Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE Press Monograph Vol. PM193)) [Pap/Cdr ed.] 0819476587, 9780819476586

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Light Propagation through Biological Tissue and Other Diffusive Media THEORY, SOLUTIONS, AND SOFTWARE

Light Propagation through Biological Tissue and Other Diffusive Media THEORY, SOLUTIONS, AND SOFTWARE

Fabrizio Martelli Samuele Del Bianco Andrea Ismaelli Giovanni Zaccanti

Bellingham, Washington USA

Library of Congress Cataloging-in-Publication Data Light propagation through biological tissue and other diffusive media : theory, solutions, and software / Fabrizio Martelli ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-0-8194-7658-6 1. Light--Transmission--Mathematical models. 2. Tissues--Optical properties. I. Martelli, Fabrizio, 1969QC389.L54 2009 535'.3--dc22 2009049647

Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: +1 360.676.3290 Fax: +1 360.647.1445 Email: [email protected] Web: http://spie.org Copyright © 2010 Society of Photo-Optical Instrumentation Engineers (SPIE) All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America.

To our families

Contents Acknowledgements

xiii

Disclaimer

xv

List of Acronyms

xvii

List of Symbols 1

xix

Introduction References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I THEORY

1 6

7

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 . . . . . . . . . .

9 . 9 . . 11 . 12 . 13 . 18 . 19 . 22 . 24

29 . . . . . . 29 . . . . . . . 31 . . . . . . 32 . . . . . . 33 . . . . . . 33 . . . . . . 35 . . . . . . . 37 . . . . . . 38 . . . . . . 39

viii

CONTENTS

3.7 3.8

Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 41 Properties of the Diffusion Equation . . . . . . . . . . . . . . . . 42 3.8.1 Scaling properties . . . . . . . . . . . . . . . . . . . . . . 42 3.8.2 Dependence on absorption . . . . . . . . . . . . . . . . . 42 3.9 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 43 3.9.1 Boundary conditions at the interface between diffusive and non-scattering media . . . . . . . . . . . . . . . . . . . . 43 3.9.2 Boundary conditions at the interface between two diffusive media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

II SOLUTIONS

55

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 62 66 74 80 81 83 84 89

5

Hybrid Solutions of the Radiative Transfer Equation 93 5.1 General Hybrid Approach to the Solutions for the Slab Geometry . 94 5.2 Analytical Solutions of the Time-Dependent Radiative Transfer Equation for an Infinite Homogeneous Medium . . . . . . . . . . . 97 5.2.1 Almost exact time-resolved Green’s function of the radiative transfer equation for an infinite medium with isotropic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.2 Heuristic time-resolved Green’s function of the radiative transfer equation for an infinite medium with non-isotropic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2.3 Time-resolved Green’s function of the telegrapher equation for an infinite medium . . . . . . . . . . . . . . . . . . . 99 5.3 Comparison of the Hybrid Models Based on the Radiative Transfer Equation and Telegrapher Equation with the Solution of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6

The Diffusion Equation for Layered Media

109

CONTENTS

6.1 6.2 6.3 6.4 6.5

Photon Migration through Layered Media . . . . . . . . . . . . . Initial and Boundary Value Problems for Parabolic Equations . . . Solution of the DE for a Two-Layer Cylinder . . . . . . . . . . . Examples of Reflectance and Transmittance of a Layered Medium 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 a homogeneous medium . . . . . . . 6.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

ix 109 110 112 118 . 121 . 121 123 124 125 126

Solutions of the Diffusion Equation with Perturbation Theory 131 7.1 Perturbation Theory in a Diffusive Medium and the Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.2 Perturbation Theory: Solutions for the Infinite Medium . . . . . . 135 7.2.1 Examples of perturbation for the infinite medium . . . . . . 137 7.3 Perturbation Theory: Solutions for the Slab . . . . . . . . . . . . . 141 7.3.1 Examples of perturbation for the slab . . . . . . . . . . . 148 7.4 Perturbation Approach for Hybrid Models . . . . . . . . . . . . . 154 7.5 Perturbation Approach for the Layered Slab and for Other Geometries155 7.6 Absorption Perturbation by Use of the Internal Pathlength Moments 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

III SOFTWARE AND ACCURACY OF SOLUTIONS

163

8

Software 165 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.2 The Diffusion&Perturbation Program . . . . . . . . . . . . . . . 166 8.3 Source Code: Solutions of the Diffusion Equation and Hybrid Models170 8.3.1 Solutions of the diffusion equation for homogeneous media 171 8.3.2 Solutions of the diffusion equation for layered media . . . 175 8.3.3 Hybrid models for the homogeneous infinite medium . . . . 177 8.3.4 Hybrid models for the homogeneous slab . . . . . . . . . 180 8.3.5 Hybrid models for the homogeneous parallelepiped . . . . 182 8.3.6 General purpose subroutines and functions . . . . . . . . 182 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9

Reference Monte Carlo Results 187 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.2 Rules to Simulate the Trajectories and General Remarks . . . . . . 187 9.3 Monte Carlo Program for the Infinite Homogeneous Medium . . . . 191

x

CONTENTS

9.4 9.5 9.6

Monte Carlo Programs for the Homogeneous and the Layered Slab Monte Carlo Code for the Slab Containing an Inhomogeneity . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 195 . 197 . 197 198 199 199 . 201

10 Comparisons of Analytical Solutions with Monte Carlo Results 203 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.2 Comparisons Between Monte Carlo and the Diffusion Equation: Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . 204 10.2.1 Infinite homogeneous medium . . . . . . . . . . . . . . . 204 10.2.1.1 Time-resolved results . . . . . . . . . . . . . . 204 10.2.1.2 Continuous wave results . . . . . . . . . . . . . 209 10.2.2 Homogeneous slab . . . . . . . . . . . . . . . . . . . . . 213 10.2.2.1 Time-resolved results . . . . . . . . . . . . . . 213 10.2.2.2 Continuous wave results . . . . . . . . . . . . . . 217 10.3 Comparison Between Monte Carlo and the Diffusion Equation: Homogeneous Slab with an Internal Inhomogeneity . . . . . . . . . 217 10.4 Comparisons Between Monte Carlo and the Diffusion Equation: Layered Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 10.5 Comparisons Between Monte Carlo and Hybrid Models . . . . . . 225 10.5.1 Infinite homogeneous medium . . . . . . . . . . . . . . . 226 10.5.2 Slab geometry . . . . . . . . . . . . . . . . . . . . . . . 230 10.6 Outgoing Flux: Comparison between Fick and Extrapolated Boundary Partial Current Approaches . . . . . . . . . . . . . . . . . . . . . 232 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 10.7.1 Infinite medium . . . . . . . . . . . . . . . . . . . . . . . 234 10.7.2 Homogeneous slab . . . . . . . . . . . . . . . . . . . . . . 237 10.7.3 Layered slab . . . . . . . . . . . . . . . . . . . . . . . . 238 10.7.4 Slab with inhomogeneities inside . . . . . . . . . . . . . 238 10.7.5 Diffusive media . . . . . . . . . . . . . . . . . . . . . . . 238 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Appendices 243 Appendix A: The First Simplifying Assumption of the Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Appendix B: Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 246 Appendix C: Boundary Conditions at the Interface Between Diffusive and Non-Scattering Media . . . . . . . . . . . . . . . . . . . . . . . . 250

CONTENTS

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index

xi

252 255 . 261 262 265 268 269 271

Acknowledgements We have to thank many people for their valuable contributions to a growing body of knowledge from which this book originated. We are very thankful to Piero Bruscaglioni, who introduced us to the study of light propagation through turbid media, for his constant help and fruitful discussions. A great amount of the research that led to this book has been contributed by the many students who worked in our laboratory to prepare their dissertations. Their questions, the fruitful discussions with them, their work, and the new ideas they proposed helped us very much in developing new models and methodologies for investigating photon migration through turbid media. Thanks a lot to Enrico Battistelli, Adriana Taddeucci, Michele Bassani, Daniele Contini, Lucia Alianelli, Silvia Carraresi, and others! We would like also to thank our numerous colleagues all over the world for their suggestions and stimulating discussions. A special thanks to the colleagues involved in the nEUROPt project, and in particular to friends Antonio Pifferi, Lorenzo Spinelli, and Alessandro Torricelli from Politecnico di Milano: We are very grateful for their suggestions and the fruitful cooperation. We thank our colleague and friend Danilo Marcucci not only for his help to solve practical problems but also for his cheerful companionship, and Ricardo Lepore, with whom we recently started a promising collaboration. We express our gratitude to Yukio Yamada for inviting Fabrizio Martelli to Japan’s Mechanical Engineering Laboratory (MEL) with the STA fellowship program: The research activity carried out at MEL by Fabrizio Martelli has been precious for the studies on photon migration through layered diffusive media. We wish to thank our friend Angelo Sassaroli from Tufts University for his advice and opinions on the work done. The discussions and the exchange of views on several subjects have been of constant help with writing this book. The research leading to these results received funding from the European Community’s Seventh Framework Programme [FP7/2007-2013] under grant agreement n. HEALTH-F5-2008-201076 (nEUROPt project).

Disclaimer While the authors have used their best efforts in preparing this book and the enclosed CD-ROM, they make no warranties, express or implied, that the formulas of the book and the software and results contained in the enclosed CD-ROM are free of error, or are consistent with any standard of merchantability, or will meet the requirements for any particular application. They should not be relied upon for solving a problem whose incorrect solution could result in injury to a person or loss of property. Use of the programs in such a manner is at the user’s own risk. The authors disclaim all liability for direct or consequential damages resulting from use of the programs.

List of Acronyms Acronym CW DE DLL DOT EBC EBPC FWHM HG MC NIRS PCBC RTE TE TCSPC TPSF TR ZBC

Description Continuous Wave Diffusion Equation Dynamic-Link Library Diffuse Optical Tomography Extrapolated Boundary Condition Extrapolated Boundary Partial Current Full Width Half Maximum Henyey and Greenstein Monte Carlo Near-Infrared Spectroscopy Partial Current Boundary Condition Radiative Transfer Equation Telegrapher Equation Time-Correlated Single-Photon Counting Temporal Point Spread Function Time-Resolved Zero Boundary Condition

List of Symbols Latin-based Notation a ′ a A c Ca Cs C

g2  dk dE dΣ dV dΩ dP D E2 , E3 , E4

f1 (θ), f2 (ϕ), f3 (z) F F (θe ) g G h I In J~ Jn Kn hki lmax

Description Radius of the sphere Extrapolated radius of the sphere Coefficient for the extrapolated boundary condition Speed of light in vacuum Absorption cross section Scattering cross section Geometrical cross section Mean square distance from the source after k scattering events Elementary energy Elementary area Elementary volume Elementary solid angle Elementary power Diffusion coefficient Coefficients for the boundary condition between two diffusive media Probability distribution functions for the photon’s scattering Cumulative probability function Angular dependence of the outgoing radiance Asymmetry factor Green’s function Planck’s constant Radiance or specific intensity Modified Bessel function of order n Flux vector Bessel functions of order n Modified Bessel function of order n Mean number of scattering events undergone by photons Maximum length of a photon trajectory

xx

List of Symbols

Latin-based Notation ℓs = 1/µs ℓa = 1/µa ℓt = 1/µt ′ ′ ℓ = 1/µs hli hlR i hlT i Lx , Ly , Lz L ′ L n ni ne nr N Nln p(θ) p(z) P Pn q0 Qa = Ca /Cg Qs = Cs /Cg ′ Qs = Qs (1 − g) r ~r → r0 → r2 → r3 − → − → + , r− rm m → rs →′ r R RF s sˆ S virt

Description Scattering mean free path Absorption mean free path Extinction mean free path Transport mean free path Mean pathlength of photons Mean pathlength for the total reflectance Mean pathlength for the total transmittance Dimensions for the parallelepiped Radius of the cylinder Extrapolated radius of the cylinder Refractive index Refractive index of the diffusive medium Refractive index of the external medium Relative refractive index Particle concentration Normalizing factor for the two-layer cylinder solution Scalar scattering function Penetration depth of photons Impinging power of a light beam or detected power Legendre Polynomials Source term of the diffusion equation Absorption efficiency Scattering efficiency Reduced scattering efficiency Radius of particles Position of the receiver Position of the source Position of the inhomogeneity Position of the detector Sources positions with the method of images for the slab Position vector of the real source Position of the source Reflectance Fresnel reflection coefficient for unpolarized light Thickness of the slab Unit vector for the direction of observation of radiance Virtual source term

xxi

List of Symbols

Latin-based Notation t ′ t hti htn i t0 hti i T u v wmp x = 2πr/λ hxk i , hyk i , hzk i ze = 2AD

Description Current observation time Emission time of the source Mean time of flight of photons nth –order moment Time of flight for ballistic photons Mean time of flight spent inside a layer i Transmittance Energy density Speed of light inside the medium Contribution to the TPSF of the mth trajectory Size parameter Mean photon’s coordinates after k scattering events Extrapolated distance

xxii

List of Symbols

Greek-based Notation α βm Γ δΦ = Φpert − Φ0 δΦa δΦD δRa δRD δT a δT D ∂V ε θ Θ(x) λ Λ λn µa µs µt = µa + µs ′ µs = µs (1 − g) µef f ξn ρ ρv σ Σ τ ϕ Φ Φ0 Φpert Ωd

Description Angular range of the field of view of the receiver ′ Positive roots of the equation Jm (βm L ) = 0 Gamma function Perturbation for the fluence rate Absorption perturbation for the fluence rate Scattering perturbation for the fluence rate Absorption perturbation for the reflectance Scattering perturbation for the reflectance Absorption perturbation for the transmittance Scattering perturbation for the transmittance External physical boundary of a domain V Source term of the radiative transfer equation Polar angle Step function (0 for x < 0 and 1 for x > 0) Wavelength Single-scattering albedo Eigenvalues Absorption coefficient Scattering coefficient Extinction coefficient Reduced scattering coefficient Effective attenuation coefficient Eigenfunctions Distance of the receiver from the pencil light beam Volume fraction of particles Standard deviation Cross section of the light beam Optical thickness Azimuthal angle Fluence rate Unperturbed fluence rate Perturbed fluence rate Acceptance solid angle of the detection system

Chapter 1

Introduction The purpose of this book and the enclosed software is to provide a tool with dual functionality. The text summarizes the theory of light propagation through diffusive media, while the software serves as a computational tool giving users an interactive approach to understanding the link between theory and real calculations. Through this two-pronged approach, the authors believe that this tool will be valuable both for research investigations and educational purposes. The software on the CD-ROM is designed to calculate the solutions of the diffusion equation (DE) — solutions that can be verified by comparing against the enclosed set of reference Monte Carlo (MC) results. The text describes the basic theory of photon transport along with analytical solutions of the diffusion equation for several geometries. The book also includes a description of the software, related materials, and their use. We will show that propagation of light through turbid media (i.e., media with scattering and absorption properties) can be accurately described with the radiative transfer equation (RTE), a complex integro-differential equation of which analytical solutions are not available for geometries of practical interest. The DE is an approximation that can be obtained from the RTE by making some simplifying assumptions. For the DE, analytical solutions are available in many geometries, but it is necessary to stress that these solutions are approximate. Therefore, each application should be checked to verify that the accuracy of these approximations is sufficient. This check can be performed by comparing the approximate solutions against reference solutions of the RTE. For this purpose, the CD-ROM also includes examples of numerical solutions of the RTE obtained with MC simulations for different geometries. Diffusive media are turbid media for which the solutions of the DE provide a sufficiently accurate description of light propagation. Through these media, photons propagate in a diffusive regime, i.e., the path followed by any photon migrating from the source to the detector looks like a random walk (zigzag trajectory). This occurs when photons undergo a sufficiently high number of scattering events that their trajectories become randomized. Section 2.5 lists a number of media common in daily life for which a diffusive regime of propagation can be assumed. This list includes highly scattering media like biological tissues, agricultural products, wood, 1

2

Chapter 1

paper, plastic materials, sugar, salt, and milk, for which the diffusive regime can be reached even when the volume of the medium is of a few cubic centimeters. The list also includes slightly scattering media like clouds of gas and dust in the interstellar medium, in which case an extremely large volume is necessary. In the book, a review is presented of the theories and the formulae based on the DE that have been widely used for biomedical applications. These theories and formulae are the basis of the programs provided with the CD-ROM. The contents of the book are divided into three parts. The first part (Chapters 2 and 3) presents the basic theory of photon transport. In Chapter 2, the general concepts and the physical quantities necessary to describe light propagation through absorbing and scattering media are introduced. In Chapter 3, the RTE and the DE are described and discussed. In the second part (Chapters 4, 5, 6 and 7), various solutions of the RTE are presented. Chapter 4 is devoted to solutions of the DE for homogeneous media. Chapter 5 is dedicated to hybrid solutions for the homogeneous slab based on solutions of the RTE and the telegrapher equation. In Chapter 6, a solution of the DE for a two-layer medium is described. In Chapter 7, solutions of the perturbed DE when small defects are introduced into the medium are obtained with the Born approximation. In the third part (Chapters 8, 9 and 10), the software provided with the CD-ROM is described, and the accuracy of the solutions presented is investigated and discussed. In Chapter 8, the software included in the CD-ROM is described. Chapters 9 and 10 are dedicated to the validation of the solutions and of the software described in the previous chapters. The validation is done by means of comparisons with the results of MC simulations. In Chapter 9, a detailed description of the MC programs employed is provided, and in Chapter 10 the results of the comparisons are described and discussed. In the CD-ROM, all the programs for calculating the solutions presented in the book are provided with the results of MC simulations that can be used as a standard reference. The CD-ROM is divided into three parts: the first includes the software named Diffusion&Perturbation; the second contains all the source code for the solutions found in the book; and the third part contains the results of MC simulations that can be used as a standard reference. The Diffusion&Perturbation software is dedicated to the calculation and the visualization of the solutions of the DE together with its perturbation theory for a homogeneous medium. This software package has a user-friendly interface developed in Visual C and can be easily installed on any PC with a WINDOWS operating system. With this software, the calculations of typical data types—such as steady-state and time-resolved reflectance, transmittance and fluence rate, both for the case of homogeneous and inhomogeneous media—are implemented. For inhomogeneous media, the calculations are based on the solutions of perturbation theory of the DE. The second part of the software provides the subroutines to estimate almost all the solutions of photon transport presented in the book: the solutions of the

Introduction

3

DE for homogeneous media, the hybrid solutions based on the RTE and on the telegrapher equation, the solutions of the DE for a two-layer medium, and the perturbed solutions for absorbing and scattering inclusions. The source code is distributed with a list of examples and drivers that lead the reader to use this software for photon migration studies. The intent of this second part of the software is to involve and to stimulate the readers to build their own programs to make investigations on photon migration through diffusive media. A set of subroutines is provided that can be composed like a mosaic to perform various calculations. Detailed calculations of the physical quantities used in photon migration (e.g., flux, radiance, etc.) are carried out in various geometries of the medium. The source code has been written using the FORTRAN language. The FORTRAN subroutines can be converted to a dynamic-link library (DLL) interface for processes that use them. In this way, the subroutines can be employed in programs written in other languages like C or, although it is not a straightforward procedure, in other environments like MATLAB.1 For a correct utilization of this second part of the CD-ROM, even though the software is accompanied by comments and examples of use, some basic programming skills are required; however, programming skills are not necessary for using the Diffusion&Perturbation software. The third part of the CD-ROM provides a set of results of MC simulations that can be used as a standard reference. The results are for three different geometries: the homogeneous infinite medium, the homogeneous slab (with and without inhomogeneities inside), and the layered slab. All the results are presented in Excel files (*.xls). The theories and computations presented here are relevant to near-infrared spectroscopy (NIRS) and diffuse optical tomography (DOT). NIRS and DOT are optical techniques that use near-infrared light to probe biological tissues. Nearinfrared light can penetrate deeply into tissues (some centimeters) and is sensitive to several tissue constituents. For this reason, NIRS techniques provide important physiological information on living tissues. Biological tissue is a complex random medium where light undergoes many scattering events, and its propagation is similar to a diffusion process. Although the interaction of near-infrared light with tissue is dominated by scattering effects (the distance between two subsequent scattering events is on the order of ≈ 100 µm), most of the physiological information is led by the absorption of chromophores like oxy- and deoxy-hemoglobin. Since biological tissue behaves as a diffusive medium in the near-infrared region, many applications in biomedical optics require an appropriate knowledge of photon migration through diffusive media. Thus, in NIRS and DOT the diffusion equation is widely used to predict measurements of data types at different detectors’ sites. The main challenge of NIRS and DOT is to achieve an accurate map of the optical properties (absorption and reduced scattering coefficients) from measurements taken at the boundary of a target organ (inverse problem). Usually, the solution to the inverse problem in DOT or NIRS relies on accurate and possibly fast models of photon migration in tissues

4

Chapter 1

(forward problem) able to provide the light distribution inside the medium, given the geometry, the distribution of sources, and the optical properties of the medium. This book examines the forward problem, and the software provided within the enclosed CD-ROM can thus be used as a package of forward solvers for the study of photon migration in tissue. Thousands of papers on the diffusion of light have been published, and a review of this huge amount of work would require a review of several research fields, but that is beyond the scope of this book. For these reasons, the bibliography does not exhaustively cover all of the work done on the diffusion equation but refers mainly to publications in the field of biomedical optics and more precisely to the field of NIRS and DOT. In order to have a more complete view of photon transport, we suggest the reader refer to other classical books on light propagation.2–6 Although the theoretical and computational tools provided with this book and CD-ROM have their primary use in the field of biomedical optics, there are many other applications in which they can be used. In fact, many media (agricultural products, wood, food, plastic materials, paper, pharmaceutical products, etc.) have optical properties at visible and/or near-infrared wavelengths for which light propagation in the diffusive regime can be established, even in a few cubic centimeters of material. Therefore, the same techniques used to study biological tissues can be used in many practical applications to monitor industrial processes or for quality control,7–12 or in completely different fields. In fact, the theories used in this book (i.e., the radiative transfer equation and the diffusion equation) arise from the more general transport theory. Transport theory concerns the transport of particles through a background medium and is used in several applications where the transported particles and the host medium can have very different quantities. The advent of personal computers has made several numerical methods affordable to solve transport theory, and the availability of numerical solutions has further encouraged the use of this theory to solve practical problems. The list of applications is surprisingly long. Duderstadt and Martin6 summarized some of the most relevant applications of transport theory. Transport theory has been applied to the study of • neutron transport in nuclear reactors, • penetration of light through the atmosphere, • diffusion of molecules in gases, • diffusion of holes and electrons in semiconductors, and • photon transport through biological tissues.

Despite the different kinds of particles (neutrons, gas molecules, electrons, photons) that may be involved in the transport processes, all these phenomena can be studied and described by using the same basic equation. When the transport process becomes diffusive, the transport equation can be simplified through the diffusion

5

Introduction

equation. Given a physical quantity u representative of the physical process studied (for instance, the particle density), whenever u is described by the equation ∂ r, t) ∂t u(~

− k1 ∇2 u(~r, t) + k2 u(~r, t) = 0,

(1.1)

we are dealing with a diffusive process. The coefficient k1 is related to the spatial and temporal scale of the diffusive phenomenon studied, and the coefficient k2 is related to the probability that the transported particles will be absorbed. For radiative transfer processes, k1 will be related to the transport coefficient or diffusion coefficient of photons through the medium. For neutron transport processes, k1 will be related to the transport coefficient of neutrons through the medium. For the diffusion of electrons and holes in semiconductors, k1 will be related to the electrical conductivity. For the diffusion of molecules in gases, k1 will be related to the transport coefficient of the molecules through the gas. The above equation with k2 = 0 can be also used to describe the conduction of heat in solid isotropic materials, where u will be the temperature of the medium, and k1 will be the thermometric conductivity of the substance, i.e., a material-specific quantity depending on the thermal conductivity, the density, and the specific heat of the substance.13 Then, with the same mathematical tool, very different physical processes can be studied where very different physical interpretations are required. Theories and software presented in the book and in the CD-ROM can therefore, in principle, be re-adapted to other fields where the same basic equation is used, i.e., where a diffusive phenomenon is involved. We finally point out that the theories and solutions presented in this book have been obtained with reference to media illuminated by unpolarized light. However, it should be stressed that the solutions are also applicable to media illuminated by polarized light, which commonly occurs when laser sources are used. In fact, multiple scattering randomly changes polarization of scattered light, so that light detected after a sufficiently large number of scattering events is completely depolarized. Memory of polarization only remains near the source where photons arrive after a small number of scattering events. It has been shown with numerical simulations14, 15 and experiments15 that when propagation occurs in diffusive regime (i.e., when the solutions presented in this book become applicable), received photons have lost memory of the initial state of polarization, and the results for polarized light become almost identical to those obtained for unpolarized light.

6

Chapter 1

References 1. The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, MATLAB 7 External Interfaces (2008). 2. H. C. van de Hulst, Light Scattering by Small Particles, John Wiley & Sons, New York (1957). 3. S. Chandrasekhar, Radiative Transfer, Oxford University Press, London/Dover, New York (1960). 4. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Publishing Company, Massachusetts, Palo Alto, London (1967). 5. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1, Academic Press, New York (1978). 6. J. J. Duderstadt and W. R. Martin, Transport Theory, John Wiley & Sons, New York (1979). 7. P. Tatzer, M. Wolf, and T. Panner, “Industrial application for inline material sorting using hyperspectral imaging in the NIR range,” Real-Time Imaging 11, 99–107 (2005). 8. M. Blanco, J. Coello, H. Iturriaga, S. Maspoch, and C. de-la Pezuela, “Nearinfrared spectroscopy in the pharmaceutical industry,” Analyst 123, 135R–150R (1998). 9. Y. Roggo, P. Chalus, L. Maurer, C. Lema-Martinez, A. Edmond, and N. Jent, “A review of near infrared spectroscopy and chemometrics in pharmaceutical technologies,” J. Pharm. Biomed. Anal. 44, 683–700 (2007). 10. G. Reich, “Near-infrared spectroscopy and imaging: Basic principles and pharmaceutical applications,” Appl. Opt. 57, 1109–1143 (2005). 11. M. C. Cruz and J. A. Lopes, “Quality control of pharmaceuticals with NIR: from lab to process line,” Vib. Spectrosc. 49, 204–210 (2009). 12. M. Zude (Ed.), Optical Monitoring of Fresh and Processed Agricultural Crops, (Contemporary Food Engineering Series), CRC Press, Boca Raton, Florida (2009). 13. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Ely House, London (1959). 14. P. Bruscaglioni, G. Zaccanti, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, 6142–6150 (1993). 15. J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).

Chapter 2

Scattering and Absorption Properties of Diffusive Media 2.1 Approach Followed in this Book The collection of analytical solutions reported in this book and the related software in the enclosed CD-ROM concern propagation of light through diffusive media. Diffusive media are turbid media where propagation occurs in the diffusive regime, i.e., propagation is dominated by multiple scattering, and photons undergo many scattering events before being detected. The solutions describe how the energy propagates through turbid media in which light-matter interaction can be modeled with elastic scattering and absorption. Scattering interaction deflects photons along new directions of propagation, but the energy, and thus the wavelength and the frequency, of scattered photons remains unchanged. Absorption interaction causes the disappearance of photons. Therefore, the wavelength of the radiation that outlives absorption and propagates through the medium remains unchanged. The turbid media considered in this book are random media, i.e., media where the amplitude and phase of the waves propagating through such media fluctuate randomly in time and space.1 The simple model used in this book to represent light-matter interaction is a crude assumption that avoids getting inside the actual complexity of the phenomena of light-matter interaction, and it has the advantage of leading to relatively simple analytical solutions for photon migration valid for many real media of everyday life. Light-matter interaction is characterized by several phenomena that will be ignored in this book since they are not going to affect the results of our investigations. Some light-matter interaction phenomena are mentioned below. Absorption and scattering are the main phenomena of light-matter interaction. Absorption is a phenomenon related to the absorption bands of molecules. When an electron within a molecule is raised by photon absorption to an excited state, the relaxation to the ground state can occur following a non-radiative decay process with the emission of heat or/and a radiative process with the emission of a photon at a different wavelength (luminescence effects). The non-radiative processes precede and/or compete with the luminescence processes. When the electron relaxes to the 9

10

Chapter 2

ground state with the emission of one photon, the involved processes are called fluorescence and phosphorescence. These photons are re-emitted at a longer wavelength (their energy is smaller with respect to absorbed photons) and can continue their migration through the medium. In this book, absorbed photons will be considered lost for propagation, and these effects will not be considered. For details on fluorescence and phosphorescence, see Ref. 2. Only the effects of linear absorption (where the strength of absorption depends linearly on the intensity of light) will be considered, while effects like two-photon absorption (where the strength of absorption depends on the square of the light intensity) will be neglected. We stress that this effect is many orders of magnitude weaker than linear absorption and therefore is not a common phenomenon.3 For the same reasons, the effects of multi-photon absorption are not discussed. Light scattering originates from the interaction of photons with structural heterogeneities present inside material bodies at the wavelength scale. The interaction between a photon and a molecule results in a photon moving in a different direction and a molecule that may maintain, increase, or decrease its energy. If the energy of the scattered photon is the same as the incident photon, the photon-molecule interaction is denoted as elastic scattering; while if the energy of the scattered photon is lower or higher than the incident photon, the interaction is denoted as inelastic scattering. Rayleigh scattering is an example of elastic scattering, which occurs when light propagates through gases, while Raman scattering is an example of inelastic scattering. Light scattering can be also influenced by the movement of scattering particles (e.g., the Doppler effect,4, 5 dynamic light scattering6, 7 ). None of these effects will be covered in this book. In this chapter, we introduce the optical properties (absorption and scattering coefficients, scattering function) used to describe the interaction of light with the turbid medium, and we summarize their dependence on the physical and chemical properties of the medium. The statistical meaning of the optical properties is discussed in Sec. 2.3. In Sec. 2.4, the reduced scattering coefficient, used to describe the scattering properties of a diffusive medium, is introduced together with a similarity relation useful to understanding propagation in the multiple-scattering regime. In Sec. 2.5, examples of diffusive media are reported. We point out that the simple modeling assumed for the interaction of the radiation with the turbid medium is applicable to many problems of practical interest and has the advantage of significantly reducing both the complexity of the problem and the mathematical formalism used for describing light propagation. However, this modeling is not suitable for studying light propagation that is dominated by inelastic scattering (e.g., fluorescence, phosphorescence, etc.). For a deeper study of light-matter interaction and of more general theories for describing light propagation, the interested reader is referred to Refs. 1, 8–16.

Scattering and Absorption Properties of Diffusive Media

11

2.2 Optical Properties of a Turbid Medium Light-matter interaction due to absorption is described by the absorption coefficient µa , defined as the ratio between the power absorbed in the unit volume and the power incident per unit area. The interaction due to scattering is described by the scattering coefficient µs and by the scattering phase function p(ˆ s, sb′ ). The scattering coefficient is defined as the ratio between the power scattered in the unit volume and the power incident per unit area. The scattering phase function, or simply the scattering function, is defined as the probability that a photon traveling in direction sˆ is scattered within the unit solid angle around the direction sb′ . The scattering function has the dimensions of sr−1 . Since, as mentioned in Chapter 1, theories and solutions presented in this book pertain to media illuminated by unpolarized light, the scattering properties of the medium are introduced with reference to unpolarized radiation. In this case, if we consider isotropic scatterers (spherical particles or randomly oriented non-spherical particles), the scattering function only depends on the scattering angle θ, i.e., the angle between directions sˆ and sb′ . So p(ˆ s, sb′ ) = p(θ). The following normalization for the scattering function is thus assumed: Z

p(ˆ s, sb′ )dΩ =2π ′

4π

Zπ

p(θ) sin θdθ = 1.

(2.1)

0

When propagation is dominated by multiple scattering, a single number can be sufficient to characterize the scattering function: It is the asymmetry factor g, defined as the average cosine of the scattering angle: g = hcos θi=2π

Zπ

cos θp(θ) sin θdθ.

(2.2)

0

The total power extracted by absorption or scattering by the unit volume is described by the extinction coefficient µt , defined as µt = µa + µs .

(2.3)

The coefficients µa , µs , and µt are measured in mm−1 . To describe the fraction of the total power extracted by scattering, the single-scattering albedo Λ is also used: Λ=

µs . µt

(2.4)

The extinction coefficient describes the attenuation due to absorption and scattering. With reference to a light beam that propagates along the z direction, the fraction of power extracted by the volume element of thickness dz is, according to (z ) the definitions, µt (z )Σdz P Σ , where Σ is the cross section of the beam, and P (z )

12

Chapter 2

is the impinging power. The variation of power passing through the volume element is thus dP (z) = −µt (z )P (z )dz . (2.5) Integrating Eq. (2.5), the Lambert-Beer law is obtained: h

P (z) = Pe exp −

Zz 0


i µt z ′ dz ′ ,

(2.6)

where Pe = P (z = 0) is the power emitted at z = 0. The Lambert-Beer law describes the exponential decay of the beam as it travels through the turbid medium. The fraction P (z) of the emitted power that propagates through the medium is often indicated as the ballistic component of the beam. The exponential factor in the Lambert-Beer law is the optical thickness τ (z) (a dimensionless quantity) of the medium crossed by the light beam. For a homogeneous medium, it is simply τ (z) = µt z. 2.2.1 Absorption properties In general, a turbid medium can be described as a background medium in which particles with different refractive indices are suspended. Absorption comes from the interaction of light with molecules both of the background medium and of the dispersed particles. Each molecule has a spectrum of absorption that can quickly change even for small variations of the wavelength due to its characteristic lines and bands of absorption. To describe absorption, it is useful to distinguish absorption due to molecules of the background medium µam and absorption due to dispersed particles µap . Depending on the type and concentration of molecules, we can have µam > µap or vice versa. Absorption caused by molecules within the background medium can be evaluated from the absorption cross section Ci (measured in mm2 ) and the number of molecules per unit volume Ni (mm−3 ) as X µam = Ni Ci , (2.7) i

where summation is on the different types of absorbing molecules,17 or from the imaginary part of the refractive index of the background medium Im (nm ) as µam =

4πIm (nm ) , λ

(2.8)

where λ is the wavelength in the medium.14, 15 Absorption due to particles can be described through the absorption cross section Ca of each particle. Ca is defined as the ratio between the power absorbed by the particle and the incident power per unit area. For particles randomly distributed and non-spherical particles randomly oriented, if the concentration of particles is

Scattering and Absorption Properties of Diffusive Media

13

sufficiently low, the absorption of each particle is not affected by its neighboring particles, and the power absorbed per unit volume is obtained by adding the power absorbed by each particle. With reference to spherical particles, µap can be written as Z ∞ Ca (r) f (r) dr, (2.9) µap = N 0

where N is the particle concentration (mm−3 ), and N f (r) dr is the concentration of particles with a radius in the interval (r, r + dr). The efficiency of a particle to absorb radiation can be also expressed by the absorption efficiency Qa = Ca /Cg , where Cg = πr2 is the geometric cross section of the particle. 2.2.2 Scattering properties Elastic light scattering originates from the heterogeneity of the refractive index inside the medium.15 It is also possible to distinguish the effect of molecules of the background medium from the effect of dispersed particles. Scattering from molecules of the background medium originates from density fluctuations; in fact, even in a homogeneous medium (e.g., gases, pure water), the number of molecules inside a given volume element fluctuates in time, although the average number of molecules is constant. These fluctuations give rise to fluctuations in the refractive index and then to scattering. For media in which propagation is dominated by multiple scattering, the effect of the background is usually negligible, so we only consider the effect of particles. Even for scattering, if the concentration of randomly oriented particles is sufficiently low (independent scattering approximation), interference effects can be disregarded, and the power scattered by the unit volume is simply obtained by adding the power scattered by each particle. The scattering coefficient and the scattering function for the volume element can thus be obtained as Z ∞ Cs (r) f (r) dr (2.10) µs = N 0

and p (θ) =

N

R∞

p (θ, r) Cs (r) f (r) dr R∞ , N 0 Cs (r) f (r) dr 0

(2.11)

where Cs (r) and p (θ, r) are respectively the scattering cross section and the scattering function of particles with radius r. Cs (r) is defined as the ratio between the power scattered by the particle and the incident power per unit area. The efficiency of a particle to scatter radiation can be expressed by the scattering efficiency: Qs = Cs /Cg .

