Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, Second Edition (SPIE...

Library of Congress Cataloging-in-Publication Data
Tuchin, V. V. (Valerii Viktorovich) Tissue optics : light scattering methods and instruments for medical diagnosis / Valery V. Tuchin. -- 2nd ed. p. ; cm. Includes bibliographical references and index. ISBN-13: 978-0-8194-6433-0 ISBN-10: 0-8194-6433-3 1. Tissues--Optical properties. 2. Light--Scattering. 3. Diagnostic imaging. 4. Imaging systems in medicine. I. Society of Photo-optical Instrumentation Engineers. II. Title. [DNLM: 1. Diagnostic Imaging. 2. Light. 3. Optics. 4. Spectrum Analysis. 5. Tissues-- radiography. WN 180 T888t 2007] QH642.T83 2007 616.07'54--dc22 2006034872 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 © 2007 The Society of Photo-Optical Instrumentation Engineers 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 My Grandkids
Preface to the Second Edition xxxix
PART I: AN INTRODUCTION TO TISSUE OPTICS 1
1 Optical Properties of Tissues with Strong (Multiple) Scattering 3
1.1 Propagation of continuous-wave light in tissues 3 1.1.1 Basic principles, and major scatterers and absorbers 3 1.1.2 Theoretical description 11 1.1.3 Monte Carlo simulation techniques 17
1.2 Short pulse propagation in tissues 22 1.2.1 Basic principles and theoretical background 22 1.2.2 Principles and instruments for time-resolved spectroscopy and
imaging 25 1.2.3 Coherent backscattering 26
1.3 Diffuse photon-density waves 28 1.3.1 Basic principles and theoretical background 28 1.3.2 Principles of frequency-domain spectroscopy and imaging of
tissues 31 1.4 Propagation of polarized light in tissues 34
1.4.1 Introduction 34 1.4.2 Tissue structure and anisotropy 35 1.4.3 Light scattering by a particle 38 1.4.4 Polarized light description and detection 40 1.4.5 Light interaction with a random single scattering media 43 1.4.6 Vector radiative transfer equation 47 1.4.7 Monte Carlo simulation 50 1.4.8 Strongly scattering tissues and phantoms 60
1.5 Optothermal and optoacoustic interactions of light with tissues 67 1.5.1 Basic principles and classification 67 1.5.2 Photoacoustic method 71
vii
1.5.3 Time-resolved optoacoustics 74 1.5.4 Grounds of OA tomography and microscopy 76 1.5.5 Optothermal radiometry 80 1.5.6 Acoustooptical interactions 85 1.5.7 Thermal effects 91 1.5.8 Sonoluminescence 93 1.5.9 Prospective applications and measuring techniques 95 1.5.10 Conclusion 103
1.6 Discrete particle model of tissue 104 1.6.1 Introduction 104 1.6.2 Refractive-index variations of tissue 104 1.6.3 Particle size distributions 106 1.6.4 Spatial ordering of particles 108 1.6.5 Scattering by densely packed particle systems 110
1.7 Fluorescence and inelastic light scattering 116 1.7.1 Fluorescence 116 1.7.2 Multiphoton fluorescence 124 1.7.3 Vibrational and Raman spectroscopies 127
1.8 Tissue phantoms 132 1.8.1 Introduction 132 1.8.2 Concepts of phantom construction 133 1.8.3 Examples of designed tissue phantoms 137 1.8.4 Examples of whole organ models 142
2 Methods and Algorithms for the Measurement of the Optical Parameters of Tissues 143
2.1 Basic principles 143 2.2 Integrating sphere technique 192 2.3 Kubelka-Munk and multiflux approach 193 2.4 The inverse adding-doubling (IAD) method 195 2.5 Inverse Monte Carlo method 198 2.6 Spatially resolved and OCT techniques 202 2.7 Direct measurement of the scattering phase function 207 2.8 Estimates of the optical properties of human tissue 209 2.9 Determination of optical properties of blood 212 2.10 Measurements of tissue penetration depth and light dosimetry 222 2.11 Refractive index measurements 226
3 Optical Properties of Eye Tissues 257
3.1 Optical models of eye tissues 257 3.1.1 Eye tissue structure 257 3.1.2 Tissue ordering 264
3.2 Spectral characteristics of eye tissues 276 3.3 Polarization properties 281
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis ix
4 Coherent Effects in the Interaction of Laser Radiation with Tissues and Cell Flows 289
4.1 Formation of speckle structures 289 4.2 Interference of speckle fields 298 4.3 Propagation of spatially modulated laser beams in a scattering medium 299 4.4 Dynamic light scattering 302
4.4.1 Quasi-elastic light scattering 302 4.4.2 Dynamic speckles 303 4.4.3 Full-field speckle technique—LASCA 305 4.4.4 Diffusion wave spectroscopy 310
4.5 Confocal microscopy 315 4.6 Optical coherence tomography (OCT) 319 4.7 Second-harmonic generation 325
5 Controlling of the Optical Properties of Tissues 329
5.1 Fundamentals of tissue optical properties controlling and a brief review 329 5.2 Tissue optical immersion by exogenous chemical agents 335
5.2.1 Principles of the optical immersion technique 335 5.2.2 Water transport 340 5.2.3 Tissue swelling and hydration 341
5.3 Optical clearing of fibrous tissues 343 5.3.1 Spectral properties of immersed sclera 343 5.3.2 Scleral in vitro frequency-domain measurements 359 5.3.3 Scleral in vivo measurements 361 5.3.4 Dura mater immersion and agent diffusion rate 364
5.4 Skin 365 5.4.1 Introduction 365 5.4.2 In vitro spectral measurements 367 5.4.3 In vivo spectral reflectance measurements 372 5.4.4 In vivo frequency-domain measurements 378 5.4.5 OCT imaging 380 5.4.6 OCA delivery, skin permeation, and reservoir function 383
5.5 Optical clearing of gastric tissue 390 5.5.1 Spectral measurements 390 5.5.2 OCT imaging 391
5.6 Other prospective optical techniques 392 5.6.1 Polarization measurements 392 5.6.2 Confocal microscopy 397 5.6.3 Fluorescence detection 397 5.6.4 Two-photon scanning fluorescence microscopy 399 5.6.5 Second-harmonic generation 402
5.7 Cell and cell flows imaging 404 5.7.1 Blood flow imaging 404 5.7.2 Optical clearing of blood 405
x Contents
5.7.3 Cell studies 423 5.8 Applications of the tissue immersion technique 428
5.8.1 Glucose sensing 428 5.8.2 Precision tissue photodisruption 435
5.9 Other techniques of tissue optical properties control 437 5.9.1 Tissue compression and stretching 437 5.9.2 Temperature effects and tissue coagulation 442 5.9.3 Tissue whitening 446
5.10 Conclusion 446
PART II: LIGHT-SCATTERING METHODS AND INSTRUMENTS FOR MEDICAL DIAGNOSIS 449
6 Continuous Wave and Time-Resolved Spectrometry 451
6.1 Continuous wave spectrophotometry 451 6.1.1 Techniques and instruments for in vivo spectroscopy and imag-
ing of tissues 451 6.1.2 Example of a CW imaging system 455 6.1.3 Example of a tissue spectroscopy system 456
6.2 Time-domain and frequency-domain spectroscopy and tomography of tissues 458 6.2.1 Time-domain techniques and instruments 458 6.2.2 Frequency-domain techniques and instruments 463 6.2.3 Phased-array technique 470 6.2.4 In vivo measurements, detection limits, and examples of clini-
cal study 475 6.3 Light-scattering spectroscopy 483
7 Polarization-Sensitive Techniques 489
7.2 Polarized reflectance spectroscopy of tissues 497 7.2.1 In-depth polarization spectroscopy 497 7.2.2 Superficial epithelial layer polarization spectroscopy 500
7.3 Polarization microscopy 501 7.4 Digital photoelasticity measurements 508 7.5 Fluorescence polarization measurements 509 7.6 Conclusion 514
8 Coherence-Domain Methods and Instruments for Biomedical Diagnostics and Imaging 517
8.1 Photon-correlation spectroscopy of transparent tissues and cell flows 517
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xi
8.1.1 Introduction 517 8.1.2 Cataract diagnostics 517 8.1.3 Blood and lymph flow monitoring in microvessels 522
8.2 Diffusion-wave spectroscopy and interferometry: measurement of blood microcirculation 526
8.3 Blood flow imaging 531 8.4 Interferometric and speckle-interferometric methods for the measure-
ment of biovibrations 540 8.5 Optical speckle topography and tomography of tissues 546 8.6 Methods of coherent microscopy 556 8.7 Interferential retinometry and blood sedimentation study 561
9 Optical Coherence Tomography and Heterodyning Imaging 565
9.1 OCT 565 9.1.1 Introduction 565 9.1.2 Conventional (time-domain) OCT 565 9.1.3 Two-wavelength fiber OCT 566 9.1.4 Ultrahigh resolution fiber OCT 567 9.1.5 Frequency-domain OCT 569 9.1.6 Doppler OCT 571 9.1.7 Polarization-sensitive OCT 571 9.1.8 Differential phase-sensitive OCT 574 9.1.9 Full-field OCT 575 9.1.10 Optical coherence microscopy 577 9.1.11 Endoscopic OCT 579 9.1.12 Speckle OCT 581
9.2 Optical heterodyne imaging 583 9.3 Summary 589
Conclusion 591
References 735
Index 825
Nomenclature
2l separation between two point light sources formed in the nodal plane
