Numerical Methods in Biomedical Engineering

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ISBN: 9780121860318 | 0121860310
Cover: Hardcover
Copyright: 11/7/2005

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This book shows how to use the principles and techniques of numerical analysis to solve problems in the rapidly evolving field of biomedical engineering. Covering the important aspects of bioengineering specialties from biomechanical phenomena to organ, cell and molecular systems, it is an essential tool for those studying biomedical instrumentation, biomedical transport, biomedical thermodynamics and kinetics, as well as biomechanics.

Preface	p. xiii
Fundamentals
Modeling Biosystems	p. 1
Biomedical Engineering	p. 1
Fundamental Aspects of Biomedical Engineering	p. 3
Constructing Engineering Models	p. 3
A framework for problem solving	p. 4
Formulating the mathematical expression of conservation	p. 5
Using balance equations	p. 7
How conservation laws lead to the Nernst equation	p. 7
Examples of Solving Biomedical Engineering Models by Computer	p. 9
Modeling rtPCR efficiency	p. 9
Modeling transcranial magnetic stimulation	p. 13
Modeling cardiac electrophysiology	p. 14
Using numerical methods to model the response of the cardiovascular system to gravity	p. 15
Overview of the Text	p. 17
Part I: Fundamentals	p. 18
Part II: Steady-state behavior (algebraic models)	p. 19
Part III: Dynamic biosystems (differential equations)	p. 19
Part IV: Modeling tools and applications	p. 19
Lessons Learned in this Chapter	p. 20
Problems	p. 20
References	p. 21
Introduction to Computing	p. 23
Introduction	p. 23
The Role of Computers in Biomedical Engineering	p. 24
Programming Language Tools and Techniques	p. 27
Sequences of statements	p. 27
Programs that are sequences of statements	p. 28
Conditional execution	p. 28
Simple control flow using if...then...else	p. 29
Use of the switch statement	p. 31
Iteration	p. 34
The use of while loops	p. 34
Using for...end loops	p. 36
Encapsulation	p. 37
Using scripts and functions	p. 37
Fundamentals of Data Structures for MATLAB	p. 39
Number representation	p. 39
Number representation in MATLAB	p. 40
Complex numbers	p. 40
Arrays	p. 41
Indexing arrays in MATLAB	p. 41
Characters and strings	p. 43
Character strings as arrays	p. 43
Logical or Boolean data types	p. 44
Logical indexing in MATLAB	p. 44
Cells and cell arrays	p. 46
Cell arrays and mixed data types	p. 46
Structure arrays and mixed data types	p. 48
Data structures not explicitly found in MATLAB	p. 48
Data structures in MATLAB: implementing a stack	p. 49
Data type conversion	p. 50
Data type conversion	p. 51
An Introduction to Object-Oriented Systems	p. 52
Simple object-oriented programs that are sequences of statements	p. 55
Analyzing Algorithms and Programs	p. 57
Polynomial complexity	p. 57
Operation counting	p. 57
Measuring execution time as a function of the amount of data	p. 59
Lessons Learned in this Chapter	p. 61
Problems	p. 62
Concepts of Numerical Analysis	p. 65
Scientific Computing	p. 65
Numerical Algorithms and Errors	p. 66
Taylor Series	p. 67
How truncation errors and roundoff errors arise	p. 68
Keeping Errors Small	p. 72
An ill-posed problem	p. 72
Floating-Point Representation in MATLAB	p. 74
The IEEE 754 standard for floating-point representation	p. 75
IEEE 754 floating-point representation	p. 75
Floating-point arithmetic, truncation, and rounding	p. 76
Propagation of floating-point errors	p. 77
Machine precision in MATLAB	p. 77
Roundoff error accumulation and cancellation error	p. 78
Avoiding overflow	p. 78
Avoiding cancellation errors	p. 79
Using Taylor series expansions to avoid cancellation errors	p. 79
Lessons Learned in this Chapter	p. 80
Problems	p. 81
References	p. 83
Steady-State Behavior
Linear Models of Biological Systems	p. 85
Introduction	p. 85
Examples of Linear Biological Systems	p. 86
Force balance in biomechanics	p. 86
Biomedical imaging and image processing	p. 88
Metabolic engineering and cellular biotechnology	p. 89