(2.12)

For sufficiently low concentrations of particles for which the independent scattering approximation is applicable, the scattering coefficient is therefore proportional to the particle concentration N ; thus, µs increases proportionally to the volume

14

Chapter 2

fraction ρv of particles. Conversely, the scattering function for the volume element is obtained as the average of the scattering function of different particles and is independent of the particle concentration. For a dispersion of identical particles (monodispersion), the scattering function for the volume element is identical to the scattering function for the single particle, and µs can be simply expressed as a function of ρv and Qs . With N = ρv /( 34 πr3 ), we obtain from Eqs. (2.10) and (2.12) 3 Qs µs = N Qs Cg = ρv . 4 r

(2.13)

As mentioned before, absorption is mainly related to the types of molecules in the background medium and in the particles. Therefore, the absorption properties of the turbid medium mainly depend on its chemical composition. Instead, scattering originates from the different refractive index of particles with respect to the background medium, and it mainly depends on the microphysical properties of the particles. In particular, the scattering properties depend on the relative refractive index nr of the particles with respect to the background medium, on the size and shape of the particles, and on the wavelength of the radiation. More precisely, the dependence on size and wavelength is through the size parameter x, which for spherical particles is defined as x = 2πr/λ, where λ is the wavelength in the background medium. Small particles, also referred to as Rayleigh scatterers, are particles with r/λ ≤ 0.05 (or x ≤ π/10).14 These particles have a small efficiency to scatter light. Their scattering efficiency, and thus their scattering coefficient, strongly depends on the size parameter. Given that Qs ∝ x4 , as the wavelength increases, µs decreases as λ−4 . However, their scattering function does not depend on wavelength and is simply given by
3 p (θ) = 1 + cos2 θ . (2.14) 16π The scattering function for Rayleigh scatterers is thus symmetric with respect to θ = π/2 and has the asymmetry factor g = 0 even if it is not exactly isotropic. For large particles, i.e., when r ≫ λ (or x ≫ 1), the scattering efficiency is almost independent of the size parameter, and the scattering function strongly depends on it. For these particles, Qs ∼ = 2 (which is the extinction paradox in light scattering).1, 11, 14 For the scattering function, the value in the forward direction, i.e., p (θ = 0), is proportional to x, and thus, as the size parameter increases, the scattering function becomes more forward peaked. However, the total power scattered in the forward direction (i.e., with θ < π/2) remains more or less unchanged, and thus the asymmetry factor remains almost constant, with a value that depends on the relative refractive index nr .14 Examples of results for the scattering properties of spherical non-absorbing particles obtained with Mie theory18 are reported in Figs. 2.1–2.4. The results have been generated with a code based on the BHMIE subroutine reported by Bohren and

15

Scattering and Absorption Properties of Diffusive Media

Huffman.15 Figures 2.1 and 2.2 show the scattering efficiency and the asymmetry factor, respectively, as a function of the size parameter for different values of the relative refractive index nr of the spheres. Figures 2.3 and 2.4 show how the

6

3 n = 1.04

n = 0.96

n = 1.1

n = 0.9

n = 1.2

n = 0.8

r

5

r

r

r

r

r

n = 1.33

4

n = 0.7

2

r

n = 1.5

r

n = 0.6

r

r

n = 0.5

r

Q

s

r

Q

s

n = 2

3

2

1

1

0

-1

0

10

10

1

10

0

2

10

-1

0

10

10

Size Parameter

1

10

2

10

Size Parameter

Figure 2.1 Scattering efficiency versus the size parameter for different values of nr .

1.2

1.2 n = 1.04

n = 0.96

r

r

n = 1.1

1.0

n = 0.9

1.0

r

r

n = 1.2

n = 0.8 r

Asymmetry Factor

Asymmetry Factor

r

n = 1.33

0.8

r

n = 1.5 r

n = 2 r

0.6

0.4

0.2

0.0

-1

10

0

10

1

10

Size Parameter

2

10

n = 0.7

0.8

r

n = 0.6 r

n = 0.5 r

0.6

0.4

0.2

0.0

-1

10

0

10

1

10

2

10

Size Parameter

Figure 2.2 Asymmetry factor versus the size parameter for different values of nr .

scattering coefficient for a fixed volume fraction of monodispersed spheres depends on r and λ. The results pertain to ρv = 0.01 (i.e., spheres occupy 1% of the volume) and have been obtained using the independent scattering approximation. Figure 2.3 reports the results as a function of r for λ = 0.6328 µm and different values of nr . There is a strong dependence on r: According to Eq. (2.13) and to the dependence of Qs on the size parameter, µs increases proportionally to r3 for r ≪ λ, whereas µs decreases as r−1 for r ≫ λ. A maximum is reached for r ≈ λ. Figure 2.4 reports µs as a function of λ for different values of r and for nr = 1.04 and 1.33. There is a strong dependence also on wavelength: In particular, µs decreases as λ−4 for values

16

Chapter 2

λ ≫ r, whereas it is almost independent of λ for λ ≪ r. 3

2

10

10

2

10

1

10

) (mm

-1

)

10

-1

(mm

1

10

n = 1.04

0

r r

r

n = 0.8 r

10

r

-1

r

n = 0.9

-1

n = 1.2

10

n = 0.96

s

s

n = 1.1

0

10

n = 0.7

n = 1.33

r

r

n = 0.6

n = 1.5

r

r

10

n = 0.5

n = 2

-2 -2

10

-1

0

10

10

1

10

r

-2

r

10

-2

-1

10

0

10

r ( m)

1

10

10

r ( m)

Figure 2.3 Scattering coefficient versus the radius for monodispersions of spheres with volume fraction 0.01 for different values of nr . The wavelength in the background medium is 0.6328 µm.

2

3

10

10 n = 1.04 r

1

m

r = 0.5

m

m

r = 5

m

r

2

10

m

r = 0.5

m

r = 1

m

r = 5

m

-1

(mm

1

10

s

-1

0

10

s

(mm

r = 0.1

)

r = 1

n = 1.33

)

10

r = 0.1

-1

0

10

10

-2

10

-1

-1

10

0

10

Wavelength ( m)

1

10

10

-1

10

0

10

1

10

Wavelength ( m)

Figure 2.4 Scattering coefficient versus the wavelength in the background medium for monodispersions of spheres with volume fraction 0.01. The different curves correspond to different values of the radius of the spheres for nr = 1.04 and 1.33.

We point out that the scattering and absorption coefficients and the scattering function previously defined do not depend on the direction of the incident unpolarized light. This is true for spherical particles for which, due to the symmetry, there are not preferred directions; however, for non-spherical particles, this is true only if particles are randomly oriented. Otherwise, both the scattering coefficient and the scattering function can be dependent on the direction of the incident light. This is the case, for instance, for cylindrical scatterers oriented in the same direction, for which the optical properties also depend on the direction of the incident light

Scattering and Absorption Properties of Diffusive Media

17

with respect to the axis of the cylinders. In this case, the optical properties must be re-evaluated each time the incident angle changes. As long as the independent scattering approximation is applicable, i.e., when the inter-particle separation is large enough to make particles independent scatterers, the optical properties of turbid media are simply related to the optical properties of single particles that can be evaluated with the Mie theory for spherical particles15, 18 or with more complicated theories for non-spherical particles (for example, see Mishchenko et al.16 ). For practical applications, it may be useful to know how far the independent scattering approximation can be applied. From theoretical investigations, Mishchenko et al.16 suggested a simple approximated condition: For particles significantly smaller than the wavelength of the radiation, the interparticle distance should be larger than one wavelength in the medium. For large particles (x > 5), the mutual distance should be larger than three times their radius. Therefore, the volume occupied by a particle should be larger than λ3 for small particles and larger than (3r)3 for large particles. For small particles, the volume fraction should be smaller than ρvmax = 34 π( λr )3 . As an example, for λ = 0.6328 µm, ρvmax ∼ = 0.002 for r = 0.05 µm (x ∼ = 0.5) and ρvmax ∼ = 0.01 for r = 0.085 4π ∼ µm. For large particles, ρvmax = 81 = 0.155. Therefore, the effect of dependent scattering is expected to be larger for small particles. This has been confirmed by experiments.19, 20 Measurements of the scattering coefficient on small particles showed that with respect to the value expected from Eq. (2.13), attenuation decreases sharply as ρv is increased above 0.01. For large particles, variations are significantly smaller and in the opposite direction, i.e., attenuation increases.19 In many experiments and practical applications in which propagation of light dominated by multiple scattering is involved, the volume fraction can be significantly higher than 0.01, and the independent scattering approximation is not applicable. For such media, the optical properties cannot be estimated with the simple rules based on the optical properties of single particles. In particular, the simple proportionality of the scattering coefficient with the particle concentration can be lost. Nevertheless, the Lambert-Beer law [Eq. (2.6)] may remain valid to describe the attenuation of the light beam through the medium. Thus, it remains possible to use the absorption coefficient to characterize the interaction due to absorption, and to use the scattering coefficient and the scattering function to characterize the interaction due to scattering. For example, experimental results have been reported for the optical properties of dense scattering media with a volume fraction of particles of about 0.23.20 The definitions of absorption and scattering coefficients and other physical parameters adopted in this book are also usually valid for media so densely packed that suspended particles and background media can be confused. This is the case for biological tissues, paper, many food products and drugs, and other media. However, if the scattering coefficient becomes so high that the scattering mean free path ℓs = 1/µs is comparable with the wavelength (see next section), due to the wavelike nature of photons, interference effects occur that may completely change the

18

Chapter 2

regime of light propagation. If kℓs . 1 , with k = 2π λ (this is possible for particles with very high refractive index and x ≈ 1), the electric field cannot even perform one oscillation before the wave is scattered again, and interference effects can be so strong that transition to the Anderson localization of light can occur.21 This is beyond the scope of this book since, for the most part, natural media have ℓs ≫ λ.

2.3 Statistical Meaning of the Optical Properties of a Turbid Medium The optical properties of turbid media have a simple statistical meaning.22, 23 The scattering function p(ˆ s, sb′ ) is defined as the probability that a photon traveling in direction sˆ is scattered within the unit solid angle around the direction sb′ . With the assumption of unpolarized radiation and of isotropic scatterers (spherical or randomly oriented non-spherical particles), the scattering function only depends on the scattering angle θ. The probability distribution functions f1 (θ) and f2 (ϕ), for the angles θ and ϕ that specify the direction sb′ , are therefore simply given by f1 (θ) = 2πp(θ) sin θ

(2.15)

and

1 . (2.16) 2π The Lambert-Beer law [Eq. (2.6)] can be used to obtain the probability distribution function for the pathlength followed by photons before interacting with the turbid medium. In fact, the ratio between the power µt (z)P (z)dz extracted by the volume element of thickness dz at depth z and the emitted power Pe can be seen as the probability that a photon emitted by the source will be scattered or absorbed within (z, z + dz). For the probability distribution function f3 (z), we therefore obtain h Zz
i (2.17) f3 (z) = µt (z) exp − µt z ′ dz ′ . f2 (ϕ) =

0

For a homogeneous medium, f3 (z) becomes

f3 (z) = µt exp(−µt z).

(2.18)

The probability distribution function f3 (z) can be used to evaluate the mean free path ℓt traveled by photons before being scattered or absorbed. With reference to a homogeneous infinitely extended medium we have Z ∞ Z ∞ 1 zµt exp(−µt z)dz = . zf3 (z) dz = (2.19) ℓt = µt 0 0 If we consider a non-absorbing medium, we obtain for the scattering mean free path traveled between two subsequent scattering events ℓs = 1/µs , and µs can be defined as the number of scattering events undergone by a photon per unit length. For a non-scattering medium, the absorption mean free path, i.e., the mean path traveled by photons before being absorbed, is ℓa = 1/µa .

19

Scattering and Absorption Properties of Diffusive Media

2.4 Similarity Relation and Reduced Scattering Coefficient For a non-absorbing turbid medium in which the interaction of light can be described with the scattering coefficient and the scattering function, the propagation of photons can be represented as a random walk in which they frequently change direction due to scattering. The rules that govern photon migration are given by the probability distribution functions for θ, ϕ, and z, which were introduced in the previous section. These functions are the starting point for developing MC routines to simulate light propagation (see Chapter 9). They can be also used to obtain some exact analytic relations for photon migration. With reference to photons emitted in an infinite non-absorbing homogeneous medium at z = 0 along the z-axis, if we indicate with (xk , yk , zk ) the coordinates of the point in which the k th scattering event occurs, it has been shown22 that k−1 1 1 − gk 1 X i g = hxk i = hyk i = 0, hzk i = µs µs 1 − g

(2.20)

i=0

and

2 2  2 k − (k + 1)g + g k+1 dk = xk + yk2 + zk2 = 2 . µs (1 − g)2

(2.21)

Since −1 ≤ g ≤ 1, for large values of k the mean value of zk reaches the value lim hzk i =

k→∞

1 1 1 = ′ , µs 1 − g µs

(2.22)

and the mean value of the square distance from the source after k scattering events becomes

2 2 2 k(1 − g) = ′ 2 k(1 − g) , dk ∼ (2.23) = 2 2 µs (1 − g) µs

where

′

µs = µs (1 − g)

(2.24)

is the reduced scattering or transport coefficient of the medium. The quantity ′

ℓ =

1 µ′s

(2.25)

is the transport mean free path. Equation (2.22) shows that for a homogeneous non-absorbing medium the transport mean free path represents the mean distance traveled by photons along the initial direction of propagation before they have effectively “forgotten” their original direction of motion. Equation (2.23) suggests a similarity relation for propagation in two media having different single-scattering properties—µs1 , p1 (θ) and µs2 , p2 (θ)—but with µs1 (1 − g1 ) = µs2 (1 − g2 ), i.e., with the same reduced scattering coefficient: The mean distance from the source after k1 scattering events in medium 1 is almost

20

Chapter 2

the same as for photons that have undergone k2 scattering events in medium 2 provided k1 (1 − g1 ) = k2 (1 − g2 ). This similarity relation can be used to evaluate the number of scattering events necessary to randomize the direction of propagation of photons propagating through a medium with non-isotropic scattering (g 6= 0). For the medium with isotropic scattering (g = 0), one scattering event is sufficient to randomize propagation; for non-isotropic scattering, we expect that k = 1/(1 − g) ′ scattering events are necessary. The corresponding mean pathlength is k/µs = 1/µs . These results suggest that when propagation is dominated by multiple scattering, the most important parameter to describe the scattering properties of the medium is ′ µs . This is confirmed by the results obtained when the diffusion approximation becomes applicable: In Sec. 3.6, we will show that photons undergoing many scattering events migrate in a diffusive regime that is fully characterized by the ′ diffusion coefficient D = 1/(3µs ). ′ It may be interesting to study how µs depends on the properties of scattering particles. For this purpose, it is useful to introduce the reduced scattering efficiency ′ Qs for the single particle as ′

Qs = Qs (1 − g) ,

(2.26)

from which the reduced scattering coefficient can be obtained as Z ∞ ′ ′ Qs (r) Cg (r) f (r) dr. µs = N

(2.27)

0

1

0

10

10

n = 1.04 r

n = 1.1 r

0

n = 1.2

10

r

-1

10

n = 1.33 r

n = 1.5 r

-1

10

n = 2 r

' s

10

n = 0.96

Q

Q

s

'

-2

-2

10

r

n = 0.9 r

n = 0.8

-3

r

10

-3

10

n = 0.7 r

n = 0.6 r

-4

10

n = 0.5 r

-4

-1

10

0

10

1

2

10

10

Size Parameter

10

-1

10

0

10

1

10

2

10

Size Parameter

Figure 2.5 Reduced scattering efficiency versus the size parameter for different values of nr . ′

Examples of results for Qs as a function of the size parameter are reported in Fig. 2.5 for different values of the relative refractive index of the spheres. Q′s is strongly affected by nr , even if small variations are observed on the shape of the ′ curves. Figures 2.6 and 2.7 report µs as a function of r and of λ, respectively, for

21

Scattering and Absorption Properties of Diffusive Media

monodispersions of spheres with volume fraction ρv = 0.01. Figure 2.6 pertains ′ to λ = 0.6328 µm and to the same values of nr as in Fig. 2.5. µs increases proportionally to r3 for r ≪ λ, reaches a maximum for r ≈ λ, and decreases as r−1 ′ for r ≫ λ. Also, for µs the shape of the curves is not much influenced by nr . In Fig. 2.7, the results are reported for different values of r for nr = 1.04 and 1.33. 3

3

10

10

n = 1.04

n = 0.96

r

r

n = 1.1 r

2

10

n = 0.9 r

2

10

n = 1.2 r

n = 0.8 r

n = 1.33

10

10

r

n = 0.6 r

n = 0.5

)

n = 2

-1

r

0

'

-1

10

s

(mm

10

-2

r

0

10

-1

10

-2

10

10

-3

10

r

1

n = 1.5

)

-1

(mm '

s

n = 0.7

r

1

-3

-2

10

-1

0

10

10

1

10

10

-2

10

-1

0

10

r ( m)

1

10

10

r ( m)

Figure 2.6 Reduced scattering coefficient versus the radius for monodispersions of spheres with volume fraction 0.01 for different values of nr . The wavelength in the background medium is 0.6328 µm.

0

2

10

10

n = 1.04 r

-1

m

r = 0.5

m

m

r = 5

m

r

1

10

-1

' (mm

-1

m

r = 0.5

m

r = 1

m

r = 5

m

-2

10

0

10

s

' (mm s

r = 0.1

)

r = 1

n = 1.33

)

10

r = 0.1

-1

10

-3

10

-2

-1

10

0

10

1

10

10

-1

0

10

Wavelength ( m)

10

1

10

Wavelength ( m)

Figure 2.7 Examples of reduced scattering coefficient versus the wavelength in the background medium for monodispersions of spheres with volume fraction 0.01. The different curves correspond to different values of the radius of the spheres for nr = 1.04 and 1.33. The values of the radius are indicated in the legend. ′

′

Only the case r ≪ λ has a strong dependence of µs on λ (µs proportional to λ−4 ). ′ In the range of visible and near-infrared wavelengths, appreciable variations of µs are expected only for small scatterers (r < 0.5µm).

22

Chapter 2

2.5 Examples of Diffusive Media In this section, we provide some examples of diffusive media in daily life. In order to discuss diffusive media, we need a definition of the diffusive regime of propagation. As will be shown in Chapter 3, light propagation through turbid media can be described by the RTE. For many turbid materials, propagation can be described by a simpler but more approximate equation, the diffusion equation, for which relatively simple analytical solutions are available. Turbid media in which propagation can be described with good accuracy by the DE are named diffusive media. As will be shown in subsequent chapters, the conditions necessary to establish a diffusive regime of propagation for photons migrating from the source to the receiver depend both on the optical properties of the turbid medium and on the geometry. In Sec. 10.7, these conditions will be indicated for establishing a diffusive regime of propagation: ′ ′ 1) µa /µs . 0.1, 2) the volume of turbid medium & (10ℓ )3 , and 3) photons should ′ have traveled paths with length & 4ℓ , and the corresponding number of scattering events should be & 4/(1 − g). When these conditions are satisfied, the largest part ′ of radiation detected at distances & 2/µs is due to diffused photons, and solutions of the RTE based on the diffusion approximation can be therefore properly used for the analysis of experimental results. ′ At visible and/or near-infrared wavelengths, the condition µa /µs . 0.1 is satisfied for many turbid materials, and a diffusive regime of propagation can therefore be established, provided their volume is sufficiently wide. By means of examples, we report here typical values for the optical properties of some turbid media observable in daily life. Solid materials are often strongly scattering media and have a reduced scattering coefficient ≈ 1 mm−1 . This is the case of biological tissues and agricultural products. Measurements of the optical properties of biological tissue show a significant intersubject variability and a dependence on the spectral range considered. Typical ′ values of µs reported at near-infrared wavelengths range between ∼ 0.5 and 1 mm−1 for muscle;24, 25 ∼ 1 and 1.6 mm−1 for subcutaneous adipose tissue;24, 26 ∼ 0.5 and 2.5 mm−1 for breast;27–29 ∼ 1 and 1.6 mm−1 for bone;30 ∼ 0.3 and 2 mm−1 for forehead;31, 32 ∼ 0.8 and 4 mm−1 for brain (grey and white matter);33–35 and so on. For 650 nm < λ < 1000 nm, the absorption coefficient is . 0.04 mm−1 .24, 25, 27–32 There are also tissues like tendon36 and tooth37 in which scattering is due to approximately cylindrical particles (dental tubules and collagen fibers, respectively) with a pronounced alignment. For these media, the scattering properties strongly depend on the photon’s direction. As an example, in Ref. 36 it has been shown that for a bovine achilles tendon the reduced scattering coefficient changes about one order of magnitude (from about 0.25 to about 2.5 mm−1 ) when measured along the parallel or the perpendicular direction with respect to the alignment of the fibers. Another example of a scattering medium with aligned fibers is wood. For wood, ′ significantly different results have been reported for µs measured along the parallel ′ or the perpendicular direction (µs ≃ 2 and 20 mm−1 ).38, 39 For these media, theories

Scattering and Absorption Properties of Diffusive Media

23

presented in this book are not applicable. Scattering properties of many agricultural products are in the range indicated for biological tissues. Spectral measurements of reduced scattering coefficients in the near infrared showed values between 1.5 and 2.5 mm−1 for apples, pears, and peaches; between 1 and 1.5 mm−1 for kiwifruits; and between 0.5 and 1 mm−1 for carrots and tomatoes; with absorption coefficients < 0.05 mm−1 .40–42 Other examples of solid, highly scattering materials with low absorption at visible and/or near-infrared wavelengths are plastic materials20, 43 and many others observable in daily life, like sugar, salt, wheat flour, cheese, paper, and many pharmaceutical products. For sugar, wheat flour, and corn meal in particular, measurements44 showed that the reduced scattering coefficient at near-infrared wavelengths spans in the range 1-70 mm−1 . For cane sugar it is around 1 mm−1 , white sugar is around 3 mm−1 , icing sugar is around 70 mm−1 , corn meal is around 5 mm−1 , and wheat flour is around 30 mm−1 . For 650 nm < λ < 1000 nm, the absorption coefficient remains . 0.04 mm−1 . For pharmaceutical tablets, a very ′ high value of µs (∼ 50 mm−1 ) has been reported.45 Examples of liquids with high scattering and low absorption in which propagation in the diffusive regime can be established are white paints, milk and fat emulsions, etc. A fat emulsion like Intralipid, a soybean oil emulsion in water that is used clinically as an intravenously administrated nutrient and is frequently used as a tissue phantom for calibrating NIRS and DOT instrumentation, has a reduced scattering coefficient between 1 and 2 mm−1 for 400 < λ < 900 nm when the concentration of fat is 1%.46–48 Propagation through gases and aerosols can also occur in the diffusive regime, even if the reduced scattering coefficient can be very low. Examples of aerosols with relatively high scattering are fogs and clouds. Dense fogs with visibility < 50 m have a scattering coefficient & 0.08 m−1 [the practical relation between ′ the horizontal visibility R and µs is R = 3.912/µs (λ = 0.55µm)]49 and µs & 0.01 m−1 . Solutions based on the diffusion equation can thus be used to describe propagation of photons that traveled paths longer than some hundreds of meters. The clear atmosphere is also a turbid medium, but it is not sufficiently extended for establishing a diffusive regime of propagation. Its scattering coefficient at sea level ranges from about 0.04 km−1 at 400 nm to 0.004 km−1 at 700 nm, and the corresponding values for the vertical optical thickness are 0.35 and 0.035.50 But propagation in the diffusive regime can be reached even in media with a smaller scattering coefficient, provided their extension is sufficiently large. This may occur, for instance, on an extremely large scale for clouds of gas and dust in the interstellar medium.

24

Chapter 2

References 1. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1, Academic Press, New York (1978). 2. J. R. Lakowicz, Principles of Fluorescence Spectroscopy, Springer (1999). 3. A. Diaspro (Ed.), Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances, Wiley-Liss, Inc., New York (2002). ˙ 4. A. Liebert, N. Zołek, and R. Maniewski, “Decomposition of a laser-doppler spectrum for estimation of speed distribution of particles moving in an optically turbid medium: Monte Carlo validation study,” Phys. Med. Biol. 51, 5737–5751 (2006). ˙ 5. S. Wojtkiewicz, A. Liebert, H. Rix, N. Zołek, and R. Maniewski, “Laser-doppler spectrum decomposition applied for the estimation of speed distribution of particles moving in a multiple scattering medium,” Phys. Med. Biol. 54, 679– 697 (2009). 6. B. J. Berne and R. Pecora, Dynamic Light Scattering With Applications to Chemistry, Biology, and Physics, Dover Publications, Inc., Mineola, New York (2000). 7. W. Brown, Dynamic Light Scattering: The Method and Some Applications, Oxford Science Publications, Clarendon Press-Oxford (1993). 8. S. Chandrasekhar, Radiative Transfer, Oxford University Press, London/Dover, New York (1960). 9. K. S. Shifrin, Scattering of Light in Turbid Media, NASA Technical Translation, NASA TT F-447, National Aeronautics and Space Administration, Washington, D.C. (1951). 10. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, Elsevier, New York (1969). 11. H. C. van de Hulst, Light Scattering by Small Particles, John Wiley & Sons, New York (1957). 12. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications, Academic Press, New York (1980). 13. E. P. Zege, A. I. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium, Springer-Verlag, New York (1991). 14. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York (1969). 15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York (1983).

Scattering and Absorption Properties of Diffusive Media

25

16. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press (2002). 17. B. H. Armstrong and R. W. Nicholls, Emission, Absorption and Transfer of Radiation in Heated Atmospheres, Pergamon Press, New York (1972). 18. G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908). 19. A. Ishimaru and Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982). 20. G. Zaccanti, S. D. Bianco, and F. Martelli, “Measurements of optical properties of high density media,” Appl. Opt. 42, 4023–4030 (2003). 21. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997). 22. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994). 23. L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging, John Wiley & Sons, New York (2007). 24. A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. 44, 2689–2702 (1999). 25. P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. 2, 124–129 (2003). 26. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of the subcutaneous adipose tissue in the spectral range 400-2500 nm,” Opt. Spectrosc. 99, 836–842 (2005). 27. A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9, 1143–1151 (2004). 28. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: An application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999). 29. T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2002).

26

Chapter 2

30. A. Pifferi, A. Torricelli, P. Taroni, A. Bassi, E. Chikoidze, E. Giambattistelli, and R. Cubeddu, “Optical biopsy of bone tissue: a step toward the diagnosis of bone pathologies,” J. Biomed. Opt. 9, 474–480 (2004). 31. D. Comelli, A. Bassi, A. Pifferi, P. Taroni, A. Torricelli, R. Cubeddu, F. Martelli, and G. Zaccanti, “In vivo time-resolved reflectance spectroscopy of the human forehead,” Appl. Opt. 46, 1717–1725 (2007). 32. A. Torricelli, A. Pifferi, P. Taroni, E. Giambattistelli, and R. Cubeddu, “In vivo optical characterization of human tissues from 610 to 1010 nm by time-resolved reflectance spectroscopy,” Phys. Med. Biol. 46, 2227–2237 (2001). 33. F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, and C. Depeursinge, “In vivo local determination of tissue optical properties: Applications to human brain,” Appl. Opt. 38, 4939–4950 (1999). 34. M. Johns, C. A. Giller, D. C. German, and H. Liu, “Determination of reduced scattering coefficient of biological tissue from a needle-like probe,” Opt. Exp. 13, 4828–4842 (2005). 35. A. Sassaroli, F. Martelli, Y. Tanikawa, K. Tanaka, R. Araki, Y. Onodera, and Y. Yamada, “Time-resolved measurements of in vivo optical properties of piglet brain,” Opt. Rev. 7, 420–425 (2000). 36. A. Kienle, C. Wetzel, A. Bassi, D. Comelli, P. Taroni, and A. Pifferi, “Determination of the optical properties of anisotropic biological media using an isotropic diffusion model,” J. Biom. Opt. 12, 014026 (2007). 37. A. Kienle, F. K. Forster, R. Diebolder, and R. Hibst, “Light propagation in dentin: influence of microstructure on anisotropy,” Phys. Med. Biol. 48, N7– N14 (2003). 38. A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet softwood,” Opt. Express 16, 9895–9906 (2008). 39. C. D’Andrea, A. Farina, D. Comelli, A. Pifferi, P. Taroni, G. Valentini, R. Cubeddu, L. Zoia, M. Orlandi, and A. Kienle, “Time-resolved optical spectroscopy of wood,” Appl. Spectrosc. 62, 569–574 (2008). 40. M. Zude (Ed.), Optical Monitoring of Fresh and Processed Agricultural Crops, (Contemporary Food Engineering Series), CRC Press, Boca Raton, Florida (2009). 41. B. M. Nicolaï, B. E. Verlinden, M. Desmet, S. Saevels, W. Saeys, K. Theron, R. Cubeddu, A. Pifferi, and A. Torricelli, “Time-resolved and continuous wave NIR reflectance spectroscopy to predict soluble solids content and firmness of pear,” Postharvest Biol. Tec. 47, 68–74 (2008).

Scattering and Absorption Properties of Diffusive Media

27

42. R. Cubeddu, C. D’Andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, C. Dover, D. Johonson, M. Ruiz-Altisent, and C. Valero, “In vivo time-resolved reflectance spectroscopy of the human forehead,” Appl. Opt. 40, 538–543 (2001). 43. H. O. Di Rocco, D. I. Iriarte, J. A. Pomarico, H. F. Ranea-Sandoval, R. Macdonald, and J. Voigt, “Determination of optical properties of slices of turbid media by diffuse CW laser light scattering profilometry,” J. Quant. Spectrosc. Ra. 105, 68–83 (2007). 44. A. Pifferi, A. Torricelli, P. Taroni, D. Comelli, A. Bassi, and R. Cubeddu, “Fully automated time domain spectrometer for the absorption and scattering characterization of diffusive media,” Rev. Sci. Instrum. 78, 053103 (2007). 45. J. Johansson, S. Folestad, M. Josefson, A. Sparén, C. Abrahamsson, S. AndersonEngels, and S. Svanberg, “Time-resolved NIR/Vis spectroscopy for analysis of solids: Pharmaceutical tablets,” Appl. Spectrosc. 56, 725–731 (2002). 46. H. van Staveren, C. Moes, J. van Marle, S. Prahl, and M. Gemert, “Light scattering in intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30, 4507–4514 (1991). 47. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). 48. F. Martelli and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method,” Opt. Express 15, 486–500 (2007). 49. E. J. McCartney, Optics of the Atmosphere, Wiley & Sons, New York (1976). 50. C. Fröhlich and G. E. Shaw, “New determination of Rayleigh scattering in the terrestrial atmosphere,” Appl. Opt. 19, 1773–1775 (1980).

Chapter 3

The Radiative Transfer Equation and Diffusion Equation As noted by Ishimaru,1 two distinct theories have been developed to deal with multiple-scattering problems. One is called the analytical theory or multiplescattering theory and is based on Maxwell’s equations; the other is the radiative transport theory. The analytical theory starts from Maxwell’s equations governing the field quantities.1 This theory is mathematically rigorous since, in principle, one can account for all the effects of multiple scattering, diffraction, and interference. However, this rather complex method has not yet produced practical models of general use to describe multiple-scattering problems. The radiative transfer equation (RTE) is a phenomenological and heuristic theory describing the transport of energy through a scattering medium that lacks a rigorous mathematical formulation able to account for all the physical effects involved in light propagation. However, it has been demonstrated that under certain simplifying assumptions, the RTE can be derived from the electromagnetic theory of multiple scattering in discrete random media.2, 3 Despite the fact that the RTE does not rigorously represent the real physics of light propagation, it has led to useful models for many practical problems.1 The RTE and the DE represent the most widely used approaches to the study of photon migration through highly scattering media, i.e., media where propagation is dominated by multiple scattering. In this chapter, the main framework of these theories is reviewed.

3.1 Quantities Used to Describe Radiative Transfer The basic quantities used in transport theory to describe energy propagation are defined starting from the spectral radiance, or spectral specific intensity Is (~r, sˆ, t, ν). This is measured in Wm−2 sr−1 Hz−1 , and is defined as the average power that at position ~r and time t flows through the unit area oriented in the direction of the unit vector sˆ, due to photons within a unit frequency band centered at ν, moving within 29

30

Chapter 3

the unit solid angle around sˆ. Knowledge of the spectral specific intensity provides more detailed information on photons that are moving inside the medium. With reference to Fig. 3.1, the power dP that at time t flows within the solid angle dΩ through the elementary area dΣ (oriented along the direction n ˆ ) placed at ~r in the frequency interval (ν, ν + dν) is given by dP = Is (~r, sˆ, t, ν)|ˆ n · sˆ|dΣdΩdν.

n

d

(3.1)

s

dP I s ( r , s, t , )

d

Figure 3.1 Relationship between the power dP and the spectral radiance Is (~r, sˆ, t, ν) given in Eq. (3.1).

