© 2007, lv wang tutorial on biomedical...

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© 2007, LV WANG

Tutorial on Biomedical OpticsTutorial on Biomedical Optics

Lihong V. Wang, PhD

Gene K. Beare Distinguished Professor

Department of Biomedical Engineering

Washington University in St. Louis

Presented at IMI Workshop

Montreal, Canada

May 4, 2007

Publisher: WileyPub. date: June, 2007ISBN-10: 0471743046 ISBN-13: 978-0471743040Homework solutions available to instructors


© 2007, LV WANG

ChaptersChapters

1. Introduction to biomedical optics2. Single scattering: Rayleigh theory and Mie theory 3. Monte Carlo modeling of photon transport4. Convolution for broad-beam responses5. Radiative transfer equation and diffusion theory6. Hybrid model of Monte Carlo method and diffusion theory7. Sensing of optical properties and spectroscopy8. Ballistic imaging and microscopy9. Optical coherence tomography10.Mueller optical coherence tomography11.Diffuse optical tomography12.Photoacoustic tomography13.Ultrasound-modulated optical tomography


© 2007, LV WANG

Motivation for Biomedical OpticsMotivation for Biomedical Optics

1. Optical photons provide nonionizing and safe radiation for medical applications.

2. Optical spectra--based on absorption, fluorescence, or Raman scattering--provide biochemical information because they are related to molecular conformation.

3. Optical absorption, in particular, reveals angiogenesis and hyper-metabolism, both of which are hallmarks of cancer; the former isrelated to the concentration of hemoglobin and the latter to theoxygen saturation of hemoglobin. Therefore, optical absorption provides contrast for functional imaging.

4. Optical scattering spectra provide information about the size distribution of optical scatterers, such as cell nuclei.

5. Optical polarization provides information about structurally anisotropic tissue components, such as collagen and muscle fiber.


© 2007, LV WANG

Motivation for Biomedical Optics (Cont’d)Motivation for Biomedical Optics (Cont’d)

6. Optical frequency shifts due to the optical Doppler effect provide information about blood flow.

7. Optical properties of targeted contrast agents provide contrast for the molecular imaging of biomarkers.

8. Optical properties or bioluminescence of products from gene expression provide contrast for the molecular imaging of gene activities.

9. Optical spectroscopy permits simultaneous detection of multiple contrast agents.

10. Optical transparency in the eye provides a unique opportunity for high-resolution imaging of the retina.


© 2007, LV WANG

Trajectories of Optical Photons in Biological TissueTrajectories of Optical Photons in Biological Tissue

Tissue

Laser beam

1 mm

Reflectometry

Photoacoustics


© 2007, LV WANG

Optical Properties of Biological TissueOptical Properties of Biological Tissue

• Basic properties• n [–]: index of refraction; e.g., 1.37• µa [cm–1]: absorption coefficient; e.g., 0.1• µs [cm–1]: scattering coefficient; e.g., 100• g [–]: scattering anisotropy, <cosθ>; e.g., 0.9

• Derived properties• µt [cm–1]: total interaction (extinction) coefficient, µa + µs

• lt [cm]: mean free path, 1/ µt; e.g., 0.1 mm• µs’ [cm–1]: reduced scattering coefficient, µs(1 – g)• µt’ [cm–1]: transport interaction coefficient, µa + µs’• lt’ [cm]: transport mean free path, 1/ µt’; e.g., 1 mm• µeff [cm–1]: effective attenuation coefficient, (3µa µt’)1/2

• δ [cm]: penetration depth, 1/(3µa µt’)1/2; e.g., 5 mm


© 2007, LV WANG

Beer’s LawBeer’s Law

( ) ( ) ( ) ( ) ( )

tcoefficien n)(extinction interactio total:pathlength:

intensity ballistic:

t

tt

t

xI

lxIxIxI

dxIdI

μ

μ

μ

−=−=

=−

exp0exp0


© 2007, LV WANG

102

103

104

105

10−4

10−2

100

102

104

106

Wavelength (nm )

Abs

orpt

ion

coef

ficie

nt (c

m−1

)

