Dunn / Constantinides / Moghe Numerical Methods in Biomedical Engineering

1. Auflage 2005
ISBN: 978-0-08-047080-1
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 632 Seiten

Numerical Modeling in Biomedical Engineering brings together the integrative set of computational problem solving tools important to biomedical engineers. Through the use of comprehensive homework exercises, relevant examples and extensive case studies, this book integrates principles and techniques of numerical analysis. Covering biomechanical phenomena and physiologic, cell and molecular systems, this is an essential tool for students and all those studying biomedical transport, biomedical thermodynamics & kinetics and biomechanics.Supported by Whitaker Foundation Teaching Materials Program; ABET-oriented pedagogical layoutExtensive hands-on homework exercises

Dr. Dunn joined Rensselaer Polytechnic Institute in 2008 as Vice Provost and Dean of Graduate Education and full Professor in the School of Engineering. Dunn's experience includes developing university-wide initiatives in such areas as packaging engineering, water resource management, and homeland security. He also has extensive experience building academic programs, including overseeing the country's first engineering-based clinical training program in prosthetics and orthotics. Dunn has mentored 14 Ph.D. students, 23 M.S. students, and many undergraduate students. These students have come from biomedical engineering, electrical and computer engineering, computer science, mathematics, dentistry, as well as the M.D./Ph.D. program. The author of three books and 150 papers on different subjects including digital subtraction radiography, Dunn is a fellow of the American Institute of Medical and Biological Engineering. He is the founding editor-in-chief of the Journal of Applied Packaging Research, and has served as an editor and officer of several journals and professional organizations.

Zielgruppe

Academic/professional/technical: Research and professional

Autoren/Hrsg.

Dunn, Stanley

Constantinides, Alkis

Moghe, Prabhas V.

