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 sepa- rated 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 proper- ties 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 tis- sues, 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 ini- tially
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 irradia- tion 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 cor- responding phase shift at modulation frequencies (the
frequency-domain or phase
xxxv
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 scatter- ing 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 con- ditions when a tissue is probed with an ultrashort laser
pulse.1,3,73,74 The co- herence 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 coher- ent tomography, diffractometry,
holography, photon-correlation spectroscopy, laser Doppler
anemometry, and speckle interferometry of tissues and fluxes of
biologi- cal 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 pene- trates
into a tissue for several centimeters, which is important for the
transillumina- tion 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
tomogra- phy 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 numeri-
cal 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 sys- tems 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 promis- ing 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 pho- ton density waves
(intensity waves), we will discuss optical methods and instru-
ments offering much promise for biomedical applications. Among
these are spec- trophotometry and polarimetry; time-domain and
frequency-domain spectroscopy and imaging systems;
photon-correlation spectroscopy; speckle interferometry; co- herent
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 prepa- ration 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, es- pecially 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 ma-
terials. 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 dis- cussions 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 bio- medical 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 as- sistance 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 re- search and
development in tissue optics, particularly in the field of tissue
diag- nostics 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 exper-
imental investigations into light transport in tissues and methods
for solving di- rect 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 ap- plications. 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 polar- ization imaging and
spectroscopy techniques (Chapter 7), as well as on OCT devel-
opments 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
xxxix
xl Preface to the Second Edition
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 mate- rials. 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
inter- national 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 col- leagues 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
de- scribed. 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 aver- age refractive index is higher than that of air. This
is responsible for partial reflec- tion 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 develop- ment 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 tis- sue
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 sec- ond 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 in- dex 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 dis- crete 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 cylin- ders 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 parame- ters. 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.
6 Optical Properties of Tissues with Strong (Multiple)
Scattering
Figure 1.2 Microphotograph of the isolated normal intestinal
epithelial cells (a) and intesti- nal 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 or- ganelles. 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 mus- cle, 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 pro- tein. 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 es- sentially monodispersive, and for others it may be quite
broad. Two opposing
Tissue Optics: Light Scattering Methods and Instruments for Medical
Diagnosis 7
examples are a transparent eye cornea stroma, which has a sharply
monodisper- sive distribution, and a turbid eye sclera, which has a
rather broad distribution of collagen fiber diameters.129,130 There
is no universal distribution size func- tion 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 de- pends 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 ultravio- let
(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 vis- ible
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 wave- length 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 com- plicated 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
re- flected light results from backscattering in different skin
layers (stratum corneum,
8 Optical Properties of Tissues with Strong (Multiple)
Scattering
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 pig- ments is a measure of bilirubin content, hemoglobin
concentration, and its satura- tion 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-
Tissue Optics: Light Scattering Methods and Instruments for Medical
Diagnosis 9
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
10 Optical Properties of Tissues with Strong (Multiple)
Scattering
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, trypto- phane,
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 pho- todetector 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 surround- ing
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 coeffi- cient, 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
Tissue Optics: Light Scattering Methods and Instruments for Medical
Diagnosis 11
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 po- larization
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 char- acterized by a
clearly apparent direction of photons undergoing single scattering,
which may be due to the presence of large cellular organelles
[mitochondria, lyso- somes, 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 func- tion, 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 wave- length, 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
12 Optical Properties of Tissues with Strong (Multiple)
Scattering
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 fol- lowing 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
irradi- ance (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)
Tissue Optics: Light Scattering Methods and Instruments for Medical
Diagnosis 13
where g is the scattering anisotropy parameter (mean cosine of the
scattering an- gle 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 dif- ferential equation of order (N + 1). For
example, four connected differential equa- tions 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)
14 Optical Properties of Tissues with Strong (Multiple)
Scattering
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 scat- tering 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