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
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
Introduction
Monte Carlo method
Incident Light
Emitted
Light
Absorption
Event
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
I. Introduction
Output
r
Large Number of Photons
Reflectance
z
Fluence
Transmission
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
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
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)
I. Introduction
II. Selecting Variables
I. Selecting step size
II. Selecting direction:
deflection angle
azimuthal angle
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
II. Selecting the Variables
The variables that govern Monte Carlo
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
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 a<x<b, such that:
b
 p(x dx  1
a
II. Selecting the Variables
Random variable, “x”
• The probability that x will fall in the interval [a,x1] such that a < x<
x1, is given by a probability distribution function, F(x1) which is
defined by:
x1
F(x1    p(x dx
a
II. Selecting the Variables
Random variable, “x”
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
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]:
1
 p(rnd d(rnd )  1;
0
p(rnd   1 over [0,1]
II. Selecting the Variables
Random number, rnd
• The probability that rnd will fall in the interval [0, rnd1] such that 0 <
rnd<rnd1, is given by the probability distribution function, F(rnd1)
which is defined by:
F(rnd 1  
rnd1
(

p
rnd
drnd  rnd 1

0
for
0  rnd  rnd 1
II. Selecting the Variables
Random number, rnd
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
II. Selecting the Variables
Random number, rnd
II. Selecting the Variables
x1
F(x1    p(x dx
a
F(rnd 1  
rnd 1

0
p(rnd  drnd  rnd 1
II. Selecting the Variables
Basic equation
• For generality, we replace the variables, rnd1 and x1 by the
continuous variables, rnd and x:
x
rnd   p(x dx
a
• This is the basic equation for sampling x from p(x) using a randomly
generated number, rnd over the interval [0,1]
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
x
rnd   p(x dx
a
II. Selecting the Variables
Three equations
• Step size
• Deflection angle
•
 ln (rnd 
s
ma  ms
2

2


1 
1

g
2 
 
m
1 g 
 1  g  2g(rnd )  
2g 

 
Azimuthal angle

y  2 (rnd)
II. Selecting the Variables
The known parameters
Incident Light
Emitted
Light
nm
ma (l), ms (l, g(l), nt
Absorption
Event
III. Photon Propagation
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):
(
nt  nm 
R sp 
2
(n t  nm 
2
• The resulting photon weight, W is decreased to:
W  1  R sp
III. Photon Propagation
III. Photon Propagation
Generating the step size
• The computer’s random number generator yields a random
variable, rnd, in the interval [0,1] and the sampling of the step size,
s is given by:
 ln( rnd )
s
ma  ms
III. Photon Propagation
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:
•
ma
DQ  W
m

m
a
s
The new photon weight, W is:
ms
W
W
ma  ms
III. Photon Propagation
III. Photon Propagation
Terminating a photon
• A technique called Roulette is used to terminate the photon when
W < 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.
III. Photon Propagation
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
II. Selecting the Variables
Three equations
• Step size
• Deflection angle
•
 ln (rnd 
s
ma  ms
2

2


1 
1

g
2 
 
m
1 g 
 1  g  2g(rnd )  
2g 

 
Azimuthal angle

y  2 (rnd)
III. Photon Propagation
III. Photon Propagation
Internal reflection or escape
•
The internal reflectance is also calculated using Fresnel’s 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