monte carlo simulations: efficiency improvement techniques and ...drogers/pubs/papers/sb06.pdf · 2...

Monte Carlo Simulations:Efficiency Improvement Techniques

and Statistical Considerations

Daryoush Sheikh-Bagheri, Ph.D.1, Iwan Kawrakow, Ph.D.2,Blake Walters, M.Sc.2, and D. W. O. Rogers, Ph.D.3

1Department of Radiation Oncology, Allegheny General HospitalPittsburgh, Pennsylvania

2Ionizing Radiation Standards, National Research Council,Ottawa, Ontario, Canada

3Physics Department, Carleton University, Ottawa, Ontario, Canada

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1The Metrics of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2The Condensed History Technique (CHT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Efficiency Improvement Techniques Used in Treatment Head Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Range Rejection and Transport Cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Splitting and Russian Roulette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Uniform Bremsstrahlung Splitting (UBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Selective Bremsstrahlung Splitting (SBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Directional Bremsstrahlung Splitting (DBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Efficiency Improvement Techniques Used in Patient Simulations . . . . . . . 9Macro Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11History Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Boundary-Crossing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Precalculated Interaction Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Woodcock Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Photon Splitting Combined with Russian Roulette . . . . . . . . . . . . . . . . . . . . . 13Simultaneous Transport of Particle Sets (STOPS) . . . . . . . . . . . . . . . . . . . . . . 13Quasi-Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Correlated Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Introduction

A large number of general-purpose Monte Carlo (MC) systems have been developedfor simulating the transport of electrons and photons. One of the most popular ones,which has been extensively used and benchmarked against experimental measurementsin a large number of medical physics situations, is the EGS code system (Ford andNelson 1978; Nelson, Hirayama, and Rogers 1985; Kawrakow and Rogers 2000).

1

Published in "Integrating New Technologies into the Clinic: MonteCarlo and Image-Guided Radiation Therapy - Proc. of 2006 AAPM Summer School: Pages 71-91: Published by Medical PhysicsPublishing,( Madison WI)

Other codes systems such as ITS (Halbleib and Melhorn 1984; Halbleib 1989; Halbleibet al. 1992); MCNP (Briesmeister 1986, 1993; Brown 2003); PENELOPE (Baró et al.1995) and GEANT4 (Agostinelli et al. 2003) also have popularity among their usergroups. The EGS and ITS/ETRAN systems are roughly of the same efficiency forcalculations when no variance reduction techniques are used, whereas the othersystems tend to be considerably slower. For special-purpose applications the use ofsophisticated variance reduction techniques has made some of these codes substan-tially more efficient than others. For example, the BEAMnrc code (Rogers, Walters,and Kawrakow 2005) is an optimized EGSnrc user-code for modeling radiotherapyaccelerators that employs several techniques which significantly enhance its overallefficiency. Nonetheless, it is still not fast enough for routine treatment planningpurposes without a substantial farm of computers. Thus, to make routine clinical usea possibility, there have been a number of attempts to develop high-efficiency MCcodes. Among the most well known of such codes are: the Macro Monte Carlo (MMC)code (Neuenschwander and Born 1992; Neuenschwander, Mackie, and Reckwerdt1995); the PEREGRINE code (Schach von Wittenau et al. 1999; Hartmann Siantar etal. 2001); Voxel Monte Carlo (VMC/xVMC) (Kawrakow, Fippel, and Friedrich 1996;Kawrakow 1997; Fippel 1999; Kawrakow and Fippel 2000b; Fippel et al. 2000),VMC++ (Kawrakow and Fippel 2000a; Kawrakow 2001); MCDOSE (Ma et al. 2000);the Monte Carlo Vista (MCV) code system (Siebers et al. 2000); the Dose PlanningMethod (DPM) (Sempau, Wilderman, and Bielajew 2000) and other codes (Keall andHoban 1996a,b; Wang, Chui, and Lovelock 1998).

In another chapter, “Monte Carlo Methods for Accelerator Simulation and PhotonBeam Modeling,” Ma and Sheikh-Bagheri have discussed the effects on the accuracyof MC-calculated dose distributions of various linac modeling parameters, such as thespecifications of the initial electron beam-on-target and accelerator components. Ourdiscussion of accuracy assumes all such influencing factors are properly addressed.

In this chapter we discuss the fundamental methods of improving the efficiencyof MC simulations that are used both in therapy beam simulations and in simulationsof the dose distribution in the patient. Throughout the chapter we explicitly distinguishbetween techniques that do not alter the physics in any way when they increase theefficiency—i.e., true variance reduction techniques (VRT)—and techniques thatachieve the improved efficiency through the use of approximations; i.e., approximateefficiency improving techniques (AEIT).

