MonteCarlo-1.pptx

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Biomedical Optics II II

Monte Carlo Modeling of

Photon Transport

2011/09/28


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Radiative Transfer in LivingTissue

2College of Engineering, Peking University II

How to

solve the

problem?

Maxwells

equations?

Modelingbased onscattering

and

absorption

Monte Carlosimulation

Radiativetransferequation


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Monte Carlo

College of Engineering, Peking University II


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!College of Engineering, Peking University II


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"College of Engineering, Peking University II


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The simulations treat photons as neutral particles

rather than as a wave phenomenon.

It is assumed that the photons are multiply

scattered by tissues. Sometimes, phase and

polarization are assumed to be randomized andcan be ignored.

Monte CarloSimulations

#College of Engineering, Peking University II


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Photon transport in biological tissue can benumerically simulated by the Monte Carlo method.

The trajectory o a photon is modeled as a

persistent random wal!, with the direction o eachstep depending on that o the previous step.

"y contrast, the directions o all o the steps in a

simple random wal! are independent.

"y trac!ing a suicient number o photons, we can

estimate physical #uantities such as diuse

relectance.

Introduction

$College of Engineering, Peking University II


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The medium is random $se %dice& to build it

' system with !nown probability distributions

(scattering and absorption)

Monte Carlo



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*rom +u and -oblingerIn all applications o the Monte Carlo method, a stochastic

model is constructed in which the epected value o a

certain random variable (or o a combination o several

variables) is e#uivalent to the value o a physical #uantityto be determined.

This epected value is then estimated by the average o

multiple independent samples representing the random

variable introduced above.*or the construction o the series o independent samples,

random numbers ollowing the distribution o the variable

to be estimated are used.

Denition of Monte CarloMethod



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The photon propagation is random and determined by tissue optical properties

/. 0istance to net scattering event1 'pply 2dice2 with properties set by the

mean ree3path3length (determined by the scattering and absorption

coeicients) to set the path length beore a scattering or an absorption

event occur.

4. 0irection ater scattering1 'pply 2dice2 with phase unction 5 anisotropy

o scattering to set the scattering angles.



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Main advantage6o limitation concerning boundary conditions or

spatial localisation o inhomogeneities in the tissue

77 *leiblilityMain disadvantage

Problem o getting good statistics, particularly i the

point o interest is located ar away rom the point oentry o the light and the scattering and absorption

coeicients are high 77 long CP$ time



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It is important to realize that the Monte Carlo method estimatesensemble3averaged #uantities.

'n ensemble o biological tissues is modeled or the average

characteristics o photon transport8 the ensemble consists o all

instances o the tissues that are microscopically dierent but

macroscopically identical.

9ules are deined or photon propagation rom the probability

distributions o, or eample, the angles o scattering and the step

sizes.

The statistical nature re#uires trac!ing a large number o photons,

which is computationally time3consuming.

Multiple physical #uantities can be simultaneously estimated,

however.

Ensemble veraging



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In this chapter, photons are treated as waves ateach scattering site but as classical particles

elsewhere.

Coherence, polarization, and nonlinearity areneglected.

Structural anisotropy33not to be conused with

scattering angular anisotropy33in tissuecomponents, such as muscle ibers or collagens, is

neglected as well.

Simplications



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0etermine the spatial interval between twosuccessive interaction events

0etermine the scattering angle

0etermine the survival o the photon

Three Ma!or SamplingProcedures

1!College of Engineering, Peking University II


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The mean ree path or an absorption orscattering event

Step size (unction o :aand :s)

The scattering angle

0election angle, ; (unction o anisotropy,

g)

'zimuthal angle, The step size o the photon is calculated based on

sampling the probability or the photonDs mean ree

path

> I step size too small, MC is ineicient, but i step

size is too large, poor approimation o real photon

travel

> Choose step size rom probability density unction



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&he total attenuation coefficient

9ecall that the probability o interaction o a photon

with a medium per unit path length is

PGs/,s/5ds/HJLtds/



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The probability o interaction o a photon with a

medium per unit path length is related to the gradient

o transmission


/

//

( )

( )t

dT sds

T s =


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The probability distribution unction (rom "eerDs

law) is deined as

College of Engineering, Peking University II

/ / /

/ / /

( ) ( ) ep( )

( ) / ( ) / ep( )

t

t

P s s T s s

P s s T s s

> = =

< = =


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'inal expression

Solving or %s& yields

!College of Engineering, Peking University II

ln

t

s

=


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Sampling the Step Si)e

( )

( ) ( )

tt

t

t

ss

s

ssP

ln/ln=

=

=

=

or

si$estep#ampled

)exp(methodondistributiInverse

)exp(

si$estepoffunctionondistributi*umulative



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#College of Engineering, Peking University II


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$College of Engineering, Peking University II


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li h i l


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Sampling the Scattering ngle

[ ]4

4 BM4

44

4

=enyey3reenstein phase unction./

(cos ) , @,4(/ 4 cos )

Sampled scattering angle.

