2. Other codes systems such as ITS (Halbleib and Melhorn 1984; Halbleib 1989; Halbleib
et al. 1992); MCNP (Briesmeister 1986, 1993; Brown 2003); PENELOPE (Baró et al.
1995) and GEANT4 (Agostinelli et al. 2003) also have popularity among their user
groups. The EGS and ITS/ETRAN systems are roughly of the same efficiency for
calculations when no variance reduction techniques are used, whereas the other
systems tend to be considerably slower. For special-purpose applications the use of
sophisticated 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 radiotherapy
accelerators that employs several techniques which significantly enhance its overall
efficiency. Nonetheless, it is still not fast enough for routine treatment planning
purposes without a substantial farm of computers. Thus, to make routine clinical use
a possibility, there have been a number of attempts to develop high-efficiency MC
codes.Among the most well known of such codes are: the Macro Monte Carlo (MMC)
code (Neuenschwander and Born 1992; Neuenschwander, Mackie, and Reckwerdt
1995); the PEREGRINE code (Schach von Wittenau et al. 1999; Hartmann Siantar et
al. 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 Planning
Method (DPM) (Sempau, Wilderman, and Bielajew 2000) and other codes (Keall and
Hoban 1996a,b; Wang, Chui, and Lovelock 1998).
In another chapter, “Monte Carlo Methods forAccelerator Simulation and Photon
Beam Modeling,” Ma and Sheikh-Bagheri have discussed the effects on the accuracy
of MC-calculated dose distributions of various linac modeling parameters, such as the
specifications of the initial electron beam-on-target and accelerator components. Our
discussion of accuracy assumes all such influencing factors are properly addressed.
In this chapter we discuss the fundamental methods of improving the efficiency
of MC simulations that are used both in therapy beam simulations and in simulations
of the dose distribution in the patient. Throughout the chapter we explicitly distinguish
between techniques that do not alter the physics in any way when they increase the
efficiency—i.e., true variance reduction techniques (VRT)—and techniques that
achieve the improved efficiency through the use of approximations; i.e., approximate
efficiency 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 required
to obtain this variance. The goal is to reduce the time it takes to obtain a sufficiently
small 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.
ε
σ
= 1
2
T
3. tion is proportional to the number N of statistically independent particles.At the same
time σ2
decreases as 1/N according to the central limit theorem (Lux and Koblinger
1991).As a result, the product T · σ2
is a constant and the efficiency ε = 1/T σ2
expresses
how fast a certain simulation algorithm can calculate a given quantity of interest at a
desired level of statistical accuracy.
The above metric does not define how to calculate the variance of the quantity of
interest. 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 emerging
from the treatment head of the linear accelerator, etc. One therefore must define a
measure of the overall uncertainty in order to evaluate the efficiency of a particular
simulation 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. Using
the above ICCR (International Conference on the Use of Computers in Radiation
Therapy) benchmark criteria, the most commonly used MC codes in radiotherapy
applications 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 primarily
attributable 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 the
geometry packages.
Before we get into a detailed discussion of VRTs, we will first discuss one single
most 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 it
sets in motion undergoes of the order of 106
elastic and inelastic collisions until locally
absorbed. It is therefore impossible to simulate each individual collision as this would
result in prohibitively long simulation times. To overcome this difficulty, Berger
(1963) introduced the condensed history technique (CHT). The CHT makes use of
the fact that most electron interactions result in extremely small changes in energy
and/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 packages
such 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
D
i
ii
n
∆
,
4. (Nelson, Hirayama, Rogers 1985), EGS4 with PRESTA (Bielajew and Rogers 1987),
EGSnrc (Kawrakow 2000, Kawrakow and Rogers 2000), and PENELOPE (Salvat et
al. 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 the
CHT the results depend on the choice of the step-size (Rogers 1984, Bielajew and
Rogers 1987, Seltzer 1991, Rogers 1993). The CHT is therefore an AEIT according
to the definition adopted earlier in this chapter. Although every MC algorithm uses
the CHT for charged particle transport, it is rarely considered as a technique to
improve the efficiency of MC simulations. Yet, the two main aspects of the CHT
implementation: (1) the “electron-step algorithm” also called “transport mechanics”
and (2) the boundary-crossing algorithm very strongly influence the simulation speed
and accuracy. For more detailed discussion on the condensed history technique the
reader 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
4 Daryoush Sheikh-Bagheri et al.
Table 1. Summary of Timing and Accuracy Results from the ICCR Benchmark. Timing
comparisons were performed using 6 MV photons, 10×10 cm2
field size, and those for the
accuracy 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 take
running 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 to
differences 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 a
correlated-histogram source model and transport through the field-defining collimators.