The Metrics of Efficiency

The efficiency, ε, of a Monte Carlo calculation is defined as: , where σ2 is

an estimate of the variance on the quantity of interest and T is the CPU time requiredto obtain this variance. The goal is to reduce the time it takes to obtain a sufficientlysmall statistical uncertainty σ on the quantity of interest. The shorter the time needed,the higher the efficiency of the simulation and vice versa. The time T for the simula-

2 Daryoush Sheikh-Bagheri et al.

εσ

= 12T

tion is proportional to the number N of statistically independent particles. At the sametime σ2 decreases as 1/N according to the central limit theorem (Lux and Koblinger1991). As a result, the product T · σ2 is a constant and the efficiency ε = 1/T σ2 expresseshow fast a certain simulation algorithm can calculate a given quantity of interest at adesired level of statistical accuracy.

The above metric does not define how to calculate the variance of the quantity ofinterest. In the case of MC simulations related to radiation treatment planning (RTP),one is typically interested in a distributed quantity such as the three-dimensional (3-D) dose distribution in the patient, or the spectrum or fluence of particles emergingfrom the treatment head of the linear accelerator, etc. One therefore must define ameasure of the overall uncertainty in order to evaluate the efficiency of a particularsimulation algorithm. Rogers and Mohan (2000) proposed to use an average uncer-tainty as a measure of the uncertainty of a MC patient calculation, given by:

where Di is the dose in the i’th voxel, ∆Di is its statistical uncertainty, and the summa-tion runs over all voxels (n) with a dose greater than 50% of the maximum. Usingthe above ICCR (International Conference on the Use of Computers in RadiationTherapy) benchmark criteria, the most commonly used MC codes in radiotherapyapplications were compared against each other (Chetty et al. 2006) and the compi-lation of results is depicted in Table 1. The differences in speed are primarilyattributable to the aggressiveness in the employment of various VRTs in the differ-ent codes, the inherent speeds of the transport algorithms, and the efficiency of thegeometry packages.

Before we get into a detailed discussion of VRTs, we will first discuss one singlemost important technique that has made MC simulations of electron transport possi-ble using our present-day computers: the condensed history technique.

The Condensed History Technique (CHT)

A typical RTP energy range electron or positron together with secondary particles itsets in motion undergoes of the order of 106 elastic and inelastic collisions until locallyabsorbed. It is therefore impossible to simulate each individual collision as this wouldresult in prohibitively long simulation times. To overcome this difficulty, Berger(1963) introduced the condensed history technique (CHT). The CHT makes use ofthe fact that most electron interactions result in extremely small changes in energyand/or direction and thus groups them into single “steps” that account for the aggre-gate effects of scattering on the path of the electron. All general purpose MC packagessuch as the original code by Berger called ETRAN (Berger 1963, Seltzer 1988), ITS(Halbleib and Melhorn 1984; Halbleib 1989; Halbleib et al. 1992), MCNP (Bries-meister et al. 1993) (which both use the ETRAN electron transport algorithms), EGS4

MC Simulations: Efficiency Improvement/Statistical Considerations 3

σ 2

2

1

1=

=∑

n

D

Di

ii

n ∆,

(Nelson, Hirayama, Rogers 1985), EGS4 with PRESTA (Bielajew and Rogers 1987),EGSnrc (Kawrakow 2000, Kawrakow and Rogers 2000), and PENELOPE (Salvat etal. 1996) use the CHT in one way or another. The CHT introduces an artificial para-meter called the “step-size.” It is well known that for many implementations of theCHT the results depend on the choice of the step-size (Rogers 1984, Bielajew andRogers 1987, Seltzer 1991, Rogers 1993). The CHT is therefore an AEIT accordingto the definition adopted earlier in this chapter. Although every MC algorithm usesthe CHT for charged particle transport, it is rarely considered as a technique toimprove the efficiency of MC simulations. Yet, the two main aspects of the CHTimplementation: (1) the “electron-step algorithm” also called “transport mechanics”and (2) the boundary-crossing algorithm very strongly influence the simulation speedand accuracy. For more detailed discussion on the condensed history technique thereader is referred to the original Berger work (Berger 1963), the article by Larsen(1992), where a formal mathematical proof is provided that any CHT implementa-tion converges to the correct result in the limit of short steps, the paper by Kawrakow


Table 1. Summary of Timing and Accuracy Results from the ICCR Benchmark. Timingcomparisons were performed using 6 MV photons, 10×10 cm2 field size, and those for theaccuracy test, using 18 MV photons and a 1.5×1.5 cm2 field size, as detailed in the ICCR

benchmark (Rogers and Mohan 2000). All times have been scaled to the time it would takerunning on a single, Pentium IV, 3 GHz processor. Readers should be aware that the timing

results, as well as the method used to scale the times, are subject to large uncertainties due todifferences in compilers, memory size, cache size, etc.

(Reprinted from Chetty et al. 2006 with permission from AAPM).

