Table of Contents

Acknowledgments

Disclaimer

List of Acronyms

List of Symbols

Preface

PART I THEORY

1 Scattering and Absorption Properties of Turbid Media

1.1 Approach Followed in This Manual

1.2 Optical Properties of a Turbid Medium

1.3 Statistical Meaning of the Optical Properties of a Turbid Medium

1.4 Similarity Relation and Reduced Scattering Coefficient

1.5 Ballistic Photons

1.6 Examples of Diffusive Media

1.7 Conclusion

References

2 The Radiative Transfer Equation

2.1 Quantities Used to Describe Radiative Transfer

2.2 The Radiative Transfer Equation

2.3 The Green's Function Method

2.4 Probabilistic Interpretation of the Solutions

2.5 Boundary Conditions for the Radiative Transfer Equation

2.6 Uniform Lambertian Illumination: A Special Reference Case

2.7 Properties of the Radiative Transfer Equation

2.8 The Radiative Transfer Equation in Transformed Domains

2.9 Numerical and Analytical Solutions of the RTE

2.10 Anisotropic Media and Anomalous Radiative Transport

2.11 Conclusion

References

3 The Diffusion Equation for Light Transport

3.1 Diffusion Equation and History

3.2 The Diffusion Approximation: Physical Assumptions

3.3 Derivation of the Diffusion Equation

3.4 Diffusion Coefficient

3.5 Properties of the Diffusion Equation

3.6 Diffusion Equation in Transformed Domains

3.7 Boundary Conditions

3.8 Conclusion

Reference

4 Anisotropic Light Propagation

4.1 The CW Anisotropic Diffusion Equation

4.2 Two Classical Cases

4.3 Conclusion

References

PART II SOLUTIONS

5 Solutions of the Diffusion Equation for Homogeneous Media

5.1 Solution of the Diffusion Equation for an Infinite Medium: Separation of Variables and Fourier Transform Method

5.2 Improved Solution for the CW Domain: Infinite Medium and Isotropic Scattering

5.3 Solution of the Diffusion Equation for a Slab: Method of Images

5.4 Solution of the Diffusion Equation for a Slab: Separation of Variables, Fourier Transform, and Eigenfunction Method

5.5 Moments of the Temporal Point Spread Function for a Slab

5.6 Solution of the Diffusion Equation for a Semi-infinite Medium

5.7 Other Solutions for the Outgoing Flux

5.8 Solution of the Diffusion Equation for an Infinite Medium: Separation of Variables and Fourier Transform Method

5.9 Analytical Green's Function for an Infinite Cylinder

5.10 Analytical Green's Function for a Sphere

5.11 Solution of the Diffusion Equation for a Pencil Beam Source Impinging on a Finite Cylinder Geometry

5.12 Ohm’s Law for Light

5.13 Solutions for a Slab Illuminated by Infinitely Extended Sources

5.14 Solutions of the DE in Transformed Domains

5.15 Angular Dependence of Radiance Exiting a Diffusive Medium

5.16 Comment: The Angular Dependence of Reflectance

5.17 Anisotropic Media

5.18 Summary Comments on Applications

5.19 Conclusion

References

6 Ballistic and Quasi-Ballistic Radiation

6.1 Solution of the RTE for Ballistic Radiation

6.2 Heuristic Hybrid Model for Ballistic Photon Detection in Collimated Transmittance CW Measurements

6.3 Conclusion

References

7 Statistics of Photon Penetration Depth in Diffusive Media

7.1 Statistics of Photon Penetration Depth inside an Infinite Laterally Extended Slab

7.2 Scaling Relationships for the Penetration Depth

7.3 Heuristic Formula for the Mean Average Penetration Depth in a Homogeneous Medium

7.4 Solutions for f and <z_max> for a Slab in the Diffusion Approximation

7.5 Heuristic Model for a Semi-infinite Medium

7.6 Frequency-Domain Penetration Depth

7.7 Summary Comments on Applications

7.8 Conclusion

References

8 Statistics of Transversal Penetration Depth in the TD

8.1 Statistics for the Radial Penetration Depth in a Laterally Infinite Slab

8.2 Statistics for the Lateral Penetration Depth in a Laterally Infinitely Extended Slab

8.3 Statistics of the Radial Penetration Depth in an Infinite Medium

8.4 Comparisons of the Different Formulas for the Maximum Penetration Depth

8.5 Summary Comments on Applications

8.6 Conclusion

References

9 Average Photon Distance from Source and Relative Moments

9.1 Statistical Relationships: Displacement of Photons from the Source in an Infinite Homogeneous Medium

9.2 Penetration Depth for all Photons Propagating in an Infinite Medium

9.3 Penetration Depth for all Photons Propagating through a Slab

9.4 Conclusion

References

10 Hybrid Solutions of the Radiative Transfer Equation

10.1 General Hybrid Approach to the Solutions for the Slab Geometry

10.2 Analytical Solutions of the Time-Dependent RTE for an Infinite Homogeneous Medium

10.3 Comparison of the Hybrid Models Based on the RTE and Telegrapher Equation with the Solution of the Diffusion Equation

