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Preface xiii Part I: Fundamentals Chapter 1 Modeling Biosystems 1 1.1 Biomedical Engineering 1 1.2 Fundamental Aspects of Biomedical Engineering 3 1.3 Constructing Engineering Models 3 1.3.1 A framework for problem solving 4 1.3.2 Formulating the mathematical expression of conservation 5 1.3.3 Using Balance Equations 7 Example 1.1 How conservation laws lead to the Nernst equation 7 1.4 Examples of Solving Biomedical Engineering Models by Computer 9 1.4.1 Using numerical methods to model rtPCR 9 1.4.2 Modeling transcranial magnetic stimulation 13 1.4.3 Modeling cardiac electrophysiology 14 1.4.4 Using numerical methods to model the response of the cardio-vascular system to gravity 15 1.5 Overview of the Text 17 1.5.1 Part I: Fundamentals 19 1.5.2 Part II: Algebraic biosystems (steady state behavior) 19 1.5.3 Part III: Dynamic behavior (differential equations) 20 1.5.4 Part IV: Modeling tools and applications 20 1.6 Lessons Learned in this Chapter 20 1.7 Problems 21 1.8 References 21 Chapter 2 Introduction to Computing 23 2.1 Introduction 23 2.2 The Role of Computers in Biomedical Engineering 24 2.3 Programming Language Tools and Techniques 26 2.3.1 Sequences of statements 27 Example 2.1 Programs that are sequences of statements 27 2.3.2 Conditional execution 28 Example 2.2 Simple control flow using if¿then¿else 29 Example 2.3 Use of the switch statement 31 2.3.3 Iteration 34 Example 2.4 The use of while loops 34 Example 2.5 Using for¿end loops 35 2.3.4 Encapsulation 36 Example 2.6 Using scripts and functions 37 2.4 Fundamentals of Data Structures for MATLAB 38 2.4.1 Number representation 39 Example 2.7 Number representation in MATLAB 39 Example 2.8 Complex numbers 40 2.4.2 Arrays 40 Example 2.9 Indexing arrays in MATLAB 41 2.4.3 Characters and strings 42 Example 2.10 Character strings as arrays 42 2.4.4 Logical or Boolean data types 43 Example 2.11 Logical indexing in MATLAB 44 2.4.5 Cells and cell arrays 45 Example 2.12 Cell arrays and mixed data types 45 Example 2.13 Structure arrays and mixed data types 47 2.4.6 Data structures not explicitly found in MATLAB 47 Example 2.14 Data structures in MATLAB: Implementing a stack 48 2.4.7 Data type conversion 50 Example 2.15 Data type conversion 50 2.5 An Introduction to Object-oriented Systems 51 Example 2.16 Simple object-oriented programs that are sequences of statements 54 2.6 Analyzing Algorithms and Programs 56 2.6.1 Polynomial complexity 56 2.6.2 Operation counting 56 Example 2.17 Measuring execution time as a function of the amount of data 58 2.7 Lessons Learned in this Chapter 60 2.8 Problems 61 Chapter 3 Numerical Analysis 65 3.1 Scientific Computing 65 3.2 Numerical Algorithms and Errors 66 3.3 Taylor Series 67 Example 3.1 How truncation errors and roundoff errors arise 68 3.4 Keeping Errors Small 72 Example 3.2 An ill-posed problem 72 3.5 Floating-point Representation in MATLAB 74 3.5.1 The IEEE 754 standard for floating-point representation 75 Example 3.3 IEEE 754 floating-point representation 75 3.5.2 Floating-point arithmetic, truncation and rounding 76 Example 3.4 Propagation of floating-point errors 77 Example 3.5 Machine precision in MATLAB 77 3.5.3 Roundoff error accumulation and cancellation error 78 Example 3.6 Avoiding overflow 78 Example 3.7 Avoiding cancellation errors 79 Example 3.8 Using Taylor series expansions to avoid cancellation errors 79 3.6 Lessons Learned in this Chapter 80 3.7 Problems 81 3.8 References 83 Part II: Algebraic Biosystems ¿ Chapter 4 Linear Biological Systems 85 4.1 Introduction 85 4.2 Examples of Linear Biological Systems 86 4.2.1 Force balance in biomechanics 86 4.2.2 Biomedical imaging and image processing 88 4.2.3 Metabolic engineering and cellular biotechnology 89 4.3 Simultaneous Linear Algebraic Equations 90 4.3.1 Illustration of simple Gauss elimination for a 3x3 matrix 90 4.3.2 Matrix notation of Gaussian elimination 91 Example 4.1 Application of the Gauss elimination method 98 4.4 The Gauss-Jordan Reduction Method 100 Example 4.2 Application of the Gauss-Jordan reduction method 102 4.5 Iterative Approach for Solution of Linear Systems 104 4.5.1 The