Monte Carlo code Time estimate % max. diff. relative to (minutes) ESG4/PRESTA/DOSXYZ

ESG4/PRESTA/DOSXYZ 42.9 0, benchmark calculation

VMC++ 0.9 + 1

MCDOSE (modified ESG4/PRESTA) 1.6 + 1

MCV (modified ESG4/PRESTA) 21.8 + 1

RT_DPM (modified DPM) 7.3 + 1

MCNPX 60.0 max. diff. of 8% at Al/lung interface (on average + 1% agreement)

Nomos (PEREGRINE) 43.3* + 1*

GEANT 4 (4.6.1) 193.3** ± 1 for homogeneous water and water/air interfaces**

*Note that the timing for the PEREGRINE code also includes the sampling from acorrelated-histogram source model and transport through the field-defining collimators.** See Poon and Verhaegen (2005) for further details.

and Bielajew (1998), which gives a detailed theoretical comparison betweendifferent electron-step algorithms, and to Kawrakow (2000a), where the variousdetails of a CHT implementation and their influence on the accuracy are investigated.The article by Siebers, Keall, and Kawrakow (2005) discusses CHT details relevantfor fast algorithms used in RTP class simulations.

Efficiency Improvement Techniques Used in Treatment Head Simulations

Range Rejection and Transport Cutoffs

Use of AEITs, such as range rejection and transport cutoffs, can improve the efficiencyof treatment head simulations substantially, without significantly changing the results(Rogers et al. 1995). In range rejection, an electron’s history is terminated wheneverits residual range is so low that it cannot escape from the current region or reach theregion of interest. Since this ignores the possible creation of bremsstrahlung or anni-hilation photons while electrons (negatrons or positrons) slow down, it is not anunbiased technique. However, as long as it is only applied to electrons below a certainenergy threshold, it has been shown to have little effect on the results in many situa-tions. Similarly, by increasing the low-energy cutoff for electron transport, one can savea lot of time, but this may have an effect on the dose distribution if too high a thresh-old is used. Playing Russian roulette with particles at energies below a relatively hightransport cutoff or with particles that would be range-rejected is a comparable vari-ance reduction technique for reducing the simulation time. However, itsimplementation is typically more difficult and this has favored the simpler use of rangerejection and high transport cutoffs in situations where it is easy to demonstrate thatthe resulting error is sufficiently small.

The particle production and transport cutoff energies and the threshold energy fordoing range rejection must both be carefully investigated to avoid significant errors inthe results of the simulations. In order to determine an optimum value for range-rejec-tion cutoff in linear accelerator (linac) photon beam simulations, Sheikh-Bagheri etal. (2000) modified the BEAM code (Rogers et al. 1995) to allow tagging ofbremsstrahlung production anywhere outside the target. They concluded that outsidethe target, values of the upper energy for doing range rejection of 0.6 MeV for 4 MVbeams, and 1.5 MeV for 6 MV and higher energy beams, provide the largest savingsin central processing unit (CPU) time (about a factor of 3 for the 10 and 20 MV beamsstudied), with negligible (less than 0.2%) underestimation of the calculated photonfluence from the linac.

In the following sections we discuss the much larger increases in efficiency thatcan be achieved for photon beams by employing true VRTs, although for electronbeams the AEITs may still be the best options available.


Splitting and Russian Roulette

Two very commonly used VRTs are splitting and Russian roulette, which were bothoriginally proposed by J. von Neumann and S. Ulam (Kahn 1956). These are espe-cially useful in simulating an accelerator treatment head (Rogers et al. 1995;Sheikh-Bagheri 1999; Kawrakow, Rogers, and Walters 2004). In the various forms ofbremsstrahlung splitting (see Figure 1), each time an electron undergoes a bremsstrah-lung interaction, a large number of secondary photons with lower weights are set inmotion, the number possibly depending on a variety of factors related to the likelihoodof them being directed toward the field of interest (FOI). The energy of the electronis reduced by an amount equal to the energy of one of the emitted photons, which is


Figure 1. A simplistic schematic of the splitting routines discussed in this chapter for linacmodeling. Without bremsstrahlung splitting many electron tracks (red dashed curved lines)

have to be simulated to get a single photon emitted toward the field of interest (FOI). Inuniform bremsstrahlung splitting (UBS) (Rogers et al. 1995) any bremsstrahlung event leads

to the sampling and transport of N (a constant splitting number) photons, increasing thelikelihood of emission toward the FOI. Selective Bremsstrahlung Splitting (SBS) (Sheikh-Bagheri 1999) treats the splitting number as a variable which is calculated as a function ofthe probability of bremsstrahlung emission into the FOI, and transports all (black arrows)sampled photons. In Directional Bremsstrahlung Splitting (DBS) (Kawrakow, Rogers, and

Walters 2004) the photons emitted toward the FOI are transported, but the many that are notemitted toward the FOI are subjected to a game of Russian roulette that many will not

survive (gray dotted arrows) and only the few that survive get transported, all leading tofurther increases in efficiency. (Note that the schematic ignores, for the sake of illustration,that bremsstrahlung emission at high electron energies and shallow depths is more intense

and more forward peaked.)

in violation of conservation of energy on an individual interaction basis but results incorrect fluctuations in energy loss for electrons and correct expectation values forphoton energy and angular distributions. Splitting can save a large amount of timebecause photon transport is fast, whereas it takes a long time to track an electron inthe target. Thus, splitting bremsstrahlung interactions makes optimal use of each elec-tron track. If the number of photons created is selected to minimize those that can neverreach the patient plane, then there is a further time savings. Russian roulette can beplayed whenever a particle resulting from a class of events is of little interest. The low-interest particles are eliminated with a given probability but to ensure an unbiasedresult, the weight of the surviving particles is increased by the inverse of that proba-bility. A common example is to play Russian roulette with secondary electrons createdin a photon beam.

