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P R E S I D E N T A N D C H I E F EXECUTIVE OFFICER P E T E R B O O T H W I L E Y
C H A I R M A N D F T H E B O A R D
BIOMEDICAL OPTICS
PRINCIPLES AND IMAGING
Lihong V. Wang
Hsin-i Wu
B I C E N T E N N I A L ·
; | 1 8 0 7
©WILEY 2 O O 7
B I C E N T E N N I A L
WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
Wang, Lihong V. Biomedical optics : principles and imaging / Lihong V. Wang, Hsin-i
Wu. p. ; cm.
Includes bibliographical references and index. ISBN: 978-0-471-74304-0 (cloth) 1. Imaging systems in medicine. 2. Lasers in medicine. 3. Optical
detectors. I. Wu, Hsin-i. II. Title. [DNLM: 1. Optics. 2. Diagnostic Imaging—methods. 3. Light.
4. Models, Theoretical. 5. Tomography, Optical—methods. WB 117 W246b 2007] R857.06W36. 2007 616.0754-dc22
2006030754
Printed in the United States of America.
1 0 9 8 7
To our families, mentors, students, and friends
CONTENTS
Preface xiii
1. Introduction 1
1.1. Motivation for Optical Imaging 1
1.2. General Behavior of Light in Biological Tissue 2
1.3. Basic Physics of Light-Matter Interaction 3
1.4. Absorption and its Biological Origins 5
1.5. Scattering and its Biological Origins 7
1.6. Polarization and its Biological Origins 9
1.7. Fluorescence and its Biological Origins 9
1.8. Image Characterization 10
Problems 14
Reading 15
Further Reading 15
2. Rayleigh Theory and Mie Theory for a Single Scatterer 17
2.1. Introduction 17
2.2. Summary of Rayleigh Theory 17
2.3. Numerical Example of Rayleigh Theory 19
2.4. Summary of Mie Theory 20
2.5. Numerical Example of Mie Theory 21
Appendix 2A. Derivation of Rayleigh Theory 23
Appendix 2B. Derivation of Mie Theory 26
Problems 34
Reading 35
Further Reading 35
vii
VIII CONTENTS
Monte Carlo Modeling of Photon Transport in Biological Tissue
3.1. Introduction
3.2. Monte Carlo Method
3.3. Definition of Problem
3.4. Propagation of Photons
3.5. Physical Quantities
3.6. Computational Examples
Appendix 3A. Summary of MCML
Appendix 3B. Probability Density Function
Problems
Reading
Further Reading
Convolution for Broadbeam Responses
4.1. Introduction
4.2. General Formulation of Convolution
4.3. Convolution over a Gaussian Beam
4.4. Convolution over a Top-Hat Beam
4.5. Numerical Solution to Convolution
4.6, Computational Examples
Appendix 4A. Summary of CONV
Problems
Reading
Further Reading
Radiative Transfer Equation and Diffusion Theory
5.1. Introduction
5.2. Definitions of Physical Quantities
5.3. Derivation of Radiative Transport Equation
5.4. Diffusion Theory
5.5. Boundary Conditions
5.6. Diffuse Reflectance
37
37
37
38
39
50
55
58
60
60
62
62
67
67
67
69
71
72
77
77
80
81
81
83
83
83
85
88
101
106
CONTENTS IX
5.7. Photon Propagation Regimes 114
Problems 116
Reading 117
Further Reading 118
Hybrid Model of Monte Carlo Method and Diffusion Theory 119
6.1. Introduction 119
6.2. Definition of Problem 119
6.3. Diffusion Theory 119
6.4. Hybrid Model 122
6.5. Numerical Computation 124
6.6. Computational Examples 125
Problems 132
Reading 133
Further Reading 133
Sensing of Optical Properties and Spectroscopy 135
7.1. Introduction 135
7.2. Collimated Transmission Method 135
7.3. Spectrophotometry 139
7.4. Oblique-Incidence Reflectometry 140
7.5. White-Light Spectroscopy 144
7.6. Time-Resolved Measurement 145
7.7. Fluorescence Spectroscopy 146
7.8. Fluorescence Modeling 147
Problems 148
Reading 149
Further Reading 149
Ballistic Imaging and Microscopy 153
8.1. Introduction 153
8.2. Characteristics of Ballistic Light 153
X CONTENTS
8.3. Time-Gated Imaging 154
8.4. Spatiofrequency-Filtered Imaging 156
8.5. Polarization-Difference Imaging 157
8.6. Coherence-Gated Holographic Imaging 158
8.7. Optical Heterodyne Imaging 160
8.8. Radon Transformation and Computed Tomography 163
8.9. Confocal Microscopy 164
8.10. Two-Photon Microscopy 169
Appendix 8A. Holography 171
Problems 175
Reading 177
Further Reading 177
