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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