Notice that Is (~r, sˆ, t, ν) is related in a simple way to the energy density. In fact, the energy dE, that in the time interval dt crosses the elementary area dΣ oriented along the direction sˆ per unit solid angle and per unit frequency interval, occupies the volume dV = dΣvdt, where v is the speed of light in the medium. The energy density per unit frequency interval and unit solid angle is therefore dE(~r, sˆ, t, ν) Is (~r, sˆ, t, ν)dΣdt Is (~r, sˆ, t, ν) = = . (3.2) dV dΣvdt v The spectral radiance Is (~r, sˆ, t, ν) is thus proportional to the number of photons in the unit volume with frequency ν that at time t are moving along the direction sˆ. Since, as mentioned in Chapter 2, we refer to media in which the frequency of the radiation does not change during propagation, and if we consider quasimonochromatic radiation, then we can use the term “radiance” or “specific intensity” I(~r, sˆ, t) (Wm−2 sr−1 ) defined as the integral of Is (~r, sˆ, t, ν) over a narrow range of frequency. To measure the radiance I(~r, sˆ, t), a receiver with a small area and a small field of view should be placed at ~r, pointed in the direction −ˆ s. The fluence rate (or irradiance or simply fluence) Φ(~r, t) is obtained integrating the radiance over the entire solid angle Z Φ(~r, t) = I(~r, sˆ, t)dΩ. (Wm−2 ) (3.3) 4π

31

The Radiative Transfer Equation and Diffusion Equation

The quantity u(~r, t) = Φ(~r, t)/v (J m−3 ) represents the energy density at position ~r at time t. The quantity u(~r, t)/(hν) (m−3 ), where h is Planck’s constant and hν is the energy of a single photon, is therefore equal to the photon density. The fluence is thus proportional to the number of photons in the unit volume regardless of their direction of motion. To measure the fluence, we need a small receiver capable of collecting photons with the same efficiency over the entire solid angle (i.e., an isotropic probe). We can measure a quantity proportional to the fluence by using a small sphere of highly scattering material attached to the tip of an optical fiber. The quantity Φ(~r, t)/(4π), i.e., the fluence divided by the whole solid angle, represents the average radiance. Another quantity useful in describing propagation is the flux vector, defined as ~ r, t) = J(~

Z

I(~r, sˆ, t)ˆ sdΩ ,

(Wm−2 )

(3.4)

4π

which represents the amount and the direction of the net flux of power. All the quantities introduced to describe the energy propagation through turbid media have been defined with reference to the more general case of a time-dependent source. Therefore, these quantities are suitable to describe the time-resolved (TR) response when the energy emitted by the source is a function of time. In many applications, the medium is illuminated by Continuous Wave (CW) sources, which are steady-state sources emitting a constant power. CW response for the radiance, the fluence, and the flux do not depend on time and can be simply indicated as ~ r). These symbols will be used in equations and solutions I(~r, sˆ), Φ(~r), and J(~ referred to CW sources. Radiation impinging on a volume element of a turbid medium is partially absorbed and scattered. If I(~r, sˆ, t) is the incident radiance, then, from the definition of absorption and scattering properties, the fraction of power extracted in the volume element of thickness ds oriented along sˆ is given by µt I(~r, sˆ, t)ds. Similarly, the fraction of power re-emitted within the solid angle dΩ′ around the direction sb′ due to scattering is µs I(~r, sˆ, t)p(ˆ s, sb′ )dsdΩ′ . Furthermore, the fraction of the total power absorbed in the unit volume is µa Φ(~r, t).

3.2 The Radiative Transfer Equation The RTE1, 4–7 is an integro-differential equation that represents the balance of energy (principle of conservation of energy) for light propagation through a volume element of an absorbing and scattering medium. The RTE can be obtained by balancing the various mechanisms by which the radiance at a given wavelength, I(~r, sˆ, t), can increase or decrease inside an arbitrary volume of the medium. In the more general

32

Chapter 3

case of time-dependent sources, the RTE is written as ∂ r, sˆ, t) v∂t I(~

= µs

R

4π

+ ∇ · [ˆ sI(~r, sˆ, t)] + µt I(~r, sˆ, t)

p(ˆ s, sb′ )I(~r, sb′ , t)dΩ′ + ε(~r, sˆ, t),

(3.5)

where v is the speed of light inside the medium, ε(~r, sˆ, t) is the source term that is the power emitted at time t per unit volume and unit solid angle along sˆ, and dΩ′ is the elementary solid angle in the direction sb′ . Considering the volume element dV identified by the position vector ~r, the terms of Eq. (3.5) represent the following: •

∂ r, sˆ, t)dV v∂t I(~

dΩdt represents the total temporal change of energy that is propagating along sˆ within dV , dΩ, and dt. This quantity is proportional to the total temporal variation of the number of photons moving along sˆ within dV , dΩ, and dt.

• ∇ · [ˆ sI(~r, sˆ, t)] dV dΩdt = sˆ · [∇I(~r, sˆ, t)] dV dΩdt represents the net flux of energy that is propagating along sˆ through the volume dV , within dΩ, and dt. The term sˆ · [∇I(~r, sˆ, t)] is often indicated as ∂I(~r∂s, sˆ, t) . • µt I(~r, sˆ, t)dV dΩdt represents the fraction of energy that is propagating along sˆ within dV , dΩ, and dt extracted by scattering and absorption phenomena. R • µs p(ˆ s, sb′ )I(~r, sb′ , t)dΩ′ dV dΩdt represents the energy coming from any 4π

direction sb′ that, within dV , dΩ, and dt, is scattered in direction sˆ.

• ε(~r, sˆ, t)dV dΩdt represents the energy generated along sˆ in dΩ and dt by sources inside dV . For steady-state sources, the RTE is1 ∇ · [ˆ sI(~r, sˆ)] + µt I(~r, sˆ) = µs

Z

4π

p(ˆ s, sb′ )I(~r, sb′ )dΩ′ + ε(~r, sˆ).

(3.6)

3.3 The Green’s Function Method The Green’s function method is a powerful approach for solving radiative transfer problems with complex source distributions and complex boundary conditions.6, 8, 9 It is convenient to express the solution of the RTE with arbitrary sources and boundary conditions as a superposition of the solutions of “basilar” problems employing the formalism of the Green’s function. The superposition principle can be applied since the RTE is a linear equation. If the source term of Eq. (3.5) is assumed to be a Dirac delta function of unitary strength, i.e., → ε(~r, sˆ, t) = δ 3 (~r − r′ )δ(ˆ s − sb′ )δ(t − t′ ), (3.7)

The Radiative Transfer Equation and Diffusion Equation

33

the TR solution of Eq. (3.5) represents the Green’s function of the problem. Since the actual source can be expressed as a superposition of elementary Dirac delta sources, the solution of the RTE can be represented as a superposition of solutions for elementary sources. This approach leads to the Green’s function method.6, 8, 9 The generic solution of Eq. (3.5) can be represented by the superposition integral, where the Green’s function is multiplied by the actual spatial, angular, and temporal distribution of the source and integrated over the whole spatial, angular, and temporal domain I(~r, sˆ, t) =

Z Z Z∞

V 4π −∞

→ → I(~r, r′ , sˆ, sb′ ,t, t′ ) ε( r′ , sb′ , t′ )dV ′ dΩ′ dt′ ,

(3.8)

→ where I(~r, r′ , sˆ, sb′ ,t, t′ ) indicates the Green’s function of Eq. (3.5), i.e., the solution of Eq. (3.5) when the source term is Eq. (3.7), and where I(~r, sˆ, t) is the solution → for the actual source term ε( r′ , sb′ , t′ ). For the sake of simplicity, the symbol I will be used to denote either the Green’s function of the RTE or the solution of the RTE → with arbitrary sources and boundary conditions. The variables r′ , sb′ , t′ represent the source’s variables, while ~r, sˆ, t represent the variables of the field point. In this book, the time-dependent Green’s function will be also denoted as Temporal Point Spread Function (TPSF). → From Eq. (3.8), the CW Green’s function I(~r, r′ , sˆ, sb′ ) [i.e., the solution of → Eq. (3.6) for a steady-state source of unitary strength ε(~r, sˆ) = δ 3 (~r − r′ )δ(ˆ s − sb′ )] is → I(~r, r′ , sˆ, sb′ ) R R R∞ → → → = I(~r, r′′ , sˆ, sb′′ ,t, t′′ )δ 3 (r′′ − r′ )δ(sb′′ − sb′ )dV ′′ dΩ′′ dt′′ =

=

V 4π −∞ R∞

−∞ R∞ 0

→ I(~r, r′ , sˆ, sb′ ,t, t′′ )dt′′

(3.9)

→ I(~r, r′ ,ˆ s, sb′ ,t)dt.

3.4 Properties of the Radiative Transfer Equation 3.4.1 Scaling properties If I(~r, sˆ, t) is the Green’s function of Eq. (3.5), where (for the sake of simplicity) the variables of the source are not shown, and a source is implicitly assumed at r~′ = 0, s, sb′ ), the sb′ = sˆ0 , t′ = 0 for a homogeneous medium characterized by µt and p(ˆ function I(~r, sˆ, t) is defined as I(~r, sˆ, t) =

µt
3 I(~r, sˆ, t), µt

(3.10)

34

Chapter 3

with ~r = ~r

µt
, µt

t=t

µt
, µt

(3.11)

is the Green’s function for a homogeneous medium having the extinction coefficient µt , the same scattering function p(ˆ s, sb′ ), and the same single-scattering albedo Λ = µs /µt (see Fig. 3.2 for a representation of the scaled domain). This property is demonstrated by observing that, after introducing in the RTE the dimensionless ~r = µt~r and e t = µt v t, [that is equivalent to considering 1/µt as the unit variables e to measure lengths and 1/(µt v) as the unit to measure time] the following equation is obtained: h i ∂ e e e e I( ~ r , s ˆ , t ) + ∇ · s ˆ I( ~ r , s ˆ , t ) + I(e ~r, sˆ, e t) ∂e t R (3.12) ′ 3 3 = Λ p(ˆ s, sb′ )I(e ~r, sb′ , e t)dΩ + vµt δ (e ~r)δ(ˆ s)δ(e t). 4π

1 δ(e t) and To obtain Eq. (3.12), the following properties have been used:9 δ(αe t) = |α| ~r). In two turbid media having different µt , but the same scattering δ 3 (αe ~r) = 1 3 δ 3 (e |α|

function and single-scattering albedo, photon migration is governed by the same homogeneous equation, and the difference is only in the source term. Furthermore, substituting the scaled distances and time of Eq. (3.11) in the RTE written for a medium with extinction coefficient µt and assuming µs /µt =µs /µt , the following equation is obtained: h i + ∇ · sˆI(~r, sˆ, t) + µt I(~r, sˆ, t)  3 R s)δ(t), = µs p(ˆ s, sb′ )I(~r, sb′ , t)dΩ′ + µµtt δ 3 (~r)δ(ˆ ∂ r, sˆ, t) v∂t I(~

(3.13)

4π

 3 from which I can be represented as the Green’s function I multiplied by µµtt . This scaling property means that the energy flowing within the time interval (t, t + ∆t) through the surface element ∆Σ, normal to the direction sˆ, at the point ~r of a medium with extinction coefficient µt , is the same energy that, within the scaled time interval (t, t + ∆t)(µt /µt ), passes through the scaled surface element ∆Σ(µt /µt )2 at the point ~r(µt /µt ) for a medium with extinction coefficient µt .
3 Thus, the scale coefficient µt /µt in Eq. (3.10) takes into account the energy redistribution inside the medium determined by a variation of µt . This property, also known as the similarity principle,10 is very useful since it enables us to use the results obtained in a certain geometry for any scaled geometry, providing Λ and p(ˆ s, sb′ ) are kept unchanged. On the basis of this principle, when studying radiative transfer problems, it is possible to choose the most convenient scale for laboratory experiments instead of making experiments in their actual scale, in which it is often difficult to carry out measurements.

35

The Radiative Transfer Equation and Diffusion Equation

detector source

t

I ( r , s, t )

2

/

t

I ( r , s, t )

r

r l

vt

t

t

t

l

vt

r

t

/

t

/

t

t

t

Figure 3.2 Schematic of two scaled geometries for two turbid media with different extinction coefficients and the same scattering function and the same albedo.

3.4.2 Dependence on absorption For media with homogeneous absorption [µa (~r) = constant], if I(~r, sˆ, t, µa = 0) is the solution of Eq. (3.5) for a non-absorbing medium with the source term described by a temporal Dirac delta function ε(~r, sˆ, t) = S(~r, sˆ)δ(t), then I(~r, sˆ, t, µa ) = I(~r, sˆ, t, µa = 0) exp(−µa vt)

(3.14)

is the solution of the same equation when the absorption coefficient of the medium is µa . The property (3.14) can be simply demonstrated by substituting Eq. (3.14) in the RTE. This property is a generalization of the Lambert-Beer law:11, 12 Regardless of the location of a source point, radiation received at time t after traveling along paths of length l = vt is attenuated by a factor exp(−µa l). This property allows us to reduce the study of radiative transfer problems in which µa is independent of ~r in the case of a non-absorbing medium and to generalize the solution by simply using Eq. (3.14). From Eq. (3.9) and Eq. (3.14), the solution for the steady-state source S(~r, sˆ) can be expressed as I(~r, sˆ, µa ) =

Z∞ 0

I(~r, sˆ, t, µa )dt =

Z∞

I(~r, sˆ, t, µa = 0) exp(−µa vt)dt.

(3.15)

0

Equation (3.15) states that in a medium with constant absorption and with stationary sources, the received power is related to the time-dependent solution for the nonabsorbing medium, I(~r, sˆ, t, µa = 0), by means of the Laplace transform.9, 13 In principle, measurements carried out with a steady-state source for different values of µa can be used for reconstructing the temporal response. The above property for the radiance can be extended to the power measured by a detector of area ΣR and with an acceptance solid angle ΩR . In fact, denoting with

36

Chapter 3

qˆ the unit vector normal to ΣR , the Green’s function for the detected power P (t, µa ) can be calculated, using Eq. (3.14), as Z Z P (t, µa ) = dΣ I(~r, sˆ, t, µa ) |ˆ s · qˆ| dΩ = P (t, µa = 0) exp(−µa vt), ΣR

ΩR

(3.16) and the Green’s function for the steady-state source P (µa ) can be consequently calculated as Z∞ P (µa ) = P (t, µa = 0) exp(−µa vt)dt . (3.17) 0

Using Eq. (3.17), it is possible to obtain a useful expression for the mean time of flight hti spent by photons that migrate from the source to the detector. According to the definition, hti is obtained as hti (µa ) =

R∞

tP (t, µa )dt

0 R∞

P (t, µa )dt

=−

∂P (µa ) ∂ 1 =− ln P (µa ). vP (µa ) ∂µa v∂µa

(3.18)

0

The mean pathlength followed by photons inside the medium is obtained as hli = v hti. Using Eq. (3.18), it is possible to obtain accurate measurements for the mean time of flight (or for the mean pathlength), even from measurements of relative attenuation with steady-state sources. The expression of Eq. (3.18) can be further generalized for the nth –order moment, htn i, as htn i (µa ) = (−1)n

∂ n P (µa ) 1 . v n P (µa ) ∂ n µa

(3.19)

The property of Eq. (3.14) can be further generalized to obtain the dependence of the radiance on the absorption coefficient µai of a generic volume Vi of a homogeneous or non-homogeneous medium. Due to the intrinsic meaning of the absorption coefficient, the probability of a photon being detected when the absorption coefficient in the region i is µai is obtained by multiplying the probability of being detected when µai = 0 by exp(−µai vi ti ) (vi and ti are the speed of light and the time of flight in the volume Vi ). Thinking in terms of the RTE, this property can be written using the radiance detected at a certain point ~r. If I0 (~r, sˆ, t) is the solution of Eq. (3.5) for µai = 0, and with the source term described by a temporal Dirac delta function, then the solution when an absorption coefficient µai is introduced in Vi can be represented as I(~r, sˆ, t, µai ) = I0 (~r, sˆ, t)

Zt 0

f0i (ti , t) exp(−µai vi ti )dti ,

(3.20)

The Radiative Transfer Equation and Diffusion Equation

37

where f0i (ti , t) indicates the probability density function (normalized to 1) for the time of flight ti . From Eq. (3.20), the expression for the mean time of flight spent inside the volume Vi , by photons contributing to I(~r, sˆ, t, µai ), is hti i (~r, sˆ, t, µai ) =

Rt 0

ti f0i (ti ,t) exp(−µai vi ti )dti

Rt

f0i (ti ,t) exp(−µai vi ti )dti

0

,ˆ s,t,µai ) = − vi I(~r,ˆs1,t,µai ) ∂I(~r∂µ ai

(3.21)

∂ = − vi ∂µ ln I(~r, sˆ, t, µai ) . ai

In all generality, the region i represents any region of the medium, and thus this result has no restrictions in its validity. The same property stated above for the radiance is also valid for the power P detected by a receiver of area ΣR and with acceptance angle ΩR , since P is obtained by an integral of the radiance [see Eq. (3.1)]. The mean time of flight spent by photons received at time t inside the volume Vi can thus be obtained similarly to Eq. (3.21) as hti i (µai , t) = −

1 ∂P (µai , t) ∂ =− ln P (µai , t). vi P (µai , t) ∂µai vi ∂µai

(3.22)

Similar relations can be obtained for steady-state sources. In particular, the mean time of flight spent by received photons inside the volume Vi of the medium is hti i (µai ) = −

1 ∂P (µai ) ∂ =− ln P (µai ). vi P (µai ) ∂µai vi ∂µai

(3.23)

This equation can be generalized for the nth –order moment htni i as htni i (µai ) = (−1)n

∂ n P (µai ) 1 . vin P (µai ) ∂ n µai

(3.24)

3.5 Diffusion Equation The RTE is an integro-differential equation, and the retrieval of solutions is an extremely expensive computational process.7, 14 Solutions of the RTE are usually based on numerical methods15–22 like the finite element method7, 17, 23–30 or others like the discrete ordinates method,7, 31–33 the spherical harmonics method,7, 19, 34, 35 the finite difference method,36 the integral transport methods,7 and the path integral method.37 Among the numerical procedures there are also stochastic methods like the MC, which is largely used to reconstruct solutions of the RTE.7, 38 The several numerical methods used to treat the RTE are a consequence of the high complexity of this equation. No general analytical (closed-form) solutions of the RTE are available,7 and simpler approximate models are usually

38

Chapter 3

sought.25, 26, 39–41 When propagation is dominated by multiple scattering, the most widely and successfully used model employs the diffusion approximation to yield a variety of solutions for both steady-state and time-dependent sources.42–45 The diffusion equation is a parabolic-type partial-differential equation largely applied in several physics fields. In this section, its derivation from the RTE will be reviewed and discussed. 3.5.1 The diffusion approximation In the more general case of time-dependent sources, the diffusion approximation consists of two simplifying assumptions. The first one assumes the radiance inside a diffusive medium to be almost isotropic: The diffuse intensity I(~r, sˆ, t) is approximated by the first two terms (isotropic and linearly anisotropic terms) of a series expansion in spherical harmonics:1, 7, 46 I(~r, sˆ, t) =

1 3 ~ Φ(~r, t) + J(~r, t) · sˆ. 4π 4π

(3.25)

A spherical harmonic expansion truncated at the second term is usually denoted as the P1 approximation.7, 25 Equation (3.25) is a good approximation for the radiance if the contribution of the higher-order spherical harmonics is negligible. This is usually true if the second term of the expansion is small with respect to the first, ~ r, t) · sˆ. A complete derivation of Eq. (3.25) can be found in i.e., Φ(~r, t) ≫ 3J(~ Ref. 46, where a spherical harmonic expansion is introduced for the radiance. In Appendix A, Eq. (3.25) is introduced using an intuitive physical reasoning. The second simplifying assumption assumes that the time variation of the diffuse ~ r, t) over a time range ∆t = 1/(vµ′ ) is negligible with respect to the flux vector J(~ s vector itself and can be expressed as7 ~ r, t) 1 ∂ J(~ ~ r, t) . (3.26) ≪ J(~ ′ vµs ∂t

With Eq. (3.26), “slow” time variations of the flux are therefore assumed. When steady-state sources are considered, the diffusion approximation is simply summarized by the expansion of the radiance in spherical harmonics, i.e., I(~r, sˆ) =

1 3 ~ Φ(~r) + J(~r) · sˆ. 4π 4π

(3.27)

In general, Eqs. (3.25), (3.26), and (3.27) are well fulfilled when photons have undergone many scattering events, since scattering tends to randomize the direction of light propagation. Conversely, absorption obstructs the diffusive regime. In fact, absorption selectively extinguishes photons with long pathlengths; consequently, in a strongly absorbing medium only photons with few scattering events will not be absorbed. Therefore, the diffusive regime of light propagation can be established when scattering effects are predominant on absorption.

39

The Radiative Transfer Equation and Diffusion Equation

Making use of these simplifying assumptions, it is possible to obtain from the RTE a simpler equation, the Diffusion Equation (DE), for which several analytical solutions are available.

3.6 Derivation of the Diffusion Equation The RTE is an integro-differential equation for the radiance, while the DE is a partial-differential equation for the fluence rate. Thus, an integration procedure is required to obtain the DE from the RTE. In order to obtain the diffusion equation for time-dependent sources, Eq. (3.5) is integrated over all directions as follows: o R n ∂ I(~ r , s ˆ , t) + ∇ · [ˆ s I(~ r , s ˆ , t)] + µ I(~ r , s ˆ , t) dΩ t v∂t 4π h i (3.28) R R = µs p(ˆ s, sˆ′ )I(~r, sˆ′ , t)dΩ′ + ε(~r, sˆ, t) dΩ . 4π

4π

From this equation, simply exchanging the order of derivatives and integrals and the orders of integration, the continuity equation is obtained without any need for simplifying assumptions: R ∂ ~ r, t) + µa Φ(~r, t) = ε(~r, sˆ, t)dΩ . r, t) + ∇ · J(~ (3.29) v∂t Φ(~ 4π

To obtain the diffusion equation, the flux vector needs to be expressed as a function of the fluence rate. For this purpose, the RTE [Eq. (3.5)] is multiplied by sˆ and then integrated over all the directions as follows: o R n ∂ I(~ r , s ˆ , t) + ∇ · [ˆ s I(~ r , s ˆ , t)] + µ I(~ r , s ˆ , t) sˆdΩ t v∂t 4π h i (3.30) R R = µs p(ˆ s, sˆ′ )I(~r, sˆ′ , t)dΩ′ + ε(~r, sˆ, t) sˆdΩ . 4π

4π

From Eq. (3.30), making use of the simplifying assumptions of the diffusion approximation [Eqs. (3.25) and (3.26)], Fick’s law is obtained (see Appendix B): Z h i ~ r, t) = −D ∇Φ(~r, t) − 3 ε(~r, sˆ, t)ˆ J(~ sdΩ , (3.31) 4π

where D is the diffusion coefficient defined as D= ′

1 , 3(µa + µ′s )

(3.32)

and µs = µs (1 − g) is the reduced scattering coefficient. For media without sources ~ r, t) = −D∇Φ(~r, t), which is the or with isotropic sources, Fick’s law becomes J(~ usual form used to represent the flux inside diffusive media.7

40

Chapter 3

The physical meaning of Fick’s law is that photons tend to migrate toward those regions of the medium where the photon density is smaller. Substituting Eq. (3.31) in the continuity equation [Eq. (3.29)], the time-dependent DE is obtained:  h i R ∂ r, t) − ∇ · D ∇Φ(~r, t) − 3 ε(~r, sˆ, t)ˆ sdΩ + µa Φ(~r, t) v∂t Φ(~ 4π (3.33) R = ε(~r, sˆ, t)dΩ, 4π

which can be written as  
1 ∂ − ∇ · D∇ + µa Φ(~r, t) = q0 (~r, t), v ∂t

where

q0 (~r, t) =

Z

4π

h Z i ε(~r, sˆ, t)dΩ − 3∇ · D ε(~r, sˆ, t)ˆ sdΩ

(3.34)

(3.35)

4π

is the source term. For a homogeneous medium, the DE becomes   1 ∂ − D∇2 + µa Φ(~r, t) = q0 (~r, t). v ∂t For an isotropic source, q0 (~r, t) becomes Z q0 (~r, t) = ε(~r, sˆ, t)dΩ = 4πε(~r, t).

(3.36)

(3.37)

4π

Equation (3.34) shows that photon migration in the diffusive regime is fully described by the knowledge of the absorption coefficient and of the reduced scattering coefficient. Therefore, the detailed knowledge of the scattering function is not necessary: It is sufficient to know the asymmetry factor g. For steady-state sources, the continuity equation (obtained from the RTE without any need for simplifying assumptions) becomes Z ~ ∇ · J(~r) + µa Φ(~r) = ε(~r, sˆ)dΩ . (3.38) 4π

Since the term µa Φ(~r) represents the power absorbed in the unit volume, Eq. (3.38) has an immediate rigorous physical interpretation: The net flux emerging from the unit volume is equal to the generated power minus the absorbed power. Fick’s law for steady-state sources is Z h i ~ r) = −D ∇Φ(~r) − 3 ε(~r, sˆ)ˆ J(~ sdΩ , (3.39) 4π

and the DE for a homogeneous medium with isotropic sources is

The Radiative Transfer Equation and Diffusion Equation

    − D∇2 + µa Φ(~r) = q0 (~r), R  q0 (~r) = ε(~r, sˆ)dΩ = 4πε(~r).

41

(3.40)

4π

We point out that the DE can be also obtained following a significantly different approach, starting from different simplifying assumptions. For steady-state sources, it can be shown (see Ref. 47) that expressing the flux through a generic elementary area in an infinite homogeneous medium as the sum of contributions due to radiation scattered in each volume element, Fick’s law can be obtained making only the assumption that the fluence is a slowly varying function of position: It is sufficient to consider the Taylor series expansion of the fluence requiring a negligible contribution from the terms subsequent to the second derivatives. For time-dependent sources, the further assumption that the fractional change of the fluence is small during the time r,t) required for radiation to travel about three reduced mean free paths Φ1 ∂Φ(~ ∂t ≪  ′ vµs is also required.47 3

3.7 Diffusion Coefficient In the last few years, a debate about the dependence of the diffusion coefficient on absorption has arisen.48–51 It is not our purpose to get inside the details of this debate since its implications are of minor interest for the aim of this book. Undoubtedly, when the DE is derived from the RTE in complete generality, as it has been shown in the previous section, the diffusion coefficient shows a dependence on absorption in accordance with Eq. (3.32). In this book, a different approach to the problem is proposed. It can be summarized in two steps: I. The DE is derived for the non-absorbing medium (lossless diffusion equation), and the DE with a diffusion coefficient independent of absorption is obtained:  with

 1 ∂ 2 − D∇ Φ(~r, t) = q0 (~r, t) v ∂t ′
D = 1/ 3µs .

(3.41)

(3.42)

II. The dependence on absorption is assumed in accordance with the RTE, i.e., in accordance with Eq. (3.14) for homogeneous media and with Eq. (3.20) for inhomogeneous media. This procedure is equivalent to adding the term µa Φ(~r, t, µa ) to the lossless DE. The meaning of this procedure can be clarified. The lossless DE is an approximate equation of balance for Φ for non-absorbing media. It is stressed that for the intensity coming from all directions, the leakage of power within the unit volume due to absorption

42

Chapter 3

is, according to the RTE, µa Φ(~r, t, µa ). As a consequence, adding the term µa Φ(~r, t, µa ) to the lossless DE allows us to describe the effect of absorption using the same rules derived from the RTE. It can be argued that the equation obtained with this procedure is slightly different from the DE as it is known from the literature, since a diffusion coefficient independent of µa is assumed. This is actually true. But this further assumption allows us to treat the dependence on absorption in accordance with the RTE without the additional intrinsic approximations of the diffusion approximation. Thus, to study photon migration in an absorbing medium, no further approximations with respect to the non-absorbing case are introduced. Therefore, from now onward, we will assume that the diffusion coefficient is defined by Eq. (3.42). From a practical ′ point of view, since the DE is applicable with good
accuracy only when  µa ≪ ′µs, ′ the solutions of the DE expressed with D = 1/ 3µs or with D = 1/ 3(µa + µs ) remain almost indistinguishable. Results of experiments49, 52 have shown that, for ′ an infinite medium and for high absorption values up to µa /µs = 1/3, the DE with the diffusion coefficient independent of absorption [Eq. (3.42)] provides a better description of measurements.

3.8 Properties of the Diffusion Equation The DE shows some general properties like the RTE. 3.8.1 Scaling properties Following the same procedure used to obtain Eq. (3.10), it is possible to show that if Φ(~r, t) is the solution of the DE with a Dirac delta source q0 (~r, t) = δ 3 (~r) δ(t) for a ′ homogeneous diffusive medium characterized by µs and µa , then Φ(~r, t), defined as ′

Φ(~r, t) =

µs µ′s

!3

Φ(~r, t)

(3.43)

with ′

µ ~r = ~r s′ , µs

′

µ t = t s′ , µs

(3.44) ′

is the solution for a homogeneous medium with the transport coefficient µs and the ′ ′ absorption coefficient µa , provided that µa /µa = µs /µs . 3.8.2 Dependence on absorption The postulated independence of the diffusion coefficient from absorption keeps the actual physical meaning to the absorption coefficient. This means that all the properties related to the dependence on absorption of RTE solutions (see Sec. 3.4.2)

43

The Radiative Transfer Equation and Diffusion Equation

can be extended also to any DE solution. In particular, for a homogeneous nonabsorbing medium, if Φ(~r, t, µa = 0) is the solution of Eq. (3.36) for a source term q0 (~r, t) = S(~r)δ(t), then Φ(~r, t, µa ) = Φ(~r, t, µa = 0) exp(−µa vt)

(3.45)

is the solution when the absorption coefficient of the medium (independent of ~r) is µa . Making use of Fick’s law and of the first assumption of the diffusion approximation, this property can be also extended to the flux and the radiance. Consequently, the property is also valid for the detected power. Finally, the properties stated for the RTE about the dependence on absorption [Eqs. (3.20) and (3.24)] remain valid within the DE. It is important to point out that if the diffusion coefficient were that of Eq. (3.32),  ′ i.e., D = 1/ 3(µa + µs ) , the absorption coefficient in diffusion theory would not keep the actual physical meaning derived from the RTE, and all the above-mentioned properties would not be exactly valid for the DE.

3.9 Boundary Conditions The DE is a partial-differential equation, and any general solution will contain unknown constants that need to be determined by using proper boundary conditions originated from the physics of the problem investigated. The first boundary condition to be mentioned is the behavior at infinity of any intensity concerning a radiative process. It is trivial to require that any intensity vanishes at infinity. The boundary conditions are much more complex when the diffusive medium has finite boundaries. Two main situations will be considered: diffusive/non-scattering interfaces and diffusive/diffusive interfaces.

3.9.1 Boundary conditions at the interface between diffusive and non-scattering media For a diffusive medium bounded by a convex or flat surface Σ at the interface with a non-scattering region, the exact boundary condition for the radiance I(~r, sˆ, t) is that there should be no diffuse light entering the medium from the external nonscattering region at the interface Σ. The diffuse intensity I(~r, sˆ, t) (~r ∈ Σ) entering towards any direction sˆ pointing toward the diffusive medium can only originate from reflections at the boundary. With reference to Fig. 3.3, if I(~r, sb′ , t) is the radiance along sb′ , the mirror image direction (with respect to Σ) to sˆ results in I(~r, sˆ, t) = RF (sb′ · qˆ)I(~r, sb′ , t),

(3.46)

44

Chapter 3

where qˆ is the unit vector normal to Σ, and RF ( sb′ · qˆ) is the Fresnel reflection coefficient for unpolarized light for the light emerging from the diffusive medium:53 RF (sb′ · qˆ) =

    

1 2



ni cos θ−ne cos θr ni cos θ+ne cos θr

2

+

1 2



ne cos θ− ni cos θr ne cos θ+ni cos θr

2

for 0 < θ < θc

1

for θ > θc = arcsin(ne /ni ). (3.47) In Eq. (3.47), ni is the refractive index of the diffusive medium, ne is the refractive index of the external medium, θ is the angle of incidence (cos θ = sb′ · qˆ), and θr {θr = arcsin [(ni /ne ) sin θ]} is the angle of refraction. The ratio ni /ne = nr is denoted as the relative refractive index of the medium.

External medium Diffus ive medium

I ( r , s, t )

s

r

r

s

'

ni

q

u

ne

Figure 3.3 Schematic of a diffusive/non-scattering interface and the notation used.

With the simple angular distribution assumed for I(~r, sˆ, t) [Eq. (3.25)], the behavior of Eq. (3.46) cannot be satisfied exactly, and an approximate boundary condition must be considered. It is assumed that the condition of Eq. (3.46) is true on average for all directions sˆ inwardly directed,1 and the following relation is obtained: Z Z − I(~r, sˆ, t)(ˆ s · qˆ)d Ω = RF (ˆ s · qˆ)I(~r, sˆ, t)(ˆ s · qˆ)dΩ. (3.48) sˆ·ˆ q 0

This is a boundary condition for the total radiation coming from the boundary surface. Making use of the angular distribution for the radiance obtained from the diffusion approximation [Eq. (3.25)], and calculating the integrals of Eq. (3.48), the

45

The Radiative Transfer Equation and Diffusion Equation

boundary condition for the fluence rate can be written as (see Appendix C for the derivation) h i ~ =0 (3.49) Φ(~r, t) − 2AJ(~r, t) · qˆ ~ r∈Σ

with

1+3 A= 1−2

π/2 R

RF (cos θ) cos2 θ sin θdθ

0 π/2 R

.

(3.50)

RF (cos θ) cos θ sin θdθ

0

The coefficient A depends on the relative refractive index nr = ni /ne . Figure 3.4 shows how A depends on nr for values of refractive index mismatch of practical interest. A = 1 if nr = 1, and A > 1 if nr 6= 1. 8

Coefficient A

7

6

5

4

3

2

1 0.5

1.0

nr

1.5

2.0

Figure 3.4 Dependence of the coefficient A on the relative refractive index.

This boundary condition is denoted as the Partial Current Boundary Condition (PCBC) and represents the most accurate boundary condition for photon diffusion at a finite boundary.1, 54–60 The PCBC states a simple relationship between the outgoing ~ r, t) · qˆ and the fluence Φ(~r, t) on the interface of the medium: flux J(~ i h Φ(~r, t) ~ J(~r, t) · qˆ = . (3.51) 2A ~r∈Σ ~ r∈Σ

The PCBC involves both the flux and the fluence. It is possible to obtain an equation in which only the fluence is involved making use of Fick’s law [Eq. (3.25)] and assuming that sources on the boundary have an isotropic emission. With these assumptions, the PCBC can be expressed as   ∂ Φ(~r, t) + 2AD Φ(~r, t) = 0. (3.52) ∂q ~ r∈Σ

46

Chapter 3

This boundary condition is also known as the Robin boundary condition.25, 61, 62 According to this condition, the derivative of Φ(~r, t) along the direction normal to the boundary is proportional to Φ(~r, t) itself. An extrapolated decrement of Φ(~r, t) inside the non-scattering region is obtained if the derivative of Φ(~r, t) is assumed to remain constant in the non-scattering region to the value on the boundary. The distance from the geometrical boundary at which Φ(~r, t) is extrapolated to zero is denoted as the extrapolated distance ze and results in ze = 2AD.