9. Optical Coherence Tomography 181
9.1. Introduction 181
9.2. Michelson Interferometry 181
9.3. Coherence Length and Coherence Time 184
9.4. Time-Domain OCT 185
9.5. Fourier-Domain Rapid-Scanning Optical Delay Line 195
9.6. Fourier-Domain OCT 198
9.7. Doppler OCT 206
9.8. Group Velocity Dispersion 207
9.9. Monte Carlo Modeling of OCT 210
Problems 213
Reading 215
Further Reading 215
10. Mueller Optical Coherence Tomography 219
10.1. Introduction 219
10.2. Mueller Calculus versus Jones Calculus 219
10.3. Polarization State 219
10.4. Stokes Vector 222
CONTENTS XI
10.5. Mueller Matrix 224
10.6. Mueller Matrices for a Rotator, a Polarizer, and a Retarder 225
10.7. Measurement of Mueller Matrix 227
10.8. Jones Vector 229
10.9. Jones Matrix 230
10.10. Jones Matrices for a Rotator, a Polarizer, and a Retarder 230
10.11. Eigenvectors and Eigenvalues of Jones Matrix 231
10.12. Conversion from Jones Calculus to Mueller Calculus 235
10.13. Degree of Polarization in OCT 236
10.14. Serial Mueller OCT 237
10.15. Parallel Mueller OCT 237
Problems 243
Reading 244
Further Reading 245
11. Diffuse Optical Tomography 249
11.1. Introduction 249
11.2. Modes of Diffuse Optical Tomography 249
11.3. Time-Domain System 251
11.4. Direct-Current System 252
11.5. Frequency-Domain System 253
11.6. Frequency-Domain Theory: Basics 256
11.7. Frequency-Domain Theory: Linear Image Reconstruction 261
11.8. Frequency-Domain Theory: General Image Reconstruction 267
Appendix 11 A. ART and SIRT 275
Problems 276
Reading 279
Further Reading 279
12. Photoacoustic Tomography 283
12.1. Introduction 283
12.2. Motivation for Photoacoustic Tomography 283
XII CONTENTS
12.3. Initial Photoacoustic Pressure 284
12.4. General Photoacoustic Equation 287
12.5. General Forward Solution 288
12.6. Delta-Pulse Excitation of a Slab 293
12.7. Delta-Pulse Excitation of a Sphere 297
12.8. Finite-Duration Pulse Excitation of a Thin Slab 302
12.9. Finite-Duration Pulse Excitation of a Small Sphere 303
12.10. Dark-Field Confocal Photoacoustic Microscopy 303
12.11. Synthetic Aperture Image Reconstruction 307
12.12. General Image Reconstruction 309
Appendix 12A. Derivation of Acoustic Wave Equation 313
Appendix 12B. Green Function Approach 316
Problems 317
Reading 319
Further Reading 319
13. Ultrasound-Modulated Optical Tomography 323
13.1. Introduction 323
13.2. Mechanisms of Ultrasonic Modulation of Coherent Light 323
13.3. Time-Resolved Frequency-Swept UOT 326
13.4. Frequency-Swept UOT with Parallel-Speckle Detection 329
13.5. Ultrasonically Modulated Virtual Optical Source 331
13.6. Reconstruction-Based UOT 332
13.7. UOT with Fabry-Perot Interferometry 335
Problems 338
Reading 339
Further Reading 339
Appendix A. Definitions of Optical Properties 343
Appendix B. List of Acronyms 345
Index 347
PREFACE
Biomedical optics is a rapidly growing area of research. Although many universi- ties have begun to offer courses on the topic, a textbook containing examples and homework problems has not been available. The need to fill this void prompted us to write this book.
This book is based on our lecture notes for a one-semester (45 lecture hours) entry-level course, which we have taught since 1998. The contents are divided into two major parts: (1) fundamentals of photon transport in biological tissue and (2) optical imaging. In the first part (Chapters 1 – 7 ) , we start with a brief introduc- tion to biomedical optics and then cover single-scatterer theories, Monte Carlo modeling of photon transport, convolution for broadbeam responses, radiative transfer equation and diffusion theory, hybrid Monte Carlo method and diffusion theory, and sensing of optical properties and spectroscopy. In the second part (Chapters 8-13), we cover ballistic imaging, optical coherence tomography, dif- fuse optical tomography, photoacoustic tomography, and ultrasound-modulated optical tomography.