Inhaltsverzeichnis

1;Front cover;1
2;Title page;4
3;Copyright page;5
4;Table of contents;6
5;Preface;14
5.1;Organization and Outline of the Book;17
6;Part I: Fundamentals;20
6.1;Chapter 1 Modeling Biosystems;20
6.1.1;1.1 Biomedical Engineering;20
6.1.2;1.2 Fundamental Aspects of Biomedical Engineering;22
6.1.3;1.3 Constructing Engineering Models;22
6.1.3.1;1.3.1 A framework for problem solving;23
6.1.3.2;1.3.2 Formulating the mathematical expression of conservation;24
6.1.3.3;1.3.3 Using balance equations;26
6.1.4;1.4 Examples of Solving Biomedical Engineering Models by Computer;28
6.1.4.1;1.4.1 Modeling rtPCR efficiency;28
6.1.4.2;1.4.2 Modeling transcranial magnetic stimulation;32
6.1.4.3;1.4.3 Modeling cardiac electrophysiology;33
6.1.4.4;1.4.4 Using numerical methods to model the response of the cardiovascular system to gravity;34
6.1.5;1.5 Overview of the Text;36
6.1.5.1;1.5.1 Part I: Fundamentals;37
6.1.5.2;1.5.2 Part II: Steady-state behavior (algebraic models);38
6.1.5.3;1.5.3 Part III: Dynamic behavior (differential equations);38
6.1.5.4;1.5.4 Part IV: Modeling tools and applications;38
6.1.6;1.6 Lessons Learned in this Chapter;39
6.1.7;1.7 Problems;39
6.1.8;1.8 References;40
6.2;Chapter 2 Introduction to Computing;41
6.2.1;2.1 Introduction;41
6.2.2;2.2 The Role of Computers in Biomedical Engineering;42
6.2.3;2.3 Programming Language Tools and Techniques;45
6.2.3.1;2.3.1 Sequences of statements;45
6.2.3.2;2.3.2 Conditional execution;46
6.2.3.3;2.3.3 Iteration;52
6.2.3.4;2.3.4 Encapsulation;55
6.2.4;2.4 Fundamentals of Data Structures for MATLAB;57
6.2.4.1;2.4.1 Number representation;57
6.2.4.2;2.4.2 Arrays;59
6.2.4.3;2.4.3 Characters and strings;61
6.2.4.4;2.4.4 Logical or Boolean data types;62
6.2.4.5;2.4.5 Cells and cell arrays;64
6.2.4.6;2.4.6 Data structures not explicitly found in MATLAB;66
6.2.4.7;2.4.7 Data type conversion;68
6.2.5;2.5 An Introduction to Object-Oriented Systems;70
6.2.6;2.6 Analyzing Algorithms and Programs;75
6.2.6.1;2.6.1 Polynomial complexity;75
6.2.6.2;2.6.2 Operation counting;75
6.2.7;2.7 Lessons Learned in this Chapter;79
6.2.8;2.8 Problems;80
6.3;Chapter 3 Concepts of Numerical Analysis;83
6.3.1;3.1 Scientific Computing;83
6.3.2;3.2 Numerical Algorithms and Errors;84
6.3.3;3.3 Taylor Series;85
6.3.4;3.4 Keeping Errors Small;90
6.3.5;3.5 Floating-Point Representation in MATLAB;92
6.3.5.1;3.5.1 The IEEE 754 standard for floating-point representation;93
6.3.5.2;3.5.2 Floating-point arithmetic, truncation, and rounding;94
6.3.5.3;3.5.3 Roundoff error accumulation and cancellation error;96
6.3.6;3.6 Lessons Learned in this Chapter;98
6.3.7;3.7 Problems;99
6.3.8;3.8 References;101
7;Part II: Steady-State Behavior;102
7.1;Chapter 4 Linear Models of Biological Systems;102
7.1.1;4.1 Introduction;102
7.1.2;4.2 Examples of Linear Biological Systems;103
7.1.2.1;4.2.1 Force balance in biomechanics;103
7.1.2.2;4.2.2 Biomedical imaging and image processing;105
7.1.2.3;4.2.3 Metabolic engineering and cellular biotechnology;106
