relevance of accurate monte carlo modeling in nuclear medical

Relevance of accurate Monte Carlo modeling in nuclear medical imagingHabib Zaidia)

Division of Nuclear Medicine, Geneva University Hospital, CH-1211 Geneva, Switzerland

~Received 23 June 1998; accepted for publication 3 February 1999!

Monte Carlo techniques have become popular in different areas of medical physics with advantageof powerful computing systems. In particular, they have been extensively applied to simulateprocesses involving random behavior and to quantify physical parameters that are difficult or evenimpossible to calculate by experimental measurements. Recent nuclear medical imaging innova-tions such as single-photon emission computed tomography~SPECT!, positron emission tomogra-phy ~PET!, and multiple emission tomography~MET! are ideal for Monte Carlo modeling tech-niques because of the stochastic nature of radiation emission, transport and detection processes.Factors which have contributed to the wider use include improved models of radiation transportprocesses, the practicality of application with the development of acceleration schemes and theimproved speed of computers. In this paper we present a derivation and methodological basis forthis approach and critically review their areas of application in nuclear imaging. An overview ofexisting simulation programs is provided and illustrated with examples of some useful features ofsuch sophisticated tools in connection with common computing facilities and more powerfulmultiple-processor parallel processing systems. Current and future trends in the field are alsodiscussed. ©1999 American Association of Physicists in Medicine.@S0094-2405~99!01904-5#

Key words: Monte Carlo simulations, SPECT, PET, scatter, detectors

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I. INTRODUCTION AND OVERVIEW

Recent developments in nuclear medicine instrumentaand multiple-processor parallel processing systems haveated a need for a review of the opportunities for Monte Casimulation in nuclear medicine imaging. One of the aimsthe medical physicist involved in nuclear medical imagiresearch is to optimize the design of imaging systems animprove the quality and quantitative accuracy of recostructed images. Several factors affect the image qualitythe accuracy of the data obtained from a nuclear medicscan. These include the physical properties of the deteccollimator and gantry design, attenuation and scatter cpensation and reconstruction algorithms.1,2 Integrating im-provements in these with current tracers and sensitivespecific tracers under development will provide major advtages to the general nuclear medicine clinician and reseinvestigator~Fig. 1!. Mathematical modeling is necessary fthe assessment of various parameters in nuclear medicaaging systems since no analytical solution is possible wsolving the transport equation describing the interactionphotons with nonuniformly attenuating body structures acomplex detector geometries.

The Monte Carlo method is widely used for solving prolems involving statistical processes and is very usefulmedical physics due to the stochastic nature of radiaemission, transport and detection processes. The methvery useful for complex problems that cannot be modeledcomputer codes using deterministic methods or when expmental measurements may be impractical. The Monte Cmethod was named by Von Neumann3 because of the similarity of statistical simulation to games of chance, andcause the city in the Monaco principality was a center

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gambling and similar pursuits. Von Neumann, Ulam aFermi applied the method towards neutron diffusion prolems in the Manhattan Project at Los Alamos during WoWar II. But even at an early stage of these investigatiovon Neumann and Ulam refined this particular ‘‘Russiroulette’’ and ‘‘splitting’’ methods. However, the systematdevelopment of these ideas had to await the work of Kaand Harris in 1948.4 During the same year, Fermi, Metropolis and Ulam obtained Monte Carlo estimates for the eigvalues of the Schrodinger equation. Uses of Monte Camethods have been many and varied since that time.applications of the Monte Carlo method in medical physwere few before the review paper by Raeside.5 Since thattime, there has been an increasing number of applicationMonte Carlo techniques to problems in this field thanksthe several books6–9 and comprehensive review papers5,10–12

describing the principles of the Monte Carlo method andapplications in medical physics.

There has been an enormous increase and interest inuse of Monte Carlo techniques in all aspects of nuclearaging, including planar imaging,13 single-photon emissioncomputed tomography~SPECT!,14–18 positron emission to-mography ~PET!19–22 and multiple emission tomograph~MET!.23 However, due to computer limitations, the methhas not yet fully lived up to its potential. With the adventhigh-speed supercomputers, the field has received increattention, particularly with parallel algorithms which havmuch higher execution rates. Our main purpose in this pais to present a framework for applying Monte Carlo simutions for a wide range of problems in nuclear medical imaing. Emphasis is given to applications where photon andelectron transport in matter is simulated. Some compu

574/574/35/$15.00 © 1999 Am. Assoc. Phys. Med.

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575 Habib Zaidi: Relevance of accurate Monte Carlo modeling in nuclear medical imaging 575

tional aspects of the Monte Carlo method, mainly relatedrandom numbers, sampling and variance reduction arecussed. Basic aspects of nuclear medicine instrumentaare reviewed, followed by the presentation of potentialplications of Monte Carlo techniques in different areasnuclear imaging such as detector modeling and systemssign, image reconstruction and scatter correction techniqinternal dosimetry and pharmacokinetic modeling. Wideused Monte Carlo codes in connection with computing facties, vectorized and parallel implementations are describCurrent trends and some strategies for future developmethe field are also discussed.

II. THE MONTE CARLO METHOD: THEORY ANDCOMPUTATIONAL ISSUES

Numerical methods that are known as Monte Carlo meods can be loosely described as statistical simulation mods, where statistical simulation is defined in quite geneterms to be any method that utilizes sequences of rannumbers to perform the simulation. A detailed descriptionthe general principles of the Monte Carlo method is givena number of publications,5,11,24,25and will not be repeatedhere. Figure 2 illustrates the idea of Monte Carlo or stati

FIG. 1. Scientific and technical strategy for recording accurate functioimages. In bold, the parts where Monte Carlo simulation plays an imporrole ~adapted from an illustration by Professor Terry Jones, MRC!.

Medical Physics, Vol. 26, No. 4, April 1999

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cal simulation as applied to an imaging system. Assumthat the behavior of the imaging system can be describedprobability density functions~pdf’s!, then the Monte Carlosimulation can proceed by sampling from these pdf’s, whnecessitates a fast and effective way to generate rannumbers uniformly distributed on the interval@0, 1#. Photonemissions are generated within the phantom and are trported by sampling from pdf’s through the scattering mdium and detection system until they are absorbed or escthe volume of interest without hitting the crystal. The oucomes of these random samplings, or trials, must be acculated or tallied in an appropriate manner to produce thesired result, but the essential characteristic of Monte Carlthe use of random sampling techniques to arrive at a soluof the physical problem.

The major components of a Monte Carlo method abriefly described below. These components comprisefoundation of most Monte Carlo applications. The followinsections will explore them in more detail. An understandiof these major components will provide a sound foundatfor the developer to construct his own Monte Carlo methalthough the physics and mathematics of nuclear imagingwell beyond the scope of this paper. The primary compnents of a Monte Carlo simulation method include the flowing.

~i! Probability density functions~pdf’s!: the physical sys-tem must be described by a set of pdf’s.

~ii ! Random number generator: a source of random nubers uniformly distributed on the unit interval must bavailable.

~iii ! Sampling rule: a prescription for sampling from thspecified pdf’s.

~iv! Scoring: the outcomes must be accumulated into ovall tallies or scores for the quantities of interest.

~v! Error estimation: an estimate of the statistical er~variance! as a function of the number of trials another quantities must be determined.

~vi! Variance reduction techniques: methods for reducthe variance in the estimated solution to reducecomputational time for Monte Carlo simulation.

~vii ! Parallelization and vectorization algorithms to allo

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Monte Carlo methods to be implemented efficienon advanced computer architectures.

A. Random numbers generation

Computational studies requiring the generation of randnumbers are becoming increasingly common. All randnumber generators~RNG! are based upon specific matematical algorithms, which are repeatable. As such, the nbers are just pseudo-random. Here, for simplicity, we shterm them just ‘‘random’’ numbers. Formally, random is dfined as exhibiting ‘‘true’’ randomness, such as the timetween ‘‘tics’’ from a Geiger counter exposed to a radioactelement. Pseudo-random is defined as having the appeaof randomness, but nevertheless exhibiting a specific, repable pattern. Quasi-random is defined as filling the solutspace sequentially~in fact, these sequences are not atrandom, they are just comprehensive at a preset levegranularity!. Monte Carlo methods make extensive userandom numbers to control the decision making whenphysical event has a number of possible results. The RNalways one of the most crucial subroutines in any MoCarlo-based simulation code.24 A large number of generatorare readily available,26 and many of these are suitable for thimplementation on any computer system,27 since today thereis no significant distinction in floating point processing capbilities between a modern desktop and a mainframe cputer. A typical simulation uses from 107 to 1012 randomnumbers, and subtle correlations between these numcould lead to significant errors.28 The largest uncertaintieare typically due more to approximations arising in the fmulation of the model than those caused by the lack of rdomness in the RNG. Mathematically speaking, the sequeof random numbers used to effect a Monte Carlo moshould possess the following properties.29

~i! Uncorrelated sequences: the sequences of rannumbers should be serially uncorrelated. Most escially, n-tuples of random numbers should be indpendent of one another.

~ii ! Long period: ideally, the generator should not repepractically, the repetition should occur only after thgeneration of a very large set of random numbers

~iii ! Uniformity: the sequence of random numbers shobe uniform, and unbiased. That is, suppose we den-tuples m i

n5(ui 11 ,....,ui 1n) and divide then-dimensional unit hypercube into many equal suvolumes. A sequence is uniform if in the limit of ainfinite sequence all the sub-volumes have an eqnumber of occurrences of randomn-tuples.

~iv! Reproducibility: when debugging programs, it is neessary to repeat the calculations to find out howerrors occurred. The feature of reproducibility is alhelpful while porting the program to a different machine.

~v! Speed: It is of course desirable to generate the randnumbers fast.

~vi! Parallelization: The generator used on vector mchines should be vectorizable, with low overhead.


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massively parallel architectures, the processors shonot have to communicate among themselves, excperhaps during initialization.

Although powerful RNGs have been suggested includshift register, inversive congruentional, combinatorial a‘‘intelligent’’ methods such as those implemented in tMCNP code,30 the most commonly used generator is the linecongruential RNG~LCRNG!.31 Recently, Monte Carlo re-searchers have become aware of the advantages of laFibonacci series~LFRNG!. With extremely long periodsthey are generally faster than LCRNG and have excelstatistical properties.32 Those generators are briefly describbelow.