(3.53)

The boundary condition that assumes Φ = 0 on the surface at the extrapolated distance ze is denoted as the extrapolated boundary condition (EBC).57, 58, 60 We will assume this boundary condition for deriving almost all the analytical solutions reported in the next chapters. A simpler, but more approximated, boundary condition that has been used in the literature to obtain analytical solutions of the DE is the socalled zero boundary condition (ZBC).42, 43 The ZBC simply assumes Φ(~r, t) = 0 at the physical boundary of the medium. These boundary conditions are also valid for steady-state sources, as can be seen in the procedure followed in Appendix C. It is worth making a final remark about the behavior of the flux vector at the interface Σ. From the continuity equation [Eq. (3.29)], it is possible to demonstrate a general condition of continuity of the component of the flux normal to the interface Σ. The derivation is straightforward through Gauss’s theorem. This boundary condition is also valid both for time-dependent sources and for steady-state sources, and it is consistent with the RTE. 3.9.2 Boundary conditions at the interface between two diffusive media When there is no refractive index mismatch at a diffusive/diffusive interface, both the fluence and the normal component of the flux must be continuous on the boundary. Conversely, when Fresnel reflections occur, as a general statement, the continuity of the normal component of the flux vector still holds at the interface, while reflections determine a discontinuity of the fluence rate. In this section, these properties will be shown making an energy balance at the interface of the two media. For medium 1 [refractive index n1 , radiance I1 (~r, sˆ, t), and flux J~1 (~r, t)] and medium 2 [refractive index n2 , radiance I2 (~r, sˆ, t), and flux J~2 (~r, t)], in contact at a boundary Σ (see Fig. 3.5), Eq. (3.48) needs to be rewritten. In this case, the energy balance at the interface has to take into account both the reflected light from Σ and the transmitted light through Σ of the two diffusive media. Following the procedure used by Faris,63 a couple of equations can be written: R

sˆ·ˆ q >0R

I1 (~r, sˆ, t)(ˆ s · qˆ)d Ω = −

+

sˆ·ˆ q >0

R

sˆ·ˆ q 0 results in ~ r, t) = J(~

r 16 (πDv)3/2 t5/2

  r2 exp − − µa vt rˆ, 4Dvt

(4.4)

and the Green’s functions for the radiance, using Eqs. (3.25), (4.3) and (4.4), can be written as h i   3r r2 1 v + (ˆ r · sˆ) exp − − µa vt . (4.5) I(~r, sˆ, t) = 2t 4Dvt (4π)5/2 (Dvt)3/2 The Green’s function for the flux can be calculated with these subroutines:

Solutions of the Diffusion Equation for Homogeneous Media

59

• Homogeneous_Flux_Infinite • Homogeneous_Flux_Perturbation_Infinite The Green’s function for the radiance can be calculated with the following subroutines: • Homogeneous_Radiance_Infinite • Homogeneous_Radiance_Perturbation_Infinite In Fig. 4.1, the TR expressions for the fluence, flux, and radiance of Eqs. (4.3), ′ (4.4), and (4.5) are plotted versus time for an infinite medium with µa = 0, µs = 1 mm−1 , and refractive index n = 1. The results for the fluence and flux are shown for r = 1, 5, 10 and 20 mm. The results for the radiance are shown for r = 1 and 10 mm and for different values of the angle α between the directions sˆ and rˆ. The figure shows that the fluence, and thus the energy density, decreases as r increases. A different behavior can be noticed at early and late times for the flux: At early times, the flux decreases as r increases and vice versa at late times. This means that at late times the energy gradient is larger for the larger distances. We point out that at r = 1 mm the radiance takes negative values for large values of α and early times. Integrating Eqs. (4.3), (4.4), and (4.5) over the whole time range (see Appendices E and F), the following expressions for the Green’s functions for steady-state ~ r), and I(~r, sˆ) are obtained: sources Φ(~r), J(~ 1 exp (−µef f r) , 4πrD   1 1 ~ + µef f exp (−µef f r) rˆ, J(~r) = 4πr r     1 D + µef f D (ˆ 1+3 I(~r, sˆ) = 2 2 r · sˆ) exp (−µef f r) , 4 π rD r Φ(~r) =

where µef f =

p

µa /D =

q

3µa µ′s

(4.6) (4.7) (4.8)

(4.9)

is the effective attenuation coefficient. It is important to stress that the analytical expression for the CW flux [Eq. (4.7)] is the exact solution of the RTE for the case of a non-absorbing medium (see Appendix E). The Green’s functions for CW fluence, flux, and radiance can be calculated with these subroutines: • DE_Fluence_Perturbation_Infinite_CW • Homogeneous_Flux_Perturbation_Infinite

60

Chapter 4

• Homogeneous_Radiance_Perturbation_Infinite Alternatively, the CW Green’s function can be calculated integrating the TR Green’s function using the Trap subroutine provided with the enclosed CD-ROM.

-2

-2

10

10 r = 1 mm

)

-1 -2

ps

r = 20 mm

-4

Flux (mm

ps

-2

Fluence (mm

r = 10 mm

10

-5

10

-6

10

-7

10

r = 5 mm r = 10 mm

-4

10

r = 20 mm

-5

10

-6

10

-7

10

-8

10

-8

10

r = 1 mm

-3

10

r = 5 mm

-1

)

-3

10

-9

0

500

1000

1500

2000

10

2500

0

Time (ps)

500

1000

) sr

-1

-1

180 degrees

-2

90 degrees

-4

10

r = 1 mm

-5

135 degrees

ps

135 degrees

-3

10

45 degrees

Radiance (mm

-1

0 degrees

180 degrees

-2

) sr

90 degrees

-1

45 degrees

ps

2500

10 0 degrees

Radiance (mm

2000

-5

-2

10

10

1500

Time (ps)

-6

10

r = 10 mm

-7

0

20

40

60

Time (ps)

80

100

10

0

500

1000

1500

Time (ps)

Figure 4.1 Time-dependent fluence, flux, and radiance for an infinite medium with µa = 0, ′ µs = 1 mm−1 , and n = 1. The results for the fluence and the flux are shown for several distances r. The results for the radiance are shown for several values of the angle between rˆ and sˆ for r = 1 and 10 mm.

In Fig. 4.2, the CW fluence, flux, and radiance of Eqs. (4.6), (4.7), and (4.8) are ′ plotted for a medium with µs = 1 mm−1 . Fluence and flux are plotted versus r for µa = 0, 0.005, 0.01, 0.05, and 0.1 mm−1 . The radiance is plotted versus α for µa = 0 and r = 0.5, 1, 2, and 5 mm. The figure shows how the CW fluence and the CW flux decrease when the absorption coefficient or the distance increase. For the CW radiance, the figure shows an almost isotropic behavior, as expected in the diffusive regime, for large values of r (r ≥ 5 mm). Finally, the expression for the mean pathlength traveled by photons detected at a

61

Solutions of the Diffusion Equation for Homogeneous Media

distance r from the source hl(r)i is: r r 1 ∂ = ln Φ(~r) = √ hl(r)i = − ∂µa 2 µa D 2

s

3µ′s . µa

(4.10)

For a non-absorbing medium, the expression of the mean pathlength diverges. 1

1

10

a

= 0 mm

10

-1

a

= 0 mm

-1

-1

0

10

a

= 0.005 mm

10

-1

a

-1

0

a

= 0.005 mm -1

= 0.01 mm

a

= 0.01 mm

a

= 0.05 mm = 0.1 mm

-2

a

Flux (mm )

-2

Fluence (mm )

-1

-1

10

-1

-2

10

-3

10

-4

a

-1

a

-2

10

-3

10

-5

0

5

10

15

10

20

0

5

10

r (mm)

20

0.015 r = 2 mm

r = 1 mm

r = 5 mm -1

sr

-1

)

r = 0.5 mm

)

0.09

-2

0.06

Radiance (mm

sr

15

r (mm)

0.12

-2

= 0.1 mm

10

-5

Radiance (mm

= 0.05 mm

-4

10

10

-1

-1

10

0.03

0.00

-0.03

-0.06 0

30

60

90

120

Angle (degrees)

150

180

0.010

0.005

0.000 0

30

60

90

120

150

180

Angle (degrees)

Figure 4.2 Fluence, flux, and radiance for steady-state sources in an infinite homogeneous ′ medium with µs = 1 mm−1 . The fluence and flux are plotted versus r for several values of µa . The radiance is plotted versus the angle between sˆ and rˆ for some values of r for µa = 0.

Physically, this means that a photon inside an infinite non-absorbing medium travels for an infinite time. Similarly, the second-order moment hl2 (r)i can be calculated as hl2 (r)i =

1 + µef f r r 1 1 + µef f r 1 ∂2 Φ(~r) = √ = hl(r)i . 2 Φ(~r) ∂ µa 4 µa D µa 2µa

(4.11)

Averaging hl(r)i [given by Eq. (4.10)], multiplied by the weighting factor Φ(~r) over the whole medium, the following expression for the total mean pathlength hli

62

Chapter 4

is obtained: hli =

+∞ R 0

hl(r)iΦ(~r)r2 dr

+∞ R

Φ(~r)r2 dr

=

1 . µa

(4.12)

0

The total mean pathlength traveled by photons in a scattering and absorbing infinite medium before being absorbed is therefore equal to the mean free path followed in an infinite non-scattering medium with the same absorption [Eq. (2.19)]. However, while trajectories in the non-scattering medium are straight lines, in the scattering medium they are chaotic broken lines in accordance with the scattering properties. It is interesting to know the mean number of scattering events hki undergone by photons before being absorbed in an infinite medium. Since the mean free path followed between two subsequent scattering events is given by ℓs = 1/µs , it results in hli µs hki = = µs hli = . (4.13) ℓs µa The information on hki can be useful in evaluating the average square distance from the source at which photons migrating through the medium are absorbed. Using Eq. (2.23) for the average distance at which photons are located after k scattering events, and using Eq. (4.13), we obtain 2 6 hd2hki i ∼ = ′ 2 hki(1 − g) = 2 . µs µef f

(4.14)

The same result can be also obtained averaging the square distance r2 over the whole medium with the probability density function µa Φ(r) [µa Φ(r)4πr2 dr is the probability to have absorption of photons within the spherical shell of radius r, r + dr].6

4.2 Solution of the Diffusion Equation for the Slab Geometry The problem of light propagation through random media bounded by parallel planes has been a subject of interest for decades because many physical systems are likely to be represented in this way.1, 7–11 The slab is used for applications of the radiative transfer in the atmosphere7, 9, 12–15 and in biological tissues.2–4, 10, 11, 16–23 For this reason, the solutions for the slab are widely used in photon migration studies. To obtain the solutions reported in this section and in Sec. 4.3, we follow the method of images (or mirror images)24 that is commonly used for the slab.2–5, 17 The method makes use of boundary conditions (ZBC2, 3 or EBC4, 5, 17, 25 ) that assume the fluence equal to 0 on the physical boundary of the diffusive medium (ZBC) or on the extrapolated surface at distance 2AD [Eq. (3.53)] from the physical boundary (EBC). For regular, bounded geometries like the slab and the parallelepiped, the method of images allows us to reconstruct the solution for the fluence inside the

63

Solutions of the Diffusion Equation for Homogeneous Media

medium as a superposition of solutions for the infinite medium. The flux exiting the diffusive medium (i.e., in the slab geometry the reflectance and the transmittance) is obtained by applying Fick’s law at the boundary of the medium [see Eqs. (4.23) and (4.24)]. The method of images leads relatively simple explicit analytical solutions expressed as summations of infinite terms. A very few of these terms is sufficient for the convergence of the solution in almost all applications of practical interest. Solutions of the DE have been also obtained following other approaches. In particular, solutions for the semi-infinite medium and slab have been obtained using the PCBC expressed as the Robin boundary condition,1, 8, 25–30 a boundary condition less approximated with respect to the ZBC or the EBC. However, these solutions have been implemented for the time domain in integral form using the Laplace transformation method, and their use is significantly more complicated. For the CW plane wave incident upon a slab, a simple closed-form solution based on the Robin boundary condition is also available.8, 27–30 Other hybrid solutions, especially devoted to obtaining improved solutions for the outgoing flux, have been also proposed. These solutions will be presented and discussed in Sec. 4.4. The geometry together with a description of notations used is shown in Fig. 4.3. The solutions reported in this section were obtained for an isotropic point source of unitary strength, i.e., a spatial and temporal Dirac delta source q(~r, t) = δ 3 (~r − → rs )δ(t) at → rs = (0, 0, zs ) with 0 ≤ zs ≤ s and s geometrical thickness of the slab. According to the EBC (Sec. 3.9.1), the fluence is assumed equal to zero at two

qˆ

z = - ze z=0 y

z0- = - 2 ze- zs

x

z0+= zs

real source

s

z=s z = s + ze qˆ Positive sources Negative sources

Extrapolated boundary

Extrapolated boundary

z z1- = 2 s + 2 ze - zs z1+ = 2 s + 4 ze+ zs

Figure 4.3 Model of the slab geometry and image sources.

64

Chapter 4

extrapolated flat surfaces outside the turbid medium at the extrapolated distance ze = 2AD from the physical boundaries of the slab. By the method of images,24 the correct boundary conditions are obtained by using, in addition to the real source rs , an infinite number of pairs of positive and negative sources in an infinite in → − → + diffusive medium having the same optical properties of the slab. The locations rm − → − of the negative sources are shown in Fig. 4.3. The of the positive sources and rm + real source is that at z0 ; all the other sources are image sources. In this way, the fluence inside the slab is composed by terms of fluence from an infinite medium: − → + and the negative ones from the sources at The positive terms from the sources at rm − → − . The sources are placed along the z-axis at rm  +  zm = 2m(s + 2ze ) + zs z − = 2m(s + 2ze ) − 2ze − zs  m m = 0, ±1, ±2, ......... ± ∞ .

(4.15)

Adding the contributions of all of the source pairs, the Green’s function for the TR fluence rate at ~r = (x, y, z) results in Φ(~r, t) =

 − µa vt h i h i + 2 − 2 (z−zm ) (z−zm ) , exp − 4Dvt − exp − 4Dvt

(4.16)

~ r, t) = −D∇Φ(~r, t) J(~

(4.17)

v exp (4πDvt)3/2 m=+∞ P

×

m=−∞



−

ρ2 4Dvt

p with ρ = x2 + y 2 and 0 ≤ z ≤ s. The flux and the radiance, making use of Fick’s law [Eq. (3.31)] and of the first assumption of the diffusion approximation [Eq. (3.25)], can be calculated as

and I(~r, sˆ, t) =

1 3 Φ(~r, t) − D∇Φ(~r, t) · sˆ. 4π 4π

(4.18)

The expressions of Eqs. (4.16)–(4.18) can be calculated with the following subroutines provided with the enclosed CD-ROM: • DE_Fluence_Perturbation_Slab_TR (for the fluence) • Homogeneous_Fluence_Slab (for the fluence) • Homogeneous_Flux_Slab (for the flux) • Homogeneous_Radiance_Slab (for the radiance)

65

Solutions of the Diffusion Equation for Homogeneous Media

The Green’s function for the TR reflectance R(ρ, t), i.e., the power crossing the surface at z = 0, per unit area, at distance ρ from the z-axis with any exit angle, can be evaluated as R R(ρ, t) = [1 − RF (ˆ s · qˆ)] I(ρ, z = 0, sˆ, t)(ˆ s · qˆ)dΩ sˆ·ˆ q >0 (4.19) ~ z = 0, t) · qˆ , = J(ρ, where qˆ is the unit vector normal to the boundary surface outwardly directed, and RF is the Fresnel reflection coefficient for unpolarized light. Making use of Fick’s law results in ∂ (4.20) R(ρ, t) = D Φ(ρ, z = 0, t). ∂z With a similar procedure, it is possible to evaluate the TR transmittance T (ρ, t): T (ρ, t) = −D

∂ Φ(ρ, z = s, t). ∂z

(4.21)

The functions R(ρ, t) and T (ρ, t) of Eqs. (4.20) and (4.21), since they correspond to a unitary energy point-like source, also represent the probability per unit rs and t = 0, exits at time t and at time and unit area that a photon, emitted at → distance ρ from the z-axis. The reflectance and the transmittance can be obtained by performing the derivative of Eqs. (4.20) and (4.21) with a numerical evaluation. This can be done using the subroutine • Homogeneous_Flux_Slab provided with the enclosed CD-ROM. Alternatively, explicit analytical expressions for R(ρ, t) and T (ρ, t) can be obtained from the same equation, as will be shown in the next section. The Green’s function for CW sources, i.e., the solution for an isotropic point source of unitary strength [q(~r) = δ 3 (~r − → rs )] is ( h i √ + 2 m=+∞ exp −µef f ρ2 +(z−zm ) P 1 √ Φ(~r) = 4πD + 2 ρ2 +(z−zm ) m=−∞ h i) √ (4.22) −

exp −µef f

√

− 2 ρ2 +(z−zm )

− 2 ρ2 +(z−zm )

.

Using Fick’s law, the Green’s function for the CW reflectance R(ρ) for steadystate sources results in ∂ (4.23) R(ρ) = D Φ(ρ, z = 0), ∂z and for the CW transmittance T (ρ) = −D

∂ Φ(ρ, z = s). ∂z

(4.24)

66

Chapter 4

The CW Green’s functions for the flux, radiance, reflectance, and transmittance can be calculated integrating the corresponding TR Green’s functions using, for instance, the Trap subroutine provided with the enclosed CD-ROM. Alternatively, the fluence can be calculated with the subroutine • DE_Fluence_Perturbation_Slab_CW provided with the enclosed CD-ROM and the reflectance and the transmittance with the software that implements the analytical expressions reported in Sec. 4.3. The whole procedure described here provides the Green’s function for a slab illuminated by an isotropic light source placed in zs . When a pencil light beam is normally impinging the medium along the z-axis, the solutions can be derived with the superposition integral [Eq. (3.8)] of the above Green’s functions with the real source. The pencil beam inside the medium can be represented as a line of point-like isotropic sources with decreasing intensity, according to the Lambert-Beer law. Alternatively, the real source term can be substituted by a single isotropic point source located at ~r = (0, 0, zs ). The coordinate zs is usually obtained by imposing that the line of isotropic point sources and the single point source have the same first moment.20, 31, 32 This assumption results in ′

zs = 1/(µa + µs ).

(4.25)

In this book, we argue that the physical meaning of the absorption coefficient in diffusion and transport theory should be equivalent [Eqs. (3.14) and (3.20)]; therefore, the coordinate zs is assumed as ′

zs = 1/µs .

(4.26)

With this assumption, the DE solutions maintain the same dependence on the absorption coefficient of the RTE solutions. From a practical point of view, since the ′ DE is applicable with good accuracy only when µa ≪ µs , the solutions of the DE ′ ′ expressed with zs = 1/(µa + µs ) or with zs = 1/µs are almost indistinguishable. We stress that all the figures of this chapter for the slab pertain to to an isotropic ′ source at zs = 1/µs . The approximation of a semi-infinite medium has often been used to describe problems that involve highly diffusive media. The approximation is valid for slabs sufficiently thick so that the probability that the radiation will escape from the surface opposite the source is negligible. In this case, the boundary conditions are fulfilled, retaining only the first of the dipole sources (that with m = 0) necessary for the slab.

4.3 Analytical Green’s Functions for Transmittance and Reflectance In this section, the analytical expressions for the reflectance and the transmittance of an infinite slab obtained with Fick’s law are reported both for time-dependent sources and for steady-state sources.

67

Solutions of the Diffusion Equation for Homogeneous Media

The analytical expressions for the TR reflectance and transmittance, making use of Eqs. (4.16), (4.20), and (4.21), respectively, result in   2

R(ρ, t) = − ×

T (ρ, t) =

3/2 2(4πDv)  t5/2 m=+∞ P

m=−∞



 2   2  z4m z3m − z4m exp − 4Dvt , z3m exp − 4Dvt

2

ρ exp − 4Dvt −µa vt

× with

ρ −µa vt exp − 4Dvt

3/2 5/2 2(4πDv) t m=+∞ P

m=−∞

 z1m    z2m  z3m   z4m



z1m exp



2 z1m − 4Dvt



− z2m exp



2 z2m − 4Dvt

= (1 − 2m)s − 4mze − zs = (1 − 2m)s − (4m − 2)ze + zs = −2ms − 4mze − zs = −2ms − (4m − 2)ze + zs ,



(4.27)

(4.28) ,

(4.29)

p and ρ = x2 + y 2 . Equations (4.27) and (4.28) are infinite series and should be truncated for practical applications. Since the distance from the boundaries [on which R(ρ, t) and T (ρ, t) are evaluated] of the sources that correspond to the index m increases when m increases, the contribution of sources corresponding to high values of m is expected to be significant only for large values of ρ and/or t. The expressions of Eqs. (4.27) and (4.28) can be calculated with the following subroutines provided on the enclosed CD-ROM: • DE_Reflectance_Perturbation_Slab_TR • DE_Transmittance_Perturbation_Slab_TR In Fig. 4.4, the TR reflectance and the TR transmittance from Eqs. (4.27) and ′ (4.28) are plotted versus time for a slab with s = 40 mm, µa = 0, µs = 1 mm−1 , ′ zs = 1/µs = 1 mm, and refractive index ni = 1.4. The reflectance is shown for ρ = 5 and 20 mm and the transmittance for ρ = 0 and 40 mm. Both are plotted for the relative refractive index nr equal to 1 and 1.4. nr is the ratio between the refractive indexes inside (ni ) and outside (ne ) the diffusive medium, respectively. The differences between the curves of Fig. 4.4 corresponding to different values of nr can be explained by the redistribution, due to reflection, of photons inside the slab. Reflection removes a consistent fraction of photons that, in the case of nr = 1, would exit near the source at short times toward longer distances. We note that with nr = 1.4, more than 50% of photons are reflected if an isotropic distribution for the radiance is assumed. These photons continue to migrate through the slab, and thus the probability of exiting at longer distances and longer times increases as is shown in Fig. 4.4.

68

Chapter 4

-4

-8

10

10 ) ps

-1

r

-5

10

= 20 mm; n = 1

-2

r

= 5 mm; n = 1.4

-6

Transmittance (mm

Reflectance (mm

-2

ps

-1

)

= 5 mm; n = 1

r

10

= 20 mm; n = 1.4 r

-7

10

-8

10

-9

10

-9

10

-10

10

= 0 mm; n = 1 r

-11

= 40 mm; n = 1

10

r

= 0 mm; n = 1.4 r

= 40 mm; n = 1.4 -10

10

r

-12

0

1000

2000

10

3000

Time (ps)

0

1000

2000

3000

4000

5000

Time (ps)

Figure 4.4 Time-dependent reflectance and transmittance from a homogeneous slab with ′ s = 40 mm , µa = 0 , µs = 1 mm−1 , zs = 1 mm, and ni = 1.4. The reflectance is shown for ρ = 5 and 20 mm and the transmittance for ρ = 0 and 40 mm. Both are plotted for nr = 1 and 1.4.

By integrating Eqs. (4.27) and (4.28) over the entire exit surface, the total TR reflectance R(t) and the total TR transmittance T (t) are obtained:
+∞ R exp −µa vt R(t) = R(ρ, t)2πρdρ = − 2(4πDv)1/2 t3/2 0  (4.30)   2   m=+∞ 2 P z4m z3m z3m exp − 4Dvt − z4m exp − 4Dvt × m=−∞

T (t) =

+∞ R




exp −µa vt

T (ρ, t)2πρdρ = 2(4πDv)1/2 t3/2    2   m=+∞ 2 P z1m z2m z1m exp − 4Dvt − z2m exp − 4Dvt . × 0

(4.31)

m=−∞

The functions R(t) and T (t) represent the Green’s functions for an infinitely extended receiver. For the reciprocity principle7, 13 the functions R(t) and T (t) can be used to describe the time-resolved reflectance and transmittance when an infinitely wide beam, having constant radiance, impinges perpendicularly on the surface of the slab. The Green’s functions R(t) and T (t) can be calculated making use of the following subroutines provided on the enclosed CD-ROM: • DE_Total_Reflectance_Slab_TR • DE_Total_Transmittance_Slab_TR In Fig. 4.5, the expressions for R(t) and T (t) of Eqs. (4.30) and (4.31) are plotted ′ ′ versus time for a slab with s = 40 mm, µa = 0 , µs = 1 mm−1 , zs = 1/µs = 1

69

Solutions of the Diffusion Equation for Homogeneous Media

mm, ni = 1.4, and nr = 1.4. For late times, the two curves tend to the same value and thus photons have the same probability to escape from the two sides of the slab. This is because at late times the photon density inside the slab tends to have a spatial distribution almost symmetric with respect to the middle of the slab. In contrast, at early times R(t) and T (t) are significantly different because of the large amount of photons contributing to R(t) that escape from the slab at z = 0, after few scattering events close to the source. -2

10

Total Reflectance -3

Total Transmittance

R, T (ps

-1

)

10

-4

10

-5

10

-6

10

-7

10

0

2500

5000

7500

10000

Time (ps)

Figure 4.5 Time-dependent total reflectance and total transmittance from a slab with s = ′ 40 mm, µa = 0, with µs = 1 mm−1 , zs = 1 mm, ni = 1.4, and nr = 1.4.

With a similar integration, the expressions for the average TR reflectance and transmittance received by a ring delimited by ρ1 and ρ2 can be obtained: ρ R2

R(ρ,t)2πρdρ

ρ1

= − 3/2 2 21 2 2) 1/2 3/2 π 4(ρ2 −ρ1 )(Dv) t 1 π(ρ2 −ρ     2 2 ρ2 ρ1 exp(−µa vt) − exp − 4Dvt × exp − 4Dvt   2   2  m=+∞ P z3m z4m z3m exp − 4Dvt − z4m exp − 4Dvt ×

hR(ρ1 , ρ2 , t)i =

(4.32)

m=−∞

hT (ρ1 , ρ2 , t)i =

ρ R2

T (ρ,t)2πρdρ

ρ1

= 3/2 2 21 π 4(ρ2 −ρ1 )(Dv)1/2t3/2      2 ρ1 ρ22 × exp − 4Dvt exp(−µa vt) − exp − 4Dvt   2   2  m=+∞ P z2m z1m z1m exp − 4Dvt − z2m exp − 4Dvt × π(ρ22 −ρ21 )

(4.33)

m=−∞

These expressions are important for analyzing experimental data since a real receiver is characterized by a finite area and an integral over this area is required for a correct evaluation of the detected light.

70

Chapter 4

The functions hR(ρ1 , ρ2 , t)i and hT (ρ1 , ρ2 , t)i can be calculated making use of the following subroutines, respectively, provided with the enclosed CD-ROM: • DE_Reflectance_Slab_Receiver_TR • DE_Transmittance_Slab_Receiver_TR The effect of the finite area of the receiver is shown in Fig. 4.6. The expression of Eq. (4.32) for the TR reflectance is plotted versus time for a ring delimited by ρ1 = 4 mm and ρ2 = 6 mm and for a ring delimited by ρ1 = 19 mm and ρ2 = 21 mm and compared with the reflectance obtained for ρ = 4, 5 and 6 mm and for ρ = ′ ′ 19, 20, and 21 mm. A slab with s = 40 mm, µa = 0, µs = 1 mm−1 , zs = 1/µs = 1 mm, ni = 1.4, and nr = 1.4 has been considered. The comparison shows that the response for the receiver of finite area is almost identical to the response for a receiver element at a distance ρ = (ρ1 + ρ2 )/2. This result indicates that for many practical applications, even for a receiver with a large area, the response can be accurately evaluated using Eqs. (4.27) and (4.28) referring to elementary receiver elements.

-4

-7

10

10

-1

ps

-1

= 4 mm

ps

-2

= 5 mm -5

= 6 mm

10

Reflectance (mm

-2

Reflectance (mm

< 6 mm

)

)

4 mm
, (mm)

Total Transmittance 2

10

T

R, T

-2

10

-3

1

10

1) takes into account the different ways photons can visit the volume occupied by the defect. Equations (7.65) and (7.67) allow us to make a direct comparison between the Born and the Rytov approximations. The Born approximation consists of retaining only the term with n = 1 in the series of Eqs. (7.65) and (7.67). The Rytov method for steady-state sources is equivalent to looking for a perturbed solution of the kind13, 29 hli i(δµa = 0) = −

Φpert (~r) = Φ0 (~r) exp(−hli iδµa ) .

(7.69)

158

Chapter 7

By using the Taylor expansion for the exponential and comparing Eq. (7.69) with Eq. (7.67), it is possible to verify that the Rytov method consists of the assumption hlin i = hli in .13 In this case, it is assumed that all photons spend the same time inside the volume Vi .

Solutions of the Diffusion Equation with Perturbation Theory

159

References 1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999). 2. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1, Academic Press, New York (1978). 3. S. Feng, F. A. Zeng, and B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995). 4. S. R. Arridge, “Photon measurements density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995). 5. M. R. Ostermeyer and S. L. Jacques, “Perturbation theory for diffuse light transport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255–261 (1997). 6. M. Morin, S. Verreault, A. Mailloux, J. Frechette, S. Chatigny, Y. Painchaud, and P. Beaudry, “Inclusion characterization in a scattering slab with timeresolved transmittance measurements: perturbation analysis,” Appl. Opt. 39, 2840–2852 (2000). 7. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration,” Appl. Opt. 40, 4622–4632 (2001). 8. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, and R. Cubeddu, “Experimental test of a perturbation model for time-resolved imaging in diffusive media,” Appl. Opt. 42, 3145–3153 (2003). 9. S. De Nicola, R. Esposito, and M. Lepore, “Perturbation model to predict the effect of spatially varying absorptive inhomogeneities in diffusing media,” Phys. Rev. E 68, 021901 (2003). 10. B. Wassermann, “Limits of high-order perturbation theory in time-domain optical mammography,” Phys. Rev. E 74, 031908 (2006). 11. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time-domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007). 12. F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model for light propagation through diffusive layered media,” Phys. Med. Biol. 50, 2159–2166 (2005). 13. A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. I. Theory,” J. Opt. Soc. Am. A 23, 2105–2118 (2006).

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14. A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. II. Continuous-wave results,” J. Opt. Soc. Am. A 23, 2119–2131 (2006). 15. A. Sassaroli, F. Martelli, and S. Fantini, “Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries,” Appl. Opt. 48, D62–D73 (2009). 16. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express 1, 404–413 (1997). 17. A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, and R. Nossal, “Timedependent contrast functions for quantitative imaging in time-resolved transillumination experiment,” Appl. Opt. 37, 1973–1981 (1998). 18. I. N. Polonsky and M. A. Box, “General perturbation technique for the calculation of radiative effects in scattering and absorbing media,” J. Opt. Soc. Am. A 19, 2281–2292 (2002). 19. M. A. Box, “Radiative perturbation theory: a review,” Environ. Modell. Softw. 17, 95–106 (2002). 20. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998). 21. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001). 22. P. K. Yalavarthy, K. Karlekar, H. S. Patel, R. M. Vasu, M. Pramanik, P. C. Mathias, B. Jain, and P. K. Gupta, “Experimental investigation of perturbation monte-carlo based derivative estimation for imaging low-scattering tissue,” Opt. Express 13, 985–997 (2005). 23. J. Heiskala, P. Hiltunen, and I. Nissil¨a, “Significance of background optical properties, time-resolved information and optode arrangement in diffuse optical imaging of term neonates,” Phys. Med. Biol. 54, 535–554 (2009). 24. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992). 25. D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomoheneities within turbid media: Analytical solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887– 4891 (1994).

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161

26. D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal to noise analysis,” Appl. Opt. 36, 75–92 (1997). 27. P. N. den Outer, T. M. Nieuwenhuizen, and A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993). 28. V. N. Pustovit and V. A. Markel, “Propagation of diffuse light in a turbid medium with multiple spherical inhomogeneities,” Appl. Opt. 43, 104–112 (2004). 29. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three-dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002). 30. G. Arfken and H. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, London (2001). 31. G. F. Roach, Green’s Functions: Introductory Theory with Applications, Van Nostrand Reinhold Company, London (1970). 32. V. Tuchin, Tissue Optics, SPIE Press, Bellingham, WA (2000). 33. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989). 34. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992). 35. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997). 36. A. Torricelli, L. Spinelli, A. Pifferi, P. Taroni, R. Cubeddu, and G. M. Danesini, “Use of a nonlinear perturbation approach for in vivo breast lesion characterization by multiwavelength time-resolved optical mammography,” Opt. Express 11, 853–867 (2003). 37. P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, and R. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm: clinical study on the detection and identification of breast lesions,” Phys. Med. Biol. 50, 2469–2488 (2005). 38. T. Svensson, J. Swartling, P. Taroni, A. Torricelli, P. Lindblom, C. Ingvar, and S. Andersson-Engels, “Characterization of normal breast tissue heterogeneity using time-resolved near-infrared spectroscopy,” Phys. Med. Biol. 50, 2559– 2571 (2005).

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39. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. Danesini, and R. Cubeddu, “Characterization of female breast lesions from multi-wavelength time-resolved optical mammography,” Phys. Med. Biol. 50, 2489–2502 (2005). 40. D. Grosenick, K. T. Moesta, H. Wabnitz, J. Mucke, C. Stroszczynski, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Time-domain optical mammography: initial clinical results on detection and characterization of breast tumors,” Appl. Opt. 42, 3170–3186 (2003). 41. D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. Möller, C. Stroszczynski, J. Stößel, B. Wassermann, P. M. Schlag, and H. Rinneberg, “Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography,” Phys. Med. Biol. 49, 1165–1181 (2004). 42. H. Obrig, R. Wenzel, M. Kohl, S. Horst, P. Wobst, J. Steinbrink, F. Thomas, and A. Villinger, “Near-infrared spectroscopy: does it function in functional activation studies of the adult brain?,” Int. J. Phychophysiol. 35, 125–142 (2000). 43. A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. Barbour, and A. Hielscher, “Three-dimentional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001). 44. A. M. Siegel, J. P. Culver, J. B. Mandeville, and D. A. Boas, “Temporal comparison of functional brain imaging with diffuse optical tomography and fMRI during rat forepaw stimulation,” Phys. Med. Biol. 48, 1391–1403 (2003). 45. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95, 078101 (2005). 46. A. Pifferi, A. Torricelli, P. Taroni, D. Comelli, A. Bassi, and R. Cubeddu, “Fully automated time domain spectrometer for the absorption and scattering characterization of diffusive media,” Rev. Sci. Instrum. 78, 053103 (2007).

