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 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

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

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 extinction scattering source Diffusion Approximation: 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 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: http://www.phy.ornl.gov/csep/CSEP/GIFFIGS/MCF12.GIF

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, 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 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)

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 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 Map of fluence Array containing accumulated weight of absorbed photons Concentration of Photons

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: 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, 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 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. 10:824-830, 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 art http://omlc.ogi.edu/software/mc/ Photon MC, MCML and others http://omlc.ogi.edu/software/polarization Photon MCML (Vector)