The ant colony method has been proven to be effective in simulations of clinical linear electron accelerators [22, 23], in a study of the response of MOSFET dosimeters [24], in dosimetry calculations of radiosurgery treatments [25], in the optimization of certain radiotherapy procedures [26], and in calculations of specific absorption fractions in Nuclear Medicine [20]. As a final comment, we would like pointing out that wery frequently, an effective increase of efficiency may be obtained by simply avoiding unnecessary calculations. Rodriguez M, Sempau J, Brualla L. A Combined Approach of Variance-Reduction Techniques for the Efficient Monte Carlo Simulation of Linacs. For non-forced interactions of types other than A, the particle weight w is used. 12. Sempau J, Snchez-Reyes A, Salvat F, Tahar HOb., Jiang SB, Fernndez-Varea JM. In class I or condensed simulation schemes, the trajectory of a charged particle is split into segments of predefined length and the cumulative energy loss and angular deflection resulting from the interactions along each segment are sampled from approximate multiple scattering theories. In the case of simple (e.g., planar, spherical, cylindrical) geometries the program may be substantially simplified and this may speed up the simulation appreciably. Global strategies for automatically determining importance functions have been developed, mostly for photon and neutron transport because these particles have relatively large mean free paths. From now on, the application volume and its cell partition are assumed to be defined; to simplify the formulas, each cell is denoted by a single index i and, accordingly, the value of the importance of a cell is written as Ii. A simple VRT that is very effective in these cases consists in forcing an interaction in a restricted path length interval, say between 0 and a given maximum length L. This is accomplished by sampling the path length s to the next interaction from the PDF [2]. This situation is encountered, e.g., in simulations of the response to photon beams of ionization chambers, where the active gas is almost completely transparent to photons. Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Methods in Computational Physics, Vol. Notice that the analog run was ten times longer than the variance-reduced one. As already mentioned, using a VRT allows increasing the efficiency of the calculation of a certain quantity or a set of related quantities, at the expense of worsening the efficiencies of other quantities. Over 100 numerical examples are presented as part of the appendix to illustrateuseful real worldapplications. As the building of the importance map continues during all the simulation, its quality improves progressively. Secondly, Monte Carlo simulation can easily follow particles through material systems with complex geometries, where deterministic methods would find great difficulties even to define the appropriate boundary conditions. In: Tech. Essentials of Monte Carlo Simulation: Statistical Methods for Building Simulation Models 2013th Edition by Nick T. Thomopoulos (Author) 4 ratings See all formats and editions In this paper, we will briefly describe the nature and relevance of Monte Carlo simulation, the way to perform these simulations and analyze results, and the underlying mathematical techniques required for performing these simulations. p. 135215. In: Proc. Features of Particle and Heavy Ion Transport Code System (PHITS) Version 3.02. Let w be the weight of the transported particle. No use, distribution or reproduction is permitted which does not comply with these terms. In the limit N , the quantity. Mathematical Verification of a Certain Monte Carlo Sampling Technique and Applications of the Technique to Radiation Transport Problems. Splitting is also useful to favor the emission of secondary particles by taking advantage of the emission symmetries of these particles. 29. Wagner JC, Haghighat A. Rare Event Simulation Using Monte Carlo Methods. (A) Result from an analog simulation, after 83min of CPU time. A simpler, and easier to implement, procedure to progressively build an importance function from information acquired from the simulation itself is provided by the ant colony method [20]. Notice that the PDF p(x) is ultimately determined by the interaction DCSs of the transported particles, and does not need to be specified. To identify the approaches for solving the problems regarding risk. We consider that the PDFs for the energy loss and the angular deflections (and the directions of emitted secondary particles, if any) in the forced interactions are the same as for real interactions, i.e., the VRT does not affect the interaction PDFs. Our practical implementation of interaction forcing consists of replacing the mean free path A of the real process by a shorter one, A,f=A/F with F>1. Splitting should be applied when the particle approaches the RoI. Geometrical setup adopted in the simulation of dose profiles from photon beams in water. He has more than 100 published papers and presentations to his credit, and for many years, he has consulted in a wide variety of industries in the United States, Europe, and Asia. Ant colony Method to Control Variance Reduction Techniques in the Monte Carlo Simulation of Clinical Electron Linear Accelerators of Use in Cancer Therapy. Clark FH. Azimuthal Particle Redistribution for the Reduction of Latent Phase-Space Variance in Monte Carlo Simulations. Generally, a shower consists of a number n of particles, and each of these particles may contribute to the score, that