2015-04-01
LightandMatter
PhotonTransportTheory
MD6305Laser‐TissueInteractions
Class5
JaeGwan Kim
[email protected] ,X2220
DepartmentofMedicalSystemEngineering
Gwangju InstituteofSciencesandTechnology
Copyright.Mostfigures/tables/textsinthislecturearefromthetextbook“Laser‐Tissue
InteractionsbyMarkolf H.Niemz 2007”andthismaterialisonlyforthosewhotakethis
classandcannotbedistributedtoanyonewithoutthepermissionfromthelecturer.
LightandBulkMatter(tissue)
• Inopaquemedia,therefractionishardtomeasure
duetoabsorptionandscattering
loss
Iinc
Transmittance(%)=Itrans/Iinc
loss
loss
Itrans
• Inlasersurgery,knowledgeofabsorbingand
scatteringpropertiesofaselectedtissueisessential
forthepurposeofpredictingsuccessfultreatment
1
2015-04-01
LightTransportinTissue
•
•
•
•
Scatteringandabsorptionoccursimultaneouslyandarewavelengthdependent
·
,b=0.5~4
Scatteringmonotonicallydecreaseswithwavelength
AbsorptionislargeinUV,nearvisible,andIR
AbsorptionislowinredandNIRTherapeuticwindow
LightTransportinTissue
• Modelingoflighttransportintissuesareoftengovernedby
therelativemagnitudesofopticalabsorptionandscattering
– μa >>μs’:Lambert‐BeerLaw(λ ≤300nm; λ≥2000nm)
– μs’>>μa :DiffusionApproximation(600nm~1000nm)
– μs’~μa:EquationofRadiativeTransfer,MonteCarlo(300nm~600nm;
1200nm~2000nm)
2
2015-04-01
PhotonPropagationinTissues
• AbsorptionCoefficient:μa
• ScatteringCoefficient:μs
• PhysicalPathlength:Lp
• OpticalPathlength:Lo
Scatteringandabsorbingtissue
BiologicalTissue
Lo/Lp =4or↑
UseMonteCarlo,RadiativeTransferTheory,or
DiffusionTheory!!!!
ModelingPhotonPropagation
a, s, g, phase function S
“Stochastic(=random)” Description
3
2015-04-01
PhotonTransportTheory
• Amathematicaldescriptionoftheabsorptionand
scatteringpropertiesoflightcanbeperformedin
twoways
– Analyticaltheory:basedonMaxwell’sequations,most
fundamentalapproach
However,verycomplextoderiveexactanalyticalsolutions
– Transporttheory:directlyaddressesthetransportof
photons,itlacksofstrictnessofanalyticaltheories
Nevertheless,thishasbeenusedextensivelywhendealing
withlaser‐tissueinteractionsandtheexperimentalresults
weresatisfactoryinmanycases
RadiativeTransportTheory
• ThedirectapplicationofEMtheoryiscomplicated
• RTTdealswiththetransportoflightenergyandignoreswave
phenomena(polarization,interference)ofEMT
• IntheRTEapproachlightistreatedascomposedofdistinct
particles(photons)propagatingthroughamedium.The
modelisrestrictedtointeractionsbetweenlightparticles
themselvesandisderivedbyconsideringchangesinenergy
flowduetoincoming,outgoing,absorbedandemitted
photonswithinaninfinitesimalvolumedV inthemedium
(energybalance).
• Themodelconsidersasmallpacketoflightenergydefinedby
→
itspositionr,directionofpropagationŝ,overatimeinterval
dt,andwithpropagationspeedc
4
2015-04-01
Steady‐stateRadiativeTransferEq.
• Whentheradianceisconstantwithtime,radiance
, isexpressedinunitsofWcm‐2sr‐1
• Powerfluxdensityinaspecificdirections withinaunitsolid
angled
• Thegoverningdifferentialequationforradianceiscalled
radiativetransferequationandisgivenby
d ,
,
,
, d d
4
where , :phasefunctionofaphotontobescattered
fromdirection into ,
d isaninfinitesimalpathlength,
d :elementarysolidangleaboutthedirection
Steady‐stateRadiativeTransferEq.
• Ifscatteringissymmetricabouttheopticalaxis,wemayset
,
with beingthescatteringangle