HbO2

Water

Melanin

Spectra of Major Biological AbsorbersSpectra of Major Biological Absorbers

Near IR window: ~700 nm

2.95 µm

~1 µm penetration


© 2007, LV WANG

Near Infrared Window Near 700 nmNear Infrared Window Near 700 nm

200 400 600 800 100010

-4

10-2

100

102

104

Wavelength (nm)

Abs

orpt

ion

Coe

ffic

ient

(cm

-1) 7% Blood

75% WaterTotal

700 nm


© 2007, LV WANG

200 400 600 800 100010

0

101

102

103

104

Wavelength (nm)

Abs

orpt

ion

coef

ficie

nt (

cm-1

)

Absorption Spectra of Pure BloodAbsorption Spectra of Pure Blood

100% deoxygenated

100% oxy

~3 µm penetrationSoret band (420 nm)

~30 µm penetrationQ-band (540–580 nm)


© 2007, LV WANG

Chapter 2Chapter 2



© 2007, LV WANG

Rayleigh Theory: Small Scatterer (ka << 1)Rayleigh Theory: Small Scatterer (ka << 1)

( )

scatterers ofdensity :tcoefficien Scattering:anisotropy Scattering

:section cross Scattering

:efficiency Scattering

scatterer of radius:background and spherebetween index refractive relative:

litypolarizabi:

constantn propagatio:

pointn observatio of anglepolar and distance radial:

:ed)(unpolarizintensity Scattered

====

+−

==

=

=+−

=

=

+=

ssss

ss

rel

relss

bsrel

rel

rel

b

NNg

aQ

nnx

aQ

kaxa

nnn

ann

nk

r

Ir

krI

,0

21

38

/21

2,

2cos1

),(

2

2

2

24

2

32

2

02

242

σμ

πσ

πσ

α

λπ

θ

αθθ


© 2007, LV WANG

Mie Theory: Scatterer of Any SizeMie Theory: Scatterer of Any Size

( )[ ]

( ) ( ) ( ) ( )...

Re112Re

124

:anisotropy Scattering

122:efficiency Scattering

1

**1

*12

1

222

∑

∑

∞

=++

∞

=

⎥⎦

⎤⎢⎣

⎡++

++++

=

++=

lllllll

s

llls

balllbbaa

lll

xQg

balx

Q


© 2007, LV WANG

Mie Theory: Plot of Scattering EfficiencyMie Theory: Plot of Scattering Efficiency

10−1

100

101

102

103

10−3

10−2

10−1

100

101

x = ka

Scat

terin

g ef

ficie

ncy

Qs

Rayleighapproximation


© 2007, LV WANG

Mie Theory: Plot of Scattering AnisotropyMie Theory: Plot of Scattering Anisotropy

10−1

100

101

102

103

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x = ka

Ani

sotro

pyg


© 2007, LV WANG

Chapter 3Chapter 3



© 2007, LV WANG

Sampling Random Variable: Inverse Distribution MethodSampling Random Variable: Inverse Distribution Method

PDF

CD

F

( )

( ) ξχ

ξχχχ

=

=∫P

dpa


© 2007, LV WANG

Sampling Step SizeSampling Step Size

( )

( ) ( )tt

t

t

ss

sµ

sµsP

μξ

μξ

ξ

ln1ln−=

−−=

=−−

−−=

or

:size step Sampled

)exp(1:methodon distributi Inverse

)exp(1 :size step offunction on distributi Cumulative


© 2007, LV WANG

Sampling Scattering AngleSampling Scattering Angle

⎪⎩

⎪⎨

⎧

=−

≠⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡+−−

−+=

−+−

=

012

021

1121

cos

)cos21(21

222

2/32

2

g

ggg

ggg

gggp

if

if

:angle scattering Sampled

)(cos

:function phase Greenstein-Henyey

ξξθ

θθ


© 2007, LV WANG

Trac

king

Pho

tons

Trac

king

Pho

tons


© 2007, LV WANG

Physical Meaning of Penetration DepthPhysical Meaning of Penetration Depth

0.1 1 10

0

1

2

Internal Fluence [J/cm2]

Dep

th [

cm]

