LIGHT TRANSPORT IN TISSUE - Oregon Medical Laser Center

221 Pages·1999·0.67 MB·English

LIGHT TRANSPORT IN TISSUE
APPROVED BY
SUPERVISORY COMMITTEE:
Supervisor:
Supervisor:
Dedicated to
My Grandfather
Arthur Isaac Johnson
LIGHT TRANSPORT IN TISSUE
by
SCOTT ALAN PRAHL, B.S.
DISSERTATION
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulflllment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT AUSTIN
December 1988
Acknowledgements
I would like to express my gratitude to Dr. Welch who provided funding, much
needed perspective, and an introduction to the art of the absence of politics; to
Dr. Valvano for the Macintosh II used for all parts of this dissertation, to Dr.
Jacques for \rolling up his sleeves and getting his hands dirty;" to Dr. Pearce for
stimulating (nay exhilarating) Friday afternoon research meetings and general
harassing; to Dr. van Gemert who insisted that what I was doing was \not
unimportant;" to Dr. Yoon for arguments about optical stufi and philosophy;
and to almost doctor Cheong for always being too kind. Finally, I must thank
Anne for loving me through the evolution of each chapter as we (the chapter
and I) metamorphosed from a glimmer of an idea, into equations, graphs, flgures
and tables, into a few typeset pages, and flnally into a monster. Without Anne
I would not be flnished yet, and may not have ever flnished.
iv
LIGHT TRANSPORT IN TISSUE
Publication No.
Scott Alan Prahl, Ph.D.
The University of Texas at Austin, 1988
Supervisors: A. J. Welch
J. W. Valvano
Two numerical solutions for radiative transport in tissue are presented: the
Monte Carlo and the adding-doubling methods. Both methods are appropriate
for tissues with internal re�ection at boundaries and anisotropic scattering pat-
terns. The adding-doubling method yields accurate solutions in one-dimension.
The slower Monte Carlo method is the only exact solution available for flnite
beam irradiance of tissue. Convolution formulas for calculation of �uence rates
for circularly symmetric �at and Gaussian irradiances using the Monte Carlo
impulse response are presented.
The delta-Eddington method is extended to include many boundary con-
ditions appropriate for tissue optics. The delta-Eddington method is compared
with exact methods. Delta-Eddington re�ection and transmission are least accu-
rate for thin tissues and mismatched boundary conditions. Fluence calculations
obtained with the delta-Eddington approximation are inaccurate (>50% error)
for tissues with both mismatched boundaries and high albedos.
A method and theory for the measurement of the phase function of tis-
sue is presented. The method is shown to have a tendency to overestimate the
v
isotropic scattering component in tissues with mismatched boundaries. A graph
is presented to correct the overestimate. The backscattered peak in goniopho-
tometer measurements is shown to result from re�ection of the forward peak
and not from a backward peak in the phase function. Measurements on hu-
man dermis indicate that the phase function can be described by a modifled
Henyey-Greenstein phase function.
A practical method for measuring the optical properties of tissue as a
function of wavelength is presented. Evaluation of the technique indicates that
the method is accurate to 10% for all optical properties of tissue when sample
thicknesses exceed one optical depth. This technique is applied to bloodless
human dermis as a function of wavelength and to bloodless human aorta dur-
2
ing moderate power (»100 mW/mm ) argon laser irradiation as a function of
irradiation time.