/ // i @cos 4 / 4

4 / i @

gp

g g

gg gg g g

g

=

+

+ = +

=

Sampling the )imuthal


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Sampling the )imuthalScattering ngle

( )

G@,4

/

4

4

p

)

=

=

!1College of Engineering, Peking University II

M i h Ph P *


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Moving the Photon Pac*et

iz

iy

ix

szz

syy

sxx

+

+

+


b ti


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bsorption

WWW

W

W

t

a

=

+pdate of Photon Propagation


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+pdate of Photon PropagationDirection

( )

( )

( ) ( )

( ) ( )( ) ( )

( ) ( ) ( )

eists.solutione'lternativ

.N,N,Nin,anglesazimuthalandpolarhasalongNnpropagatiophotonringPostscatte

.N,N,Ngettoorabout,,9otate(4)

.,,scoordinateteintermediagettoorabout,,9otate(/)

,N,N,Nscoordinatelocalto,,scoordinateglobalromtransormTo

.,anglesazimuthalandpolarhasnpropagatiophotonringPrescatte6ote

.cossgnsinsincossinthen,/I

.coscossin/

,cos/

)sincos(sin

,cos

/

)sincos(sin

@

OOOO

OOO

@

@@

4

4

4

zyxzk

zyxyzyx

zyxzzyx

zyxzyx

k

,,

z

'

z

'

y

'

xz

zz

'

z

y

z

xzy'

y

x

z

yzx'

x

===

+=

+

+=

+

=

!!College of Engineering, Peking University II


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Please go to ianan QuDs ppt /Bb.pd and /Bc.pd.

!"College of Engineering, Peking University II


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int main ()

Ralbedo J mus (mus 5 mua)8

rs J (n3/.@)O(n3/.@)(n5/.@)(n5/.@)8 O specular relection O

critangle J s#rt(/.@3/.@nn)8 O cos o critical angle O

binspermp J /emicronsperbin(mua5mus)8

or (i J /8 i KJ photons8 i55)R

launch ()8

while (weight 7 @) R

move ()8

absorb ()8

scatter ()8U

U

printresults()8

return @8

U

Monte Carlo Program

!#College of Engineering, Peking University II


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void launch() O Start the photon O

R J @.@8 y J @.@8 z J @.@8

u J @.@8 v J @.@8 w J /.@8

weight J /.@ 3 rs8

U

void bounce () O Interact with top surace O

R

double t, temp, temp/,r8

w J 3w8

z J 3z8

i (w KJ critangle) return8 O total internal relection O

t J s#rt(/.@3nOnO(/.@3wOw))8 O cos o eit angle Otemp/ J (w 3 nOt)(w 5 nOt)8

temp J (t 3 nOw)(t 5 nOw)8

r J (temp/Otemp/5tempOtemp)4.@8 O *resnel relection O

rd 5J (/.@3r) O weight8

weight 3J (/.@3r) O weight8

U

Monte Carlo Program

!$College of Engineering, Peking University II


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void move() O move to net scattering or absorption event O

Rdouble d J 3log((rand()5/.@)(9'60M'V5/.@))8

5J d O u8

y 5J d O v8

z 5J d O w8

i ( zKJ@ ) bounce()8

U

void absorb () O 'bsorb light in the medium O

R

int binJzObinspermp8

i (bin 7J "I6S) bin J "I6S3/8

heatGbinH 5J (/.@3albedo)Oweight8weight OJ albedo8

i (weight K @.@@/)R O 9oulette O

bit 3J weight8

i (rand() 7 @./O9'60M'V) weight J @8 else weight J @./8

bit 5J weight8

U

U

Monte Carlo Program



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void scatter() O Scatter photon and establish new direction O