is. They may not be optimal, i.e., the efficiency could be improved further by varying those parameter values. J Comput Appl Maths (2010) 233:153441. Essentials of Monte Carlo Simulation Statistical Methods for Building Simulation Models Home Book Authors: Nick T. Thomopoulos Offers a "fundamentals" approach to developing Monte Carlo computer simulations Illustrates the best ways to select input distributions and parameters with or without sample data For example, Q may be the average energy deposited into a certain volume V of the geometry, in which case q(x) is the energy deposited by the set of particles in a shower (not only the primary particle). Tech. The state variables of a particle are the particles kind k (defined by its mass and charge), the kinetic energy E (energy in the case of photons), position r and direction of motion d. Ant colony Algorithm Implementation in Electron and Photon Monte Carlo Transport: Application to the Commissioning of Radiosurgery Photon Beams. Error bars represent statistical uncertainties with a coverage factor k = 3. Initially, no variance reduction technique is applied, and the importance map is built from the progressing analog simulation until the gathered information is sufficient to switch on the VRT. Mathematics and Statistics, Mathematics and Statistics (R0), Copyright Information: Springer Science+Business Media New York 2013, Hardcover ISBN: 978-1-4614-6021-3Published: 19 December 2012, Softcover ISBN: 978-1-4899-8608-5Published: 28 January 2015, eBook ISBN: 978-1-4614-6022-0Published: 19 December 2012, Topics: The central limit theorem then implies that, for sufficiently large N, the probability distribution of f is the normal distribution with variance var(f)/N. 2.X-5 Monte Carlo Team. This amounts to removing one spatial coordinate (the azimuthal angle) and leads to an effective reduction of the variance of the calculated local dose. The ant colony method is applied to particles within a limited volume of the particle-state space (k,r,E,d), which is split into cells. The difficulty of simulations of photon emission by electrons with energies of the order of, or lower than 100keV is the low probability with which these electrons induce the emission of photons. In the absence of symmetries, we may use Cartesian space coordinates, r = (x, y, z) and represent direction vectors by means of the polar angle and the azimuthal angle , d=(sincos,sinsin,cos). It takes advantage of the high penetration of photons to simplify their tracking through material systems with complex geometries. METHODOLOGY In this paper, general focus has been created on the danger factors. Nucl Instrum Meth A (2016) 835:186225. A Jaw Calibration Method to Provide a Homogeneous Dose Distribution in the Matching Region when Using a Monoisocentric Beam Split Technique. The basic ideas leading to the formulation of VRTs for transport simulations are presented in Section 3, followed by the description of various VRTs that are applicable to any Monte Carlo transport algorithm. Because it is very difficult to avoid this kind of bias, many simulations of energy-deposition spectra are purely analog. They perform detailed event-by-event simulation for photons, while charged particles are simulated by means of a combination of class I and class II schemes (see Ref. The Monte Carlo estimator of the quantity Q is obtained by generating a sufficiently large number N of showers and setting. Because of the assumed isotropy of the material, the DCSs depend only on the polar angle of scattering, , and the energy transfer W. The DCSs are conveniently expressed as. 4.Decrease variability of simulation output that we use to estimate . Nick T. Thomopoulos is a professor emeritus at the Illinois Institute of Technology. We recall that more uniform weights normally have associated a smaller variance. As compared with alternative deterministic finite-difference methods, Monte Carlo simulation has several distinct advantages. Our aim here is to offer a general perspective of the VRTs and of their capabilities. [1] and references therein). We correct for the introduced distortion of the mean free path as follows: A weight wf=w/F is assigned to the deposited energy, the released secondary particles, and to any other alteration of the medium (such as, e.g., charge deposition) that result from A interactions (real and forced) of the transported particle. Rep. SLAC-R-730 (KEK 2005-8). The VRT of interaction forcing has been proven to be effective to cope with this problem. FIGURE 2. (2). Interaction forcing can be activated locally, at any stage of the trajectory of a particle. Monte Carlo Simulation Using the PENELOPE Code with an Ant colony Algorithm to Study MOSFET Detectors. IEEE Trans Syst Man Cybern B (1996) 26:2941. Garca-Pareja S, Vilches M, Lallena AM. The length s of the free flight of a particle to its next interaction is a random variable with PDF. It is in these cases where the ant colony method proves to be effective, mostly when combined with interaction forcing. (22) yield values of the exponent [i] between 5 and 12. Ant colony Method to Control Variance Reduction Techniques in the Monte Carlo Simulation of Clinical Electron Linear Accelerators. RTI2018-098117-B-C22 and PID2019-104888GB-I00) and the Junta de Andaluca (projects nos. As a rule of thumb, the variance 2(QVRT) is reduced when the number of contributions to the score increases and their weights become more uniform. doi:10.2172/877507. Generally, secondary particles inherit the weight of the parent particle that induced their