Uniform Bremsstrahlung Splitting (UBS)

When using UBS, in each interaction that produces photons, Nsplit bremsstrahlung or2 Nsplit annihilation photons are sampled instead of one or two photons. To make thegame fair, a statistical weight of w0 /Nsplit is assigned to the photons, where w0 is thestatistical weight of the incident electron or positron. In this way the Nsplit or 2 Nsplit

photons count statistically for as much as the one or two photons that would beproduced in a normal simulation that does not use splitting. Many of the photons setin motion in interactions of the primary electrons impinging on the bremsstrahlungtarget will undergo additional collisions before emerging from the treatment head orbeing locally absorbed. Electrons and positrons set in motion in such interactions willinherit the statistical weight of the photons, i.e., they will have a weight of w0 /Nsplit ifUBS is used. If one would split bremsstrahlung and annihilation events of suchsecondary charged particles one would have photons with weights w0 /N2

split, w0 /N3split,

etc. This is not desirable and therefore such higher order interactions are not split. Ifone is only interested in the dose beyond the depth of maximum dose in the phantom,where the contribution of contaminant electrons is negligible, or if one wants to obtainanother photon-only quantity such as the photon (energy) fluence or spectrum, one canfurther improve the efficiency of the treatment head simulation by employing Russ-ian roulette with p = 1/Nsplit for charged particles set in motion in photon collisions. Inthis case, the secondary electrons that survive the Russian roulette game have again aweight of w0 and their bremsstrahlung and annihilation interactions must be split toavoid the production of “fat” photons (Sheikh-Bagheri 1999). It is important to keepin mind that a UBS simulation that uses Russian roulette for secondary charged parti-cles results in a poor estimate of the contaminant electrons and is therefore not suitablefor full dose calculations in the patient. UBS was implemented in the original BEAMversion (Rogers et al. 1995) and it was refined in Sheikh-Bagheri (1999) and Sheikh-Bagheri et al. (2000). UBS was shown in the article by Kawrakow, Rogers, and Walters(2004) to improve the efficiency of photon treatment head simulations by up to a factorof 8 (without Russian roulette) or 25 (with Russian roulette).


Selective Bremsstrahlung Splitting (SBS)

Bremsstrahlung photons can be emitted in all directions by an electron, but those emit-ted by the electrons aiming toward the FOI at the time of emission have a higher chanceof reaching it (see Figure 1). In a typical photon beam treatment head simulation mostphotons produced in electron and positron interactions are absorbed by the primarycollimator, the photon jaws, and the linac shielding. For instance, for a 6 MV beamand a 10×10 cm2 field size only about 2% to 3% of the photons reach the plane under-neath the jaws. With this realization, Sheikh-Bagheri and Rogers developed a techniqueknown as selective bremsstrahlung splitting (SBS) (Sheikh-Bagheri 1999). The differ-ence between SBS and UBS is that SBS uses a variable splitting number forbremsstrahlung that depends on the probability to emit a photon directed towards theFOI. The probability is precalculated for different incident electron directions assum-ing that the electron position is on the beam axis, and the splitting number is selected,during the simulation and according to the probability, between a maximum (electronmoving forward) and a minimum number (electron moving backwards). The minimumsplitting number is typically 1/10 of the maximum. Although SBS substantially reducesthe time needed to simulate photons not reaching the FOI, it introduces a non-uniformdistribution of statistical weights, which leads to a lower efficiency than theoreticallypossible. Nevertheless, SBS improves the efficiency by a factor of 2.5 to 3.5 comparedto UBS for a total efficiency gain compared to a simulation without splitting of ~20when not using Russian roulette for secondary electrons or ~65 when Russian rouletteof secondary electrons is used (see Figure 2).

Directional Bremsstrahlung Splitting (DBS)

As with SBS, the goal of DBS (Kawrakow, Rogers, and Walters 2004) is to reduce thenumber of transported photons not reaching the FOI. But unlike SBS, only those splitphotons that are aimed into the FOI are transported. The remaining photons are imme-diately subjected to a game of Russian roulette where, on average, only a single photonsurvives (see Figure 1). This approach leads to uniform statistical weights within theFOI, which improves the efficiency further compared to SBS. DBS uses a complexcombination of interaction splitting, Russian roulette for secondary electrons andphotons, particle splitting for electrons, and directional biasing together with the factthat the probability for bremsstrahlung, Compton scattered, annihilation, and fluores-cence photons reaching the FOI can be calculated in advance, thus reducing the numberof actual interactions being sampled. DBS improves the efficiency of photon beamtreatment head simulations by up to a factor of 8 compared to SBS for a total gain inefficiency compared to a simulation without any splitting of ~150 when electron split-ting is employed and therefore good statistics are achieved for contaminant electrons,or of ~500 when electron splitting is not employed and therefore only useful forphoton-only quantities (see Figure 2).


