Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Validations

Light Propagation through Biological Tissue
Contents
Acknowledgments
Disclaimer
List of Acronyms
List of Symbols
Preface
Chapter 1 Scattering and Absorption Properties of Turbid Media
1.1 Approach Followed in This Manual
1.2 Optical Properties of a Turbid Medium
1.2.1 The basic definitions
1.2.2 Lambert-Beer law
1.2.3 Absorption properties
1.2.4 Scattering properties
1.2.5 Limitations of the parameter and function definitions presented in this manual
1.2.6 Anomalous light transport
Laboratory phantoms and theoretical media models
Real biological tissues
1.3 Statistical Meaning of the Optical Properties of a Turbid Medium
1.3.1 Mean free paths between scattering and absorption events
1.3.2 Photon extinction due to absorption or scattering events along general photons' paths
1.4 Similarity Relation and Reduced Scattering Coefficient
1.5 Ballistic Photons
1.6 Examples of Diffusive Media
1.7 Conclusion
References
Chapter 2 The Radiative Transfer Equation
2.1 Quantities Used to Describe Radiative Transfer
2.2 The Radiative Transfer Equation
2.2.1 RTE for the general case
2.2.2 RTE for a problem with planar symmetry
2.3 The Green's Function Method
2.3.1 Time-resolved Green's function
2.3.2 Continuous-wave Green's function
2.3.3 Relation between TR and CW Green's functions
2.4 Probabilistic Interpretation of the Solutions
2.4.1 Probability density function for a photon to be detected
2.4.2 Probability density function for a photon to be absorbed
2.4.3 Probability density function to find a photon in the medium
2.5 Boundary Conditions for the RTE
2.5.1 Physical phenomena at the interface of two media with different optical properties
2.5.2 Boundary conditions at the interface of two scattering media
2.5.3 Boundary conditions at the interface between scattering and non-scattering media
2.6 Uniform Lambertian Illumination: A Special Reference Case
2.7 Properties of the Radiative Transfer Equation
2.7.1 Scaling properties
2.7.2 Reciprocity theorem for the CW RTE
2.7.3 Dependence on absorption
2.7.4 Absorbed power: useful equations
2.7.5 Ballistic photons and mean chord theorem
2.7.6 Invariance property of the mean pathlength ⟨L⟩ in scattering media
2.8 The RTE in Transformed Domains
2.8.1 Temporal frequency domain
2.9 Numerical and Analytical Solutions of the RTE
2.10 Anisotropic Media and Anomalous Radiative Transport
2.10.1 Anisotropic media
2.10.2 Anomalous radiative transport
2.11 Conclusion
References
Chapter 3 The Diffusion Equation for Light Transport
3.1 Diffusion Equation and History
3.2 The Diffusion Approximation: Physical Assumptions
3.3 Derivation of the Diffusion Equation
3.3.1 Fick's law and diffusion equation
3.3.2 Fick's law in the history
3.4 Diffusion Coefficient
3.4.1 Diffusion coefficient compatible with the μa-dependence law of the RTE
3.4.2 Diffusion coefficient for the case of μ′s≈μa
3.4.3 Theoretically exact diffusion coefficient and the related DE
3.5 Properties of the Diffusion Equation
3.5.1 Scaling properties
3.5.2 Dependence on absorption
3.5.3 Reciprocity theorem for the CW DE
3.6 Diffusion Equation in Transformed Domains
3.7 Boundary Conditions
3.7.1 Boundary conditions at the interface between diffusive and non-scattering media
3.7.2 Boundary conditions at the interface between two diffusive media
3.8 Conclusion
References
Chapter 4 Anisotropic Light Propagation
4.1 The CW Anisotropic Diffusion Equation
4.2 Two Classical Cases
4.2.1 Anisotropic medium with azimuthal symmetry and isotropic phase function
4.2.2 Isotropic medium: a test case
4.3 Conclusion
References
Chapter 5 Solutions of the Diffusion Equation for Homogeneous Media