** See Poon and Verhaegen (2005) for further details.
5. and Bielajew (1998), which gives a detailed theoretical comparison between
different electron-step algorithms, and to Kawrakow (2000a), where the various
details of a CHT implementation and their influence on the accuracy are investigated.
The article by Siebers, Keall, and Kawrakow (2005) discusses CHT details relevant
for fast algorithms used in RTP class simulations.
Efficiency Improvement Techniques Used
in Treatment Head Simulations
Range Rejection and Transport Cutoffs
Use ofAEITs, such as range rejection and transport cutoffs, can improve the efficiency
of treatment head simulations substantially, without significantly changing the results
(Rogers et al. 1995). In range rejection, an electron’s history is terminated whenever
its residual range is so low that it cannot escape from the current region or reach the
region of interest. Since this ignores the possible creation of bremsstrahlung or anni-
hilation photons while electrons (negatrons or positrons) slow down, it is not an
unbiased technique. However, as long as it is only applied to electrons below a certain
energy 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 save
a 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 high
transport cutoff or with particles that would be range-rejected is a comparable vari-
ance reduction technique for reducing the simulation time. However, its
implementation is typically more difficult and this has favored the simpler use of range
rejection and high transport cutoffs in situations where it is easy to demonstrate that
the resulting error is sufficiently small.
The particle production and transport cutoff energies and the threshold energy for
doing range rejection must both be carefully investigated to avoid significant errors in
the 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 et
al. (2000) modified the BEAM code (Rogers et al. 1995) to allow tagging of
bremsstrahlung production anywhere outside the target. They concluded that outside
the target, values of the upper energy for doing range rejection of 0.6 MeV for 4 MV
beams, and 1.5 MeV for 6 MV and higher energy beams, provide the largest savings
in central processing unit (CPU) time (about a factor of 3 for the 10 and 20 MV beams
studied), with negligible (less than 0.2%) underestimation of the calculated photon
fluence from the linac.
In the following sections we discuss the much larger increases in efficiency that
can be achieved for photon beams by employing true VRTs, although for electron
beams the AEITs may still be the best options available.
MC Simulations: Efficiency Improvement/Statistical Considerations 5
6. Splitting and Russian Roulette
Two very commonly used VRTs are splitting and Russian roulette, which were both
originally 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 of
bremsstrahlung splitting (see Figure 1), each time an electron undergoes a bremsstrah-
lung interaction, a large number of secondary photons with lower weights are set in
motion, the number possibly depending on a variety of factors related to the likelihood
of them being directed toward the field of interest (FOI). The energy of the electron
is reduced by an amount equal to the energy of one of the emitted photons, which is
6 Daryoush Sheikh-Bagheri et al.
Figure 1. A simplistic schematic of the splitting routines discussed in this chapter for linac
modeling. 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). In
uniform bremsstrahlung splitting (UBS) (Rogers et al. 1995) any bremsstrahlung event leads
to the sampling and transport of N (a constant splitting number) photons, increasing the
likelihood 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 of
the 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 not
emitted 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 to
further 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.)
7. in violation of conservation of energy on an individual interaction basis but results in
correct fluctuations in energy loss for electrons and correct expectation values for
photon energy and angular distributions. Splitting can save a large amount of time
because photon transport is fast, whereas it takes a long time to track an electron in
the 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 never
reach the patient plane, then there is a further time savings. Russian roulette can be
played 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 unbiased
result, 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 created
in a photon beam.
Uniform Bremsstrahlung Splitting (UBS)
When using UBS, in each interaction that produces photons, Nsplit bremsstrahlung or
2 Nsplit annihilation photons are sampled instead of one or two photons. To make the
game fair, a statistical weight of w0 /Nsplit is assigned to the photons, where w0 is the
statistical 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 be
produced in a normal simulation that does not use splitting. Many of the photons set
in motion in interactions of the primary electrons impinging on the bremsstrahlung
target will undergo additional collisions before emerging from the treatment head or
being locally absorbed. Electrons and positrons set in motion in such interactions will
inherit the statistical weight of the photons, i.e., they will have a weight of w0 /Nsplit if
UBS is used. If one would split bremsstrahlung and annihilation events of such
secondary charged particles one would have photons with weights w0 /N2
split, w0 /N3
split,
etc. This is not desirable and therefore such higher order interactions are not split. If
one 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 obtain
another photon-only quantity such as the photon (energy) fluence or spectrum, one can
further 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. In
this case, the secondary electrons that survive the Russian roulette game have again a
weight of w0 and their bremsstrahlung and annihilation interactions must be split to
avoid the production of “fat” photons (Sheikh-Bagheri 1999). It is important to keep
in 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 suitable
for full dose calculations in the patient. UBS was implemented in the original BEAM
version (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 factor
of 8 (without Russian roulette) or 25 (with Russian roulette).