• Whenperformingtheopticalpropertiesmeasurement,
observablequantityistheintensity whichistheradianceby
integrationoverthesolidangle
,
d (unit:Wcm−2
• Radiancecanbeexpressedas
,
whereis
issolidangledeltafunctionpointinginto
thedirectiongivenbys
5
2015-04-01
Steady‐stateRadiativeTransferEq.
• Whenalaserbeamisincidentonaturbidmedium,the
radianceinsidethemediumcanbedividedintoacoherent
andadiffusetermaccordingtotherelation
• Coherentradianceisattenuatedbyabsorptionandscattering
ofthedirectbeamandthereforecanbedescribedas
anditssolution
exp
where :incidentintensity,d:opticaldepth
 coherentintensityinturbidmediumdecaysasan
exponentialfunction
• Themainproblemisinthediffuseradiance,sincescattered
photonsdonotfollowadeterminedpath
Steady‐stateRadiativeTransferEq.
• Dependingonthevalueofthealbedo,adequate
approximationsandstatisticalapproachesmustbechosento
evaluatethediffuseradiance
–
–
–
–
–
First‐orderscattering
Kubelka‐Munk theory
Diffusionapproximation
MonteCarlosimulations
Inverseadding‐doubling
• Morecomplex moreaccurate,butmore
calculationtimeneeded
6
2015-04-01
First‐OrderScattering
• Assumption:diffuseradianceisverysmallcomparedto
coherentradiance
≃ :firstorderscattering,sincescatteredlightcan
•
betreatedinasimilarmannerasabsorbedlight
• Theintensityatadistancez fromthetissuesurface:
exp
wherez denotestheaxisoftheincidentbeam
 first‐orderscatteringislimitedtoplaneincidentwaves
andmultiplescatteringisnotcounted
• Thisworkswhend<<1ora<<0.5(verylowscattering)
Kubelka‐Munk Theory
•
•
•
•
•
Assumption:coherentradianceisnegligible
≃
Flux :incidentradiationdirection
Backscatteredflux :oppositedirection
AKM andSKM:Kubelka‐Munk absorptionandscattering
coefficientsofdiffuseradiation
7
2015-04-01
Kubelka‐Munk Theory
,
•
•
• Thisstatesthatradianceineitherdirectionencounterslosses
duetoabsorptionandscatteringandonegaindueto
oppositedirectionscattering
Kubelka‐Munk Theory
• Thegeneralsolutionsare
exp
exp
with
exp
exp
,
2
• Keyquestionishowtoconvert
AKM andSKM into and
• d :infinitesimalpathlengthof
ascatteredphoton
• d :infinitesimalpathlengthof
acoherentphoton
8
2015-04-01
Kubelka‐Munk Theory
• Wecanwritetheaveragevalues
withb asanumericalfactorandis>1
• WiththegeometryshowninFig.2.12,weobtain
1
cos
cos
cos
cos
cos
• Since doesnotdependon (assumption:purelydiffuse
scattering)
cos
2
cos
cos
• Therefore,
2
Kubelka‐Munk Theory
• BecauseonlybackscatteringisassumedasshowninFig.2.11,
thecorrespondingrelationfor
isgivenby
• Kubelka‐Munk theory(twoflux)isoneofmanyfluxtheory,
wherethetransportequationisconvertedintoamatrix
differentialequationbyconsideringtheradianceatmany
discreteangles
• Yoonetal.(1987):7fluxes
• Mudgett andRichards(1971):22fluxes
• However,theseareallstillrestrictedtoonedimensional
geometryandassumethattheincidentlightisalreadydiffuse
• Extensivecomputercalculationisanotherdisadvantage
9
2015-04-01
DiffusionApproximation
• Foralbedosa>>0.5,scatteringismuchgreaterthan
absorption,thediffuseradiancetendstobealmostisotropic
• Thendiffuseradiance inaseriesby
3
⋯ , ( diffusionapproximation)
where isthediffuseintensityand
:thevectorflux
• Thediffuseintensity
equation
where
•
3
•
3
,
itselfsatisfiesthefollowingdiffusion
:diffusionparameter, :sourceofscatteredphotons
3
exp
, :incidentfluxamplitude
