Analysis of Some Mathematical Models of Cell Dynamics in Hematology

DESPRE CARTE

Bringing new contributions to science might be challenging, however, this is our path to evolution. We cannot assume that we know everything, but it is only our curiosity that can lead us to answers. Therefore, we can and we should keep trying, seeking new and different paths. There is always a starting point: asking yourself „Why?”, and the answer will follow...

The purpose of the present book is the study of some mathematical models of cell dynamics and convex optimization problems applied to chronic myeloid leukemia, a type of leukemia also known as chronic myelogenous leukemia. We take into consideration basic concepts, methods and results from the theory of differential equations, such as: existence, uniqueness, monotonicity, boundedness, continuous dependence on data and stability of solutions.

CONTENTS

INTRODUCTION 9

CHAPTER 1. MEDICAL BACKGROUND 15
1.1. Hematopoiesis 15
1.2. What is leukemia? 17
1.3. Introduction to chronic myeloid leukemia 17

CHAPTER 2. MATHEMATICAL MODELING CHRONIC MYELOID LEUKEMIA 21
2.1. The basic mathematical model 22
2.2. Existence, uniqueness and continuous dependence on data 24
2.3. Steady states and local stability 31
2.4. Global stability 34
2.5. Numerical simulations 37
2.6. The extended model to terminally differentiated cells 45

CHAPTER 3. OPTIMIZATION PROBLEMS IN CHRONIC MYELOID LEUKEMIA 49
3.1. The optimization problem 50
3.2. Optimal personalized dosing of a selective drug 53
3.3. Non-specific drug optimization problems 56
3.4. Optimal personalized dosing of a selective drug based on the extended model to terminally differentiated cells 63

CHAPTER 4. MATHEMATICAL MODELING OF STEM CELL TRANSPLANTATION IN CHRONIC MYELOID LEUKEMIA 69
4.1. The mathematical model 70
4.2. Existence, uniqueness and continuous dependence on data 71
4.3. Steady states and stability 79
4.4. Numerical simulations 94

APPENDIX 103

BIBLIOGRAPHY 109

INDEX 119