MC Simulations: Efficiency Improvement/Statistical Considerations 7
8. 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 chance
of reaching it (see Figure 1). In a typical photon beam treatment head simulation most
photons produced in electron and positron interactions are absorbed by the primary
collimator, the photon jaws, and the linac shielding. For instance, for a 6 MV beam
and 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 technique
known as selective bremsstrahlung splitting (SBS) (Sheikh-Bagheri 1999). The differ-
ence between SBS and UBS is that SBS uses a variable splitting number for
bremsstrahlung that depends on the probability to emit a photon directed towards the
FOI. 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 (electron
moving forward) and a minimum number (electron moving backwards). The minimum
splitting number is typically 1/10 of the maximum.Although SBS substantially reduces
the time needed to simulate photons not reaching the FOI, it introduces a non-uniform
distribution of statistical weights, which leads to a lower efficiency than theoretically
possible. Nevertheless, SBS improves the efficiency by a factor of 2.5 to 3.5 compared
to UBS for a total efficiency gain compared to a simulation without splitting of ~20
when not using Russian roulette for secondary electrons or ~65 when Russian roulette
of 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 the
number of transported photons not reaching the FOI. But unlike SBS, only those split
photons 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 photon
survives (see Figure 1). This approach leads to uniform statistical weights within the
FOI, which improves the efficiency further compared to SBS. DBS uses a complex
combination of interaction splitting, Russian roulette for secondary electrons and
photons, particle splitting for electrons, and directional biasing together with the fact
that the probability for bremsstrahlung, Compton scattered, annihilation, and fluores-
cence photons reaching the FOI can be calculated in advance, thus reducing the number
of actual interactions being sampled. DBS improves the efficiency of photon beam
treatment head simulations by up to a factor of 8 compared to SBS for a total gain in
efficiency 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 for
photon-only quantities (see Figure 2).
8 Daryoush Sheikh-Bagheri et al.
10. 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 direct
MC-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 and
semi-empirical measurement-based beam models. Therefore, the routine utilization of
a MC code in the clinic will very strongly depend on the efficiency of the simulation
for each patient.
As discussed earlier, the CHT implementation is the most important factor for
improved efficiency. The following sections provide a brief discussion of additional
VRTs andAEITs employed by MC algorithms for dose calculations in a patient. Dose
distribution post-processing (i.e., denoising), which is another approach to significantly
reduce CPU time, is discussed in detail in the chapter by Kawrakow and Bielajew
(“Monte Carlo Treatment Planning: Interpretation of Noisy Dose Distributions and
Review of Denoising Method”). The combined efficiency enhancements thus achieved
have made MC patient dose calculations viable for routine clinical use.
10 Daryoush Sheikh-Bagheri et al.
Figure 3. An example of the speed of the calculation of photon spectra in a 10×10 cm2
field
from 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.
11. 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 a
fast MC code for use in RTP. The idea behind it is simple: one performs simulations
of electrons impinging on homogeneous spheres of varying radii and media using a
general purpose package such as EGS4 (used in the original MMC implementation)
or EGSnrc (used in the current commercial implementation in the Eclipse treatment
planning system), and stores the probability distribution of particles emerging from
the sphere in a database. In the actual patient simulation electrons are transported using
this database. For each electron step the medium and size of the spheres employed for
the transport is determined from the properties of the surrounding voxels. The MMC
approach is an AEIT because (1) it uses two-dimensional histograms to represent the
database probability distributions (in reality the distributions are five-dimensional but
this would require too much data) and (2) the energy is distributed along a straight line
between the initial and final electron position. Neuenschwander, Mackie, and Reck-
werdt (1995) recognized that because of item (2) above, the radius of the spheres must
be limited to about 5 mm to achieve sufficient accuracy for typical RTP electron beam
simulations.
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, and
Friedrich 1996) codes. It is also used by xVMC (Fippel 1999; Kawrakow and Fippel
2000b) 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 at
the patient surface (electron beams) or from different interaction sites (photon beams).