DiffusionApproximation
• Effectivediffusionlength
1
1
3
• Effectiveattenuationcoefficient
1
3
• Ingeneral,thediffusionapproximationstatesthat
exp
exp
withA+B=I0
10
2015-04-01
DiffusionApproximation
• Therearedifferentsetsofvaluesfor , ,andg andtheycan
beexpressedintermsofeachotherbyso‐calledsimilarity
relationsgivenby
1
1
whiletildesindicatetransformedparameters
• Themotiveofapplyingsimilarityrelationsisthe
transformationofanisotropicscatteringintoisotropic
scattering byusing
=0,
,
1
• Bythistransformation,computercalculationsare
significantlyfacilitatedsinceonly and areneededfor
characterizingopticaltissueproperties
DiffusionApproximation
• Fig.2.13showsthedependenceofthediffuseradianceon
opticaldepthinthecaseofisotropicscatteringanddifferent
albedos
• Whena=0,itfollowslambert’slawofabsorption
• Fora=1,theradianceapproachesanasymptoticvalue
• Noticethatbelowthesurface,
thediffuseintensityincreases
duetothebackscattered
photons
11
2015-04-01
RadiativeTransportTheory
• ThedirectapplicationofEMtheoryiscomplicated
• RTTdealswiththetransportoflightenergyandignoreswave
phenomena(polarization,interference)ofEMT
• IntheRTEapproachlightistreatedascomposedofdistinct
particles(photons)propagatingthroughamedium.The
modelisrestrictedtointeractionsbetweenlightparticles
themselvesandisderivedbyconsideringchangesinenergy
flowduetoincoming,outgoing,absorbedandemitted
photonswithinaninfinitesimalvolumedV inthemedium
(energybalance).
• Themodelconsidersasmallpacketoflightenergydefinedby
→
itspositionr,directionofpropagationŝ,overatimeinterval
dt,andwithpropagationspeedc
TDRadiativeTransferEquation
• Thechangeinenergyradiance , , ̂ isequalto
thelossinenergyduetoabsorptionandscattering
outofŝ(unitdirectionvector),plusthegainsin
energyfromlightscatteredintotheŝ‐directed
packetfromotherdirectionsandfromanylocal
sourceofthelightat .
• Radianceisdefinedasenergyflowperunitnormala
reaperunitsolidangleperunittime.
12
2015-04-01
TDRadiativeTransferEquation
• c isthespeedoflightinthetissue
•
̂ , ̂ ’)isthephasefunction,representingtheprobabilityoflightwith
propagationdirection ̂ ′ beingscatteredintosolidangledΩ around ̂ .
→
→
→
→
→
TDRadiativeTransferEquation
→
→
→
→
→
13
2015-04-01
TDRadiativeTransferEquation
• Severalotherimportantphysicalquantitiesbasedon
thedefinitionofradiance
– Photondensityordiffusephotonfluence rate(insidethe
element)
– Fluence
Φ
,
/
– Photonfluxor radiantfluxdensity(atitsboundary):
measurableparameterandallowsRTEtobesolvedforμa
andμa’respectively.
DiffusionApproximation
• InRTE,sixdifferentindependentvariablesdefineth
eradianceatanyspatialandtemporalpoint(x,yand
zfrom ,polarangle andazimuthalangle from ̂ ,
andt).
• Bymakingappropriateassumptionsaboutthebehav
iorofphotonsinascatteringmedium,thenumberof
independentvariablescanbereduced.
14
2015-04-01
DiffusionApproximation
• SimplifiedformofRTTat“diffusionlimit”
1. Relativetoscatteringevents,thereareveryfewabsorpti
onevents(μs’>>μa)
• thenumberofphotonundergoingtherandomwalkislarge
• Radiancewillbecomenearlyisotropic
2. Thetimeforsubstantialphotondensitychangeismuchl
ongerthanthetimetotraverseonetransportmeanfree
path
• Assumetissueis“macroscopicallyhomogeneous”
• Over1mfp’,thefractionalchangeinphotonflux,
J(r,t)
<<1
DiffusionApproximation
isotropic and anisotropic terms
 

