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Lecture 12 Monte Carlo Simulations

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Title: Lecture 12 Monte Carlo Simulations


1
Lecture 12 Monte Carlo Simulations
Useful web sites http//kurslab-atom.fysik.lth.se
/FED4Medopt/MonteCarloSim.pdf http//omlc.ogi.edu
/classroom/ece532/class4/index.html
2
Introduction
  • Monte Carlo method
  • Simulates photon transport in turbid medium such
    as tissue, that contains absorption and
    scattering
  • This model is based on a random walk, where a
    photon or a photon package is traced through the
    tissue until it exits or until it gets entirely
    absorbed
  • The method is based on a set of rules that govern
    the movement of the photon in tissue (transport
    equation) the 2 key decisions
  • 1) the mean free path for an absorption or
    scattering event
  • 2) the scattering angle

3
Introduction
  • Monte Carlo method

Incident Light
Emitted Light
Absorption Event
4
Introduction
  • Input - photons
  • The rules of photon propagation are expressed as
    probability distributions (for incremental steps)
  • The method is statistical in nature and requires
    the propagation of a large number of photons
  • Computational time is proportional to the number
    of photons launched in the simulation (typically,
    1E6) number of photons determine precision

5
I. Introduction
  • Output

r
Large Number of Photons
Reflectance
z
Fluence
Transmission
6
I. Introduction
  • Assumptions
  • The simulations treat photons as neutral
    particles rather than as a wave phenomenon
  • It is assumed that the photons are multiply
    scattered by tissues
  • Therefore, phase and polarization are assumed to
    be randomized and can be ignored

7
I. Introduction
  • Limitations
  • The Monte Carlo method is based on macroscopic
    optical properties that are assumed to extend
    uniformly over the tissue volume
  • Cannot handle small heterogeneities such as
    photon transport in cells or organelles
  • Significant computational time not ideal for
    real-time analysis
  • Does not provide intuition on the dependence of
    reflectance on optical properties

8
Monte Carlo Simulations
  • Why?
  • Perform simulations for conditions under which
    the accuracy of the diffusion approximation is
    limited
  • Simulate more complicated tissue geometries
    (multilayer media)
  • Simulate more complicated light illumination
    geometries (collimated, angled, Gaussian beam
    illumination)

9
I. Introduction

10
II. Selecting Variables
  • I. Selecting step size
  • II. Selecting direction
  • deflection angle
  • azimuthal angle

11
II. Selecting the Variables
  • The variables that govern Monte Carlo
  • The mean free path for an absorption or
    scattering event
  • Step size (function of ma and ms)
  • The scattering angle
  • Deflection angle, q (function of anisotropy, g)
  • Azimuthal angle, y
  • Random number generators will be used for the
    selection of step size, deflection angle and
    azimuthal angle for sampling from known
    probability density functions

12
II. Selecting the Variables
  • The variables that govern Monte Carlo

13
II. Selecting the Variables
  • Basis for the Monte Carlo method
  • The Monte Carlo method as its name implies
    (throwing the dice) relies on the random
    sampling of a probability density function based
    on a computer generated random number
  • Need to understand definitions for probability
    density functions and probability distributions

14
II. Selecting the Variables
  • Random variable, x
  • A probability density function is a function
    defined on a continuous interval so that the area
    under the curve described by the function is
    unity
  • The probability density function p(x) for some
    random variable, x, defines the distribution of x
    over the interval altxltb, such that

15
II. Selecting the Variables
  • Random variable, x
  • The probability that x will fall in the interval
    a,x1 such that a lt xlt x1, is given by a
    probability distribution function, F(x1) which is
    defined by

16
II. Selecting the Variables
  • Random variable, x

17
II. Selecting the Variables
  • Random Numbers
  • Now we want to utilize a uniform random number
    generator of the computer to generate a random
    variable, rnd between 0 and 1
  • The uniform probability density function (for the
    random number) will be mapped to a specific
    probability density function p(x) corresponding
    to step size, deflection angle and azimuthal angle