WhereasVMC/xVMC generate the reference track on-the-fly using their own physics
implementation, MCDOSE and SMC use EGS4 for this purpose. The difference
between 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 the
reference track to the heterogeneous patient geometry requires the appropriate scal-
ing of the step lengths and the multiple elastic scattering angles. The scaling of multiple
elastic scattering angles is done within a small angle approximation. In addition, it is
not possible to obtain the correct number of discrete interactions. History repetition
is therefore anAEIT. For materials typically found in the patient anatomy the error can
be 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 not
possible if one wants to ensure an acceptable accuracy.
It was originally believed that history repetition is the main reason for the much
higher simulation speed of VMC/xVMC, SMC, and MCDOSE compared to general-
MC Simulations: Efficiency Improvement/Statistical Considerations 11
12. purpose MC codes. However, a more careful analysis reveals that history repetition
only results in a modest gain in efficiency that depends on the voxel size and the details
of 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 theVMC/xVMC code, for instance, the efficiency gain
varies 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 effect
on the efficiency of a MC calculation. As an example, BEAMnrc uses the PRESTA
boundary crossing algorithm since it is three to four times faster in phantom or accel-
erator calculations than the EGSnrc boundary crossing algorithm, and in these
simulations gives the same result. All of the very fast dose-in-phantom codes do not
even 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, and
Bielajew 2000)
Precalculated Interaction Densities
Use of precalculated interaction densities is a true VRT that has been employed in
SMC, (Keall and Hoban 1996b), MCPAT (Wang, Chui, and Lovelock 1998) and in
xVMC (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 all
voxels are calculated in advance for all voxels and these interaction densities are then
used 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 not
using any VRT. The main limitation of this method is the fact that relatively simple
source models are required to calculate the interaction densities accurately.
Woodcock Tracing
Woodcock tracing, also known as the “delta scattering method” (Lux and Koblinger
1991) is employed in DPM (Sempau, Wilderman, and Bielajew 2000) and the PERE-
GRINE code (Hartmann Siantar et al. 2001) for photon transport. This technique is
also known as the fictitious cross-section method and can also be used to handle the
energy-dependent cross section in electron transport (Nelson, Hirayama, and Rogers
1985). 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 possible
photon interactions. The cross section for this fictitious interaction is selected to make
the total photon cross section constant in the entire geometry, i.e., a smaller fictitious
interaction cross section is used in voxels with a larger cross section and vice versa.
12 Daryoush Sheikh-Bagheri et al.
13. One can then transport the photon immediately to the interaction site without the need
for tracing through the geometry. The correct number of real interactions is obtained
by selecting a fictitious interaction with the appropriate probability that depends only
on the fictitious cross section at the point of interaction. Woodcock tracing is a true
VRT. In a detailed investigation of various photon transport VRTs, Kawrakow and
Fippel (2000b) reported approximately 20% efficiency gain when Woodcock tracing
was implemented in xVMC; however no quantitative efficiency gain is reported in
DPM 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 Russian
roulette introduced by Kawrakow and Fippel (2000b), and denoted “SPL.” SPL is
employed in recent versions of VMC /xVMC (Kawrakow and Fippel 2000b; Fippel
et al. 2000), andVMC++ (Kawrakow and Fippel 2000a; Kawrakow 2001), and is also
used in DOSXYZnrc (Walters, Kawrakow, and Rogers 2005). In this true VRT, Ns
photon interaction sites are sampled for each incident photon using a single pass
through the geometry. Secondary photons, resulting from Compton scattering,
bremsstrahlung, and annihilation, are subjected to a Russian roulette game with a
survival probability of 1/Ns. Surviving secondary photons are transported in the same
way as primary photons. For a typical patient anatomy the optimum splitting number
Ns is around 40. Efficiency gains from SPL vary between a factor of 5 and a factor of
9 (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 time
spent simulating particle interactions as in history repetition, while not introducing
any approximations. STOPS is a true VRT. To accomplish this task, particles are
transported in sets. The particles in a set have the same energy and charge but differ-
ent positions and directions. STOPS saves time by calculating the
material-independent quantities such as mean free paths, interpolation indices,
azimuthal scattering angles and cross sections (e.g., Møller and Bhabha) once for all
the 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. If
one or more particles in a set undergo a discrete interaction different from the other
particles, the set is separated into subsets and each subset is transported individually.
