Principles of Diffuse Light Propagation : Light Propagation in Tissues with Applications in Biology and Medicine

[BOOK DESCRIPTION]

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.

[TABLE OF CONTENTS]

Foreword                                           vii
Preface                                            ix
  Part I Light Propagation in Tissues              1  (150)
    1 Light Absorbers, Emitters, and               3  (50)
    Scatterers: The Origins of Color in Nature
      1.1 Introduction                             3  (5)
      1.2 The Classical Picture of Light           8  (2)
      Interaction With Matter
      1.3 Light Absorbers in Nature                10 (11)
        1.3.1 Tissue Absorption                    13 (8)
      1.4 Light Emitters in Nature                 21 (17)
        1.4.1 Coherent and Incoherent Light        23 (2)
        Sources
        1.4.2 Fluorescence                         25 (12)
        1.4.3 Bioluminescence                      37 (1)
      1.5 Light Scatterers in Nature               38 (7)
        1.5.1 Tissue Scattering                    41 (4)
      1.6 Optical Molecular Imaging                45 (8)
    2 Scattering and Absorption                    53 (36)
      2.1 Definition of Scattering                 53 (2)
      2.2 Poynting's Theorem and Energy            55 (6)
      Conservation
        2.2.1 The Time-Averaged Expressions        58 (3)
      2.3 Single Scattering                        61 (5)
        2.3.1 The Scalar Theory of Scattering      62 (2)
        2.3.2 Far-Field Approximation              64 (2)
      2.4 Main Optical Parameters of a Particle    66 (9)
        2.4.1 The Absorption Cross-Section         66 (2)
        2.4.2 The Scattering Cross-Section         68 (1)
        2.4.3 The Total or Extinction              69 (1)
        Cross-Section and the Optical Theorem
        2.4.4 The Phase Function                   70 (3)
        2.4.5 The Anisotropy Factor                73 (2)
      2.5 Multiple Scattering                      75 (6)
        2.5.1 The Scattering and Absorption        78 (3)
        Coefficients
      2.6 Extinction by a Slab of Absorbing        81 (2)
      Particles
      2.7 Polarization Effects                     83 (3)
      2.8 Self-Averaging                           86 (3)
    3 The Radiative Transfer Equation (RTE)        89 (38)
      3.1 Radiative Transfer                       89 (5)
        3.1.1 Volume Averaged Flow of Energy       92 (2)
      3.2 Specific Intensity, Average Intensity    94 (4)
      and Flux
        3.2.1 The Specific Intensity               94 (1)
        3.2.2 The Average Intensity                95 (1)
        3.2.3 The Energy Density                   96 (1)
        3.2.4 The Total Flux Density               97 (1)
      3.3 The Detected Power                       98 (4)
        3.3.1 The Numerical Aperture               100(2)
      3.4 Isotropic Emission and its Detection     102(3)
      3.5 Reflectivity and Transmissivity          105(5)
      3.6 Derivation of the Radiative Transfer     110(9)
      Equation
        3.6.1 The Source Term                      113(2)
        3.6.2 The Equation of Energy               115(1)
        Conservation
        3.6.3 Summary of Approximations: How       116(3)
        Small is `Small Enough'?
      3.7 Some Similarity Relations of the RTE     119(1)
      3.8 The RTE and Monte Carlo                  120(7)
        3.8.1 Photon Density                       122(5)
    4 Fick's Law and The Diffusion Approximation   127(24)
      4.1 Historical Background                    127(4)
      4.2 Diffuse Light                            131(5)
        4.2.1 Reduced and Diffuse Intensity        132(2)
        4.2.2 Angular Distribution of Diffuse      134(2)
        Light
      4.3 Derivation of the Diffusion Equation     136(8)
        4.3.1 The Diffusion Coefficient            141(2)
        4.3.2 The Diffusion Coefficient In         143(1)
        Absorbing Media
      4.4 The Diffusion Equation                   144(1)
      4.5 The Mean Free Path                       145(3)
      4.6 Limits of Validity of the Diffusion      148(3)
      approximation
  Part II Diffuse Light                            151(148)
    5 The Diffusion Equation                       153(24)
      5.1 The Diffusion Equation in Infinite       153(1)
      Homogeneous Media
      5.2 Green's Functions and Green's Theorem    154(4)
        5.2.1 The Diffusion Equation and           156(2)
        Green's Theorem
      5.3 The Time-dependent Green's Function      158(5)
      5.4 The Constant Illumination Green's        163(3)
      Function
      5.5 Waves of Diffuse Light                   166(3)
      5.6 The Diffusion Equation in                169(3)
      Inhomogeneous Media
      5.7 Summary of Green's Functions             172(5)
        5.7.1 1D Green's functions                 172(1)
