Principles of Diffuse Light Propagation: Light Propagation in Tissues with Applications in Biology and Medicine (354 Pages)

The main idea behind this book is to present a rigorous derivation of the equations that govern light propagation in highly scattering media, with an emphasis on their applications in imaging in biology and medicine. The equations and formulas for diffuse light propagation are derived from the very beginning, and all the necessary analytical expressions needed to complete a complex imaging or characterization problem are presented step by step. This book provides postgraduate and PhD students with the basic framework and sufficient knowledge in light transport and the related mathematical methods to solve most complex problems that may appear in biomedical applications involving multiple scattered light. All results presented are formal analytical derivations from the complete problem, presenting, in those cases which are relevant, approximations to these expressions. Readership: Students and professionals working in the field of optical biomedical applications. Contents 14 Foreword 8 Preface 10 Part I: Light Propagation in Tissues 20 1. Light Absorbers, Emitters, and Scatterers: The Origins of Color in Nature 22 1.1 Introduction 22 1.2 The Classical Picture of Light Interaction With Matter 27 1.3 Light Absorbers in Nature 29 1.3.1 Tissue Absorption 32 1.3.1.1 Blood: The Main Absorber In Tissue 33 1.3.1.2 The effect of Water 36 1.3.1.3 The effect of Skin 37 1.4 Light Emitters in Nature 40 1.4.1 Coherent and Incoherent Light Sources 42 1.4.2 Fluorescence 44 1.4.2.1 Fluorescence Lifetime 47 1.4.2.2 Steady-state Fluorescence Intensity 49 1.4.2.3 Quantum Yield 50 1.4.2.4 Tissue Auto-fluorescence 53 1.4.3 Bioluminescence 56 1.5 Light Scatterers in Nature 57 1.5.1 Tissue Scattering 60 1.6 Optical Molecular Imaging 64 The definition further elaborates 64 Key Points 67 Further Reading 68 2. Scattering and Absorption 72 2.1 Definition of Scattering 72 2.2 Poynting’s Theorem and Energy Conservation 74 2.2.1 The Time-Averaged Expressions 77 2.3 Single Scattering 80 2.3.1 The Scalar Theory of Scattering 81 2.3.2 Far-Field Approximation 83 2.4 Main Optical Parameters of a Particle 85 2.4.1 The Absorption Cross-Section 85 2.4.2 The Scattering Cross-Section 87 2.4.3 The Total or Extinction Cross-Section and the Optical Theorem 88 2.4.4 The Phase Function 89 2.4.5 The Anisotropy Factor 92 2.5 Multiple Scattering 94 2.5.1 The Scattering and Absorption Coefficients 97 2.6 Extinction by a Slab of Absorbing Particles 100 2.7 Polarization Effects 102 2.8 Self-Averaging 105 Key Points 106 Further Reading 107 3. The Radiative Transfer Equation (RTE) 108 3.1 Radiative Transfer 108 3.1.1 Volume Averaged Flow of Energy 111 3.2 Specific Intensity, Average Intensity and Flux 113 3.2.1 The Specific Intensity 113 3.2.2 The Average Intensity 114 3.2.3 The Energy Density 115 3.2.4 The Total Flux Density 116 3.3 The Detected Power 117 3.3.1 The Numerical Aperture 119 3.4 Isotropic Emission and its Detection 121 3.5 Reflectivity and Transmissivity 124 3.6 Derivation of the Radiative Transfer Equation 129 3.6.1 The Source Term 132 3.6.2 The Equation of Energy Conservation 134 3.6.3 Summary of Approximations: How Small is ‘Small Enough’? 135 3.7 Some Similarity Relations of the RTE 138 3.8 The RTE and Monte Carlo 139 3.8.1 Photon Density 141 Key points 141 Further Reading 143 4. Fick’s Law and The Di usion Approximation 146 4.1 Historical Background 146 4.2 Diffuse Light 150 4.2.1 Reduced and Diffuse Intensity 151 4.2.2 Angular Distribution of Diffuse Light 153 4.3 Derivation of the Diffusion Equation 155 4.3.1 The Diffusion Coefficient 160 4.3.2 The Diffusion Coefficient In Absorbing Media 162 4.4 The Diffusion Equation 163 4.5 The Mean Free Path 164 4.6 Limits of Validity of the Diffusion approximation 167 Key points 168 Further Reading 169 Part II: Diffuse Light 170 5. The Diffusion Equation 172 5.1 The Diffusion Equation in Infinite Homogeneous Media 172 5.2 Green’s Functions and Green’s Theorem 173 5.2.1 The Diffusion Equation and Green’s Theorem 175 5.3 The Time-dependent Green’s Function 177 5.4 The Constant Illumination Green’s Function 182 5.5 Waves of Diffuse Light 185 5.6 The Diffusion Equation in Inhomogeneous Media 188 5.7 Summary of Green’s Functions 191 5.7.1 1D Green’s functions 191 5.7.2 2D Green’s functions 192 5.7.3 3D Green’s functions 193 Key points 193 Further Reading 194 6. Propagation and Spatial Resolution of Diffuse Light 196 6.1 Propagation of Diffuse Light 196 6.1.1 The Diffusion Wavenumber 198 6.2 The Angular Spectrum Representation 200 6.2.1 Angular spectrum of a point source: The Green Function in K-space 202 6.3 Spatial Transfer Function and Impulse Response 205 6.3.1 Spatial Transfer Function and Impulse Response 207 6.4 Spatial Resolution 211 6.4.1 Resolution of Propagating Scalar Waves 212 6.4.2 Resolution of Diffuse Waves 213 6.5 Backpropagation of Diffuse Light 216 Key points 218 Further Reading 219 7. The Point Source Approximation 220 7.1 General Solution 220 7.1.1 Solution for a