Due to the similarity of interaction properties of materials typically found in the
patient 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
MC Simulations: Efficiency Improvement/Statistical Considerations 13
14. 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 more
frequently encountered in MC simulations, quasi-random numbers are generated with
emphasis 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 that
the computation converges faster compared to the same problem using a sequence of
pseudo-random numbers. Quasi-random sequences are used in VMC/xVMC for
photon transport (Kawrakow and Fippel 2000b) and in VMC++ for electron and
photon transport (Kawrakow 2001) and were reported to result in an efficiency gain
of 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 gains
but 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 in
external beam RTP calculations that we are aware of is Holmes et al. (1993), where
modest efficiency gains were observed for relatively simple heterogeneous geometries.
In addition to the papers cited above the reader is referred to Siebers, Keall, and
Kawrakow (2005) and for more theoretical considerations to the book by Lux and
Koblinger (1991).
Statistical Considerations
A reliable estimate for the efficiency of a MC simulation requires a correct estimate
of the statistical uncertainties. The following sections discuss the different methods
of 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 probability
distribution to observe a certain result in a MC simulation will approach a Gaussian
in 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 can
estimate the uncertainty σX of the quantity X using
14 Daryoush Sheikh-Bagheri et al.
15. , (1)
where Xi is the result of the i’th batch, is the mean of the Xi, and N is the number of
batches. Equation (1) is the usual way to estimate the variance of a set of normally
distributed observations. The batch method has been widely employed due to its
conceptual and numerical simplicity and has been used, for instance, in many EGS user
codes prior to 2002. Its main disadvantage is that there is a relatively large uncertainty
on 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 employ
history-by-history statistical analysis. The use of a history-by-history method is based
on 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 single
history observations xi, i.e.,
. (3)
Implementation of the above equation in a MC code is trivial if one is interested in no
more than a few quantities. However, for a distributed quantity such as a 3-D dose
distribution with hundreds of thousands of voxels needed in RTP, a straightforward
implementation of the summations at the end of each history would result in a very
long calculation time. This problem was resolved only recently due to a computational
trick attributed to Salvat (Sempau, Wilderman, and Bielajew 2000); see also Walters,
Kawrakow, and Rogers (2002). For general-purpose codes the computational penalty
due 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 can
lead to a 10% to 20% penalty in terms of CPU (central processing unit) time compared
to batch analysis (Kawrakow 2001). The main advantage of the history-by-history
method is that it provides a more precise estimate of the uncertainty. This is illustrated
in Figure 4, which shows the fractional dose uncertainty along the central axis of an
18 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
MC Simulations: Efficiency Improvement/Statistical Considerations 15
X
σX
i
i
N
X X
N N
=
−
−
=
∑( )
( )
1
2
1
σX
X X
N
2
2 2
1
=
−
−
X
N
x X
N
xi
i
N
i
i
N
= =
= =
∑ ∑
1 1
1
2 2
1
,
18. Summary
The design and special-purpose implementation of several efficiency enhancement
techniques such as condensed history technique (CHT), splitting, and Russian roulette
have facilitated the development of faster and faster MC codes used in beam model-
ing and in patient dose simulation. Variance reduction techniques (VRTs), in contrast
to approximate efficiency enhancement techniques (AEITs), do not bias the physics
of the simulations and, when implemented properly, can confidently be used to
enhance 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 the
results. Due to the increased popularity of such aggressive implementations of VRTs,
it has become more important to take correlations between phase-space particles into
account when calculating uncertainties. These correlations can be properly taken into
account 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 substantial
improvements achieved in the efficiency of MC calculations are paving the way for
routine clinical use.
References
Agostinelli, S., et al. (Geant4 Collaboration). (2003). “GEANT4-a simulation toolkit.” Nucl
Instrum Methods Phys Res A 506:250–303.
Baró, J., J. Sempau, J. M. Fernandez-Varea, and F. Salvat. (1995). “Penelope—an algorithm
for Monte-Carlo simulation of the penetration and energy-loss of electrons and positrons
in matter.” Nucl Instrum Methods Phys Res B 100:31–46.
Berger, M. J. “Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Parti-
cles” in Methods in Computational Physics,Vol 1. B.Alder, S. Fernbach, and M. Rotenberg
(eds.). New York: Academic Press, pp. 135–215, 1963.
Bielajew, A. F., and D. W. O. Rogers. (1987). “PRESTA: The parameter reduced electron-step
transport algorithm for electron Monte Carlo transport.” Nucl Instrum Methods Phys Res
B 18:165–181.
