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Lectu re Notes in
Biomathematics
Managing Editor: S. Levin
30
Martin Eisen
Mathematical Models in
Cell Biology
and Cancer Chemotherapy
Springer-Verlag
Berlin Heidelberg New York 1979
Editorial Board
W. Bossert· H. J. Bremermann . J. D. Cowan' W. Hirsch
S. Karlin' J. B. Keller' M. Kimura' S. Levin (Managing Editor)
R. C. Lewontin . R. May· G. F. Oster' A S. Perelson
T. Poggio . L. A Segel
Author
Martin Eisen
Department of Mathematics
Temple University
Philadelphia, PA 19122
USA
AMS Subject Classifications (1970): 92A07
ISBN-13: 978-3-540-09709-9 e-ISBN-13: 978-3-642-93126-0
001: 10.1007/978-3-642-93126-0
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re
printing, re-use of illustrations, broadcasting, reproduction by photocopying
machine or similar means, and storage in data banks. Under § 54 of the
German Copyright Law where copies are made for other than private use,
a fee is payable to the publisher, the amount of the fee to be determined by
agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1979
2141/3140-543210
To my mother, Sarah.
PREFACE
The purpose of this book is to show how mathematics can be applied to improve
cancer chemotherapy. Unfortunately, most drugs used in treating cancer kill both
normal and abnormal cells. However, more cancer cells than normal cells can be
destroyed by the drug because tumor cells usually exhibit different growth kinetics
than normal cells. To capitalize on this last fact, cell kinetics must be studied
by formulating mathematical models of normal and abnormal cell growth. These
models allow the therapeutic and harmful effects of cancer drugs to be simulated
quantitatively. The combined cell and drug models can be used to study the effects
of different methods of administering drugs. The least harmful method of drug
administration, according to a given criterion, can be found by applying optimal
control theory.
The prerequisites for reading this book are an elementary knowledge of ordinary
differential equations, probability, statistics, and linear algebra. In order to
make this book self-contained, a chapter on cell biology and a chapter on control
theory have been included. Those readers who have had some exposure to biology may
prefer to omit Chapter I (Cell Biology) and only use it as a reference when
required. However, few biologists have been exposed to control theory. Chapter 7
provides a short, coherent and comprehensible presentation of this subject. The
concepts of control theory are necessary for a full understanding of Chapters 8 and
9. For readers not already familiar with control theory, the time required for the
mastery of Chapter 7 will be well spent since this powerful tool is applicable to
many other branches of biology.
The appendices provide a brief description of topics which are not treated in
detail in this book. These short topic outlines are more helpful in choosing a new
research direction than scattered references throughout the book.
Biology has become a quantitative science. It is hoped that this book will
interest biologists in mathematics and mathematicians in biology. A biologist will
find that mathematical models are absolutely essential for research in modern cell
kinetics. A mathematician will discover that there are many exciting, unsolved
mathematical problems in cell biology.
The author is grateful to Tom Slook and George Swan for their helpful comments
on Chapter 7 and 8 respectively, to Werner Duchting for contributing Appendix G,
to Simon Levin for his editorial work and Alan Perelson for his constructive sug
gestions for improving the manuscript. I am glad to record my thanks to Gerry
Sizemore and Mittie Davis for typing the book and Rae Ballou for tracing many
obscure references and obtaining copies of many papers. The author is indebted to
his wife, Carole, for helping with all phases of the manuscript and his daughter,
Debby, for proofreading the book and preparing the index.
Martin Eisen
University of Maryland School
of Medicine
Temple University
TABLE OF CONTENTS
INTRODUCTION 1
References 3
CHAPTER I CELLS 5
1.1 Introduction 5
1.2 Cell Organization - Protoplasm 6
1.3 Cellular Structures and their Function 8
1.4 The Life Cycle of Cells 20
1.5 Control of Cell Proliferation 27
1.6 Cancer 34
1.7 Metastasis and Invasion 38
References 41
CHAPTER II MODELLING AND CELL GROWTH 44
2.1 Introduction 44
2.2 Modelling Philosophy 44
2.3 Growth Laws 56
2.4 Two Compartment Growth 66
References 71
CHAPTER III SOME KINETIC CELL MODELS 73
3.1 Introduction 73
3.2 A Discrete Differential Model 73
3.3 Continuous Versions of the Takahashi-Kendall Equations 80
3.4 Solutions of Continuous Models 95
3.5 Another General Approach to Continuous Models 111
3.6 Trucco's Model 114
References 119
CHAPTER IV AUTORADIOGRAPHY 122
4.1 Introduction 122
4.2 Fractional Labelled Mitosis Curve 122
4.3 Mathematical Models for FLM Curves: Pulse Labelling 128
4.4 FLM Curves for Continuous Labelling 141
4.5 The Labelling Index 143
4.6 Discussion 147
References 148
VIII
CHAPTER V CELL SYNCHRONY 152