2Ra diameter of circular aperture
A = log 1/Rd apparent absorbance a numerical coefficient depending on the form of the diffusion
equation a radius of a scatterer (particle), nm or μm A signal amplitude in the frequency-domain measuring technique A acoustic amplitude A = i2 square of the mean value of the photocurrent (the base line of
the autocorrelation function) a′ the largest dimension of a nonspherical particle, nm or μm A0 initial amplitude due to the instrumental response Aac ac component of the amplitude of the photon-density wave Adc dc component of the amplitude of the photon-density wave am more probable scatterer radius, μm an and bn Mie coefficients A(r) describes the optical absorption properties of the tissue at r aT thermal diffusivity of the medium, m2/s Bd detection bandwidth bs accounts for additional irradiation of upper layers of a tissue
due to backscattering (photon recycling effect) c velocity of light in the medium, cm/c c0 velocity of light in vacuum, cm/c C1 and C2 concentrations of molecules in two spaces separated by a
membrane Ca(x, t) concentration of the agent Ca0 initial concentration of the agent cab concentration of absorber in μmol, mmol, or mol cb blood specific heat, J/kg K CHb hemoglobin concentration Cf (x, t) fluid concentration cP specific heat capacity for a constant pressure, J/kg K cs relative concentration of the scattering centers CS average concentration of dissolved matter in two interacting
solutions
xiii
xiv Nomenclature
cV specific heat capacity for a constant volume, J/kg K Cα
n Gegenbauer polynomials C average blood concentration CVrms blood flux or perfusion D = zλ/πL2
φ wave parameter
D photon diffusion coefficient, cm2/c DA diattenuation (linear dichroism) Da agent diffusion coefficient, cm2/c DB coefficient of Brownian diffusion, cm2/c Df fluid coefficient of diffusion, cm2/c d sample (tissue layer or slab) thickness, cm D−1 inverse of the measurement matrix D⊥ dimension of incident light beam across the area where the total
radiant energy fluence rate is maximal (determined from the 1/e2 level), cm
D dimension of incident light beam along the area where the total radiant energy fluence rate is maximal (determined from the 1/e2 level), cm
d unit solid angle about a chosen direction, sr dav average size of a speckle in the far-field zone Df fractal (volumetric) dimension DI structure function of the fluctuation intensity component dp length of the space where the exciting and the probe laser
beams are overlapped, cm ds mean distance between the centers of gravity of the particles DT coefficient of translation diffusion DTf coefficient of translation diffusion for fast process DTs coefficient of translation diffusion for slow process DV diameter of a microvessel dn/dλ material dispersion, 1/nm dn/dT medium (tissue) refractive index temperature gradient, 1/C DPF differential path length factor accounting for the increase in
photon migration paths due to scattering dS thermoelastic deformation, cm E incident pulse energy, J e electron charge E0 incident laser pulse energy at the sample surface (J/cm2)
E0j scattering amplitude of an isolated particle, V/m E⊥i electric field component of the incident light perpendicular to
the scattering plane, V/m Ei electric field component of the incident light parallel to the
scattering plane, V/m Es electric field component of the scattered light parallel to the
scattering plane, V/m
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xv
E⊥s electric field component of the scattered light perpendicular to the scattering plane, V/m
Es scattered electric field vector, V/m Es amplitude of a scattered wave, V/m ET absorbed pulse energy, J E(0) subsurface irradiance (J/cm2)
F (Hct) packing function of RBC F(r) radiant flux density or irradiance, W/cm2
f (t, t ′) describes the temporal deformation of a δ-shaped pulse following its single scattering
f1,2 volume fractions of tissue components fa frequency of acoustic oscillations, Hz fc volume fraction of the collagen in tissue fcp volume fraction of the fluid in the tissue contained inside the
cells fcyl surface fraction of the cylinders’ faces fD Doppler frequency fDs Doppler frequency shift ff volume fraction of the fibers in the tissue fge oscillator strength of transition between the ground and excited
states Fint(θ) interference term taking into account the spatial correlation
of particles fn = gn n’th order moment of the phase function fnc volume fraction of the nuclei in the tissue contained inside the
cells for volume fraction of the organelles in the tissue contained inside
the cells fp pulse repetition rate fr fixed reference (lock-in) frequency fRBCi volume fraction of RBCs fs volume fraction of scatterers fT focal length of the “thermal lens,” cm Fv total volume fraction of the particles fσ material fringe value g1(τ) first-order autocorrelation function (normalized autocorrelation
function of the optical field) g2(ξ) autocorrelation function of intensity fluctuations G domain where radiative transport is examined g scattering anisotropy parameter (the mean cosine of the
scattering angle θ, cos(θ)) G1(τ) autocorrelation function of the scalar electric field, E(t), of the
scattered light G(f ) power spectrum with a Gaussian envelope
xvi Nomenclature
G(r) binary density-density correlation function g2 autocorrelation function of the fluctuation intensity component gd scattering anisotropy factor of dermis ge scattering anisotropy factor of epidermis Gs attenuation factor accounting for scattering and geometry of the
tissue GV gradient of the flow rate H or Hct blood hematocrit H tissue hydration h Planck’s constant h apparent energy transfer coefficient H(r, t) heating function defined as the thermal energy per time and
volume deposited by the light source in the close proportion to the optical absorption coefficient of interest
hν photon energy h(x, y) spatial variations in the thickness of the RPS I (θ)/I (0) normalized scattering indicatrix, 1/sr
≡ p(θ)
I (θ) scattering indicatrix (angular dependence of the scattered light intensity), W/cm2 sr i = (−1)1/2
IAS, IS intensity of the anti-Stokes and Stokes Raman lines for a given vibration state
IF fluorescence intensity Ii irradiance or intensity of the incident light beam, W/cm2
I mean value of the intensity fluctuations I refers to the irradiance or intensity of the light, W/cm2
I⊥(t) intensity of the scattered light polarized orthogonal to the incident light
I (r, s) radiance (or the specific intensity)—average power flux density at a point r in the given direction s, W/cm2 sr
I (r, s, t) time-dependent radiance (or the specific intensity), W/cm2 sr I (0) intensity at the center of the beam I (d) intensity of light transmitted by a sample of thickness d
measured using a distant photodetector with a small aperture (on line or collimated transmittance), W/cm2
I,Q,U , and V Stokes parameters IH , IV , I+45, are the light intensities measured with a horizontal linear I−45, IR , and IL polarizer, a vertical linear polarizer, a +45 linear polarizer,
a −45 linear polarizer, a right circular analyzer, and a left circular analyzer in front of the detector, respectively
Iin(ηc) incident radiance angular distribution
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xvii
I(θ) angular distribution of the scattered intensity of a system of N
particles I(x, y) intensity of light transmitted by an RPS I and I⊥ intensities of the transmitted (scattered) light polarized in
parallel or perpendicular to linear polarization of the incident light, respectively
I (θ) angular distribution of the scattered light by a particle, W/cm2 sr I (2ω) SHG signal intensity I0(λ) spectrum of the incident light I0 incident light intensity, W/cm2
Ib intensity of the uniform background light Ic(x, y) intensity of light transmitted in the forward direction (the
specular component) IF and IF⊥ fluorescence intensities of light polarized in parallel or
perpendicular to the exciting electric field vector Ipar and Iper intensity images for light polarized in parallel or perpendicular
to linear polarization of the incident light, respectively Ir(r) and Is(r) intensity distributions of the reference and signal fields Irest and Itest light intensity detected when an object is at rest (brain tissue,
skeletal muscle, etc.) and test (induced brain activity, cold or visual test, training, etc.)