Briesmeister, J. F. (Editor). MCNP—A General Purpose Monte Carlo Code for Neutron and
Photon Transport, Version 3A. Los Alamos National Laboratory Report LA-7396-M, Los
Alamos, NM, 1986.
Briesmeister, J. F. (Editor). MCNP—A General Monte Carlo N-Particle Transport Code,Version
4ª. Los Alamos National Laboratory Report LA-12625-M, Los Alamos, NM, 1993.
Brown, F. B. (Editor). MCNP—A General Monte Carlo N-Particle Transport Code, Version 5.
Los Alamos National Laboratory Report LA-UR-03 1987. Los Alamos, NM, 2003.
Buckley, L.A., I. Kawrakow, and D. W. O. Rogers. (2004). “CSnrc: Correlated sampling Monte
Carlo calculations using EGSnrc.” Med Phys 31:3425–3435.
Chetty, I. J., B. Curran, J. Cygler, J. J. DeMarco, G. Ezzell, B. A. Faddegon, I. Kawrakow, P. J.
Keall, H. Liu, C.-M. Ma, D. W. O. Rogers, D. Sheikh-Bagheri, J. Seuntjens, and J. V.
Siebers. (2006). “Issues associated with clinical implementation of Monte Carlo-based treat-
ment planning: Report of the AAPM Task Group No. 105. Med Phys (Submitted).
18 Daryoush Sheikh-Bagheri et al.
19. Fippel, M. (1999). “Fast Monte Carlo dose calculation for photon beams based on the VMC
electron algorithm.” Med Phys 26:1466–1475.
Fippel, M., I. Kawrakow, F. Nüsslin, and D. W. O. Rogers. “Implementation of Several Vari-
ance Reduction Techniques into the XVMC Monte Carlo Algorithm for Photon Beams” in
XIIIth International Conference on the Use of Computers in Radiation Therapy (XIIIth
ICCR). W. Schlegel and T. Bortfeld (eds.). Heidelberg: Springer-Verlag, pp. 406–408, 2000.
Halbleib, J. A. “Structure and Operation of the ITS Code System” in Monte Carlo Transport of
Electrons and Photons. W. R. Nelson, T. M. Jenkins, A. Rindi, A. E. Nahum, and D. W. O.
Rogers (eds.). New York: Plenum Press, pp. 249–262, 1989.
Halbleib, J. A., and T. A Melhorn. ITS: The Integrated TIGER Series of Coupled
Electron/Photon Monte Carlo Transport Codes. Sandia National Laboratory Report
SAND84-0073. Albuquerque, NM, 1984.
Halbleib, J. A., R. P. Kensek, T. A. Mehlhorn, G. D. Valdez, S. M. Seltzer, and M. J. Berger.
ITS Version 3.0: The Integrated TIGER Series of Coupled Electron/Photon Monte Carlo
Transport Codes, Sandia National Laboratory Report SAND91-1634. Albuquerque, NM,
1992.
Hartmann Siantar, C. L., R. S. Walling, T. P. Daly, B. Faddegon, N. Albright, P. Bergstrom, A.
F. Bielajew, C. Chiang, D. Garnet, R. K House, D. Knapp, D. J. Wieczorek, and L. J.Verhey.
(2001). “Description and dosimetric verification of the PEREGRINE Monte Carlo dose
calculation system for photon beams incident on a water phantom.” Med Phys
28:1322–1337.
Hedtjärn, H., G.A. Carlsson, and J. F. Williamson. (2002). “Accelerated Monte Carlo based dose
calculations for brachytherapy planning using correlated sampling.” Phys Med Biol
47:351–376.
Holmes, M. A., T. R. Mackie, W. Söhn, P. J. Reckwerdt, T. J. Kinsella, A. F. Bielajew, and D.
W. O. Rogers. (1993). “The application of correlated sampling to the computation of elec-
tron beam dose distributions in heterogeneous phantoms using the Monte Carlo method.”
Phys Med Biol 38:675–688.
Kahn, H. “Use of Different Monte Carlo Sampling Techniques” in Symposium on Monte Carlo
Methods. H. A. Meyer (ed.). New York: John Wiley and Sons, pp. 146–190, 1956.
Kawrakow, I. (1997). “Improved modeling of multiple scattering in the voxel Monte Carlo
model.” Med Phys 24:505–517.
Kawrakow, I. (2000a). “Accurate condensed history Monte Carlo simulation of electron trans-
port. I. EGSnrc, the new EGS4 version.” Med Phys 27:485–498.
Kawrakow, I. (2000b). “Accurate condensed history Monte Carlo simulation of electron trans-
port. II. Application to ion chamber response simulations.” Med Phys 27:499–513.