emission. Nick T. Thomopoulos Stuart School of Business Illinois Institute of Technology Chicago, Illinois, USA ISBN 978-1-4614-6021-3 ISBN 978-1-4614-6022- (eBook) DOI 10.1007/978-1-4614-6022- 14. A Monte Carlo simulation of radiation transport can be regarded as the simultaneous evaluation of a number of integrals of the type. Thisbook features11 comprehensive chapters, and discusses such keytopics as random number generators, multivariate random variates, and continuous random variates. The Woodcock method is also applicable to fast neutrons, and to any kind of particles that have relatively large mean free paths. It is to be expected that Monte Carlo calculations with different PDFs would yield estimates f with different statistical uncertainties (f). Evidently, this protection is unnecessary when the only VRT applied is the ant colony method in a single cell structure. Available in PDF, EPUB and Kindle. Of course, the importance map can be stored and reused in subsequent runs. must be used in all subsequent contributions to the scores. An additional index, kp, designated the particle kind (= 1 for electrons and positrons, and = 2 for photons). Allison J, Amakoca K, Apostolakisd J, Arce P, Asaif M, Asog T, et al. doi:10.1118/1.3456108, 26. and we express the integral in the form of an expectation value: Then, the integral can be evaluated by generating a large number N of random values xi from p(x) and, by virtue of the law of large numbers, we have. Interaction forcing has been efficiently used in simulations of electron-probe microanalysis [31, 32], photon beams from medical electron accelerators [33], the response of ionization chambers [34], and the calculation of doses absorbed in small organs in Nuclear Medicine treatments [20]. Exploitation of local symmetries present in the simulation is often very useful in reducing the variance [33]. Firstly, it can describe arbitrary interaction processes, including those with cross sections that are rapidly varying functions of the physical variables (e.g., the atomic photoelectric effect, whose total cross section presents sharp absorption edges). where T is the computer (CPU) time spent in the calculation. Monte Carlo Calculation of Specific Absorbed Fractions: Variance Reduction Techniques. The text also contains an easy to read presentation with minimal use of difficult mathematical concepts. The effectiveness of these techniques is mostly determined by the edequacy of the adopted VRTs and their defining parameters. The VRT of interaction forcing (Section 3.4), has been shown to be useful in simulations with penelope of processes with intrinsic small probabilities, such as calculations of absorbed doses in thin material bodies and the emission of photons from samples irradiated by electron beams. The variance-reduction techniques (VRTs) are strategies aimed at increasing the efficiency of the calculation of the integral without modifying its expectation, i.e., aimed at reducing the relative statistical uncertainty attained after a given CPU time. Thus, the S split particles are assigned different positions and directions, and this gives a net information gain and an increase in efficiency [35]. via simulation. Bar J, Sempau J, Fernndez-Varea JM, Salvat F. PENELOPE: An Algorithm for Monte Carlo Simulation of the Penetration and Energy Loss of Electrons and Positrons in Matter. Particles undergo discrete interactions with the material, in which they lose energy, change their direction of motion, and occasionally, may release secondary particles with lower energies1. Petit O, Lee Y-K, Diop CM. The considered techniques are 1) splitting and Russian roulette, with the ant colony method as builder of importance maps, 2) exponential transform and interaction-forcing biasing, 3) Woodcock or delta-scattering method, 4) interaction forcing, and 5) proper use of symmetries and combinations of different techniques. In such cases, the effectiveness of VRTs is best appreciated from a plot of the simulated histogram with statistical uncertainties displayed as error bars. Over 100 numerical examples are presented as part of the appendix to illustrate useful real world applications. Essentials of Monte Carlo Simulation focuses on the fundamentals of Monte Carlo methods using basic computer simulation techniques. We would like to show you a description here but the site won't allow us. Primary particles emitted from unbiased sources are usually assigned a weight equal to unity. 15. Class II schemes describe hard interactions accurately (i.e., according to the adopted DCSs) and also provide a better description of soft events (because multiple scattering approximations are more accurate when applied to soft collisions only). All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Detailed appendices provide background material on probability theory, stochastic processes, and mathematical statistics as well as the key optimization concepts and techniques that are relevant to Monte Carlo simulation. Ferrari A, Sala PR, Fass A, Ranft J. Fluka: a Multi-Particle Transport Code. Brun E, Damian F, Diop CM, Dumonteil E, Hugot FX, Jouanne C, et al. where Q is the quantity of interest, q(x) is the contribution of an individual shower, the random variable x (usually an array of random variables) characterizes each individual shower, and p(x) is the PDF for the occurrence of that particular shower. Another difficult calculation is that of the deposited energy in bodies having thicknesses much less than the total mean free paths of the transported particles. Received: 01 June 2021; Accepted: 06 October 2021;Published: 27 October 2021. Very little has been published in the area of computer Monte Carlo simulation methods, and this book will appeal to students and researchers in the fields of Mathematics and Statistics. doi:10.1016/j.nima.2007.05.217, 23. Coleman WA. The upper left plot (A) is the result from a 30min analog simulation. This new and updated deals with all aspects of Monte Carlo simulation ofcomplex physical systems encountered in condensed-matterphysics and sta-tistical mechanics as well as in related elds, for example polymer science,lattice gauge theory and protein folding. [9]). doi:10.1109/3477.484436, 22. 