Is(x, y) intensity of the scattered component Isp mean intensity of speckles J flux of matter, mol/s/cm2
J0 zero-order Bessel function J1 first-order Bessel function JS dissolved matter flux JW water flux k = 2π/λ wavenumber ka acoustic wave vector kF rate constant of the fluorescence transition to the ground state
S0 (including its vibrational states) kET rate constant of non-radiative energy transfer to adjacent
molecules K,S Kubelka–Munk parameters Kφ(x) correlation coefficient of phase fluctuations of the boundary
field kB Boltzmann constant kbvo modification factor for reducing the crosstalk between changes
of blood volume and oxygenation kG gas heat conductivity, W/K ki(ω) imaginary part of the photon-density wave vector, 1/cm kr(ω) real part of the photon-density wave vector, 1/cm kIC rate constant of internal conversion to the ground state S0
xviii Nomenclature
kISC rate constant of intersystem crossing from the singlet to the triplet state T1
kT heat conductivity, W/K L total mean path length of a photon L tissue slab thickness L = Dλ/2l period of interferential fringes (D is the mean distance between
eye nodal plane and retina) LD phenomenological coefficient characterizing the interchange
flux induced by osmotic pressure Lφ correlation length of the phase fluctuations of the scattered field l0 amplitude of longitudinal harmonic vibrations Lc correlation length of the inhomogeneities (random relief) lc coherence length of a light source ld = μ−1
eff diffusion length, cm le depth of light penetration into a tissue Lp phenomenological coefficient indicating that the volumetric
flux can be induced by rising hydrostatic pressure Lpd phenomenological coefficient indicating on the one hand the
volumetric flux that can be induced for the membrane by the osmotic pressure, and on the other, the efficiency of the separation of water molecules and dissolved matter
lph = μ−1 t photon mean free path, cm
ls = l/μs scattering length, cm lt = (μ′
s + μa) −1 photon transport mean free path (MFP), cm
lT length of thermal diffusivity (thermal length), cm M molecule weight m ≡ ns/n0 relative refractive index of the scatterers M = I1/I0 intensity modulation depth defined as the ratio between the
intensity at the fundamental frequency I1 and the unmodulated intensity I0
M normalized 4 × 4 scattering matrix (intensity or Mueller’s matrix) (LSM)
M0 zero-moment of the power density spectrum S(ν) of the intensity fluctuations
M1 first-moment of the power density spectrum S(ν) of the intensity fluctuations
mI intensity modulation depth of the incident light Mij LSM elements, i, j = 1–4, 16 elements
Mij LSM element normalized to the first one
M0 ij LSM elements of an isolated particle
mRBC relative index of refraction of RBC mt amount of dissolved matter at the moment t
m∞ amount of dissolved matter at the equilibrium state
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xix
mU ≡ ACdetector/ modulation depth of scattered light intensity DCdetector
n relative mean refractive index of tissue and surrounding media n′′ imaginary part of index of refraction n mean refractive index of the scattering medium N number of scatterers (particles) N = θ/2π fringe order (θ is the optical phase) N0 number of scatterers in a unit volume N1(z) = z × μex
s average number of scattering events experienced by the excitation light before it reached the fluorophore (z is the distance of fluorophore location)
N2(z) = z × μem s average number of scattering events experienced by the emitted
light before it exited the medium (z is the distance of fluorophore location)
N outside vector normal to ∂G
n2f rate of two-photon excitation n0 refractive index of the ground matter n0 average background index of refraction nc refractive index of collagen fibers ncp refractive index of the cytoplasm ne extraordinary refractive index nf refractive index of tissue fibers (collagen and elastin) ng0 refractive index of the ground material of a tissue ng1 effective (mean) group refractive index of a tissue ng2 group refractive index of the homogeneous reference medium
(air) ng group refractive index ngs group refractive index of the scatterers nH2O refractive index of water Ni = fRBCi/ number of RBC in a unit volume of blood VRBCi
Nint = density of interferential fringes per a degree of the view angle [arcsin(λ/ (an angular resolving power of the eye or retinal visual acuity) 2l)]−1
nis refractive index of the interstitial fluid nnc refractive index of cell nucleus no ordinary refractive index nor refractive index of cell organelles Np number of particle diameters ns refractive index of the scattering centers ns refractive index of a scattering particle received by averaging of
refractive indices of tissue components nsc average refractive index of eye sclera Nsp number of speckles within the receiving aperture
xx Nomenclature
NA numerical aperture of the objective or fiber n(x, y) spatial variations in the refractive index of the RPS nt average refractive index of the tissue OD optical density osm osmolarity p packing dimension p porosity coefficient P laser beam power, W P induced polarization Pa coefficient of permeability P0 average incident power, W PC = V/I = degree of circular polarization
[Q2 + U2]1/2
(IF + IF⊥)
P r L(λ) residual polarization degree spectra
Pmin minimal detectable signal power p(I) intensity probability density distribution function p(s) distribution function of photon migration paths in the medium p(s, s′) = p(θ) scattering phase function (the probability density function for
scattering in the direction s′ of a photon travelling in the direction s), 1/sr
pgk(θ) Gegenbauer kernel phase function (GKPF) phg(θ) Henyey-Greenstein phase function (HGPF) PI polarization degree image P 1
n (cosθ) Legendre polynomials p(L) probability density distribution function of relief variations p(r, t) the acoustic wave q scattering vector |q| value of scattering vector q(r) source function (i.e., the number of photons injected into the
unit volume) Q,U , and V represent the extent of horizontal liner, 45 linear, and circular
polarization, respectively Qa asymmetry parameter of the intensity fluctuations qb blood perfusion rate (1/s), defined as the volume of blood
flowing through unit volume of tissue in one second Qs factor of scattering efficiency r transverse spatial coordinate
r = I−I⊥ I+2I⊥ polarization anisotropy
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxi
rF = (IF − IF⊥) fluorescence polarization anisotropy /(IF + 2IF⊥)
R(φ) Stokes rotation matrix for angle φ
r radius vector of a scatterer or of a given point where the radiance is evaluated, cm
r⊥||(τ) cross-correlation function (correlation coefficient) for two polarization states
R(λ) and R⊥(λ) reflectance spectra at in parallel and perpendicular orientations of polarization filters
R reflection operator R 4 × 1 response vector corresponding to the four retarder/
analyzer settings Ra reflectance from the backward surface of the sample
impregnated by an agent Rθ(λ) spectrum of light scattered under the angle (θ+ dθ)
r0 radius of the incident light beam, cm Rbd distance between the axis of exciting laser beam and the
acoustic detector, cm Rd diffuse reflectance RF = [(n − l)/ coefficient of Fresnel reflection
(n + l)]2
RG gas cell radius, cm rh hydrodynamic radius of a particle Ro dimension (radius for a cylinder form) of a bioobject, cm rp radius of the pinhole rRBC radius of RBC rs radius of the scattered beam in the observation plane Rs reflectance from the backward surface of the control sample rsd distance between light source and detector at the tissue surface
(source-detector separation), cm R(η′
c,ηc) reflection redistribution function RTCS osmotic pressure R(z) optical backscattering or reflectance s total photon path length (or mean path length of a photon) S hemoglobin oxygen saturation S heat source term, W/m3
S sample area SD surface of detection S Stokes vector S Stokes vector calculated on the basis of experimental data Ss Stokes vector of the scattered light Si Stokes vector of the incident light s and s′ directions of photon travel or unit vectors for incident and
scattered waves
xxii Nomenclature
|s| = 2k sin(θ/2) magnitude of the scattering wave vector k = 2πn/λ0 S0 unit vector of the direction of the incident wave S1 unit vector of the direction of the scattered wave S(r, s) incident light distribution at ∂G
S(f ) power spectrum of intensity fluctuations of the speckle field S(q) structure factor S3(θ) 3-D structure factor S2(θ) 2-D structure factor S(ω) spectrum of intensity fluctuations S1-4 elements of the amplitude scattering matrix (S-matrix) or Jones
matrix Sr(t) surface radiometric signal S(t) describes the shape of the irradiating pulse Ta acoustic period Tθ(λ) transmission spectrum when a measuring system with a finite
angle of view is used (the collimated light beam with some addition of a forward-scattered light in the angle range 0 to θ is detected)
t0 spatially independent amplitude transmission of the RPS t1 the first moment of the distribution function f (t, t ′); time
interval of an individual scattering act, s t2 = 1/(μtc) average interval between interactions, s T absolute temperature T exposure time, s T (r) change in tissue temperature at point r T (η′
c,ηc) transmission redistribution function Ta arterial blood temperature, K tb blood temperature Tc(λ) collimated transmission spectrum Tc collimated transmittance Td diffuse transmittance Ts and Te temperature of the tissue surface and environment, respectively ts(x, y) amplitude transmission coefficient of an RPS Tt = Tc + Td total transmittance Tt(λ) total transmission spectrum t time, s U(r) total radiant energy fluence rate, W/cm 2
U averaged amplitude of the output signal of the homodyne interferometer
Um maximum of the total radiant energy fluence rate, W/cm2
V illuminated volume V volume of the tissue sample v velocity of motion of the object with respect to the light beam VC volume of collagen fibers
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxiii
Ve volume of an erythrocyte VM molecular volume
V (z) contrast of average-intensity fringes V phase velocity of a photon-density wave, cm/s V0 contrast of the interference pattern in the initial laser beam va velocity of acoustic waves in a medium, m/s VI contrast of the intensity fluctuations vp radius (in optical units) of conjugate pinholes of a confocal
microscopic system VP contrast of the polarization image VRBC RBC volume, μm3
Vrms root-mean-square speed of moving particles Vs velocity of a moving particle
V S partial mole volumes of dissolved matter vsh shear rate VV parameter directly proportional to the flow velocity V W partial mole volumes of water w laser (Gaussian) beam radius (or radius of a cylinder
illuminated by a laser beam), cm wp probing laser beam radius, cm w0 radius of the Gaussian beam waist x0 fixed point at the plane where speckles are observed x = 2πa/λ size (diffraction) parameter z linear coordinate (depth inside the medium), cm Z normalized phase matrix z0 = (μ′
s) −1, cm
Greek α(z) reflectivity of the sample at the depth of z
αHb spectrally-dependent coefficient of proportionality of hemoglodin imaginary refractive index on its concentration
αi incidence angle of the beam, angular degrees β coefficient of volumetric expansion, 1/K β modulation depth of photoelectric signal of the interferometer β orientation averaged first molecular hyperpolarizability βsb parameter of self-beating efficiency Grüneisen parameter (dimensionless, temperature-dependent
factor proportional to the fraction of thermal energy converted into mechanical stress)
eff effective shear rate T relaxation parameter γ = cP /cV ratio of specific heat capacities γ11(t) degree of temporal coherence of light ψ phase shift in a measuring interferometer, degrees
xxiv Nomenclature
a halfwidth of the radii distribution Evib = hνvib energy of the molecular vibration state F width of the averaged spectrum k wavenumber shift L = (nh) optical length (relief) variations n refractive indices difference noe refractive indices difference due to birefringence of form p change of pressure, Pa p hydrostatic pressure, Pa Rr(λ) differential residual polarization spectra V change of illuminated volume caused by local temperature
increase, m3
w change of radius of a cylinder illuminated by a laser beam caused by local temperature increase, cm
x linear shift of the center of maximal diffuse reflection, cm z longitudinal displacement of the object T local temperature increase, C T optical clearing (enhancement of transmittance) xT amplitude of mechanical oscillations, cm n mean refractive index variation 0 initial phase due to the instrumental response θ angular width of the coherent peak in backscatter, angular
degrees λ bandwidth of a light source ξ change in variable I(r) deterministic phase difference of the interfering waves phase shift relative to the incident light modulation phase (phase
lag), degrees r2(τ) mean-square displacement of a particle within time interval τ
I(r) random phase difference TS temperature change of a sample, C TG temperature change of a surrounding gas, C t time shift of the transmitted pulse peak I(r) time-dependent phase difference related to the motion of an
object δ = 2πdn/λ0 phase delay (retardance) of optical field δn and δd parameters related to the average contributions per photon free
path and per scattering event, respectively, to the ultrasonic modulation of light intensity
δoe = 2πdnoe/ phase delay of optical field due to birefringence λ0
δp(ω) amplitude of harmonically modulated pressure, Pa δp(t) time-dependent change of pressure, Pa ∂G boundary surface of the domain G
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxv
∂n/∂p adiabatic piezo-optical coefficient of the tissue zopt optical path length εab absorption coefficient measured in mol−1 cm−1
εd λ
extinction coefficient of deoxyhemoglobin measured in mol−1 cm−1
εo λ
extinction coefficient of oxyhemoglobin measured in mol−1 cm−1
ελ extinction coefficient at the wavelength λ in mol−1 cm−1
η absolute viscosity of the medium η(a) or η(2a) radii (a) or diameter (2a) distribution function of scatterers ηc cosine of the polar angle ηF fluorescence quantum yield ηq quantum efficiency of the detector η′(2a) correlation-corrected distribution η(2a)
θ scattering angle, angular degrees θI angle between the wave vectors of the interfering fields θGK
rnd GKPFrandom scattering angle
= σsca σext
= μs μt
albedo for single scattering (characterizes the relation of scattering and absorption properties of a tissue)
′ = μ′ s
transport albedo
photon-density wavelength, cm I spacing of interference fringes λ = λ0/n wavelength in the scattering medium, nm λ0 wavelength of the light in vacuum, nm λp wavelength of the probe beam, nm μ′
a absorption coefficient at the thermal radiation emission wavelength, 1/cm
μa absorption coefficient, 1/cm μb volume-averaged backscattering coefficient, 1/cm sr μeff = [3μa(μ
′ s+ effective attenuation coefficient or inverse diffusion length,
μa)]1/2 1/cm μge change in dipole moment between the ground and excited states μn norder statistical moment (n = 1,2,3 . . .)
