1. A Monte Carlo model of light propagation in tissue
An introduction to Monte Carlo techniques
ENGS168
Ashley Laughney
November 13th, 2009

2. 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

3. Ref: Venkat Krishnaswamy
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

4. 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

5. 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
extinction
scattering
source
Diffusion Approximation:
with diffusion coefficient,
Ref: Ambrocio, Master Thesis,

6. Tissue optical properties

7. 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

8. 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

9. 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

10. * CDFs and Random Sampling*
The CDF is always uniformly distributed on the interval [0,1].
Sampling by inversion of the CDF
Sample a random number ξ from U[0,1]
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:

11. Ref: Jacques, Prahl, http://omlc.ogi.edu/software/mc/
Pseudo random numbers
Computers can generate random numbers, , with a uniform PDF
The associated CDF is given by
Ref: Jacques, Prahl,

12. 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)

13. 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:

14. 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)

15. 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
// angle of incidence on the boundary
// angle of transmission given by Snell’s law
Ref: Prahl,  A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)

16. 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)

17. 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
Ref: Prahl,  A Monte Carlo Model of Light Propagation in Tissue, SPIE (1989)

18. 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)

19. 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

20. Abbreviated review

21. 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:
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
Map of fluence
Array containing accumulated weight of absorbed photons
Concentration of Photons

22. Implementing Monte Carlo: Definitions of variables and arrays

23. Implementing Monte Carlo: User input

24. Implementing Monte Carlo: Launching photons

25. Implementing Monte Carlo: Moving photons ~ HOP
Beer’s Law 

26. Implementing Monte Carlo: Moving photons ~ DROP

27. Implementing Monte Carlo: Moving photons ~ SPIN

28. Implementing Monte Carlo: Moving photons ~ SPIN

29. Implementing Monte Carlo: Moving photons ~ CHECK ROULETTE

30. Implementing Monte Carlo: Output bin arrays as fluence rate

31. Implementing Monte Carlo: Example Output
Ref: Jacques, Prahl,

32. Advance Application: Radiation Treatment Planning
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
Diagnosed with life-threatening forms of cancer annually
Receive radiation treatment
Die anyway
Are considered curable
Ref: Venkat Krishnaswamy,

33. 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

34. 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

35. PEREGRINE 3D Monte Carlo Treatment Planing
Ref:

36. 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:

37. 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: Venkat

38. Suggested Reading Source Code:
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: , 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: , 1995.
L Wang, SL Jacques, "Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C,” Download as 177-page manual in pdf format.
Source Code:
Radiation MC, state of the art
Photon MC, MCML and others
Photon MCML (Vector)