To give an idea of the power of these techniques, Figure 3 presents calculated spec-tra averaged over a 10×10 cm2 field from a typical high-energy photon accelerator,using BEAMnrc on a 1.8 GHz CPU.

Efficiency Improvement Techniques Used in Patient Simulations

An efficient linac simulation algorithm plays an important role in the process of beamcommissioning for a MC-based RTP, by facilitating the often-iterative process of deter-mining the phase-space of the incident electron beam (at the location of thebremsstrahlung target in photon beams or at the vacuum exit window in electronbeams). A substantial improvement in the efficiency of beam modeling for MC treat-


Figure 2. Relative efficiency for calculating photon fluence within the 10×10 cm2 field of asimulated Elekta SL25 6 MV photon beam as a function of bremsstrahlung splitting number

(NBRSPL). Efficiencies shown are relative to total photon fluence efficiency with nosplitting. For UBS and SBS, efficiencies are shown with Russian roulette on (open circles)

and off (solid circles). The field size parameter, FS, used with SBS was 30 cm (Sheikh-Bagheri 1999; Sheikh-Bagheri and Rogers 2002a,b). For DBS, results are shown withelectron splitting off (open circles) and electron splitting on with the splitting plane at

Z=15.46 cm and the Russian roulette plane at Z=15.2 cm (closed circles). For UBS theminimum splitting number is 20 and for SBS and DBS it is 50. Note the y axis is

logarithmic. (Reprinted from Kawrakow, I., D. W. O. Rogers, and B.Walters,“Large efficiency improvements in BEAMnrc using directional bremsstrahlung

splitting,” Med Phys 31:2883–2898. © 2004, with permission from AAPM.)

ment planning does not necessarily depend on direct high-efficiency MC methods,since the beam modeling is feasible through the use of a variety of other (and not directMC-based) methods such as (see chapter by Ma and Sheikh-Bagheri); recycling phase-space files, design of MC-based beam models, or the employment of empirical andsemi-empirical measurement-based beam models. Therefore, the routine utilization ofa MC code in the clinic will very strongly depend on the efficiency of the simulationfor each patient.

As discussed earlier, the CHT implementation is the most important factor forimproved efficiency. The following sections provide a brief discussion of additionalVRTs and AEITs employed by MC algorithms for dose calculations in a patient. Dosedistribution post-processing (i.e., denoising), which is another approach to significantlyreduce CPU time, is discussed in detail in the chapter by Kawrakow and Bielajew(“Monte Carlo Treatment Planning: Interpretation of Noisy Dose Distributions andReview of Denoising Method”). The combined efficiency enhancements thus achievedhave made MC patient dose calculations viable for routine clinical use.


Figure 3. An example of the speed of the calculation of photon spectra in a 10×10 cm2 fieldfrom a 16 MV beam from a realistic linac with the DBS technique, with electron splitting

(blue histogram) and without electron splitting (red histogram), using the BEAMnrc system.

Macro Monte Carlo

The Macro Monte Carlo (MMC) approach (Neuenschwander and Born 1992; Neuen-schwander, Mackie, and Reckwerdt 1995) was perhaps the first attempt to develop afast MC code for use in RTP. The idea behind it is simple: one performs simulationsof electrons impinging on homogeneous spheres of varying radii and media using ageneral purpose package such as EGS4 (used in the original MMC implementation)or EGSnrc (used in the current commercial implementation in the Eclipse treatmentplanning system), and stores the probability distribution of particles emerging fromthe sphere in a database. In the actual patient simulation electrons are transported usingthis database. For each electron step the medium and size of the spheres employed forthe transport is determined from the properties of the surrounding voxels. The MMCapproach is an AEIT because (1) it uses two-dimensional histograms to represent thedatabase probability distributions (in reality the distributions are five-dimensional butthis would require too much data) and (2) the energy is distributed along a straight linebetween the initial and final electron position. Neuenschwander, Mackie, and Reck-werdt (1995) recognized that because of item (2) above, the radius of the spheres mustbe limited to about 5 mm to achieve sufficient accuracy for typical RTP electron beamsimulations.