μ′ s = (1 − g)μs reduced (transport) scattering coefficient, 1/cm
μs scattering coefficient, 1/cm μex
s scattering coefficient of the excitation light, 1/cm μem
s scattering coefficient of the emitting light, 1/cm μt = μa + μs extinction coefficient (interaction or total attenuation
coefficient), 1/cm μ′
t = μa + μ′ s transport coefficient
|μ(z)| modulus of the transverse correlation coefficient of the complex amplitude of the scattered field
xxvi Nomenclature
νI exponential factor of the spatial intensity fluctuations ξ ≡ x or t spatial or temporal variable ξI characteristic depolarization length for linearly (i = L) and
circularly (i = C) polarized light ρ medium density, kg/m3
ρ polarization azimuth ρa volume density of absorbers, 1/cm3
ρb blood density (kg/m3)
ρG gas density, kg/m3
ρs volume density of the scatterers, 1/cm3
ρ(s) probability density function of the optical paths σ halfwidth of the particle size distribution σ = −(Lpd/Lp) molecular reflection coefficient (σ1 − σ2) difference in the in-plane principle stress σabs absorption cross-section of a particle, cm2
σabs specific absorption coefficient, cm−1
σext extinction cross section of a particle, cm2
σf photon absorption cross-section σh standard deviation of the altitudes (depths) of inhomogeneities σI standard deviation of the intensity fluctuations σL standard deviation of relief variations (in optical lengths) σm width of the skewed logarithmic distribution function for the
volume fraction of particles of diameter 2a
σs(2ai) optical cross-section of an individual particle with diameter 2ai
and volume vi , cm2
σsca specific scattering coefficient, cm−1
sca scattering cross-section for the system of particles, cm σφ standard deviation of the phase fluctuations of the scattered field
σ2 I variance of the intensity fluctuations
σ2 s spatial variance of the intensity in the speckle pattern
σ2 U variance of the output signal of the homodyne interferometer
τ delay time τ lifetime of the excited state τ = ∫ s
0 μtds optical thickness τa = 1/μac average travel time of a photon before being absorbed, s τc correlation time of intensity fluctuations in the scattered field τd time delay between optical and acoustical pulses, s τL duration of a laser pulse, s τp pulse duration τr time constant of rotational diffusion τth time delay for the “thermal lens” technique, s τT thermal relaxation time of the photoacoustic cell, s
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxvii
τ−1 B ≡ T characterizes the random (Brownian) flow
τ−1 S
∼= characterizes the directed flow 0.18GV |q|lt
(x, y) random phase shift introduced by the RPS at the (x, y) point p(ω) phase-lag of harmonically modulated pressure, degrees φ(t) phase shift defined by a scatterer position angle of observation and azimuthal angle, angular degrees d deflection angle of a probe laser beam, angular degrees solid angle, sr v frequency of harmonic vibrations ω = 2πf modulation frequency, 1/s ωa fundamental acoustic frequency ωge energy difference between the ground and excited states ωp packing factor of a medium filled with a volume fraction fs
of scatterers (ωt − θ) phase of the photon-density wave χ(n) the nth order nonlinear susceptibility
Acronyms
ac alternating current ADC amplitude-digital convertor AF autocorrelation function AF autofluorescence AHA α-hydroxy acid AO acoustooptical AOM acoustooptic modulator AOT AO tomography APD avalanch photodetector ALA δ-aminolevulenic acid ATR-FTIR attenuated total reflectance Fourier transform infrared AW acoustic waves BEM boundary-element method BSA bovine serum albumin BW birefringent wedges CBF cerebral blood flow CCD charge-coupled device CDI coherent detection imaging CFD constant-fraction discriminator CIE Commission Internationale de l’Eclairage which is the French
title of the International Commission on Illumination CIN cervical intraepithelial neoplasia CIS carcinoma in situ CM confocal microscopy CMOS complementary metal-oxide-semiconductor CP OCT cross-polarization OCT CPU central processing unit CSF cerebrospinal fluid CT computed tomography CW continuous wave DBM double-balanced mixer dc direct current DG delay generator DIS double integrating sphere DMSO dimethyl sulfoxide DNA deoxyribonucleic acid
xxix
DOCP degree of circular polarization DOLP degree of linear polarization DOP degree of polarization DOPA 3,4-dihydroxyphenylalanine DOPE dioleylphosphatidylethanolamine DPF differential path length factor DPS OCT differential phase-sensitive OCT DT diffusion theory DWS diffusion wave spectroscopy EDL extensor digitorum longus EDTA ethylenediaminetetraacetic acid EEM excitation-emission map ESR erythrocyte sedimentation rate FAD flavin dinucleotide FD frequency domain FDA Food Drug Administration FD-LUM frequency-domain luminescence FD-OTR frequency-domain OTR FDPM frequency-domain photon migration FDTD finite-difference time-domain FFT fast Fourier transform FG function generator FMN flavin mononucleotide FRAP fluorescence recovery after photobleaching FWHM full width half maximum GHb glycated hemoglobin GK Gergenbauer kernel GKPF Gegenbauer kernel phase function GPM goniophotometric measurements GRIN gradient index Hb hemoglobin HEM human epidermal membrane HCM human cervical mucus Hct hematocrit HPD hematoporphirin derivative HG Henyey-Greenstein HGPF Henyey-Greenstein phase function HWHM half width half maximum IAD inverse adding–doubling ICG indocyanine green IF intermediate frequency IFS interfibrillar spacing IMC inverse Monte Carlo IMS intermolecular spacing
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxxi
IC25 Infracyanine 25 IQ in-phase quadrature IR infrared IS integrating sphere KDP kalium dihydrophosphate KMM Kubelka-Munk model LASCA laser speckle contrast analysis LD laser diode LDA laser Doppler anemometer LDI laser Doppler imaging LDM laser Doppler microscope LED light-emitting diode LID lattice of islet damage LIPT laser-induced pressure transient LITT laser-induced interstitial thermal therapy LO local oscillator LPF low-pass filter LSI laser speckle imaging LSM light-scattering matrix LSMM laser scattering matrix meter LSS light scattering spectroscopy LVDS low-voltage differential signaling MAR modified amino resin MB methylene blue MBG mean blood glucose MC Monte Carlo MCA multichannel analyzer MCP-PMT multichannel plate-photomultiplier tube MED minimal erythema dose MFP mean free path length MIM multispectral imaging micropolarimeter MIR middle infrared MO microobjective MONSTIR multichannel optoelectronic near-infrared system for
time-resolved image reconstruction MPS maximum permissible exposure MR magnetic resonance MRI MR imaging MTT meal tolerance test NA numerical aperture NAD nicotinamide adenine dinucleotide NAD+ oxidized form of NAD NADH reduced form of NAD NIR near infrared
xxxii Acronyms
OA optoacoustic OAT OA tomography OCA optical clearing agent OCI optical coherence interferometry OCM optical coherence microscopy OCP optical clearing potential OCT optical coherence tomography OD optical density OGTT oral glucose tolerance test OMA optical multichannel analyzer OT optothermal OTR optothermal radiometry PA photoacoustic PAM photoacoustic microscopy PBS phosphate buffered saline PC personal computer PD photodetector PDF probability distribution function PDMD phase-delay measurement device PDT photodynamic therapy PDWFCS photon-density wave-fluctuation correlation spectroscopy PEG polyethylene glycol PG propylene glycol PHA pulse-height analysis PM polarization-maintaining PMT photomultiplier tube POS polyorganosiloxane PPG polypropylene glycol PRS polarized reflectance spectroscopy PS OCT polarization-sensitive OCT PS-OLCR phase-sensitive optical low-coherence reflectometer PT photothermal PTFC PT flow cytometry PTM PT microscopy PTR PT radiometry PVDF polyvinyldenefluoride PY Percus-Yevick QELS quasi-elastic light scattering RBC red blood cell RC relative contrast RCM reflection confocal microscopy RF radio frequency RGA Rayleigh-Gans approximation rms root mean square
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxxiii
RNA ribonucleic acid RNFL retinal nerve fiber layer ROI region of interest RPS random phase screen RSODL rapid scanning optical delay line RTE radiative transfer equation RTT radiation transfer theory SC stratum corneum SEM standard error of the mean SERS surface-enhanced Raman scattering SHG second harmonic generation SMF skeletal muscle fibers SL sonoluminescence SLD superluminescent diode SLT SL tomography SMLB spatially-modulated laser beam SNR signal-to-noise ratio SPR spatially resolved reflectance SSB single sideband SRR spatially resolved reflectance ST Staphylococcus toxin TAC time-to-amplitude convertor TD time-domain TDM time division multiplex TDM transillumination digital microscopy TEWL transepidermal water loss TGS thermal gradient spectroscopy THb total hemoglobin TMP trimethylolpropanol TOAST time-resolved optical absorption and scattering tomography TRS time-resolved spectroscopy US ultrasound UV ultraviolet VOA variable optical attenuator WP Wollaston prism VRTE vector radiative transfer equation VTW virtual transparent window WDM wavelength division multiplex WHO World Health Organization
Preface to First Edition
Many up-to-date medical technologies are based on recent progress in physics, including optics.1–102 An interesting example relevant to the topic of this tutorial is provided by computer tomography.1,4 X-ray, magnetic resonance, and positron- emission imaging techniques are extensively used in high-resolution studies of both anatomical structures and local metabolic processes. Another safe and technically simple tool currently in use is diffuse optical tomography.1,3,4,6,15,28,71
From the viewpoint of optics, biological tissues and fluids (blood, lymph, saliva, mucus, gastric juice, urine, aqueous humor, semen, etc.) can be separated into two large classes.1–69,92–97,101 The first class includes strongly scattering (opaque) tissues and fluids, such as skin, brain, vessel walls, eye sclera, blood, and lymph. The optical properties of these tissues and fluids can be described within the framework of the model of multiple scattering of scalar or vector waves in a randomly nonuniform absorbing medium. The second class consists of weakly scattering (transparent) tissues and fluids, such as cornea, crystalline lens, vitreous humor, and aqueous humor of the front chamber of the eye. The optical properties of these tissues and fluids can be described within the framework of the model of single scattering (or low-step scattering) in an ordered isotropic or anisotropic medium with closely packed scatterers with absorbing centers.