1. Linear congruential generators

The LCRNG has the form31

un115a~un1c!mod~m!, ~1!

where m is the modulus,a is the multiplier andc is theadditive constant or addend. The size of the modulus cstrains the period, and is usually chosen to be either prima power of 2.33 An important subset of LCRNG is obtaineby settingc50 in Eq. ~1!, which defines the multiplicativelinear congruential RNG~MLCRNG!. This generator~with ma power of 2 andc50) is the de facto standard included wiFORTRAN and C compilers.34 One of the biggest disadvantages to using a power of 2 modulus is that the least signcant bits of the integers produced by these LCRNGs hextremely short periods. For example,mnmod(2j ) will havea period of 2j .33 In particular, this means the least-significabit of the LCRNG will alternate between 0 and 1. Somcautions to the programmer are in order:~i! the bits ofmn

should not be partitioned to make several random numbsince the higher order bits are much more random thanlower order bits;~ii ! the power of 2 modulus in batches opowers of 2 should be avoided;~iii ! RNGs with large modu-lus are preferable to ones with small modulus. Not onlythe period longer, but the correlations are lower. In partilar, one should not use a 32 bit modulus for applicatiorequiring a high resolution in the random numbers. In spof this known defect of power of 2 LCRNGs, 48 bit multpliers ~and higher! have passed many very stringent randoness tests.

The initial seed should be set to a constant initial valsuch as a large prime number~it should be odd, as this willsatisfy period conditions for any modulus!. Otherwise, theinitial seed should be set to a ‘‘random’’ odd valuAnderson35 recommends setting the initial seed to the folowing integer:

u05 iyr11003„imonth211123~ iday21131

3„ihour1243~ imin1603 isec!…!…, ~2!

where the variables on the right-hand side are the intevalues of the date and time. Note that the year is 2 diglong, i.e., the domain ofiyr is @0–99#. However, it may bepreferable to introduce the maximum variation in the se

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into the least significant bits by using the second of tcentury, rather than the most significant bits. The followiequation is preferable:

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Generally, LCRNGs are best parallelized by parameterizthe iteration process, either through the multiplier or theditive constant. Based on the modulus, different paramezations have been tried.35

2. Lagged-Fibonacci generators

The lagged-Fibonacci series RNG~LFRNG! have the fol-lowing general form:26

un5un2 l ^ un2k mod~m!, l .k, ~4!

where ^ may be one of the following binary arithmetic operators1, 2,* , l and k are the lags andm is a power of2(m52P). In recent years the additive lagged-FibonaRNG ~ALFRNG! has become a popular generator for seas well as scaleable parallel machines36 because it is easy toimplement, it is cheap to compute and it does well on stdard statistical tests, especially when the lagk is sufficientlyhigh ~such ask51279). The maximal period of the ALFRNG is (2k21)2p21 and has 2(k21)(p21) different full-period cycles.37 Another advantage of the ALFRNG is thaone can implement these generators directly in a floatpoint to avoid the conversion from an integer to a floatinpoint that accompanies the use of other generators. Howesome care should be taken in the implementation to avfloating-point round-off errors.

Instead, the ALFRNG can be parameterized throughinitial values because of the tremendous number of differcycles. Different streams are produced by assigning estream a different cycle. An elegant seeding algorithm taccomplishes this is described by Mascagni.36 An interestingcousin of the ALFRNG is the multiplicative laggedFibonacci RNG ~MLFRNG!. While this generator hasmaximal-period (2k21)2p23, which is a quarter the lengthof the corresponding ALFRNG, it has empirical properticonsidered to be superior to ALFRNGs.26 Of interest forparallel computing is that a parameterization analogousthat of the ALFRNG exists for the MLFRNG. This lattealgorithm was used for generating uniformly distributed radom numbers on a parallel computer based on the MIMprinciple.38 The sequence of 24 bit random numbers haperiod of about 2144 and has passed stringent statistical tefor randomness and independence.32

B. Photon transport

For radiation transport problems, the computatiomodel includes geometry and material specifications.39 Ev-ery computer code contains a database of experimentallytained quantities, known as cross-sections, that determineprobability of a particle interacting with the medium througwhich it is transported. Every cross-section is peculiar to


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type and energy of the incident particle and to the kindinteraction it undergoes. These partial cross-sectionssummed to form the total cross-section; the ratio of the ptial cross-section to the total cross-section gives the probaity of this particular interaction occurring. Cross-section dafor the interaction types of interest must be supplied for ematerial present. The model also consists of algorithms uto compute the result of interactions~changes in particle energy, direction, etc.! based on the physical principles thdescribe the interaction of radiation with matter and tcross-section data provided. Therefore, it is extremely imptant to use an accurate transport model as the Monte Cresult is only as valid as the data supplied.

When a photon~having an energy below 1 MeV! passesthrough matter, any of the three interaction processes~pho-toelectric, incoherent scattering, coherent scattering! may oc-cur. The probability of a photon of a given energyE under-going absorption or scattering when traversing a layermaterialZ can be expressed quantitatively in terms of a lear attenuation coefficientm ~cm21! which is dependent onthe material’s density,r ~g.cm23!,

m5mphoto1m incoh1mcoh. ~5!

In the case of photoelectric absorption, the total photonergy is transferred to an atomic electron and the random wis terminated. In an incoherent photon interaction, a fractof the photon energy is transferred to the atomic electrThe direction of the scattered photon is changed to consthe total momentum of the interaction. The Klein–Nishiexpression for the differential cross-section per electronan incoherent interaction is used to sample the energythe polar angle of the incoherently scattered photon.40 Thecoherent scattering only results in a change in the direcof the photon since the momentum change is transferrethe whole atom. The kinetic energy loss of the photonnegligible. Coherent scattering of a photon could be genated using the random number composition and rejectechnique4 to sample the momentum of the scattered phoand the scattering angle according to the form factor disbution.

It is common to neglect coherent scattering in PET MoCarlo simulation of photon transport because of its low cotribution to the total cross-section at 511 keV. In the following examples, the relative importance of the various pcesses involved in the energy range of interest~below 1MeV! are considered for some compounds and mixtuused in nuclear medicine to justify some of the approximtions made in Monte Carlo codes. Figure 3 illustratesrelative strengths of the photon interactions versus energywater, cortical bone, sodium iodide~NaI! and bismuth ger-manate~BGO!, respectively. For water, a moderately low-Zmaterial, we note two distinct regions of single interactidominance: photoelectric below and incoherent abovekeV. The almost order of magnitude depression of the cohent contribution is some justification for the approximatiodiscussed. The coherent contribution to the total crosection is less than 1% for energies above 250 keV. Hoever, this contribution is in the order of 7% for high-Z ma-

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FIG. 3. Components of photon crosssections for different tissues (H2O andcortical bone! and detector materials~NaI and BGO! of interest in nuclearimaging, illustrating the relative con-tribution of each process.

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terials like BGO. Therefore, efforts should be made to trthe coherent scattering process adequately for detector mrials. In a recent investigation, photon cross-section libra~NIST, PHOTX!41,42 and parametrizations implementedsimulation packages~GEANT, PETSIM!43,44were comparedto the recent library provided by the Lawrence LivermoNational Laboratory~EPDL97!45 for energies from 1 keV to1 MeV for a few human tissues and detector materialsinterest in nuclear imaging.46 The cross-section data for mixtures and compounds are obtained from the equation:

m5r(i

wi~m/r! i , ~6!

where r is the density of the material,wi the fraction byweight of the i th atomic constituent, as specified in ICRReport 4447 and (m/r) i the mass attenuation coefficientDifferent photon cross-section libraries show quite larvariations as compared to the most recent EPDL97 data fiIt is recommended that Monte Carlo developers only usemost recent version of this library.46

A calculation of the distances between interactions inmedium are performed by sampling from the exponenattenuation distribution@Eq. ~10! below#. Different tech-niques have been proposed to improve the computa


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speed when sampling from the probability distributionThey are described in more detail in Secs. II D and II E.

C. Electron transport

In principle, electron transport should be included whsimulating the complete electromagnetic cascade~micro-scopic techniques!. However, the large number of interactions that may occur during electron slowing down makeunrealistic to simulate all the physical interactions~macro-scopic techniques!.10 Secondary electrons are generally asumed to deposit all their energy at the point of interactbecause of the low energies involved in nuclear mediciand therefore the short ranges of the electrons generatedtheir negligible bremsstrahlung production. Therefore, eltron transport has not received particular attention in nucimaging applications of the Monte Carlo method. Howeva number of investigators considered this effect mainlydosimetry calculations.48–50

Most existing electron transport algorithms are basedthe multiple collision models for scattering and energy loThe complexity of the techniques used in microscopic mels varies considerably, although a common approach ineglect bremsstrahlung interactions. Simple models

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based on the simulation of all the scattering events, calcuing the step length between consecutive collisions withelastic mean-free path. Energy losses are determinedthe Bethe theory of stopping power and an approximatincluded to account for the energy-loss straggling. Tmodel has been improved later by taking inelastic collisiointo account.10 Macroscopic techniques classify the physicinteractions of electrons into groups that provide an ovepicture of the physical process. Berger51 divided electrontransport algorithms into two broad classes~class I and classII ! distinguished by how they treat individual interactiothat lead to the energy losses of the primary electrons andproduction of bremsstrahlung photons and/or knock-on etrons. The condensed-history technique for electron transhas been reviewed51 and comparisons of class I with classalgorithms and of Goudsmit and Saunderson multipscattering theory have also been made.10,40 The Molieretheory contains a small-angle approximation52 and requires acertain minimum number of scattering events to occwhereas the Goudsmit and Saunderson theory is exactsingle-scattering cross-section. It has been shown, howethat the effects of the small-angle approximation cancompensated.51 An improved model for multiple scatterininto the voxel Monte Carlo algorithm comparable in accracy with the Parameter Reduced Electron-Step TransAlgorithm ~PRESTA!53 has been developed recently.54

A systematic error is introduced in low energy transpwhen the algorithm does not account for the change idiscrete interaction cross-section with energy.55 To over-come this problem, Ma56 developed an algorithm to accouproperly for the change in an electron discrete interactcross-section as a function of energy for low energy electtransport.

D. Analog sampling

Analog Monte Carlo attempts to simulate the full statisdevelopment of the electromagnetic cascade. If we assthat a large number of particle histories,N, are included in abatch, the individual batch estimates can be consideredrawn from a normal distribution. For a given calculatiothe estimated uncertainty is proportional to the inverse ofsquare root of the number of histories simulated. The eciency e of a Monte Carlo calculation can therefore be dfined as57

e51

s2T, ~7!

whereT is the calculation time to obtain a variance estims2. For largeN, e should be constant as long as the calcution technique remains the same.

As described earlier, the imaging system can be descrin terms of pdf’s. These pdf’s, supplemented by additiocomputations, describe the evolution of the overall systwhether in space, energy, time or even some higher dimsional phase space. The goal of the Monte Carlo methodsimulate the imaging system by random sampling from thpdf’s and by performing the necessary supplementary c


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putations needed to describe the system evolution. Insence, the physics and mathematics are replaced by ransampling of possible states from pdf’s that describe the stem. Thus, it is frequently necessary to sample some physevent, the probability of which is described by a known pExamples include the distance to the next interaction andenergy of a scattered photon. Letx be the physical quantityto be selected andf (x) the pdf. Among the properties of thpdf is that it is integrable and non-negative. Assume thatdomain of f (x) is the interval@xmin ,xmax# and that it is nor-malized to unit area. The cumulative distribution functioF(x) of the frequency functionf (x) gives the probabilitythat the random variablet is less or equal tox. It is definedas

F~x![probability~t<x!5Exmin

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A stochastic variable can be sampled by the use of unifordistributed random numbersR in the range@0–1# using oneof the techniques described below.

