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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 altxltb, such that

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

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

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

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

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

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

II. Selecting the Variables

- Three equations
- Step size
- Deflection angle
- Azimuthal angle

II. Selecting the Variables

- The known parameters

Incident Light

Emitted Light

nm

Absorption Event

ma (l), ms (l), g(l), nt

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) - The resulting photon weight, W is decreased to

III. Photon Propagation

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

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 - The new photon weight, W is

III. Photon Propagation

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.

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
- Azimuthal angle

III. Photon Propagation

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