optimization. The solution of this model is computationally prohibitive. Therefore, we develop an iterative MIP-based heuristic algorithm which solves the model multiple times on a reduced set of decision variables. We introduce valid inequalities that decrease solution times, and, more importantly, that identify higher quality integer solutions within specified time limits. Computational studies on a spinal tumor and a prostate tumor case produce clinically acceptable results.

The articles collected in this volume are based on lectures given at the IMA Workshop, "Computational Radiology and Imaging: Therapy and Diagnostics", March 17-21, 1997. Introductory articles by the editors have been added. The focus is on inverse problems involving electromagnetic radiation and particle beams, with applications to X-ray tomography, nuclear medicine, near-infrared imaging, microwave imaging, electron microscopy, and radiation therapy planning. Mathematical and computational tools and models which play important roles in this volume include the X-ray transform and other integral transforms, the linear Boltzmann equation and, for near-infrared imaging, its diffusion approximation, iterative methods for large linear and non-linear least-squares problems, iterative methods for linear feasibility problems, and optimization methods. The volume is intended not only for mathematical scientists and engineers working on these and related problems, but also for non-specialists. It contains much introductory expository material, and a large number of references. Many unsolved computational and mathematical problems of substantial practical importance are pointed out.