Download A Monte Carlo model of light propagation in tissue

A Monte Carlo model of light
propagation in tissue
An introduction to Monte Carlo techniques
ENGS168
Ashley Laughney
November 13th, 2009
Overview of Lecture
• Introduction to the Monte Carlo Technique
– Stochastic modeling
– Applications (with a focus on Radiation Transport)
– Random sampling and Probability Distribution Functions
• Monte Carlo Treatment of the Radiation Transport Problem
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Photon initialization
Generating the propagation distance
Internal reflection
Photon absorption
Photon termination
Photon scattering
Calculation of observable quantities
• Implementing Monte Carlo - Sample Code
• Radiation treatment planning
Stochastic Modeling
• Launch a photon (or particle)
• Sample physical properties using random variables (random
sampling)
• Let the photon evolve through the system
• Keep track of all important parameters until the photon dies or
exits the problem space
• Summarize and infer results after ENOUGH photons interact.
Ref: Venkat Krishnaswamy
Applications (a few)
• Modeling photon transport in tissue
• Calculating dose distributions for radiation therapy
• Solving the problem of neutron diffusion in a fissionable
material
• Calculating financial derivatives and evaluating investment
value and market behavior
• Wireless network design
• Numerical integration
Ref: Wikipedia
The radiation transport problem
Symbol
Description
Units
N(r , sˆ)
Number density of photons at a point r,
moving along s
m-3Sr-1
Lυ
Spectral Radiance
Energy flow per unit normal area per unit solid
angle per unit time per unit temporal frequency
bandwidth
L
Radiance, quantity used to describe
propagation of photon power
Spectral radiance integrated over a narrow
frequency range [υ ,υ + Δυ ]
Φ
Fluence rate, indicates net radiant energy
Energy flow per unit area per unit time
Radiative Transfer Equation (RTE):
divergence
Diffusion Approximation:
extinction
scattering
source
with diffusion
coefficient,
Ref: Ambrocio, Master Thesis, http://appliedmath.ucmerced.edu/theses/ambrocio_2009.pdf
Tissue optical properties
Probability density functions (PDF)
• A probability distribution
describes the range of possible
values that a random variable
can attain and the probability
that the value of the random
variable is within any subset of
that range.
– i.e., what is the chance of getting
a value for every possible
outcome of the random variable
– Coin toss
• A PDF is the functional form of
a probability distribution
Ref: Venkat Krishnaswamy
Beer’s Law - a probability distribution
• The fraction of photons that will survive after a
distance d’<d can be seen as a probability
distribution over all d’.
• How far will a photon travel in an absorbing
medium without an interaction?
-100% chance it will travel 0cm
~40% chance it will travel 1cm
Ref: Venkat Krishnaswamy
Cumulative distribution function (CDF)
• The probability that a measurement yielding a value of x will
lie in the interval [0,x1] is given by the cumulative distribution
function.
• Where,
, represents the
probability density distribution of a set size, x є [0, Inf] , that
a photon takes between any two scattering events.
 Beer’s
law as a cumulative distribution
* CDFs and Random Sampling*
• The CDF is always uniformly
distributed on the interval [0,1].
• Sampling by inversion of the CDF
1) Sample a random number ξ
from U[0,1]
2) Equate ξ with the CDF,
F(x) = ξ
3) Invert the CDF and solve for x
Cumulative distribution functions
associated with physical processes
can be sampled using random
numbers via direct inversion.
Ref: http://www.phy.ornl.gov/csep/CSEP/GIFFIGS/MCF12.GIF
Pseudo random numbers
• Computers can generate random numbers, ,
with a uniform PDF
• The associated CDF is given by
Ref: Jacques, Prahl,
http://omlc.ogi.edu/software/mc/
Flowchart for variable stepsize MC
• Implicit Capture ~ each
photon launched into the
medium is thought to
represent a photon packet,
where each packet enters
the medium carrying the
photon weight (i.e. 1J)
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
1. Photon initialization
• N photons are launched, each with a "photon weight" initially set
to 1 (computationally efficient)
• Start coordinates for each photon are identical
• Photon’s initial direction chosen via convolution with the beam
shape
Example of
Convolution
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
Image: http://support.svi.nl/wikiimg/ft_1.png
2. Propagation distance
• A fixed stepsize, Δs, must be small relative to the average
mean free path of a photon in tissue.
• It is more efficient to choose a different stepsize for each
photon step; the PDF for the Δs follows Beer’s law.
• A function of a random variable, ξ : [0,1], that is distributed
uniformly and yields a random variable with this distribution:
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
3. Internal reflection
• The probability that the photon is internally reflected is determined
by the Fresnel reflection coefficient
// angle of incidence on the boundary
// angle of transmission given by Snell’s law
• A random number uniformly distributed between 0 and 1 is used to
determine if the photon is reflected or transmitted.
 Internal reflection
• i.e. For a semi-infinite slab, the internally reflected position is
updated by only changing the z-component of photon coordinates
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
4. Photon absorption
• After each propagation step, a fraction of the photon packet is
absorbed and the remainder is scattered.
a = the single particle albedo (fraction scattered)
// New weight assigned to surviving photon packet
• Or generate photon absorption (weight) according to randomly
generated step size and Beer’s law
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
5. Photon termination
• Propagating a photon packet with minimal weight yields little
information. How is the packet terminated?
– Roulette is used to terminate a photon when its weight falls below some
minimum
– The roulette gives a photon of weight w, one chance in m, of surviving
with weight mw. Otherwise, w  0
– Unbiased elimination, conservation of energy
mw if   1 / m
w
others
0
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
6. Photon scattering
• The PDF for the scattered cosine of the deflection angle (cosθ) in
tissue is characterized by the Henyey-Greenstein phase function.