History Repetition

The history repetition technique was introduced almost simultaneously in the SMC(Super Monte Carlo) (Keall and Hoban 1996a,b), and VMC (Kawrakow, Fippel, andFriedrich 1996) codes. It is also used by xVMC (Fippel 1999; Kawrakow and Fippel2000b) and MCDOSE (Ma et al. 2000). In history repetition, one simulates an elec-tron track in an infinite, homogeneous medium (typically water) and then “applies”the track to the actual patient geometry starting at different positions and directions atthe patient surface (electron beams) or from different interaction sites (photon beams).Whereas VMC/xVMC generate the reference track on-the-fly using their own physicsimplementation, MCDOSE and SMC use EGS4 for this purpose. The differencebetween SMC and MCDOSE is that in MCDOSE the track is generated on the fly,whereas SMC uses precalculated tracks stored in a data file. The application of thereference track to the heterogeneous patient geometry requires the appropriate scal-ing of the step lengths and the multiple elastic scattering angles. The scaling of multipleelastic scattering angles is done within a small angle approximation. In addition, it isnot possible to obtain the correct number of discrete interactions. History repetitionis therefore an AEIT. For materials typically found in the patient anatomy the error canbe made very small, of the order of 1% to 2%, as discussed in more detail in Kawrakow(1997). Use of history repetition in arbitrary materials, such as high-Z elements, is notpossible if one wants to ensure an acceptable accuracy.

It was originally believed that history repetition is the main reason for the muchhigher simulation speed of VMC/xVMC, SMC, and MCDOSE compared to general-


purpose MC codes. However, a more careful analysis reveals that history repetitiononly results in a modest gain in efficiency that depends on the voxel size and the detailsof the CHT implementation (Kawrakow 2001; Siebers, Keall, and Kawrakow 2005).The more accurate the CHT implementation (fewer steps required), the less the effi-ciency gain and vice versa. For the VMC/xVMC code, for instance, the efficiency gainvaries between a factor of ~1.5 (1 mm voxels) and ~3 (1 cm voxels).

Boundary-Crossing Algorithms

The algorithm used for geometrical boundary crossing can have a substantial effecton the efficiency of a MC calculation. As an example, BEAMnrc uses the PRESTAboundary crossing algorithm since it is three to four times faster in phantom or accel-erator calculations than the EGSnrc boundary crossing algorithm, and in thesesimulations gives the same result. All of the very fast dose-in-phantom codes do noteven stop at the boundaries and use various other techniques to avoid loss of accuracy(Kawrakow, Fippel and Friedrich 1996; Ma et al. 2000; Sempau, Wilderman, andBielajew 2000)

Precalculated Interaction Densities

Use of precalculated interaction densities is a true VRT that has been employed inSMC, (Keall and Hoban 1996b), MCPAT (Wang, Chui, and Lovelock 1998) and inxVMC (Fippel 1999) for photon beam calculations. Instead of tracing the photons inci-dent on the patient again and again, in this technique the interaction densities in allvoxels are calculated in advance for all voxels and these interaction densities are thenused to start the appropriate number of electrons from the different voxels. Fippel(1999) reports about a factor of 2 gain in efficiency compared to photon transport notusing any VRT. The main limitation of this method is the fact that relatively simplesource models are required to calculate the interaction densities accurately.

Woodcock Tracing

Woodcock tracing, also known as the “delta scattering method” (Lux and Koblinger1991) is employed in DPM (Sempau, Wilderman, and Bielajew 2000) and the PERE-GRINE code (Hartmann Siantar et al. 2001) for photon transport. This technique isalso known as the fictitious cross-section method and can also be used to handle theenergy-dependent cross section in electron transport (Nelson, Hirayama, and Rogers1985). In Woodcock tracing for photon transport, one adds a fictitious interaction,which leaves the energy and direction of the photon unaltered, to the list of possiblephoton interactions. The cross section for this fictitious interaction is selected to makethe total photon cross section constant in the entire geometry, i.e., a smaller fictitiousinteraction cross section is used in voxels with a larger cross section and vice versa.


One can then transport the photon immediately to the interaction site without the needfor tracing through the geometry. The correct number of real interactions is obtainedby selecting a fictitious interaction with the appropriate probability that depends onlyon the fictitious cross section at the point of interaction. Woodcock tracing is a trueVRT. In a detailed investigation of various photon transport VRTs, Kawrakow andFippel (2000b) reported approximately 20% efficiency gain when Woodcock tracingwas implemented in xVMC; however no quantitative efficiency gain is reported inDPM or in PEREGRINE.

Photon Splitting Combined with Russian Roulette

The single most significant gain in efficiency (about a factor of 5) for photon trans-port in the patient is obtained from a combination of particle splitting and Russianroulette introduced by Kawrakow and Fippel (2000b), and denoted “SPL.” SPL isemployed in recent versions of VMC /xVMC (Kawrakow and Fippel 2000b; Fippelet al. 2000), and VMC++ (Kawrakow and Fippel 2000a; Kawrakow 2001), and is alsoused in DOSXYZnrc (Walters, Kawrakow, and Rogers 2005). In this true VRT, Ns

photon interaction sites are sampled for each incident photon using a single passthrough the geometry. Secondary photons, resulting from Compton scattering,bremsstrahlung, and annihilation, are subjected to a Russian roulette game with asurvival probability of 1/Ns. Surviving secondary photons are transported in the sameway as primary photons. For a typical patient anatomy the optimum splitting numberNs is around 40. Efficiency gains from SPL vary between a factor of 5 and a factor of9 (Kawrakow and Fippel 2000b) when coupled with history repetition (VMC/XVMC)or STOPS (VMC++) to transport the resulting electrons and positrons.