5.1 Introduction 152
5.2 Definition of Synchrony 152
5.3 Instantaneous Indices of Synchrony 153
5.4 Time-interval Indices of Synchrony 156
5.5 The Decay of Synchronization 160
5.6 Discussion 165
References 166
CHAPTER VI FLOW MICROFLUOROMETRY 167
6.1 Introduction 167
6.2 DNA Histogram: Steady-state and Constant Phase Length 168
6.3 Generation of DNA Histogram for Random Phase Lengths 172
6.4 Definition of an Asynchronous Population 177
6.5 Graphical Analysis of Asynchronous Populations 177
6.6 Analytic Analysis of Asynchronous Populations 178
6.7 Estimation of Mean Phase Lengths for Asynchronous Populations 184
6.8 Methods of Estimating Mean Cycle and Mitotic Time for
Asynchronous Populations 185
6.9 Analysis of Synchronous Populations. Single Histogram 191
6.10 Analysis of Synchronous Populations from Multiple Histograms 195
6.11 Rate of DNA Synthesis 209
6.12 Determination of Percentage of Cells in GO 210
6.13 Generalization of the Degree of Synchrony 212
6.14 Discussion 213
References 216
CHAPTER VII CONTROL THEORY 219
7.1 Introduction 219
7.2 External Description of Systems (input output relations) 222
7.3 Internal Description of Systems (State Space Description) 243
7.4 Optimal Control Theory 251
References 271
CHAPTER VIII TOWARDS MATHEMATICAL CHEMOTHERAPY 275
8.1 Introduction 275
8.2 Growth Laws and Cycle Nonspecific Cancer Chemotherapy 278
8.3 Cycle Specific Chemotherapy 295
8.4 Pharmacokinetics 307
8.5 Remarks 322
References 327
CHAPTER IX MATHEMATICAL MODELS OF LEUKOPOIESIS AND LEUKEMIA 333
9.1 Introduction 333
9.2 The Hemopoietic System and its Neoplasms 333
9.3 Steady State Models of the Hemopoietic System 343
9.4 Kinetic Model of Neutrophil Production 350
9.5 Acute Myeloblastic Leukemia 361
9.6 A Chemotherapy Model of AML 366
9.7 Models of Chronic Granulocytic Leukemia (CGL) 368
9.8 A Discrete Mathematical Model of Acute Lymphoblastic Leukemia 370
IX
9.9 A Comprehensive Computer Model of Granulopoiesis and Cancer
Chemotherapy 372
9.10 Discussion 381
References 382
APPENDIX A CHEMISTRY OF GENES. PROTEIN SYNTHESIS 386
1 • Introduction 386
2 . The Building Blocks of DNA and RNA 386
3. The Chemical Structure of DNA and RNA 387
4. The Replication of DNA 390
5 . The Genetic Code 391
6. Synthesis of RNA 392
7. Formation of Proteins 396
8. Defining the Gene 398
References 399
APPENDIX B VIRUSES 400
1 . Introduction 400
2. Structure of Viruses 400
3. Replication of Viruses 401
4. Oncogenic Viruses 402
References 403
APPENDIX C CELLULAR ENERGY 404
1 . Introduction 404
2 . Adenosine Triphosphate (ATP) 404
3. Formation of ATP 405
References 406
APPENDIX D IMMUNOLOGY 407
1 . The Immune System 407
2 . The Immune System and Cancer 409
References 413
APPENDIX E MATHEMATICAL THEORIES OF CARCINOGE~SIS 415
References 417
APPENDIX F RADIOLOGY AND CANCER 419
References 420
APPENDIX G APPLICATIONS OF CONTROL THEORY TO NORMAL AND
MALIGNANT CELL GROWTH 422
References 424
INDEX 426
INTRODUCTION
This year, approximately 350,000 Americans will die from cancer. The number
of new cases which develop yearly is twice this figure. Cancer strikes 7,000
children each year. The purpose of this monograph is to indicate how
mathematicians can play an important role in irradicating this disease.
Two well known methods used in cancer treatments are removal of the tumor
(surgery) and destruction of the tumor in situ by an external attacking agent
(chemotherapy, radiation) or an internal attacking agent (immunotherapy,
endocrinotherapy). A third, more subtle method, described in Chapter 1, tumor
regression, has not yet been used. However, sometimes a tumor which remains after
surgery, chemotherapy or immunotherapy disappears. There have even been instances
of tumor regression without any form of treatment. These spontaneous tumor
regressions may be the reason for the successes cited in controversial formsl of
therapy such as laetrile, vitamin C, meditation and so on.
The most frequently used treatment for cancer is surgery. Surgery is most
successful when combined with early diagnosis. Perhaps the most spectacular
example of surgical success is the reduction in deaths from cancer of the cervix
due to early detection by the Pap test.
Unfortunately, surgery has its limitations. The primary tumor may be
inoperable or may be so large that it is only partially removable. Moreover, if
metastases have occurred it is usually impossible to remove all of the secondary
2
growths. Several forms of cancer are disseminated and therefore cannot be
surgically removed (e.g. the leukemias which cause about 10% of all cancer deaths).
Radiation treatment is also limited for the same reasons.
Utilizing the body's own defense mechanism is a theoretically possible
treatment for disseminated cancer. However, much basic research is still required
in immunotherapy before this form of treatment becomes practical (See Appendix D.).
1
These treatments are controversial since they have not been tested in properly
designed trials on a single form of cancer.
2 More than 50% of present cancer deaths are due to nonlocalized or diffuse
neoplasms.