* For isotropic scattering,
• The azimuth angle is uniformly distributed between [0,2π], and
may be generated by multiplying a random number ξ:[0,1] by 2π
*Assumes phase function has
no azimuth dependence
• The photon is scattered at an angle (θ,
)
Ref: Prahl, A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)
7. Observable quantities
• Analytic Solution to RTE
 fluence rate resulting from photons launched at a
single point corresponds to the Green’s function for
the medium.
• Monte Carlo Solution to RTE
 defines grid over solution space and scores physical quantities (reflection,
transmission, absorption = energy deposited) at each grid element as the
program traces N photons.
Ref: Venkat Krishnaswamy
Abbreviated review
Implementing Monte Carlo:
A steady state example (“mc321.c”)
• The following slides walk through a steady-state Monte Carlo simulation by
Steve Jacques and Scott Prahl at the Oregon Medical Laser Center. All code
discussed is available online: http://omlc.ogi.edu/software/mc/
• Problem Definition:
– Photons are launched from an isotropic point source of unit power, P = 1W into an infinite,
homogeneous medium with no boundaries
– The medium has the optical properties of absorption, scattering and anisotropy
– N Photons are launched, each initialized with weight, W = 1.
– Solution space is divided into an array of bins (position is defined by distance r from
source), 3 options – spherical shell, cylindrical shell, planar shell
– Each bin accumulates photon weights deposited due to absorption by N photons
Array containing accumulated
weight of absorbed photons
Concentration
of Photons
Map of fluence
Implementing Monte Carlo:
Definitions of variables and arrays
Implementing Monte Carlo:
User input
Implementing Monte Carlo:
Launching photons
Implementing Monte Carlo:
Moving photons ~ HOP
Beer’s Law 
Implementing Monte Carlo:
Moving photons ~ DROP
Implementing Monte Carlo:
Moving photons ~ SPIN
Implementing Monte Carlo:
Moving photons ~ SPIN
Implementing Monte Carlo:
Moving photons ~ CHECK ROULETTE
Implementing Monte Carlo:
Output bin arrays as fluence rate
Implementing Monte Carlo:
Example Output
Ref: Jacques, Prahl, http://omlc.ogi.edu/software/mc/
Advance Application:
Radiation Treatment Planning
• Motivation:
Diagnosed with
life-threatening
forms of cancer
annually
Receive
radiation
treatment
Die anyway
Are considered
curable
– Mortality caused by (1) providing too little radiation to the tumor for cure,
or (2) providing too much radiation to nearby healthy tissue.
• Need: Improved radiation therapy planning
• Ionizing and non-ionizing radiation can be used
– Photon therapy accounts for 90% of all radiation treatment in US
– Photons, electrons, neutrons, heavy charged particles (protons)
• Dose distribution is key parameter of interest in treatment
planning
Ref: Venkat Krishnaswamy,
https://www.llnl.gov/str/Moses.html
Current dose estimation techniques
• Generate 3D electron-density map of body using stack of CTs
– Model the body as a homogeneous bucket of water
• One way to estimate dose in tissue is to use a water phantom
– use ionization chambers/chamber arrays to detect dose
distribution
– currently used in clinical treatment planning
– complicated experiments
– heterogeneities hard to model
Ref: Venkat Krishnaswamy
Monte Carlo-based treatment planning
• Voxelize medium of interest using CT/MR patient images
– Compute solution space geometry
– Assign material data to each voxel (from atomic and
nuclear-interaction databases)
• Launch radiation particles one at a time and let them evolve
• Store accumulated dose per voxel for N radiation particles
• After following many particle histories, an accurate estimation
of dose is obtained.
Ref:Venkat Krishnaswamy
PEREGRINE
3D Monte Carlo Treatment Planing
Ref:https://www.llnl.gov/str/Moses.html
PEREGRINE
Defining the radiation source and patient
•3D Transport mesh of patient
generated from stack of CT
images
• Radiation Source
- upper portion of accelerator does
not vary between treatments, but the
lower portion is modified by
collimators, blocks and wedges to
customize patient treatment.
- PEREGRINE library accounts for
modification in lower half of
accelerator
Ref: https://www.llnl.gov/str/Moses.html
PEREGRINE
Calculating Dose
Five-field treatment for a lung
tumor; 6 MV photon beam
Seven-field conformal boost
to the prostate; 18MV photon
beam
Predicted dose build up for
treatment of a brain tumor
Ref: https://www.llnl.gov/str/Moses.html, Venkat
Suggested Reading
Literature:
ED Cashwell, CJ Everett, "A Practical Manual on the Monte Carlo Method for Random Walk
Problems,“ Pergammon Press, New York, 1959.
BC Wilson, G Adams, A Monte Carlo model for the absorption and flux distributions of light in
tissue, Med. Phys. 10:824-830, 1983.
L Wang, SL Jacques, L Zheng, MCML - Monte Carlo modeling of light transport in multi-layered
tissues, Computer Methods and Programs in Biomedicine 47:131-146, 1995.
L Wang, SL Jacques, "Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard
C,” 1992-1998. Download as 177-page manual in pdf format.
Source Code:
http://mcnp-green.lanl.gov/index.html
http://omlc.ogi.edu/software/mc/
http://omlc.ogi.edu/software/polarization
Radiation MC, state of the art
Photon MC, MCML and others
Photon MCML (Vector)