The vector nature of light waves is especially important for transparent tissues, although much attention has been recently focused also on the inves- tigation of polarization properties of light propagating in strongly scattering media.3,5,6,8–10,23,28,43,59–64,69,70 In scattering media, the vector nature of light waves is manifested as polarization of an initially nonpolarized light beam or as depolarization (generally, the change in the character of polarization) of an initially polarized beam propagating in a medium. Similar to coherence properties of a light beam reflected from or transmitted through a biological object, polarization parameters of light can be employed as a selector of photons coming from different depths in an object.
The problems of optical diagnosis and spectroscopy of tissues are concerned with two radiation regimes: continuous wave and time- resolved.1,3,4,6,12,14,15,28,31,71,92 The latter is realized by means of the exposure of a scattering object to short laser pulses (∼10−10 to 10−12 s) and the subsequent recording of scattered broadened pulses (the time-domain method), or by irradiation with modulated light, usually in the frequency range 50 MHz to 1000 MHz and recording the depth of modulation of scattered light intensity and the corresponding phase shift at modulation frequencies (the frequency-domain or phase
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xxxvi Preface to First Edition
method). The time-resolved regime is based on the excitation of the photon-density wave spectrum in a strongly scattering medium, which can be described in the framework of the nonstationary radiation transfer theory (RTT). The continuous radiation regime is described by the stationary RTT.
Many modern medical technologies employ laser radiation and fiber-optic devices.1–7 Since the application of lasers in medicine has both fundamental and technical purposes, the problem of coherence is very important for the analysis of the interaction of light with tissues and cell ensembles. On the one hand, this problem can be considered in terms of the loss of coherence due to the scattering of light in a randomly nonuniform medium with multiple scattering, or the change in the statistics of speckle structures of the scattered field. On the other hand, this problem can be interpreted in terms of the appearance of an ampli- fied, coherent, sharply directed component in backscattered radiation under conditions when a tissue is probed with an ultrashort laser pulse.1,3,73,74 The coherence of light is of fundamental importance for the selection of photons that have experienced a small number of scattering events or none, as well as for the generation of speckle-modulated fields from scattering phase objects with single and multiple scattering.1,3,75–77 Such approaches are important for coherent tomography, diffractometry, holography, photon-correlation spectroscopy, laser Doppler anemometry, and speckle interferometry of tissues and fluxes of biological fluids.1,3,5,15,22,28,76–83 The use of optical sources with a short coherence length opens up new opportunities in coherent interferometry and tomography of tissues, organs, and blood flows.1,3,8,17,18,77,84
The transparency of tissues reaches its maximum in the near infrared (NIR), which is associated with the fact that living tissues do not contain strong intrinsic chromophores that would absorb radiation within this spectral range. Light penetrates into a tissue for several centimeters, which is important for the transillumination of thick human organs (brain, breast, etc.). However, tissues are characterized by strong scattering of NIR radiation, which prevents one from obtaining clear images of localized inhomogeneities arising in tissues due to various pathologies, e.g., tumor formation, a local increase in blood volume caused by a hemorrhage or the growth of microvessels. Strong scattering of NIR radiation also imposes certain requirements on the power of laser radiation, which should be sufficient to ensure the detection of attenuated fluxes. Special attention in optical tomography and spectroscopy is focused on the development of methods for the selection of image-carrying photons or detection of photons providing the information con- cerning the optical parameters of the scattering medium. These methods employ the results of fundamental studies devoted to the propagation of laser beams in scattering media.1,3,4,6,15,28,31,71,92
Another important area in which deep tissue probing is practiced is reflecting spectroscopy, e.g., optical oxymetry for the evaluation of the degree of hemoglobin oxygenation in working muscular tissue, the diseased neonatal brain, or the active brain of adults.1,3,4
This tutorial is primarily concerned with light-scattering techniques recently developed for quantitative studies of tissues and optical cell ensembles. It discusses
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis xxxvii
the results of theoretical and experimental investigations into photon transport in tissues and describes methods for solving direct and inverse scattering problems for random media with multiple scattering and quasi-ordered media with single scattering, in order to model different types of tissue behavior. The theoretical con- sideration is based on stationary and nonstationary radiation transfer theories for strongly scattering tissues, the Mie theory for transparent tissues, and the numerical Monte Carlo method, which is employed for the solution of direct and inverse problems of photon transport in multilayered tissues with complicated boundary conditions.
These are general approaches extensible to the examination of a large number of abiological scattering media. It is worthwhile to note that many known methods of scattering media optics (e.g., the integrating sphere technique) were brought to perfection when used in biomedical research. Concurrently, new measuring systems and algorithms for the solution of inverse problems have been developed that are useful for scattering media optics in general. Moreover, the improvement of certain methods was undertaken only because they were needed for tissue studies; this is especially true of the diffuse photon-density waves method, which is promising for the examination of many physical systems: aqueous media, gels, foams, air, aerosols, etc.
Based on such fundamental optical phenomena as elastic and quasi-elastic (sta- tic and dynamic) scattering, diffraction, and interference of optical fields and photon density waves (intensity waves), we will discuss optical methods and instruments offering much promise for biomedical applications. Among these are spectrophotometry and polarimetry; time-domain and frequency-domain spectroscopy and imaging systems; photon-correlation spectroscopy; speckle interferometry; coherent topography and tomography; phase, confocal, and heterodyne microscopy; and partial coherence interferometry and tomography.
I am grateful to Terry Montonye, Donald O’Shea, Alexander Priezzhev, Barry Masters, and Rick Hermann for their valuable suggestions and comments on preparation of this tutorial.
I am very thankful to Andre Roggan, Lihong Wang, and Alexander Oraevsky for their valuable comments and constructive criticism of the manuscript.
I greatly appreciate the cooperation and contribution of all my colleagues, especially D. A. Zimnyakov, V. P. Ryabukho, S. S. Ul’yanov, I. L. Maksimova, V. I. Kochubey, S. R. Uts, I. V. Yaroslavsky, A. B. Pravdin, G. G. Akchurin, I. L. Kon, E. I. Zakharova, A. A. Bednov, A. A. Chaussky, S. Yu. Kuz’min, K. V. Larin, I. V. Meglinsky, A. A. Mishin, I. S. Peretochkin, and A. N. Yaros- lavskaya.
I am very thankful to attendees of my short courses on biomedical optics, which I have giving during SPIE Photonics West International Symposia since 1992, for their good questions, fruitful discussions, and critical evaluations of presented materials. Their responses were very valuable for preparation of this volume. I am especially grateful to Michael DellaVecchia, Hatim Carim, Sandor Vari, M. Pais Clemente, Haishan Zeng, Leon Sapiro, and Zachary Sacks, who became my good friends and colleagues for many years.
xxxviii Preface to First Edition
Prolonged collaboration with the University of Pennsylvania, my fruitful discussions with Britton Chance, Shoka Nioka, Arjun Yodh, David Boas, and many others were very helpful in writing this book.
My joint chairing with Halina Podbielska, Ben Ovryn, and Joe Izatt of the SPIE Conference on Coherence Domain Optical Methods in Biomedical Science and Clinical Applications also was very helpful.
The original part of this work was supported within the program “Leading Sci- entific Schools” of the Russian Foundation for Basic Research (Project No. 96-15- 96389), USA–Russia CRDF grant RB1-230, and ISSEP grants p97-372, p98-768, and p99-703 within the program “Soros Professors.”