1
3  

I (r , s , t ) 
j (r , t )  s
 (r , t ) 
4
4

, , ̂ with I (r , s, t ) 
• Byreplacing
• RTEbecomes
•
,
·
,

1
3  

j (r , t )  s
 (r , t ) 
4
4
,
,
• FromFick’slaw,
,
•
,
·
,
,
,
3
,
,
15
2015-04-01
MeasurementStrategies
“BlackBox”
Optical
Source
‘input’
TISSUE
H(μa,μs)
Detector
‘output’
H:SystemFunction
• Goal:Tofindout H(μa,μs)
• RequiresNon‐Staticsystem Perturba
tionsineitheropticalsourceortissue
MeasurementSchemes
• CW(ContinuousWave)Measurement
–
–
–
–
Simplestformofmeasurement
Static,continuouswaveinput
requiresdynamictissuepropertychanges
pulseoximetry
• Time‐ResolvedMeasurements
– Temporalchangesinopticalsources
• TimeDomainPhotonMigration(TDPM)
• FrequencyDomainPhotonMigration(FDPM)
• Spatially‐ResolvedMeasurement
– Spatialchangesinopticalpath
16
2015-04-01
attenuation μt‐total
CW(continuouswave)
μt‐oxy
arterial
pulsatile
venous(Hb‐O2)
μt‐background non‐pulsatile
tissue
?
time
μt =
μa + μs’
CWExample,pulse‐oximetry
pulseoximetrylocksintopulse
healthyadultcalibrationaccountsfortissuescatter(μs’)
~μa‐oxy
= μa‐total ‐
μa‐background
typicallyat2wavelengths(660,940nm)
17
2015-04-01
TimeDomainPhotonMigration(I)
Impulse Function,

TimeDomainPhotonMigration(II)
• Directlymeasureμa andμs fromTPSFusingDiffu
sionEquation
• Complicatedandexpensivedetectionsystem
• ratherlowSNR
TemporalPointSpreadFunction
(TPSF)
18
2015-04-01
Source
Continued
Detector
25 mm
0
10 mm
25
t1 = 200 ps
t2 = 600 ps
t3 = 1000 ps
t4 = 1400 ps
Laser pulse
Moving
time gate
Detected pulse
t = 200 ps
0
100
200
300
400
500
600
700
t = 200 ps
800
900
1000 1100 1200 1300 1400
time (picoseconds)
Source
Continued
Detector
25 mm
0
10 mm
25
t1 = 200 ps
t2 = 600 ps
t3 = 1000 ps
t4 = 1400 ps
Laser pulse
t
slope of ln  -ac
0
100
200
300
400
500
600
700
800
900
1000 1100 1200 1300 1400
time (picoseconds)
19
2015-04-01
FrequencyDomainPhotonMigration
SOURCE
DETECTED
TISSUE
stuffhappens

ACsrc AC
SRC
AMPLITUDE
ACdet
DC
DCSRC
src
DC
DC det
ACDET
DET
 ~ TIME
M = AC/DC
TIME
FrequencyDomainPhotonMigration
1
,
·
,
,
D  1 /[3(  a   s ' )]
,
Diffusion constant
Intensity modulating the light source gives :
 ( r , t )  e i t  ( r )
Plugging this into time dependent diffusion equation and
approximating tissue as homogeneous gives the frequency domain
equation:
D 2 (r )  (  a 
i n
) (r )   S (r )
c
Tromberg et al, Appl. Opt., 32(4), 607-616,1993.
20
2015-04-01
FrequencyDomainPhotonMigration
Frequency-Domain
Instrument
Pham, Tromberg, et al., Rev. Sci. Instr., 71, 2500, (2000)
Experimental Response