Simultaneous Linear Algebraic Equations	p. 90
Illustration of simple Gauss elimination for a 3 x 3 matrix	p. 90
Matrix notation of Gaussian elimination	p. 91
Application of the Gauss elimination method	p. 98
The Gauss-Jordan Reduction Method	p. 100
Application of the Gauss-Jordan reduction method	p. 102
Iterative Approach for Solution of Linear Systems	p. 105
The Jacobi method	p. 105
Application of the iterative Jacobi method	p. 107
The Gauss-Seidel method	p. 110
Application of the iterative Gauss-Seidel method	p. 111
Lessons Learned in this Chapter	p. 114
Problems	p. 114
References	p. 116
Nonlinear Equations in Biomedical Engineering	p. 117
Introduction	p. 117
General Form of Nonlinear Equations	p. 118
Examples of Nonlinear Equations in Biomedical Engineering	p. 120
Molecular bioengineering	p. 120
Cellular and tissue engineering	p. 121
Bioheat transport: photothermal therapy	p. 122
Biomedical flow transport dynamics	p. 123
The Method of Successive Substitution	p. 124
The Method of False Position (Linear Interpolation)	p. 125
The Newton-Raphson Method	p. 127
Cardiovascular physiology	p. 130
Solution of the Colebrook equation using Newton-Raphson	p. 136
Successive substitution method for solution of nonlinear equation	p. 141
Solution of the Colebrook equation using linear interpolation	p. 145
Solution of a Michaelis-Menten kinetics equation using the Newton-Raphson method	p. 150
Newton's Method for Simultaneous Nonlinear Equations	p. 151
Determination of receptor occupancy during receptorligand dynamics	p. 153
Lessons Learned in this Chapter	p. 157
Problems	p. 157
References	p. 161
Dynamic Behavior
Finite Difference Methods, Interpolation and Integration	p. 163
Introduction	p. 163
Symbolic Operators	p. 164
Backward Finite Differences	p. 167
Express the first-order derivative in terms of backward finite differences with error of order h	p. 169
Express the first-order derivative in terms of backward finite differences with error of order h[superscript 2]	p. 170
Forward Finite Differences	p. 172
Express the first-order derivative in terms of forward finite differences with error of order h	p. 173
Express the second-order derivative in terms of forward finite differences with error of order h	p. 173
Central Finite Differences	p. 176
Express the first-order derivative in terms of central finite differences with error of order h[superscript 2]	p. 176
Express the second-order derivative in terms of central finite differences with error of order h[superscript 2]	p. 177
Interpolating Polynomials	p. 178
Interpolation of Equally Spaced Points	p. 182
Gregory-Newton interpolation	p. 182
Gregory-Newton method for interpolation of equally spaced data	p. 186
Interpolation of Unequally Spaced Points	p. 191
Lagrange polynomials	p. 191
Spline interpolation	p. 192
Integration Formulas	p. 193
The Newton-Cotes Formulas of Integration	p. 194
The trapezoidal rule	p. 195
Simpson's 1/3 rule	p. 197
Simpson's 3/8 rule	p. 198
Summary of Newton-Cotes integration	p. 200
Integration formulas-Trapezoidal and Simpson's 1/3 rules	p. 201
Lessons Learned in this Chapter	p. 205
Problems	p. 206
References	p. 208
Dynamic Systems: Ordinary Differential Equations	p. 209
Introduction	p. 209
Pharmacokinetics: the dynamics of drug absorption	p. 210
Tissue engineering: cell differentiation, cell adhesion and migration dynamics	p. 211
Metabolic Engineering: Glycolysis pathways of living cells	p. 212
Transport of molecules across biological membranes	p. 213
Classification of Ordinary Differential Equations	p. 214
Transformation to Canonical Form	p. 216
Transformation of ordinary differential equations into their canonical form	p. 218
Nonlinear Ordinary Differential Equations	p. 221
The Euler and modified Euler methods	p. 221
The Runge-Kutta methods	p. 224
Simultaneous differential equations	p. 227
MATLAB functions for nonlinear equations	p. 227
Solution of enzyme catalysis reactions	p. 229
Linear Ordinary Differential Equations	p. 233