1. Direct method

This method can be used if the inverse of the cumulatdistribution functionF21(x) is easily obtainable. SinceF(x)is uniformly distributed in@0–1#, the sampled value ofxcould be obtained by substitutingF(x) in Eq. ~8! by a uni-form random numberR, that is, x5F21(R). A practicalexample of using this technique is the calculation of the dtance to the next interaction vertex. The inversion is notways possible, but in many important cases the inversreadily obtained.

2. Rejection method

Another method of performing this when it is too compcated to obtain the inverse of the distribution function isuse the rejection technique,4 which follows the followingsteps:~i! define a normalized functionf 8(x)5 f (x)/ f max(x),wheref max(x) is the maximum value off (x); ~ii ! sample twouniformly distributed random numbersR1 andR2; ~iii ! cal-culatex using the equationx5xmin1R1(xmax2xmin); and~iv!if R2 is less than or equal tof 8(x), thenx is accepted as asampled value; otherwise a new value ofx is sampled.

Over a large number of samples, this technique will yiea set of values ofx within the required distribution. It doeshowever, require two random numbers per trial and matrials may be required depending on the area under ofcurve of f (x). A typical example of using this techniquethe photon energy and scattering angle resulting from inherent scattering.

3. Mixed methods

When the previous two methods are impractical, tmixed method that combines the two may be used.57 Assumethat the pdf can be factored as follows:

f ~x!5h~x!•g~x!, ~9!

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g(x) producing g8(x) such that g8(x)<1 for x in@xmin ,xmax#; ~iii ! use the direct method to select anx usingh8(x) as the pdf;~iv! usex and apply the rejection methousingg8(x), i.e., choose a random numberR, if g8(x)<R,acceptx; otherwise go back to step~iii !.

E. Nonanalog sampling ‘‘variance reductiontechniques’’

A direct Monte Carlo simulation using true probabilifunctions may require an unacceptable long time to prodstatistically relevant results. Photons emission is isotropicdirectional parameters may be sampled uniformly withtheir individual ranges. Nuclear imaging systems have ageometrical efficiency because of the small solid anglefined by the collimator and/or the small axial apertuTherefore, the calculation would be very ineffective in termof required computing time.58 It is thus desirable to bias thsampling ~nonanalog sampling! by introducing differenttypes of importance sampling and other variance reductechniques to improve the computational efficiency ofMonte Carlo method.59 The results obtained by nonanalosimulation are, however, biased by the variance reductechnique and a correction for this is required. A partihistory weight,W, is introduced, which describes the proability of the particle following the current path. This weigis calculated for each particle history, and used in the calation of the results. If an event occurs, the weightW isadded to the counter rather than incrementing the counteone unit. Bielajew and Rogers57 divided variance reductiontechniques in three categories: those that concern phtransport only, those that concern electron transport oand other more general methods. The most useful techniare described below.

1. Photon-specific methods

Interaction forcing.In an analog Monte Carlo simulationphotons are tracked through the object until they eithercape the object, are absorbed or their energy drops beloselected threshold. The probability function for a photonteraction is given by

p~x!5me2mx. ~10!

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d52log~12R!

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Since the maximum distancedmax, the photon travels beforeinteraction is infinite, and the number of photon mean frpaths across the geometry in any practical situation is finthere is a large probability that photons leave the geomof interest without interacting. To increase the statisticalcuracy in the imparted energy calculation, we force the ptons to interact by assigningdmax a finite distance, e.g., thethickness of the detector being simulated.57 A true distrib-uted photon pathlengthd within dmax can be sampled fromthe equation

d521

mln~12R@12e2mdmax# !. ~13!

The photon’s weight must be multiplied by the interactiprobability,

Wn115Wn@12e2mdmax#. ~14!

In emission computed tomography, the photon is allowedinteract through coherent or incoherent interactions owithin the phantom since photoabsorption does not contute to energy imparted in the crystal. The weight is thmultiplied by the probability for the photon being scattere

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wherem incoh andmcoh are the cross-section data for incoheent and coherent scattering, respectively, andm is the totallinear attenuation coefficient.

Stratification.Stratification refers to the process of detemining the frequencies with which the various regionsstate space are used to start a particle.60 The solid angle ofacceptance of the detector array,Vmax, is small due to col-limation and to the size of the detector array itself. Thresults in significant computational inefficiencies with analMonte Carlo simulation, because only a few percent ofphotons generated and tracked will actually be detected.goal of stratification is to simulate only photons that aemitted in directions within the solid angle, which cancalculated from the maximum acceptance angle,umax,which, in turn, can be estimated from the dimensions ofphantom and the detection system. The solid angle doeschange in magnitude when simulating source locationscenter. The photon escaping from the phantom is eithermary or scattered. If the photon happens to be a primphoton, its direction within the solid angle could be sampfrom

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efficiency of the calculation for those regions. To implemethis method, the distance to the next interaction in numbemean free paths,dl , should be sampled from57

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Russian roulette and splitting are often used together wthe exponential transform although they are still effectwhen used independently. In Russian roulette, a randnumber is selected and compared to a threshold,l. If therandom number turns out to be smaller thanl, the particle isallowed to survive but the weight should be updated accoingly, Wn115Wn /l. In particle splitting, a particle comingfrom a region of interest can be divided intoN particles, eachhaving a new weighting,Wn115Wn /N.

2. Electron-specific methods

Electron range rejection.A fundamental difference between the transport of photons and electrons in a condenhistory simulation code is that photons travel relatively lodistances before interacting while electron tracks are inrupted not only by geometrical boundaries but also by mtiple scattering ‘‘steps.’’ A large amount of simulation timis spent on checking boundaries and selecting deflecangles and so on. Electron range rejection means thatelectrons with their residual range smaller than the distato the nearest boundary or to the region of interest insimulation will be terminated to save computing time. Dferent methods have been suggested for electron range rtion. The reduced interrogation of geometry~RIG! methodcalculates the distance to the nearest boundary and comit to the maximum multiple-scattering step length. If thelectron cannot reach any of the boundaries during this sthe boundary checking routine will not be called and this wsave computing time. Another method called ‘‘disregawithin a zone’’ is usually used with RIG to further speedthe simulation. It consists of disregarding electrons whenergies are so low that they cannot reach the nearest boary. Those methods are, however, inefficient for simulatioinvolving curved surfaces,57 where the time required to caculate the distance to the closest boundary may be consable. An alternative way is to use a range-related ‘‘reg


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rejection’’ technique. In this method, different energy cuoffs are chosen for the regions surrounding the region whenergy deposition is to be scored, each energy cut-off bechosen according to the distance to the nearest boundathe region of interest.

Parameter reduced electron step.This algorithm allowsus to use small electron steps in the vicinity of interfaces aboundaries and large steps elsewhere.53 Its components arethe following: a path-length correction algorithm whichbased on the multiple scattering theory of Moliere and whtakes into account the differences between the straight plength and the total curved pathlength for each electron sa lateral correlation algorithm which takes into account leral transport; and a boundary crossing algorithm whichsures that electrons are transported accurately in the vicof interfaces. The algorithm has been implemented inEGS4 code system and proved that substantial savingscomputing time may be realized when using this method

3. General methods

Correlated sampling.The correlated sampling techniqucan be used in the transport of both photons and electronis especially effective for calculating ratios or differencestwo quantities which are nearly equal. The basic idea is tthe simulations of the geometries of interest are keptclosely correlated as possible so that most of the statisfluctuations will cancel in the ratios and differences. The rdifference between the two geometries will be betterflected in the ratios and the differences obtained. The calational uncertainties in the ratios and the differencestained with correlated sampling are, in general, smaller tthose obtained from uncorrelated simulations.

There are several ways of doing correlated samplingradiation transport. In coupled photon–electron transporsimple method has been used in which random number sof the particle histories, for which a primary particle or anof the secondaries has deposited energy in the region oterest for one geometry, is stored and used for the simtions of the alternative geometry.57 A new correlated sam-pling method for the transport of electrons and photonsbeen developed in which a main particle history is splitwhenever a particle meets the boundary of the region whthe medium differs between the two or more cases.61 Thisparticle is then followed separately for each case until it aall its descendants terminate. Holmes62 described a corre-lated sampling technique which forces histories to havesame energy, position, direction and random number seeincident on both a heterogeneous and homogeneous wphantom. This ensures that a history that has, by chatraveled through only water in the heterogeneous phanwill have the same path as it would have through the homgeneous phantom, resulting in a reduced variance wheratio of the heterogeneous dose to the homogeneous doformed.

Use of geometry symmetry.The use of some of the inherent symmetry of the geometry may realize a consideraincrease in efficiency. If both the source and target confi

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III. NUCLEAR MEDICAL IMAGING TECHNIQUES

Recent advances in detector design focus on enhasensitivity and spatial and temporal resolution, and onpossibility of using conventional photon and coincidencetection ~two back-to-back photons, each with an energy511 keV! simultaneously. In this section, we describe instmentation advances in nuclear medical imaging.

A. Planar gamma camera imaging

Gamma camera imaging requires the collimationgamma rays emitted by the radiopharmaceutical distribuwithin the body. Collimators are typically made of leadtungsten and are about 4 to 5 cm thick and 20 by 40 cma side. The collimator contains thousands of squares, roor hexagonal parallel channels through which gamma rare allowed to pass. Although quite heavy, these collimaare placed directly on top of a very delicate single crystaNaI~Tl!. Any gamma camera so equipped with a collimais called an Anger camera.63 Gamma rays traveling alongpath that coincides with one of the collimator channels wpass through the collimator unabsorbed and interact withNaI~Tl! crystal creating light. Behind the crystal, a gridlight sensitive photomultiplier tubes collect the light for prcessing. It is from an analysis of these light signals thatages are produced. Depending on the size of the Anger cera, whole organs such as the heart and liver can be imaLarge Anger cameras are capable of imaging the entire band are used, for example, for bone scans.

