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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 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)