δ

Wide Source

1/e = 36.7%


© 2007, LV WANG

Scattering Enhanced Internal Light FluenceScattering Enhanced Internal Light Fluence

0 1 2 3

0

1

2

Internal Fluence [J/cm 2]

Dep

th [

cm]

Sourcefluence

Wide Source

Enhancedfluence

~lt’


© 2007, LV WANG

Transition from Ballistic to Diffusive RegimesTransition from Ballistic to Diffusive Regimes2

mm

Simulation software MCML available fromhttp://oilab.seas.wustl.edu [ ])/exp(1 ''

ttc lctlz −−=

:center Cloud


© 2007, LV WANG

Chapter 4Chapter 4



© 2007, LV WANG

Convolution: FormulationConvolution: Formulation

• Convolution is applicable to a system that is

Stationary (time-invariant)LinearTranslation-invariant (shift-invariant).

( ) ( ) ( )

response beam-broad:source beam-broad:

beam pencil toresponse impulse:

''',',',',,

CSG

dydxyxSzyyxxGzyxC ∫ ∫∞

∞−

∞

∞−

−−=


© 2007, LV WANG

Fluence Distribution: Pencil Beam vs Gaussian BeamFluence Distribution: Pencil Beam vs Gaussian Beam

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Radius r (cm)D

epth

z (c

m)

30.010.0

3.01.0

0.3

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Radius r (cm)

Dep

thz (

cm) 0 .3

1.0

10.030.0

100.0


© 2007, LV WANG

Chapter 5Chapter 5



© 2007, LV WANG

Physical QuantitiesPhysical Quantities

[ ]( )

[ ]( ) ( )

[ ]( ) ( )

[ ]( ) ( ) Ω=

Φ=

Ω=Φ

Ω⋅=

∫

∫

∫

−

∞+

∞−

−

−

−−

dtsrLstrJ

Wm

dttrrF

Jm

dtsrLtr

Wm

dtdAdnsdEtsrL

srWm

,ˆ,ˆ,

:density Current

,

: Fluence

,ˆ,,

: rate Fluence

)ˆˆ(,ˆ,

: Radiance

4

2

24

2

12

rrr

rr

rr

r

π

π


© 2007, LV WANG

Radiative Transfer Equation (Boltzmann Equation)Radiative Transfer Equation (Boltzmann Equation)

( )

{ }( )

( )( ) ( )

( ) Source,ˆ,

oncontributi Scattering'ˆ'ˆ,'ˆ,

Extinction,ˆ,Divergence,ˆ,ˆ

srm J ofDimension

densityenergy of rate Change/,ˆ,

4

13

tsrS

dssPtsrL

tsrLtsrLs

cLt

ctsrL

s

t

r

r

r

r

r

+

Ω⋅+

−∇⋅−

==∂

∂

∫

−−

π

μ

μ


© 2007, LV WANG

Diffusion TheoryDiffusion Theory

( ) ( ) ( )

( ) ( )

( )

( )

{ }( )[ ]

( )( ) Source,

Absorption,Divergence,

m J ofDimension

densityenergy of rate Change,:equationDiffusion

31 :tcoefficienDiffusion

,, :law sFick'

ˆ,43,

41,ˆ,

:radiance ofexpansion Diffusion

3

'

trStr

trDc

tctr

D

trDtrJ

strJtrtsrL

a

sa

r

r

r

r

rrr

rrrr

+Φ−

Φ∇−⋅∇−=Φ=

∂Φ∂

+=

Φ∇−=

⋅+Φ=

−

μ

μμ

ππ


© 2007, LV WANG

Impulse ResponsesImpulse Responses

( )( )

( ) ( )

( ) ( )

( ) ( )zz

D

rDr

r

ctcrdtr

ctDctr

Dctctr

effa

eff

aeff

eff

a

a

μμμ

μμ

μπ

μ

μπ

−=Φ

=

−=Φ

−=Φ

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=Φ

∫

exp2

exp4

1

exp,

4exp

4,

2

2/3

1D

:tcoefficienn attenuatio Effective

r

rr

r


© 2007, LV WANG

Boundary ConditionBoundary Condition

( )