A typical Anger camera equipped with a low-energy climator detects roughly one in every ten thousand gammaphotons emitted by the source in the absence of attenuaThis number depends on the type of collimator used. Tsystem spatial resolution also depends on the type of cmator and the intrinsic resolution of the Anger camera.typical modern Anger camera has an intrinsic resolution oto 9 millimeters. Independent of the collimator, system relution cannot get any better than intrinsic resolution. Tsame ideas also apply to sensitivity: system sensitivityalways worse than intrinsic~crystal! sensitivity. A collimatorwith thousands of straight parallel lead channels is calleparallel-hole collimator, and has a geometric or collimaresolution that increases with the distance from the gamray source. The geometric sensitivity, however, is inversrelated to geometric resolution, which means improving climator resolution decreases collimator sensitivity, and vversa. High resolution and great sensitivity are two pamount goals of gamma camera imaging. Therefore, reseaers must always consider this trade-off when workingnew collimator designs. There have been several collimdesigns in the past fifteen years, which optimizedresolution/sensitivity inverse relation for their particuldesign.64 Converging hole collimators, for example, fa


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beam and cone-beam65 have been built to improve the tradeoff between resolution and sensitivity by increasing tamount of the Anger camera that is exposed to the radioclide source. This increases the number of counts, whimproves sensitivity. More modern collimator designs, suas half-cone-beam and astigmatic, have also been conceSensitivity has seen an overall improvement by the introdtion of multi-camera SPECT systems. A typical triplcamera SPECT system equipped with ultra-high resoluparallel-hole collimators can achieve a resolution of fromto 7 millimeters.2 Other types of collimators with only one oa few channels, called pinhole collimators, have beensigned to image small organs and human extremities, sucthe wrist and thyroid gland, in addition to research animsuch as rats.66,67

B. Single-photon emission computed tomography

SPECT has, in recent years, become one of the mtools for thein vivo localization of radiopharmaceuticals inuclear medicine studies. SPECT systems are now widavailable and important clinical areas of SPECT imagingclude cardiology, neurology, psychiatry and oncology.conjunction with new and existing radiopharmaceuticaquantitative SPECT may be used to noninvasively measblood-flow, metabolic function, receptor density and drdelivery. In oncology, it is important in radiation dosimetand treatment planning for internal radionuclide therapygeneral and radioimmunotherapy~RIT!, in particular.2

Transverse tomographic images can be reconstructedprojection data acquired at discrete angles around the obMany mathematical approaches have been used for imreconstruction in SPECT. Two broad categories haemerged, which we refer to as analytic and iterative alrithms. The common characteristic of analytic methodsthat they utilize exact formulas for the reconstructed imadensity. The most popular method is filtered backprojectwhere the acquired projection data are filtered with a rafilter before being backprojected. The iterative approachbased on the process of matching the measured projectiothe calculated projections. The calculated projections aretermined from an initial reconstruction and are comparedthe measured data. The difference between the two datais used to correct the calculated projections. This procedis repeated until some predefined error level has breached. Statistical reconstruction techniques such asmaximum-likelihood expectation-maximization~ML-EM !algorithm seek a source distribution which will maximize tML function relating the estimated and the measured protions.

The quantitative determination of the radioactivity contein tissues is required in both diagnostic and therapenuclear medicine. Planar scintillation camera imaging hbeen used to estimate activity in tumors and varioorgans.68 The drawback with this technique is, however, tlack of information regarding the variation of activity witdepth. The acquired images are, furthermore, distorted byactivity content in overlapping structures. In contra

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SPECT has great potential for the quantitation of activdistributions in vivo due to its three-dimensional imagincapability. There are, however, several factors that musconsidered in quantitative imaging. Some of these factorsthe system sensitivity and spatial resolution, dead-timepulse pile-up effects, the linear and angular sampling invals of the projections, the choice of reconstruction filter athe size of the object and attenuation and scatter.1 Since im-age quality in nuclear medicine is limited by statistics, tadministered dose and the imaging time are extremelyportant. In practice, the limited count statistics in most clical studies affect the accuracy and precision of quantitaSPECT. However, the two most significant effects arephoton attenuation in the object and the contribution inimages of events arising from photons scattered in the obThese effects limit the accuracy of quantitative measuments and result in decreased contrast and blurred edgthe reconstructed activity distribution in the image.2

C. Positron emission tomography

Measurement of the tissue concentration of a positremitting radionuclide is based on coincidence detectionthe two photons arising from positron annihilation. Following the administration of a positron-emitting radioisotopdetector arrays surrounding the patient detect the emerannihilation photons. After being sorted into parallel projetions, the lines of response~LORs! defined by the coinci-dence channels are used to reconstruct the three-dimens~3D! distribution of the positron-emitter tracer within the ptient. In two-dimensional~2D! PET, each 2D transverse setion of the tracer distribution is reconstructed independenof adjacent sections. In fully three-dimensional~3D! PET,the data are sorted into sets of LORs, where each separallel to a particular direction, and is therefore a 2D palel projection of the 3D tracer distribution. Coincidences acollected for each LOR and stored in a 2D array, or singram. In each sinogram, there is one row containingLORs for a particular azimuthal angle; each such row cosponds to a 1D parallel projection of the tracer distributiona different coordinate along the scanner axis. An evenregistered if both crystals detect an annihilation phowithin a coincidence time window of the order of 10 ndepending on the timing properties of the scintillator. A pof detectors is sensitive only to events occurring in the tujoining the two detectors, thereby registering direction infmation~electronic collimation!. Coincidence detection offersignificant advantages over single-photon detection: etronic collimation eliminates the need for physical collimtion, thereby significantly increasing sensitivity. Accuracorrections can be made for the self-absorption of photwithin the patient so that absolute measurements of tistracer concentration can be made.

While the physics of positron annihilation limits the sptial resolution to, at best, 2–3 mm, the statistical accuracrelated to the sensitivity of the detection system. In the ptwenty years, there has been a significant evolution in Pinstrumentation from a single ring of bismuth german


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~BGO! detectors with a spatial resolution of 15 mm,19 tomultiple rings of small BGO crystals offering a spatial reslution of 5 mm.69 The spatial resolution improvements habeen achieved through smaller crystals and the efficientof photomultipliers and position readout based on Anglogic. The tomograph design which has proved successfurecent years represents a compromise between maximsensitivity while keeping detector dead time and contamition from scattered and random coincidences at a reasonlevel. To achieve this performance, multi-ring tomograpincorporate collimators~or septa! between the detector ringswith coincidences acquired only within a ring or betweadjacent rings.70 Thus, in the interest of maximizing thsignal-to-noise ratio and quantitative accuracy, compatively little use has been made of electronic collimation, oof the main advantages of coincidence counting. Conquently, an increase in sensitivity by a factor of 4–5 has bachieved by removing the septa and acquiring coincidenbetween detectors in any two rings71 ~Fig. 4!. It is also foundthat tomographs without septa can be operated more etively with lower activity levels in the field-of-view.

A modern tomograph with inter-ring septa detects arecords only 0.5% of the photon pairs emitted from thetivity within the tomograph field-of-view. This increases

FIG. 4. ~a! Schematic representation of a volume-imaging multi-ring Pscanner.~b! A block detector consists of a set of crystals having cutsdifferent depths acting as light guides and segmenting the block into(838) detection elements in this example. The block is coupled to fphotomultiplier tubes at the back, and the crystal in which photoabsorpoccurs is identified by comparing the outputs of the four photomultiptubes.

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over 3% when the septa are removed.69 However, even if thedetector system is 100% efficient for the detection of anhilation photons, the angular acceptance of modern scanwould record only 4.5% of the coincidences. The sparesolution obtained with modern tomographs is about 5mm in all three directions. Most use detectors based5cm35cm blocks of BGO. Each BGO block is cut into 8 b8 individual detector cells and read out by four photomuplier tubes. Light sharing schemes are used to identifyactive detector cell. Energy resolution at 511 keV of suBGO detector blocks is decreased from an intrinsic valueabout 15% FWHM to around 23% up to 44%, dependingthe cell, because of scintillation light losses resulting frothe cuts applied to the detector block.

Compton scatter in the field-of-view is another effectfluencing sensitivity and represents more than 30% ofdata acquired with a 3D scanner.70 Increasingly sophisticatedscatter correction procedures are under investigation, parlarly those based on accurate scatter models, andsubtraction–convolution approaches.72,73 Monte Carlo meth-ods give further insight and might in themselves offer a psible correction procedure. The development of fully 3Dconstruction algorithms has been necessary in order toadvantage of the acquisition of PET data without septa.74–76

In the most widely used 3D filtered backprojection~FBP!algorithm of Kinahan and Rogers,75 unmeasured oblique projection data, not accessible within the finite axial extensof the scanner, are estimated by forward-projecting througlow-statistics image reconstructed by 2D-FBP from trasaxial projections. The completed 2D projections are threconstructed by the FBP technique: each 2D projectioconvolved with a 2D filter kernel, and then backprojected3D through the image volume.

D. Multiple emission tomography

In recent years, there has been an increased intereusing conventional SPECT scintillation cameras for PETaging, however, the count rate performance is a limiting ftor. A sandwich-like construction of two different crystaallows the simultaneous use of gamma and positron radharmaceuticals referred to as multiple emission tomogra~MET!.77 This may be implemented with solid-state photdiode readouts, which also allows electronically collimatcoincidence counting~Fig. 5!. The resultant images will provide finer imaging resolution~less than 5 mm!, better con-trast and a ten-fold improvement in coincidence sensitivwhen compared to what is currently available. Althoughphotodiode noise might be a major problem, this cansolved to some extent but with a significant increase in c

The performance of a detector block design which wohave high resolution and high count rate capabilities in bdetection modes was recently evaluated.23 The high lightoutput of LSO~approximately 5–6 times BGO! allows theconstruction of a detector block that would have similartrinsic resolution characteristics at 140 keV as a conventiohigh resolution BGO block detector at 511 keV. Howevthe intrinsic radioactivity of LSO prevents the use of th


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scintillator in a single-photon counting mode. YSO is a sctillator with higher light output than LSO but worse absortion characteristics than LSO. YSO and LSO could be cobined in a phoswich detector block, where YSO is placeda front layer and is used for low energy SPECT imaging aLSO in a second layer is used for PET imaging.23 Events inthe two detector materials can be separated by pulse sdiscrimination, since the decay times of the light in YSO aLSO are different~70 and 40 ns, respectively!.

IV. APPLICATIONS OF THE MONTE CARLOMETHOD IN NUCLEAR MEDICAL IMAGING

A. Detector modeling

Monte Carlo simulation of detector responses and eciencies is one of the areas which has received considerattention.5–10 The critical component of emission tomogrphy is the scintillation detector. Increased light per gamray interaction, faster rise and decay times, greater stoppower and improved energy resolution are the desired cacteristics. Table I summarizes these properties for selescintillators under development and currently in use. Iprovements in these characteristics enable detectors tdivided into smaller elements, thus increasing resolutionminimizing dead-time losses.