I


source
light
time (ns)
detected
light
Nonlinear
Least
Square Fits
Theoretical Response
a 

a 

light scattering tissues
a, s’
NIR Tissue
Spectroscopy
Bulk Tissue
Function &
Structure
Spectroscopic
Analysis
FrequencyDomainPhotonMigration
AMPLITUDE
-4
4.5x10
180
PHASE
160
-4
140
-4
3.5x10
120
-4
100
-4
80
3.0x10
2.5x10
60
-4
2.0x10
PHASE (deg)
AMPLITUDE (a.u.)
4.0x10
40
-4
1.5x10
20
100
200
300
400
500
600
700
FREQUENCY (MHz)
21
2015-04-01
FrequencyDomainPhotonMigration
simultaneous fit
AMPLITUDE
-4
4.5x10
180
PHASE
160
-4
140
-4
3.5x10
120
-4
100
-4
80
3.0x10
2.5x10
60
-4
2.0x10
PHASE (deg)
AMPLITUDE (a.u.)
4.0x10
40
-4
1.5x10
20
100
200
300
400
500
600
700
FREQUENCY (MHz)
Monte CarloSimulations
• AnumericalapproachtothesolutionoftheRTEis
basedonMonteCarloSimulations
• Itrunsacomputersimulationoftherandomwalkof
anumberN ofphotons statisticalapproach
 large
• Thestatisticalaccuracydependson
numberofphotonsrequired time‐consuming
• ItwasfirstproposedbyMetropolisandUlam in
1949
22
2015-04-01
Monte CarloSimulations
• It isfollowingtheopticalpathofaphotonthroughtheturbid
medium
• Thedistancebetweentwocollisionsisselectedfroma
logarithmicdistributionusingarandomnumbergenerated
bythecomputer
• Absorption:giveaweighttoeachphotonandpermanently
reducingthisweightduringpropagation
• Whenscatteringistooccur,anewdirectionofpropagationis
chosenaccordingtoagivenphasefunctionandanother
randomnumber
• Theprocesscontinuesuntilthephotonescapesfromthe
volumeoritsweightreachesagivencutoffvalue
Monte CarloSimulations
• Meieretal.(1978)andGroenhuis etal.(1983)statedthat
thereare5principalstepsinMonteCarlosimulations
1.
2.
3.
4.
5.
Sourcephotongeneration
Pathwaygeneration
Absorption
Elimination
Detection
1) Sourcephotongeneration
–
–
Photonsaregeneratedatasurfaceoftheconsideredmedium
Theirspatialandangulardistributioncanbefittedtoagivenlight
sourcee.g.Gaussianbeam
23
2015-04-01
Monte CarloSimulations
2) Pathwaygeneration
–
–
–
–
–
Afteraphotongeneration,thedistancetothefirstcollisionis
determined
Absorbingandscatteringparticlesshouldberandomlydistributed
Thus,themeanfreepathis1⁄ ,where isthedensityof
particlesand istheirscatteringcross‐section
Arandomnumber0
1 isgeneratedbythecomputer
Thedistance
tothenextcollisioniscalculatedfrom
ln
–
Since
ln
1,theaveragevalueof
–
Hence,ascatteringpointhasbeenobtained
isindeed1⁄
Monte CarloSimulations
2) Pathwaygeneration
–
–
Thescatteringangleisdeterminedbyasecondrandomnumber
inaccordancewithacertainphasefunction(e.g.Henyey‐Greenstein
phasefunction)
Thecorrespondingazimuthangle ischosenas
2
where isthethirdrandomnumberbetween0and1
24
2015-04-01
Monte CarloSimulations
3) Absorption
Toaccountforabsorption,aweightisattributedtoeachphoton
Photonsenteringthemediumhaveweightasaunity
Duetoabsorptiontheweightisreducedbyexp
Alternativewayistoimplementaweightbyassigningafourth
randomnumber rangingfrom0and1
– TheninsteadofassumingonlyscatteringeventsinStep2,scattering