vi
Table of Contents
Acknowledgements iv
Abstract v
Chapter 1. Introduction and Background 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 General assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Deflnitions and nomenclature . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Dimensional quantities . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Dimensionless quantities . . . . . . . . . . . . . . . . . . . . . . 5
1.3.3 Phase functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.4 The transport equation . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 2. Monte Carlo 15
2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Fixed stepsize method . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 Variable stepsize method . . . . . . . . . . . . . . . . . . . . . 20
2.1.3 Variance reduction techniques . . . . . . . . . . . . . . . . . . . 22
2.2 Mechanics of photon propagation . . . . . . . . . . . . . . . . . . . . 25
2.3 Phase function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Photon absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Internal re�ection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Veriflcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
vii
Chapter 3. The Adding-Doubling Method 40
3.1 Deflnition of re�ection and transmission operators . . . . . . . . . . . 43
3.2 Derivation of the adding-doubling method . . . . . . . . . . . . . . . 46
3.3 The redistribution function . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Re�ection and transmission of thin layers . . . . . . . . . . . . . . . . 49
3.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 Tabulated values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4. The Delta-Eddington Approximation 63
4.1 Derivation of the difiusion equation . . . . . . . . . . . . . . . . . . . 63
4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Index matching, no incident difiuse light . . . . . . . . . . . . . 69
4.2.2 Index matching, difiuse light incident . . . . . . . . . . . . . . 72
4.2.3 Index mismatch, no incident difiuse light . . . . . . . . . . . . 73
4.2.4 Index mismatch, difiuse light incident . . . . . . . . . . . . . . 76
4.2.5 Glass slide | no incident difiuse light . . . . . . . . . . . . . . 77
4.2.6 Glass slide | difiuse light incident . . . . . . . . . . . . . . . . 78
4.2.7 Index matching, no incident difiuse light, both media scattering 79
4.2.8 Index mismatch, no incident difiuse light, both media scattering 79
4.3 Dimensionless form of the difiusion equation . . . . . . . . . . . . . . 81
4.4 Solution of the one-dimensional difiusion equation . . . . . . . . . . . 84
0
4.4.1 Non-conservative scattering (a < 1), flnite slab . . . . . . . . . 84
0
4.4.2 Non-conservative scattering (a < 1), semi-inflnite slab . . . . . 85
4.4.3 Conservative scattering a = 1, flnite slab . . . . . . . . . . . . . 86
4.4.4 Conservative scattering (a = 1), semi-inflnite slab . . . . . . . . 86
4.5 Re�ection, transmission, and �uence rates in one-dimension . . . . . . 87
4.6 Three-dimensional solution of the difiusion equation . . . . . . . . . . 89
4.6.1 Formal solution of `d(r) in terms of Green’s functions . . . . . 89
4.6.2 The Green’s Function for an Inflnite Slab . . . . . . . . . . . . 92
4.6.3 Explicit Expressions for `d(r) . . . . . . . . . . . . . . . . . . . 93
4.6.4 Flux, Re�ection, and Transmission in Three Dimensions . . . . 95
viii
4.7 Evaluation of the Delta-Eddington Approximation . . . . . . . . . . . 95
4.7.1 Comparison of total re�ection and transmission . . . . . . . . . 95
4.7.2 Comparison of Fluence Rates . . . . . . . . . . . . . . . . . . . 96
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 5. Goniophotometry 104
5.1 Single scattering approximation . . . . . . . . . . . . . . . . . . . . . 105
5.2 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 Tissue preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.1 Corrections for internal re�ection and refraction . . . . . . . . 113
5.4.2 Least squares flt . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.5 Evaluation of the method . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Chapter 6. Spectrophotometry 131
6.1 Inverse method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1.1 Evaluation of the inverse method . . . . . . . . . . . . . . . . . 140
6.1.2 Experimental measurements . . . . . . . . . . . . . . . . . . . 145
6.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Chapter 7. Conclusions 157
7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Appendices 161
Appendix A1. Random Variables with Non-Uniform Density Func-
tions 162
A1.1Analytic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
A1.2Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A1.3Discrete form of the Analytic Method . . . . . . . . . . . . . . . . . . 166
ix
Appendix A2. Internal Re�ection 167
A2.1Basic Re�ection Formulas . . . . . . . . . . . . . . . . . . . . . . . . 167
A2.2Fresnel Re�ection in a Glass Slide . . . . . . . . . . . . . . . . . . . . 170
A2.3Re�ection Moments R0, R1, and R2 . . . . . . . . . . . . . . . . . . . 171
A2.4Star’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
A2.5Keijzer’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 174
A2.6Walsh’s Analytic Solution for R1 . . . . . . . . . . . . . . . . . . . . . 175
A2.7Egan Polynomial Approximation for R1 . . . . . . . . . . . . . . . . . 175
A2.8Polynomial Approximations to R0, R1, and R2 . . . . . . . . . . . . . 181
A2.9Approximations for the Boundary Coe–cient A . . . . . . . . . . . . 181
A2.10The Boundary Condition Parameter in the Presence of a Glass Slide . 185
Appendix A3. Solid Angle Integrals and Dirac-Delta Functions 187
A3.1Integrals over entire spheres . . . . . . . . . . . . . . . . . . . . . . . 187
A3.2Integrals over hemispheres . . . . . . . . . . . . . . . . . . . . . . . . 190
A3.3Delta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A3.4Examples of Delta Functions . . . . . . . . . . . . . . . . . . . . . . . 195
Appendix A4. Numerical Details of the 3D Difiusion Solution 196
A4.1Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
0
A4.1.1One eigenvalue less than …=2¿ . . . . . . . . . . . . . . . . . . 197
0
A4.1.2More than one eigenvalue less than …=2¿ . . . . . . . . . . . . 199
A4.2Summation of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Bibliography 203
Vita 211
x