An early contribution to the field providing a detailed dscription of the techniques used was due to Zerby.78 Tabula-tions of the response of NaI~Tl! detectors were performebetween 100 keV and 20 MeV,79 and the simulations of in-cident photons above 300 keV impinging on cylindrical dtectors of different materials due to Rogers.80 Simulations ofNaI~Tl! detectors with different shapes and volumes bel300 keV have also been reported.81 A detailed investigationof energy responses of germanium detectors and the usMonte Carlo simulations to correct the measured spectrabeen performed by Chan82 and comparisons of efficienccalculations for BGO scintillators between Monte Carlo ameasurements reported.83 The detection efficiency of a highpressure, gas scintillation proportional chamber, designedmedical imaging in the 30–150 keV energy range, has binvestigated through measurement and Monte Carlo simtion with the aim to design an optimized detector for usespecialized nuclear medicine studies.84 An approximate ex-

FIG. 5. The possible detector design of a multiple emission tomogracamera. The detector blocks employ two separate crystals: one for siphoton emitters~yttrium oxyorthosilicate, YSO! and one for positron emit-ters ~lutenium oxyorthosilicate, LSO!. Rectangular photomultiplier tube~PMTs! are preferred because they reduce the dead spaces betweePMTs when compared to those of the circular ones.

37


TABLE I. Characteristics of scintillator crystals under development and currently used in nuclear medicine imaging systems.

Scintillator NaI~Tl! BGO BaF2 LSO GSO LuAP YAP

Formula NaI~Tl! Bi4Ge3O12 BaF2 Lu2SiO5:Ce Gd2SiO5:Ce LuAlO3:Ce YAlO3:CeDensity ~g/cc! 3.67 7.13 4.89 7.4 6.71 8.34 5.37Light yield ~%! 100 15–20 3–20 75 20–25 25–50 40Effective Z 51 75 53 66 60 65 34Decay constant~ns! 230 300 1/700 42 30–60 18 25Peak wavelength~nm! 410 480 195–220 420 440 370 370index of refraction 1.85 2.15 1.56 1.82 1.95 1.95 1.56Photofraction~%!a/b 17.3/7.7 41.5/88 18.7/78.6 32.5/85.9 25/82.3 30.6/85.1 4.5/48.Mean free path~cm!a/b 2.93/0.4 1.04/0.08 2.19/0.27 1.15/0.1 1.4/0.16 1.05/0.1 2.17/0.Hygroscopic Yes No No No No No No

aAt 511 keV.bAt 140 keV.

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pression for the count rate characteristics of Anger scintition cameras has been derived, and validated by Monte Csimulations85 while the EGS4 Monte Carlo code has beeused to evaluate the response of HgI2 crystal in terms ofefficiency, energy and space resolutions versus photonergy in the diagnostic energy range~20–100 keV!.86

Many detector modeling applications were developedthe PET field, including the pioneering work of Derenzo87

who simulated arrays of detectors of different materials asizes to study the effect of the inter-crystal septa and lateto optimize the optical coupling between BGO crystals aPMTs88 by taking into account the reflection and scatterialong the detection system. The search for an appropdetector for this imaging modality was conducted in a coparative study of several crystals including BGO, CsF aNaI~Tl!,89 BaF2 used in time-of-flight PET,90 and liquidXenon.91 Binkley92 modeled the impulse response of a PMfront-end amplifier, and constant fraction discriminatorevaluate the effects of front-end bandwidth and consfraction delay and fraction for timing-system optimizatioof BGO scintillation detectors.

The penetration of annihilation photons into the detecmaterial before interaction is a statistical process which leto significant displacement and anisotropy of the pospread function. Compensation for crystal penetration is tan important issue to recover the spatial resolution in PE93

Comanor94 investigated algorithms to identify and correct fdetector Compton scatter in hypothetical PET modules w333330 mmBGO crystals coupled to individual photosesors. The true crystal of first interaction was determinedthe simulation for eventual comparison with the crystal idetified by a given algorithm. They reported a misidentificatifraction of 12% if the detector has good energy and positresolution when using position of interaction to identify foward scatter.

Numerous strategies have been proposed for construcdetector modules that measure the depth of interac~DOI!, but most of them proved impractical to implementprovided insufficient DOI measurement resolution. Two iportant questions can be addressed through Monte Csimulation:~i! what fraction of events will be mis-identifiebecause of noise fluctuations in the photomultiplier tub


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~PMT’s! or photodiode array and~ii ! how will the DOI mea-surement resolution affect the reconstructed resolution oPET camera? The position-dependent light distributionbeen used to measure the 511 keV photon interaction ption in the crystal on an event by event basis to reduce raelongation.95 Different geometrical modifications were alssimulated, leading to a proposal of a 2.235330 mmBGOcrystal, for which a 2.2 mm FWHM light distribution is predicted, which should yield a PET detector module with Dmeasurement resolution of 3.6 mm FWHM. A test moduwith one 333330 mmBGO crystal, one 3 mm square PIphotodiode and one PMT operated at220 °C with an ampli-fier peaking time of 4ms, and a measured DOI resolution5 to 8 mm FWHM has been proposed by Moses.96 Simula-tions predicted that this virtually eliminates radial elongatiin a 60 cm diameter BGO tomograph. The performance osingle detector element must be extrapolated using MoCarlo simulations to predict the performance of a muelement module or a complete PET camera.

The Triumph PET group has developed a simulation tto model position encoding multicrystal detectors for PEthat treats the interactions of energetic photons in a scinttor, the geometry of the multi-crystal array, as well as tpropagation and detection of individual scintillatiophotons.97 Design studies of a whole-body PET tomograwith the capacity to correct for the parallax error inducedthe DOI of gamma-rays were also performed.98 The experi-mental energy, depth and transverse position resolutionBGO block detectors were used as main inputs to the silations to avoid extensive light transport in position encodblocks. An improved model for energy resolution which icludes the nonproportionality of the scintillation responseBGO and the statistical noise from photoelectron amplifition in the PMT’s was also proposed.99 Simulation studieshave also been carried out to investigate the feasibilityusing a triangular detection module for PET with neutnetworks to reconstruct the coordinates of the photon abstion point and thus recover the DOI information.100 Anotherexciting application is the use of a PET imaging systemmonitoring the dose delivered by proton and gamma-raydiotherapy beams.101 By measuring the amount and ratio o

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the beam-induced positron-emitting activity, the dose disbution and tissue composition may be determined.

B. Imaging systems and collimators design

Image modeling was employed by Schulz,102 who deviseda computer program simulating a rectilinear scanner whwas used to study the influence of different imaging procols on the detectability of lesions. Simulation of gammcamera imaging to assess qualitatively and quantitativelyimage formation process and interpretation103 and to assistdevelopment of collimators104 using deterministic methodand simplifying approximations have been developed mato improve speed of operation.

In gamma camera imaging, the choice of collimatorvolves a compromise between sensitivity and sparesolution.64,65 The proper choice of collimator is especialdifficult at the cut-off energy level of low-energy collimato~e.g.,123I; 159 keV! and in multiple tracer studies. The relationships between sensitivity, spatial resolution and sepenetration of a given set of collimators have tostudied.1,105 The physicist has to determine which of thavailable collimators provides superior image quality forgiven acquisition time.106 To that end, in addition to its quantitative clinical applications, Monte Carlo simulation may ba useful research tool for tasks such as evaluating collimdesign and optimizing gamma camera motion. In recyears, there is an increased use of specialized collimasuch as fan-beam,65 convergent-beam,64 concave,107 variablefocus ~cardiofocal! and long-bore collimators. The improvement in image quality results from the fact that the increin resolution is greater than the loss of sensitivity. The effof collimation in a Compton-scatter tissue densitomescanner has been studied in a detailed paper.108

Monte Carlo techniques were extensively used to anathe performance of new collimators design for planar gamcamera,109,110 SPECT111 and PET imaging.112,113 Practicalguidance could be offered for understanding trade-offsmust be considered for clinical imaging. Selective compasons among different collimators could also be presentedillustrative and teaching purposes. Approaches to the cmator optimization problem, as well as more sophistica‘‘task-dependent’’ treatments and important consideratifor collimators design have been performed.114 The well-known imaging performance parameters of parallel-hole climators could be compared with those of fan- and cobeam collimators106,115 which have enjoyed considerabsuccess in recent years, particularly for brain SPECT.duced noise and higher sensitivity was reported for cobeam collimators compared to other collimators having silar geometric resolutions. Webb109 proposed a rotating-slicollimator which collects one-dimensional projections frowhich the planar image may be reconstructed by the theof computed tomography. A spatial resolution of 6 mm adistance of 100 mm from the collimator with seven timessensitivity of a parallel-hole collimator was achieved. Timaging properties of optimally designed planar-concacollimators were evaluated by means of Monte Ca


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simulations.111 The authors showed that the image noise dtribution along the object radius became more uniform whthe curved collimator was used and that the spatial resoluof the lateral cortex when using the curved collimator wsignificantly improved due to improved radial resolutioMonte Carlo calculations were also used to aid in the devopment of a method for imaging therapeutic doses of131I byusing thick Pb sheets to the front face of a high-eneparallel-hole collimator.116

There has been renewed interest in pinhole collimationhigh resolution imaging of small organs such as the thyrsince it provides an improved spatial resolution and ancrease in sensitivity as the distance between the sourcethe pinhole aperture decreases.67 Wang117 simulated pointresponse functions for pinhole apertures with various apture span angle, hole size and materials. The point respowere parameterized using radially circularly symmetric twdimensional exponential functions which can be incorprated into image reconstruction algorithms that compensfor the penetration effect. The effect of pinhole aperturesign parameters on angle-dependent sensitivity for high relution pinhole imaging was also investigated using MonCarlo modeling.118 Simulated131I SPECT studies for uni-form cylinders showed that activity concentrations were uderestimated toward the outside of the cylinders whensin3 u rather than the correct sinx u sensitivity correction wasapplied in image reconstruction, wherex is a parameter anduis the angle of the incident ray with the surface of the dettor crystal.

In a similar way in the PET field, Monte Carlo techniquwere used to determine the effects of crystals with straiand pointed tips and septa on spatial resolution aefficiency,119 to compare the singles to true coincident everatios in well collimated single, multi-slice and open colmator 3D configurations,112 to evaluate tungsten inter-plansepta of different thicknesses and geometries113 and to assessthe effect of collimation on the scatter fraction.120

The design of SPECT and PET systems using the MoCarlo method has received considerable attention and a lnumber of applications were the result of suinvestigations.121,122 Bradshaw121 used this tool for the de-sign of a detector suitable for use in a SPECT cylindricashaped scintillation camera. Detection characteristics ofscintillator materials and the optical performance of sevegeometric configurations were studied. The design of protype systems that utilize solid-state detectors and low-noelectronics to achieve improved energy resolution were cried out using simulated SPECT projections of a simmyocardial perfusion phantom.123 The results showed thatFWHM energy resolution of 3–4 keV is sufficient to rendthe error due to scatter insignificant compared to the untainty due to photon statistics. Monte Carlo simulations haalso been performed to evaluate the design of collimadetectors used to measure125I or 131I in the thyroid gland.124

Two detector sizes were simulated for each radioisotopeactivity was placed in both the gland and the remainderthe body in varying amounts to assess the efficacy of comation. This study showed that a wide angle of accepta

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The Monte Carlo method has also been used in the deof single-slice19,122 and multi-slice PET scanners.70,71 AVME bus-based microcomputer system has been useimplement a model for simulation of the flux of gamma rain cylindrical PET detector systems.125 The program is ca-pable of tracing over one million photons per hour and hbeen used to explore some of the effects of ‘‘opening uplanar detector geometries into volumetric imageRogers126 compared some of the performance parametera tomograph based on large area NaI~Tl! detectors to similarparameters of conventional small crystal machines. Mich70

used theGEANT package from CERN43 to study the responsfunction and the scatter fraction in two PET scanners wand without inter-plane septa. The simulation of a larmulti-plane PET camera named HISPET127 and a planar im-aging system made of two matrices, each one consistin400 (232330 mm3) crystals of YAP:Ce128 using theEGS4

system have also been reported. Thompson129 investigatedthe effects of detector material and structure on PET sparesolution and efficiency in terms of the number of interations and tangential component of the mean square distbetween the centroid and the point of first interaction.