Rdouble /, 4, B, t, mu8

or(88) R Onew directionO

/J4.@Orand()9'60M'V 3 /.@8

4J4.@Orand()9'60M'V 3 /.@8

i ((BJ/O/54O4)KJ/) brea!8

U

i (gJJ@) R O isotropic O

u J 4.@ O B 3/.@8

v J / O s#rt((/3uOu)B)8

w J 4 O s#rt((/3uOu)B)8

return8

U

Monte Carlo Program



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mu J (/3gOg)(/3g54.@OgOrand()9'60M'V)8mu J (/ 5 gOg3muOmu)4.@g8

i ( abs(w) K @.W ) R

t J mu O u 5 s#rt((/3muOmu)(/3wOw)B) O (/OuOw34Ov)8

v J mu O v 5 s#rt((/3muOmu)(/3wOw)B) O (/OvOw54Ou)8

w J mu O w 3 s#rt((/3muOmu)O(/3wOw)B) O /8

U else R

t J mu O u 5 s#rt((/3muOmu)(/3vOv)B) O (/OuOv 5 4Ow)8

w J mu O w 5 s#rt((/3muOmu)(/3vOv)B) O (/OvOw 3 4Ou)8

v J mu O v 3 s#rt((/3muOmu)O(/3vOv)B) O /8

U

u J t8

U

Monte Carlo Program

"0College of Engineering, Peking University II

l


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void printresults() O Print the results O

Rint i8

print(2XsYnXsYnYnScattering J X?.BcmYn'bsorption J X?.BcmYn2,t/,t4,mus,mua)8

print(2'nisotropy J X?.BYn9er Inde J X?.BYnPhotons J X?ld2,g,n,photons)8

print(2YnYnSpecular 9el J X/@.ZYn"ac!scattered 9el J X/@.Z2,rs,rd(bit5photons))8

print(2YnYn 0epth =eatYnGmicronsH GAcm[BHYn2)8

or (iJ@8iK"I6S3/8i55)R

print(2X\.@ X/4.ZYn2,iOmicronsperbin, heatGiHmicronsperbinO/e(bit5photons))8

U

print(2 etra X/4.ZYn2,heatG"I6S3/H(bit5photons))8

U

Monte Carlo Program


M C l P


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char t/G?@H J 2Small Monte Carlo by Scott Prahl (httpomlc.ogi.edu)28

char t4G?@H J 2/ Acm[4 $niorm Illumination o Semi3Ininite Medium28

]include Kstdio.h7

]include Kstdlib.h7

]include Kmath.h7

]deine "I6S /@/

double mua J Z8 O 'bsorption Coeicient in /cm O

double mus J WZ8 O Scattering Coeicient in /cm O

double g J @.4Z8 O Scattering 'nisotropy 3/KJgKJ/ O

double n J /.Z8 O Inde o reraction o medium O

double micronsperbin J 4@8O Thic!ness o one bin layer O

long i, photons J /@@@@@8

double ,y,z,u,v,w,weight8double rs, rd, bit, albedo, critangle, binspermp, heatG"I6SH8

Monte Carlo Program


3


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Monte Carlo can simulate photon transport inbiological tissue

Three steps move, absorb, scatter

Aeight deines its alive or dead

The trajectory o a photon is modeled as a

persistent random wal!. The directions are

independent.

3ummar(


4 t& di


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httpomlc.ogi.educlassroomeceZB4classinde.html

Tinymc.c, Smallmc.c, mcB4/.c

httpomlc.ogi.edusotwaremc

MCM+ download

httplabs.seas.wustl.edubmeAangmc.html

4urt&er readings

"!College of Engineering, Peking University II

# *
http://omlc.ogi.edu/classroom/ece532/class4/index.htmlhttp://omlc.ogi.edu/classroom/ece532/class4/index.htmlhttp://labs.seas.wustl.edu/bme/Wang/mc.htmlhttp://labs.seas.wustl.edu/bme/Wang/mc.htmlhttp://labs.seas.wustl.edu/bme/Wang/mc.htmlhttp://omlc.ogi.edu/classroom/ece532/class4/index.htmlhttp://omlc.ogi.edu/classroom/ece532/class4/index.html


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/. Plot the =enyey reenstein *unction with dierent g.

4. Arite a program (in C or M'T+'") to describe a simple

random wal! in 3y plane. The direction is uniormly

random in B\@ degrees, and the wal!ing step size ollows

the same way o a photon transportation in a mediumwith LJ/@@ cm. Plot your result and hand in your

program. (up to 4@ steps)

B. 9ewrite the tinymc.c program to Matlab.

#ome,or*

""College of Engineering, Peking University II

E-periment. Monte Carlo


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pSimulation

"#College of Engineering, Peking University Biomedical Optics II

Fiber

5= 633 nm



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pSimulation

"$College of Engineering, Peking University Biomedical Optics II

Fiber

5= 633 nm



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pSimulation

"8College of Engineering, Peking University Biomedical Optics II

Fiber

5= 633 nm

E i t t


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$sing g J @.W, and the measured scattering coeicient aswell as other parameters, apply Monte Carlo simulation

(net eperiment), then compare your simulated result

with your measured data.

E-periment report

MonteCarlo-1.pptx

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