Biomathematical Problems in Optimization of Cancer Radiotherapy

A.Y. Yakovlev, L. Pavlova, L.G. Hanin

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000142396
• Topic: Medicine
• Subtopic: Mathematics General

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.

1

Mathematical Models of Irradiated Cell Survival

1.1 INTRODUCTION

Cell radiobiology has proven to be a fertile field for the application of mathematical, especially stochastic, models. Probabilistic methods of data analysis have been inseparably linked with experimental research in this field during the whole of its history. It was the need to explain the diverse character of cell response to radiation damage that caused the ideas of “hit and target” theory. Contributions to the quantitative formulation of target principle influenced by the ideas of quantum mechanics are dated as early as the 1920s. But as it was pointed out by Turner,⁸⁶ the first significant attempt at a comprehensive theory was due to Danzer¹³ in 1934. The target theory made it possible to express in mathematical form the contribution of the discrete nature of ionizing radiation and the stochastic character of radiation energy scattering on cell structures to the ultimate biological effect. Classical hit and target models will be considered in Section 1.2.

With the accumulation of experimental evidence there appeared an ever-growing need for a probabilistic description of the stochastic responses of a cell to radiation damage formed according to the hit and target principle. This has given rise to the randomized models which will be briefly discussed in Section 1.3. Section 1.5 is devoted to the problem of parametric identification of a simple randomized version of the “multihit-one target” model. In the subsequent chapters this model will serve as a basis for all considerations associated with the optimal strategies in fractionated irradiation of tumors. To give a complete picture of the principal trends in the target theory, some Markovian discrete models will be described in Section 1.4.

1.2 CLASSICAL HIT AND TARGET MODELS

The contents of this section relate largely to the history of stochastic radiobiology, but it seems necessary to follow the evolution of ideas in the field. Consider a single cell exposed to an acute (short-term) irradiation. We begin with enlisting, following Turne...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Acknowledgments
• The Authors
• Table of Contents
• Introduction
• Chapter 1 Mathematical Models of Irradiated Cell Survival
• Chapter 2 Theoretical Bounds for the Efficiency of Cancer Treatment
• Chapter 3 Optimal Fractionation Problem
• Conclusion
• Appendix 1 The Relationship Between Sets UM>0 LM and W
• Appendix 2 Proof of Relation 3.5, Section 2.3
• References
• Subject Index