Chapter 8

Software This chapter describes the software in the enclosed CD-ROM. The software requires ′ use of the following units: µa and µs as measured in mm−1 , all distances measured in mm, and time measured in ps. For the speed of light in vacuum, the value c = 0.299792458 mm ps−1 has been used.

8.1 Introduction The contents of the CD-ROM can be visualized by means of the enclosed index.html file and comprises three main parts with the following subcategories: • The Diffusion&Perturbation Program – Install the Diffusion&Perturbation Program – The Diffusion&Perturbation User Manual – Examples of Results • Source Code: Solutions of the Diffusion Equation and Hybrid Models – Solutions of the Diffusion Equation for Homogeneous Media – Solutions of the Diffusion Equation for Layered Media – Hybrid Models for the Homogeneous Infinite Medium – Hybrid Models for the Homogeneous Slab – Hybrid Models for the Homogeneous Parallelepiped – General Purpose Subroutines and Functions • Reference Monte Carlo Results In this chapter, the Diffusion&Perturbation program and the source code are described. The reference MC results and examples of comparisons will be described in the last two chapters. 165

166

Chapter 8

8.2 The Diffusion&Perturbation Program The Diffusion&Perturbation program is dedicated to the calculation and the visualization of the solutions of the DE and its perturbation theory for a homogeneous medium. The program is thus a forward solver for the calculation of the solutions of the DE in homogeneous media. This software has a user-friendly interface, which was developed in a Visual C environment. The program is accompanied by a user manual. The Diffusion&Perturbation program may be installed by using the link "Install Diffusion&Perturbation Program" on the index.html or by using the setup.exe file that is in the directory \Diffusion&Perturbation\cvidistkit.Diffusion&Perturbation. The main panel of the program guides the user through the Diffusion&Perturbation software. From the main panel (see Fig. 8.1), the user can proceed to: • Set up the simulation • Run the simulation • Display the simulated data • Save the simulated data • Stop the program Simulations are set up in a corresponding panel where all the relevant quantities for the simulation can be set by the user (see Fig. 8.2). The program can perform simulations with three different geometries: infinite medium, semi-infinite medium, and slab. The output of Diffusion&Perturbation includes classical quantities like reflectance, transmittance, and fluence rate. Simulations can be either carried out in the time domain or in the CW domain. The selected output can be visualized in different ways. For the time domain, the program can display the temporal response of the medium, a receiver map for a fixed value of time, or an inhomogeneity map for a fixed value of time. For the CW domain, two types of output can be displayed by the program: a receiver map or an inhomogeneity map. For temporal response, the TPSF is meant for the selected output (reflectance, transmittance, fluence) together with the corresponding absorption and scattering perturbation. The receiver map is obtained by varying the position of the receiver, while an inhomogeneity map is obtained for a fixed position of the receiver by varying the position of a unitary perturbation (scattering or absorption with δDVi = 1 mm4 or δµa Vi = 1 mm2 ) inside the medium. In the set simulation panel, the optical and geometrical characteristics of the simulation are selected by the user. The range (temporal or spatial) of the quantities displayed by the program are selected in the same panel (see Fig. 8.2). After the set simulation panel has been filled with the necessary information it is possible to proceed with the simulation. Pressing the OK button returns the

167

Software

Figure 8.1 Main panel of the program Diffusion&Perturbation.

program to the main menu (Fig. 8.1) from which it is possible to run the simulation by pressing the RUN button. The computation time is negligible for most of the simulations that can be performed with the program. The quantities calculated by the program can be displayed and saved as an ASCII file using the comma separated values (CSV) format. The data thereby can be further analyzed and elaborated by the user. The use of the program is supported by an included HELP function that can be accessed by clicking the right button of the mouse on the enabled buttons. Examples of results generated and displayed with the Diffusion&Perturbation program are shown in Figs. 8.3, 8.4, and 8.5. Figure 8.3 shows an example of TR ′ reflectance from a homogeneous slab (s = 40 mm, µa = 0.01 mm−1 , µs = 1 mm−1 , zs = 1 mm, ni = 1.2 and nr = 1.2, ρ = 20 mm). Figure 8.4 shows how the perturbation on transmittance at a certain time (t = 1000 ps) changes as a function of the position of an absorbing inhomogeneity (inhomogeneity map obtained by varying the coordinate xi and yi of the inhomogeneity) (s = 10 mm, µa = 0.05 ′ mm−1 , µs = 1 mm−1 , zs = 1 mm, ni = 1.2 and nr = 1.2, ρ = 0, depth of the inhomogeneity zi = 5 mm). Figure 8.5 shows the CW fluence in an infinite medium with an internal absorbing inhomogeneity as a function of the position of the receiver (receiver map obtained by varying the coordinate xr and zr of the receiver) [µa = ′ 0.001 mm−1 , µs = 1 mm−1 , ni = 1.2, inhomogeneity at (0, 0, 6) mm, and receiver at (xr , 5, zr ) mm].

168

Chapter 8

Figure 8.2 Set up panel of the program Diffusion&Perturbation.

Diffusion&Perturbation employs several subroutines based on the Green’s functions presented in Chapters 4 and 7. For the sake of the reader, we have summarized the links between the subroutines employed by the program and the Green’s functions presented in the book in Table 8.1. The source code of these subroutines is written in the FORTRAN language and used by the Diffusion&Perturbation program as DLL libraries. These subroutines evaluate the Green’s function at a single time for the TR Green’s functions and at a single position of the receiver for the CW Green’s functions. The source code of the subroutines employed by the program is available in the enclosed CD-ROM inside the directory Source_Code\Diffusion&Perturbation. More details on these subroutines will be given in the next section. We point out that solutions for reflectance and transmittance are implemented using solutions based on Fick’s law. The solutions based on the hybrid EBPC approach can be obtained by dividing the solution provided by the Diffusion&Perturbation code for the fluence evaluated on the external surface by 2A [A is the coefficient for the EBC, Eq. (3.50)].

Software

169

Figure 8.3 Example of a TPSF generated with the Diffusion&Perturbation program for the reflectance from a diffusive slab.

Figure 8.4 Example of an inhomogeneity map for the transmittance generated with the Diffusion&Perturbation program for an absorption perturbation inside a diffusive slab.

170

Chapter 8

Figure 8.5 Example of a receiver map for the fluence generated with the Diffusion&Perturbation program for an absorption perturbation inside an infinite medium.

8.3 Source Code: Solutions of the Diffusion Equation and Hybrid Models In this section, all the source code provided with the CD-ROM is briefly introduced and described. The source code is divided into five main environments. All the code is written in the FORTRAN language. The FORTRAN subroutines can be converted as a dynamic-link library (DLL) interface for processes that use them. In this way, the subroutines can be employed in programs written in other languages like C or, although it is not a straightforward procedure, in other environments like MATLAB.1 For a correct utilization of this second part of the CD-ROM, even if the software is accompanied by comments and examples of use, some basic programming skills that are not necessary for using the Diffusion&Perturbation software are required here. The name of the subroutines have been built using acronyms like DE, RTE, TE, TR, and CW and keywords like Fluence, Flux, Radiance, Reflectance, Transmittance, Perturbation, Infinite, Slab, Parallelepiped, Cylinder, Sphere, and Layered. The intent is to help the reader identify the subroutine and the type of Green’s function associated with the subroutine. Thus, the name of the subroutine represents a code that gives information on the task performed by the subroutine itself. For instance, the subroutine DE_Fluence_Perturbation_Infinite_TR calculates the TR fluence and TR perturbations for the fluence in an infinite medium obtained with the DE.

171

Software

The solutions provided by the subroutines have physical meaning only when the field point is inside the diffusive medium or on its boundary. 8.3.1 Solutions of the diffusion equation for homogeneous media The environment of the solutions of the DE for homogeneous media is based on the subroutines used by the Diffusion&Perturbation program to evaluate the fluence, reflectance, transmittance, and perturbation due to scattering and absorbing inhomogeneities. Table 8.1 shows a link between each subroutine and the corresponding equations in Chapters 4 and 7. Table 8.1 In this table, the equations of Chapters 4 and 7 and the name of the corresponding subroutines used by the program Diffusion&Perturbation are summarized.

Equations Eqs. (4.3), (7.21), and (7.22) Eqs. (4.6), (7.23), and (7.24) Equations Eqs. (4.16), (7.25), and (7.26) Eqs. (4.22), (7.28), and (7.29) Eqs. (4.27), (7.36), and (7.37) Eqs. (4.28), (7.38), and (7.39) Eqs. (4.34), (7.44), and (7.45) Eqs. (4.35), (7.46), and (7.47)

Infinite Medium Subroutines DE_Fluence_Perturbation_Infinite_TR DE_Fluence_Perturbation_Infinite_CW Semi-Infinite Medium and Slab Subroutines DE_Fluence_Perturbation_Slab_TR DE_Fluence_Perturbation_Slab_CW DE_Reflectance_Perturbation_Slab_TR DE_Transmittance_Perturbation_Slab_TR DE_Reflectance_Perturbation_Slab_CW DE_Transmittance_Perturbation_Slab_CW

Other subroutines for evaluating the response for a receiver with a finite area, the total reflectance, and the total transmittance are summarized in Table 8.2, where a link between each subroutine and the corresponding equation of Chapter 4 is given. In the names used for the subroutines, the keyword "Receiver" will identify the response for a receiver with a finite area, and the keyword "Total" will identify the response for an infinitely extended receiver. A brief description of each subroutine of Tables 8.1 and 8.2 is provided below: • DE_Fluence_Perturbation_Infinite_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1 ) for the infinite medium, and the perturbations (mm−2 ps−1 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

172

Chapter 8

Table 8.2 In this table, the equations of Chapter 4 and the name of the corresponding subroutines of the software package for homogeneous media are summarized.

Equation Eq. (4.63)

Slab Geometry Subroutines DE_Reflectance_Slab_Receiver_TR DE_Reflectance_Slab_Receiver_CW DE_Transmittance_Slab_Receiver_TR DE_Transmittance_Slab_Receiver_CW DE_Total_Reflectance_Slab_TR DE_Total_Reflectance_Slab_CW DE_Total_Transmittance_Slab_TR DE_Total_Transmittance_Slab_CW DE_Fluence_Slab_Receiver_TR DE_Fluence_Slab_Receiver_CW DE_Total_Fluence_Slab_TR DE_Total_Fluence_Slab_CW Cylindrical Geometry Subroutine DE_Fluence_Cylinder_TR

Equation Eq. (4.64)

Spherical Geometry Subroutine DE_Fluence_Sphere_TR

Equations Eq. (4.32) Eq. (4.38) Eq. (4.33) Eq. (4.39) Eq. (4.30) Eq. (4.36) Eq. (4.31) Eq. (4.37) Eq. (4.53) Eq. (4.54) Eq. (4.59) Eq. (4.60)

• DE_Fluence_Perturbation_Infinite_CW

Calculates the CW fluence per unit of emitted power (mm−2 ) for the infinite medium, and the perturbations (mm−2 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• DE_Fluence_Perturbation_Slab_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1 ) for the slab, and the perturbations (mm−2 ps−1 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• DE_Fluence_Perturbation_Slab_CW

Calculates the CW fluence per unit of emitted power (mm−2 ) for the slab, and the perturbations (mm−2 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

Software

173

• DE_Reflectance_Perturbation_Slab_TR

Calculates the TR reflectance per unit of emitted energy (mm−2 ps−1 ) for the slab, and the perturbations (mm−2 ps−1 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• DE_Transmittance_Perturbation_Slab_TR

Calculates the TR transmittance per unit of emitted energy (mm−2 ps−1 ) for the slab, and the perturbations (mm−2 ps−1 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• DE_Reflectance_Perturbation_Slab_CW

Calculates the CW reflectance per unit of emitted power (mm−2 ) for the slab, and the perturbations (mm−2 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• DE_Transmittance_Perturbation_Slab_CW

Calculates the CW transmittance per unit of emitted power (mm−2 ) for the slab, and the perturbations (mm−2 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• DE_Reflectance_Slab_Receiver_TR Calculates the TR average reflectance over the receiver area per unit of emitted energy (mm−2 ps−1 ) for the slab. • DE_Reflectance_Slab_Receiver_CW Calculates the CW average reflectance over the receiver area per unit of emitted power (mm−2 ) for the slab. • DE_Transmittance_Slab_Receiver_TR Calculates the TR average transmittance over the receiver area per unit of emitted energy (mm−2 ps−1 ) for the slab. • DE_Transmittance_Slab_Receiver_CW Calculates the CW average transmittance over the receiver area per unit of emitted power (mm−2 ) for the slab. • DE_Total_Reflectance_Slab_TR

Calculates the TR total reflectance per unit of emitted energy (ps−1 ) for the slab.

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• DE_Total_Reflectance_Slab_CW

Calculates the CW total reflectance per unit of emitted power for the slab.

• DE_Total_Transmittance_Slab_TR

Calculates the TR total transmittance per unit of emitted energy (mm−2 ) for the slab.

• DE_Total_Transmittance_Slab_CW

Calculates the CW total transmittance per unit of emitted power for the slab.

• DE_Fluence_Slab_Receiver_TR

Calculates the TR average fluence over the receiver area per unit of emitted energy (mm−2 ps−1 ) for the slab.

• DE_Fluence_Slab_Receiver_CW

Calculates the CW average fluence over the receiver area per unit of emitted power (mm−2 ) for the slab.

• DE_Total_Fluence_Slab_TR

Calculates the TR total fluence per unit of emitted energy (mm−2 ) for the slab.

• DE_Total_Fluence_Slab_CW

Calculates the CW total fluence per unit of emitted power for the slab.

• DE_Fluence_Cylinder_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1 ) for the infinite cylinder.

• DE_Fluence_Sphere_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1 ) for the sphere.

Two drivers, Program1 and Program2, are also available to show examples of how to use the subroutines DE_Fluence_Cylinder_TR and DE_Fluence_Sphere_TR. The computation time of DE_Fluence_Cylinder_TR and DE_Fluence_Sphere_TR is significantly longer compared to that of the other subroutines for homogeneous geometries. A faster version of these subroutines can be obtained using the routines in Ref. 2 to calculate the Bessel functions. Our priority in making these two subroutines was to guarantee a double precision accuracy for the calculation and more flexibility, while minor importance was given to the computation time. Thus, with minor changes, it is possible to have faster routines like those implemented in Ref. 3, which were used in fitting procedures to retrieve the optical properties of the medium.

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8.3.2 Solutions of the diffusion equation for layered media The environment of the solutions of the DE for layered media is based on following subroutines: • Root • DE_Fluence_Layered_Slab_TR The subroutine Root calculates the longitudinal eigenvalues for the solution of the DE for a two-layer cylinder [roots of Eq. (6.28)]. The subroutine DE_Fluence_Layered_Slab_TR calculates the TR fluence rate per unit of emitted energy (mm−2 ps−1 ) in the two-layer cylinder [Eqs. (6.30) and (6.32)], and it uses the longitudinal eigenvalues calculated by the subroutine Root and the radial eigenvalues [roots of Eq. (6.19), i.e., roots of J0 (Kρl L) = 0] provided inside the file j0.dat. All the other subroutines of this environment make use of these subroutines to calculate the quantities of interest for photon migration through a twolayer medium. In particular, the quantities that can be evaluated with this software package are summarized in Table 8.3. A brief description of each subroutine is provided below. Table 8.3 The equations of Chapter 6 and the name of the corresponding subroutines of the software package for the solution of the diffusion equation for layered media are summarized.

Quantity calculated Eqs. (6.30) and (6.32) Gradient Eq. (3.31) Eq. (3.25) Longitudinal eigenvalues Eqs. (7.11) and (7.12) Eqs. (7.11) and (7.12)

Two-layer medium Subroutine used DE_Fluence_Layered_Slab_TR Layered_Fluence_Gradient Layered_Flux Layered_Radiance Root Layered_Fluence_Perturbation Layered__Fluence_Perturbation_2

• Layered_Fluence_Gradient

Numerically calculates the gradient of the TR fluence (mm−3 ps−1 ) for the two-layer cylinder.

• Layered_Flux

Calculates the TR flux (mm−2 ps−1 ) for the two-layer cylinder.

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• Layered_Radiance

Calculates the TR radiance (mm−2 ps−1 sr−1 ) for the two-layer cylinder.

• Layered_Fluence_Perturbation

Calculates the TR perturbations (mm−2 ps−1 ) for a two-layer cylinder, due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ). The perturbation is calculated by numerically evaluating Eqs. (7.11) and (7.12), using the fluence of Eqs. (6.30) and (6.32) obtained with the subroutine DE_Fluence_Layered_Slab_TR.

• Layered_Fluence_Perturbation_2 This is a faster version of the Layered_Fluence_Perturbation, which was developed to calculate the perturbation for an array of values of time. For each call, the subroutine Layered_Fluence_Perturbation calculates the perturbation for a single time, which makes the computation of the convolution integrals of Eqs. (7.11) and (7.12) time consuming. In contrast, Layered_Fluence_Perturbation_2 calculates with a single call the whole TPSF of the scattering and of the absorption perturbation. For this reason, the computation time is shorter since the information on the temporal response is stored for each convolution integral of Eqs. (7.11) and (7.12). Although the subroutine Layered_Fluence_Perturbation is slower than the subroutine Layered_Fluence_Perturbation_2, it is included for its greater flexibility, which may be preferred for some kinds of calculations. As an example to help the reader get accustomed to this environment, we show some lines of the enclosed driver Program3 that, using the subroutines of Table 8.3, can be used to calculate the TR reflectance from the two-layer cylinder: C C C C C C

C C

................................ Calculation of the TR reflectance ................................ Maximum number of radial eigenvalues n_kj=100 Maximum number of longitudinal eigenvalues n_k0=200 Calculation of the longitudinal Eigenvalues CALL Root(mua,musp,s,R,n0,AA,n_k0,n_kj, + kappa_z0,kappa_j,I_bound,Nl) DO i=1,NPOINT Definition of the temporal scale t(i)=0.d0+DBLE(i-1)*dTIME ! Time (ps) Calculation of the fluence with the DE

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CALL DE_Fluence_Layered_Slab_TR( + mua,musp,s,R,t(i),xs,ys,zs,xr,yr,zr,n0,AA, + n_k0,n_kj,kappa_z0,kappa_j,I_bound,Nl,flu) C Calculation of the reflectance (Eq. 6.33) DE_reflectance(i)=flu/2.d0/AA(1) END DO As is shown in the program, the initial requirement for the calculation is evaluation of the longitudinal eigenvalues with the subroutine Root. The radial eigenvalues are provided inside the file j0.dat, which must be inserted in the same directory where the driver program is run. Two important parameters that need to be set on the driver program are the number of radial (n_kj) and longitudinal (n_k0) eigenvalues used by the program. The values selected for n_kj and n_k0 affect the computation time and the level of convergence of the calculated solution. Increasing n_kj and n_k0 will increase both the computation time and the convergence of the calculation to the actual solution. The number of radial eigenvalues required for convergence of the solution depends proportionally to the product of the radius of the cylinder with the reduced scattering coefficient of the medium. It is possible that the number selected for n_k0 will not be enough to provide all the imaginary longitudinal roots for the situation investigated. In this case, the program will provide a warning, and a higher number for n_k0 will need to be inserted to perform the calculation. Note that the eigenfunction method has a fast convergence of the solution at late times but a slow convergence at early times. For this reason, due to convergence problems related to the number and to the accuracy of the calculated eigenvalues, it is also possible to have negative values for the temporal response at early times. 8.3.3 Hybrid models for the homogeneous infinite medium The environment of the hybrid models for the infinite homogeneous medium is based on the following subroutines: • RTE_Fluence_Infinite_TR Calculates the solution of the RTE for the TR fluence per unit of emitted energy (mm−2 ps−1 ) for the infinite medium. • TE_Fluence_Infinite_TR Calculates the solution of the TE for the TR fluence per unit of emitted energy (mm−2 ps−1 ) for the infinite medium. • DE_Fluence_Infinite_TR Calculates the solution of the DE for the TR fluence rate per unit of emitted energy (mm−2 ps−1 ) for the infinite medium.

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These subroutines calculate the Green’s function of the fluence for the RTE [Eq. (5.12)], the TE [Eq. (5.16)], and the DE [Eq. (4.2)], respectively. All the other subroutines of this environment call these subroutines to calculate the quantities of interest for photon migration through a homogeneous infinite medium like flux, radiance, perturbation, etc. All the quantities that can be evaluated with this software package are summarized in Table 8.4 and are briefly described below: Table 8.4 Software package for hybrid models for the homogeneous infinite medium: Summary of equations and quantities calculated with the corresponding subroutines.

Quantity calculated Eq. (5.12) Eq. (5.16) Eq. (4.3) Gradient Eq. (3.31) Eq. (3.25) Eqs. (7.11) and (7.12) Gradient and Perturbation Eqs. (3.31) and (7.14) Eqs. (3.25) and (7.15) Fluence and perturbation with RTE Fluence and perturbation with TE Fluence and perturbation with DE

Infinite Medium Subroutine used RTE_Fluence_Infinite_TR TE_Fluence_Infinite_TR DE_Fluence_Infinite_TR Homogeneous_Fluence_Gradient _Infinite Homogeneous_Flux_Infinite Homogeneous_Radiance_Infinite Homogeneous_Fluence_Perturbation _Infinite Homogeneous_Fluence_Gradient _Perturbation_Infinite Homogeneous_Flux_Perturbation _Infinite Homogeneous_Radiance_Perturbation _Infinite RTE_Fluence_Perturbation_Infinite_TR TE_Fluence_Perturbation_Infinite_TR DE_Fluence_Perturbation_Infinite_TR_2

• Homogeneous_Fluence_Gradient_Infinite

Numerically calculates the gradient of the TR fluence (mm−3 ps−1 ) for the infinite medium.

• Homogeneous_Flux_Infinite

Calculates the TR flux (mm−2 ps−1 ) for the infinite medium.

• Homogeneous_Radiance_Infinite

Calculates the TR radiance (mm−2 ps−1 sr−1 ) for the infinite medium.

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• Homogeneous_Fluence_Perturbation_Infinite

Calculates the perturbations of TR fluence (mm−2 ps−1 ) due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium.

• Homogeneous_Fluence_Gradient_Perturbation_Infinite:

Calculates the gradient of the TR fluence (mm−3 ps−1 ) or of the CW fluence (mm−3 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium.

• Homogeneous_Flux_Perturbation_Infinite

Calculates the TR flux (mm−2 ps−1 ) or the CW flux (mm−2 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium.

• Homogeneous_Radiance_Perturbation_Infinite

Calculates the TR radiance (mm−2 ps−1 sr−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium.

• RTE_Fluence_Perturbation_Infinite_TR

Calculates the TR fluence (mm−2 ps−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium, making use of the subroutines RTE_Fluence_Infinite_TR and Homogeneous_Perturbation_Infinite.

• TE_Fluence_Perturbation_Infinite_TR

Calculates the TR fluence (mm−2 ps−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium, making use of the subroutines TE_Fluence_Infinite_TR and Homogeneous_Perturbation_Infinite.

• DE_Fluence_Perturbation_Infinite_TR_2

Calculates the TR fluence (mm−2 ps−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the infinite medium, making use of the subroutines DE_Fluence_Infinite_TR and Homogeneous_Perturbation_Infinite.

180

Chapter 8

8.3.4 Hybrid models for the homogeneous slab The environment of the hybrid models for the homogeneous slab is based on the subroutine Homogeneous_Fluence_Slab_TR, which calculates the Green’s function of the hybrid models of Eq. (5.1). This subroutine calculates the Green’s function for the slab using the subroutines of the Green’s functions for the infinite medium based on RTE, TE, and DE. The other subroutines of this environment make use of the Homogeneous_Fluence_Slab_TR subroutine to calculate the other quantities of interest for photon migration through a homogeneous slab. In particular, the quantities that can be evaluated with this software package are summarized in Table 8.5 and are briefly described below. • Homogeneous_Fluence_Gradient_Slab

Numerically calculates the gradient of the TR fluence (mm−3 ps−1 ) for the slab.

• Homogeneous_Flux_Slab

Calculates the TR flux (mm−2 ps−1 ) for the slab.

• Homogeneous_Radiance_Slab

Calculates the TR radiance (mm−2 ps−1 sr−1 ) for the slab.

• Homogeneous_Fluence_Perturbation_Slab

Calculates the perturbations of the TR fluence (mm−2 ps−1 ) for the slab due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ).

• RTE_Fluence_Perturbation_Slab_TR

Calculates the TR fluence (mm−2 ps−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the slab using the subroutines RTE_Fluence_Infinite_TR and Homogeneous_Perturbation_Slab.

• TE_Fluence_Perturbation_Slab_TR

Calculates the TR fluence (mm−2 ps−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the slab using the subroutines TE_Fluence_Infinite_TR and Homogeneous_Perturbation_Slab.

• DE_Fluence_Perturbation_Slab_TR_2

Calculates the TR fluence (mm−2 ps−1 ), and the perturbations due to both a scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµa Vi = 1 mm2 ) for the slab using the subroutines DE_Fluence_Infinite_TR and Homogeneous_Perturbation_Slab.

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Table 8.5 Software package for hybrid models for the homogeneous slab: Summary of equations and quantities calculated with the corresponding subroutines.

Quantity calculated Eq. (5.1) Gradient Eq. (3.31) Eq. (3.25) Eqs. (7.11) and (7.12) Fluence and perturbation with RTE Fluence and perturbation with TE Fluence and perturbation with DE

Homogeneous slab Subroutine used Homogeneous_Fluence_Slab_TR Homogeneous_Fluence_Gradient_Slab Homogeneous_Flux_Slab Homogeneous_Radiance_Slab Homogeneous_Fluence_Perturbation_Slab RTE_Fluence_Perturbation_Slab_TR TE_Fluence_Perturbation_Slab_TR DE_Fluence_Perturbation_Slab_TR_2

As an example to help the reader, below are some lines of the enclosed driver Program4 that, using the subroutines of Tables 8.4 and 8.5, calculates the TR reflectance with the different models: C ................................ C Calculation of the TR reflectance C ................................ CALL A_parameter(n0,ne,AA) DO i=1,NPOINT C Definition of the temporal scale t(i)=0.d0+(i-1.d0)*1.d0 ! Time (ps) C Calculation of the fluence with the Hybrid DE CALL Homogeneous_Fluence_Slab( + mua,musp,s,t(i),xs,ys,zs,xr,yr,zr, + n0,AA,DE_Fluence_Infinite_TR,flu) C Calculation of the reflectance with the DE (Eq. 5.5) DE_reflectance(i)=flu/2.d0/AA C Calculation of the fluence with the TE CALL Homogeneous_Fluence_Slab( + mua,musp,s,t(i),xs,ys,zs,xr,yr,zr, + n0,AA,TE_Fluence_Infinite_TR,flu) C Calculation of the reflectance with the TE (Eq. 5.5) TE_reflectance(i)=flu/2.d0/AA C Calculation of the fluence with the RTE CALL Homogeneous_Fluence_Slab( + mua,musp,s,t(i),xs,ys,zs,xr,yr,zr, + n0,AA,RTE_Fluence_Infinite_TR,flu) C Calculation of the reflectance with the RTE (Eq. 5.5) RTE_reflectance(i)=flu/2.d0/AA

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Chapter 8

END DO C .................................. This driver shows how it is possible to switch from one hybrid model to another simply by changing the name of the subroutine used to calculate the fluence for the infinite medium in the input variable of the subroutine Homogeneous_Flux_Slab. The same calculation can also be carried out for the perturbation on the TR reflectance using the driver Program5. 8.3.5 Hybrid models for the homogeneous parallelepiped The environment of the hybrid models for the homogeneous parallelepiped is based on the subroutine Homogeneous_Fluence_Parallelepiped, which calculates the Green’s function of the hybrid models of Eq. (5.7). This subroutine calculates the Green’s function for the parallelepiped using the subroutines for the infinite medium based on RTE, TE, and DE. The number of terms (dipoles) that need to be retained to have convergence in Eq. (5.7) in the temporal range of some ns is usually . 4. 8.3.6 General purpose subroutines and functions All the subroutines described in previous sections make use of some service routines that are necessary to implement their calculations. These are subroutines and functions used by the subroutines that evaluate the solutions of the diffusion equation for homogeneous and layered media and by the subroutines used to evaluate the hybrid models. These routines are grouped together inside the directory \Source_Code\General_Purpose_Subroutines&Functions. The calculations of the Bessel functions and of the Legendre polynomials are implemented with the following subroutines: IKNA, JY01A, JYNA, JYV, and LPN. The copyright of this software belongs to Professor Jianming Jin of the University of Illinois at Urbana-Champaign. These subroutines have been downloaded from the web site http://jin.ece.uiuc.edu/routines/routines.html, with the kind permission of Professor Jianming Jin. Some more information on these subroutines can be found in Ref. 4. All the quantities that can be evaluated with this software package are summarized in Table 8.6 and are briefly described below. • A_parameter

Numerically calculates parameter A of the extrapolated boundary condition [Eq. (3.50)].

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Table 8.6 General purpose subroutines and functions: Quantities calculated with the corresponding subroutines or functions.

Quantity calculated Equation (3.50) Equation (3.47) Integration with the trapezoidal method Scalar product of two vectors Equation (7.41) Equation (7.49) Equation (7.42) Equation (7.43) Equation (7.48) Modified Bessel functions In and Kn Bessel functions J0 , J1 , Y0 , Y1 Bessel functions Jn , Yn Bessel functions Jv , Yv Legendre Polynomials Pn

Subroutine or function used A_parameter Fresnel Trap Sca f h p q w IKNA JY01A JYNA JYV LPN

• Fresnel Calculates the Fresnel reflection coefficient for unpolarized light given by Eq. (3.47). • Trap Calculates the integral of a function y on the variable x using the trapezoidal method. • Sca Calculates the scalar product of two vectors. • f Calculates Eq. (7.41). • h Calculates Eq. (7.49). • p Calculates Eq. (7.42). • q Calculates Eq. (7.43).

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• w Calculates Eq. (7.48). • IKNA

Calculates the modified Bessel functions In and Kn of order n, and their derivatives.

• JY01A

Calculates the Bessel functions J0 , J1 , Y0 , and Y1 of integer order, and their derivatives.

• JYNA

Calculates the Bessel functions Jn and Yn of integer order, and their derivatives.

• JYV

Calculates the Bessel functions Jv , Yv of real order, and their derivatives.

• LPN Calculates the Legendre polynomials.

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References 1. The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, MATLAB 7 External Interfaces (2008). 2. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge (1988). 3. A. Sassaroli, F. Martelli, G. Zaccanti, and Y. Yamada, “Performance of fitting procedures in curved geometry for retrieval of the optical properties of tissue from time-resolved measurements,” Appl. Opt. 40, 185–197 (2001). 4. S. Zhang and J. Jin, Computation of Special Functions, John Wiley & Sons, New York (1996).

Chapter 9

Reference Monte Carlo Results 9.1 Introduction Almost all the analytical solutions reported in Chapters 4–7 and the related software described in Chapter 8 are approximate solutions of the RTE. Before using them, it is useful to know the level of their accuracy in order to establish their applicability in different fields of interest. For this purpose, the enclosed CD-ROM reports examples of solutions of the RTE obtained with MC simulations. With MC simulations, the RTE can be solved without the need of simplifying assumptions and with an accuracy only limited by the statistical fluctuations related to the stochastic nature of the method. The MC results can therefore be used as a standard reference for comparison. This chapter is mainly devoted to describe the MC results reported in the CD-ROM (Sec. 9.6). The MC results have been obtained using several different MC programs. For a better understanding and a proper use of the MC results, a description of the MC programs we used is reported in Secs. 9.2–9.5. Examples of comparisons will be shown in Chapter 10.

9.2 Rules to Simulate the Trajectories and General Remarks The MC method provides a physical simulation of light propagation. For each launched energy packet (photon packet or simply photon for brevity), the trajectory is numerically generated using the probability functions that govern propagation through the turbid medium. A photon is received if the simulated trajectory enters the receiver within the acceptance angle. With conventional MC simulations, the impulse response or temporal point spread function (TPSF), i.e., the probability (per unit time and per unit area of the receiver) of receiving emitted photons, is reconstructed by classifying the received photons in a histogram on the basis of their time of flight. The temporal resolution is only limited by the length of the intervals chosen for the temporal histogram and has a feature similar to the one obtained with a time-resolved experimental setup in which photons are detected using the timecorrelated single-photon counting (TCSPC) technique.1 As for the experiment, the simulated response is noisy: Owing to the central limit theorem, the standard error decreases proportionally to the inverse of the square root of the number of received photons. To obtain results with the required accuracy, the simulation of a large 187

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number of trajectories may be necessary with computation times that sometimes become prohibitively long. Techniques of variance reduction can be sometimes successfully used to reduce the computation time. The key point for MC simulations is the generation of the trajectories: They must be generated using the same statistical rules that govern propagation in real experiments. The basis for generating photon trajectories are the probability distribution functions describing the interaction of photons with the turbid medium reported in Sec. 2.2.1 [Eqs. (2.15)–(2.17)]. Recall that with the assumptions of isotropic scatterers (spherical particles or randomly oriented non-spherical particles) and unpolarized light, the probability distribution functions f1 (θ) and f2 (ϕ) for the scattering angle θ and the azimuthal angle ϕ specifying the new direction after a scattering event, are given by f1 (θ) = 2πp(θ) sin θ

(9.1)

and

1 , (9.2) 2π where p(θ) is the scattering function of the medium. The probability density function f3 (ℓ) for the path followed between two subsequent interactions with the turbid medium is given by f2 (ϕ) =



f3 (ℓ) = µt (ℓ) exp −

Zℓ

′




µt ℓ dℓ

0

′



.

(9.3)

The sampling of the variables ℓ, θ, and ϕ is obtained using the inverse distribution method (see Ref. 2), starting from pseudo-random numbers ξ with uniform distribution between 0 and 1 generated by computer, and making use of the cumulative probability function F (x) defined as F (x) =

Zx 0


f x′ dx′ .

(9.4)

Since the probability distribution function is normalized to 1, F (x) assumes all the values between 0 and 1. The value of the variable x associated to ξ is obtained posing F [x(ξ)] = ξ , (9.5) from which x(ξ) = F −1 (ξ) .