I would like to thank all my numerous colleagues and friends all over the world who kindly sent me reprints of their papers, which were used in this tutorial and made my work much easier, especially Y. Aizu, J. D. Briers, Z. Chen, B. Devaraj, A. F. Fercher, M. Ferrari, J. G. Fujimoto, M. J. C. van Gemert, E. Gratton, J. Greve, A. H. Hielscher, S. L. Jacques, R. G. Johnston, G. W. Kattawar, M. Keijzer, S. M. Khanna, A. Ya. Khairulllina, A. Knuettel, J. R. Lakowicz, M. W. Lindner, Q. Luo, R. L. McCally, W. P. van de Merwe, G. Mueller, F. F. M. de Mul, M. S. Pat- terson, B. Pierscionek, H. Rinneberg, P. Rol, W. Rudolph, B. Ruth, J. M. Schmitt, W. M. Star, R. Steiner, H. J. C. M. Sterenborg, L. O. Svaasand, J. E. Thomas, B. J. Tromberg, A. J. Welch, and J. R. Zip.
I would like to say a few words in memory of Pascal Rol, my good friend and colleague with whom I have organized many SPIE meetings. Pascal died suddenly on January 10, 2000. The reader will find many of his excellent results on scleral tissue optics in this tutorial. He has made many outstanding contributions to biomedical optics, and I will always remember him as a good scientist and friendly person.
I am very thankful to Ruth Haas, Erika Wittmann, and Sue Price for their assistance in editing and production of the book, and to S. P. Chernova and E. P. Sav- chenko for their help in the preparation of the figures.
Last, but not least, I express my gratitude to my wife, Natalia, and all my family for their support, understanding, and patience.
Valery Tuchin April 2000
Preface to the Second Edition
This is the second edition of the tutorial Tissue Optics: Light Scattering Meth- ods and Instruments for Medical Diagnosis first published in 2000. The last seven years, since the printing of the first edition, have seen intensive growth of research and development in tissue optics, particularly in the field of tissue diagnostics and imaging.103–147 Further developments of light-scattering techniques for the quantitative evaluation of optical properties of normal and pathological tissues and cell ensembles have occurred. New results on theoretical and experimental investigations into light transport in tissues and methods for solving direct and inverse scattering problems for random media with multiple scattering and quasi-ordered media have been found. A few specific fields, such as optical coherence tomography (OCT)108–111,115,116,126,127,129,130,136,142 and polarization- sensitive technologies,129,130,135,136,138,139 which are very promising for optical medical diagnostics and imaging, have developed rapidly over the last few years. The optical clearing method, based on reversible reduction of tissue scattering due to refractive index matching of scatterers and ground matter, has also been of great interest for research and application since the last edition.129,132,136,139,140 Fur- ther developments of Raman and vibrational spectroscopies104,105,123,130,132,136,143
and multiphoton microscopy114,119,122,130,132,136,137 applied to morphology and the functioning of living cells and tissues have been provided by many research groups.
This new edition of this book is conceptually the same as the first one. It is also divided into two parts: Part I describes tissue optics fundamentals and basic research, and Part II presents optical and laser instrumentation and medical applications. The author has corrected misprints, updated the references, and added some new results mostly on tissue optical properties measurements (Chapter 2) and polarized light interaction with turbid tissues (Section 1.4). Recent results on polarization imaging and spectroscopy techniques (Chapter 7), as well as on OCT developments and applications (Chapter 9) are also overviewed. Materials on controlling tissue optical properties (Chapter 5) and optothermal and optoacoustic interactions of light with tissues (Section 1.5) are updated. Brief descriptions of fluorescent, nonlinear, and inelastic light scattering spectroscopies are provided in Chapter 1.
I am grateful to Sharon Streams for her suggestion to prepare the second edition of the tutorial and for her assistance in editing of the book. I also would like to thank Merry Schnell for her assistance on the final stage of book editing and production.
I am very thankful to attendees of my short courses “Coherence, Light Scat- tering, and Polarization Methods and Instruments for Medical Diagnosis,” “Tissue Optics and Spectroscopy,” “Tissue Optics and Controlling of Tissue Optical Prop- erties,” and “Optical Clearing of Tissues and Blood,” which I have given during
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SPIE Photonics West Symposia, SPIE/OSA European Conferences on Biomedical Optics, and OSA CLEO/QELS Conferences over last seven years, for their stim- ulating questions, fruitful discussions, and critical evaluations of presented materials. Their responses were very valuable for preparation of this edition. My joint chairing with Joseph A. Izatt and James G. Fujimoto of the SPIE Conference on Coherence Domain Optical Methods and Optical Coherence Tomography in Bio- medicine also was very helpful.
The original part of this work was supported within the Russian and international research programs by grant N25.2003.2 of President of Russian Fed- eration “Supporting of Scientific Schools,” grant N2.11.03 “Leading Research- Educational Teams,” contract No. 40.018.1.1.1314 “Biophotonics” of the Ministry of Industry, Science and Technologies of RF, grant REC-006 of CRDF (U.S. Civil- ian Research and Development Foundation for the Independent States of the For- mer Soviet Union) and the Russian Ministry of Education, the Royal Society grants for a joint projects between Cranfield University (UK) and Saratov State Univer- sity, grants of National Natural Science Foundation of China (NSFC), grant of Fed- eral Agency of Education of RF No. 1.4.06, RNP.2.1.1.4473, CRDF grants BRHE RUXO-006-SR-06 and RUB1-570-SA-04, and by Palomar Medical Technologies Inc., MA, USA.
I greatly appreciate the cooperation, contributions, and support of all my colleagues from Optics and Biomedical Physics Division of Physics Department and Research-Educational Institute of Optics and Biophotonics of Saratov State Uni- versity, especially A. N. Bashkatov, I. V. Fedosov, E. I. Galanzha, E. A. Genina, I. L. Maksimova, I. V. Meglinski, V. I. Kochubey, V. P. Ryabukho, A. B. Pravdin, G. V. Simonenko, Yu. P. Sinichkin, S. S. Ul’yanov, D. A. Yakovlev, and D. A. Zim- nyakov.
I would like to thank all my numerous colleagues and friends all over the world for collaboration and sending materials which were used in this tutorial and made my work much easier, especially P. E. Andersen, J. F. de Boer, Z. Chen, P. M. W. French, J. G. Fujimoto, V. M. Gelikonov, P. Gupta, C. K. Hitzenberger, J. A. Izatt, S. L. Jacques, A. Kishen, S. J. Kirkpatrick, A. Knüttel, J. R. Lakow- icz, K. V. Larin, G. W. Lucassen, Q. Luo, B. R. Masters, K. Meek, G. Mueller, F. F. M. de Mul, L. T. Perelman, A. Podoleanu, A. V. Priezzhev, F. Reil, J. Ro- driguez, H. Schneckenburger, A. M. Sergeev, A. N. Serov, N. M. Shakhova, B. J. Tromberg, L. V. Wang, R. K. Wang, A. J. Welch, A. N. Yaroslavskaya, I. V. Yaroslavsky, P. Zhakharov, and V. P. Zharov, R. Myllylä, S. A. Boppart, M. Meinke, A. Mahadevan-Jansen, T. Troy, L. Oliveira, M. Pais Clemente, and X. H. Hu.
I express my gratitude to my wife, Natalia, and all my family, especially to my daughter, Nastya, and grandchildren, Dasha, Zhenya, and Stepa, for their in- dispensable support, understanding, and patience during my writing this book.
Valery Tuchin June 2007
This first chapter introduces the problem of light (laser beams) transport within strongly (multiple) scattering tissues such as skin, breast, brain, and vessel walls. Basic principles and theoretical descriptions using radiation transfer theory or Monte Carlo (MC) simulation are considered. The propagation of short pulses and photon-density diffusion waves in scattering and absorbing media is analyzed, and the prospects of these methods for tissue spectroscopy and tomography are discussed. Tissue structure and anisotropy, polarization phenomena, optothermal, optoacoustic, and acoustooptical interactions in strongly scattering tissues are described. A discrete-particle model of soft tissue is presented. Fluorescence and inelastic light scattering, including multiphoton fluorescence and vibrational and Raman spectroscopies, are discussed. The design and characterization of tissuelike phantoms for optical diagnostics and light dosimetry are described.
1.1 Propagation of continuous-wave light in tissues
1.1.1 Basic principles, and major scatterers and absorbers
Biological tissues are optically inhomogeneous and absorbing media whose average refractive index is higher than that of air. This is responsible for partial reflection of the radiation at the tissue/air interface (Fresnel reflection), while the remain- ing part penetrates the tissue. Multiple scattering and absorption are responsible for laser beam broadening and eventual decay as it travels through a tissue, whereas bulk scattering is a major cause of the dispersion of a large fraction of radiation in the backward direction. Therefore, light propagation within a tissue depends on the scattering and absorption properties of its components: cells, cell organelles, and various fiber structures.1–3,15,129,130,134,138 The size, shape, and density of these structures; their refractive index relative to the tissue ground substance; and the polarization states of the incident light all play important roles in the propagation of light in tissues.1–3,15,129,130,134,138,145–153
In view of the great diversity and structural complexity of tissues, the development of adequate optical models that account for the scatter and absorption of light is often the most complex step of a study. Two approaches are currently used for tissue modeling. In the framework of the first one, tissue is modeled as a medium with a continuous random spatial distribution of optical parameters;3,129,154,155 the second one considers tissue as a discrete ensemble of scatterers.1–3,15,129,130,134,138,156
3
4 Optical Properties of Tissues with Strong (Multiple) Scattering
The choice of the approach is dictated by both the structural specificity of the tissue under study and the kind of light scattering characteristics that are to be obtained.