5.1 Solution of the Diffusion Equation for an Infinite Medium: Separation of Variables and Fourier Transform Method
5.2 Improved Solution for the CW Domain: Infinite Medium and Isotropic Scattering
5.3 Solution of the Diffusion Equation for a Slab: Method of Images
5.3.1 The diffusion equation and the choice of light sources
5.3.2 Analytical Green's function for transmittance and reflectance
5.4 Solution of the Diffusion Equation for a Slab: Separation of Variables, Fourier Transform, and Eigenfunction Method
5.5 Moments of the Temporal Point Spread Function for a Slab
5.6 Solution of the Diffusion Equation for a Semi-infinite Medium
5.7 Other Solutions for the Outgoing Flux
5.8 Analytical Green's Function for a Parallelepiped
5.8.1 Time domain
5.8.2 CW domain
5.9 Analytical Green's Function for an Infinite Cylinder
5.10 Analytical Green's Function for a Sphere
5.11 Solution of the Diffusion Equation for a Pencil Beam Source Impinging on a Finite Cylinder Geometry
5.12 Ohm's Law for Light
5.12.1 Isotropic source case
5.12.2 Pencil beam source case
5.13 Solutions for a Slab Illuminated by Infinitely Extended Sources
5.13.1 Uniform distribution of isotropic sources inside a slab
5.13.2 Spatially uniform illumination with sources on the external surfaces of a slab
5.14 Solutions of the DE in Transformed Domains
5.14.1 Solutions of the DE in the temporal-frequency domain
5.14.2 Solutions of the DE in the spatial-frequency domain
5.14.3 Solution of the DE in the spatial-frequency domain: Laplace transform approach and semi-infinite medium
5.15 Angular Dependence of Radiance Exiting a Diffusive Medium
5.16 Comment: The Angular Dependence of Reflectance
5.17 Anisotropic Media
5.17.1 Solution for a slab and a pencil beam source: method of images
5.18 Summary Comments on Applications
5.18.1 Isotropic media
5.18.2 Anisotropic media
5.19 Conclusion
References
Chapter 6 Ballistic and Quasi-Ballistic Radiation
6.1 Solution of the RTE for Ballistic Radiation
6.1.1 Pencil beam source
6.1.2 Isotropic source
6.2 Heuristic Hybrid Model for Ballistic Photon Detection in Collimated Transmittance CW Measurements
6.2.1 Preliminary definition of the model
6.2.2 Model in the ballistic regime: small optical thickness
6.2.3 Model in the diffusive regime: large optical thickness
6.2.4 Model for the intermediate regime
6.2.5 Heuristic hybrid model for d = 0
6.3 Conclusion
Chapter 7 Statistics of Photon Penetration Depth in Diffusive Media
7.1 Statistics of Photon Penetration Depth inside an Infinite Laterally Extended Slab
7.2 Scaling Relationships for the Penetration Depth
7.3 Heuristic Formula for the Mean Average Penetration Depth ⟨z¯⟩ in a Homogeneous Medium
7.4 Solutions for f and ⟨zmax⟩ for a Slab in the Diffusion Approximation
7.5 Heuristic Model for ⟨zmax|t⟩ and ⟨z¯|t⟩ for a Semi-infinite Medium
7.6 Frequency-Domain Penetration Depth
7.7 Summary Comments on Applications
7.8 Conclusion
References
Chapter 8 Statistics of Transversal Penetration Depth in the TD
8.1 Statistics for the Radial Penetration Depth in a Laterally Infinite Slab
8.1.1 Heuristic relation for the average radial penetration depth
8.1.2 Scaling relationships for the radial penetration depth
8.1.3 Calculation of f(r|ρ, t) and ⟨rmax|ρ, t⟩ with DE solutions
8.1.4 Properties of f(r|ρ, t)|DE and ⟨rmax|ρ, t⟩|DE
8.1.5 Heuristic formula for ⟨rmax|ρ, t⟩|DE Slab at ρ = 0
8.1.6 Comparison of the radial versus longitudinal penetration depth in a semi-infinite medium
8.2 Statistics for the Lateral Penetration Depth in a Laterally Infinitely Extended Slab
8.2.1 Heuristic relation for ⟨|y| |ρ, t⟩|slab
8.2.2 Scaling relationships for the lateral penetration depth
8.2.3 Formulas with the DE and invariant properties
