Towards a Mathematical Theory of Complex Biological Systems

by Bianca, c.; Bellomo, N.

• ISBN13: 9789814340533
• ISBN10: 9814340537
• Format: Hardcover
• Copyright: 2011-01-12
• Publisher: World Scientific Pub Co Inc

Summary

This monograph has the ambitious aim of developing a mathematical theory of complex biological systems with special attention to the phenomena of ageing, degeneration and repair of biological tissues under individual self-repair actions that may have good potential in medical therapy.The approach to mathematically modeling biological systems needs to tackle the additional difficulties generated by the peculiarities of living matter. These include the lack of invariance principles, abilities to express strategies for individual fitness, heterogeneous behaviors, competition up to proliferative and/or destructive actions, mutations, learning ability, evolution and many others.Applied mathematicians in the field of living systems, especially biological systems, will appreciate the special class of integro-differential equations offered here for modeling at the molecular, cellular and tissue scales. A unique perspective is also presented with a number of case studies in biological modeling.

Table of Contents

Preface	p. v
Acknowledgments	p. vii
List of Figures	p. xiii
List of Tables	p. xvii
Looking for a Mathematical Theory of Biological Systems	p. 1
Introduction	p. 1
On the Concept of Mathematical Theory	p. 2
Plan of the Monograph	p. 3
On the Complexity of Biological Systems	p. 7
Ten Common Features of Living Systems	p. 7
Some Introductory Concepts of Systems Biology	p. 10
Reducing Complexity	p. 13
Immune System, Wound Healing Process, and System Biology	p. 15
The Immune System: A Phenomenological Overview	p. 17
Introduction	p. 17
Bacteria and Viruses	p. 18
The Immune System Components	p. 19
The Lymphatic System	p. 19
The White Blood Cells	p. 21
Antibodies and Hormones	p. 24
The Immune Response	p. 25
Innate Immunity	p. 26
Adaptive Immunity	p. 29
Immune System Diseases	p. 32
Critical Analysis	p. 35
Wound Healing Process and Organ Repair	p. 37
Introduction	p. 37
Genes and Mutations	p. 38
The Phases of Wound Healing	p. 43
Hemostasis Phase	p. 44
Inflammation Phase	p. 47
Proliferation Phase	p. 48
Maturation or Remodeling Phase	p. 49
The Fibrosis Disease	p. 50
Critical Analysis	p. 54
From Levels of Biological Organization to System Biology	p. 55
Introduction	p. 55
From Scaling to Mathematical Structures	p. 56
Guidelines to the Modeling Approach	p. 60
Mathematical Tools	p. 65
Mathematical Tools and Structures	p. 67
Introduction	p. 67
Mathematical Frameworks of the Kinetic Theory of Active Particles	p. 68
Guidelines Towards Modeling at the Molecular and Cellular Scales	p. 78
Additional Analysis Looking at the Immune Competition	p. 80
Critical Analysis	p. 85
Multiscale Modeling: Linking Molecular, Cellular, and Tissues Scales	p. 89
Introduction	p. 89
On the Phenomenological Derivation of Macroscopic Tissue Models	p. 91
Cellular-Tissue Scale Modeling of Closed Systems	p. 94
Asymptotic Methods for a Single Subsystem	p. 95
Asymptotic Methods for Binary Mixtures of Subsystems	p. 99
Cellular-Tissue Scale Modeling of Open Systems	p. 108
On the Molecular-Cellular Scale Modeling	p. 111
Critical Analysis	p. 113
Applications and Research Perspectives	p. 117
A Model for Malign Keloid Formation and Immune System Competition	p. 119
Introduction	p. 119
The Mathematical Model	p. 121
Simulations and Emerging Behaviors	p. 131
Sensitivity Analysis of the Progression Rate ¿	p. 132
Sensitivity Analysis of the Proliferation Rate ßI	p. 144
Sensitivity Analysis of the Initial Distributions	p. 147
Critical Analysis and Perspectives	p. 154
Macroscopic Models of Chemotaxis by KTAP Asymptotic Methods	p. 157
Introduction	p. 157
Linear Turning Kernels: Relaxation Models	p. 159
The Case of a Single Subsystem	p. 160
The Case of a Binary Mixture of Subsystems	p. 162
Cellular-Tissue Scale Models of Chemotaxis	p. 163
Classical Keller-Segel Type Models	p. 165
Optimal Drift Following the Chemoattractant	p. 165
Nonlinear Flux-Limited Model by the Mixed Scalings	p. 166
Critical Analysis	p. 168
Looking Ahead	p. 171
Introduction	p. 171
Some Challenges for Applied Mathematicians and Biologists	p. 172
How Far is the Mathematical Theory for Biological Systems	p. 173
Closure	p. 177
Mathematical Modeling of Space and Velocity-Dependent Systems	p. 179
Introduction	p. 179
Mathematical Tools for Homogeneous Activity Systems	p. 179
Mathematical Tools for Heterogeneous Activity Systems	p. 182
Glossary	p. 187
Bibliography	p. 195
Index	p. 205
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