( ) ( )

( )

( ) ( )

0

,2,0

,2:boundary edExtrapolat

0,2,

:ionapproximatdiffusion In

0ˆˆ,ˆ,

0

0ˆˆ

=∂

Φ∂−=Φ=

−=Φ

=∂

Φ∂−Φ

=Ω⋅

=

>⋅∫

z

ns

ztrDtz

tDz

ztrDtr

dnstsrL

r

rr

r


© 2007, LV WANG

Diffuse ReflectanceDiffuse Reflectance

Pencil beam

Similarityrelation

Isotropicsource

Extrapolatedboundary &Method of image

'tl


© 2007, LV WANG

Photon Propagation RegimesPhoton Propagation Regimes

• Mean free path: lt = 0.1 mm (0.2 ps)• Transport mean free path: lt' = 1 mm (2 ps)

• Ballistic regime: Pathlength ct < lt = 0.1 mmProbability of no scattering > exp(–1) = 37%

• Quasi-ballistic regime: Pathlength ct = lt – lt' = 0.1 – 1 mmProbability of no scattering = exp(–1) – exp(–10) = 0.37 – 0.45E–4

• Quasi-diffusive regime: Pathlength ct = lt' – 10lt' = 1 – 10 mmPhoton-cloud center distance to final position = [exp(–1) – exp(–10)] lt'

• Diffusive regime: Pathlength ct > 10lt' = 10 mmPhoton-cloud center distance to final position < exp(–10) lt'

)/exp(

)/exp()('''ttct

t

lctlzl

lctctP

−=−

−=


© 2007, LV WANG

Chapter 6Chapter 6



© 2007, LV WANG

Hybrid Model for a SlabHybrid Model for a Slab


© 2007, LV WANG

Accuracy and Speed of Hybrid Model Accuracy and Speed of Hybrid Model

d (cm) aμ (cm-1) MCT (s) HT (s) HMC TT10 0.01 6684 23 291 10 0.1 2589 23 113 10 1 679 23 30 3 0.01 2095 23 91 3 0.1 1961 23 85 3 1 679 23 30 1 0.01 696 23 30 1 0.1 698 23 30 1 1 583 23 25

37.1=reln


© 2007, LV WANG

Chapter 7Chapter 7



© 2007, LV WANG

Ballistic Transmission Method: SpectrophotometryBallistic Transmission Method: Spectrophotometry

( )

( )dA

dA

II

d

IIA

dII

st

s

ts

303.210lnln1 :tcoefficien Extinction

log :dB) 10OD density, (optical Absorbance

exp :ion transmissballisticfor law sBeer'

0

010

0

==−=

−==

−=

μ

μ


© 2007, LV WANG

Molar Extinction Spectra of HemoglobinMolar Extinction Spectra of Hemoglobin

[nm]259.93339.54390.01422.05452.36500.11529.24545.26570.18584.09796.80

Isosbesticpoint


© 2007, LV WANG

OximetryOximetry

[ ][ ]

deoxHbdeox

ox

oxdeoxde

aoxaoxde

oxdeoxde

adeadeox

dedeoxoxa

dedeoxoxa

CCCCC

CSO

C

C

CCCC

+=+

=

−−

=

−−

=

+=+=

,

:hemoglobin ofion concentrat totaland saturationn Oxygenatio)()()()()()()()(

10ln1

)()()()()()()()(

10ln1

:hemoglobin eddeoxygenat and oxygenated of ionsConcentrat)()(10ln)(

)()(10ln)(:tscoefficienabsorption Measured

2

2112

1221

2112

2112

222

111

λελελελελμλελμλελελελελελμλελμλε

λελελμλελελμ


© 2007, LV WANG

Measurement of Tissue Optical PropertiesMeasurement of Tissue Optical Properties

1

10

100

0 0.1 0.2 0.3 0.4 0.5

Monte CarloDiffusion

Ref

lect

ance

(1/

cm2)

Horizontal Axis, x (cm)

Determinedby δ


© 2007, LV WANG

Normal Versus Oblique Incidence ReflectometryNormal Versus Oblique Incidence Reflectometry