When the book is used as the textbook in a course, the instructor may request a solution manual containing homework solutions from the publisher. To ben- efit from this text, students are expected to have a background in calculus
® and differential equations. Experience in MATLAB or C/C++ is also helpful. Source codes and other information can be found at ftp://ftp.wiley.com/public/ scLtech_med/biomedical_optics.
Although our multilayered Monte Carlo model is in the public domain, we have found that students are able to better grasp the concept of photon transport in biological tissue when they implement simple semiinfinite versions of the model. For this reason, we encourage the use of simulations whenever appropriate.
Because a great deal of material beyond our original lecture notes has been added, two semesters are recommended to cover the complete textbook. Alterna- tively, selected chapters can be covered in a one-semester course. In addition to serving as a textbook, this book can also be used as a reference for professionals and a supplement for trainees engaged in short courses in the field of biomedical optics.
We are grateful to Mary Ann Dickson for editing the text and to Elizabeth Smith for redrawing the figures. We appreciate Sancy Wu’s close reading of
xiii
XIV PREFACE
the manuscript. We are also thankful to the many students who contributed to the homework solutions. Finally, we wish to thank our students Li Li, Manojit Pramanik, and Sava Sakadzic for proofreading the book.
LIHONG V. WANG, P H . D . HSIN-I Wu, P H . D .
CHAPTER 1
Introduction
1.1. MOTIVATION FOR OPTICAL IMAGING
The most common medical imaging modalities include X-ray radiography, ultra- sound imaging (ultrasonography), X-ray computed tomography (CT), and mag- netic resonance imaging (MRI). The discovery of X rays in 1895, for which Roentgen received the first Nobel Prize in Physics in 1901, marked the advent of medical imaging. Ultrasonography, which is based on sonar, was introduced into medicine in the 1940s after World War II. The invention of CT in the 1970s, for which Cormack and Hounsfield received the Nobel Prize in Medicine in 1979, initiated digital cross-sectional imaging (tomography). The invention of MRI, also in the 1970s, for which Lauterbur and Mansfield received the Nobel Prize in Medicine in 2003, enabled functional imaging with high spatial resolution. Optical imaging, which is compared with the other modalities in Table 1.1, is currently emerging as a promising new addition to medical imaging.
Reasons for optical imaging of biological tissue include
1. Optical photons provide nonionizing and safe radiation for medical appli- cations.
2. Optical spectra—based on absorption, fluorescence, or Raman scatter- ing—provide biochemical information because they are related to molec- ular conformation.
3. Optical absorption, in particular, reveals angiogenesis and hyperme- tabolism, both of which are hallmarks of cancer; the former is related to the concentration of hemoglobin and the latter, to the oxygen satura- tion of hemoglobin. Therefore, optical absorption provides contrast for functional imaging.
4. Optical scattering spectra provide information about the size distribution of optical scatterers, such as cell nuclei.
5. Optical polarization provides information about structurally anisotropic tissue components, such as collagen and muscle fiber.
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
1
INTRODUCTION
TABLE 1.1. Comparison of Various Medical Imaging Modalities
Characteristics
Soft-tissue contrast Spatial resolution Maximum imaging depth Function Nonionizing radiation Data acquisition Cost
X-ray Imaging
Poor Excellent Excellent None No Fast Low
Ultrasonography
Good Good Good Good Yes Fast Low
MRI
Excellent Good Excellent Excellent Yes Slow High
Optical Imaging
Excellent Mixed” Good Excellent Yes Fast Low
“High in ballistic imaging (see Chapters 8-10) and photoacoustic tomography (see Chapter 12); low in diffuse optical tomography (see Chapter 11).
6. Optical frequency shifts due to the optical Doppler effect provide infor- mation about blood flow.
7. Optical properties of targeted contrast agents provide contrast for the molecular imaging of biomarkers.
8. Optical properties or bioluminescence of products from gene expression provide contrast for the molecular imaging of gene activities.
9. Optical spectroscopy permits simultaneous detection of multiple contrast agents.
10. Optical transparency in the eye provides a unique opportunity for high- resolution imaging of the retina.
1.2. GENERAL BEHAVIOR OF LIGHT IN BIOLOGICAL TISSUE
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