8.2.4 Heuristic formula for ⟨ymax|t⟩|DE
8.3 Statistics of the Radial Penetration Depth in an Infinite Medium
8.3.1 Heuristic formula for maximum penetration depth in an infinite medium
8.4 Comparisons of the Different Formulas for the Maximum Penetration Depth
8.5 Summary Comments on Applications
8.6 Conclusion
References
Chapter 9 Average Photon Distance from Source and Relative Moments
9.1 Statistical Relationships: Displacement of Photons from the Source in an Infinite Homogeneous Medium
9.1.1 Time domain
9.1.2 CW domain
Solutions for ⟨rn⟩(μa) with the DE
9.2 Penetration Depth for all Photons Propagating in an Infinite Medium
9.2.1 Mean penetration depth in the TD
9.2.2 Mean penetration depth in the CW domain
9.3 Penetration Depth for all Photons Propagating through a Slab
9.3.1 Mean penetration depth in the TD
9.3.2 Mean penetration depth in the CW domain
9.4 Conclusion
References
Chapter 10 Hybrid Solutions of the Radiative Transfer Equation
10.1 General Hybrid Approach to the Solutions for the Slab Geometry
10.2 Analytical Solutions of the Time-Dependent RTE for an Infinite Homogeneous Medium
10.2.1 Almost exact time-resolved Green's function of the RTE for an infinite medium with isotropic scattering
10.2.2 Heuristic time-resolved Green's function of the RTE for an infinite medium with non-isotropic scattering
10.2.3 Time-resolved Green's function of the telegrapher equation for an infinite medium
10.3 Comparison of the Hybrid Models Based on the RTE and Telegrapher Equation with the Solution of the Diffusion Equation
10.4 Conclusion
References
Chapter 11 The Diffusion Equation for a Two-Layered Cylinder
11.1 Photon Migration through Layered Media
11.2 Initial and Boundary Value Problems for Parabolic Equations
11.3 Solution of the DE for a Two-Layer Cylinder
11.4 Examples of Reflectance and Transmittance of a Layered Medium
11.5 General Properties of Light Re-emitted by a Diffusive Medium
11.5.1 Mean time of flight in a generic layer of a homogeneous cylinder
11.5.2 Mean time of flight in a two-layer cylinder
11.5.3 Penetration depth in a homogeneous medium
11.5.4 Light re-emitted by a diffusive medium: summary
11.6 Summary Comments on Applications
11.7 Conclusion
References
Chapter 12 The Diffusion Equation for an N-Layered Cylinder
12.1 Photon Migration through an N-Layered Cylinder
12.1.1 Solution for an N-layered cylinder in the FD and CW domain
12.1.2 Solution for an N-layered cylinder in the TD via Fourier transform
12.1.3 Solution for an N-layered cylinder in the TD via Laplace transform
12.2 Conclusion
References
Chapter 13 Solutions of the Diffusion Equation with Perturbation Theory
13.1 Perturbation Theory in a Diffusive Medium and the Born Approximation
13.2 Perturbation Theory: Solutions for the Infinite Medium
13.2.1 Examples of perturbation for an infinite medium
13.3 Perturbation Theory: Solutions for the Slab
13.3.1 Examples of perturbation for a slab
13.4 Perturbation Approach for Hybrid Models
13.5 Perturbation Approach for a Layered Slab and for Other Geometries
13.6 Absorption Perturbation by Using the Internal Pathlength Moments
13.7 Closed-Form CW Perturbative Solutions of the DE with Absorbing Inclusions
13.7.1 Perturbation theory to the DE: iterative solutions for the CW domain
13.8 Summary Comments on Applications
13.9 Conclusion
References
Chapter 14 Time-Domain Raman and Fluorescence Analytical Solutions
14.1 Theoretical Approach and General Definitions
14.2 Heuristic Model
14.3 Raman Analytical Solutions Based on the Time-Dependent Diffusion Equation
14.3.1 Solution of the DE for the Raman signal in a parallelepiped
14.3.2 Solution of the DE for the Raman signal in a finite cylinder
14.4 Solution of the DE for Time-Resolved Fluorescence in an Infinite Medium