© 2007, LV WANG

Diffuse Reflectance: Normal vs Oblique IncidenceDiffuse Reflectance: Normal vs Oblique Incidence

0.01

0.1

1

10

100

-1.5 -1 -0.5 0 0.5 1 1.5

0o

45 o45 o Shifted

Ref

lect

ance

[cm

–2]

Horizontal Axis x [cm]

1 mfp'


© 2007, LV WANG

Spectroscopic Oblique Incidence ReflectometrySpectroscopic Oblique Incidence Reflectometry

Q-band


© 2007, LV WANG

Chapter 8Chapter 8



© 2007, LV WANG

Time-gated (Early-photon) Imaging Time-gated (Early-photon) Imaging

Diffuse

Quasi-ballistic

BallisticTime of arrival

Diffuse

Quasi-ballisBallistic

Filter

Laser pulse

1. 100 fs × (3 × 108 m/s) = 30 µm2. exp(–µtd) = exp(–100 × 0.3) = exp(–30) = 120 dB

Quasi-ballistic


© 2007, LV WANG

Spatial-frequency Filtered ImagingSpatial-frequency Filtered Imaging


© 2007, LV WANG

Polarization-difference ImagingPolarization-difference Imaging

( ) ( ) ( )

( ) ( )

( ) ( ) ( )yxIyxIyxI

I

I

yxIyxI

yxIyxIyxI

b

nb

b

nb

nbb

,,,

d.unpolarize Assumed intensity. ballistic-Non:

polarized. Assumed intensity. Ballistic:

,21,

,21,,

//

//

⊥

⊥

−=

=

+=


© 2007, LV WANG

Confocal Microscopy Confocal Microscopy


© 2007, LV WANG

Two-photon Microscopy: SchematicTwo-photon Microscopy: Schematic


© 2007, LV WANG

Two-photon Microscopy in Comparison to Confocal Microscopy

Two-photon Microscopy in Comparison to Confocal Microscopy

• A more localized excitation volume leads to reduced photo-bleaching.

• A longer excitation wavelength leads to increased penetration because both the absorption and the reduced scattering coefficients are decreased in the typical spectral region.

• No pinhole is needed.

• An ultrashort pulsed laser is used.• Scattering contrast is not directly measured.


© 2007, LV WANG

Chapter 9Chapter 9



© 2007, LV WANG

Principle of Time-domain Optical Coherence TomographyPrinciple of Time-domain Optical Coherence Tomography

SLD

PD

LensLens

Mirror

SampleNBS

SLD: Superluminescent diodeNBS: Non-polarizing beam splitter

Reference armSample arm

Photodiode


© 2007, LV WANG

Michelson Interferometer with a Monochromatic SourceMichelson Interferometer with a Monochromatic Source

( )[ ]( )[ ]

( )[ ]

( )

.n with oscillatio Sustained :differencelength -Arm

2/22 :difference Phase

space freeIn 2cos2

2exp2exp

002

02

0

20

0

ll

lllk

lklkEEEEI

EEI

tlkiEEtlkiEE

RS

RRSSSRSR

SR

SSSS

RRRR

ΔΔ

Δ=−=Δ

−++=

+=

−−=−−=

λπφ

ωω


© 2007, LV WANG

Coherence LengthCoherence Length

• Defined as the spatial full width at half maximum (FWHM) of the autocorrelation function of the electric field.

{ }

λλ

π

τ

Δ=

+= ∫∞+

∞−

202ln4

:spectrumGaussian afor length Coherence

)()()(

:functionation Autocorrel

cl

dttEtEtEC


© 2007, LV WANG

Michelson Interferometer with a Low-coherence Source:Basic OCT

Michelson Interferometer with a Low-coherence Source:Basic OCT

( )

mlnmnm

Dfx

lz

l

lkllI

c

R

cR

c

cAC

μλλ

πλ

λλ

π

30,20,830:exampleFor

4 :resolution Lateral

2 :resolution Axial

2ln4 :length Coherence

2cos2ln16exp

0

0

20

0

2

==Δ=

=Δ

=Δ

Δ=

Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−∝


© 2007, LV WANG

Demodulation of OCT Signals: SystemDemodulation of OCT Signals: System


© 2007, LV WANG

Demodulation of OCT Signals: SimulationDemodulation of OCT Signals: Simulation

• MATLAB program:

• % Use SI units throughout

• % center wavelength• lambda0 = 830E-9; • % bandwidth (delta lambda)• dlambda = 60E-9; • % speed of light• c = 3E8; •• % coherence length• lc = 4*log(2)/pi*lambda0^2/dlambda • % number of sampling points• N = 2^12; • % array for Delta_l• dl = lc*linspace(-2,2, N); • % propagation constant• k0 = 2*pi/lambda0; • ...


© 2007, LV WANG

MATLAB Program: Original InterferogramMATLAB Program: Original Interferogram

• …• subplot(4, 1, 1) % interferogram• Iac = exp(-16*log(2)*(dl/lc).^2) .* cos(2*k0 * dl);• plot(dl/lc, Iac, 'k')• title('(a) Interferogram')• xlabel('\Deltal/l_c')• ylabel('Signal')• axis([-0.6, 0.6, -1, 1])• …


© 2007, LV WANG

MATLAB Program: Rectified InterferogramMATLAB Program: Rectified Interferogram

• …• subplot(4, 1, 2) % rectified interferogram• Irec = abs(Iac); • plot(dl/lc, Irec, 'k')• title('(b) Rectified interferogram')• xlabel('\Deltal/l_c')• ylabel('Signal')• axis([-0.6, 0.6, -1, 1])• …


© 2007, LV WANG

MATLAB Program: Spectrum of Rectified InterferogramMATLAB Program: Spectrum of Rectified Interferogram

• …• subplot(4, 1, 3)

% spectrum of the rectified interferogram • Frec1 = fft(Irec)/sqrt(N);• % order of frequencies: • % 0,1...(N/2-1),-N/2,-(N/2-1)...-1

• Frec2 = fftshift(Frec1); • % shifted order of frequencies: • % -N/2,-(N/2-1)...-1, 0,1...(N/2-1)

• dfreq = 1/(4*lc); % bin size = 1/sampling range• freq = dfreq*(-N/2:N/2-1); % frequency array• plot(freq*lambda0, abs(Frec2), 'k')• title('(c) Spectrum of the rectified

interferogram')• xlabel('Frequency (1/\lambda_0)')• ylabel('Amplitude')• axis([-10, 10, 0, 5])• …


© 2007, LV WANG

MATLAB Program: EnvelopesMATLAB Program: Envelopes

• …• subplot(4, 1, 4) % envelope• % cut-off frequency for filtering• freq_cut = 1/lambda0/2; • % convert freq_cut to an array index• i_cut = round(freq_cut/dfreq); • Ffilt = Frec1; % initialize array• Ffilt(i_cut:N-i_cut+1) = 0; % filter• % inverse FFT then take the amplitude• Ifilt = abs(ifft(Ffilt))*sqrt(N); • plot(dl/lc, Ifilt/max(Ifilt), 'k')

• Iac_en = exp(-16*log(2)*(dl/lc).^2); % envelope• hold on;• plot(dl(1:N/32:N)/lc, Iac_en(1:N/32:N), 'ko')• hold off;• title('(d) Envelopes')• xlabel('\Deltal/l_c')• ylabel('Signals')• axis([-0.6, 0.6, -1, 1])• legend('Demodulated','Original')• …


© 2007, LV WANG

Fourier-domain OCT: TheoryFourier-domain OCT: Theory

( ) ( ) ( )( )[ ]( ) ( ) ( )( )[ ]

( )

( )( )[ ]

( )[ ] ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−

+−+=

+=

===

−−=

−−=

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

2

2

2

0

0

2exp)('

2cos)('2)(

)()(

/// free, dispersion If

2exp)('

2exp

SSSSS

SRSSSSRR

SR

SSRR

SSSSSS

RRRR

ldlnkilr

ldllnklrrrkSkI

kcEkcEkI

cknknk

ldtlkilrEE

tlkirEE

ω

ωωωω

ωωωω


© 2007, LV WANG

Simulated Fourier-domain OCT: Two Back-scatterersSimulated Fourier-domain OCT: Two Back-scatterers