        5.7.2 2D Green's functions                 173(1)
        5.7.3 3D Green's functions                 174(3)
    6 Propagation and Spatial Resolution of        177(24)
    Diffuse Light
      6.1 Propagation of Diffuse Light             177(4)
        6.1.1 The Diffusion Wavenumber             179(2)
      6.2 The Angular Spectrum Representation      181(5)
        6.2.1 Angular spectrum of a point          183(3)
        source: The Green Function in K-space
      6.3 Spatial Transfer Function and Impulse    186(6)
      Response
        6.3.1 Spatial Transfer Function and        188(4)
        Impulse Response
      6.4 Spatial Resolution                       192(5)
        6.4.1 Resolution of Propagating Scalar     193(1)
        Waves
        6.4.2 Resolution of Diffuse Waves          194(3)
      6.5 Backpropagation of Diffuse Light         197(4)
    7 The Point Source Approximation               201(10)
      7.1 General Solution                         201(3)
        7.1.1 Solution for a point source          203(1)
      7.2 Solution for a collimated source         204(2)
      7.3 Point Source Approximation to a          206(2)
      collimated source
        7.3.1 Limits of Validity                   208(1)
      7.4 Accounting for the Source Profile        208(3)
    8 Diffuse Light at Interfaces                  211(44)
      8.1 Diffusive/Diffusive (D-D) Interfaces     211(11)
        8.1.1 D-D Boundary Conditions              212(3)
        8.1.2 D-D Reflection and Transmission      215(6)
        Coefficients
        8.1.3 Frequency independent coefficients   221(1)
      8.2 Diffusive/Non-diffusive (D-N)            222(7)
      Interfaces
        8.2.1 D-N Boundary Conditions              223(3)
        8.2.2 D-N Reflection and Transmission      226(3)
        Coefficients
      8.3 Layered Diffusive Media                  229(6)
        8.3.1 Expression for a Slab in a           229(3)
        Diffusive medium
        8.3.2 Expression for a Slab in a           232(3)
        Non-Diffusive medium
      8.4 Multiple layered media                   235(3)
      8.5 The Detected Power in Diffuse Media      238(4)
        8.5.1 Accounting for the Detector          240(2)
        Profile
      8.6 Non-contact Measurements                 242(13)
        8.6.1 Free-space source                    243(3)
        8.6.2 Free-space detector                  246(9)
    9 Fluorescence and Bioluminescence in          255(26)
    Diffuse Media: An ill-posed problem
      9.1 Fluorescence in Diffuse Media            255(4)
      9.2 Bioluminescence in Diffuse Media         259(1)
      9.3 Why is imaging in diffuse media an       260(8)
      ill-posed problem?
        9.3.1 Recovering size and position in      262(6)
        diffuse media
      9.4 Reducing Ill-posedness                   268(13)
        9.4.1 Introducing a spatial dependence     268(1)
        on the emission
        9.4.2 Normalized measurements              269(2)
        9.4.3 Multispectral imaging                271(2)
        9.4.4 Phase Information                    273(2)
        9.4.5 Background Signal                    275(2)
        9.4.6 Prior Information                    277(4)
    10 Imaging in Diffusive Media: The Inverse     281(18)
    Problem
      10.1 The Forward and Inverse Problem         281(1)
      10.2 The Born Approximation                  282(1)
      10.3 The Rytov Approximation                 283(4)
      10.4 The Normalized Born Approximation       287(3)
      and the Sensitivity Matrix
      10.5 Direct Inversion Formulas               290(9)
  Appendix A Useful Formulas                       299(6)
      A.1 The Fourier Transform                    299(1)
      A.2 The Hankel Transform                     300(1)
      A.3 The Laplace Transform                    301(1)
      A.4 The Delta Function                       301(1)
      A.5 Gaussian Function                        302(2)
      A.6 Vector Identities                        304(1)
  Appendix B The Solid Angle                       305(6)
      B.1 The solid angle delta function           307(1)
      B.2 The solid angle and the unit             307(4)
      direction vector
  Appendix C An Alternative Derivation of the      311(10)
  Radiative Transfer Equation
      C.1 Derivation of the Radiative Transfer     311(1)
      Equation
        C.1.1 Volume Averaged Change in Energy     312(1)
        Density
        C.1.2 Volume Averaged Absorbed Power       313(1)
        C.1.3 Volume Averaged Change in Energy     314(2)
        Flow
        C.1.4 The Scattering Contribution          316(1)
        C.1.5 The Radiative Transfer Equation      317(1)
        C.1.6 Summary of Approximations            318(3)
Bibliography                                       321(10)
Index                                              331