18
II. Selecting the Variables
  • Random number, rnd
  • Now apply the same logic to the probability
    density function for a random number, rnd
  • The probability density function for a random
    number, rnd is a constant over 0,1

19
II. Selecting the Variables
  • Random number, rnd
  • The probability that rnd will fall in the
    interval 0, rnd1 such that 0 lt rndltrnd1, is
    given by the probability distribution function,
    F(rnd1) which is defined by

20
II. Selecting the Variables
  • Random number, rnd

21
II. Selecting the Variables
  • Relationship between x and rnd
  • The key to the Monte Carlo selection of x using
    rnd is to equate the probability that rnd is in
    the interval 0, rnd1 with the probability that
    x is in the interval a,x1
  • In other words, we can equate F(x) to F(rnd)
    i.e., we are equating the shaded areas under the
    curves

22
II. Selecting the Variables
  • Random number, rnd

23
II. Selecting the Variables
24
II. Selecting the Variables
  • Basic equation
  • For generality, we replace the variables, rnd1
    and x1 by the continuous variables, rnd and x
  • This is the basic equation for sampling x from
    p(x) using a randomly generated number, rnd over
    the interval 0,1

25
II. Selecting the Variables
  • Random Numbers
  • Need pdfs for step size, deflection angle and
    azimuthal angle
  • Need to relate rnd to pdf using the
    transformation expression
  • Express step size, deflection angle and azimuthal
    angle in terms of rnd

26
II. Selecting the Variables
  • Three equations
  • Step size
  • Deflection angle
  • Azimuthal angle

27
II. Selecting the Variables
  • The known parameters

Incident Light
Emitted Light
nm
Absorption Event
ma (l), ms (l), g(l), nt
28
III. Photon Propagation

29
III. Photon Propagation
  • Initializing and launching the photon
  • Each photon is initially assigned a weight, W
    equal to unity, before it is injected into the
    tissue at the origin
  • If there is a mismatched boundary between the
    outside medium (m) and tissue (t), then specular
    reflectance will occur (Fresnel law)
  • The resulting photon weight, W is decreased to

30
III. Photon Propagation

31
III. Photon Propagation
  • Generating the step size
  • The computers random number generator yields a
    random variable, rnd, in the interval 0,1 and
    the sampling of the step size, s is given by

32
III. Photon Propagation

33
III. Photon Propagation
  • Photon absorption
  • Once the photon has taken a step, some
    attenuation of the photon weight due to
    absorption must occur the amount of deposited
    photon weight, DQ is
  • The new photon weight, W is

34
III. Photon Propagation

35
III. Photon Propagation
  • Terminating a photon
  • A technique called Roulette is used to terminate
    the photon when W lt Wthreshold
  • This technique gives the photon one chance in m
    of surviving with a weight, mW.
  • This is also carried out using a random number
    generator.

36
III. Photon Propagation

37
III. Photon Propagation
  • Change photon direction
  • If the photon survived, it is ready to be
    scattered
  • Photon deflection angle, q equals cos-1(m) (Here
    m is related to anisotropy)
  • Photon azimuthal angle is y
  • Next the new trajectory of the photon (mx, my,
    mz), can be calculated from the old trajectory

Where, mx, my, and mz are the direction
cosines specified by taking the angle that the
photon direction makes with each axis
38
II. Selecting the Variables
  • Three equations
  • Step size
  • Deflection angle
  • Azimuthal angle

39
III. Photon Propagation

40
III. Photon Propagation
  • Internal reflection or escape
  • The internal reflectance is also calculated using
    Fresnels law
  • A fraction, 1-R(q) of the current photon weight
    successfully escapes the tissue as observable
    reflectance and updates the total reflectance
  • A fraction R(q) of the current photon weight is
    internally reflected. The new photon weight is
    R(q)W
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