Numerical Methods in Biomedical Engineering
Stanley M. Dunn
 
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