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Applied Radiation and Isotopes 68 (2010) 536–541

Contents lists available at ScienceDirect

Applied Radiation and Isotopes

0969-80

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/apradiso

Monte Carlo methods and techniques status and prospects forfuture evolution

Pedro Vaz

Instituto Tecnologico e Nuclear, Estrada Nacional 10, P-2686-953 Sacavem, Portugal

a r t i c l e i n f o

Keywords:

Monte Carlo

Simulation

Medical

Nuclear energy

43/$ - see front matter & 2009 Elsevier Ltd. A

016/j.apradiso.2009.10.020

ail address: pedrovaz@itn.pt

a b s t r a c t

Over the last two decades, the utilization of Monte Carlo methods and techniques for modeling and

simulating increasingly complex and sophisticated systems has grown at a sustained pace. This paper

focuses on the applications of Monte Carlo simulation methods and techniques in the medical and

nuclear energy fields, two of the sectors that impose stringent computational requirements. The

prospects for the future evolution of Monte Carlo methods and techniques are addressed. Specific

needs, features and capabilities are discussed.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Over the last two decades, a sustained growth in the utilizationof Monte Carlo methods and techniques for modeling andsimulating increasingly complex and sophisticated systemswas observed. This evolution was made possible by the adventof powerful processors and computer architectures that allowedthe accurate specification of the systems’ geometries andmaterials and of the detailed characteristics of radiation sources(particle beams, radioactive sources and radiation fields). Inparallel, an unprecedented effort was made to collect anddisseminate existing cross-section data sets as well as to performnew measurements of cross-section data for energy ranges,particle types and reactions of interest for different applications.These efforts, combined with major international exercises tovalidate and benchmark the predictions of the theoretical physicsmodels used by the available Monte Carlo simulation tools,allowed a significant improvement in the accuracy of thesimulation of the underlying physics processes.

In recent years, a boom was observed in the medical andindustrial applications of ionizing radiations, in emerging andinnovative technological applications, in the space- and in thesecurity-related applications, among others. The renewed interestin the utilization of nuclear power as a sustainable, safe,proliferation resistant and economically competitive source ofenergy allowing to solve the rapidly increasing energy demands ofmankind, triggered significant efforts in the design of newgenerations of reactors, exploiting new technologies, featuringhigher efficiency, using advanced fuels and exhibiting improvedsafety. The detailed description of these reactor designs and

ll rights reserved.

operation, namely the neutronics calculations, the safetysequences, the transient behaviors and the burnup and kineticscalculations do impose stringent computational requirementsthat are being addressed by the available Monte Carlo codes.

This paper focuses on the applications of Monte Carlosimulation methods and techniques in the medical and nuclearenergy fields, two of the sectors that impose stringent computa-tional requirements.

Firstly, the vision and the basic components of Monte Carlosimulations are discussed, together with a concise presentation ofthe Monte Carlo approach to solve the Boltzmann neutrontransport equation. Afterwards, Monte Carlo and deterministicmethods used to perform radiation physics and perform radiationtransport simulation are compared. Following, major challengesposed to the utilization of Monte Carlo codes by the medicalapplications of radiation and by advanced nuclear energy systemsare discussed. Finally, the prospects for the future evolution ofMonte Carlo methods and techniques are addressed. Specificneeds, features and capabilities are discussed.

2. The vision

The vision for Monte Carlo simulations consists in performingdetailed three-dimensional, time-dependent, neutral andcharged-particle transport calculations both efficiently and accu-rately.

The main components of the Monte Carlo method applied toperform radiation physics and particle transport simulation formultiple applications are:

�

Probability distribution functions (pdfs)—the physical (or mathematical)system must be described by a set of pdfs.

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P. Vaz / Applied Radiation and Isotopes 68 (2010) 536–541 537

�

Q ð~r

Random number generators—a source of random numbersuniformly distributed on the interval [0,1] must be avail-able.

� Sampling rules—a (set of) prescription(s) for sampling from

the specified pdfs must be given.

� Scoring (or tallying)—the outcomes must be accumulated into

overall tallies or scores for the quantities of interest.

� Error estimation—assessment of the statistical error (variance)

as a function of the number of trials and other quantities.

� Variance reduction techniques—methods for reducing the

variance in the estimated solution to reduce the computationaltime for Monte Carlo simulation.