Jacobi Method 105 Example 4.3 Application of the iterative Jacobi method 107 4.5.2 The Gauss-Seidel Method 110 4.6 Lessons Learned in this Chapter 113 4.7 Problems 114 4.8 References 116 Chapter 5 Nonlinear Equations in Biomedical Engineering 117 5.1 Introduction 117 5.2 General Form of Nonlinear Equations 118 5.3 Examples of Nonlinear Equations in Biomedical Engineering 120 5.3.1 Molecular bioengineering 121 5.3.2 Cellular and tissue engineering 121 5.3.3 Bioheat transport: Photothermal therapy 122 5.3.4 Biomedical flow transport dynamics 123 5.4 The Method of Successive Substitution 124 5.5 The Method of False Position (Linear Interpolation) 125 5.6 The Newton-Raphson Method 127 Example 5.1 Cardiovascular Physiology 130 Example 5.2a Solution of the Colebrook equation using Newton-Raphson 136 Example 5.2b Successive substitution method for solution of nonlinear equation 141 Example 5.2c Solution of the Colebrook equation using linear interpolation 145 Example 5.3 Solution of a Michaelis-Menten kinetics equation using the Newton-Raphson method 150 5.7 Newton's Method for Simultaneous Nonlinear Equations 151 Example 5.4 Determination of receptor occupancy during receptor-ligand dynamics 153 5.8 Lessons Learned in this Chapter 157 5.9 Problems 157 5.10 References 160 Part III: Dynamic Biosystems ¿ Chapter 6 Finite Difference Methods 163 6.1 Introduction 163 6.2 Symbolic Operators 164 6.3 Backward Finite Differences 169 Example 6.1 Express the first-order derivative in terms of backward finite differences with error of order h 174 Example 6.2 Express the first-order derivative in terms of backward finite differences with error of order h2 175 6.4 Forward Finite Differences 177 Example 6.3 Express the first-order derivative in terms of forward finite differences with error of order h 181 Example 6.4 Express the second-order derivative in terms of forward finite differences with error of order h 182 6.5 Central Finite Differences 184 Example 6.5 Express the first-order derivative in terms of central finite differences with error of order h2 190 Example 6.6 Express the second-order derivative in terms of central finite differences with error of order h2 190 Example 6.7 Express the first-order derivative in terms of central finite differences with error of order h4 191 6.6 Interpolating Polynomials 193 6.7 Interpolation of Equally Spaced Points 195 6.7.1 Gregory-Newton interpolation 195 Example 6.8 Gregory-Newton method for interpolation of equally spaced data 200 6.7.2 Stirling's interpolation 204 6.8 Interpolation of Unequally Spaced Points 207 6.8.1 Lagrange polynomials 207 6.8.2 Spline interpolation 208 6.9 Spline Differentiation 211 6.10 Integration Formulas 212 6.11 The Newton-Cotes Formulas of Integration 213 6.11.1 The trapezoidal rule 213 6.11.2 Simpson's 1/3 rule 216 6.11.3 Simpson's 3/8 rule 217 6.11.4 Summary of Newton-Cotes integration 219 Example 6.9 Integration formulas - Trapezoidal and Simpson¿s 1/3 rules 221 6.12 Lessons Learned in this Chapter 225 6.13 Problems 225 6.14 References 228 Chapter 7 Dynamic Systems: Ordinary Differential Equations 229 7.1 Introduction 229 7.1.1 Pharmacokinetics: The dynamics of drug absorption 230 7.1.2 Tissue engineering: Stem cell differentiation, cell migration, adhesion 231 7.1.3 Glycolysis pathways of living cells 232 7.1.4 Transport of molecules in biological membranes 233 7.2 Classification of Ordinary Differential Equations 234 7.3 Transformation to Canonical Form 236 Example 7.1 Transformation of ordinary differential equations into their canonical form 238 7.4 Nonlinear Ordinary Differential Equations 241 7.4.1 The Euler and modified Euler methods 241 7.4.2 The Runge-Kutta methods 245 7.4.3 Simultaneous differential equations 249 7.4.4 MATLAB functions for nonlinear equations 250 Example 7.2 Solution of enzyme catalysis reactions 252 7.5 Linear Ordinary Differential Equations 255 7.5.1 Method using eigenvalues and eigenvectors 255 7.5.2 MATLAB functions for linear equations 257 Example 7.3 The dynamics of drug absorption 258 7.6 Steady State Solutions and Stability Analysis 263 7.7 Numerical Stability and Error Propagation 268 7.8 Advanced Examples 270 