14.4.1 Theoretical approach and general definitions
14.5 Solution of the DE for a Raman Signal with Background Fluorescence
14.5.1 Time-resolved reflectance with the EBPC
14.5.2 Time-resolved reflectance with Fick's law
14.5.3 Improved numerical calculation
14.6 Examples of Raman Re-emission Calculated with Raman Forward Solvers
14.7 Summary Comments on Applications
14.8 Conclusion
References
Chapter 15 Elementary Monte Carlo Methods in Turbid Media
15.1 Photon Packets
15.2 Photon Trajectories
15.3 Photon Detection
15.4 Statistical Error in MC Results
15.5 MC Methods for Handling Photon Packet Weight
15.5.1 Microscopic Lambert-Beer law (mLBL) method
15.5.2 Alternative methods to the mLBL method
15.6 Boundary Conditions in MC: Compatibility between Classical and Anomalous Photon Transport
15.7 Interruption of the Propagation of a Photon Packet: Russian Roulette
15.7.1 Russian roulette applied to the mLBL and AW
15.8 Comparison of the Different Methods
15.8.1 General features
15.9 Conclusion
References
Chapter 16 Reference Monte Carlo Results
16.1 General Remarks
16.2 MC for an Infinite Homogeneous Medium
16.3 MC for a Homogeneous and a Layered Slab
16.4 Monte Carlo Code for a Slab Containing an Inhomogeneity
16.5 Description of the Monte Carlo Program Calculating the Maximum Mean Penetration Depth of Detected Photons
16.6 Description of the Monte Carlo Program Simulating the Raman Signal and the Fluorescence Signal
16.7 Conclusion
References
Chapter 17 Comparisons of Analytical Solutions with Monte Carlo Results
17.1 Introduction
17.2 Comparisons between MC and DE: Homogeneous Medium
17.2.1 Infinite homogeneous medium
17.2.2 Laterally infinite homogeneous slab
17.3 Validation of the DE Solutions for the Mean Maximum and Mean Average Penetration Depth
17.4 Comparison between MC and DE: Homogeneous Slab with an Internal Inhomogeneity
17.5 Comparisons between MC and DE: N-Layered Slab and N-Layered Cylinder
17.5.1 Two-layered slab
17.5.2 Four-layered cylinder
17.6 Comparisons between MC and Hybrid Models
17.6.1 Infinite homogeneous medium
17.6.2 Slab geometry
17.7 Comparisons between the MC and Heuristic Model for Ballistic Photon Detection
17.8 Outgoing Flux: Comparison between Fick and Extrapolated Boundary Partial Current Approaches
17.9 Validation of the DE Solutions for the Raman Signal
17.10 Conclusions
17.10.1 Infinite medium
17.10.2 Homogeneous slab
17.10.3 Layered slab
17.10.4 Slab with inhomogeneities inside
17.10.5 Finite diffusive media
17.10.6 Diffusion approximation: from a theoretical to a practical world
References
Chapter 18 Numerical Implementations and Reference Database
18.1 Numerical Implementation of the Solutions
18.1.1 MATLAB® functions
18.1.2 Previous FORTRAN codes
18.2 Reference Database: Monte Carlo Simulations
18.2.1 Description of the MC-generated data files
References
Appendix A Intuitive Justification of theDiffusion Approximation
References
Appendix B Fick’s Law
Reference
Appendix C Boundary Conditions between Diffusive and Non-Scattering Media
Appendix D Boundary Conditions between Two Diffusive Media
References
Appendix E Diffusion Equation with anInfinite Homogeneous Medium: Separation of Variables and Fourier Transform Methods
E.1 Time-Dependent Source
E.2 Steady-State Source
E.3 Time-Dependent Source: Alternative Quick Method
E.4 CW Photon Flux for an Infinite Non-Absorbing Medium
Reference
Appendix F Anisotropic CW DiffusionEquation with an Infinite Homogeneous Medium:Separation of Variables and Fourier Transform Methods
Appendix G
The Reciprocity Principle for a
Plane Wave and a Pencil Beam
Impinging on a Slab
References
Appendix H
Temporal Integration of the
Time-Dependent Green’s