Several researchers used Monte Carlo simulation methto study potential designs of dedicated small animal posittomographs.130,131 An important conclusion drawn fromthese studies is that unlike human imaging where both ssitivity and spatial resolution limitations significantly affethe quantitative imaging performance of a tomograph,imaging performance of dedicated animal tomographs ismost solely based upon its spatial resolution limitations132

Recently, a conceptual design for a PET camera designeimage the human brain and small animals has bpresented.133 The authors performed a Monte Carlo simution to predict the spatial resolution for a single plane Pcamera with 3 mm LSO crystals. They concluded thatdetector modules must be able to measure the DOI onevent by event basis in order to eliminate radial elongatartifacts, and that such depth information can be incorporainto the reconstruction algorithm in an artifact free way wa simple rebinning method.

C. Image reconstruction algorithms

Monte Carlo simulations have been shown to be very uful for validation and comparative evaluation of image rconstruction techniques since it is possible to obtain a reence image to which reconstructed images shouldcompared. Three different algorithms for performing PEimage reconstruction have been compared using MoCarlo phantom simulations.134 The results demonstrate thimportance of developing a complete 3D reconstructiongorithm to deal with the increased gamma detection sangle and the increased scatter fraction that result wheninterslice septa are removed from a multi-ring tomograThe Eidolon Monte Carlo package135 was used to simulateprojection data of the cold rod phantom both with and wi


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out scatter simulation. Figure 6 shows transaxial slices ofphantom reconstructed using four analytic algorithms:reprojection algorithm~PROMIS!,75 the fast volume recon-struction algorithm~FAVOR!,76 the Fourier rebinning algo-rithm ~FORE!136 and the single-slice rebinning algorithm~SSRB!.74 Using simulated data, Hanson137 validated amethod of evaluating image recovery algorithms basedthe numerical computation of how well a specified visutask can be performed on the basis of the reconstructedages. Task performance was rated on the basis of the deability index derived from the area under the receiver opating characteristic curve. Three-dimensional phodetection kernels characterize the probabilities that photemitted by radioisotopes in different parts of the sourcegion will be detected at particular projection pixels of thprojection images.138 Smith139 used Monte Carlo modeling tostudy these kernels for the case of parallel-hole collimatoThe authors also proposed a reconstruction method using

FIG. 6. Reconstructions of Monte Carlo data sets of the Jaszczack’s coldphantom generated without~left! and with ~right! scatter simulation usingfrom top to bottom the PROMIS, FAVOR, FORE, and SSRB algorithmThe cold rod diameters are from top right counter clockwise: 4.8, 6.4,9.5, 11.1 and 12.7 mm. Approximately 25 Mcounts were recorded for btypes of simulations.

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3D kernels in which projection measurements in three acent planes are used simultaneously to estimate the soactivity of the center plane. The matrix equations for imareconstruction are solved using generalized matrix inverKing140 conducted a Monte Carlo study to investigate tartifacts caused by liver uptake in SPECT perfusion imagand to verify the hypothesis that the cardiac count chanare due to the inconsistencies in the projection data inpureconstruction. A correction of the causes of these incontencies before reconstruction, or including knowledge ofphysics underlying them in the reconstruction algorithwould virtually eliminate these artifacts.

Floyd141 evaluated convergence properties of the ML-Ealgorithm for SPECT image reconstruction as a functionPoisson noise, precision of the assumed system resolumodel and iteration number. It was also shown that lescontrasts and signal-to-noise ratios in ML-EM estimatesSPECT images can be improved by considering Compscattering when calculating the photon detection probabmatrix.142 Bayesian reconstruction methods introduce prinformation, often in the form of a spatial smoothness relarizer. More elaborate forms of smoothness constraints mbe used to extend the role of the prior beyond that ostabilizer in order to capture actual spatial information abthe object.143 In recent years, many investigators proposGibbs prior models to regularize images reconstructed fremission computed tomography data. Unfortunately, theperparameters used to specify Gibbs priors can greatly inence the degree of regularity imposed by such priors anda result, numerous procedures have been proposed tomate hyperparameter values from observed image dHigdon144 used recent results in Markov chain Monte Casampling to estimate the relative values of Gibbs partitfunctions. Using these values, sampling was performed fjoint posterior distributions on image scenes. This allowsa fully Bayesian procedure which does not fix the hyperrameters at some estimated or specified value, but enauncertainty about these values to be propagated throughestimated intensities.

Maximum a posteriori ~MAP! reconstruction has beeshown to have significant advantages over traditionalmethods in terms of noise performance, but these advantare highly dependent on the choice of the distribution usemodel the prior knowledge about the solution image. A MAapproach for iterative reconstruction based on a weighleast-squares conjugate gradient~WLS-CG! algorithm wasproposed and validated using simulated hot-sphere phanSPECT data and patient studies.145 The ill-posed nature oftomography leads to slow convergence for standard gradbased iterative approaches such as the steepest descentconjugate gradient algorithm. Chinn and Huang146 proposeda preconditioned conjugate gradient~PCG! iterative algo-rithm for WLS reconstruction in order to accelerate the covergence rate of iterative reconstruction. Using simulaPET data of the Hoffman brain phantom, the authors hshown that the convergence rate of PCG can reduce the nber of iterations of the standard conjugate gradient algori


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The search for unified reconstruction algorithms led todevelopment of inverse Monte Carlo~IMC! reconstructiontechniques.147 The concept of IMC was introduced in 1981an attempt to describe a numerical method for solving a cof inverse problems.148 The IMC method converts the inverse problem, through a noniterative simulation techniqinto a system of algebraic equations that can be solvedstandard analytical or numerical techniques. The princimerits of IMC are that, like direct Monte Carlo, the methocan be applied to complex and multivariable problems, avariance reduction procedures can be applied. The nonittive IMC is strongly related to the variance reduction tecnique in direct simulation called importance sampling whethe sampling process is altered by using random numbfrom a modified distribution. Floyd149 used IMC to performtomographic reconstruction for SPECT with simultaneocompensation for attenuation, scatter and distance-depencollimator resolution. A detection probability matrix iformed by Monte Carlo solution to the photon transpequation for SPECT acquisition from a unit source activityeach reconstruction source voxel. The measured projecvectorpj will equal the product of this detection probabilitmatrix Ai j with the unknown source distribution vectorsi :

@pj #5@Ai j #@si #. ~20!

The resulting large, nonsparse system of equationssolved for the source distribution using an iterative ML-Eestimator. The IMC technique proved to provide compention for the collimator effects in addition to providing higheresolution.150 It is worth noting that although the techniquwas developed for SPECT, it is also valid for other imagitechniques like PET and transmission CT.

The ability to theoretically model the propagation of phton noise through emission computed tomography recstruction algorithms is crucial in evaluating the reconstrucimage quality as a function of parameters of the algorithWilson151 used a Monte Carlo approach to study the noproperties of the ML-EM algorithm and to test the predtions of the theory. The ML-EM statistical properties wecalculated from sample averages of a large number of imawith different noise realizations. The agreement betweenmore exact form of the theoretical formulation and tMonte Carlo formulation was better than 10% in most caexamined, and for many situations the agreement was withe expected error of the Monte Carlo experiments. The samethodology was also followed to analyze a MAP-EM algrithm incorporating an independent gamma prior, and a ostep-late~OSL! version of a MAP-EM algorithm incorporating a multivariate Gaussian prior, for which familiasmoothing priors are special cases.152

D. Attenuation and scatter correction techniques

The presence of scatter and attenuation in the imagesits the accuracy of quantification of activity.1 With no cor-rections, the uncertainty could be as high as 50–1002

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Scatter does not produce major artifacts comparable totenuation but reduces image contrast by including a lofrequency blur in the image. The impact of scatter generdepends on the photon energy, camera energy resolutionenergy window settings, besides the object shape andsource distribution.153,154Many of these parameters are nostationary which implies a potential difficulty when develoing proper scatter and attenuation correction techniquHowever, correction for scatter remains essential, not ofor quantification, but also for lesion detection and imasegmentation.155 For the latter case, if the boundary of aactivity region is distorted by scatter events, then the acracy in the calculated volume will be affected.156 MonteCarlo calculations have been found to be powerful toolsquantify and correct for photon attenuation and scatteringnuclear medicine imaging since the user has the abilityseparate the detected photons into their components: primevents, scatter events, contribution of down-scatter eveetc. Monte Carlo modeling thus allows a detailed investition of the spatial and energy distribution of Compton scawhich would be difficult to perform using present expemental techniques, even with very good energy resoludetectors.157

In gamma camera imaging and SPECT, simulation pgrams have been used to obtain information on the diffeprocesses occurring within the phantom and the detecFor example, energy pulse-height distribution, point-sprfunction and the scatter fraction can be obtained.158 The scat-tered events in the energy–pulse-height distribution canseparated according to the number of scattering events inphantom~Fig. 7!. It is clearly shown that a significant number of scattered events will be accepted by the photopenergy window. The scatter fraction is of great importanfor quantitative estimation of the scattering contribution.159 Itis defined as the ratio between the number of scatteredtons and the total number of photons~scattered and unscatered!. The scatter fraction is generally measured by scning a line source placed at the center of a water-filcylinder. Line spread functions~LSFs! are generated and thscatter fraction determined by fitting the scatter tails ofLSFs to a mono-exponential function~Fig. 8!. The scatter


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fraction is calculated as scatter/total where total and scaare calculated as the integral of the LSF and the fit withindiameter of the field-of-view. Figure 9 compares scatter frtions for different source depths and energy window sicalculated with different Monte Carlo codes simulating sctillation camera characteristics.13–15Beekman160 developed afast analytic simulator of tomographic projection data takiinto account attenuation, distance-dependent detectorsponse and scatter based on an analytical point spreadtion ~PSF! model. Several simplifying approximations wealso adopted to improve the speed of operation; restrictiothe extent of the primary and scatter PSFs, coarse sampof the PSFs in the direction perpendicular to the camera fand use of a circularly symmetric scatter function.161

A study of the factors mostly responsible for spectral cotamination~overlapping of unscattered and scattered evethroughout the energy spectrum! including nuclear medicineimaging instrumentation itself has been performed.162

Frey163 generated scatter response functions~SRFs! usingMonte Carlo techniques and investigated the characteris

FIG. 8. Experimental determination of the scatter fraction by fitting tscatter tails to a monoexponential function. The scatter fraction is calculas the integral of the scatter tails~gray area! to the integral of the LSF,within the diameter of the FOV.