Example 7.4 Modeling the glycolysis pathways of living cells 270 Example 7.5 The dynamics of membrane and nerve cell potentials 277 Example 7.6 The dynamics of stem cell differentiation 286 Example 7.7 Tissue engineering: models of epidermal cell migration 293 7.9 Lessons Learned in this Chapter 301 7.10 Problems 301 7.11 References 308 Chapter 8 Dynamic Systems: Partial Differential Equations 311 8.1 Introduction 311 8.2 Examples of PDEs in Biomedical Engineering 312 8.2.1 Diffusion across biological membranes 312 8.2.2 Diffusion of macromolecules and controlled release of drugs 314 8.2.3 Cell migration on vascular prosthetic materials 315 8.2.4 Fluid flow in physiological and extracorporeal vessels 315 8.3 Classification of Partial Differential Equations 316 8.4 Initial and Boundary Conditions 318 8.5 Solution of Partial Differential Equations 321 8.5.1 Elliptic partial differential equations 327 Example 8.1 Solution of the Laplace and Poisson equations 332 8.5.2 Parabolic partial differential equations 339 Example 8.2 Migration of human leukocytes on prosthetic materials 343 8.5.3 Hyperbolic partial differential equations 349 8.6 Polar Coordinate Systems 351 8.7 Stability Analysis 353 8.8 PDE Toolbox in MATLAB 353 Example 8.3 Solution of Fick¿s second law of diffusion using the PDE toolbox 354 8.9 Lessons Learned in this Chapter 360 8.10 Problems 360 8.10 References 365 Part IV: Modeling Tools and Applications Chapter 9 Measurements, Models and Statistics 367 9.1 The Role of Numerical Methods 367 9.2 Measurements, Errors and Uncertainty 368 9.3 Descriptive Statistics 371 Example 9.1 Computing statistics of MRI and CT image intensities 373 9.4 Inferential Statistics 378 Example 9.2 Estimating the mean value of a population from a sample 379 Example 9.3 Hypothesis testing in DNA microarray analysis 383 9.5 Least Squares Modeling 385 Example 9.4 Least square fit of a first-order polynomial (straight line) 387 Example 9.5 Least squares fit of a cubic polynomial 388 Example 9.6 Least squares fit of a nonlinear model 389 Example 9.7 Least squares fit of a multivariate model 390 9.6 Curve Fitting 390 9.6.1 Lagrange interpolating polynomials 391 9.6.2 Newton divided difference interpolating polynomials 392 9.6.3 Splines 393 Example 9.8 Re-sampling and baseline correction of MALDI-TOF mass spectra data 396 9.7 Fourier Transforms 401 Example 9.9 Separating EEG frequency components 404 9.8 Lessons Learned in the Chapter 408 9.9 Problems 409 9.10 References 409 Chapter 10 Modeling Biosystems 411 10.1 Numerical Modeling of Bioengineering Systems 411 10.2 PhysioNet, PhysioBank and PhysioToolkit 413 10.2.1 ECG simulation 413 Example 10.1 Using the MATLAB script ECGwaveGen to synthesize ECG data 413 10.2.2 Reading PhysioBank data 416 Example 10.2 Read and visualize PhysioBank signals and annotations 417 10.3 Signal Processing: EEG data 419 Example 10.3 Differential brain activity in the left and right hemispheres 422 10.4 Diabetes and Insulin Regulation 426 Example 10.4 Simulink model of glucose regulation 427 10.5 Renal Clearance 433 Example 10.5 Renal clearance 434 10.6 Correspondence Problems and Motion Estimation 436 Example 10.6 Estimating motion from features on a rigid body 437 10.7 PHYSBE Simulations 441 Example 10.7 Normal PHYSBE operation 442 10.7.1 Coarctation of the aorta 444 Example 10.8 Simulink model of coarctation of the aorta 445 10.7.2 Aortic stenosis 448 Example 10.9 Simulink model of aortic valve stenosis 449 10.7.3 Ventricular septal defect 452 Example 10.10 Ventricular septal defect 452 10.7.4 Left ventricular hypertrophy 456 Example 10.11 Left ventricular hypertrophy 457 Example 10.12 Pressure-volume loops 460 10.8 References 461 Appendices A: Introduction to MATLAB 463 B: Introduction to Simulink 507 C: Review of Linear Algebra and Related MATLAB Commands 537 D: Analytical Solution of Differential Equations 547 E: Numerical Stability and Other Topics 595 Index 613

Library of Congress Subject Headings for this publication:

Biomedical engineering -- Mathematics.

Biomedical engineering -- Mathematical models.

Biomedical Engineering -- methods.

Mathematical Computing.

Models, Theoretical.