� Parallelization and vectorization—algorithms to allow Monte

Carlo methods to be implemented efficiently on advancedcomputer architectures.

; E; ~OÞ ¼

Sð~r ; E; ~OÞ 3Fixed source

Sð~r ; E; ~OÞþRCð~r ; E0~O 0; ÞFðE0; ~O 0-E; ~OjrÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

fission ‘‘operator’’

dE0d2O0 3Fixed sourceþ fission

1

k

ZCð~r ; E0; ~O 0ÞFðE0; ~O 0-E; ~OjrÞdE0d2O0 3 Eigen value

8>>>>>>><>>>>>>>:

As noticed by James (1968), Monte Carlo problems are essentiallyintegrations. As pointed out by Lux and Koblinger (1991)

the idea of estimating an integral over many-dimensionalspace by evaluating the function at one random point in thespace is far-fetched.

These considerations emphasize the importance ofunderstanding evaluation of integrals using the Monte Carlomethod.

Indeed, if r1, r2,y,rM are selected random numbers, F(r1,r2,y,rM) is an unbiased estimator of

I¼

Z 1

0dx1

Z 1

0dx2

Z 1

0dx3 � � �

Z 1

0dxMFðx1; x2; . . . ; xMÞ

How to compute the M-dimensional integral I?

Fork¼ 1; . . . ;N:

Form¼ 1; . . . ;M : chose xðkÞm randomly in ðam;bmÞ

Then

I� ðb1 � a1Þðb2 � a2Þðb3 � a3Þ � � � ðbM � aMÞ1

N

XN

k ¼ 1

FðxðkÞ1 ; xðkÞ2 ; . . . ; xðkÞM Þ

In a simplified one-dimensional approach, considering thefunction g(x), integrable, piecewise continuous and everywherefinite in the interval [a,b]:

(a)

Generate N numbers xi randomly with a uniform probabilitydensity function in [a, b].

(b)

For each generated random number xi, compute g(xi).

Then, for large N, the following relation holds:

1

N

XN

i ¼ 1

gðxiÞ

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}Monte Carlo estimator

of the integral

�!N-1

1

b� a

Z b

agðxÞdx

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}expectation of the

function g

Meaning that for large N, the sum is an estimator of the integral inthe right side of the equation.

3. Solving the neutron transport equation using the MonteCarlo approach

A general integral form of the time-independent, linearBoltzman transport equation in terms of the particle collisiondensities, collision and transition kernels formalism is (Lux andKoblinger 1991; Brown, 2005):

Cð~r ; E; ~OÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}particle collision

density

¼

Z ZCð~r 0; E0;O0Þ � CðE0; ~O 0-E;Oj~r 0Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

collision kernel

dE0 d2O024

þQ ð~r 0; E; ~OÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}source term

35 � Tð~r 0-~r jE;OÞ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

transition kernel

d3r0

the source term being defined as

Defining the transport kernel K (P0,P) as

KðP0; PÞ � Kð~r 0; E0;O0-~r ; E;OÞ ¼ CðE0;O0-E;Oj~r 0Þ � Tð~r 0-~r jE;OÞ

A history (trajectory) can then be understood as a sequence oftransition between states (P0, P1, P2, P3,y) as shown in the Fig. 1.At each collision point Pi, the change of energy and directionsoccurs.

Then, the transport equation can recursively be written as

CiðPÞ ¼

Z Z Z� � �

ZC0ðP0ÞKðP0-P1ÞKðP1-P2Þ � � �KðPi�1-PÞdP0 dP1 dP2 � � �dPi�1

And the integral can be solved using the methodologypreviously described for solving an M-dimensional integral.

4. Monte Carlo vs deterministic

Monte Carlo methods applied to radiation transport simulationhave been compared to the deterministic methods that can alsobe used to solve the transport equations. Table 1 extracted from(Haghighat, 2006) succinctly summarizes in a qualitative way themain differences, advantages and drawbacks of both methods.

5. Medical applications

During the last years a substantially increase was observed inthe medical applications of ionizing radiation, both for diagnosticand therapy purposes. This trend was also due to the availabilityof new technologies, equipments and methodologies, namely inComputer Tomography but also in radiation therapy. Thesophistication of the medical treatments using ionizing radiation,namely in radiation therapy, has benefited from the dissemina-tion of the utilization of Monte Carlo methods, imposing at thesame time stringent requirements in the needed features,computing time, algorithms.