Function
References
Appendix I
The Diffusion Equation:
Separation of Variables and
Eigenfunction Methods
References
Appendix J
The Diffusion Equation with a
Homogeneous Parallelepiped:
Separation of Variables and
Eigenfunction Methods
Reference
Appendix K
Mean Square Displacement of
the Light Penetration in Turbid
Media Based on the RTE
K.1 Elastically Scattered Light without Inelastic Interaction
K.2 Elastically Scattered Light Including Fluorescence or Raman Scattering
References
Appendix L
Expression for the Normalizing
Factor
References
Appendix M
Finite Integral Transforms
M.1 Finite Hankel Transform of Order n over the Interval [0, a]
M.1.1 Finite Hankel transform of S
x = f 00
x+1x
f 0
x– n2
x2 f
x
M.2 Inverse Finite Hankel Transform
M.3 Finite ``Shifted'' Cosine Transform of a Periodic Function f(y)
M.3.1 Finite ``shifted'' cosine transform of f″(y)
M.4 Inverse Finite ``Shifted'' Cosine Transform
References
Appendix N
Relationship between the
Inverse Fourier Transform and
Inverse Laplace Transform
N.1 Inverse Fourier Transform Expressed as an Inverse Laplace Transform
N.2 Numerical Inverse Laplace Transform
References
Appendix O Equivalence of the MC Methods
O.1 Probability of Detecting a Trajectory Γm with the AW
O.2 Probability of Detecting a Trajectory Γm with the AR
O.3 Probability of Detecting a Trajectory Γm with the mLBL
O.4 Probability of Detecting a Trajectory Γm with the ASPR
O.5 Comparison of the AW, AR, mLBL, and ASPR
Reference
Index
Ch05_online.pdf
Chapter 5 Solutions of the Diffusion Equation for Homogeneous Media
5.1 Solution of the Diffusion Equation for an Infinite Medium: Separation of Variables and Fourier Transform Method
5.2 Improved Solution for the CW Domain: Infinite Medium and Isotropic Scattering
5.3 Solution of the Diffusion Equation for a Slab: Method of Images
5.3.1 The diffusion equation and the choice of light sources
Pencil beam source
Beam source
Lambertian source
Summary comments on applications
5.3.2 Analytical Green's function for transmittance and reflectance
5.4 Solution of the Diffusion Equation for a Slab: Separation of Variables, Fourier Transform, and Eigenfunction Method
5.5 Moments of the Temporal Point Spread Function for a Slab
5.6 Solution of the Diffusion Equation for a Semi-infinite Medium
5.7 Other Solutions for the Outgoing Flux
5.8 Analytical Green's Function for a Parallelepiped
5.8.1 Time domain
5.8.2 CW domain
5.9 Analytical Green's Function for an Infinite Cylinder
5.10 Analytical Green's Function for a Sphere
5.11 Solution of the Diffusion Equation for a Pencil Beam Source Impinging on a Finite Cylinder Geometry
5.12 Ohm's Law for Light
5.12.1 Isotropic source case
Non-absorbing slab
Absorbing slab
5.12.2 Pencil beam source case
Absorbing slab
Non-absorbing slab
5.13 Solutions for a Slab Illuminated by Infinitely Extended Sources
5.13.1 Uniform distribution of isotropic sources inside a slab
5.13.2 Spatially uniform illumination with sources on the external surfaces of a slab
5.14 Solutions of the DE in Transformed Domains
5.14.1 Solutions of the DE in the temporal-frequency domain
5.14.2 Solutions of the DE in the spatial-frequency domain
5.14.3 Solution of the DE in the spatial-frequency domain: Laplace transform approach and semi-infinite medium
Normally incident, sinusoidally modulated light source
Obliquely incident sinusoidally modulated light source
5.15 Angular Dependence of Radiance Exiting a Diffusive Medium
5.16 Comment: The Angular Dependence of Reflectance
5.17 Anisotropic Media
5.17.1 Solution for a slab and a pencil beam source: method of images
5.18 Summary Comments on Applications
5.18.1 Isotropic media
5.18.2 Anisotropic media
5.19 Conclusion
References