(9.6)

In all the MC programs we used, the trajectories are simulated for the nonabsorbing medium. As will be shown in subsequent sections, the results for the

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Reference Monte Carlo Results

medium with absorption can be easily obtained from the results for the non-absorbing medium making use of the general property for the RTE described in Sec. 3.4.2. We summarize here the rules for sampling the trajectories for the homogeneous non-absorbing medium. For the homogeneous non-absorbing medium, the cumulative probability function for the step size between two subsequent scattering events results in

F3 (ℓ) =

Zℓ 0

f3

Zℓ


ℓ′ dℓ′ =

0

µs exp(−µs ℓ′ )dℓ′ = 1 − exp(−µs ℓ) .

(9.7)

The length ℓ(ξ) associated with the pseudo-random number ξ is obtained as 1 − exp[−µs ℓ(ξ)] = ξ ,

(9.8)

from which ℓ(ξ) = −

ln(1 − ξ) . µs

(9.9)

From Eq. (9.1) and (9.2), the relations for θ(ξ) and ϕ(ξ) result in θ(ξ) Z F1 [θ(ξ)] = 2π p(θ′ ) sin θ′ dθ′ = ξ

(9.10)

ϕ(ξ) Z

(9.11)

0

and F2 [ϕ(ξ)] =

1 dϕ′ = ξ . 2π

0

Equation (9.11) can be easily inverted to obtain ϕ(ξ) = 2πξ. On the contrary, Eq. (9.10) can be inverted analytically only when simplified analytical models for p(θ), like the Henyey and Greenstein (HG) model,3 are considered. For more realistic scattering functions, like those obtained from Mie theory, Eq. (9.10) should be inverted numerically. The MC programs that we used can work with scattering functions generated both with the HG model and with Mie theory. The HG scattering function, defined as p(θ) =

1 − g2 1 , 4π (1 + g 2 − 2g cos θ)3/2

(9.12)

is completely characterized by the parameter g. It is possible to show that g corresponds to the asymmetry factor of the function, i.e., hcos θi = g, and that 2 hcos2 θi = 1+2g 3 . For g = 0, the function is isotropic. For g > 0, the function

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decreases monotonously as θ increases. The corresponding cumulative probability function is i h ( 1−g 2 1 1 − if g 6= 0 , 1/2 2g 1−g (1+g 2 −2g cos θ) (9.13) F1 (θ) = (1 − cos θ) /2 if g = 0 . This function can be inverted analytically to obtain the relation for θ(ξ). To work with Mie scattering functions, our MC programs make use of a table of scattering angles θi , for i = 1 to 8001, with 2π

Zθi 0

p(θ′ ) sin θ′ dθ′ = (i − 1)/8000

(9.14)

evaluated numerically by a program based on Mie theory. The scattering angle θ(ξ) is obtained by interpolating between the values of θi and θi+1 with i − 1 ≤ 8000 ξ ≤ i. The procedure used to simulate trajectories is based on a uniform random number generator within the interval (0, 1). Any actual algorithm for computer random number generators is characterized by a period, so that after long sequences of random numbers, the algorithm repeats the same sequence. As a consequence, there is a possibility that the generation of trajectories can get into a loop when the number of simulated trajectories is large. For our MC simulations, we used the algorithm ran2 proposed in Ref. 4 with a period greater than 2·1018 . For all the MC simulations we carried out, the sequence of random numbers used never exceeded the period of the random number generator. As long as the photons travel within the medium, their trajectories are simulated by iterating the aforementioned rules that associate three pseudo-random numbers with the pathlengths between two subsequent scattering events and with the new direction of propagation after each scattering event. Trajectories are followed until their length l either becomes longer than a limit value lmax fixed by the user or they leave the turbid medium. The choice of lmax determines the temporal range on which the TPSF can be reconstructed. Details of the rules used for non-homogeneous media, to include the effect of boundaries, and on the specific programs used for different geometries are given in subsequent sections. As mentioned before, the TPSF is obtained by classifying in a histogram the contributions of each trajectory according to its time of flight t = l/v (v is the speed of light in the turbid medium). The CW response for a steady-state source has been obtained by adding the contribution of all trajectories independently of the time of flight. Since trajectories have been broken when their length becomes longer than lmax , a careful check must be done to verify that the contribution of trajectories with l > lmax is negligible.

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For the validation of a MC simulator, it would be very useful to compare the results of simulations with results obtained from exact theories. Unfortunately, there are very few examples of exact solutions for photon migration, and they pertain to the total transmittance and total reflectance of a homogeneous slab obtained for simplified models of the scattering function.5, 6 For the validation of the programs, since the core of a MC program is actually the generation of the trajectories, we compared MC results, for the statistics of the position where different orders of scattering occur, with exact analytical expressions.7 Comparisons showed an excellent agreement: For all the scattering functions that we used (both from the HG model and from Mie theory), discrepancies were within the standard deviation of MC results, even when a large number of trajectories (108 ) was simulated, and the relative error was as small as 0.01%. As will be shown in Secs. 10.2.1 and 10.5.1, further validations have been provided by comparisons of MC results for the CW flux and the TR fluence with almost exact solutions of the RTE for the infinite medium. In order to develop a MC code, it is sufficient to know the probability laws for the trajectories. These laws come from the intrinsic physical meaning of the optical properties of the medium, i.e., the extinction coefficient and the scattering function. With a MC simulation, the RTE can be solved without the need to know the corresponding integro-differential equation, since the equation itself is not implemented in the MC code.

9.3 Monte Carlo Program for the Infinite Homogeneous Medium To study propagation through the infinite homogeneous medium illuminated by an isotropic point source that emits a light pulse at time t = 0, we simulated the response of receivers at different distances from the source. The code we used is basically the same described in Refs. 8 and 9. Both the TR and the CW response have been evaluated for the radiance, fluence, and flux. Since the source is isotropic, the problem has spherical symmetry, i.e., the response is identical in each point of a sphere concentric with the light source and only depends on the distance r from the source. Without losing any information, the response for an elementary receiver at distance rk can thus be obtained considering a spherical receiver of radius rk concentric with the source. To evaluate the radiance, the solid angle has been divided into 100 intervals. To obtain similar statistics in each angular interval, the intervals have been chosen to have the same number of received photons when an isotropic radiance is assumed. The j th interval of solid angle ∆Ωj is identified by αj , the angle averaged over ∆Ωj , between the direction sˆ and the unit vector outwardly directed rˆ normal to the surface. To obtain the radiance from the MC simulation, recall that the power flowing through the elementary surface dΣ of the sphere within the elementary solid angle dΩ around the direction sˆ is given [Eq. (3.5)] by I(r, sˆ, t)|ˆ r · sˆ|dΣdΩ. The TPSF for the radiance in the ith time

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interval and j th angular interval is obtained as Nk,j,i X 1 1 I (rk , αj , ti , µa = 0) = , 2 | cos βl | Ntot 4πrk ∆Ωj ∆ti

(9.15)

l=1

where Ntot is the total number of drawn trajectories, Nk,j,i is the number of trajectories that intersect the sphere of radius rk within the solid angle ∆Ωj with time of flight ti − ∆ti /2 ≤ t < ti + ∆ti /2, and βl is the angle between rˆ and the direction of motion of the photon when it intersects the sphere. Precautions have been taken to avoid division by a small number when βl approaches π/2. We observe that each trajectory can give multiple contributions to the radiance at different times and distances. According to the definition, the TPSF for the fluence is obtained adding the response for the radiance over the 100 angular intervals: Φ (rk , ti , µa = 0) =

100 X

I (rk , αj , ti , µa = 0) ∆Ωj .

(9.16)

j=1

To obtain the TPSF for the flux, again we exploit the symmetry of the problem, for which the only non-null component of the flux is the one along the radial direction that is obtained as Nk,i X cos βl 1 Jr (rk , ti , µa = 0) = , 2 | cos βl | Ntot 4πrk ∆ti

(9.17)

l=1

where Nk,i is the number of trajectories that intersect the sphere of radius rk within the time interval ∆ti . Being the trajectories evaluated for the non-absorbing medium, Eqs. (9.15), (9.16), and (9.17) represent the response when µa = 0. The response for the absorbing medium can be obtained, according to a general property of the RTE [Eq.(3.16)], by multiplying the response for the non-absorbing medium by the scaling factor exp(−µa vti ). According to Eq. (3.17), the CW response has been obtained integrating the TPSF. As an example, the CW radiance is I (rk , αj , µa ) =

Nt X

I (rk , αj , ti , µa = 0) exp(−µa vti )∆ti ,

(9.18)

i=1

where Nt is the number of temporal intervals, and ti is the average time over the ith interval. As mentioned in Sec. 9.2, due to the stochastic nature of the method, the results of MC simulations are noisy. Similar to a real experiment, we carried out independent MC simulations to evaluate the error (10 simulations for the results reported in the

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CD-ROM), all with the same number of simulated trajectories, which we used to evaluate both the average and the standard error (standard deviation of the average) for each quantity of interest. The TPSF for the radiance, fluence, and flux obtained from MC simulations can be directly compared with the Green’s functions provided by different analytical models.

9.4 Monte Carlo Programs for the Homogeneous and the Layered Slab The programs we used are basically that described in Refs. 10 and 11. To study propagation through the homogeneous or the layered slab, the rules to draw the trajectories described in Sec. 9.2 must be modified to include both the effect of the boundaries and of the non-homogeneity of the medium. Simulations have been carried out for a pencil beam source on the surface of the infinite slab at z = 0 that emits a light pulse at time t = 0 along the z-axis (the z-axis is perpendicular to the surface). To include the effect of boundaries, the incidence angle θi , the refraction angle θt , and the corresponding Fresnel reflection coefficient for unpolarized light RF (θi ) [Eq. (3.47)] are first calculated each time the trajectory hits the boundary of the slab (or of the current layer when a layered slab is considered). RF (θi ) represents the probability that the impinging light will be reflected. To determine whether reflection or refraction occurs, a pseudo-random number ξ is generated and compared with RF (θi ). If ξ ≤ RF (θi ), a reflection occurs, and the trajectory goes on in the same layer. The direction is updated according to the reflection law, and a new step size is calculated. If ξ > RF (θi ), a refraction occurs, and the new direction is calculated with Snell’s law. If refraction occurs on the external surface of the slab, the trajectory is terminated, and its contribution to the reflectance or to the transmittance is calculated; otherwise, a new step size is generated using the new scattering coefficient, and the trajectory goes on in the new layer. When a layered slab is considered, Eq. (9.9), which is used to generate the step size ℓ, should be modified to include the effect of the non-homogeneity of the medium. According to Eq. (9.3), when the step intersects volumes with a different scattering coefficient, the cumulative probability function for ℓ is  X  F3 (ℓ) = 1 − exp − µsi ℓi , (9.19) i

where ℓi and µsi are the length of the segment and theP scattering coefficient in the th i –volume, respectively. Therefore, the step size ℓ = ℓi is obtained from i

h X i 1 − exp − µsi ℓi (ξ) = ξ . i

(9.20)

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One obtains the new direction after a scattering event by using the scattering function of the layer where the scattering occurs. Trajectories are followed until they leave the slab or their length l becomes longer than a limit value lmax . To calculate the reflectance and the transmittance, trajectories leaving the slab are classified in histograms on the base of the exit point and of the time of flight. Since the source is a pencil beam perpendicular to the slab and the turbid medium is isotropic, both the reflectance and the transmittance only depend on the distance ρ from the light beam. Therefore, the receivers we have considered are rings coaxial with the pencil beam. For the homogeneous slab, the TPSF for the reflectance (for the transmittance similar expressions have been used) in the k th spatial interval and ith time interval has been obtained as R (ρk , ti , µa ) =



Nk,i

 exp(−µa vti ) , Ntot π ρ2kmax − ρ2kmin ∆ti

(9.21)

where Ntot is the total number of drawn trajectories, Nk,i is the number of trajectories exiting at z = 0 within ρkmin ≤ ρ < ρkmax , with direction within the acceptance solid angle of the receiver, and time of flight ti − ∆ti /2 ≤ t < ti + ∆ti /2. The exponential factor accounts for the effect of absorption when an absorbing medium is considered. To evaluate the error on the TPSF, for each spatial and temporal interval, we have also calculated the variance, from which the standard deviation σ has been obtained. Since each trajectory that intersects the receiver in the ith time interval gives the same contribution, the standard deviation results in12 R (ρk , ti , µa ) p σ (ρk , ti , µa ) ∼ . = Nk,i

(9.22)

In order to obtain a relative error of 0.01, it is necessary to simulate a number of trajectories sufficient to obtain Nk,i = 104 . Equation (9.22) is applicable for large values of Nk,i . The CW response has been obtained by integrating the TPSF as shown for the infinite medium. To calculate the response for the absorbing layered medium from simulations carried out for the non-absorbing slab, for each trajectory that intersects the receiver our MC code stores the partial pathlengths followed in each layer, and the TPSF is obtained using the property described in Sec. 3.4.2. The reflectance is obtained as R (ρk , ti , µa ) =

1

  Ntot π ρ2kmax − ρ2kmin ∆ti

Nk,i NL XY

exp(−µan vn tm,n ) , (9.23)

m=1 n=1

where µan , vn , and tm,n are, respectively, the absorption coefficient, the speed of light, and the partial time of flight of the mth –trajectory in the nth –layer, and summation is over the Nk,i trajectories that intersect the k th –receiver with total time of flight in the ith –time interval. NL is the number of layers.

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The partial time of flight tm,n has been also used to evaluate the TR mean time of flight spent inside the nth –layer by photons received at distance ρk and at time ti as PNk,i Q NL exp(−µan vn tm,n ) m=1 tm,n . (9.24) htn i (ρk , ti , µa ) = PN Q n=1 NL k,i m=1 n=1 exp(−µan vn tm,n ) As mentioned before, the MC results reported in the CD-ROM pertain to a pencil beam source. In the MC programs we used, the effect of the specular reflection of the pencil beam on the input surface is not considered (this is equivalent to assuming the source on the internal surface). Therefore, it is possible to directly compare the TPSF obtained from MC simulations with the corresponding Green’s function obtained from the DE or from hybrid models in which the response is provided for a ′ unit isotropic point source placed at depth z0 = 1/µs .

9.5 Monte Carlo Code for the Slab Containing an Inhomogeneity To simulate the effect of an inhomogeneity within the homogeneous or layered slab, a procedure similar to the one described for the layered slab can also be used: It is sufficient to include in the summation of Eq. (9.20) also the optical properties of the volume occupied by the inhomogeneity. The perturbation can be obtained from the comparison of two independent simulations carried out with and without the inhomogeneity. However, the expression for the standard deviation on the TPSF [Eq. (9.22)] clearly shows that to obtain the perturbation with a good accuracy a prohibitively large number of trajectories, with a prohibitively long computation time, may be necessary, especially if the perturbation is small. As an example, with reference to the reflectance, if R (ρk , ti , µa ) is evaluated with a relative error of 1%, the relative error on δR (ρk , ti , µa ) would be 20% when δR (ρk , ti , µa ) /R (ρk , ti , µa ) = 0.1, and 200% when δR (ρk , ti , µa ) /R (ρk , ti , µa ) = 0.01. To overcome this limit, we used a procedure, known as perturbation Monte Carlo,13 that makes use of scaling relationships to reduce the variance.14 With this method, we obtained, with a reasonable computation time, results suitable for comparisons with analytical solutions based on a perturbation approach. The MC procedure can be summarized in two steps: 1. In the first step, trajectories are generated with a conventional MC code for the non-absorbing slab—homogeneous or layered. For all trajectories that intersect the receiver within the acceptance angle, the information necessary to reconstruct the length of the path both within each layer and within the volume Vi of the inhomogeneity is stored together with the number of scattering events undergone within Vi . 2. In the second step, the stored information is used to obtain both the unperturbed TPSF and the perturbation. The unperturbed response is obtained as

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shown in the previous section [Eqs. (9.21) or (9.23) for the reflectance]. The absorption and the scattering perturbations are obtained taking into account that if the scattering and/or the absorption coefficients within Vi are changed from µs0 , µa0 to µsi , µai , the contribution to the TPSF of the mth trajectory changes from the unperturbed value wm0 to     µsi km wmp = wm0 exp − (µsi − µs0 + µai − µa0 ) vtm , (9.25) µs0

where v, km , and tm are the speed of light, the number of collisions undergone, and the time of flight within Vi , respectively. The terms that multiply wm0 in Eq. (9.25) represent the ratio between the probability of following the same trajectory after and before the introduction of the inhomogeneity, provided that the scattering function is not changed.14 Evaluating, for each time interval ∆ti , the average and the variance of the differences wmp − wm0 , it is possible to obtain the time-resolved perturbation together with its standard deviation. Furthermore, the time of flight tm within Vi can also be used to evaluate the mean time of flight spent by received photons inside the inhomogeneity. Although Eq. (9.25) is exact, the use of this MC procedure is limited by the noise on the results. As for absorbing perturbations, it has been shown in Ref. 14 that the procedure can be used for a wide range of values, both for the absorption coefficient and for the volume of the inhomogeneity. In contrast, due to the wide range of values that wmp − wm0 can assume when the scattering coefficient is changed, for scattering inhomogeneities the procedure provides reliable results only for sufficiently small variations of the scattering coefficient, provided that the volume of the inhomogeneity is not too small (small variations of the scattering coefficient limit the range of values for wmp − wm0 , while large volumes increase the number of involved trajectories improving the statistics). This MC procedure has an important feature that makes it particularly suitable for generating reference results to compare against analytical solutions based on a perturbation approach. In fact, since the effect of the inhomogeneity is evaluated as a variation of the probability to receive a finite set of trajectories (all those that intersect the volume of the inhomogeneity) when the inhomogeneity is introduced inside the medium, it is possible that the perturbation is statistically significant even when it is significantly smaller than statistical fluctuations on the unperturbed response. It is therefore possible to obtain reliable results for small perturbations of the optical properties for which the analytical models are expected to be accurate. The MC results reported in the CD-ROM have been obtained using two MC programs that differ in how they store trajectory information. For each received trajectory, the first program stores the coordinates of all the scattering points. The second stores only the NL + 1 lengths within the NL layers and within Vi together with the number of collisions within Vi . The first program has the advantage that, with the stored information, it is possible to evaluate the perturbation introduced by

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one or more inhomogeneities in whichever position within the slab, but the large amount of data that must be stored limits the number of simulated trajectories and then the accuracy of the results. The second program has the advantage of requiring less memory to store the information, and thus a larger number of trajectories can be simulated, but the perturbation can be evaluated only for the inhomogeneity for which the information has been stored.

9.6 Description of the Monte Carlo Results Reported in the CD-ROM In this section, a brief description is provided for the MC results reported in the M C_Results directory of the enclosed CD-ROM. In both the enclosed software and in the MC code, we assumed for the speed of light in vacuum the value c = 0.299792458 mm ps−1 .15 All the results are presented in Excel files (*.xls). A summary of the results is also provided in the index.html file, from which it is possible to open all the results files. 9.6.1 Homogeneous infinite medium The medium is illuminated by a point source at r = 0 that emits a light pulse at time t = 0. The results are summarized in the following files: • Infinite_CW_g=0.0.xls • Infinite_CW_g=0.9.xls • Infinite_TR_g=0.0.xls • Infinite_TR_g=0.9.xls The files Infinite_CW_g=0.0.xls and Infinite_CW_g=0.9.xls report the CW results for a medium with asymmetry factor g = 0.0 (isotropic scattering) and g = 0.9, respectively. Each file contains three worksheets: Fluence_CW, Flux_CW, and Radiance_CW in which are reported the responses for the fluence, flux, and radiance, respectively. The results for the fluence and for the flux are reported as a function of the source-receiver distance r for several values of the absorption coefficient. The results for the radiance are reported as a function of r and of the angle with the radial direction for µa = 0.001 mm−1 . The files Infinite_TR_g=0.0.xls and Infinite_TR_g=0.9.xls report the TR results for the scattering functions with g = 0.0 and g = 0.9. In each file, the worksheets Fluence_TR and Flux_TR report the TPSFs for the fluence and the flux, respectively, for several values of r. Three worksheets are devoted to the results for the TR radiance for r = 1, 5, and 10 mm. The TPSFs pertain to a non-absorbing medium. The

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results have been reported focusing on short times for which the larger differences are expected when comparisons with results of approximated theories are made. ′ All the results are reported for µs = 1 mm−1 and refractive index n = 1. The ′ results for different values of µs can be easily obtained making use of the scaling properties of the RTE reported in Sec. 3.4.1. 9.6.2 Homogeneous slab The results are summarized in the following files: • s=40mm,n_i=1.4,n_e=1.4,g=0.0.xls • s=40mm,n_i=1.4,n_e=1.4,g=0.9.xls • s=40mm,n_i=1.4,n_e=1.0,g=0.0.xls • s=40mm,n_i=1.4,n_e=1.0,g=0.9.xls • s=4mm,n_i=1.4,n_e=1.4,g=0.9.xls • s=4mm,n_i=1.4,n_e=1.0,g=0.9.xls The values for the thickness of the slab s for the refractive index inside the slab n_i, and in the external medium n_e, and for the asymmetry factor of the scattering function g are indicated in the name of the file. The source is a pencil beam normally incident onto the slab emitting a light pulse at time t = 0. The worksheets CW_Refl and CW_Transm report the CW reflectance and the CW transmittance as a function of the distance of the receiver from the light beam axis for some values of the absorption coefficient. The acceptance solid angle of receivers is 2π (all photons impinging the receiver are detected). With the MC simulations, trajectories have been broken when their length becomes longer than a limit value lmax , and attention must be paid when using the CW results for a slab with s = 40 mm for small values of absorption (µa = 0.001 mm−1 ) and large distances from the light beam axis (ρ & 40 mm) since they could be underestimated. The worksheets TR_Refl and TR_Transm report the TPSFs for the reflectance and the transmittance of the non-absorbing slab together with the corresponding standard deviation evaluated with Eq. (9.22). Since Eq. (9.22) cannot be used for Nk,i = 0, in the results reported in the CD-ROM, we have set σ = 1030 when Nk,i = 0 to underscore that the result is not significant from a statistical point of view. The worksheet Total Refl&Transm summarizes the results for the total reflectance and total transmittance both for the CW and the TR response. We point out that the CW response for an arbitrary value of absorption can be obtained from the TPSF for the non-absorbing medium by using Eq. (3.17).

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9.6.3 Layered slab The results are summarized in the following files: • Lay_Scat_Mus1=1_Mus2=0.5.xls • Lay_Scat_Mus1=1_Mus2=2.xls • Lay_Abs_Mua1=0_Mua2=0.01.xls • Lay_Abs_Mua1=0.01_Mua2=0.xls • Lay_Refr_n1=1_n2=1.4.xls • Lay_Refr_n1=1.4_n2=1.xls All files pertain to the TR reflectance from a two-layer slab illuminated by a pencil beam. Results are reported for three receivers with acceptance solid angle = 2π at distances ρ = 10, 20, and 30 mm. Each file contains two worksheets: The worksheet TPSF reports the TPSF for the reflectance; the worksheet ht_ii(t) reports the TR mean time of flight spent by received photons inside the layers. The files Lay_Scat_Mus1=1_Mus2=0.5.xls and Lay_Scat_Mus1=1_Mus2=2.xls ′ pertain to slabs with a layered architecture of scattering: They have µs2 = 0.5 and ′ 2 mm−1 , respectively. The other parameters are µs1 = 1 mm−1 , g1 = g2 = 0, µa1 = µa2 = 0, n1 = n2 = 1.4, ne = 1, s1 = 4 mm, and s2 = 100 mm. Slabs with a layered architecture of absorption are instead considered in the files Lay_Abs_Mua1=0_Mua2=0.01.xls and Lay_Abs_Mua1=0.01_Mua2=0.xls: They have µa1 = 0 and µa2 = 0.01 mm−1 , and µa1 = 0.01 mm−1 and µa2 = ′ ′ 0, respectively. The other parameters are µs1 = µs2 = 1 mm−1 , g1 = g2 = 0, n1 = n2 = 1.4, ne = 1, s1 = 4 mm, and s2 = 100 mm. The files Lay_Refr_n1=1_n2=1.4.xls and Lay_Refr_n1=1.4_n2=1.xls pertain to slabs with a layered architecture of refractive index: They have n1 = 1 and n2 = ′ ′ 1.4, and n1 = 1.4 and n2 = 1, respectively. The other parameters are µs1 = µs2 = 1 mm−1 , g1 = g2 = 0, µa1 = µa2 = 0.01 mm−1 , ne = 1, s1 = 8 mm, and s2 = 100 mm. 9.6.4 Perturbation due to inhomogeneities inside the homogeneous slab The results are summarized in the following files: • Pert_Refl_s=4000mm.xls • Pert_Transm_s=40mm.xls • Scan_Pert_Refl_s=4000mm.xls • Scan_Pert_Transm_s=40mm.xls

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• Pert_Refl_s=4mm.xls The file Pert_Refl_s=4000mm.xls reports the TR absorption and the TR scattering perturbations for reflectance from a slab with thickness s = 4000 mm (semi′ infinite medium), µs0 = 0.5 mm−1 , g = 0, µa0 = 0, and ni = ne = 1.4. The source is a pencil beam normally incident at (0, 0, 0). The receiver is circular with a radius of 2 mm and an acceptance solid angle of 2π centered at (30, 0, 0) mm. The inhomogeneity is spherical and centered at (15, 0, 15) mm. The results are reported for different values of the volume of the inhomogeneity: Vi = 1, 10, 100, 500, and 1000 mm3 in the worksheets V = 1, V = 10 , V = 100, V = 500, and V = 1000, respectively. Each worksheet reports both the absorption and the scattering perturbation together with the TPSF for the unperturbed medium. The file Pert_Transm_s=40mm.xls reports results for the transmittance of a slab with s = 40 mm and the same optical properties of file Pert_Refl_s=4000mm.xls. Both the receiver, circular with a radius of 3 mm, and the spherical inhomogeneity, at (0, 0, 20) mm, are coaxial with the light beam. Also, the results for transmittance are reported in different worksheets for Vi = 1, 10, 100, 500, and 1000 mm3 . The file Scan_Pert_Refl_s=4000mm.xls reports examples of perturbation for the same medium and the same receiver of file Pert_Refl_s=4000mm.xls. Results are reported for the scanning of an inhomogeneity with Vi = 500 mm3 along the xand the z-axis. The results for the absorption and for the scattering perturbation are reported in worksheets Absorption and Scattering, respectively. Similarly, the file Scan_Pert_Transm_s=40mm.xls reports the scanning for the same medium and the same receiver of file Pert_Transm_s=40mm.xls. The file Pert_Refl_s=4mm.xls reports the perturbations for reflectance from a ′ slab with s = 4 mm, µs0 = 1 mm−1 , g = 0.9, µa0 = 0, and ni = ne = 1.4. The source is at (0, 0, 0), and the receiver, circular with a radius of 0.2 mm and an acceptance solid angle of 2π, is at (3,0,0) mm. The spherical inhomogeneity with Vi = 1 mm3 is at (1.5, 0, 1.5) mm. For all the files, the absorption perturbation has been evaluated using µai such ′ ′ that Vi (µai − µa0 ) = 0.01 mm2 and µsi = µs0 , and the scattering perturbation using ′ ′ ′ µsi such that Vi (Di − D0 ) = 0.01 mm4 [with Di = 1/(3µsi ) and D0 = 1/(3µs0 )] ′ and µai = µa0 . With these values of µai and µsi , perturbations are sufficiently small to be predictable with the Born approximation. The MC results are thus suitable for comparisons with the analytical expression from the perturbation theory of Chapter 7.

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References 1. W. Becker, Advanced Time-correlated Single Photon Counting Techniques, Springer, Berlin (2005). 2. L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging, John Wiley & Sons, New York (2007). 3. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941). 4. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge (1988). 5. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications, Academic Press, New York (1980). 6. R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2, 153–162 (1955). 7. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994). 8. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point-source case,” Appl. Opt. 30, 2031–2041 (1991). 9. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: Numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373 (2000). 10. F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. 36, 4600–4612 (1997). 11. F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A 19, 71–80 (2002). 12. S. Brandt, Statistical and Computational Methods in Data Analysis, NorthHolland, Amsterdam (1970). 13. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).

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14. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998). 15. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge (1980).

Chapter 10

Comparisons of Analytical Solutions with Monte Carlo Results 10.1 Introduction In this chapter, we detail comparisons between the solutions of the RTE described in Chapters 4–7 and the reference MC results reported in the enclosed CD-ROM. All of the solutions based on the diffusion approximation used for comparisons have been calculated using the code and the subroutines included in the CD-ROM. Comparisons with MC results thus provide a validation of the enclosed software. Furthermore, since almost all the analytical solutions presented are approximate solutions of the RTE, comparisons also provide information on their accuracy and range of applicability. As validation of the Diffusion&Perturbation software, comparisons are shown in Sec. 10.2 for the homogeneous infinite medium, the homogeneous slab, and the homogeneous slab with an internal inhomogeneity. Comparisons for the layered slab and the hybrid models are reported in Secs. 10.4 and 10.5, respectively. For the infinite medium, both the MC and the analytical solutions pertain to a point-like isotropic source. For the slab, MC results pertain to a pencil beam. The corresponding analytical results, according to the discussion in Sec. 4.2, pertain to an isotropic point-like source inside the medium at a distance ′ zs = 1/µs from the entrance point of the beam. All the figures of this chapter, apart from Fig. 10.31 (the MC results used for this figure have not been reported in the CD-ROM), can be reproduced using the software and the MC results provided with the enclosed CD-ROM. Comparisons of analytical models with MC results are shown through the ratio MC/MODEL. The results are presented without any filtering even though some TR curves are very noisy. We preferred this representation because the noise gives information on the accuracy of the reference MC results, and because a filtering procedure can severely distort the response at early times where the larger discrepancies between the results from analytical models and reference results are expected. In the figures, the error bars on the MC results have not been reported, 203

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Chapter 10

although the standard deviations are available. Nevertheless, information of the accuracy of MC results is provided by the noise on the curves, which is on the order of the standard deviation. All the figures with TR results will be focused on the response at short times, i.e., times at which the larger differences between the results from the DE and the reference MC results are expected.

10.2 Comparisons Between Monte Carlo and the Diffusion Equation: Homogeneous Medium In this section, we compare MC results with solutions of the DE. 10.2.1 Infinite homogeneous medium All the results reported pertain to a point source at r = |~r| = 0 that emits a light ′ pulse at time t = 0 in a medium with µs = 1 mm−1 and refractive index n = 1. The TR results are shown only for the non-absorbing medium. However, as for the ratios between DE and MC results used for comparisons, we point out that the ratios are independent of µa since both MC and DE results depend on µa by the factor exp(−µa vt). Results and comparisons are shown for the medium both with an isotropic (g = 0) and anisotropic (g = 0.9) scattering function. The solutions of the DE for the fluence have been calculated with the Diffusion&Perturbation code. Flux and radiance have been obtained using the relevant subroutines described in Sec. 4.1. 10.2.1.1 Time-resolved results

Examples of TR results are reported in Figs. 10.1–10.4. Figures 10.1 and 10.2 show examples of TPSFs obtained from MC simulations for the fluence rate and for the flux, respectively. The corresponding results from the DE are shown in Fig. 4.1. The comparison of MC results with the solution of the DE is shown plotting the ratio MC/DE versus time. To look for a general rule to establish the accuracy of the DE, the ratio is also plotted as a function of t/t0 , where t0 = r/v is the time of flight for ballistic photons. We notice that, for the results reported in the figures, t0 varies from 3.3 ps to 66.7 ps when r varies from 1 to 20 mm. Comparison between MC results for g = 0 and 0.9 shows almost identical results, as expected by the similarity relation presented in Sec. 2.4. Differences are appreciable (the response is significantly larger for g = 0) only for early received photons (photons received after a smaller number of scattering events). Comparisons of Figs. 10.1 and 10.2 show that, apart from early received photons, both the fluence and the flux from the DE are in excellent agreement with MC results. Discrepancies also remain for early photons and large distances, for which even early photons are expected to have experienced a large number of scattering events. As an example, since the average number of scattering events for early photons is r/ℓs = rµs , about 200 scattering events are expected at r = 20 mm for the medium with g = 0.9. The agreement is slightly better with the results for g = 0.9, i.e., when

205

Comparisons of Analytical Solutions with Monte Carlo Results

-1

-1

10

10 r = 1 mm

-2

10

r = 5 mm

)

-1

)

ps

10

r = 20 mm

Fluence (mm

-4

10

-5

10

-6

10

-7

10

0

10

r = 20 mm

-4

10

-5

10

-6

10

-7

g = 0.9

10

g = 0

-8

10

r = 10 mm

-3

-2

-1

ps

r = 10 mm

-3

-2

Fluence (mm

r = 1 mm

-2

10

r = 5 mm

-8

200

400

600

10

800

0

200

1.20

1.20

1.15

1.15

1.10

1.10

1.05 1.00 0.95 0.90 0.85 0.80 0

800

600

800

15

20

1.00 0.95 0.90 0.85

200

400

600

0.80 0

800

200

400

Time (ps)

1.20

1.20

1.15

1.15

1.10

1.10

Fluence MC/DE

Fluence MC/DE

600

1.05

Time (ps)

1.05 1.00 0.95 0.90 0.85 0.80 0

400 Time (ps)

Fluence MC/DE

Fluence MC/DE

Time (ps)

1.05 1.00 0.95 0.90 0.85

5

10

15

20

t/t

0.80 0

5

10

t/t

0

0

′

Figure 10.1 Infinite medium with µs = 1 mm−1 , µa = 0, and n = 1. Examples of TPSFs for the fluence rate from MC simulations (first row) and comparisons with the DE. Comparisons are shown plotting the ratio MC/DE as a function both of t and of t/t0 (second and third row, respectively). Results are reported both for g = 0 and 0.9 (left and right column, respectively).

206

Chapter 10

-1

-1

10

10 r = 1 mm

-2

10

)

-5

10

-6

10

-7

10

-8

r = 20 mm

10

-5

10

-6

10

-7

10

-9

200

400

600

800

10

0

200

1.4

1.4

1.3

1.3

1.2

1.2

1.1 1.0 0.9

0.7

600

0.6 0

800

200

1.4

1.4

1.3

1.3

1.2

1.2

1.1 1.0 0.9

800

15

20

1.1 1.0 0.9

0.8

0.8

0.7

0.7

10

400

Time (ps)

Flux MC/DE

Flux MC/DE

Time (ps)

5

600

0.9 0.8

0.6 0

800

1.0

0.7

400

600

1.1

0.8

200

400 Time (ps)

Flux MC/DE

Flux MC/DE

Time (ps)

0.6 0

r = 10 mm

-4

10

-9

0

g = 0.9

-8

10 10

10

-1

ps

-2

10

Flux (mm

ps

-2

Flux (mm

r = 20 mm

-4

r = 5 mm

-3

r = 10 mm

-1

)

-3

10

r = 1 mm

-2

10

r = 5 mm

g = 0

15

20

0.6 0

5

t/t

0

Figure 10.2 Same as Fig. 10.1, but for the flux.