Analytical calculations for the transport of the radiationthrough media can be performed only in very simple geometriesand under severe approximations. Monte Carlo method, which isbased on the first principles, provides the only practical way ofperforming accurate calculations of 3-D dose distributions fromparticle interactions in a complex target such as the human body.

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P0

P1

P2

P3

P4

P5

P...

...

Pi−1

Fig. 1. Pictorial way of conceiving a particle history (trajectory) as a sequence of collisions (occurring at the points Pi) followed by transition to the next collision point Pi + 1

Fig. 2. Past evolution, from commercial treatment planning systems to the currently available Monte Carlo treatment planning systems, and the foreseeable evolution,

anticipated by experts, towards real time treatment planning systems.

Table 1Comparison between Monte Carlo and deterministic methods and codes used to

perform the simulation of radiation transport (from Ref. Haghighat, 2006).

Item Deterministic Monte Carlo

Geometry Discrete/approximate ‘‘Exact’’

Energy treatment (x-

section)

Discrete (multigroup) ‘‘Exact’’

Direction Discrete/truncated

series

‘‘Exact’’

Input preparation Difficult ‘‘Simple’’

Computer memory Large ‘‘Small’’

Computer time Small Large

Numerical issues Convergence Statistical

uncertainty

Amount of information Large Limited

Parallel computing Complex Trivial

P. Vaz / Applied Radiation and Isotopes 68 (2010) 536–541538

The advantages of using Monte Carlo methods for radiationtherapy result from the enhanced capabilities in the description ofthe geometry and anatomy of organs and tissues (as well as fromthe geometries of the irradiation fields), allowing a more accuratedose assessment in the tumor region and the healthy adjacenttissues and a more rigorous conformation of the dose delivered tothe tumor.

6. Geometries and modeling—phantoms

A significant evolution occurred from the analytical (e.g. MIRD-based) mathematical phantoms, used since the 1970s and1980s—namely the ADAM and EVA adult male and femalephantoms (Kramer et al., 1982)—to the currently used voxelphantoms, mostly developed using medical imaging techniques(CT and MRI). In recent years significant efforts were undertakento develop image-based whole body reference voxel phantoms(VIP Man Xu et al., 2000 and Zubal et al., 1994, the ICRP referencecomputational phantoms of the adult man and woman (ICRP,2002) based on the voxel models Golam and Laura) specifically forMonte Carlo simulations. A detailed set of phantoms developed bydifferent authors is available from CCHP.

7. Monte Carlo treatment planning (MCTP)

Another topic that could be addressed with the developmentsof state-of-art computational tools using Monte Carlo methods,was the required dose accuracy in treatment planning forradiotherapy. The importance of a more accurate assessment of

the doses in radiotherapy, was emphasized in Papanikolaou et al.(2004):

�

Section II.A.1 Slopes of dose-effect curves: ‘‘At this point, a 5%change in dose may result in a 10% to 20% change in tumorcontrol probability at a TCP of 50%. Similarly, a 5% change indose may result in a 20% to 30% impact on complication ratesin normal tissues.’’ � Section II.A.2 The level of dose differences that can be detected

clinically: ‘‘Thus it could be concluded that at least a 7%difference in dose delivered is manifested in the patient’sresponse to radiation treatment and is detectable clinically bya radiation oncologist.’’

The shortcomings and limitations of commercial treatmentplanning systems were overcome by the gradual utilization ofMonte Carlo treatment systems (MCTP). As noticed in Spezi(2008) in recent years, there has been a significant rise in theliterature in MCTP-related papers. The same author reports asurvey in the ISI Web of Science, from which it could beestablished that out of the almost 600 papers in the subject,about 85% have been published in the period 2000–2007.

The foreseeable evolution that experts anticipate for a fewyears, consisting of achieving in a not distant future real timetreatment planning, using powerful and sophisticated algorithmsimplementing Monte Carlo methods, is depicted in Fig. 2. Suchultimate goal, a very challenging one, will only be reachable usingMonte Carlo methods, by means of very powerful computers andprocessors, together with fast and accurate algorithms.