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of the scattered radiation by fitting the SRFs with fittinfunctions. The parameters of the fitting functions were stied as a function of source position in a water-filled cylindcal phantom with circular cross-section. A third-order ponomial for modeling the SRF and an approximately constfitting window was also proposed.164 SRFs were also simulated for inhomogeneous scattering media.165,166This modelhas been implemented in a projector–backprojector pairmakes iterative reconstruction based scatter compensfeasible.167

Ljungberg168 simulated both parallel and fan-beam tranmission imaging to study the effect of down-scatter fromemission99mTc radionuclide into the energy window fortransmission153Gd radionuclide. An investigation of the efects of scattered photons in gamma-ray transmission CTseveral types of data acquisition systems was also perforincluding a flood source and a parallel-hole collimator,collimated flood source and a parallel-hole collimator, a lsource and a symmetric fan-beam collimator and a comated line source and a symmetric fan-beam collimator169

The results showed that a fan-beam collimator andsource rejected most of the scattered collimated emitted ptons at the object side, and that almost all the scatteredtons could be rejected at the collimator on the detector sSpeller and Horrocks170 studied multiple scatter effects alower energies, including incident diagnostic x-ray specand obtained correction factors for clinical use in tissue dsitometry.

Much research and development has been concentratethe scatter compensation required for quantitative SPEC1,2

Floyd171 used Monte Carlo simulations to validate the baassumptions underlying the empirical implementationtheir scatter subtraction algorithm. Three scatter correctechniques for SPECT have been assessed and com

FIG. 9. A comparison of simulated scatter fractions for different soudepths and energy window sizes obtained with different Monte Carlo cosimulating scintillation camera characteristics.


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where scatter coefficients and parameters characteristieach technique have been calculated through Monte Csimulations and experimental measurements for varisource geometries.172 The Compton scatter correctiomethod based on factor analysis of dynamic structuresevaluated both on planar imaging and SPECT data usMonte Carlo simulations and real phantoms.173 A compari-son with the modified dual-window~DW! method was alsopresented. Ljungberg174 derived a method based on the combined use of 3D density information provided by computtomography to correct for attenuation and the applicationMonte Carlo simulated build-up factors to correct fbuild-up in the projection pixels. A similar method was alproposed for planar imaging.175 The effects of tissue-background activity, tumor location, patient size, uncertaiof energy windows and definition of the tumor region on taccuracy of quantification were investigated by calculatthe multiplier which yields correct activity for the volumeof-interest when using the DW method.176

A scatter correction method in which Monte Carlo simlated scatter line-spread functions~SLSF! for different depthand lateral positions has been developed.177 The uncorrectedreconstructed images are convolved with the SLSF and stracted from the projection data to yield scatter-correcprojections. The method was further validated using a clcally realistic, nonhomogeneous, computer phantom178

Naude179 studied the accuracy of the channel ratio scacorrection technique which is based on the assumptionthe ratio of the scatter components in the two windows~Hvalue! is constant and independent of the relative size ofscatter contribution. The results have shown that althothe true H value depends on both source size and deptthe source in the scattering medium, the channel ratio tenique can be applied successfully when an average H vis used. Welch180 developed a method based on the use otransmission map to define the inhomogeneous scatteringject for modeling the distribution of scattered events in emsion projection data. The probability of a photon being sctered through a given angle and being detected inemission energy window was approximated using a Gausfunction whose parameters were determined using MoCarlo generated parallel-beam SLSFs from a nonuniformattenuating phantom. A combined scatter and attenuacorrection that does not require a transmission scan wasproposed and validated using measured planar datasimulated SPECT for111In imaging.181

Hademenos182 applied a modified dual photopeak windo~DPW! scatter correction method to Monte Carlo simulat201Tl emission images. This method was also applied to tviews of an extended cardiac distribution within an anthpomorphic phantom, resulting in at least a six-fold improvment between the scatter estimate and the Monte Carlo slated true scatter. A simulation study of the triple enerwindow ~TEW! method was conducted in a multradionuclide (99mTc/201Tl) SPECT study.183 A good agree-ment between the activity distributions reconstructed frprimary photons and those from corrected data has bshown. A spill-down correction method was also propos

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for the 201Tl window image in simultaneous dual-isotop99mTc/201Tl SPECT imaging based on a single acquisitiinto three energy windows.184 Using Monte Carlo tech-niques, the fractional amount of99mTc and201Tl spill-downin the 201Tl window with respect to the total counts from thspill-down window, was calculated for simulated imagespoint sources at varying depths within a water-filled elliptictub phantom.

The DW and the convolution~CV! scatter correction techniques were compared using projection data, simulatedthe Monte Carlo method.185 The scatter distributions predicted by the CV technique were found to be consistenlower than those simulated by the Monte Carlo method inpart of the scatter distribution corresponding to the locatiof the sources while the DW technique gave lower estimaof the scatter distribution. Further comparisons of four scter correction methods: DW, DPW, TEW and CV were aperformed using simple phantoms and a clinically realissource distribution simulating brain imaging.186 The authorsconcluded that performing scatter correction is essentialaccurate quantification, and that all four methods yieldgood, but not perfect, scatter correction. Buvat187 comparednine scatter correction methods based on spectral analSimulations and physical phantom measurements wereused to compare the accuracy and noise properties oftransmission-dependent convolution subtraction andTEW scatter correction techniques.188 The TEW had theworst signal-to-noise ratio in the heart chamber of a simlated chest phantom.

In the PET imaging world, Compton scattering effectswater on profiles of activity have been simulated.189 Figure10~a! shows a comparison between measured and simul


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single energy spectra of the ECAT-953B PET scanner.energy resolution of 23% FWHM has been assumed, sithis is the typical value for BGO block detectors.135 An en-ergy pulse-height distribution obtained by simulation ofline source in the center of a water-filled cylindrical phantowhere scattered events have been separated accordingnumber of scatterings is also shown@Fig. 10~b!#. The accu-racy of experimental methodologies used for scatter fracand scatter pair spatial distribution determination were evaated using the Monte Carlo method.190 Figure 11 shows com-parisons between measured and simulated scatter fractioa function of the lower energy threshold.191 Barney192 devel-oped an analytical simulation for single and multiplscattered gamma rays in PET. The multiple-scatter moshowed good agreement with a Monte Carlo simulationtotal object scatter. The authors also proposed a scatterrection method which uses the analytical simulation andploits the inherent smoothness of the scatter distributionaccount for three-dimensional effects in scatter distributand object shapes. Scatter components in PET dividedprimaries, object scatter, gantry scatter and mixed scattertheir effects on the degradation of reconstructed images walso investigated.193 Quantification of those components forsmall animal PET prototype were also reported.194 A MonteCarlo study of the acceptance to scattered events in a dencoding large aperture camera made of position encoblocks modified to have DOI resolution through a variatiin the photopeak pulse height was performed.191 It was re-ported that the poorer discrimination of object scatters wdepth sensitive blocks does not lead to a dramatic increasthe scatter fraction.

Although several approaches have been proposed for s

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ter correction for 3D PET, six basic approaches have btaken to this correction: multi-energy window approachintegral transformation approaches, an approach relyingan auxiliary, septa extended scan, curve-fitting approacmodel-based approaches, and direct Monte Carlo techniqLevin195 developed a correction method that uses thereconstructed image volume as the source intensity distrtion for a photon-tracking Monte Carlo simulation. The htory of each annihilation photon’s interactions in the scatting medium is followed, and the sinograms for the scatteand unscattered photon pairs are generated in a simulatePET acquisition. The calculated scatter contribution is uto correct the original data set. Monte Carlo techniques wused to estimate ‘‘best possible’’ weighting functions for dferent energy-based scatter correction schemes and to eine the optimal number of energy windows for NaI~Tl! andBGO scintillators.196 Ollinger72 developed a model-basescatter correction method that uses a transmission scanemission scan, the physics of Compton scatter and a mematical model of the scanner in a forward calculation ofnumber of events for which one photon has undergonsingle Compton interaction. A single-scatter simulation tenique for scatter correction where the mean scatter contrtion to the net true coincidence data is estimated by simuing radiation transport through the object was also suggeand validated using human and chest phantom studies.73

E. Dosimetry and treatment planning

There is no doubt that the area where early Monte Cacalculations in the field have been performed is dosimemodeling and computations.10 The approach adopted by thMedical Internal Radiation Dose~MIRD! committee wasfirst proposed in 1968 and published in a series of sup

FIG. 11. A comparison between measured and simulated scatter fractiothe ECAT-953B as a function of the lower energy threshold~reprinted withpermission from Ref. 191!.


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ments to theJournal of Nuclear Medicineas differentpamphlets.197–199 Some of these pamphlets made extensuse of Monte Carlo calculations to derive specific absorbfractions for electron and photon sources uniformly distruted in organs of mathematical phantoms. Cristy200 demon-strated that the reciprocity theorem which states that forpair of regions in a uniform isotropic or uniform scatterlemodel, the specific absorbed fraction is independent of whregion is designated source and which is designated tamay also be valid for heterogeneous phantoms for cerconditions. Comparisons between measured and calculdoses when the uncertainties associated with both techniare considered validated the experimental validity of tapproach.201 Other approaches using the MIRD formalishave also been proposed.202 Poston203 calculated photon specific absorbed fractions for both the Cristy and Eckermgastrointestinal tract and their revised model and repodifferences between electron absorbed fraction values wand without electron tracking. The calculation of absorbfractions for positron emitters relevant to neurologic studwere also reported.204 Interest in Monte Carlo-based doscalculations withb-emitters has been revived with the appcation of labeled monoclonal antibodies to RIT.