Simultaneous Transport of Particle Sets (STOPS)

The goal of the STOPS technique (Kawrakow 2001) is to reduce the relative timespent simulating particle interactions as in history repetition, while not introducingany approximations. STOPS is a true VRT. To accomplish this task, particles aretransported in sets. The particles in a set have the same energy and charge but differ-ent positions and directions. STOPS saves time by calculating thematerial-independent quantities such as mean free paths, interpolation indices,azimuthal scattering angles and cross sections (e.g., Møller and Bhabha) once for allthe particles in the set. However, it still has to sample material-dependent quantities,such as multiple elastic scattering and bremsstrahlung, for each particle in the set. Ifone or more particles in a set undergo a discrete interaction different from the otherparticles, the set is separated into subsets and each subset is transported individually.Due to the similarity of interaction properties of materials typically found in thepatient anatomy, separating particle sets is a relatively rare event. Hence, the effi-ciency gain is almost the same as from history repetition, yet no approximations are


involved. This fact allows the use of STOPS in arbitrary media, not just low-Z mate-rials, as is the case with history repetition.

Quasi-Random Sequences

The successful implementation of the MC technique depends heavily on the genera-tion and use of random numbers. Unlike pseudo-random numbers, that are morefrequently encountered in MC simulations, quasi-random numbers are generated withemphasis on filling the multidimensional space of interest in as uniform a way as possi-ble. The main advantage of this uniformity in the generated random numbers is thatthe computation converges faster compared to the same problem using a sequence ofpseudo-random numbers. Quasi-random sequences are used in VMC/xVMC forphoton transport (Kawrakow and Fippel 2000b) and in VMC++ for electron andphoton transport (Kawrakow 2001) and were reported to result in an efficiency gainof about a factor of 2. For a more detailed discussion the reader is referred to Siebers,Keall, and Kawrakow (2005).

Correlated Sampling

Correlated sampling is a standard VRT that may lead to substantial efficiency gainsbut has not received sufficient attention in MC dose calculations in external beam RTP.Substantial efficiency gains from correlated sampling have been reported for the calcu-lation of ion chamber dosimetry correction factors (Ma and Nahum 1993; Buckley,Kawrakow, and Rogers 2004) and for brachytherapy dose calculation (Hedtjärn, Carls-son, and Williamson 2002). The only publication on the use of correlated sampling inexternal beam RTP calculations that we are aware of is Holmes et al. (1993), wheremodest efficiency gains were observed for relatively simple heterogeneous geometries.In addition to the papers cited above the reader is referred to Siebers, Keall, andKawrakow (2005) and for more theoretical considerations to the book by Lux andKoblinger (1991).

Statistical Considerations

A reliable estimate for the efficiency of a MC simulation requires a correct estimateof the statistical uncertainties. The following sections discuss the different methodsof estimating the variance of a quantity of interest when performing MC simulations,as well as the pitfalls associated with the recycling of particles.

According to the Central Limit Theorem (Lux and Koblinger 1991), the probabilitydistribution to observe a certain result in a MC simulation will approach a Gaussianin the limit of a large number of particle histories. Hence, if one divides the simula-tion into batches (groups), each batch containing a large number of particles, one canestimate the uncertainty σX of the quantity X using


, (1)

where Xi is the result of the i’th batch, is the mean of the Xi, and N is the number ofbatches. Equation (1) is the usual way to estimate the variance of a set of normallydistributed observations. The batch method has been widely employed due to itsconceptual and numerical simplicity and has been used, for instance, in many EGS usercodes prior to 2002. Its main disadvantage is that there is a relatively large uncertaintyon the uncertainty estimate due to the small number of batches (typically 10 to 40).An alternative way to estimate the uncertainty of a MC computed quantity is to employhistory-by-history statistical analysis. The use of a history-by-history method is basedon the fact that the uncertainty σX is related to the first (<X>) and second (<X2>)moments of the single-history probability distribution, according to

, (2)

where now N is the number of particles instead of batches. The moments <X> and <X2> can be estimated within the simulation by keeping track of the sum of the singlehistory observations xi, i.e.,

. (3)

Implementation of the above equation in a MC code is trivial if one is interested in nomore than a few quantities. However, for a distributed quantity such as a 3-D dosedistribution with hundreds of thousands of voxels needed in RTP, a straightforwardimplementation of the summations at the end of each history would result in a verylong calculation time. This problem was resolved only recently due to a computationaltrick attributed to Salvat (Sempau, Wilderman, and Bielajew 2000); see also Walters,Kawrakow, and Rogers (2002). For general-purpose codes the computational penaltydue to the use of history-by-history statistics with the Salvat trick is negligible.However, in the case of fast codes such as VMC++, history-by-history analysis canlead to a 10% to 20% penalty in terms of CPU (central processing unit) time comparedto batch analysis (Kawrakow 2001). The main advantage of the history-by-historymethod is that it provides a more precise estimate of the uncertainty. This is illustratedin Figure 4, which shows the fractional dose uncertainty along the central axis of an18 MeV electron beam obtained using a history-by-history method and batch meth-ods with different number of batches.