7.1.3;4.3 Simultaneous Linear Algebraic Equations;107
7.1.3.1;4.3.1 Illustration of simple Gauss elimination for a 3×3 matrix;107
7.1.3.2;4.3.2 Matrix notation of Gaussian elimination;108
7.1.4;4.4 The Gauss-Jordan Reduction Method;117
7.1.5;4.5 Iterative Approach for Solution of Linear Systems;122
7.1.5.1;4.5.1 The Jacobi method;122
7.1.5.2;4.5.2 The Gauss-Seidel method;127
7.1.6;4.6 Lessons Learned in this Chapter;131
7.1.7;4.7 Problems;131
7.1.8;4.8 References;133
7.2;Chapter 5 Nonlinear Equations in Biomedical Engineering;134
7.2.1;5.1 Introduction;134
7.2.2;5.2 General Form of Nonlinear Equations;135
7.2.3;5.3 Examples of Nonlinear Equations in Biomedical Engineering;137
7.2.3.1;5.3.1 Molecular bioengineering;137
7.2.3.2;5.3.2 Cellular and tissue engineering;138
7.2.3.3;5.3.3 Bioheat transport: photothermal therapy;139
7.2.3.4;5.3.4 Biomedical flow transport dynamics;140
7.2.4;5.4 The Method of Successive Substitution;141
7.2.5;5.5 The Method of False Position (Linear Interpolation);142
7.2.6;5.6 The Newton-Raphson Method;144
7.2.7;5.7 Newton’s Method for Simultaneous Nonlinear Equations;168
7.2.8;5.8 Lessons Learned in this Chapter;174
7.2.9;5.9 Problems;174
7.2.10;5.10 References;178
8;Part III: Dynamic Behavior;179
8.1;Chapter 6 Finite Difference Methods, Interpolation and Integration;179
8.1.1;6.1 Introduction;179
8.1.2;6.2 Symbolic Operators;180
8.1.3;6.3 Backward Finite Differences;183
8.1.4;6.4 Forward Finite Differences;188
8.1.5;6.5 Central Finite Differences;192
8.1.6;6.6 Interpolating Polynomials;194
8.1.7;6.7 Interpolation of Equally Spaced Points;198
8.1.7.1;6.7.1 Gregory-Newton interpolation;198
8.1.8;6.8 Interpolation of Unequally Spaced Points;207
8.1.8.1;6.8.1 Lagrange polynomials;207
8.1.8.2;6.8.2 Spline interpolation;208
8.1.9;6.9 Integration Formulas;209
8.1.10;6.10 The Newton-Cotes Formulas of Integration;210
8.1.10.1;6.10.1 The trapezoidal rule;211
8.1.10.2;6.10.2 Simpson’s 1/3 rule;213
8.1.10.3;6.10.3 Simpson’s 3/8 rule;214
8.1.10.4;6.10.4 Summary of Newton-Cotes integration;216
8.1.11;6.11 Lessons Learned in this Chapter;221
8.1.12;6.12 Problems;222
8.1.13;6.13 References;224
8.2;Chapter 7 Dynamic Systems: Ordinary Differential Equations;225
8.2.1;7.1 Introduction;225
8.2.1.1;7.1.1 Pharmacokinetics: the dynamics of drug absorption;226
8.2.1.2;7.1.2 Tissue engineering: cell differentiation, cell adhesion and migration dynamics;227
8.2.1.3;7.1.3 Metabolic Engineering: Glycolysis pathways of living cells;228
8.2.1.4;7.1.4 Transport of molecules across biological membranes;229
8.2.2;7.2 Classification of Ordinary Differential Equations;230
8.2.3;7.3 Transformation to Canonical Form;232
8.2.4;7.4 Nonlinear Ordinary Differential Equations;237
8.2.4.1;7.4.1 The Euler and modified Euler methods;237
8.2.4.2;7.4.2 The Runge-Kutta methods;240
8.2.4.3;7.4.3 Simultaneous differential equations;243
8.2.4.4;7.4.4 MATLAB functions for nonlinear equations;243
8.2.5;7.5 Linear Ordinary Differential Equations;249
