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.

Inhalt

1 General Introduction.- 1.1 Introduction.- 1.2 History of Cancer and its Treatment by Radiotherapy.- 1.3 Some Mathematical Models of Tumor Growth.- 1.4 Spatial Distribution of the Radiation Dose.- 2 Survival Curves from Statistical Models.- 2.1 Introduction.- 2.2 The Target Model.- 2.3 Single-hit-to-kill Model.- 2.4 Multitarget, Single-hit Survival.- 2.5 Multitarget, Multihit Survival.- 2.6 Single-target, Multihit Survival.- 2.7 Properties of In Vitro Survival Curves.- 3 A Molecular Model of cell Survival.- 3.1 Introduction.- 3.2 The Molecular Model.- 3.3 Interpretations of the Molecular Model.- 4 Kinetic Models of Biological Radiation Response.- 4.1 Introduction.- 4.2 Basic Postulates in the Dienes Model.- 4.3 Low LET Kinetic Models with no Recovery.- 4.4 Further Discussion of the Models.- 4.5 Low LET Kinetic Models with Recovery.- 4.6 Low LET and High LET Kinetic Model with No Recovery.- 4.7 Other Kinetic Schemes: Sparsely-ionizing Radiation.- 4.8 Kinetic Schemes with Age-specific Compartments in the Cell Cycle.- 4.9 Thermal Potentiation of Cell Killing.- 5 Cell Survival after Successive Radiation Fractions.- 5.1 Introduction.- 5.2 Some Results in Connection with Instantaneous Cell Kill and Exponential Tumor Growth.- 5.3 Instantaneous Cell Kill Followed by Logistic Growth of Normal Tissue.- 5.4 Cohen's Cell Population Kinetics Programs.- 5.5 A Model of Radiation Therapy with Resistant and Sensitive Cell Populations.- 5.6 Dose Fractionation and General Survival Curves.- 6 Optimization Models in Solid Tumor Radiotherapy.- 6.1 Introduction.- 6.2 Optimal Radiotherapy of Tumor Cells Based on Cumulative Radiation Effect and a Multitarget, Single-hit Survival Function.- 6.3 Optimal Radiotherapy of Tumor Cells Based on Cumulative Radiation Effect and an Exponential-quadratic Survival Expression.- 6.4 Fractionation Scheme with a Four Level Population Tumor Model.- 6.5 A Dynamic Programming Solution to the Problem of the Determination of Optimal Treatment Schedules.- 6.6 Optimal Treatment Schedules in Fractionated Radiation Therapy for Fischer's Tumor Model.- 6.7 Optimal Radiation Schedules with Cell Cycle Analysis.- 7 Numerical Solution of Multistage Optimal Control Problems.- 7.1 Introduction.- 7.2 Continuous Time Optimal Control.- 7.3 Optimization of Multistage Systems.- 7.4 Multi-dimensional Optimization by Gradient Methods.- 7.5 Gradient Method with Penalty Function.- 7.6 A Numerical Scheme for a Nonlinear Problem.- 7.7 The Method of Conjugate Gradients.- 7.8 Discrete Dynamic Programming.- 8 Some Optimization Criteria in Radiotherapy.- 8.1 Introduction.- 8.2 Therapeutic Policy, Strategy and Tactics.- 8.3 Optimization and Clinical Trials.- 8.4 Score Functions and Age Response Functions.- 8.5 Comparison of Models Used in Optimization Procedures.- 8.6 The Complication Probability Factor.- 9 The Optimization of External Beam Radiation Therapy.- 9.1 Introduction.- 9.2 Some Approaches for Treatment Plans.- 9.3 Linear Programming.- 9.4 Linear Programming and Radiation Treatment Planning.- 9.5 Optimization of External Beam Radiation Therapy Using Nonlinear Programming.- 9.6 Quantitative Study of Relative Radiation Effects and Isoeffect Patterns.- 10 Reconstructive Tomography.- 10.1 Introduction.- 10.2 Reconstruction Algorithm.- 10.3 Numerical Approximations for the Attenuation Coefficient.- 10.4 Cross-sectional Absorption Density Reconstruction for Treatment Planning.- 10.5 Towards the Optimization of Dose Reduction in Computerized Tomography.- 10.6 Other Imaging Technologies.- Appendix 1.- Appendix 2.- Appendix 3.