Kawrakow, I. “VMC++, Electron and Photon Monte Carlo Calculations Optimized for Radia-
tion Treatment Planning” in Advanced Monte Carlo for Radiation Physics, Particle
Transport Simulation and Applications. A. Kling, F. Barao, M. Nakagawa, L. Távora, and
P. Vaz (eds.). Proceedings of the Monte Carlo 2000 Meeting Lisbon. Berlin: Springer-
Verlag, pp. 229–236, 2001.
Kawrakow, I., and A. F. Bielajew. (1998). “On the condensed history technique for electron
transport.” Nucl Instrum Methods Phys Res B 142:253–280.
Kawrakow, I., and D. W. O. Rogers. The EGSnrc Code System: Monte Carlo Simulation of Elec-
tron and Photon Transport. Technical Report PIRS–701, National Research Council of
Canada, Ottawa, Canada, 2000.
MC Simulations: Efficiency Improvement/Statistical Considerations 19
20. Kawrakow, I., and M. Fippel. “VMC++, A Fast MC Algorithm for Radiation Treatment Plan-
ning” in XIIIth International Conference on the Use of Computers in Radiation Therapy
(XIIIth ICCR). W. Schlegel and T. Bortfeld (eds.). Heidelberg: Springer-Verlag, pp.
126–128, 2000a.
Kawrakow, I., and M. Fippel. (2000b). “Investigation of variance reduction techniques for Monte
Carlo photon dose calculation using XVMC.” Phys Med Biol 45:2163–2184.
Kawrakow, I., M. Fippel, and K. Friedrich. (1996). “3D electron dose calculation using a voxel
based Monte Carlo algorithm (VMC).” Med Phys 23:445–457.
Kawrakow, I., D. W. O. Rogers, and B. Walters. (2004). “Large efficiency improvements in
BEAMnrc using directional bremsstrahlung splitting.” Med Phys 31:2883–2898.
Keall, P. J., and P. W. Hoban. (1996a). “Superposition dose calculation incorporating Monte
Carlo generated electron track kernels.” Med Phys 23:479–485.
Keall, P. J., and P. W. Hoban. (1996b). “Super-Monte Carlo: A 3D electron beam dose calcula-
tion algorithm.” Med Phys 23:2023–2034.
Larsen, E. W. (1992). “A theoretical derivation of the condensed history algorithm.” Ann Nucl
Energy 19:701–714.
Lux, I., and L. Koblinger. Monte Carlo Particle Transport Methods: Neutron and Photon Calcu-
lations. New York: CRC Press, 1991.
Ma, C.-M., and A. E. Nahum. (1993). “Calculation of absorbed dose ratios using correlated
Monte Carlo sampling.” Med Phys 20:1189–1199.
Ma, C.-M., J. S. Li, T. Pawlicki, S. B. Jiang, and J. Deng. “MCDOSE - A Monte Carlo Dose
Calculation Tool for Radiation Therapy Treatment Planning” in XIIIth International Confer-
ence on the Use of Computers in Radiation Therapy (XIIIth ICCR). W. Schlegel and T.
Bortfeld (eds.). Heidelberg: Springer-Verlag, pp. 123–125, 2000.
Nelson, W. R., H. Hirayama, and D.W.O. Rogers. The EGS4 Code System. Stanford Linear
Accelerator Report SLAC-265, Stanford CA, 1985.
Neuenschwander, H., and E. J. Born. (1992). “A macro Monte Carlo method for electron beam
dose calculations.” Phys Med Biol 37:107–125.
Neuenschwander, H., T. R. Mackie, and P. J. Reckwerdt. (1995). “MMC—A high-performance
Monte Carlo code for electron beam treatment planning.” Phys Med Biol 40:543–574.
Poon, E., and F. Verhaegen. (2005). “Accuracy of the photon and electron physics in GEANT4
for radiotherapy applications.” Med Phys 32(6):1696–1711.
Rogers, D. W. O. (1984). “Low energy electron transport with EGS.” Nucl Instrum Methods
227:535–548.
Rogers, D. W. O. (1993). “How accurately can EGS4/PRESTA calculate ion chamber
response?” Med Phys 20:319–323.
Rogers, D. W. O., and R. Mohan. “Questions for Comparison of Clinical Monte Carlo Codes”
in XIIIth International Conference on the Use of Computers in Radiation Therapy (XIIIth
ICCR). W. Schlegel and T. Bortfeld (eds.). Heidelberg: Springer-Verlag, pp.120 – 122, 2000.
