Modelling, Analysis and Optimization of Biosystems

By: Werner Krabs

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Hardcover | 10 August 2007

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216 Pages

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23.5 x 15.5 x 1.27

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Mathematical models in biology and medicine cannot be based on natural laws as it is the case with physics and chemistry. This is due to the fact that biological and medical processes are concerned with living organisms. Mathematical models, however, can be used as a language by which certain aspects of biological or medical processes can be expressed. In general, several mathematical models can be designed in order to describe a biological or medical process and there is no unique criterion which model gives the best description. This book presents several of these models and shows applications of them to different biological and medical problems. The book shows that operations research expertise is necessary in respect to modeling, analysis and optimization of biosystems.

Industry Reviews

From the reviews:

"The impact of mathematical modeling and analysis on biological applications has been significant. But the area of optimization is just beginning to be applied to biosystems, and this book illustrates some possible applications and techniques. ... An undergraduate student interested in population models and game theory could benefit from this book. ... This book would be a valuable source for game theory and examples of optimization of parameters, and for optimal control ... ." (Suzanne Lenhart, SIAM Review, Vol. 50 (3), 2008)

"The book comprises four lengthy chapters, with liberal use of sub-sections to make the whole more easily digestible. ... The book is clearly laid out ... . Each chapter has a short list of references ... . For those who seek a detailed introduction into mathematical modelling of biological systems this book would be a most useful resource." (B. Spedding, Journal of the Operational Research Society, Vol. 60 (2), 2009)

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Non-Fiction Science Biology, Life Sciences Business & Management Operational Research Mathematics Applied Mathematics Medicine Clinical & Internal Medicine Haematology Science in General Maths for Scientists

Engineering & Technology Energy Technology & Engineering Electrical Engineering Electronics & Communications Engineering Electronics Engineering Automatic Control Engineering

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Paperback

Published: 15th October 2010

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Preface	p. V
Growth Models	p. 1
A Growth Model for one Population	p. 1
Interacting Growth of two Populations	p. 9
Interacting Growth of n [greater than or equal] 2 Populations	p. 15
Discretization of the Time-Continuous Model	p. 23
The n-Population Model	p. 23
The One-Population Model	p. 33
Determination of Model Parameters from Data	p. 36
References	p. 39
A Game-Theoretic Evolution Model	p. 41
Evolution-Matrix-Games for one Population	p. 41
The Game and Evolutionarily Stable Equilibria	p. 41
Characterization of Evolutionarily Stable Equilibria	p. 45
Evolutionarily Stable Equilibria for 2x2-Matrices	p. 50
On the Detection of Evolutionarily Stable Equilibria	p. 52
A Dynamical Treatment of the Game	p. 57
Existence and Iterative Calculation of Nash Equilibria	p. 62
Zero-Sum Evolution Matrix Games	p. 74
Evolution-Bi-Matrix-Games for two Populations	p. 79
The Game and Evolutionarily Stable Equilibria	p. 79
A Dynamical Treatment of the Game	p. 83
Existence and Iterative Calculation of Nash Equilibria	p. 88
A Direct Method for the Calculation of Nash Equilibria	p. 93
References	p. 102
Four Models of Optimal Control in Medicine	p. 103
Controlled Growth of Cancer Cells	p. 103
Optimal Administration of Drugs	p. 111
A One-Compartment Model	p. 112
A Two-Compartment Model	p. 114
Optimal Control of Diabetes Mellitus	p. 119
The Model	p. 119
On the Approximate Solution of the Model Problem	p. 121
A Time-Discrete Diabetes Model	p. 124
An Exact Solution of the Model Problem	p. 127
Optimal Control Aspects of the Blood Circulation in the Heart	p. 130
Blood Circulation in the Heart	p. 130
A Model of the Left-Ventricular Ejection Dynamics	p. 130
An Optimal Control Problem	p. 132
Another Model of the Left-Ventricular Ejection Dynamics	p. 137
References	p. 139
A Mathematical Model of Hemodialysis	p. 141
A One-Compartment Model	p. 141
The Mass Transport in the Dialyzer	p. 141
The Temporal Development of the Toxin Concentration in the Blood without Ultrafiltration	p. 143
The Temporal Development of the Toxin Concentration in the Blood with Ultrafiltration	p. 148
A Two-Compartment Model	p. 152
Derivation of the Model Equations	p. 152
Determination of the Clearance of the Cell Membranes for Urea	p. 154
Computation of Periodic Toxin Concentrations	p. 158
The General Method	p. 158
The Case of Constant Clearance of the Dialyzer	p. 162
Discretization of the Model Equations	p. 163
Numerical Results for Urea	p. 167
The Influence of the Urea Generation Rate	p. 170
Determination of the Urea Generation Rate and the Rest Clearance of the Kidneys	p. 17
A Three-Compartment Model	p. 173
Motivation and Derivation of the Model Equations	p. 173
Determination of the Clearance of the Cell Membranes of the Brain	p. 175
Computation of Periodic Urea Concentration Curves	p. 176
Numerical Results	p. 182
References	p. 183
Appendix	p. 185
A Problem of Optimal Control	p. 185
The Problem	p. 185
A Multiplier Rule	p. 186
Existence of Positive Periodic Solutions in a General Diffusion Model	p. 189
The Model	p. 189
An Existence and Unicity Theorem	p. 190
Asymptotic Stability of Fixed Points	p. 195
Index	p. 201
Table of Contents provided by Ingram. All Rights Reserved.

Modelling, Analysis and Optimization of Biosystems

At a Glance

Hardcover

Industry Reviews

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Other Editions and Formats

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