A Monte Carlo model of light propagation in tissue

An introduction to Monte Carlo techniquesENGS168

Ashley LaughneyNovember 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– 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 UnitsN(r , sˆ) Number density of photons at a

point r, moving along sm-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):

Diffusion Approximation:with diffusion coefficient, Ref: Ambrocio, Master Thesis,

http://appliedmath.ucmerced.edu/theses/ambrocio_2009.pdf

source

scattering

extinction

divergence

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

Ref: Venkat Krishnaswamy

• 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

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 CDF1) Sample a random

number ξ from U[0,1]2) Equate ξ with the CDF,

F(x) = ξ3) Invert the CDF and

solve for xRef:

http://www.phy.ornl.gov/csep/CSEP/GIFFIGS/MCF12.GIF

Cumulative distribution functions associated with physical processes can be sampled using random numbers via direct inversion.

Ref: Jacques, Prahl, http://omlc.ogi.edu/software/mc/

• Computers can generate random numbers, , with a uniform PDF

• The associated CDF is given by

Pseudo random numbers

Flowchart for variable stepsize MC

Ref: Prahl,  A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)

• 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)

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

Ref: Prahl,  A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)Image: http://support.svi.nl/wikiimg/ft_1.png

Example of Convolution

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

• 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)

// angle of incidence on the boundary// angle of transmission given by Snell’s law

4. Photon absorption• After each propagation step, a fraction of the

photon packet is absorbed and the remainder is scattered.

• Or generate photon absorption (weight) according to randomly generated step size and Beer’s law

a = the single particle albedo (fraction scattered)// New weight assigned to surviving

photon packet

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

others

mifmww

0/1

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.

• The azimuth angle is uniformly distributed between [0,2π], and may be generated by multiplying a random number ξ:[0,1] by 2π

• The photon is scattered at an angle (θ, )

* For isotropic scattering,

*Assumes phase function has no azimuth dependence

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/

Ref: Venkat Krishnaswamy, https://www.llnl.gov/str/Moses.html

• Motivation:

– 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

Advance Application: Radiation Treatment Planning

Diagnosed with life-threatening forms of cancer annually

Receive radiation treatment

Are considered curable

Die anyway

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

PEREGRINE3D Monte Carlo Treatment Planing

Ref:https://www.llnl.gov/str/Moses.html

PEREGRINE Defining the radiation source and patient

Ref: https://www.llnl.gov/str/Moses.html

•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

PEREGRINE Calculating Dose

Ref: https://www.llnl.gov/str/Moses.html, Venkat

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

Suggested ReadingLiterature: 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 Radiation MC, state of the arthttp://omlc.ogi.edu/software/mc/ Photon MC, MCML and othershttp://omlc.ogi.edu/software/polarization Photon MCML (Vector)