10

t/t

0

207

Comparisons of Analytical Solutions with Monte Carlo Results

the scattering function is forward peaked, which usually occurs for light propagation through actual turbid media. For r & 5 mm and t/t0 &4, discrepancies for g = 0.9 remain within ≃ 2%. For shorter distances, discrepancies remain within ≃2% for ′ ′ ′ t & 4ℓ /v (ℓ = 1/µs is the transport mean free path). 3

3 r = 1 mm r = 5 mm r = 10 mm

2

2

I1

I1

r = 20 mm

g = 0.9

1

0 0

1

100

200

300

0 0

400

5

10

15

20

15

20

t/t

Time (ps)

0

4

0.8

3

0.6

I2

1.0

I2

5

2

0.4

1

0.2

0 0

100

200

300

400

0.0 0

5

10

t/t

Time (ps)

0

′

Figure 10.3 Infinite medium with µs = 1 mm−1 , µa = 0, g = 0.9, and n = 1. The MC results for the ratios I1 and I2 [Eqs. (10.1) and (10.2)] are reported as a function both of t and of t/t0 .

Since the simplifying assumptions necessary to obtain the DE (Sec. 3.5.1) can be summarized by the two inequalities h i I1 = 3|J~ (~r, t) | /Φ (~r, t) ≪ 1 (10.1) and

1 I2 = vµ′s

∂ J~ (~r, t) ~ / J (~r, t) ≪ 1 , ∂t

(10.2)

208

Chapter 10

it may be interesting to show how I1 and I2 change as t increases. The MC results for I1 and I2 are reported in Fig. 10.3 as a function both of t and t/t0 . The figure reports the results for g = 0.9. For I1, the curves for the different distances are almost indistinguishable when plotted as a function of t/t0 and show that I1 . 0.3 when t/t0 & 4. For I2, the curves remain distinguishable and show that when t/t0 & 4, apart from the shortest distance, I2 . 0.1. These values of I1 and I2 can thus be indicated as limit values to fulfill the inequalities. The rapid decrease of I1 as t increases shows how more quickly the flux decreases with respect to the fluence. -5

-1

10

10

)

-1

sr

10

r = 1 mm g = 0.9 -4

10

10

-5

0

g = 0.9 -6

10

-2

= 174.3 deg -3

10

r = 10 mm

ps

= 135.6 deg

-2

Radiance (mm

-1

= 84.3 deg

ps

-1

sr

-1

= 45.6 deg -2

10

Radiance (mm

)

= 5.7 deg

= 5.7 deg -7

= 45.6 deg = 84.3 deg = 135.6 deg = 174.3 deg

-8

20

40

60

80

10

100

0

200

Time (ps)

600

800

1000

Time (ps)

2.0

2.0

t/t 1.6

t/t t/t

1.4

0 0 0 0

= 2

t/t

1.8

= 4.1

t/t

Radiance MC/DE

t/t

1.8

Radiance MC/DE

400

= 8 = 16.1

1.2

1.0 r = 1 mm

0.8

1.6

t/t t/t

1.4

0 0 0 0

= 2 = 4.1 = 8 = 16.1

r = 10 mm

1.2

g = 0.9 1.0

0.8

g = 0.9 0.6 0

45

90

135

180

(deg)

0.6 0

45

90

135

180

(deg)

′

Figure 10.4 Infinite medium with µs = 1 mm−1 , µa = 0, g = 0.9, and n = 1. Examples of TPSF for the radiance from MC simulations and comparisons with the DE. The TPSFs are reported for some values of α for r = 1 and 10 mm. Comparisons are shown plotting the ratio MC/DE as a function of α for some values of t/t0 .

Examples of MC results for the radiance and comparisons with the DE are reported in Fig. 10.4. The TPSFs are reported for some values of the angle α between the directions sˆ and rˆ, for r = 1 and 10 mm. Comparisons are shown

209

Comparisons of Analytical Solutions with Monte Carlo Results

plotting the ratio MC/DE as a function of α for some values of t/t0 . Results are reported for g = 0.9. Also, for the radiance, we observe that results from the DE are in excellent agreement with MC results provided t/t0 & 4 for r & 5 mm and t & ′ 4ℓ /v for shorter distances. We point out a feature of the figures showing comparisons between DE and MC results that is also common to figures reported in subsequent sections for other geometries: As the comparisons reported use the ratio between MC and DE results, the information on the response for t < t0 is partially lost since the corresponding MC result is 0 while the results for DE is 6= 0. 10.2.1.2 Continuous wave results

Examples of CW MC results and comparisons with the DE solutions are reported in Figs. 10.5–10.8. Results are shown both for g = 0 and g = 0.9.

1

1

10

10

0

0

10

-1

-1

-2

-2

)

g = 0

ps

10 10

Fluence (mm

Fluence (mm

-2

ps

-1

)

10

-3

10

-1

-4

10

a

-1

a

-5

10 10

= 0.005 mm

-1

a

-6

= 0.01 mm

-1

a

= 0.05 mm

0

a

10

-2

10

-3

10

-1

-4

10 10

10

20

= 0.01 mm

-1

a

= 0.05 mm

-1

-7

15

= 0.005 mm

-1

a

10

10

= 0.001 mm

-1

a

= 0.1 mm

5

a

-5 -6

-1

-7

10

= 0.001 mm

g = 0.9

-1

0

a

= 0.1 mm

5

1.10

1.10

1.05

1.05

1.00

0.95

0.90

0.85

0.80 0

10

15

20

15

20

r (mm)

Fluence MC/DE

Fluence MC/DE

r (mm)

1.00

0.95

0.90

0.85

5

10

r (mm)

15

20

0.80 0

5

10

r (mm)

′

Figure 10.5 MC results for the fluence for an infinite medium with µs = 1 mm−1 and comparison with DE results. The fluence is reported as a function of r for some values of µa both for g = 0 and 0.9.

210

Chapter 10

1

1

10

10 -1

a

0

10

a

= 0.01 mm

10 )

-2

10

= 0.05 mm

-1

a

= 0.1 mm

-3

10

-4

10

-5 -6

0

10

a

-3 -4

10

-6

g = 0.9

-7

5

10

15

10

20

0

5

10

15

20

15

20

15

20

r (mm)

1.10

1.10

1.05

1.05

Flux MC/DE

Flux MC/DE

= 0.1 mm

10

10

1.00

0.95

0.90

0.85

1.00

0.95

0.90

0.85

5

10

15

0.80 0

20

5

10

r (mm)

r (mm)

1.002

1.002

1.000

1.000

0.998

0.998

Flux MC/DE

Flux MC/DE

= 0.05 mm

-1

r (mm)

0.80 0

= 0.01 mm

-1

a

10 g = 0

-7

10

a

-2

-5

10 10

= 0.005 mm

-1

-1

-1

a

-2

= 0.001 mm

-1

a

Flux (mm

)

10 -2

= 0.005 mm

-1

-1

a

0

10

-1

a

Flux (mm

-1

= 0.001 mm

0.996 a

0.994

a

0.992 a

= 0.001 mm = 0.005 mm = 0.01 mm

-1

-1

0.996

0.994

0.992

-1

0.990

0.990

0.988 0

0.988 0

a

a

a

5

10

15

20

= 0.001 mm = 0.005 mm = 0.01 mm

5

r (mm)

Figure 10.6 Same as Fig. 10.5, but for the flux.

-1

-1

-1

10

r (mm)

211

Comparisons of Analytical Solutions with Monte Carlo Results

Figures 10.5 and 10.6 show MC results for the fluence and for the flux, respectively, as a function of r for some values of µa . The corresponding results from the DE are shown in Fig. 4.2. Also, CW results for g = 0 and 0.9 are almost identical. Differences are only appreciable observing the ratio with DE results. For the fluence, discrepancies remain within 1% for µa . 0.01 mm−1 and r & 2 mm. The agreement is slightly better with the results for g = 0.9 for small values of µa and r, and vice versa for large values of µa and r. For r & 2 mm and large values of absorption, the DE significantly overestimates the CW response. For large values of absorption, the flux is also significantly overestimated by the DE, but for small values agreement with MC results is almost exact at all the distances and for both values of g. To underscore this result, Fig. 10.6 reports separate comparisons for small values of absorption (µa ≤ 0.01 mm−1 ). From this figure, we observe that for µa = 0.001 mm−1 discrepancies remain smaller than 0.1%. This exceptionally good agreement is not surprising since, as we pointed out in Sec. 4.1 and Appendix E, for a non-absorbing
medium, the solution of the DE for the flux, which becomes 2 J (r) = 1/ 4πr , represents the exact solution of the RTE and is independent of the scattering properties of the medium. Unfortunately, the MC code we have used (see Sec. 9.3) cannot provide the CW response for µa = 0 (it would be necessary to follow each trajectory for an infinitely long time). Nevertheless, the excellent agreement of MC results for small values of absorption with the almost exact solution of the DE can be considered as a further validation of the MC code we have used. 2.0

2.0 -1

1.8

a

-1

1.6

a

1.4

a

= 0.005 mm

-1

= 0.01 mm

-1

1.8

= 0.001 mm

a

-1

1.6

a

1.4

a

g = 0

a

= 0.05 mm

0.8

0.6

0.6

0.4

0.4

0.2

0.2

5

10

r (mm)

g = 0.9

15

20

0.0 0

= 0.05 mm

-1

a

1.0

0.8

0.0 0

a

1.2

= 0.1 mm

I3

I3

1.0

= 0.01 mm

-1

-1

a

= 0.005 mm

-1

-1

1.2

= 0.001 mm

5

= 0.1 mm

10

15

20

r (mm)

Figure 10.7 The MC results for the ratio I3 [Eq. (10.3)] are reported as a function of r for an ′ infinite medium with µs = 1 mm−1 , for some values of µa , both for g = 0 and 0.9.

Also, for CW results, we report examples of MC results to understand when we can consider the inequality fulfilled: I3 = 3|J~ (~r) |/Φ (~r) ≪ 1 , (10.3) which represents the simplifying assumption necessary to obtain the DE for CW sources. Examples of results are shown in Fig. 10.7 in which I3 has been plotted

212

Chapter 10

0

0

10

10

r = 0.5 mm

r = 0.5 mm

r = 1 mm

)

-1

r = 5 mm

-1

10

sr

r = 10 mm

-2

10

-3

10

-4

10

0

r = 5 mm r = 10 mm r = 20 mm

-2

10

-3

10

-4

45

90

135

10

180

0

45

(deg)

2

r = 0.5 mm

r = 2 mm

0

10

1

10

r = 0.5 mm r = 1 mm r = 2 mm

0

10

g = 0

g = 0.9

-1

0

-1

45

90

135

10

180

0

45

(deg)

g = 0

180

135

180

g = 0.9

Radiance MC/DE

Radiance MC/DE

135

1.2

1.1

1.0

0.8 0

90 (deg)

1.2

0.9

180

2

r = 1 mm

10

135

10

Radiance MC/DE

Radiance MC/DE

1

90 (deg)

10

10

r = 2 mm

-1

10

-2

r = 20 mm

Radiance (mm

Radiance (mm

r = 1 mm

g = 0.9

r = 2 mm

-2

sr

-1

)

g = 0

r = 5 mm

1.1

1.0

0.9

r = 5 mm

r = 10 mm

r = 10 mm

r = 20 mm

r = 20 mm

45

90

135

180

(deg)

0.8 0

45

90

(deg)

′

Figure 10.8 Infinite medium with µs = 1 mm−1 and µa = 0.001 mm−1 . Examples of MC results and comparisons with the DE for the CW radiance. Results are reported as a function of α for some values of r.

Comparisons of Analytical Solutions with Monte Carlo Results

213

as a function of r for some values of µa both for g = 0 and 0.9. The figure shows that for values of µa . 0.01 mm−1 and r & 2 mm, for which discrepancies remain within 1% for the fluence and the flux, I3 . 0.6. Examples of MC results for the CW radiance and comparisons with the DE are reported in Fig. 10.8. The radiance is reported for several values of r as a function of the angle α for a medium with µa = 0.001 mm−1 for g = 0 and 0.9. Also, for the CW radiance, there is an excellent agreement with MC results provided r & 2 mm. For short distances, there are significant discrepancies, especially near α = 180 degrees, where the DE also provides negative values.

10.2.2 Homogeneous slab All the MC results reported pertain to a pencil beam source on the surface of the slab that emits a light pulse at time t = 0. The corresponding DE results pertain to ′ an isotropic point source at zs = 1/µs , and the outgoing flux has been calculated ′ with Fick’s law. Results are reported for slabs with µs = 1 mm−1 and internal refractive index ni = 1.4. Two values, 1 and 1.4, have been considered for the relative refractive index nr = ni /ne (ne is the refractive index of the external medium). Results are shown for a scattering function with asymmetry factor g = 0.9 representative of many real media.

10.2.2.1 Time-resolved results

Examples of TR results are reported in Figs. 10.9–10.11. Figure 10.9 reports comparisons between the TPSFs for the reflectance from the DE and from MC simulations for three values of the source-receiver distance ρ for a slab 40 mm thick. Figure 10.10 reports the results for the transmittance. Receivers considered in MC simulations are rings coaxial with the pencil light beam source (ρ indicates the average radius of the ring). The DE results, obtained from Eqs. (4.32) and (4.33), pertain to receivers identical to those used for MC simulations. The ratio between MC and DE results is also plotted as a function of t/t0 (t0 is the ballistic time to cover the source-receiver distance). For transmittance discrepancies between DE and MC, results are significant only for early times and involve a very small fraction of transmitted photons. For reflectance, there are discrepancies, especially at small distances and early times (notably for times shorter than the time of flight for ballistic photons t0 ), that can involve a significant fraction of received photons. For late times, there is an almost exact agreement when nr = 1.4; while for nr = 1, the DE underestimates the results of about 2%. Figure 10.11 reports results for the total TR reflectance and the total TR trans-

214

Chapter 10

-4

1.5

10

1.4

= 5.1 mm

1.3

-5

10

MC/DE n = 1.4

MC/DE

Reflectance (mm

-2

ps

-1

)

= 5.1 mm

MC n = 1.4 r

-6

10

MC n = 1

r

1.2

MC/DE n = 1 r

1.1

1.0

r

DE n = 1.4 r

0.9

DE n = 1 r

-7

10

0

100

0.8 0

200

10

20

t/t

Time (ps)

-6

30

0

1.2

10

-1

)

= 10.2 mm

= 10.2 mm

MC/DE

Reflectance (mm

-2

ps

1.1

-7

10

MC n = 1.4 r

1.0

MC n = 1 r

0.9

DE n = 1.4

MC/DE n = 1.4

DE n = 1

MC/DE n = 1

r

r

r

r

-8

10

0

100

200

300

0.8 0

400

10

20

t/t

Time (ps)

-7

30

0

1.1

10

-8

10

1.0

MC/DE

Reflectance (mm

-2

ps

-1

)

= 20.6 mm

-9

10

MC n = 1.4 r

= 20.6 mm 0.9

MC n = 1 r

-10

10

DE n = 1.4

MC/DE n = 1.4

DE n = 1

MC/DE n = 1

r

r

r

r

-11

10

0

200

400

Time (ps)

600

0.8 0

10

20

t/t

30

40

0

′

Figure 10.9 TR reflectance of a homogeneous slab with s = 40 mm, µs = 1 mm−1 , g = 0.9, µa = 0, and ni = 1.4. Comparison between DE and MC results are shown for different values of ρ and nr .

215

Comparisons of Analytical Solutions with Monte Carlo Results

mittance of slabs with s = 40 and 4 mm. For s = 4 mm, the results are reported both for nr = 1 and 1.4; for s = 40 mm results are reported only for nr = 1.4. There is an excellent agreement between DE and MC results for s = 40 mm. Discrepancies are appreciable only for early times and involve only a small fraction of reflected or transmitted photons. For s = 4 mm, there are appreciable differences in the shape of the curves, especially for nr = 1.4. However, as will be shown in Fig. 10.13, an excellent agreement remains for the CW results.

-8

1.2

-1

)

10

-9

1.0

10

MC/DE

Transmittance (mm

-2

ps

1.1

= 1.5 mm

MC n = 1.4 r

-10

10

0.9 = 1.5 mm 0.8

MC n = 1 r

DE n = 1.4 r

0.7

DE n = 1

r

-11

0

2000

4000

r

MC/DE n = 1

r

10

MC/DE n = 1.4

0.6 0

6000

10

t/t

Time (ps)

-9

20

0

1.2

-1

)

10

Transmittance (mm

-2

ps

1.1

-10

1.0

MC/DE

10

= 39.9 mm

MC n = 1.4 r

-11

10

0.9 = 39.9 mm 0.8

MC n = 1 r

DE n = 1.4 r

0.7

DE n = 1

r

-12

0

2000

4000

Time (ps)

r

MC/DE n = 1

r

10

MC/DE n = 1.4

6000

0.6 0

10

t/t

20

0

′

Figure 10.10 TR transmittance of a homogeneous slab with s = 40 mm, µs = 1 mm−1 , g = 0.9, µa = 0, and ni = 1.4. Comparison between DE and MC results are shown for different values of ρ and nr .

216

Chapter 10

-1

1.6

10

MC Total Reflectance -2

MC/DE Total Reflectance 1.4

MC Total Transmittance

10

MC/DE Total Transmittance

DE Total Reflectance

MC/DE

) R, T (ps

10

-1

DE Total Transmittance

-3

10

1.2

-4

1.0

0.8

-5

10

0.6 s = 40 mm

-6

10

s = 40 mm

n = 1.4

-7

10

0

2500

n = 1.4

0.4

r

5000

7500

0.2 0

10000

r

2500

Time (ps)

10-1

7500

10000

1.6

MC Total Reflectance MC Total Transmittance DE Total Reflectance DE Total Transmittance

10-2

1.4

s = 4 mm

1.2

n = 1.4

10-3

MC/DE

R, T (ps-1)

5000

Time (ps)

10-4

r

1.0

0.8

0.6

s = 4 mm nr = 1.4

10-5

0.4

MC/DE Total Reflectance MC/DE Total Transmittance

10-6 0

100

200

300

400

500

0.2 0

600

100

200

Time (ps)

300

400

500

600

Time (ps)

-1

1.4

10

MC Total Reflectance

s = 4 mm

MC Total Transmittance

-2

10

n = 1 r

DE Total Reflectance

1.2

-3

10

MC/DE

R, T (ps

-1

)

DE Total Transmittance

-4

10

-5

10

1.0

0.8

s = 4 mm n = 1

MC/DE Total Reflectance

r

MC/DE Total Transmittance -6

10

0

100

200 Time (ps)

300

400

0.6 0

100

200

300

400

Time (ps)

Figure 10.11 Comparison between DE and MC results for the total TR reflectance and ′ transmittance of homogeneous slabs with s = 40 and 4 mm, µs = 1 mm−1 , g = 0.9, µa = 0, and ni = 1.4. Results for s = 4 mm are shown both for nr = 1 and 1.4.

217

Comparisons of Analytical Solutions with Monte Carlo Results

10.2.2.2 Continuous wave results

-1

1.4

10

MC/DE Transmittance n = 1

MC Transmittance n = 1

r

r

-2

1.3

MC Transmittance n = 1.4

10

r

MC/DE Transmittance n = 1.4 r

MC/DE Reflectance n = 1

MC Reflectance n = 1 r

-3

r

1.2

MC Reflectance n = 1.4

10

MC/DE Reflectance n = 1.4

r

r

-4

MC/DE

Reflectance, Transmittance (mm

-2

)

Comparisons between DE and MC results are reported in Figs. 10.12 and 10.13. Figure 10.12 reports the results for the CW reflectance and transmittance as a function of ρ for µa = 0.01 mm−1 . For transmittance, discrepancies remain within 2%; for reflectance, they are within 3% only for ρ > 10 mm. Discrepancies increase as the absorption coefficient increases.

10

-5

10

1.1 1.0 0.9

-6

10

0.8

-7

10

0.7

-8

10

0

10

20

30

Distance (mm)

40

0.6 0

10

20

30

40

Distance (mm)

Figure 10.12 Comparison between DE and MC results for the CW reflectance and the CW ′ transmittance from a homogeneous slab with s = 40 mm, µs = 1 mm−1 , g = 0.9, and µa = −1 0.01 mm . Results are shown both for nr = 1 and 1.4.

Comparisons for the total reflectance R and the total transmittance T , i.e., the fraction of emitted photons exiting at z = 0 and z = s, respectively, are shown in Fig. 10.13. The figure also shows the comparison for the mean pathlengths hliR and hliT , followed by photons before exiting at z = 0 and z = s. The results are reported as a function of µa for s = 4 mm and nr = 1.4. The results from the DE are in excellent agreement with MC results for moderate values of absorption. For µa < 0.01 mm−1 , discrepancies on R and T are within 2%. For large values of µa , ′ the agreement also remains good, especially for R. Even for µa = µs = 1 mm−1 , discrepancies remain within 50%. The agreement is even better for nr = 1 or for larger values of s. Therefore, the solution of the DE for R [Eq. (4.36)] is suitable for developing inversion procedures to reconstruct the spectral dependence of µa and ′ µs from measurements of spectral reflectance.

10.3 Comparison Between Monte Carlo and the Diffusion Equation: Homogeneous Slab with an Internal Inhomogeneity As described in Sec. 9.6, MC results have been reported for inhomogeneities ′ ′ for which µsi = µs0 and Vi δµa = 0.01 mm2 for absorption perturbations, and µai = µa0 and Vi δD = 0.01 mm4 for scattering perturbations, with δµa = µai −µa0

218

Chapter 10

0

100

>, (mm)

10

-1

T

R, T

10

MC 5/µs from the source and from the receiver, and to small perturbations of scattering or absorption

222

Chapter 10

0

3.0 x = 0 mm i

x = 10 mm

2.5

i

-12

-1x10

x = 20 mm DE

a

/ T

-12

x = 0 mm

T

T

a

i

a

-2x10

x = 30 mm i

1.5

MC

(mm

-2

ps

-1

)

i

2.0

1.0

x = 10 mm

-12

i

-3x10

x = 20 mm

0.5

i

x = 30 mm i

-12

-4x10

0

1000

2000

0.0 0

3000

1000

Time (ps)

2000

3000

Time (ps)

-13

3.0

1x10

x = 0 mm

2.5

-14

x = 0 mm

2.0

x = 5 mm

x = 15 mm

D

i

-14

2x10

DE

D

i

-14

4x10

/ T

x = 10 mm

MC

-14

x = 10 mm i

i

6x10

i

x = 5 mm i

i

1.5 1.0 0.5

T

T

D

(mm

-2

ps

-1

)

8x10

0.0 0

-0.5

-14

-2x10

0

1000

2000

Time (ps)

3000

-1.0 0

500

1000

1500

Time (ps)

Figure 10.17 MC results for δT a (t) and δT D (t) for different positions of an inhomogeneity with Vi = 500 mm3 . The geometry and the optical properties are the same as in Fig. 10.15. Results are reported for different values of xi , for yi = 0 and zi = 20 mm.

(Vi δD = 0.01 mm4 , Vi δµa = 0.01 mm2 ). It is possible to show that, for shorter distances, the agreement remains good provided that the distance of the inhomogeneity both from the receiver and from the isotropic source considered for the ′ DE remains larger than about 2/µs . For large values of Vi δµa (Vi |δµa | & 4 mm2 ′ ′ for µs0 = 0.5 mm−1 ) or Vi δD (Vi |δD| & 100 mm4 for µs0 = 0.5 mm−1 ), a good agreement remains for the shape of the perturbation, but the intensity is overestimated for absorption inhomogeneities and for scattering inhomogeneities with ′ ′ ′ ′ µsi < µs0 ; whereas, for scattering inhomogeneities with µsi > µs0 , the intensity is underestimated.1

223

Comparisons of Analytical Solutions with Monte Carlo Results

10.4 Comparisons Between Monte Carlo and the Diffusion Equation: Layered Slab Comparisons between the solution of the DE for the layered slab and the MC results are reported in Figs. 10.18–10.21. All figures pertain to a two-layer slab illuminated by a pencil beam. Results are reported for the TR reflectance at distances ρ = 10, 20, and 30 mm. The figures show both the MC results and the ratio between MC and DE results. The DE outgoing flux has been evaluated using the EBPC approach.

-6

1.2

10

1.0

= 20 mm

10

= 30 mm

0.8

MC/DE

Reflectance (mm

-2

ps

-1

)

= 10 mm -7

-8

10

= 10 mm 0.6

= 20 mm = 30 mm

0.4 -9

10

' s1 ' s2

= 1 mm

-1

= 0.5 mm

'

0.2

-1

0

1000

2000

' s2

-10

10

s1

3000

4000

0.0 0

5000

= 1 mm

-1

= 0.5 mm

1000

2000

-1

3000

4000

5000

Time (ps)

Time (ps)

-6

1.4

10 )

= 10 mm -1

= 30 mm 1.0

MC/DE

ps

-2

Reflectance (mm

1.2

= 20 mm -7

10

-8

10

-9 ' s1 ' s2

-10

0

= 10 mm

0.6

= 20 mm = 30 mm

0.4

10

10

0.8

1000

2000

= 1 mm = 2 mm

3000

Time (ps)

'

-1

s1

0.2 -1

4000

' s2

5000

0.0 0

= 1 mm = 2 mm

1000

-1

-1

2000

3000

4000

5000

Time (ps)

Figure 10.18 TR reflectance from a slab with a layered architecture of scattering. The slab ′ has s1 = 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, µs1 = 1 mm−1 , and µa1 = µa2 = ′ 0. MC results are reported for µs2 = 0.5 and 2 mm−1 . Comparisons with the DE are also shown for different values of ρ.

Figure 10.18 reports comparisons for a slab with a layered architecture of ′ scattering. The slab has s1 = 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, µs1 = ′ 1 mm−1 , and µa1 = µa2 = 0. Results are reported for µs2 = 0.5 and 2 mm−1 . Figure 10.19 pertains to a slab with a layered architecture of absorption having s1 = ′ ′ 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, and µs1 = µs2 = 1 mm−1 . Results are reported for µa1 = 0 and µa2 = 0.01 mm−1 , and for µa1 = 0.01 mm−1 and

224

Chapter 10

-6

1.2

-7

= 10 mm

10

1.0

= 20 mm -8

10

= 30 mm

0.8

-9

MC/DE

Reflectance (mm

-2

ps

-1

)

10

10

-10

10

' a1

-12

10

' a2

= 0 mm

0

= 20 mm

1000

= 30 mm

-1

'

= 0.01 mm

0.2

-1

a1 ' a2

-13

10

= 10 mm

0.4

-11

10

0.6

2000

3000

4000

0.0 0

5000

= 0 mm

-1

= 0.01 mm

1000

-1

2000

3000

4000

5000

Time (ps)

Time (ps)

-6

1.4

10 )

= 10 mm -1

= 30 mm 1.0 ' a1 '

-8

10

a2

= 0.01 mm = 0 mm

-1

MC/DE

ps

-2

Reflectance (mm

1.2

= 20 mm -7

10

-1

0.8 = 10 mm

0.6

= 20 mm = 30 mm

0.4

-9

10

'

0.2

a1 ' a2

-10

10

0

1000

2000

3000

Time (ps)

4000

5000

0.0 0

= 0.01 mm = 0 mm

1000

-1

-1

2000

3000

4000

5000

Time (ps)

Figure 10.19 TR reflectance from a slab with a layered architecture of absorption. The slab ′ ′ has s1 = 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, µs1 = µs2 = 1 mm−1 . MC results −1 −1 are reported for µa1 = 0 mm and µa2 = 0.01 mm , and for µa1 = 0.01 mm−1 and µa2 = 0 mm−1 . Comparisons with the DE are also shown for different values of ρ.

µa2 = 0. Examples for a slab with a layered architecture of refractive index are ′ ′ shown in Fig. 10.20 for a slab with s1 = 8 mm, s2 = 100 mm, ne = 1, µs1 = µs2 = 1 mm−1 , and µa1 = µa2 = 0.01 mm−1 . Results are reported for n1 = 1 and n2 = 1.4, and for n1 = 1.4 and n2 = 1. All the comparisons show that the solutions of the DE for the layered slab are in excellent agreement with the reference MC results: Discrepancies are appreciable only for early received photons and are of the same order of those observed for the homogeneous slab. In Fig. 10.21, an example of comparison is shown for the TR mean times of flight ht1 i(t) and ht2 i(t) spent by received photons inside the first and the second layer, respectively. The results pertain to the slab with layered refractive index n1 = 1 and n2 = 1.4 considered in Fig. 10.20. The MC results show that both ht1 i(t) and ht2 i(t) are almost independent of the source-receiver distance ρ even when there

225

Comparisons of Analytical Solutions with Monte Carlo Results

is a significant mismatch in the refractive index of the layers. These results are in excellent agreement with the theoretical predictions reported in Sec. 6.5 on the penetration depth of light re-emitted by diffusive media. The figure also shows that the values provided by DE for ht1 i(t) and ht2 i(t) are in excellent agreement with the reference MC results. -6

1.6

= 20 mm

-8

10

-10

10

= 1

0.8 0.6

n

= 1.4

1

0.4

1000

2000

0.0 0

3000

-6

)

= 10 mm

-7

10

2000

3000

1.0

= 20 mm = 30 mm

-8

10

0.8

MC/DE

-1

ps

-2

1000

1.2

10

Reflectance (mm

= 30 mm

Time (ps)

Time (ps)

-9

10

-10

-11

10

n

= 1.4

n

= 1

1 2

1000

= 10 mm = 20 mm = 30 mm

n

= 1.4

n

= 1

1

0.2

2

-12

0

0.6

0.4

10

10

= 20 mm

0.2

-12

0

1.0

= 10 mm

n

2

10

= 1.4

2

1.2

-9

-11

= 1

n

= 30 mm

10

10

n

1

1.4

= 10 mm

-7

10

MC/DE

Reflectance (mm

-2

ps

-1

)

10

2000

Time (ps)

3000

0.0 0

1000

2000

3000

Time (ps)

Figure 10.20 TR reflectance from a slab with a layered architecture of refractive index. The ′ ′ slab has s1 = 8 mm, s2 = 100 mm, µs1 = µs2 = 1 mm−1 , µa1 = µa2 = 0.01 mm−1 , and ne = 1. MC results are reported for n1 = 1 and n2 = 1.4, and for n1 = 1.4 and n2 = 1. Comparisons with the DE are also shown for different values of ρ.

10.5 Comparisons Between Monte Carlo and Hybrid Models In this section, comparisons are shown between the solutions of the RTE obtained with the hybrid models of Chapter 5 and MC results. Comparisons are shown for TR results plotting the ratios MC/RTE, MC/TE, and MC/DE. All the figures will be focused on the response at short times and small source-receiver distances, times

Chapter 10

600

1.2

500

1.0

400

MC/DE

300

0.8

0.6

1

1

(ps)

226

200

= 10 mm

= 10 mm

0.4

= 20 mm

= 20 mm

100 0 0

= 30 mm

1000

2000

= 30 mm

0.2

0.0 0

3000

1000

Time (ps)

3000

1.0

MC/DE

= 30 mm

2

(ps)

= 20 mm

1000

0.8 0.6 ρ = 10 mm ρ = 20 mm ρ = 30 mm

0.4 0.2

0 0

3000

1.2 = 10 mm

2000

2000

Time (ps)

1000

2000

0.0 0

3000

1000

2000

3000

Time (ps)

Time (ps)

Figure 10.21 TR mean times of flight spent by received photons inside the first and the ′ ′ second layer of a slab with s1 = 8 mm, s2 = 100 mm, n1 = 1, n2 = 1.4, ne = 1, µs1 = µs2 = −1 −1 1 mm , and µa1 = µa2 = 0.01 mm . The MC results are shown together with the ratio between MC and DE results.

and distances at which the larger differences between the models are expected. 10.5.1 Infinite homogeneous medium ′

We refer to a non-absorbing medium with µs = 1 mm−1 and refractive index n = 1. Figure 10.22 reports comparisons for the fluence for r = 1, 2, and 5 mm both for g = 0 and 0.9. It is particularly interesting to compare RTE and MC results for the isotropic scattering function (g = 0) since MC results are compared with an almost exact analytical solution:2 The comparison shows that, apart from very early times, discrepancies remain within 2%, i.e., within the range of error indicated in Ref. 2 for the analytical solution of the RTE. These results can be seen as a further validation of the MC code. For g = 0, we also observe that the DE solution is faster than the TE to recover the correct values at long times. From Fig. 10.22, it is difficult to

227

Comparisons of Analytical Solutions with Monte Carlo Results

1.8

1.8

r = 1 mm

g = 0.0

Fluence MC/Model

Fluence MC/Model

r = 1 mm 1.6

MC/DE

1.4

MC/RTE MC/TE 1.2

1.0

0.8 0

50

100

150

1.6

MC/DE

1.4

MC/RTE MC/TE 1.2

1.0

0.8 0

200

g = 0.9

50

Time (ps)

1.4 MC/DE

MC/DE

MC/RTE

MC/RTE

MC/TE 1.2

1.0

MC/TE

1.0

r = 2 mm

g = 0.0 0.8 0

100

200

g = 0.9 0.8 0

300

100

Time (ps)

200

300

Time (ps)

1.1

1.1 MC/DE

MC/DE

MC/RTE

MC/RTE

Fluence MC/Model

Fluence MC/Model

200

1.2

r = 2 mm

MC/TE

1.0

MC/TE

1.0

r = 5 mm

r = 5 mm

g = 0.0 0.9 0

150

1.4

Fluence MC/Model

Fluence MC/Model

100

Time (ps)

100

200

300

Time (ps)

400

g = 0.9

500

0.9 0

100

200

300

400

500

Time (ps)

Figure 10.22 Comparison of DE, TE, and RTE models for the TR fluence rate with MC ′ results for an infinite medium with µs = 1 mm−1 and n = 1. Comparisons are shown for r = 1, 2, and 5 mm both for g = 0 and 0.9.