Rogers, D. W. O., B. Walters, and I. Kawrakow. BEAMnrc Users Manual. NRC Report PIRS-
509(a)(rev I), 2005.
Rogers, D. W. O., B. A. Faddegon, G. X. Ding, C.-M. Ma, J. Wei, and T. R. Mackie. (1995).
“BEAM: A Monte Carlo code to simulate radiotherapy treatment units.” Med Phys
22:503–524.
Salvat, F., J. M. Fernandez-Varea, J. Baro, and J. Sempau. PENELOPE, An Algorithm and
Computer Code for Monte Carlo Simulation of Electron-Photon Showers, University of
Barcelona Report, 1996.
20 Daryoush Sheikh-Bagheri et al.
21. Schach von Wittenau, A. E., L. J. Cox, P. M. Bergstrom, Jr., W. P. Chandler, C. L. Hartmann
Siantar, and R. Mohan. (1999). “Correlated histogram representation of Monte Carlo derived
medical accelerator photon-output phase space.” Med Phys 26(7):1196–1211.
Seltzer, S. M. “An Overview of ETRAN Monte Carlo Methods” in Monte Carlo Transport of
Electrons and Photons. T. M. Jenkins, W. R. Nelson, A. Rindi, A. E. Nahum, and D. W. O.
Rogers (eds.). New York: Plenum Press, pp. 153–182, 1988.
Seltzer, S. M. (1991). “Electron-photon Monte Carlo calculations: The ETRAN code.” Int J Appl
Radiat Isotopes 42:917–941.
Sempau, J., S. J. Wilderman, and A. F. Bielajew. (2000). “DPM, a fast, accurate Monte Carlo
code optimized for photon and electron radiotherapy treatment planning dose calculations.”
Phys Med Biol 45:2263–2291.
Sempau, J., A. Sánchez-Reyes, F. Salvat, H. Oulad ben Tahar, S. B. Jiang, and J. M. Fernan-
dez-Varea. (2001). “Monte Carlo simulation of electron beams from an accelerator head
using PENELOPE.” Phys Med Biol 46:1163–1186.
Siebers, J.V., P. J. Keall, J. Kim, and R. Mohan. “Performance Benchmarks of the MCV Monte
Carlo System” in XIIIth International Conference on the Use of Computers in Radiation
Therapy (XIIIth ICCR). W. Schlegel and T. Bortfeld (eds.). Heidelberg: Springer-Verlag, pp.
129–131, 2000.
Siebers, J., P. Keall, and I. Kawrakow. “Monte Carlo Dose Calculations for External Beam Radi-
ation Therapy” in The Modern Technology of Radiation Oncology. Volume 2. J. Van Dyk.
Madison, WI: Medical Physics Publishing, pp. 91–130, 2005.
Sheikh-Bagheri, D. Monte Carlo Study of Photon Beams from Medical Linear Accelerators;
Optimization, Benchmark and Spectra. Ph.D. Thesis, Carleton University, Ottawa, 1999.
Sheikh-Bagheri, D., and D. W. O. Rogers. (2002a). “Sensitivity of megavoltage photon beam
Monte Carlo simulations to electron beam and other parameters.” Med Phys 29(3):379–390.
Sheikh-Bagheri, D., and D. W. O. Rogers. (2002b). “Monte Carlo calculation of nine mega-
voltage photon beam spectra using the BEAM code.” Med Phys 29(3):391–402.
Sheikh-Bagheri, D., D. W. O. Rogers, C. K. Ross, and J. P. Seuntjens. (2000). “Comparison of
measured and Monte Carlo calculated dose distributions from the NRC linac.” Med Phys
27(10):2256–2266.
Walters, B. R. B., and D. W. O. Rogers. DOSXYZnrc Users Manual. NRC Report PIRS-794(rev
B), 2004.
Walters, B. R. B., I. Kawrakow, and D. W. O. Rogers. (2002). “History by history statistical esti-
mators in the BEAM code system.” Med Phys 29:2745–2752.
Walters, B. R. B., I. Kawrakow, and D. W. O. Rogers. DOSXYZnrc Users Manual, NRC Report
PIRS 794 (rev B), 2005.
Wang, L., C.-S. Chui, and M. Lovelock. (1998). “A patient-specific Monte Carlo dose calcula-
tion method for photon beams.” Med Phys 25:867–878.
MC Simulations: Efficiency Improvement/Statistical Considerations 21