Most tissues are composed of structures with a wide range of sizes, and most can be described as a random continuum of inhomogeneities of the refractive index with a varying spatial scale.154,155 Phase contrast microscopy has been used in particular to show that the structure of the refraction index inhomogeneities in mammalian tissues is similar to the structure of frozen turbulence in a number of cases.154 This fact is of fundamental importance for understanding the pecu- liarities of light propagation in tissue, and it may be a key to the solution of the inverse problem of tissue structure reconstruction. This approach is applicable for tissues with no pronounced boundaries between elements that feature significant heterogeneity. The process of scattering in these structures may be described under certain conditions using the model of a phase screen.75,136,155,157
The second approach to tissue modeling is its representation as a system of discrete scattering particles. In particular, this model has been advantageously used to describe the angular dependence of the polarization characteristics of scattered radiation.145,146,148,150,158 Blood is the most important biological example of a dis- perse system that entirely corresponds to the model of discrete particles.48,101,159
Biological media are often modeled as ensembles of homogeneous spherical particles, since many cells and microorganisms, particularly blood cells, are close in shape to spheres or ellipsoids. A system of noninteracting spherical particles is the simplest tissue model. Mie theory rigorously describes the diffraction of light in a spherical particle.148,160 The development of this model involves taking into account the structures of the spherical particles, namely, the multilayered spheres and the spheres with radial nonhomogeneity, anisotropy, and optical activity.145,146
Because connective tissue consists of fiber structures, a system of long cylinders is the most appropriate model for it. Muscular tissue, skin dermis, dura mater, eye cornea, and sclera belong to this type of tissue formed essentially by collagen fibrils. The solution of the problem of light diffraction in a single homogeneous or multilayered cylinder is also well understood.148
The sizes of cells and tissue structure elements vary in size from a few tenths of nanometers to hundreds of micrometers.47,58,94–96,129,130,135,138,149–153,161–180
Blood cells (erythrocytes, leukocytes, and platelets) exhibit the following parameters. A normal erythrocyte in plasma has the shape of a concave-concave disk with a diameter varying from 7.1 to 9.2 μm, a thickness of 0.9–1.2 μm in the center and 1.7–2.4 μm on the periphery, and a volume of 90 μm3. Leukocytes are formed like spheres with a diameter of 8–22 μm. Platelets in the bloodstream are bicon- vex disklike particles with diameters ranging from 2 to 4 μm. Normally, blood has about 10 times as many erythrocytes as platelets and about 30 times as many platelets as leukocytes.
Most other mammalian cells have diameters in the range of 5–75 μm. In the epidermal layer, the cells are large (with an average cross-sectional area of about 80 μm2) and quite uniform in size. Fat cells, each containing a single lipid droplet that nearly fills the entire cell and therefore results in eccentric placement of the
Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis 5
cytoplasm and nucleus, have a wide range of diameters, from a few microns to 50–75 μm. Fat cells may reach diameters of 100–200 μm in pathological cases.
There are a wide variety of structures within cells that determine tissue light scattering (see Fig. 1.1). Cell nuclei are on the order of 5–10 μm in diameter; mitochondria, lysosomes, and peroxisomes have dimensions of 1–2 μm; ribosomes are on the order of 20 nm in diameter; and structures within various organelles can have dimensions of up to a few hundred nanometers. Usually, the scatterers in cells are not spherical. The models of prolate ellipsoids with a ratio of the ellipsoid axes between 2 and 10 are more typical.
Figure 1.1 Major organelles and inclusions of the cell.129
The hollow organs of the body are lined with a thin, highly cellular surface layer of epithelial tissue, which is supported by underlying, relatively acellular connective tissue. In healthy tissues, the epithelium often consists of a single well- organized layer of cells with en face diameter of 10–20 μm and height of 25 μm (see Fig. 1.2). In dysplastic epithelium, cells proliferate and their nuclei enlarge and appear darker (hyperchromatic) when stained.150 Enlarged nuclei are primary indicators of cancer, dysplasia, and cell regeneration in most human tissues.
In fibrous tissues or tissues containing fiber layers (cornea, sclera, dura mater, muscle, myocardium, tendon, cartilage, vessel wall, retinal nerve fiber layer, etc.) and composed mostly of microfibrils and/or microtubules, typical diameters of the cylindrical structural elements are 10–400 nm. Their length is in a range from 10– 25 μm to a few millimeters.
Figure 1.2 Microphotograph of the isolated normal intestinal epithelial cells (a) and intestinal malignant cell line T84 (b). Note the uniform nuclear size distribution of the normal epithelial cell (a) in contrast to the T84 malignant cell line, which at the same magnification shows larger nuclei and more variation in nuclear size (b). Solid bars equal 20 μm in each panel (from Ref. 150 © 1999 IEEE).
The dominant scatterers in an artery may be the fibers, cells, or subcellular organelles. Muscular arteries have three main layers. The inner intimal layer consists of endothelial cells with a mean diameter of less than 10 μm. The medial layer consists mostly of closely packed smooth muscle cells with a mean diameter of 15–20 μm; small amounts of connective tissue, including elastin, collagenous, and reticular fibers, as well as a few fibroblasts, are also located in the medial. The outer adventitial layer consists of dense fibrous connective tissue that is largely made up of 1- to 12-μm-diameter collagen fibers and thinner, 2- to 3-μm-diameter elastin fibers.
Another two examples of complex scattering structures are the myocardium and the retinal nerve fiber layer. The myocardium consists mostly of cardiac muscle, which is comprised of myofibrils (about 1 μm in diameter) that in turn consist of cylindrical myofilaments (6–15 nm in diameter) and aspherical mitochondria (1–2 μm in diameter). The retinal nerve fiber layer comprises bundles of unmyeli- nated axons that run across the surface of the retina. The cylindrical organelles of the retinal nerve fiber layer are axonal membranes, microtubules, neurofilaments, and mitochondria. Axonal membranes, like all cell membranes, are thin (6–10 nm) phospholipid bilayers that form cylindrical shells enclosing the axonal cytoplasm. Axonal microtubules are long tubular polymers of the protein tubulin with an outer diameter of ≈25 nm, an inner diameter ≈15 nm, and a length of 10–25 μm. Neu- rofilaments are stable protein polymers with a diameter ≈10 nm. Mitochondria are ellipsoidal organelles that contain densely involved membranes of lipid and protein. They are 0.1–0.2 μm thick and 1–2 μm long.
For some tissues, the size distribution of the scattering particles may be essentially monodispersive, and for others it may be quite broad. Two opposing
examples are a transparent eye cornea stroma, which has a sharply monodispersive distribution, and a turbid eye sclera, which has a rather broad distribution of collagen fiber diameters.129,130 There is no universal distribution size function that would describe all tissues with equal adequacy. In optics of dispersed systems, Gaussian, gamma, or power size distributions are typical.171 Polydisper- sion for randomly distributed scatterers can be accounted for by using the gamma- distribution or the skewed logarithmic distribution of scatterers’ diameters, cross sections, or volumes.61,129,154,156,165,172 In particular, for turbid tissues such as eye sclera, the gamma radii distribution function is applicable.61,172
Absorbed light is converted to heat or radiated in the form of fluorescence; it is also consumed in photobiochemical reactions. The absorption spectrum depends on the type of predominant absorption centers and water content of tissues (see Figs. 1.3–1.7). Absolute values of absorption coefficients for typical tissues lie in the range 10−2 to 104 cm−1.1–4,6,9–15,28,29,31,37–42,56,57,72,86–91 In the ultraviolet (UV) and infrared (IR) (λ ≥ 2000 nm) spectral regions, light is readily absorbed, which accounts for the small contribution of scattering and the inability of radiation to penetrate deep into tissues (only through one or two cell layers). Short-wave visible light penetrates typical tissues as deep as 0.5–2.5 mm, whereupon it undergoes an e-fold decrease of intensity. In this case, both scattering and absorption occur, with 15–40% of the incident radiation being reflected. In the 600–1600-nm wavelength range, scattering prevails over absorption, and light penetrates to a depth of 8–10 mm. Simultaneously, the intensity of the reflected radiation increases to 35–70% of the total incident light (due to backscattering).
Figure 1.3 The absorption spectrum of water.56
Light interaction with a multilayer and multicomponent skin is a very complicated process.57 The horny-skin layer (stratum corneum) reflects about 5–7% of the incident light. A collimated light beam is transformed to a diffuse one by microscopic inhomogeneities at the air/horny-layer interface. A major part of reflected light results from backscattering in different skin layers (stratum corneum,
Figure 1.4 Molar attenuation spectra for solutions of major visible light-absorbing human skin pigments: 1, DOPA-melanin (H2O); 2, oxyhemoglobin (H2O); 3, hemoglobin (H2O); 4, bilirubin (CHCl3).57
Figure 1.5 The transmittance spectrum of a 3-mm-thick slab of female breast tissue. A spectrometer with an integrating sphere was used. The contributions of absorption bands of the tissue components are marked: 1, hemoglobin; 2, fat; and 3, water.50
epidermis, dermis, blood, and fat). The absorption of diffuse light by skin pigments is a measure of bilirubin content, hemoglobin concentration, and its saturation with oxygen, and the concentration of pharmaceutical products in blood and tissues; these characteristics are widely used in the diagnosis of various diseases (see Fig. 1.4). Certain phototherapeutic and diagnostic modalities take advantage of ready transdermal penetration of visible and near-infrared (NIR) light inside the body in the wavelength region, corresponding to the therapeutic or diagnostic window (600–1600 nm) (Fig. 1.7).