8.2.5.1;7.5.1 Method using eigenvalues and eigenvectors;249
8.2.5.2;7.5.2 MATLAB functions for linear equations;251
8.2.6;7.6 Steady-State Solutions and Stability Analysis;257
8.2.7;7.7 Numerical Stability and Error Propagation;262
8.2.8;7.8 Advanced Examples;264
8.2.9;7.9 Lessons Learned in this Chapter;295
8.2.10;7.10 Problems;295
8.2.11;7.11 References;302
8.3;Chapter 8 Dynamic Systems: Partial Differential Equations;304
8.3.1;8.1 Introduction;304
8.3.2;8.2 Examples of PDEs in Biomedical Engineering;305
8.3.2.1;8.2.1 Diffusion across biological membranes;305
8.3.2.2;8.2.2 Diffusion of macromolecules and controlled release of drugs;307
8.3.2.3;8.2.3 Cell migration on vascular prosthetic materials;308
8.3.2.4;8.2.4 Fluid flow in physiological and extracorporeal vessels;308
8.3.3;8.3 Classification of Partial Differential Equations;309
8.3.4;8.4 Initial and Boundary Conditions;311
8.3.5;8.5 Solution of Partial Differential Equations;314
8.3.5.1;8.5.1 Elliptic partial differential equations;320
8.3.5.2;8.5.2 Parabolic partial differential equations;332
8.3.5.3;8.5.3 Hyperbolic partial differential equations;342
8.3.6;8.6 Polar Coordinate Systems;344
8.3.7;8.7 Stability Analysis;346
8.3.8;8.8 PDE Toolbox in MATLAB;346
8.3.9;8.9 Lessons Learned in this Chapter;353
8.3.10;8.10 Problems;353
8.3.11;8.11 References;358
9;Part IV: Modeling Tools and Applications;359
9.1;Chapter 9 Measurements, Models and Statistics;359
9.1.1;9.1 The Role of Numerical Methods;359
9.1.2;9.2 Measurements, Errors and Uncertainty;360
9.1.3;9.3 Descriptive Statistics;363
9.1.4;9.4 Inferential Statistics;370
9.1.5;9.5 Least Squares Modeling;377
9.1.6;9.6 Curve Fitting;383
9.1.6.1;9.6.1 Lagrange interpolating polynomials;383
9.1.6.2;9.6.2 Newton divided difference interpolating polynomials;384
9.1.6.3;9.6.3 Splines;386
9.1.7;9.7 Fourier Transforms;393
9.1.8;9.8 Lessons Learned in the Chapter;400
9.1.9;9.9 Problems;401
9.1.10;9.10 References;402
9.2;Chapter 10 Modeling Biosystems;403
9.2.1;10.1 Numerical Modeling of Bioengineering Systems;403
9.2.2;10.2 PhysioNet, PhysioBank, and PhysioToolkit;405
9.2.2.1;10.2.1 ECG simulation;405
9.2.2.2;10.2.2 Reading PhysioBank data;409
9.2.3;10.3 Signal Processing: EEG Data;411
9.2.4;10.4 Diabetes and Insulin Regulation;417
9.2.5;10.5 Renal Clearance;425
9.2.6;10.6 Correspondence Problems and Motion Estimation;428
9.2.7;10.7 PHYSBE Simulations;433
9.2.7.1;10.7.1 Coarctation of the aorta;436
9.2.7.2;10.7.2 Aortic stenosis;440
9.2.7.3;10.7.3 Ventricular septal defect;444
9.2.7.4;10.7.4 Left ventricular hypertrophy;448
9.2.8;10.8 References;454
10;Appendices;457
10.1;Appendix A: Introduction to MATLAB;457
10.1.1;A.1 The MATLAB Environment;458
10.1.1.1;A.1.1 Customizing the MATLAB environment;460
10.1.1.2;A.1.2 The MATLAB path;461
10.1.1.3;A.1.3 Where to find help for MATLAB;461
10.1.2;A.2 Elementary Operations;464
10.1.3;A.3 Vectors and Matrices;467
10.1.3.1;A.3.1 MATLAB construction functions for special arrays;470