takesplaceif <a,whereaisalbedo
– For >a,thephotonisabsorbedwhichisequivalenttoStep4.
–
–
–
–
4) Elimination
–
–
–
Thisstepappliedifaweighthasbeengiventoeachphoton(step3)
Whenthisweightreachesacutoffvalue,thephotoniseliminated
Thennewphotonislaunchedandstartsfromstep1
Monte CarloSimulations
5) Detection
– Afterhavingrepeatedstep1‐4forasufficientnumberofphotons,a
mapofpathwaysiscalculatedandstoredinthecomputer
– Thus,statisticalstatementscanbemadeaboutthefractionofincident
photonsbeingabsorbedbythemediumaswellasthespatialand
angulardistributionofphotonshavingescapedfromit
• In1993,Graaff etal.proposed“condensedMonteCarlo
simulations”
– Theresultsofearliercalculationscanbestoredandused
againifneededforthesamephasefunctionbutfor
differentvaluesoftheabsorptioncoefficientandalbedo
– Thisreducesaconsiderableamountofcomputingtime
25
2015-04-01
InverseAdding‐DoublingMethod
• In1993Prahl etal.proposed“inverseadding‐doubling”
• “inverse”impliesareversaloftheusualprocessofcalculating
reflectanceandtransmittancefromopticalproperties
• “adding‐doubling”referstoearliertechniquesestablishedby
vandeHulst (1962)andPlass etal.(1973)
• Thedoublingmethodassumesthatreflectionand
transmissionoflightincidentatacertainangleisknownfor
onelayerofatissueslab
• Thesamepropertiesforalayertwiceasthickisfoundby
dividingitintotwoequalslabsandaddingthereflectionand
transmissioncontributionsfromeitherslab.
InverseAdding‐DoublingMethod
• Thus,reflectionandtransmissionforanarbitraryslabof
tissuecanbecalculatedbystartingwithathinslabwith
knownproperties,e.g.asobtainedbyabsorptionandsingle
scatteringmeasurements,anddoublingituntilthedesired
thicknessisachieved
• The addingmethodextendsthedoublingmethodto
dissimilarslabsoftissue.
• Withthissupplement,layeredtissueswithdifferentoptical
propertiescanbesimulated,aswell.
26
2015-04-01
PolarizationEffects
• Sofar,theradianceJisassumedtobeascalarand
polarizationeffectsarenegligible
• Inthe1980s,severalextensivestudiesweredoneon
transporttheorypointingouttheimportanceofadditional
polarizingeffects(agoodsummarybyIshimaru andYeh
(1984))
• Herein,theradianceisreplacedbyafour‐dimensionalStokes
vector,andthephasefunctionbya4× 4Müllermatrix
• TheStokesvectoraccountsforallstatesofpolarization.The
Müllermatrixdescribestheprobabilityofaphotontobe
scatteredintoacertaindirectionatagivenpolarization
• The transportequationthenbecomesamatrixintegro‐
differentialequationandiscalledavectortransportequation.
Summary
• InFig.2.14,theintensitydistributionsinsideaturbid
mediumcalculatedwitheithermethodarecomparedwith
eachother.
• Becauseisotropicscatteringis
assumed,ananalyticalsolution
canalsobeconsideredwhichis
labeled“transporttheory”.
• Twodifferentalbedos,a=0.9and
a=0.99,aretakenintoaccount.
• TheKubelka–Munk theoryusually
yieldshighervalues,whereas
diffusionapproximationand
MonteCarlosimulationfrequently
underestimatetheintensity.
27