Method using eigenvalues and eigenvectors	p. 233
MATLAB functions for linear equations	p. 235
The dynamics of drug absorption	p. 236
Steady-State Solutions and Stability Analysis	p. 241
Numerical Stability and Error Propagation	p. 246
Advanced Examples	p. 248
Metabolic engineering: Modeling the glycolysis pathways of living cells	p. 248
The dynamics of membrane and nerve cell potentials	p. 255
The dynamics of stem cell differentiation	p. 264
Tissue engineering: models of epidermal cell migration	p. 271
Lessons Learned in this Chapter	p. 279
Problems	p. 279
References	p. 286
Dynamic Systems: Partial Differential Equations	p. 289
Introduction	p. 289
Examples of PDEs in Biomedical Engineering	p. 290
Diffusion across biological membranes	p. 290
Diffusion of macromolecules and controlled release of drugs	p. 292
Cell migration on vascular prosthetic materials	p. 293
Fluid flow in physiological and extracorporeal vessels	p. 293
Classification of Partial Differential Equations	p. 294
Initial and Boundary Conditions	p. 296
Solution of Partial Differential Equations	p. 299
Elliptic partial differential equations	p. 305
Solution of the Laplace and Poisson equations	p. 310
Parabolic partial differential equations	p. 317
Migration of human leukocytes on prosthetic materials	p. 321
Hyperbolic partial differential equations	p. 327
Polar Coordinate Systems	p. 329
Stability Analysis	p. 331
PDE Toolbox in MATLAB	p. 331
Solution of Fick's second law of diffusion using the PDE toolbox	p. 332
Lessons Learned in this Chapter	p. 338
Problems	p. 338
References	p. 343
Modeling Tools and Applications
Measurements, Models and Statistics	p. 345
The Role of Numerical Methods	p. 345
Measurements, Errors and Uncertainty	p. 346
Descriptive Statistics	p. 349
Computing statistics of MRI and CT image intensities	p. 351
Inferential Statistics	p. 356
Estimating the mean value of a population from a sample	p. 357
Hypothesis testing in DNA microarray analysis	p. 383
Least Squares Modeling	p. 363
Least square fit of a first-order polynomial (straight line)	p. 365
Least squares fit of a cubic polynomial	p. 367
Least squares fit of a nonlinear model	p. 367
Least squares fit of a multivariate model	p. 368
Curve Fitting	p. 369
Lagrange interpolating polynomials	p. 369
Newton divided difference interpolating polynomials	p. 370
Splines	p. 372
Resampling and baseline correction of MALDI-TOF mass spectra data	p. 374
Fourier Transforms	p. 379
Separating EEG frequency components	p. 382
Lessons Learned in the Chapter	p. 386
Problems	p. 387
References	p. 388
Modeling Biosystems: Applications	p. 389
Numerical Modeling of Bioengineering Systems	p. 389
PhysioNet, PhysioBank, and PhysioToolkit	p. 391
ECG simulation	p. 391
Using the MATLAB script ECGwaveGen to synthesize ECG data	p. 391
Reading PhysioBank data	p. 395
Read and visualize PhysioBank signals and annotations	p. 395
Signal Processing: EEG Data	p. 397
Differential brain activity in the left and right hemispheres	p. 400
Diabetes and Insulin Regulation	p. 403
Simulink model of glucose regulation	p. 405
Renal Clearance	p. 411
Renal clearance	p. 412
Correspondence Problems and Motion Estimation	p. 414
Estimating motion from features on a rigid body	p. 415
Physbe Simulations	p. 419
Normal PHYSBE operation	p. 420
Coarctation of the aorta	p. 422
Simulink model of coarctation of the aorta	p. 423
Aortic stenosis	p. 426
Simulink model of aortic valve stenosis	p. 427
Ventricular septal defect	p. 430
Ventricular septal defect	p. 430
Left ventricular hypertrophy	p. 434
Left ventricular hypertrophy	p. 436
Pressure-volume loops	p. 438
References	p. 440
Appendices
Introduction to MATLAB	p. 443
Introduction to Simulink	p. 487
Review of Linear Algebra and Related MATLAB Commands	p. 517
Analytical Solutions of Differential Equations	p. 527
Numerical Stability and Other Topics	p. 575
Index	p. 593
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Numerical Methods in Biomedical Engineering

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