10.1.3.2;A.3.2 Array arithmetic;471
10.1.4;A.4 MATLAB Built-in Functions;474
10.1.5;A.5 Graphics;476
10.1.5.1;2-D graphs;476
10.1.5.2;3-D graphs;478
10.1.5.3;2½-D Graphs;480
10.1.5.4;Interactive Plot Creation;482
10.1.6;A.6 Scripts and Functions;483
10.1.7;A.7 Flow Control;486
10.1.8;A.8 Display, Export, and Import of Data;488
10.1.8.1;A.8.1 Displaying data and results;488
10.1.8.2;A.8.2 Saving and loading data;490
10.1.8.3;A.8.3 Generating data in a program and saving into a file;494
10.1.9;A.9 Symbolic Computation;495
10.1.9.1;A.9.1 Symbolic solution of algebraic equations;495
10.1.9.2;A.9.2 Symbolic solution of differential equations;497
10.1.9.3;A.9.3 Symbolic differentiation;499
10.1.9.4;A.9.4 Symbolic integration;499
10.1.10;A.10 MATLAB Toolboxes;500
10.1.11;A.11 References;500
10.2;Appendix B: Introduction to Simulink;501
10.2.1;B.1 Dynamic System Simulation;502
10.2.2;B.2 Getting Started;503
10.2.2.1;B.2.1 A Simulink model of a sine wave generator;504
10.2.2.2;B.2.2 Modifying Simulink models;508
10.2.3;B.3 The Simulink Block Libraries;513
10.2.4;B.4 Constructing Models;518
10.2.4.1;B.4.1 Algebraic operations, signal routing and MATLAB variables;518
10.2.4.2;B.4.2 Simultaneous differential equations;521
10.2.4.3;B.4.3 PHYSBE and subsystems;523
10.2.5;B.5 References;530
10.3;Appendix C: Review of Linear Algebra and Related MATLAB Commands;531
10.3.1;C.1 Matrix and Vector Operations;531
10.3.2;C.2 Matrix Factorization;535
10.4;Appendix D: Analytical Solutions of Differential Equations;540
10.4.1;D.1 Ordinary Differential Equations of First Order;541
10.4.1.1;D.1.1 Equations with separable variables;541
10.4.1.2;D.1.2 Equations with homogeneous coefficients;543
10.4.1.3;D.1.3 Exact equations;545
10.4.1.4;D.1.4 Linear equations and the integrating factor;547
10.4.1.5;D.1.5 Nonlinear equations and the integrating factor;549
10.4.2;D.2 Ordinary Differential Equations of Higher Order;550
10.4.2.1;D.2.1 Linear homogeneous equations with constant coefficients;550
10.4.2.2;D.2.2 Linear nonhomogeneous equations (constant coefficients);553
10.4.3;D.3 Partial Differential Equations with Separable Variables;556
10.4.3.1;D.3.1 The diffusion equation;556
10.4.3.2;D.3.2 The potential equation;562
10.4.3.3;D.3.3 Periodic functions and the Fourier series;569
10.4.3.3.1;D.3.3.1 Even and odd symmetric functions;570
10.4.4;D.4 Laplace Transform Methods;572
10.4.4.1;D.4.1 The Laplace transform;573
10.4.4.2;D.4.2 Solution of ordinary differential equations;577
10.4.4.3;D.4.3 Solution of partial differential equations;580
10.4.5;D.5 References;587
10.5;Appendix E: Numerical Stability and Other Topics;588
10.5.1;E.1 Stability of the Euler Methods;589
10.5.2;E.2 Stability of the Runge-Kutta Methods;596
10.5.3;E.3 Stability of Multistep Methods;598
10.5.4;E.4 Stability of Methods for Partial Differential Equations;599
10.5.5;E.5 Step Size Control;603
10.5.6;E.6 Stiff Differential Equations;604
10.5.7;E.7 References;605
11;Index;606