• % Use SI units throughout

• lambda0 = 830E-9; • dlambda = 20E-9; % FWHM• ns=1.38; % refractive index of sample• ls1 = 100E-6; % location of scatterer 1• ls2 = 150E-6; % location of scatterer 2• rs1 = 0.50; % reflectivity of scatterer 1• rs2 = 0.25; % reflectivity of scatterer 2

• k0=2*pi/lambda0; • % FWHM bandwidth of k• delta_k=2*pi*dlambda/lambda0^2; • % standard deviation of k• sigma_k = delta_k/sqrt(2*log(2)); • N=2^10; % # of sampling points• % # of SD to plot on each side of k0• nsigma = 5; • …


© 2007, LV WANG

Simulated Fourier-domain OCT: Original InterferogramSimulated Fourier-domain OCT: Original Interferogram

• …• subplot(4,1,1); % Generate the

interferogram• k = k0 + sigma_k*linspace(-

nsigma,nsigma, N); % array for k• S_k = exp(-(1/2)*(k-k0).^2/sigma_k^2);

% Gaussian source PSD• E_s1 = rs1*exp(i*2*k*ns*ls1); % sample

electric field from scatter 1• E_s2 = rs2*exp(i*2*k*ns*ls2); % sample

electric field from scatter 2• I_k1 = S_k .* abs(1 + E_s1 + E_s2).^2;

% interferogram (r_R = 1)• plot(k/k0,I_k1/max(I_k1), 'k');• title('Interferogram');• xlabel('Propagation constant k/k_0');• ylabel('Normalized intensity');• axis([0.9 1.1 0 1]);• …


© 2007, LV WANG

Simulated Fourier-domain OCT: IFT of Original Interferogram

Simulated Fourier-domain OCT: IFT of Original Interferogram

• …• subplot(4,1,2); % Inverse Fourier

transform (IFT) of the interferogram• spec1=abs(fftshift(ifft(I_k1)))/sqrt(N);• dls_prime =

1/(2*nsigma*sigma_k/(2*pi)); % freq bin size = 1/sampling range

• ls_prime = dls_prime*(-N/2:N/2-1); % frequency array

• plot(ls_prime/(2*ns),spec1/max(spec1), 'k'); % scale the frequency

• title('Inverse Fourier transform of the interferogram');

• xlabel('Depth ls (m)');• ylabel('Relative reflectivity');• axis([-2*ls2 2*ls2 0 1]); • …


© 2007, LV WANG

Simulated Fourier-domain OCT:IFT of Deconvolved InterferogramSimulated Fourier-domain OCT:

IFT of Deconvolved Interferogram

• …• subplot(4,1,3); % IFT of the

deconvolved interferogram• spec1_norm

=abs(fftshift(ifft(I_k1./S_k)))/sqrt(N);• dls_prime =

1/(2*nsigma*sigma_k/(2*pi)); % freq bin size = 1/sampling range

• ls_prime = dls_prime*(-N/2:N/2-1); % frequency array

• plot(ls_prime/(2*ns),spec1_norm/max(spec1_norm), 'k'); % scale the frequency

• title('Inverse Fourier transform of the deconvolved interferogram');

• xlabel('Depth ls (m)');• ylabel('Relative reflectivity');• axis([-2*ls2 2*ls2 0 1]); • …


© 2007, LV WANG

Simulated Fourier-domain OCT: IFT of Deconvolved Differential Interferogram

Simulated Fourier-domain OCT: IFT of Deconvolved Differential Interferogram

• …• subplot(4,1,4); % IFT of the

deconvolved differential interferogram• I_k2 = S_k .* abs(-1 + E_s1 + E_s2).^2;

% interferogram• delta_I_k = I_k1 - I_k2;• spec2=abs(fftshift(ifft(delta_I_k./S_k)))/

sqrt(N);• plot(ls_prime/(2*ns),spec2/max(spec2),

'k');• title('Inverse Fourier transform of the

deconvolved differential interferogram');