Indeed, in recent years, a number of commercial Monte Carlocodes for clinical dose calculation have emerged, being optimizedfor speed:

�

PEREGRINE (Hartman-Siantar et al.) � DPM/PENFAST (Sempau, Salvat et al.). � MCDOSE/MCSIM (Ma et al.). � VMC, XVMC, VMC++ (Kawrakow and Fippel). � MMC (Neuenschwander et al.). � Others not explicitly mentioned here.

As a very interesting and illustrative example, the ICCRbenchmark (Table 2) (Rogers and Mohan, 2000) consisting of (a)a speed test using a 30.5�30.5�30 cm phantom with 5 mm3

voxels filled either randomly with one of four materials water,aluminum, lung, and graphite or with water alone, 6 MV photonsfrom a point source at 100 cm SSD and collimated to 10�10 cm2

at the phantom surface and (b) an accuracy test using anheterogeneous phantom as defined in (a) with 5�5�2 cm3

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Table 2Summary of timings and accuracy results from the ICCR benchmark study

(extracted from Haghighat, 2006).

Monte Carlo code Time estimate

(min)

% Mean difference relative to

ESG4/PRESTA/DOSXYZ

ESG4/PRESTA/DOSXYZ 43 0, benchmark calculation

VMC++ 0.9 71

XVMC 1.1a 71

MCDOSE (modified

ESG4/PRESTA)

1.6 71

MCV (modified ESG4/

PRESTA)

22 71

DPM (modified DPM) 7.3b 71

MCNPX 60c Maximum difference of 8% at

Al/lung interface (on average

71% agreement)

PEREGRINE 43d 71

GEANT4 (4.6.1) 193e 71 for homogeneous water

and water/air interfaces

P. Vaz / Applied Radiation and Isotopes 68 (2010) 536–541 539

voxels, 18 MV photons from a point source at 100 cm SSD andcollimated to 1.5�1.5 cm2 at the phantom surface. The followingtable, extracted from Chetty et al. (2007) displays the results forthe time estimates to compute the benchmark exercise, bydifferent Monte Carlo programs, representative of the state-of-the-art.

8. Monte Carlo simulations of advanced nuclear energysystems

It is commonly accepted, considering the complexity, multi-disciplinary, leading edge and cross-cutting engineering, techno-logical and scientific topics involved, as well as financialconstrains, that any credible scientific methodology todayincludes theory, experiment, and simulation. Emerging andadvanced nuclear energy systems and nuclear reactor technolo-gies are a good example, imposing very specific requirements tothe state-of-art (Monte Carlo and other) simulation programs andtools, their features and capabilities. An excellent overview of thesimulation and modeling needs for developing advanced nuclearenergy systems can be found in Gehin (2006) and US Departmentof Energy (2006).

9. Nuclear energy ‘‘renaissance’’?

In recent years there was a regain of interest in advanced andinnovative nuclear energy and nuclear technology systems andassociated design, namely:

�

Generation III+ (evolutionary) and Generation IV nuclearreactors. � Actinide burners (GNEP concept, fast reactors). � ADS (accelerator driven systems, hybrid subcritical reactors for

the transmutation of high-level radioactive waste).

Some of these complex nuclear energy systems, consider:

�

Advanced fuel cycles. � Higher fuel burnups. � Actinide-enriched fuels. � Innovative core designs (e.g. compact cores). � Fast neutron fluxes (En41MeV). � High neutron fluxes (in excess of 1015 n cm�2 s�1).

The new generation of nuclear power reactors aims at featuringimproved safety, better economics, optimized reactor perfor-mance and enhanced proliferation-resistance.

10. Nuclear reactor simulations—‘‘New’’ requirementsand trends

From the previously stated, it results clear that Monte Carlosimulations of emerging nuclear energy systems require theutilization of cross-section data for actinides (isotopes of Pu, Am,Cm, Np, etc.) and must be performed to assess and study thestructural materials properties and behavior at high temperaturesand exposed to high neutron fluences. Detailed 3-D burnup,depletion, criticality and shielding calculations. These calculationsrequire the coupling of neutronics calculations with thermal-hydraulics, heat transfer and fluid dynamics computations,amongst others.