Leseproben

Preface The purpose of this textbook is to serve as an introductory overview of computational tools to solve numerical problems in the rapidly emerging discipline of biomedical engineering. Despite the popularity of bioengineering as a major in engineering, only a handful of textbooks have been written primarily for the instruction of undergraduates in bioengineering, none of which are in the area of numerical methods in biomedical engineering. Addressing this void was one of the driving forces for the current effort. This book is intended as the primary text for an undergraduate course in biomedical engineering. The authors have adopted the book for the Fall semester junior course on Numerical Methods in the Department of Biomedical Engineering at Rutgers University—the book could be easily adopted for either semester of the junior year as well as for the senior year in BME. If the bioengineering concepts are somewhat de-emphasized and the calculus offerings are adjusted, it could also be adopted in the sophomore class. The book assumes that students have prerequisite skills in Calculus (I-IV), freshman Chemistry and Physics, General Biology, and an Introduction to Biomedical Engineering. The Numerical Methods course using this book may be offered in parallel with the treatment of junior topics such as Biomedical Transport Phenomena, Biomedical Thermodynamics/Kinetics, Biomechanics, and Bioinstrumentation. This book is well suited to train bioengineers interested in all major subfields within biomedical engineering, because it addresses a wide range of biosystems topics. The book can additionally be used as a text for quantitative biology curriculum aimed at life scientists (cell biologists, biomaterials scientists, and biochemists). Fig. 1 illustrates our philosophy of the role that computing and numerical methods plays in the education of modern-day biomedical engineers. Placed early in the junior year, the course serves as a focal point for integrating fundamentals and problem-solving skills in the context of biomedical applications. The course satisfies two major goals: assimilating computing tools within the student’s tool-kit, and applying these tools to a wide range of numerical models encountered in modern biomedical engineering. Figure 1 Numerical methods as an enabling pathway from engineering principles to biomedical applications. A major challenge in biomedical engineering is the immense scope of biomedically relevant problems, which range from the molecular scale to the macroscopic scale and have classically been treated by different “breeds” of engineers: chemical engineers handling the molecular and tissue engineering problems; mechanical engineers tackling the biomechanics aspects; and electrical engineers treating imaging and measurement problems. We have sought to unify these traditionally disparate approaches into an integrated overview of the major problems encountered by bioengineers. Several classes of problems that may require the application of numerical methods for their solution are tabulated in Table 1. Table 1 Bioengineering problems from a diverse range of system scales can be effectively treated through the use of the numerical and modeling techniques that are discussed in this book. Organization and Outline of the Book
The organization of the book is illustrated in Table 2. The book is divided into four parts, followed by the Appendices, which should serve as a comprehensive resource. Part I: Fundamentals: Introduces the student to the nature and behavior of physiological systems, and how to apply mathematical modeling techniques to these systems in order to develop models that may be used to simulate their behavior. This section also discusses the different types of models and relates them to numerical methods that may be used for their solution. The use of computing languages and computer programs to solve such problems is explained. Part II: Steady-State Behavior: Treats the analysis of systems that result in algebraic models (both linear and nonlinear), develops the numerical methods necessary for solution of such systems, and applies the numerical methods to several examples drawn from physiological, cell, and molecular systems. The classical numerical methods, such as Gauss elimination, Gauss-Jordan reduction, and Gauss-Seidel substitution methods for linear systems, and linear interpolation, Newton-Raphson, and Newton’s methods for simultaneous nonlinear systems, are presented to the reader, discussed, and applied. Part III: Dynamic Behavior: Concentrates on the analysis of dynamic and multidimensional systems whose models contain ordinary and partial differential equations. Finite difference methods are used to develop the integration and differentiation algorithms. The classical techniques, such as quadrature and Newton-Cotes formulas for integration and Euler and Runge-Kutta methods for ordinary differential equations, are presented and applied. The stability of several finite difference methods applied to the solution of partial differential equations are analyzed and discussed. Part IV: Modeling Tools and Applications: The solution of the mathematical models of complex biosystems requires a combination of several of the methods discussed in the previous sections of the book. Appendices: Appendix A offers a tutorial introduction to MATLAB. This appendix is strongly recommended for students who need a review of the language and capabilities of MATLAB. Appendix B focuses on the Simulink environment within MATLAB. Appendix C reviews the foundations of linear algebra, which are integral to the numerical methods discussed in Chapters 4 and 6. Appendix D offers analytical approaches for solutions of ordinary and partial differential equations—a summary that will be helpful to the students and instructors to allow validation of the numerical techniques discussed in Chapters 7 and 8. Appendix E offers an overview of topics related to numerical stability. Table 2 Overview of Numerical Methods in Biomedical Engineering. Part I: Fundamentals Chapter 1. Modeling Biosystems Biomedical Engineering Fundamental Aspects of Biomedical Engineering Constructing Engineering Models Examples of Solving Biomedical Engineering Models by Computer Overview of the Text Chapter 2. Introduction to Computing Introduction The Role of Computers in Biomedical Engineering Programming Language Tools and Techniques Fundamentals of Data Structures for MATLAB An Introduction to Object-Oriented Systems Analyzing Algorithms and Programs Chapter 3. Concepts of Numerical Analysis Scientific Computing Numerical Algorithms and Errors Taylor Series, Keeping Errors Small Floating-Point Representation in MATLAB Part II: Steady-State Behavior Chapter 4. Linear Biological Systems Examples of Linear Biological Systems Simultaneous Linear Algebraic Equations Gauss elimination, Gauss-Jordan reduction, iterative methods Iterative Approach for Solution of Linear Systems Jacobi and Gauss-Seidel methods Applications Force balance in biomechanics Biomedical imaging and image processing Metabolic engineering and cellular biotechnolgy Chapter 5. Nonlinear Biological Systems General Form of Nonlinear Equations Examples of Nonlinear Equations in Biomedical Engineering The Method of Successive Substitution The Method of False Position (Linear Interpolation) The Newton-Raphson Method Newton’s Method for Simultaneous Nonlinear Equations Applications: Friction factor in a catheter Michaelis-Menten kinetics Ventricular pressure measurements Receptor-ligand dynamics Part III: Dynamic Behavior Chapter 6. Finite Difference Methods: Interpolation, Differentiation, and...