10.4 Conclusion

References

11 The Diffusion Equation for a Two-Layered Cylinder

11.1 Photon Migration through Layered Media

11.2 Initial and Boundary Value Problems for Parabolic Equations

11.3 Solution of the DE for a Two-Layer Cylinder

11.4 Examples of Reflectance and Transmittance of a Layered Medium

11.5 General Properties of Light Re-emitted by a Diffusive Medium

11.6 Summary Comments on Applications

11.7 Conclusion

References

12 The Diffusion Equation for an N-Layered Cylinder

12.1 Photon Migration through an N-Layered Cylinder

12.2 Conclusion

References

13 Solutions of the Diffusion Equation with Perturbation Theory

13.1 Perturbation Theory in a Diffusive Medium and the Born Approximation

13.2 Perturbation Theory: Solutions for the Infinite Medium

13.3 Perturbation Theory: Solutions for the Slab

13.4 Perturbation Approach for Hybrid Models

13.5 Perturbation Approach for a Layered Slab and for Other Geometries

13.6 Absorption Perturbation by Using the Internal Pathlength Moments

13.7 Closed-Form CW Perturbative Solutions of the DE with Absorbing Inclusions

13.8 Summary Comments on Applications

13.9 Conclusion

References

14 Time-Domain Raman and Fluorescence Analytical Solutions

14.1 Theoretical Approach and General Definitions

14.2 Heuristic Model

14.3 Raman Analytical Solutions Based on the Time-Dependent Diffusion Equation

14.4 Solution of the DE for Time-Resolved Fluorescence in an Infinite Medium

14.5 Solution of the DE for a Raman Signal with Background Fluorescence

14.6 Examples of Raman Re-emission Calculated with Raman Forward Solvers

14.7 Summary Comments on Applications

14.8 Conclusion

References

PART III VALIDATION OF THE SOLUTIONS

15 Elementary Monte Carlo Methods in Turbid Media

15.1 Photon Packets

15.2 Photon Trajectories

15.3 Photon Detection

15.4 Statistical Error in MC Results

15.5 MC Methods for Handling Photon Packet Weight

15.6 Boundaries Conditions in MC: Compatibility between Classical and Anomalous Photon Transport

15.7 Interruption of the Propagation of a Photon Packet: Russian Roulette

15.8 Comparison of the Different Methods

15.9 Conclusion

References

16 Reference Monte Carlo Results

16.1 General Remarks

16.2 MC for an Infinite Homogeneous Medium

16.3 MC for a Homogeneous and a Layered Slab

16.4 Monte Carlo Code for a Slab Containing an Inhomogeneity

16.5 Description of the Monte Carlo Program Calculating the Maximum Mean Penetration Depth of Detected Photons

16.6 Description of the Monte Carlo Program Simulating the Raman Signal and the Fluorescence Signal

16.7 Conclusion

References

17 Comparisons of Analytical Solutions with Monte Carlo Results

17.1 Introduction

17.2 Comparisons between MC and DE: Homogeneous Medium

17.3 Validation of the DE Solutions for the Mean Maximum and Mean Average Penetration Depth

17.4 Comparison between MC and DE: Homogeneous Slab with an Internal Inhomogeneity

17.5 Comparisons between MC and DE: N-Layered Slab and N-Layered Cylinder

17.6 Comparisons between MC and Hybrid Models

17.7 Comparisons between the MC and Heuristic Model for Ballistic Photon Detection

17.8 Outgoing Flux: Comparison between Fick and Extrapolated Boundary Partial Current Approaches

17.9 Validation of the DE Solutions for the Raman Signal

17.10 Conclusions

References

18 Numerical Implementations and Reference Database

18.1 Numerical Implementation of the Solutions

18.2 Reference Database: Monte Carlo Simulations

PART IV APPENDICES

A:  Intuitive Justification of the Diffusion Approximation

B:  Fick's Law

C:  Boundary Conditions between Diffusive and Non-Scattering Media

D:  Boundary Conditions between Two Diffusive Media

E:  Diffusion Equation with an Infinite Homogeneous Medium: Separation of Variables and Fourier Transform Methods

F:  Anisotropic CW Diffusion Equation with an Infinite Homogeneous Medium: Separation of Variables and Fourier Transform Methods

G:  The Reciprocity Principle for a Plane Wave and a Pencil Beam Impinging on a Slab

H:  Temporal Integration of the Time-Dependent Green's Function

I:  The Diffusion Equation: Separation of Variables and Eigenfunction Methods

J:  The Diffusion Equation with a Homogeneous Parallelepiped: Separation of Variables and Eigenfunction Methods

K:  Mean Square Displacement of the Light Penetration in Turbid Media Based on the RTE

L:  Expression for the Normalizing Factor

M:  Finite Integral Transforms

N:  Relationship between the Inverse Fourier Transform and Inverse Laplace Transform

O:  Equivalence of the MC Methods

Index

Preface (abridged)

This manual is intended as an in-depth introduction to light propagation through biological tissues and diffusive media. After having treated the general theory of light diffusion and its physical and biological interpretation, the text proposes the derivation of tens of already reported and newly derived analytical and/or semi-analytical solutions. These solutions are "ready to use" and represent the most employed algorithms appearing in tissue optics and related fields, where light is used to probe the optical and/or biological properties of diffusive media. By studying these examples, the readers should be able to directly apply the solutions to real laboratory problems or to develop their own specific solutions.

In a dedicated part of the manual, the solutions are tested against "gold standard" reference data, and their domain of validity is carefully discussed. This part also serves as a tutorial explaining how to generate suitable reference data and how to test new algorithms obtained, e.g., by the reader.

The text is particularly well suited for skilled master students but also for advanced scientists searching for rapid solutions, eliminating the problem of repeating cumbersome calculations in diffusive optics, and bypassing the need to search among hundreds of published papers.

Thus, to summarize, the present manual offers: (I) A general introduction to the theory of photon migration; (II) Ready-to-use analytical and/or semianalytical solutions, derived from the general theory of photon migration, associated with problems typically encountered in biomedical optics and related domains; (III) A validation of the proposed solutions by means of comparisons with Monte Carlo (MC) simulations; (IV) A tutorial software package, implementing the most representative analytical and semi-analytical solutions of the manual (see the links to the supplemental material) and (V) A set of precalculated MC data serving as a gold-standard reference and allowing the reader to personally check the presented exact/approximated solutions (see the link icons).