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Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software
Author(s): Fabrizio Martelli; Samuele Del Bianco; Andrea Ismaelli; Giovanni Zaccanti
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Book Description

This book provides foundational information on modeling light propagation through diffusive media, with special emphasis on biological tissue. A summary of the theoretical background on light propagation through diffusive media is provided with the aid of easy-to-use software designed to calculate the solutions of the diffusion equation. The book also provides:

  • The basic theory of photon transport with the analytical solutions of the diffusion equation for several geometries
  • Detailed coverage of the radiative transfer equation and the diffusion equation
  • The theories and the formulae based on the diffusion equation that have been widely used for biomedical applications
  • The general concepts and the physical quantities necessary to describe light propagation through absorbing and scattering media
  • A description of the software provided in the supplemental materials, along with the accuracy of the presented solutions.

Although the theoretical and computational tools provided with this book and have their primary use in the field of biomedical optics, there are many other applications in which they can be used, including agricultural products, forest products, food products, plastic materials, pharmaceutical products, and many others.


Book Details

Date Published: 23 December 2009
Pages: 298
ISBN: 9780819476586
Volume: PM193

Table of Contents
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Acknowledgements
Preface
List of Acronyms
List of Symbols
1. Introduction
Part I: Theory
2. Scattering and Absorption Properties of Diffusive Media
2.1 Approach Followed in this Book
2.2 Optical Properties of a Turbid Medium
2.2.1 Absorption properties
2.2.2 Scattering properties
2.3 Statistical Meaning of the Optical Properties of a Turbid Medium
2.4 Similarity Relation and Reduced Scattering Coefficient
2.5 Examples of Diffusive Media
References
3. The Radiative Transfer Equation and Diffusion Equation
3.1 Quantities Used to Describe Radiative Transfer
3.2 The Radiative Transfer Equation
3.3 The Green's Function Method
3.4 Properties of the Radiative Transfer Equation
3.4.1 Scaling properties
3.4.2 Dependence on absorption
3.5 Diffusion Equation
3.5.1 The diffusion approximation
3.6 Derivation of the Diffusion Equation
3.7 Diffusion Coefficient
3.8 Properties of the Diffusion Equation
3.8.1 Scaling properties
3.8.2 Dependence on absorption
3.9 Boundary Conditions
3.9.1 Boundary conditions at the interface between diffusive and nonscattering media
3.9.2 Boundary conditions at the interface between two diffusive media
References
Part II: Solutions
4. Solutions of the Diffusion Equation for Homogeneous Media
4.1 Solution of the Diffusion Equation for an Infinite Medium
4.2 Solution of the Diffusion Equation for the Slab Geometry
4.3 Analytical Green's Functions for Transmittance and Reflectance
4.4 Other Solutions for the Outgoing Flux
4.5 Analytical Green's Function for the Parallelepiped
4.6 Analytical Green's Function for the Infinite Cylinder
4.7 Analytical Green's Function for the Sphere
4.8 Angular Dependence of Radiance Outgoing from a Diffusive Medium
References
5. Hybrid Solutions of the Radiative Transfer Equation
5.1 General Hybrid Approach to the Solutions for the Slab Geometry
5.2 Analytical Solutions of the Time-Dependent Radiative Transfer Equation for an Infinite Homogeneous Medium
5.2.1 Almost exact time-resolved Green's function of the radiative transfer equation for an infinite medium with isotropic scattering
5.2.2 Heuristic time-resolved Green's function of the radiative transfer equation for an infinite medium with non-isotropic scattering
5.2.3 Time-resolved Green's function of the telegrapher equation for an infinite medium
5.3 Comparison of the Hybrid Models Based on Radiative Transfer Equation and Telegrapher Equation with the Solution of the Diffusion Equation
References
6. The Diffusion Equation for Layered Media
6.1 Photon Migration through Layered Media
6.2 Initial and Boundary Value Problems for Parabolic Equations
6.3 Solution of the DE for a Two-Layer Cylinder
6.4 Examples of Reflectance and Transmittance for a Layered Medium
6.5 General Property of Light Re-Emitted by a Diffusive Medium
6.5.1 Mean time of flight in a generic layer of a homogeneous cylinder
6.5.2 Mean time of flight in a two-layer cylinder
6.5.3 Penetration depth in an homogeneous medium
References
7. Solutions of the Diffusion Equation with Perturbation Theory
7.1 Perturbation Theory in a Diffusive Medium and the Born Approximation
7.2 Perturbation Theory: Solutions for the Infinite Medium
7.2.1 Examples of perturbation for the infinite medium
7.3 Perturbation Theory: Solutions for the Slab
7.3.1 Examples of perturbation for the slab
7.4 Perturbation Approach for Hybrid Models
7.5 Perturbation Approach for the Layered Slab and for Other Geometries
7.6 Absorption Perturbation by use of the Internal Pathlength Moments
References
Part III: Software and Accuracy of Solutions
8. Software
8.1 Introduction
8.2 The Diffusion&Perturbation Program
8.3 Source Code: Solutions of the Diffusion Equation and Hybrid Models
8.3.1 Solutions of the diffusion equation for homogeneous media
8.3.2 Solutions of the diffusion equation for layered media
8.3.3 Hybrid models for the homogeneous infinite medium
8.3.4 Hybrid models for the homogeneous slab
8.3.5 Hybrid models for the homogeneous parallelepiped
8.3.6 General purpose subroutines and functions
References
9. Reference Monte Carlo Results
9.1 Introduction
9.2 Rules to Simulate the Trajectories and General Remarks
9.3 Monte Carlo Program for the Infinite Homogeneous Medium
9.4 Monte Carlo Programs for the Homogeneous and the Layered Slab
9.5 Monte Carlo Code for the Slab Containing an Inhomogeneity
9.6 Description of the Monte Carlo Results Reported in the CD-ROM
9.6.1 Homogeneous infinite medium
9.6.2 Homogeneous slab
9.6.3 Layered slab
9.6.4 Perturbation due to inhomogeneities inside the homogeneous slab
References
10. Comparisons of Analytical Solutions with Monte Carlo Results
10.1 Introduction
10.2 Comparisons between Monte Carlo and the Diffusion Equation: Homogeneous Medium
10.2.1 Infinite homogeneous medium
10.2.2 Homogeneous slab
10.3 Comparison between Monte Carlo and the Diffusion Equation: Homogeneous Slabwith an Inhomogeneity Inside
10.4 Comparisons between Monte Carlo and the Diffusion Equation: Layered Slab
10.5 Comparisons between Monte Carlo and Hybrid Models
10.5.1 Infinite homogeneous medium
10.5.2 Slab geometry
10.6 Outgoing Flux: Comparison between Fick and Extrapolated Boundary Partial Current Approaches
10.7 Conclusions
References
Appendix A. The First Simplifying Assumption of the Diffusion Approximation
Appendix B. Fick's Law
Appendix C. Boundary Conditions at the Interface between Diffusive and Non-Scattering Media
Appendix D. Boundary Conditions at the Interface Between two Diffusive Media
Appendix E. Green's Function of the Diffusion Equation in an Infinite Homogeneous Medium
Appendix F. Temporal Integration of the Time-Dependent Green's Function
Appendix G. Eigenfunction Expansion
Appendix H. Green's Function of the Diffusion Equation for the Homogeneous Cube Obtained with the Eigenfunction Method
Appendix I: Expression for the Normalizing Factor
References


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