27. Med Phys (2004) 31:288398. The event at the end of each free flight is assumed to be either a real interaction or a delta interaction (which does nothing). The fraction, characterizes the relevance of the cell. For instance, if interaction forcing is the only VRT applied and if primary particles are assigned unit weights, a window (0.5,1.5) ensures that interaction forcing with F>2 will be applied only to the primary particles. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The procedure produces "simulated" data where the generating model and its parameters are known. Ann Nucl Energ (2015) 82:15160. Capote R, Jeraj R, Ma CM, Rogers DW, Snchez-Doblado F, Sempau J, et al. These electrons emit bremsstrahlung photons and x rays with quite small probabilities. random variates in section5and analyzing output of Monte Carlo simulation in section6. doi:10.1118/1.3581377. Salvat F. PENELOPE-2018: A Code System for Monte Carlo Simulation of Electron and Photon Transport. Photons are transported freely across the system using an augmented inverse mean free path, 1, which is required to be larger than the actual total inverse mean free paths in all the materials crossed by a trajectory ray. Monte Carlo Simulation, also known as the Monte Carlo Method or a multiple probability simulation, is a mathematical technique, which is used to estimate the possible outcomes of an uncertain event. The Monte Carlo Method uses random numbers to try todetermine the answer to problems. Interactions of type A (real and forced) are simulated to determine the energy loss and the possible emission of secondary particles, but the state variables of the transported particle are altered only when the interaction is real. It will then collide with the atoms in the shield and cause dierent kinds of reactions. By contrast, Russian roulette increases the variance (because it produces fewer contributions with higher weights) and reduces the CPU time. Results from illustrative simulation examples are presented in Section 4. What is a Monte Carlo Simulation? Nucl Sci Eng (2014) 176:3757. In the variance-reduced simulation we applied the ant colony method. Practical experience indicates that moderate variations of those coefficients do not produce significant improvements of the effectiveness of the method. For the sake of simplicity, we consider analog Monte Carlo simulations, in which the transport process retains its Markovian character. This method became famous after the mathematician Stanislaw Ulam considered it during the project to construct the atomic bomb. Front. In May 2023, Frontiers adopted a new reporting platform to be Counter 5 compliant, in line with industry standards. Splitting is also useful, e.g., in simulations where primary particles are read from pre-calculated phase-space files [13]; since these files are limited in size, splitting the primary particles allows reducing the statistical uncertainty, at the cost of increasing the simulation time.2. 9:718873. doi: 10.3389/fphy.2021.718873. Automated Variance Reduction of Monte Carlo Shielding Calculations Using the Discrete Ordinates Adjoint Function. Essentials of Monte Carlo Simulation Statistical Methods for Building Simulation Models. doi:10.13182/nse98-2, 17. A simulation code generates the trajectory of a particle as a sequence of free flights, each ending with an interaction where the particle changes its direction of flight, loses energy, and may induce the emission of secondary particles. The analog simulation lasted for 83min and involved the generation of 2 109 showers (a part of them involving no interactions at all). The authors propose a model to predict the values of the investment accounts IRA and 401(k . N.T. Other details are the same as in Figure 1. 1. Carvajal MA, Garca-Pareja S, Guirado D, Vilches M, Anguiano M, Palma AJ, et al. Prog Nucl Sci Tech (2014) 4:40812. 2nd ed. X-ray emission spectra from a tungsten target bombarded by 100keV electrons at normal incidence. Again, interaction forcing provides an effective practical solution in this case. After the models are run several times, in a random sample way, the data for each output variable(s) of interest is analyzed by ordinary statistical methods. However, for a given simulation time, other quantities may exhibit standard deviations larger than those of the analog simulation because of the time spent in simulating the forced interactions. Essentials of Monte Carlo Simulation Authors: Nick T. Thomopoulos Illinois Institute of Technology Abstract The Monte Carlo method originated in the 1940s, just when computer technology. The following examples show that even a blind setting of the VRT parameters may save quite a fraction of the computer time. Altmetric, Tax calculation will be finalised at checkout. The examples in the last Section evidence the usefulness of VRTs to speed up simulations of radiation transport in difficult situations.