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.