Cover image for Towards a mathematical theory of complex biological systems
Title:
Towards a mathematical theory of complex biological systems
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Series:
Series in mathematical biology and medicine ; 11
Physical Description:
xvii, 208 pages : illustrations ; 24 cm.
ISBN:
9789814340533
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30000010280476 QH323.5 B53 2011 Open Access Book Book
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Summary

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

Prefacep. v
Acknowledgmentsp. vii
List of Figuresp. xiii
List of Tablesp. xvii
1 Looking for a Mathematical Theory of Biological Systemsp. 1
1.1 Introductionp. 1
1.2 On the Concept of Mathematical Theoryp. 2
1.3 Plan of the Monographp. 3
2 On the Complexity of Biological Systemsp. 7
2.1 Ten Common Features of Living Systemsp. 7
2.2 Some Introductory Concepts of Systems Biologyp. 10
2.3 Reducing Complexityp. 13
Immune System, Wound Healing Process, and System Biologyp. 15
3 The Immune System: A Phenomenological Overviewp. 17
3.1 Introductionp. 17
3.2 Bacteria and Virusesp. 18
3.3 The Immune System Componentsp. 19
3.3.1 The Lymphatic Systemp. 19
3.3.2 The White Blood Cellsp. 21
3.3.3 Antibodies and Hormonesp. 24
3.4 The Immune Responsep. 25
3.4.1 Innate Immunityp. 26
3.4.2 Adaptive Immunityp. 29
3.5 Immune System Diseasesp. 32
3.6 Critical Analysisp. 35
4 Wound Healing Process and Organ Repairp. 37
4.1 Introductionp. 37
4.2 Genes and Mutationsp. 38
4.3 The Phases of Wound Healingp. 43
4.3.1 Hemostasis Phasep. 44
4.3.2 Inflammation Phasep. 47
4.3.3 Proliferation Phasep. 48
4.3.4 Maturation or Remodeling Phasep. 49
4.4 The Fibrosis Diseasep. 50
4.5 Critical Analysisp. 54
5 From Levels of Biological Organization to System Biologyp. 55
5.1 Introductionp. 55
5.2 From Scaling to Mathematical Structuresp. 56
5.3 Guidelines to the Modeling Approachp. 60
Mathematical Toolsp. 65
6 Mathematical Tools and Structuresp. 67
6.1 Introductionp. 67
6.2 Mathematical Frameworks of the Kinetic Theory of Active Particlesp. 68
6.3 Guidelines Towards Modeling at the Molecular and Cellular Scalesp. 78
6.4 Additional Analysis Looking at the Immune Competitionp. 80
6.5 Critical Analysisp. 85
7 Multiscale Modeling: Linking Molecular, Cellular, and Tissues Scalesp. 89
7.1 Introductionp. 89
7.2 On the Phenomenological Derivation of Macroscopic Tissue Modelsp. 91
7.3 Cellular-Tissue Scale Modeling of Closed Systemsp. 94
7.3.1 Asymptotic Methods for a Single Subsystemp. 95
7.3.2 Asymptotic Methods for Binary Mixtures of Subsystemsp. 99
7.4 Cellular-Tissue Scale Modeling of Open Systemsp. 108
7.5 On the Molecular-Cellular Scale Modelingp. 111
7.6 Critical Analysisp. 113
Applications and Research Perspectivesp. 117
8 A Model for Malign Keloid Formation and Immune System Competitionp. 119
8.1 Introductionp. 119
8.2 The Mathematical Modelp. 121
8.3 Simulations and Emerging Behaviorsp. 131
8.3.1 Sensitivity Analysis of the Progression Rate ¿p. 132
8.3.2 Sensitivity Analysis of the Proliferation Rate ß Ip. 144
8.3.3 Sensitivity Analysis of the Initial Distributionsp. 147
8.4 Critical Analysis and Perspectivesp. 154
9 Macroscopic Models of Chemotaxis by KTAP Asymptotic Methodsp. 157
9.1 Introductionp. 157
9.2 Linear Turning Kernels: Relaxation Modelsp. 159
9.2.1 The Case of a Single Subsystemp. 160
9.2.2 The Case of a Binary Mixture of Subsystemsp. 162
9.3 Cellular-Tissue Scale Models of Chemotaxisp. 163
9.3.1 Classical Keller-Segel Type Modelsp. 165
9.3.2 Optimal Drift Following the Chemoattractantp. 165
9.3.3 Nonlinear Flux-Limited Model by the Mixed Scalingsp. 166
9.4 Critical Analysisp. 168
10 Looking Aheadp. 171
10.1 Introductionp. 171
10.2 Some Challenges for Applied Mathematicians and Biologistsp. 172
10.3 How Far is the Mathematical Theory for Biological Systemsp. 173
10.4 Closurep. 177
Appendix A Mathematical Modeling of Space and Velocity-Dependent Systemsp. 179
A.l Introductionp. 179
A.2 Mathematical Tools for Homogeneous Activity Systemsp. 179
A.3 Mathematical Tools for Heterogeneous Activity Systemsp. 182
Glossaryp. 187
Bibliographyp. 195
Indexp. 205