228

Chapter 10

1.8

1.8

r = 1 mm

r = 1 mm 1.6

g = 0.0

Flux MC/Model

Flux MC/Model

1.6

MC/DE

1.4

MC/RTE MC/TE 1.2

1.0

0.8 0

g = 0.9

MC/DE

1.4

MC/RTE MC/TE 1.2

1.0

50

100

150

0.8 0

200

50

Time (ps)

100

150

1.4

1.4 MC/DE

MC/DE

MC/RTE

r = 2 mm

MC/RTE

MC/TE

g = 0.9

MC/TE

Flux MC/Model

Flux MC/Model

200

Time (ps)

1.2

1.0

1.2

1.0

r = 2 mm g = 0.0 0.8 0

100

200

0.8 0

300

100

Time (ps)

200

1.3

1.3 MC/DE

MC/DE 1.2

MC/TE

Flux MC/Model

Flux MC/Model

MC/RTE

MC/RTE

1.2

r = 5 mm 1.1

g = 0.0

1.0

0.9 0

300

Time (ps)

100

200

300

Time (ps)

400

500

MC/TE r = 5 mm g = 0.9

1.1

1.0

0.9 0

100

200

300

400

500

Time (ps)

Figure 10.23 Same as Fig. 10.22, but for the flux.e

indicate the model in better agreement with MC results when g = 0.9. However, if we also include in the comparison the response for t < t0 , for which the figure

229

Comparisons of Analytical Solutions with Monte Carlo Results

2.0

2.0 MC/DE

MC/DE 1.8

MC/RTE MC/TE

1.6

1.4

1.2

1.0 r = 1 mm

0.8

t/t 0.6 0

45

90

Radiance MC/Model

Radiance MC/Model

1.8

MC/TE 1.6

1.4

1.2

1.0 r = 5 mm

0.8

= 2.0

0

135

t/t 0.6 0

180

Angle (degrees)

45

90

135

MC/DE

MC/RTE

MC/RTE

Radiance MC/Model

MC/DE

MC/TE r = 1 mm

1.4

t/t

0

= 4.1

1.2

1.0

180

MC/TE 1.1

1.0

r = 5 mm t/t

0.8 0

45

90

135

0.9 0

180

Angle (degrees)

45

90

0

= 4.0

135

180

Angle (degrees)

1.3

1.1 MC/DE

MC/DE

MC/RTE

MC/RTE

Radiance MC/Model

Radiance MC/Model

= 2.0

1.2

1.6

MC/TE

1.2

1.1

1.0

MC/TE

1.0

r = 5 mm t/t

r = 1 mm t/t 0.9 0

0

Angle (degrees)

1.8

Radiance MC/Model

MC/RTE

0

0

= 8.0

= 8.0 45

90

135

Angle (degrees)

180

0.9 0

45

90

135

180

Angle (degrees)

Figure 10.24 Same as Fig. 10.22, but for the radiance. Comparisons with MC results for g = 0 are shown for three values of t/t0 for r = 1 and 5 mm.

230

Chapter 10

does not provide information, the RTE and the TE models become preferable. In fact, the MC, RTE, and TE responses are properly equal to zero for t < t0 , while the solution of DE predicts the arrival of photons even before the ballistic ones. Figure 10.23 reports comparisons for the flux in the same condition considered for the fluence. There is also excellent agreement between RTE and MC results for g = 0. We point out that, since the RTE flux is obtained from the almost exact solution of the RTE making use of Fick’s law [Eq. (3.31) and Sec. 5.1], the comparison with MC results provides information on the accuracy of Fick’s law itself. These results also show that Fick’s law, when applied to the correct values for the fluence, provides significantly better results with respect to the solution of the DE where Ficks’s law is used twice: the first time to obtain the solution for the fluence, and the second time to obtain the flux. This is perhaps the key point to understanding why the hybrid model based on the RTE provides results significantly better than the DE, as will be shown in next section. As seen for the fluence, for g = 0.9 it is difficult to indicate a model in better agreement with MC results. For the radiance, examples of comparisons are shown in Fig. 10.24 for r = 1 and 5 mm for the medium with g = 0. Comparisons are shown for some values of t/t0 . For the radiance, the RTE solution gives the best agreement with MC results, particularly at short distances. 10.5.2 Slab geometry ′

We refer to a non-absorbing medium with µs = 1 mm−1 and refractive index ni = 1.4. Comparisons with MC results are shown in Fig. 10.25 for the reflectance of a slab with s = 40 mm and in Fig. 10.26 for the reflectance and the transmittance of a slab with s = 4 mm. Comparisons are shown for a scattering function with g = 0.9 since it is more representative of real media. For DE, TE, and RTE models, the outgoing flux has been evaluated using the EBPC approach. Comparisons show that the hybrid model based on the RTE provides results that are significantly better with respect to the others. Apart from very early times, the RTE hybrid model results are in excellent agreement with MC simulations, both for s = 40 and 4 mm: For ′ ′ ′ t > 4ℓ /v (ℓ = 1/µs ), discrepancies remain smaller than about 5% even when the source receiver distance is < 4 mm. The hybrid RTE model can be used instead of the DE to provide an improved description of light propagation through small volumes of diffusive media. As to the TE hybrid solution, as observed for the infinite medium, it is slower than the DE to recover the correct asymptotic value at long times. In particular, as also shown in Figs. 5.3 and 5.4, due to the large number of image sources that give a significant contribution even at the short times displayed in Fig. 10.26, the correct asymptotic behavior is not reached for the slab 4 mm thick.

231

Comparisons of Analytical Solutions with Monte Carlo Results

MC/DE MC/RTE 1.4

MC/TE

1.2

1.0

0.8 0

50

100

150

Reflectance MC/Model

Reflectance MC/Model

MC/DE

= 1.0 mm

1.6

MC/RTE 1.2

1.0

= 2.02 mm

0.8 0

200

MC/TE

100

Time (ps)

400

ρ = 10.2 mm

1.0

MC/DE MC/RTE MC/TE

200

400

Time (ps)

600

800

Reflectance MC/Model

= 5.1 mm

Reflectance MC/Model

300

1.1

1.2

0.8 0

200

Time (ps)

1.0

MC/DE MC/RTE MC/TE 0.9 0

300

600

900

1200

Time (ps)

Figure 10.25 Comparison of DE, TE, and RTE models with MC results for the TR reflectance ′ from a slab with s = 40 mm, µs = 1 mm−1 , g = 0.9, ni = 1.4, and nr = 1. Comparisons are shown for four values of ρ.

The results presented pertain to nr = 1. A slightly worse agreement has been observed for nr = 1.4. In Fig. 10.27, a comparison is shown for the perturbation due to an absorbing and to a scattering inhomogeneity on the TR reflectance. The results pertain to a slab ′ with s = 4 mm, µs = 1 mm−1 , g = 0.9, µa = 0, ni = 1.4, and nr = 1. The receiver is at (3, 0, 0) mm, and the inhomogeneity with Vi = 1 mm3 , Vi δµa = 0.01 mm2 , and Vi δD = 0.01 mm4 is at (1.5, 0, 1.5) mm. We stress that between source and receiver, between inhomogeneity and source, and between inhomogeneity and receiver, there are short distances (the distance is ≃ 1.6 mm from the isotropic point source and ≃ 2.1 mm from the receiver) for which the DE is expected to give poor results. Also in this case, the RTE hybrid model provides results significantly better than other models.

232

Chapter 10

2.0

1.6

3.5 mm

Reflectance MC/Model

Reflectance MC/Model

MC/DE MC/RTE MC/TE 1.5

1.0

= 1.0 mm

1.4

1.2

1.0

MC/DE

0.8

MC/RTE MC/TE

0.5 0

20

40

60

80

0.6 0

100

50

Time (ps)

200

250

300

250

300

1.6 = 0.15 mm

Transmittance MC/Model

Transmittance MC/Model

150

Time (ps)

1.6

1.4

1.2

1.0

MC/DE

0.8

MC/RTE

= 3.5 mm 1.4

1.2

1.0

MC/DE

0.8

MC/RTE

MC/TE 0.6 0

100

50

100

150

MC/TE

200

Time (ps)

250

300

0.6 0

50

100

150

200

Time (ps)

Figure 10.26 Comparison of DE, TE, and RTE models with MC results for the TR reflectance ′ and transmittance from a slab with s = 4 mm, µs = 1 mm−1 , g = 0.9, ni = 1.4, and nr = 1. Comparisons are shown for two values of ρ.

10.6 Outgoing Flux: Comparison between Fick and Extrapolated Boundary Partial Current Approaches In Chapter 4, three different methods have been presented to evaluate the flux outgoing from a diffusive medium. The first method calculates the flux applying Fick’s law to the fluence on the boundary [Eqs. (4.20) and (4.21)]. The second integrates the radiance transmitted at the interface over the solid angle [Eq. (4.42)]. The third, the hybrid approach named EBPC, applies the PCBC to the solution for the fluence obtained using the EBC, and the outgoing flux is simply proportional to the fluence [Eq. (4.44)]. The first method, used in the Diffusion&Perturbation software (Sec. 4.3), is more widely used. All the figures in Chapters 4 and 7 and in Secs. 10.2.2 and 10.3 have been generated using this method. The third method, being the simplest one

233

Comparisons of Analytical Solutions with Monte Carlo Results

0

3.0 MC/DE MC/TE

-7

a

TE

-7

RTE

-7

-5x10

0

1.0

R

a

DE

-4x10

MC/RTE 2.0

/ R

MC

MC

-7

-3x10

MODEL

-7

-2x10

R

a

(mm

-2

ps

-1

)

-1x10

30

60

90

120

0.0 0

150

30

Time (ps)

-7

-7

4x10

120

150

-7

RTE

3x10

D

2x10

-7

R

D

1x10

MC

/ R

-7

MODEL

MC/RTE

TE

)

-1

ps

150

MC/TE

12.0

DE

-2

(mm D

R

120

MC/DE

MC

0

10.0

8.0

6.0

4.0

2.0

-7

0

90

14.0

5x10

-1x10

60

Time (ps)

30

60

90

Time (ps)

120

150

0.0 0

30

60

90

Time (ps)

Figure 10.27 Comparison of DE, TE, and RTE with MC results for the TR perturbation on ′ reflectance from a slab with s = 4 mm, µs0 = 1 mm−1 , g = 0.9, µa0 = 0, ni = 1.4, and nr = 1. The receiver is at (3, 0, 0) mm, and the inhomogeneity with Vi = 1 mm3 is at (1.5, 0, 1.5) mm.

and probably more accurate, has been suggested both for hybrid models (Sec. 5.1) and for the layered slab (Sec. 6.3) and has been used for the figures of Chapters 6 and 7 and in Secs. 10.4 and 10.5.2. As pointed out in Sec. 4.4, the three methods lead to significantly different expressions for the outgoing flux, but in spite of the differences in the formal expressions, they produce results that in many situations are almost indistinguishable (see Figs. 4.9 and 4.10). In Sec. 4.4, we also predicted that the EBPC solution would provide a more accurate description, especially for TR measurements at short sourcereceiver distances. This can be seen from comparisons with reference MC results. Examples are shown in Figs. 10.28, 10.29, and 10.30. Comparisons are shown only for the first and the third method, i.e., between the Fick and EBPC approaches. Since the solution obtained with the second method was a linear combination of

234

Chapter 10

those obtained with the Fick and EBPC approaches, the corresponding results are between those obtained with the other methods. Figures 10.28 and 10.29 show results for the TR reflectance at moderate distances (ρ = 2.53, 5.12, and 10.2 mm) from the source and pertain to the case of matched (nr = 1) and mismatched (nr = 1.4) refractive indexes, respectively. Comparisons ′ pertain to a slab with s = 40 mm, µs = 1 mm−1 , g = 0.9, and ni = 1.4 illuminated by a pencil light beam. Differences between results from the Fick and EBPC approaches are larger in the case of mismatched refractive indexes. As anticipated in Sec. 4.4, there is a slightly better agreement between the MC results and the EBPC results. In Ref. 3, it has been shown that appreciably more accurate results could be obtained when the EBPC solution is used in the fitting procedure devoted to retrieving the optical properties of the medium from TR measurements of reflectance. Figure 10.30 shows comparisons for the CW reflectance from a slab with s = ′ 40 mm, µs = 1 mm−1 , µa = 0.01 mm−1 , g = 0.9, and ni = 1.4. For both the RTE hybrid model and the DE, the CW response has been calculated, integrating over time and over the area of the receiver, the temporal response being provided by the models (we considered receivers identical to those used for MC simulations). The figure shows that the results from the RTE hybrid model are in better agreement with MC results than those from the DE, but it is difficult to draw a general conclusion from the comparison between the Fick and EBPC approaches. On the contrary, for the total reflectance (that is strongly influenced by photons exiting very near the source) and for the total transmittance, it is possible to show that results obtained with Fick’s law are in significantly better agreement with MC results.

10.7 Conclusions In this chapter, we have compared solutions of the RTE based on the diffusion approximation with MC results. Since the MC results are exact solutions of the RTE (apart from the statistical noise), the presented comparisons can be therefore used to determine the degree of accuracy of the DE solutions. In this section, we summarize the results of these comparisons also with the aim to better specify the meaning of "diffusive media". Making use of the scaling properties of Sec. 3.4.1, comparisons can be summarized as: 10.7.1 Infinite medium Comparisons in Sec. 10.2.1 (Figs. 10.1 and 10.2) showed that for TR results discrepancies between the solution of the DE and MC simulations for g = 0.9 (representa′ tive of real media) are within 2% for rµs & 5 and t & 4t0 (t0 is the time of flight for ′ ′ ′ ballistic photons), and for t & 4ℓ /v (ℓ = 1/µs is the transport mean free path) for shorter distances. Since both MC and DE have the same dependence on absorption, these results are independent of the absorption coefficient. On the contrary, for CW results, discrepancies increase as absorption increases. In fact, photons with long

235

Comparisons of Analytical Solutions with Monte Carlo Results

1.3

1.3 MC/DE

Reflectance MC/Model

MC/DE 1.2

= 2.53 mm

EBPC

MC/RTE

EBPC

1.1

1.0

0.9

0.8 0

100

200

300

400

Reflectance MC/Model

EBPC

1.2

1.0

0.9

100

200

Time (ps)

Reflectance MC/Model

Reflectance MC/Model

800

1000

= 5.12 mm

1.05

1.00

0.95

MC/DE

EBPC

MC/RTE

200

400

1.05

1.00

0.95 MC/DE

600

Fick

MC/RTE

EBPC

800

0.90 0

1000

200

Time (ps)

Fick

400

600

Time (ps)

1.10

1.10

EBPC ρ = 10.21 mm

Fick

Reflectance MC/Model

Reflectance MC/Model

500

Fick

= 5.12 mm

1.05

1.00

MC/DEEBPC

= 10.21 mm 1.05

1.00

0.95

MC/DE

MC/RTEEBPC 0.90 0

400

1.10 EBPC

0.95

300

Time (ps)

1.10

0.90 0

= 2.53 mm Fick

1.1

0.8 0

500

Fick

Fick

MC/RTE

500

1000

Fick

MC/RTE

1500

0.90 0

500

Time (ps)

Fick

1000

1500

Time (ps)

′

Figure 10.28 TR reflectance from a slab with s = 40 mm, µs = 1 mm−1 , g = 0.9, ni = 1.4, and nr = 1 illuminated by a pencil light beam: Comparison of MC results with results obtained using the Fick and EBPC approaches. Comparisons are shown for the DE and the RTE.

236

Chapter 10

1.8

1.8 EBPC

1.6 1.5 1.4 1.3

MC/DE

1.2

MC/RTE

EBPC EBPC

1.1 1.0 0.9 0.8 0

Fick

1.7

= 2.53 mm

Reflectance MC/Model

Reflectance MC/Model

1.7

= 2.53 mm 1.6 1.5 MC/DE

1.4

1.1 1.0 0.9

100

200

300

400

0.8 0

500

100

200

Reflectance MC/Model

Reflectance MC/Model

EBPC

1.3 MC/DE

1.2

EBPC

MC/RTE 1.1

EBPC

1.0

0.9

200

400

600

800

1.4

= 5.12 mm 1.3

1.2

MC/DE

Fick

MC/RTE

1.1

Fick

1.0

0.9

0.8 0

1000

200

400

600

800

1000

Time (ps)

1.15

1.15 EBPC

Fick

= 10.21 mm

Reflectance MC/Model

Reflectance MC/Model

500

Fick

Time (ps)

1.10 MC/DE

EBPC

MC/RTE

EBPC

1.00

0.95

0.90 0

400

1.5

= 5.12 mm

1.05

300

Time (ps)

1.5

0.8 0

Fick

1.2

Time (ps)

1.4

Fick

MC/RTE

1.3

500

1000

Time (ps)

1500

= 10.21 mm 1.10 MC/DE

Fick

MC/RTE

1.05

Fick

1.00

0.95

0.90 0

500

1000

1500

Time (ps)

Figure 10.29 Same as Fig. 10.28 but for the case of the mismatched refractive index (nr = 1.4).

237

Comparisons of Analytical Solutions with Monte Carlo Results

1.1

1.1 Fick

n = 1

Reflectance MC/Model

Reflectance MC/Model

EBPC r

1.0

0.9 MC/DE

EBPC

MC/RTE 0.8

0.7 0

10

20

EBPC

30

n = 1 r

1.0

0.9

MC/DE 0.8

0.7 0

40

10

20

(mm)

1.3

n = 1.4

Reflectance MC/Model

Reflectance MC/Model

30

40

1.4 EBPC r

1.2

MC/DE

1.1

EBPC

MC/RTE

EBPC

1.0 0.9 0.8 0.7 0.6 0

Fick

(mm)

1.4 1.3

Fick

MC/RTE

Fick n = 1.4

1.2

r

1.1 1.0 0.9

MC/DE

0.8

Fick

MC/RTE

Fick

0.7

10

20

30

40

0.6 0

10

(mm)

20

30

40

(mm)

′

Figure 10.30 CW reflectance from a slab with s = 40 mm, µs = 1 mm−1 , µa = 0, g = 0.9, and ni = 1.4 illuminated by a pencil light beam: Comparison of MC results with results obtained using the Fick and EBPC approaches. Comparisons are shown for the DE and the RTE, both for the case of matched (nr = 1) and mismatched (nr = 1.4) refractive indexes.

time of flight (those better described by the DE) are more attenuated with respect to ′ early photons when absorption increases. For µa /µs . 0.01, discrepancies remain ′ ′ within 1% provided rµs & 2 (Figs. 10.5 and 10.6). For larger values of µa /µs , ′ discrepancies increase as the source-receiver distance increases. For rµs < 20, they ′ ′ remain within 10% for µa /µs = 0.05 and within about 20% for µa /µs = 0.1. 10.7.2 Homogeneous slab For this geometry, disagreements in addition to those caused by the intrinsic approximations of the DE can be also due to the approximations in modeling, both the boundary conditions and the source (the pencil light beam is modeled with a point′ like isotropic source at zs = 1/µs ). Disagreements observed with respect to MC results are in general larger than those for the infinite medium, and they significantly

238

Chapter 10

depend on the refractive index mismatch. Therefore, it is more difficult to draw a general conclusion on the accuracy of the analytical solutions. Comparisons in Sec. 10.2.2, with solutions reported in Chapter 4 for a slab with reduced optical thickness ′ sµs = 40, showed that for TR results the disagreement remains within 5% for t & 4 ′ t0 only if the source receiver distance is larger than about 10/µs (Figs. 10.9 and 10.10). Disagreement is larger when there is a significant mismatch in the refractive index, and reflections play an important role on propagation. For shorter distances, discrepancies are larger and can involve a significant fraction of received photons. This is shown by comparisons for CW reflectance for which discrepancies of about ′ ′ 5% are observed also for ρµs = 10 and µa /µs = 0.01 (Fig. 10.12). Comparisons in Sec. 10.5.2 showed that more accurate results are obtained, especially at early times and at short distances, when the hybrid model based on the solution of the RTE for the infinite medium is used. It has been shown that, for ′ the case of a matched refractive index, discrepancies remain .5% for t & 4 ℓ /v ′ both for TR reflectance and for TR transmittance from a slab with sµs = 4, even for ′ distances ρµs < 4 (Figs. 10.25 and 10.26). 10.7.3 Layered slab Comparisons in Sec. 10.4 showed that the accuracy of solutions for the layered slab is similar to that for the homogeneous slab. Comparisons have been shown for ′ ′ µs1 /µs2 = 0.5 and 2, n1 /n2 = 1.4 and 0.71, µa1 = 0 and µa2 = 0.01 mm−1 , and µa1 = 0.01 mm−1 and µa2 = 0. Results reported in Ref. 4 showed that a similar accuracy is also obtained for a larger mismatch in the optical properties. 10.7.4 Slab with inhomogeneities inside For the perturbation due to inhomogeneities inside the slab, comparisons in Sec. 10.3 showed good agreements both for absorption and for scattering perturbations, provided that perturbations are sufficiently small and the distance of the inhomo′ geneity both from the receiver and from the source remains larger than about 2/µs . For large values of Vi δµa or Vi δD, a good agreement remains for the shape of the perturbation, but the intensity is overestimated for absorption inhomogeneities and for scattering inhomogeneities with a reduced scattering coefficient smaller than the ′ ′ background; whereas, for scattering inhomogeneities with µsi > µs0 , the intensity is underestimated. 10.7.5 Diffusive media All the comparisons reported in the previous chapters pertain to the infinite medium or to the infinite slab (results for the layered slab have also been reported for a cylinder with a sufficiently large radius to behave like an infinite slab). When measurements are carried out on media with a finite volume, it may be useful to know how long it is possible to use results based on the diffusion approximation. The influence of the finite volume of the medium in establishing the diffusive regime of

239

Comparisons of Analytical Solutions with Monte Carlo Results

propagation can be investigated comparing the solution of the DE with MC results for a finite parallelepiped. An example of comparison is shown in Fig. 10.31 (the MC results used for this comparison have not been reported in the CD-ROM). The ′ figure shows the TR reflectance of a cube with 10 mm sides, µs = 1 mm−1 , g = 0.9, µa = 0, ni = 1.4, and nr = 1. The pencil light beam impinges at (2.5, 5, 0) mm, and the receiver, circular with a radius of 3 mm, is centered at (7.5, 5, 0) mm (the origin of the reference system is at the corner of the cube). The temporal responses obtained with the DE and with the RTE hybrid model are compared with MC results. The finite volume of the diffusive medium strongly influences the temporal response. This is shown from the comparison between solutions of the DE for the cube and for the infinitely extended slab 10 mm thick (curves DE and DESlab , respectively). Differences increase with the time of flight. At 400 ps, the response for the slab is about one order of magnitude larger. The figure also shows the ratio MC/DE and MC/RTE as a function of t/t0 . The agreement is similar to the one observed for the reflectance from a slab 40 mm thick at the same source-receiver distance (see Figs. 10.9 and 10.25): Disagreement remains within about 5% for t & 4t0 . Therefore, we can also conclude that for a finite volume of turbid medium, solutions of the RTE based on the diffusion approximation accurately describe propagation for photons with a sufficiently long time of flight. For the CW response, since the fraction of photons received with time of flight t > 4t0 (those better described by the DE) decreases as the volume of the medium decreases, the results are a bit worse with respect to the infinite slab: For the cube of Fig. 10.31, the CW solution of the RTE is overestimated by 10% for µa = 0 and 21% for µa = 0.1 mm−1 , about 1% more than for the infinite slab in the same conditions.

1E-4

1.5

Reflectance MC/Model

)

-1

ps

DE

1E-5

RTE

-2

Reflectance (mm

1.4

MC

DE

Slab

1E-6

1E-7

MC/DE MC/RTE

1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6

1E-8 0

100

200

300

Time (ps)

400

500

0.5 0

5

10

15

20

t/t

0

Figure 10.31 Comparison of DE and RTE models for the TR reflectance with MC results for ′ a cube with 10 mm sides, µs = 1 mm−1 , g = 0.9, µa = 0, ni = 1.4, and nr = 1. The figures also show the DE model for the infinitely extended slab 10 mm thick (DESlab curve).

240

Chapter 10

From comparisons reported in this chapter and others that can be done with results reported in the CD-ROM, we can conclude that solutions of the RTE based on the diffusion approximation provide a description of photon migration that can ′ be suitable for many applications, provided the turbid medium has µa /µs . 0.1 and ′ a volume & (10ℓ )3 . In such conditions, the diffusive regime of propagation can be established even near the source provided photons traveled trajectories with length ′ ′ ′ & 4ℓ inside the medium. Since, as shown in Sec. 2.4 [to travel the length ℓ = 1/µs , the average number of scattering events is 1/(1 − g)] we can also conclude that to randomize propagation in order to reach the diffusive regime, photons should have undergone a number of scattering events & 4/(1 − g). Therefore, about 4 scattering events are sufficient for a medium with Rayleigh scatterers, but about 20–40 are required for most natural media for which 0.8 . g . 0.9. The conditions we have indicated as necessary for establishing the diffusive regime of propagation can be satisfied for many of the materials of which we reported typical values for the optical properties in Sec. 2.5. For many of these solid ′ ′ and liquid materials µs ≈ 1 mm−1 and µa /µs . 0.1 at visible and/or near-infrared wavelengths. Therefore, for such materials, the diffusive regime of propagation can also be reached when the volume is as small as ≈ 1 cm3 . For dense fogs and clouds, propagation in the diffusive regime can be established for photons that traveled paths longer than some hundreds of meters, but the volume involved should be much larger. An even larger volume is required for media with lower scattering properties as may occur for clouds of gas and dust in the interstellar medium. Instead, the clear atmosphere (the optical thickness of which is between 0.35 and 0.035 for 400 < λ < 700 nm) is not suitable for propagation in the diffusive regime since the available volume of the turbid medium is not sufficiently extended.

Comparisons of Analytical Solutions with Monte Carlo Results

241

References 1. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration,” Appl. Opt. 40, 4622–4632 (2001). 2. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997). 3. F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999). 4. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the timedependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52, 2827–2843 (2007).

Appendix A

The First Simplifying Assumption of the Diffusion Approximation For obtaining Eq. (3.25), the specific intensity I(~r, sˆ, t) can be expanded in spherical harmonics retaining only the first two terms (isotropic and linearly anisotropic) (see Ref. 1 for a detailed description). In this appendix, an intuitive physical justification of Eq. (3.25) is provided following the scheme proposed by Ishimaru.2 In the diffusive regime of propagation, due to multiple scattering, the radiance has an almost isotropic angular distribution and consequently can be represented as ~ r, t) · sˆ, I(~r, sˆ, t) = αΦ(~r, t) + β J(~

(A-1)

where I(~r, sˆ, t) is the sum of two terms: a first isotropic term and a second term that accounts for the weak anisotropy of the specific intensity. The coefficient α can be determined accordingly to the definition of Φ(~r, t) as Z 1 Φ(~r, t) = I(~r, sˆ, t)dΩ = α4πΦ(~r, t) ⇒ α = . (A-2) 4π 4π

Therefore, the first term of Eq. (A-1) is the average value of I(~r, sˆ, t) over the solid ~ r, t): If sˆJ is the angle. The value of β is determined by using the definition of J(~ ~ direction of J(~r, t), then Z ~ ~ s · sˆJ dΩ. (A-3) J(~r, t) = J(~r, t) · sˆJ = I(~r, sˆ, t)ˆ 4π

For obtaining β, Eq. (A-1) is substituted in Eq. (A-3): R R ~ 1 ~ r, t) · sˆ(ˆ Φ(~r, t)ˆ s · sˆJ dΩ + β J(~ s · sˆJ )dΩ J(~r, t) = 4π 4π 4π R R ~ 1 = 4π Φ(~r, t)ˆ sJ · sˆdΩ + β J(~ r, t) (ˆ sJ · sˆ)(ˆ s · sˆJ )dΩ 4π R 4π ~ r, t) (ˆ s · sˆJ )2 dΩ . = β J(~

(A-4)

4π

Since

Z

4π

we obtain and

2

(ˆ s · sˆJ ) dΩ =

Zπ

4 cos2 θ2π sin θdθ = π , 3

0

4 3 ~ ~ J(~ r , t) = β J(~ r , t) π =⇒ β = 3 4π I(~r, sˆ, t) =

(A-5)

3 ~ 1 Φ(~r, t) + J(~r, t) · sˆ. 4π 4π 245

(A-6) (A-7)

Appendix B

Fick’s Law In this appendix, the time-dependent Fick’s law is obtained following the main frames of the procedure used by Ishimaru for steady-state sources.2 The two simplifying assumptions of the diffusion approximation are necessary for this calculation. Multiplying the RTE, i.e., Eq. (3.5), by sˆ and integrating over the whole solid angle, a new equation is obtained where the different terms are listed below: R R  1 ∂I(~r,ˆs,t)  ∂ ∂ ~ J(~r, t) sˆdΩ = v1 ∂t I(~r, sˆ, t)ˆ sdΩ = v1 ∂t • I1 = v ∂t 4π

4π

• I2 = • I3 = • I4 = • I5 =

R

4π

R

(ˆ s · ∇I(~r, sˆ, t)) sˆdΩ ~ r, t) µt I(~r, sˆ, t)ˆ sdΩ =µt J(~

4π

R

4π

R



µs

R

4π

p(ˆ s, sb′ )

h

1 r, t) 4π Φ(~

+

3 ~ r, t) 4π J(~

ε(~r, sˆ, t)ˆ sdΩ

 i ′ ′ b · s dΩ sˆdΩ

4π

For integral I2 , making use of the first simplifying assumption [Eq. (3.25)], we obtain   Z  3 ~ 1 Φ(~r, t) + J(~r, t) · sˆ sˆdΩ. (B-1) I2 = sˆ · ∇ 4π 4π 4π

The first term of Eq. (B-1) is an integral of the form

R 

4π

 ~ indepen~ sˆdΩ with A sˆ · A

dent of sˆ. With reference to Fig. B-1, assuming the x-axis of the reference system ~ results in along A  R  ~ sˆdΩ = sˆ · A 4π

=A

R2π 0

= A2π

dϕ Rπ 0

Thus,

Rπ 0

  sin θ cos θ cos θ ˆi + sin θ cos ϕ ˆj + sin θ sin ϕkˆ dθ

~ sin θ cos2 θˆidθ = −2π A

−1 R 1

cos2 θd cos θ =

4π ~ 3 A.

  Z  Φ(~r, t) ∇Φ(~r, t) sˆ · ∇ sˆdΩ = . 4π 3

4π

246

(B-2)

(B-3)

247

Fick’s Law

y

s

A

x

z

Figure B-1 x, y, z reference system used to make the integral I2 .

u Ju s

'

Js

s

Figure B-2 Decomposition of the flux vector for the integral I4 .

~ is Since it is possible to demonstrate that, for every vector, A Z

4π

i h  ~ · sˆ dΩ = 0 , sˆ sˆ · ∇ A

(B-4)

248

Appendix B

the second term of Eq. (B-1),

3 4π

io h R n ~ r, t) · sˆ sˆdΩ, is equal to zero. In sˆ · ∇ J(~

4π

fact, using the same notation of Fig. B-1 results in   ~ · sˆ = cos2 θ ∂A + sin θ cos θ cos ϕ ∂A + sin θ cos θ sin ϕ ∂A , (B-5) sˆ · ∇ A ∂x ∂y ∂z

and thus the integral of Eq. (B-4) is an integral of trigonometric functions of θ (over the interval 0, π) and ϕ (over the interval 0, 2π). Each term of this integral vanishes. Consequently, I2 is ∇Φ(~r, t) . (B-6) I2 = 3 The fourth integral I4 is the integral of the function i h R 1 3 ~ Φ(~r, t) + 4π J(~r, t) · sb′ dΩ′ h =µs p(ˆ s, sb′ ) 4π 4π h i (B-7) R µs 3 ~ r, t) · sb′ dΩ′ . = 4π Φ(~r, t) + µs 4π p(ˆ s, sb′ ) J(~ 4π

~ r, t) along sˆ gives a not null In the second term of h, only the component of J(~ ~ r, t) can be written as contribution to the integral. With reference to Fig. B-2, J(~ ~ s, sb′ ) J(~r, t) = J u u ˆ + J s sˆ. According to this notation, taking into account that p(ˆ only depends on the angle θ between sˆ and sb′ and that u ˆ · sb′ = cos ϕ sin θ, the second term of Eq. (B-7) becomes h i R 3 ~ r, t) · sb′ dΩ′ = µs 4π p(ˆ s, sb′ ) J(~ 4π R R 3 3 (B-8) Js p(θ) cos θdΩ′ + µs 4π Ju p(θ) cos ϕ sin θdΩ′ = µs 4π 4π

3 3 ~ = µs 4π Js g = µs 4π J(~r, t) · sˆg.

4π

Finally, h results in h=

µs 3g ~ Φ(~r, t) + µs J(~ r, t) · sˆ 4π 4π

and I4 =

Z

~ r, t), hˆ sdΩ = µs g J(~

(B-9)

(B-10)

4π

since

R

4π

Φ(~r, t)ˆ sdΩ = 0 and

R h

4π

i ~ r, t)·ˆ J(~ s sˆdΩ =

4π ~ r, t) 3 J(~

[see Eq. (B-2)].

Adding I1 , I2 , I3 , I4 , and I5 , the integral over the whole solid angle of the RTE multiplied by sˆ can be written as Z 1 ∂ ~ ∇Φ(~r, t) ~ + (µt − gµs )J(~r, t) = ε(~r, sˆ, t)ˆ J(~r, t) + sdΩ. (B-11) v ∂t 3 4π

249

Fick’s Law

Applying the second simplifying assumption of the diffusion approximation [Eq. (3.27)], and then neglecting the time derivative of the photon flux, Fick’s law is obtained as " # Z 1 ∇Φ(~ r , t) ~ r, t) = − J(~ − ε(~r, sˆ, t)ˆ sdΩ . (B-12) µt − gµs 3 4π

If we assume an isotropic source ε(~r, t), then the standard form of Fick’s law is obtained: ∇Φ(~r, t) 1 ~ r, t) = − = −D∇Φ(~r, t), (B-13) J(~ µt − gµs 3 with D =

1 3(µt −gµs )

=

1 ′ 3(µa +µs )

as the diffusion coefficient.

Appendix C

Boundary Conditions at the Interface Between Diffusive and Non-Scattering Media When an interface between a diffusive and a non-scattering medium is considered (see Fig. 3.3), the boundary condition for radiative transfer can be summarized by Eq. (3.48), which we rewrite as Z Z − I(~r, sˆ, t)(ˆ s · qˆ)dΩ = RF (ˆ s · qˆ)I(~r, sˆ, t)(ˆ s · qˆ)dΩ, (C-1) sˆ·ˆ q 0

with I specific intensity at the boundary Σ of the diffusive medium, RF is the Fresnel reflection coefficient for unpolarized light, and qˆ is the unit vector normal to the boundary surface. For solving the integrals of Eq. (C-1), we make use of the first simplifying assumption of the diffusion approximation [Eq. (3.25)]. Writing the flux ~ r, t) = Ju u as J(~ ˆ unit vector tangential to the boundary surface, with ˆ + Jq qˆ with u sˆ · qˆ = cos θ and u ˆ · sˆ = cos ϕ sin θ, the first integral of Eq. (C-1) becomes R

I(~r, sˆ, t)( sˆ · qˆ)dΩ i R h1 3 ~ = r, t) + 4π J(~r, t) · sˆ (ˆ s · qˆ)dΩ 4π Φ(~

sˆ·ˆ q