The trends can be identified as: (a) most design/operatingconstraints are based on thermal and material issues, notneutronics, (b) thermal-hydraulics approaches moving to im-proved simulation with CFD which will provide opportunities formore detailed coupled calculations, (c) fuel and material perfor-mance modeling & simulation will be key for new reactor typesand fuel cycles, (d) sensitivity analysis and uncertainty propaga-tion are mandatory and a key component of the simulationstudies, and (e) licensing approaches more heavily based onsimulation will be a challenge.

Last but not least, the need to perform full three-dimensional andtime-dependent calculations, will certainly require the utilization ofsupercomputing and parallel computing. As of 2008, the identifiedcomputational resources were in the Petaflop range.

11. The future of Monte Carlo simulations—conclusions

It can be anticipated that no pure Monte Carlo computerprograms can provide in the short- and medium-term future atimely and accurate answer and perform the detailed simulationof the very complex systems previously alluded to, both in thecontext of the medical applications of radiations and of theemerging nuclear energy systems. As such, hybrid computationalmethods resulting from merging Monte Carlo and deterministictechniques are necessary and must be implemented to enhancethe strengths and suppress the individual weaknesses of theindividual Monte Carlo and deterministic approaches. Despitesome exceptions, the development of hybrid computational toolsis however, still largely unexplored.

No successful and accurate Monte Carlo modeling of thesesystems can be performed if no accurate nuclear data is available,covering a broad energy range and for a set of nuclides of interest.Emphasis should be devoted to improving actinide data for fuelcycle applications, for multiple recycling of actinides withsignificant quantities of minor actinides (Np, Am, Cm) and withimpact in safety and reactor performance.

Powerful sensitivity and uncertainty analysis tools must beimplemented and made available in order to determine what isimportant, which cross sections to measure, to perform thefundamental assessment of uncertainties from basic data and todetermine the applicability of experiments.

As experiments are expensive—but are a requirement forvalidation—high fidelity calculations can fill (some) of this gap. Thisrequires careful assessment of predictive capability of the existingtools. Special emphasis must be devoted to the validation of methods,codes and tools. New reactor types and new fuels (e.g. actinides-enriched) will require experiments to validate methods and codes.

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Fig. 4. Qualitative representation of the relationship between the complexity of

the problem been modeled (expressed in units of voxel) and simulated and the

computer resources (in units of floating operations—flop) and the types of

processors and computing architectures (extracted from Kirk, 2008).

Fig. 5. Qualitative description of the time evolution and prospective views on the

challenging problems arising from the increasing sophistication of radiation

therapy and imaging techniques for medical applications (extracted from

Kirk, 2008).Fig. 3. Time evolution of the number of transistors in the different families and

types of processors, correlated with the typical number of lines of coding

(from Kirk, 2008 using data from the Intels Corporation website).

P. Vaz / Applied Radiation and Isotopes 68 (2010) 536–541540

One of the key issues to be solved is the speedup of the existingtools and methodologies. The desirable features of future MonteCarlo codes are:

�

Capability to perform full 3-D, time-dependent calculations. � Availability of ‘‘automated’’ variance reduction techniques. � Fully parallel computations should be easily activated (by

control card?).

� Input capabilities should include the ability to easily import

geometrical data (for Medical Physics related applications,imaging data for organs) and to perform CAD driven inputpreparation.

� Output capabilities should include new tallying features and

capabilities.

As displayed in Fig. 3 (Kirk, 2008), over the years there is acorrelation between the increasing computing power (throughthe increase of the number of transistors per processor) and thesize (measured in terms of the number of lines of programminglanguage) of the available computer programs. Whether anextrapolation based on the Moore’s law is legitimate, iscurrently a big unknown. The future of Monte Carlo simulationis thus also closely related to the future evolution of the

processors manufacturing technology (semiconductor andbeyond) and of computer architectures.

In Fig. 4 (from Kirk, 2008 using data from the Intels

corporation website) a qualitative representation of therelationship between the complexity of the problem beenmodeled (expressed in units of voxel) and simulated and thecomputer resources (in units of floating point operations—flop)and the types of processors and computing architectures.

Finally, a major challenge in the utilization of computational(and Monte Carlo) methods arises for the increasingly morecomplex description of the human anatomy for application inRadiation Therapy and Imaging is displayed in Fig. 5. Besides full3-D description using voxel models of the organs and tissues, thedescription of the motion seems to be the very challengingobjective, going beyond today’s supercomputers computationaland hardware capacities and calling for new and revolutionarymethodologies, algorithms and technologies.

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