Another example of heterogeneous multicomponent tissue is a female breast (which is principally composed of adipose and fibrous tissues). The absorption bands of hemoglobin, fat, and water are clearly seen in vitro in the measured spec-
Figure 1.6 UV absorption spectra of major chromophores of human skin: 1, DOPA-melanin, 1.5 mg % in H2O; 2, urocanic acid, 104 M in H2O; 3, DNA, calf thymus, 10 mg % in H2O (pH = 4.5); 4, tryptophane, 2 × 104 M (pH = 7); 5, tyrosine, 2 × 104 M (pH = 7).57
Figure 1.7 Absorption spectra of skin and aorta; spectra of tissue components—water (75%), epidermis, melanosome, and whole blood are also presented; diagnostic lasers and their wavelengths as well as diagnostic/therapeutic window and wavelength ranges suitable for superficial and deep spectroscopy are shown (adapted from Ref. 36).
trum of a 3-mm slab of breast tissue presented in Fig. 1.5.50 Measurement was done using the integrating sphere spectrometer. There is a wide window between
700 and 1100 nm, and narrow ones at about 1300 and 1600 nm, where the lowest percentage of light is attenuated.
Solid tissues such as ribs and the skull, as well as whole blood, are also eas- ily penetrable by visible and NIR light.1–4,6,9–16,36,91,129,130 The relatively good transparency of skin for long-wave UV light (UVA) depends on DNA, tryptophane, tyrosine, urocanic acid, and melanin absorption spectra and underlies se- lected methods of photochemotherapy of skin tissues using UVA irradiation (see Fig. 1.4).3,6,10,57,86,129,130
A collimated (laser) beam is attenuated in a thin tissue layer of thickness d in accordance with the Bouguer-Beer-Lambert exponential law as37
I (d) = (1 − RF)I0 exp(−μtd), (1.1)
where I (d) is the intensity of transmitted light measured using a distant photodetector with a small aperture (on-line or collimated transmittance), W/cm2; RF is the coefficient of Fresnel reflection; at the normal beam incidence, RF = [(n − 1)/(n + 1)]2; n is the relative mean refractive index of tissue and surrounding media; I0 is the incident light intensity, W/cm2;
μt = μa + μs (1.2)
is the extinction coefficient (interaction or total attenuation coefficient), 1/cm, where μa is the absorption coefficient, 1/cm, and μs is the scattering coefficient, 1/cm. Strictly speaking, Eq. (1.1) is valid only for a highly absorbing media, when μa μs.
The extinction coefficient is connected with the extinction cross section σext as
μt = ρsσext, (1.3)
where ρs is the density of particles (tissue and cell compounds). For a system of particles with absorption,
σext = σsca + σabs, (1.4)
μs = ρsσsca, μa = ρsσabs. (1.5)
The average scattering cross section per particle can be presented in a form suitable for experimental evaluations:148
σsca = (
λ2
2π
)( 1
I0
)∫ π
0 I (θ) sinθdθ, (1.6)
where I0 is the intensity of the incident light, I (θ) is the angular distribution of the scattered light by a particle, and θ is the scattering angle. For macroscopically
isotropic and symmetric media, the average scattering cross section is independent of the direction and polarization of the incident light. The average extinction, σext, and absorption, σabs, cross sections are also independent of the direction and polarization state of the incident light.
The probability that a photon incident on a small volume element will survive is equal to the ratio of the scattering and extinction cross sections, and is called the “albedo” for single scattering, :
= σsca
σext = μs
μt . (1.7)
The albedo ranges from zero for a completely absorbing medium to unity for a completely scattering medium.
The mean free path length (MFP) between two interactions is denoted by
lph = μ−1 t . (1.8)
1.1.2 Theoretical description
To analyze light propagation under multiple scattering conditions, it is assumed that absorbing and scattering centers are uniformly distributed across the tissue. UV-A, visible, or NIR radiation is normally subject to anisotropic scattering characterized by a clearly apparent direction of photons undergoing single scattering, which may be due to the presence of large cellular organelles [mitochondria, lysosomes, and inner membranes (Golgi apparatus)].3,58,85,95,96,129,130,135,150–153
When the scattering medium is illuminated by unpolarized light and/or only the intensity of multiply scattered light needs to be computed, a sufficiently strict mathematical description of continuous wave (CW) light propagation in a medium is possible in the framework of the scalar stationary radiation transfer theory (RTT).1,3,6,12–16,129,130,135,136,145,146,181–197
This theory is valid for an ensemble of scatterers located far from one another and has been successfully used to work out some practical aspects of tissue optics. The main stationary equation of RTT for monochromatic light has the form1
∂I (r, s)
4π
∫ 4π
I (r, s′)p(s, s′)d′, (1.9)
where I (r, s) is the radiance (or specific intensity)—average power flux density at point r in the given direction s, W/cm2 sr; p(s, s′) is the scattering phase function, 1/sr; and d′ is the unit solid angle about the direction s′, sr. It is assumed that there are no radiation sources inside the medium.
The scalar approximation of the radiative transfer equation (RTE) gives poor accuracy when the size of the scattering particles is much smaller than the wavelength, but provides acceptable results for particles comparable to and larger than the wavelength.146,184 There is ample literature on the analytical and numerical solutions of the scalar radiative transfer equation.1,3,15,129,130,184–197
If radiative transport is examined in a domain G ⊂ R3, and ∂G is the domain boundary surface, then the boundary conditions for ∂G can be written in the following general form:
I (r, s) (sN)<0 = S(r, s) + RI (r, s)
(sN)>0, (1.10)
where r ∈ ∂G,N is the outside normal vector to ∂G,S(r, s) is the incident light distribution at ∂G, and R is the reflection operator. When both absorption and reflection surfaces are present in the domain G, conditions analogous to Eq. (1.10) must be given at each surface.
For practical purposes, integrals of the function I (r, s) over certain phase space regions (r, s) are of greater value than the function itself. Specifically, optical probes of tissues frequently measure the outgoing light distribution function at the medium surface, which is characterized by the radiant flux density or irradiance (W/cm2):
F(r) = ∫
(sN)>0 I (r, s)(sN)d, (1.11)
where r ∈ ∂G. In problems of optical radiation dosimetry in tissues, the measured quantity is
actually the total radiant-energy-fluence rate U(r). It is the sum of the radiance over all angles at a point r and is measured by watts per square centimeter:
U(r) = ∫
4π
I (r, s)d. (1.12)
The phase function p(s, s′) describes the scattering properties of the medium and is in fact the probability density function for scattering in the direction s′ of a photon traveling in the direction s; in other words, it characterizes an elementary scattering act. If scattering is symmetric relative to the direction of the incident wave, then the phase function depends only on the scattering angle θ (angle between directions s
and s′), i.e.,
p(s, s′) = p(θ). (1.13)
The assumption of random distribution of scatterers in a medium (i.e., the ab- sence of spatial correlation in the tissue structure) leads to normalization:∫ π
0 p(θ)2π sinθdθ = 1. (1.14)
In practice, the phase function is usually well approximated with the aid of the postulated Henyey-Greenstein function:1,3,12–16,70,129,130,164
p(θ) = 1
(1 + g2 − 2g cosθ)3/2 , (1.15)
where g is the scattering anisotropy parameter (mean cosine of the scattering angle θ):
g ≡ cosθ = ∫ π
0 p(θ) cosθ · 2π sinθdθ. (1.16)
The value of g varies in the range from −1 to 1:145,146 g = 0 corresponds to isotropic (Rayleigh) scattering, g = 1 to total forward scattering (Mie scattering at large particles), and g = −1 to total backward scattering.
The integrodifferential Eq. (1.9) is too complicated to be employed for the analysis of light propagation in scattering media. Therefore, it is frequently sim- plified by representing the solution in the form of spherical harmonics. Such sim- plification leads to a system of (N + 1)2 connected differential partial derivative equations known as the PN approximation. This system is reducible to a single differential equation of order (N + 1). For example, four connected differential equations reducible to a single diffusion-type equation are necessary for N = 1.191–197
It has the following form for an isotropic medium:
(∇2 − μ2 eff)U(r) = −Q(r), (1.17)
where
μeff = [3μa(μ ′ s + μa)]1/2 (1.18)
is the effective attenuation coefficient or inverse diffusion length, μeff = 1/ld, 1/cm;
Q(r) = (cD)−1q(r), (1.19)
where q(r) is the source function (i.e., the number of photons injected into the unit volume), and
D = 1
μ′ s = (1 − g)μs (1.21)
is the reduced (transport) scattering coefficient, 1/cm, and c is the velocity of light in the medium. The transport mean free path of a photon (cm) is defined as
lt = (1/μ′ t) = (μa + μ′
s) −1, (1.22)
It is worthwhile to note that the transport mean free path (MFP) in a medium with anisotropic single scattering significantly exceeds the MFP in a medium with isotropic single scattering, lt lph [see Eq. (1.8)]. The transport MFP lt is the distance over which the photon loses its initial direction.
Diffusion theory provides a good approximation in the case of a small scattering anisotropy factor g ≤ 0.1 and large albedo → 1. For many tissues, g ≈ 0.6–0.9, and can be as large as 0.990–0.999, for example, for blood.48,49,87,129
This significantly restricts the applicability of the diffusion approximation. It is ar- gued that this approximation can be used at g < 0.9, when the optical thickness τ
of an object is of the order 10–20:
τ = ∫ d
0 μtds, (1.23)
where d is the tissue depth (thickness) in the direction s. However, the diffusion approximation is inapplicable for beam input near the
object’s surface where single or low-step scattering prevails. When a narrow light beam is normally incident upon a semi-infinite turbid medium with anisotropic scattering, it can be considered as converted into an isotropic point source at the depth of one transport MFP lt [Eq. (1.22)] below the surface. The strength of this point source is the orig

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