• xlabel('Depth ls (m)');• ylabel('Relative reflectivity');• axis([-2*ls2 2*ls2 0 1]);


© 2007, LV WANG

Simulated Fourier-domain OCT: A Single Back-scattererSimulated Fourier-domain OCT: A Single Back-scatterer


© 2007, LV WANG

Doppler OCTDoppler OCT

( ) ( )

( ) ( )

( ) ( )RSAC

RR

RR

R

vvdt

tdf

vdt

tdf

tvltvlk

tvltl

−=Δ

=

=Δ

=

−Δ=−Δ=Δ

−Δ=Δ

θλ

φπ

λφ

πλ

λπφ

cos221

221

42

)(

0

00

00

00

0


© 2007, LV WANG

Composition of OCT SignalsComposition of OCT Signals

• Source coherence length = 15 microns• Absorption coefficient = 1.5 /cm• Scattering coefficient = 60 /cm• Anisotropy factor = 0.9


© 2007, LV WANG

Stokes Vector Measured by OCTStokes Vector Measured by OCT

H V P M R Lk

2

4

3

2

1

,

OCTxx

LR

MP

VH

Total

LR

MP

VH

VH

AI

IIIIII

I

IIIIIIII

SSSS

∝

+=+=+=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−+

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

needed. tsmeasurement independen 4

S

x: polarization states of the reference beam.


© 2007, LV WANG

Mueller MatrixMueller Matrix

inout MSS =

[ ]VHRVHPVHVH ,,, SSS2SSS2SSSS21M −−−−−+=

[ ]

30RiR

20PiP

10ViV

10HiH

3210

MMMSSMMMSSMMMSSMMMSS

M,M,M,MM

+==+==−==+==

=


© 2007, LV WANG

Raw Polarized Images and Mueller Images (750x500 microns)

Raw Polarized Images and Mueller Images (750x500 microns)

HH HV HP HR

M11 M12/M11 M13/M11 M14/M11

VH VV VP VR

M21/M11 M22/M11 M23/M11 M24/M11

PH PV PP PR

M31/M11 M32/M11 M33/M11 M34/M11

RH RV RP RR

M41/M11 M42/M11 M43/M11 M44/M11

-60 -30 0 dB -1 0 +1

H

V

P

R

Source


© 2007, LV WANG

Jones Vector and Jones MatrixJones Vector and Jones Matrix

INOUT

V

H

JJJJ

EE

JEE

J

E

=

⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡=

2221

1211

• 7 real independent parameters in a Jones matrix.


© 2007, LV WANG

Jones Reversibility Theorem Jones Reversibility Theorem

• Reduces to 5 real independent parameters in a Jones matrix.

TSI

TSISIMSB JJJJJJJ ===


© 2007, LV WANG

Mueller Images of Porcine Tendon(0.5 mm × 1 mm)

Mueller Images of Porcine Tendon(0.5 mm × 1 mm)

• 10 micron resolution• ~1 mm imaging depth

• Birefringence: (4.2 ± 0.3) × 10–3

(e.g., density of collagen)• Orientation: accurate to <5°

(e.g., direction of collagen)• Diattenuation: 0.26/mm

(e.g., property of collagen)

0 1 2 30

11

02

3 -1


© 2007, LV WANG

Modes of Diffuse Optical TomographyModes of Diffuse Optical Tomography

Mode Source light )','( trsr

Φ Re-emitted light )',';,( trtrmrr

Φ

Time domain Impulse:

)()( '' tr δδ r

Time-resolved:

)',';,( trtrmrr

Φ

Frequency domain Amplitude-modulated:

[ ])'cos(' sss tADr φωδ ++)(r

Amplitude-modulated:

( ))';(cos)';()';( rrtrrArrD mmmrrrrrr φω ++

Direct current

(DC)

DC:

)'(rDsrδ

DC:

)';( rrDmrr


© 2007, LV WANG

Visit Our Web Site:http://oilab.seas.wustl.eduClick on “Presentations”

Visit Our Web Site:http://oilab.seas.wustl.eduClick on “Presentations”

© 2007, lv wang tutorial on biomedical...

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