point source 222 7.2 Solution for a collimated source 223 7.3 Point Source Approximation to a collimated source 225 7.3.1 Limits of Validity 227 7.4 Accounting for the Source Profile 227 Key points 229 Further Reading 229 8. Diffuse Light at Interfaces 230 8.1 Diffusive/Diffusive (D-D) Interfaces 230 8.1.1 D-D Boundary Conditions 231 8.1.1.2 Index Matched Conditions 234 8.1.2 D-D Reflection and Transmission Coefficients 234 8.1.2.1 Approximate reflection and transmission coefficients 239 8.1.2.2 Snell’s law for Diffuse Waves 240 8.1.3 Frequency independent coefficients 240 8.2 Diffusive/Non-diffusive (D-N) Interfaces 241 8.2.1 D-N Boundary Conditions 242 8.2.1.1 The Extrapolated Boundary Condition 243 8.2.2 D-N Reflection and Transmission Coefficients 245 8.2.2.1 Black interface 247 8.3 Layered Diffusive Media 248 8.3.1 Expression for a Slab in a Diffusive medium 248 8.3.2 Expression for a Slab in a Non-Diffusive medium 251 8.4 Multiple layered media 254 8.5 The Detected Power in Diffuse Media 257 8.5.1 Accounting for the Detector Profile 259 8.6 Non-contact Measurements 261 8.6.1 Free-space source 262 8.6.2 Free-space detector 265 8.6.2.1 Free-space detection through a system of lenses 268 8.6.2.2 Normalized Fluorescence in Free-space 268 Key points 270 Further Reading 271 9. Fluorescence and Bioluminescence in Diffuse Media: An ill-posed problem 274 9.1 Fluorescence in Diffuse Media 274 9.2 Bioluminescence in Diffuse Media 278 9.3 Why is imaging in diffuse media an ill-posed problem? 279 9.3.1 Recovering size and position in diffuse media 281 9.3.1.1 Ill-posed nature of Propagating Scalar Waves 282 9.3.1.2 Ill-posed nature of Diffuse Waves 283 9.4 Reducing Ill-posedness 287 9.4.1 Introducing a spatial dependence on the emission 287 9.4.2 Normalized measurements 288 9.4.3 Multispectral imaging 290 9.4.3.1 Distinct background absorption features 291 9.4.4 Phase Information 292 9.4.5 Background Signal 294 9.4.6 Prior Information 296 Key points 297 Further Reading 298 10. Imaging in Diffusive Media: The Inverse Problem 300 10.1 The Forward and Inverse Problem 300 10.2 The Born Approximation 301 10.3 The Rytov Approximation 302 10.4 The Normalized Born Approximation and the Sensitivity Matrix 306 10.5 Direct Inversion Formulas 309 Key points 314 Further Reading 315 Appendix A Useful Formulas 318 A.1 The Fourier Transform 318 A.2 The Hankel Transform 319 A.3 The Laplace Transform 320 A.4 The Delta Function 320 A.5 Gaussian Function 321 A.6 Vector Identities 323 Appendix B The Solid Angle 324 B.1 The solid angle delta function 326 B.2 The solid angle and the unit direction vector 326 Appendix C An Alternative Derivation of the Radiative Transfer Equation 330 C.1 Derivation of the Radiative Transfer Equation 330 C.1.1 Volume Averaged Change in Energy Density 331 C.1.2 Volume Averaged Absorbed Power 332 C.1.3 Volume Averaged Change in Energy Flow 333 C.1.4 The Scattering Contribution 335 C.1.5 The Radiative Transfer Equation 336 C.1.6 Summary of Approximations 337 Bibliography 340 Index 350 The main idea behind this book is to present a rigorous derivation of the equations that govern light propagation in highly scattering media, with an emphasis on their applications in imaging in biology and medicine. The equations and formulas for diffuse light propagation are derived from the very beginning, and all the necessary analytical expressions needed to complete a complex imaging or characterization problem are presented step by step.This book provides postgraduate and PhD students with the basic framework and sufficient knowledge in light transport and the related mathematical methods to solve any complex problems that may appear in any biomedical applications involving multiple scattered light. All results presented are formal analytical derivations from the complete problem, presenting, in those cases which are relevant, approximations to these expressions. In this sense, numerical solutions to these expressions, such as the finite element methods, are not within the scope of this book La 4e de couverture indique : The main idea behind this book is to present a rigorous derivation of the equations that govern light propagation in highly scattering media, with an emphasis on their applications in imaging in biology and medicine. The equations and formulas for diffuse light propagation are derived from the very beginning, and all the necessary analytical expressions needed to complete a complex imaging or characterization problem are presented step by step. This book provides postgraduate and PhD students with the basic framework and sufficient knowledge in light transport and the related mathematical methods to solve most complex problems that may appear in biomedical applications involving multiple scattered light. All results presented are formal analytical derivations from the complete problem, presenting, in those cases which are relevant, approximations to these expressions Pt.1. Light Propagation In Tissues -- Pt.2. Diffuse Light. Jorge Ripoll Lorenzo. Includes Bibliographical References (p. 321-329) And Index.