Equations (1) and (2) both assume that the batch or history observations are statis-tically independent. In a “normal” Monte Carlo simulation that does not employ any


X

σX

i

i

N

X X

N N=

−

−=∑( )

( )1

2

1

σ X

X X

N2

2 2

1=

−−

XN

x XN

xi

i

N

i

i

N

= == =∑ ∑1 1

1

2 2

1

,

VRTs, the statistical independence of individual particle histories is typically auto-matically guaranteed by the use of a good quality random number generator thatproduces an uncorrelated sequence of random numbers. However, when one employsVRTs, or splits the simulation into parts as typically done in patient simulations thatuse a phase-space file from a full treatment head simulation, it is very important tocombine particles into statistically independent groups in order to obtain a reliableuncertainty estimate. For instance, when using techniques such as history repetition orSTOPS, all repetitions of the same track (history repetition) or all particles within a set(STOPS) are correlated and must therefore be considered as one event for the sake ofstatistical analysis. In a similar way, all particles originating from the same electron inci-dent on the bremsstrahlung target or vacuum exit window belong to the same initial,statistically independent, history. Therefore, when using a phase-space file for thepatient calculation, particles must be grouped according to their initial electron historyand cannot be considered to be independent. An important conclusion from theseconsiderations is that, when re-using phase-space file particles, one must recycle theparticles (i.e., use them several times before moving on to the next particle), instead ofrestarting the phase-space file several times. It is not possible to group the phase-space


Figure 4. Fractional uncertainty along the central axis for an 18 MeV electron beam from a Clinac 2100C (20×20 cm2 field at SSD=100 cm) in a water phantom. Fractional

uncertainties are estimated using the history-by-history method and the batch method with10 and 40 batches. A scaled depth-dose curve is also shown for reference. Dose was scored

in 1×1×0.5 cm3 voxels. (Reprinted with permission from Walters, B. R. B., I. Kawrakow, andD. W. O. Rogers, “History by history statistical estimators in the BEAM code system.”

Med Phys 29:2745–2752. © 2002, with permission from AAPM.)

particles into statistically independent initial histories when restarting a phase-space fileand as a result the uncertainties will be underestimated. For a more detailed discussionof these issues the reader is referred to (Walters, Kawrakow, and Rogers 2002). Anotherimportant consideration when using phase-space files is that the uncertainties in thepatient simulation cannot be reduced beyond the variance present in the phase-spacefile, even if particles are recycled a very large number of times. This is most easilyunderstood by considering a phase-space file containing a single particle. If one recy-cles this particle a very large number of times, the uncertainty in dose deposition fromthis particular particle will tend to zero, but the overall uncertainty (variance) of thesimulation will be very large since each reuse of the particle is statistically speaking,fully dependent on the other. This variance that is present in a phase-space file hasbecome known as the “latent variance” (Sempau et al. 2001).

A simple technique to demonstrate the presence of and accurately estimate thelatent variance in a phase-space calculation of dose was proposed by Sempau et al.(2001) and is depicted in Figure 5. In this technique the variance is separated into twocomponents, only one of which depends on the number of recyclings. Then by increas-ing the recycling factor (K), that term is taken to zero, at which limit the remainingterm is the latent variance.


Figure 5. The dependence of the variance on the inverse of the number of times (K)particles from a phase-space file are recycled. When 1/K approaches zero, the only

component of variance remaining is the latent variance present in the phase-space file.(Reprinted with permission from Sempau, J., A. Sánchez-Reyes, F. Salvat, H. Oulad ben

Tahar, S. B. Jiang, and J. M. Fernandez-Varea,. “Monte Carlo simulation of electron beamsfrom an accelerator head using PENELOPE.” Phys Med Biol 46:1163–1186.

© 2001, with permission from Elsevier.)

Summary

The design and special-purpose implementation of several efficiency enhancementtechniques such as condensed history technique (CHT), splitting, and Russian roulettehave facilitated the development of faster and faster MC codes used in beam model-ing and in patient dose simulation. Variance reduction techniques (VRTs), in contrastto approximate efficiency enhancement techniques (AEITs), do not bias the physicsof the simulations and, when implemented properly, can confidently be used toenhance the efficiency of the simulations. AEITs can be utilized to enhance the effi-ciency when care is taken to show that they have a small or negligible effect on theresults. Due to the increased popularity of such aggressive implementations of VRTs,it has become more important to take correlations between phase-space particles intoaccount when calculating uncertainties. These correlations can be properly taken intoaccount using either the history-by-history or the batch technique by grouping the parti-cles according to the primary original history that generated them. The substantialimprovements achieved in the efficiency of MC calculations are paving the way forroutine clinical use.

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