In a review article on tumor dosimetry forb-emitters,Leichner and Kwok205 divided the various approaches inseveral classes, namely numerical, analytical or MoCarlo. It is also necessary to consider hybrid approachnamely numerical approaches using Monte Carlo data.use of Monte Carlo codes enables the absorbed fractioenergy to be calculated directly for a given radionuclide retive to its geometry and emission spectrum. This can be dfor relatively simple geometries206 but the main trend ofMonte Carlo approaches is that they allow complex simutions involving inhomogeneities.48,207–211 Sometimes, theMonte Carlo technique is used just to simulate random dtribution of sources or targets whereas the actual dosimecalculation is performed using dose-point kernels.212,213

Mono-energetic dose-point kernels which indicate variatioin energy delivered at a distance from mono-energetic pton or electron point sources are commonly used data sThe scaled point kernelF(x/r 0 ,E0) is defined by the equation

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wherer is the density of the medium,r 0 is the range in thecontinuous slowing down approximation~CDSA! at energyE0 and F(x,E0) is the specific absorbed fraction. Thepoint kernels are calculated from Monte Carlo codes. ThMonte Carlo codes are often reported in the literatuETRAN,197,198,214ACCEPT215 and EGS4.207,216,217Although re-sults obtained withETRAN versions prior to 1986 may diffefrom those obtained withEGS4,40 due to an incorrect sampling of the energy-loss straggling inETRAN, there are noimportant differences in the results obtained with these tcodes. It must be noted that the results reported in MIpamphlet 7,198 do not share the error described above. O

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major limitation in applying these codes to dosimetryb-emitting radionuclides is that they cannot deal satisfacrily with electron energies below 10 keV.40 The EGS4 codewas also used to characterize the spatial and energy disttion of bremsstrahlung radiation from beta point sourcesportant to RIT in water.217 This study provided the initiadata required for modeling and analyzing the scatter, atteation, and image formation processes in quantitative imagof bremsstrahlung for RIT dosimetry.

Leichner218 proposed a unified approach to photon ab-particle dosimetry. This approach is based on a fit of Beer’s tables for photons197 and electrons.198 The empiricalfunction proposed is equally valid for photons ab-particles. Therefore both point-kernel and Monte Catechniques can be effectively employed to calculate absodose to tissue from radionuclides that emit photons or etrons. The latters are much computationally intensive, hoever, point-kernel methods are restricted to homogeneoussue regions that can be mathematically describedanalytical geometries, whereas Monte Carlo methods hthe advantage of being able to accommodate heterogentissue regions with complex geometric shapes. RecenFurhang219 generated photon point dose kernels andsorbed fractions in water for the full photon emission sptrum of radionuclides of interest in nuclear medicine,simulating the transport of particles using Monte Carlo teniques. The kernels were then fitted to a mathematicalpression. Figure 12 shows dose kernels generated in annite water medium for selected radionuclides of poteninterest in RIT.219

The most recently available version of theMIRDOSE3

code,220 developed by the Radiation Internal Dose Informtion Center~Oak Ridge Institute for Science and Educatio!allows the calculation of the absorbed dose as well aseffective dose and the effective dose equivalent. The pgram deals with three phantoms representing the pregwoman at 3, 6 and 9 months of gestation and allows the d


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delivered to the fetus to be calculated at different stagesgrowth. A phantom of the adult woman has also beencluded in the program which differs from that of the 15-yeaold adolescent. TheMABDOS program221 starting with a ref-erence man allows for the definition of a spherical tumtarget and ‘‘on the fly’’ Monte Carlo calculations to bmade.222 This code was also used to show that neglectingphoton contribution from131I photon spectrum underestmates the tumor dose by 10–25%.223

Akabani224 usedEGS4 Monte Carlo calculations to estimate absorbed doses to the blood and to the surface oblood vessel wall as well as to a mathematical model oHaversian canal.225 Calculation of the dose to the upper spinregion near the thyroid resulting from the administration3700 MBq of131I and assuming a thyroid uptake of 10% walso performed.226 A Monte Carlo model has also been dveloped for the simulation of dose delivery to skeletal mtastases by the bone surface-seeking radiopharmaceu186Re~Sn! HEDP to optimize treatment planning and doresponse evaluations of therapeutic bone-seekradiopharmaceuticals.227 Beta-particle dosimetry of variouradionuclides used in radiation synovectomy, an intarticular radiation therapy to treat rheumatoid arthritis walso estimated using theEGS4Monte Carlo code.228,229

It seems as if we are on the way to a more and mpersonalized human dosimetry with radiolabeled antibodosimetry one of the aims. The dose distribution patternoften calculated by generalizing a point source dodistribution,230,231 but a direct calculation by Monte Carltechniques is also frequently reported because it allowsdia of inhomogeneous density to be considered.232 The de-velopment of a 3D treatment planner based on SPECT/Pimaging is an area of considerable research interest anderal dose calculation algorithms have been developed.223 Fig-ure 13 lists the essential steps required in developing atreatment planning program for RIT. Projection data aquired from an emission tomographic imaging system

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processed to reconstruct transverse section images wyields a count density map of source regions in the boThis count density is converted to an activity map usingsensitivity derived from a calibration phantom. In the finstep, this activity distribution is converted to a dose ratedose map by convolving the activity distribution with dospoint kernels or by direct Monte Carlo calculations. Telaborate a treatment plan for an individual patient, prosptive dose estimates can be made by using a tracer activitradiolabeled antibody to obtain biodistribution informatioprior to administration of a larger therapeutic activity. Tclinical implementability of treatment planning algorithmwill depend to a significant extent on the time requiredgenerate absorbed dose estimates for a particular patien

In particular, Sgouros213 proposed real 3D treatment planning for RIT in which patient data~cumulative activity vox-els! are convolved with dose-point kernels in order to detmine the isodose distribution.233,234 This is thensuperimposed on the target visualized in 3D by computeritomography~CT! or MRI. The methodology was extendelater to develop a dosimetry algorithm based on a MoCarlo procedure that simulates photon and electron transand scores energy depositions within the patient.50,231,235

Microdosimetric approaches are required when the rtive deviations from the mean of the local dose in the tarexceed 20%. Humm236 developed a full Monte Carlo simulation of the stochastic variation of particle hits and enedeposition in cell nuclei under two extreme geometric coditions, namely, when211At is retained in the capillary andwhen it is homogeneously distributed in the tumor.method which allows dose calculations to be made to invidual target cells in different regions of mouse bone marrexposed to alpha particles emitted from bone was adeveloped.237

FIG. 13. A diagram showing the essential steps required in developinthree-dimensional internal dosimetry program based on quantitative esion computed tomography.


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F. Pharmacokinetic modeling

Pharmacokinetic modeling is a useful component forestimation of cumulated activity in various source organsthe body. A few applications of Monte Carlo techniquhave been reported in the field of pharmacokinetic modeand are discussed in this section.

Casciari238 developed a compartmental model of@F-18#fluoromisonidazole transport and metabolism to computevolume average kappa in tissue regions from@F-18# fluo-romisonidazole PET time–activity data and characterizeusing Monte Carlo simulations and PET time–activity daThis model was able to accurately determine kappa fovariety of computer generated time–activity curves, incluing those for hypothetical heterogeneous tissue regionspoorly perfused tissue regions. Compartmental models alalso the in vivo analysis of radioligand binding to recepsites in the human brain. Benzodiazepine receptor bindwas studied using a three-compartmental model.239 The va-lidity of the results of the coefficient of variation of eacparameter were verified with statistical results providedMonte Carlo simulation. Burger240 examined the possibilityof mathematical metabolite correction, which might obviathe need for actual metabolite measurements. Mathemametabolite correction was implemented by estimating theput curve together with kinetic tissue parameters. The geral feasibility of the approach was evaluated in a MoCarlo simulation using a two tissue compartment modelsimplified approach involving linear-regression straight-liparameter fitting of dynamic scan data was developedboth specific and nonspecific models.241 Monte-Carlo simu-lations were used to evaluate parameter standard deviatdue to data noise, and much smaller noise-induced biaThe authors reported good agreement between regresand traditional methods.

Welch242 investigated and quantified the effect of typicSPECT system resolution and photon counting statisticsthe bias and precision of dynamic cardiac SPECT paraeters. The simulation of dynamic SPECT projection data wperformed using a realistic human torso phantom assumboth perfect system resolution and a system resolution tcal of a clinical SPECT system. The results showed thatrate constant characterizing the washing of activity intomyocardium is more sensitive to the region of interest potion than is the washout rate constant, and that the meffect of increased photon noise in the projection data isdecrease the precision of the estimated parameters.

Computer simulations demonstrate that an estimationthe kinetic parameters directly from the projections is moaccurate than the estimation from the reconstrucimages.243 A strategy for the joint estimation of physiologcal parameters and myocardial boundaries was proposedevaluated by simulated myocardial perfusion studies baon a simplified heart model.244 A method allowing the esti-mation of kinetic parameters directly from SPECT conbeam projections was also proposed and validated witsimulated chest phantom.245 The results showed that myocadial uptake and washout parameters estimated by con

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tional analysis of noiseless simulated cone-beam databiases ranging between 3–26% and 0–28%, respectivwhile uncertainties of parameter estimates with this metranged between 0.2–9% for the uptake parameters andtween 0.3–6% for the washout parameters.

V. OBJECT MODEL AND SOFTWARE PHANTOMS

Mathematical descriptions of human bodies and anthromorphic phantoms are useful in radiation transport calcutions. They are widely used in computer calculationsdoses delivered to the entire body and to specific organs,are valuable tools in the design and assessment of imreconstruction algorithms. Software phantoms modeledimaging situations were historically limited to simple poinrod, and slab shapes of sources and attenuating media.simple geometries are useful in studying fundamental issof scatter and attenuation, but clinically realistic distributiocannot be evaluated by such simple geometries. A premodeling of the human body requires appropriate informtion on the location, shape, density and elemental comption of the organs or tissues.

A. Object modeling

Object modeling is fundamental for performing photand electron transport efficiently by means of a Monte Camethod. It consists of a description of the geometry and mterial characteristics for an object.246 The material characteristics of interest include density and energy-dependent crsections. The modeling includes simple geometry~SG!,shape-based~SB!, and voxel-based~VB! approaches. Thethree approaches use a piecewise uniform distribution ofject characteristics to model an object. With the SG modan object is composed of a simple combination of primitivsuch as cylinders and spheres. The SB approach reprethe boundaries of shapes by mathematical equations. Reshapes such as sphere, cylinder, rectangular solid, etc.been used to approximate irregularly-shaped regions.VB approach discretizes an object into tiny cubes~voxels!with uniform characteristics. An object is thus representeda union of voxels of the same size.

Extensions of SG and SB models such as the sgeometry-based~SGB! approach247 includes more primitives~ellipsoids, elliptic cylinders, tapered elliptic cylinders, recangular solids, and their subsets: half, quarter, and eig!and uses an inclusion tree data structure to provide relatships between primitives. These extensions provide simirregular shape modeling. To allow anthropomorphic moding the composite model248 which is an extension to the SGapproach adds to the primitives a voxelized rectangular sprimitive. An object model based on a combination of mofied SG, SB and VB models without restriction in the combination set was also proposed.249 The data are structured ia hierarchical and adjacence tree associated with an effictree scanning to reduce the computation time. Combinatoapproaches to solid modeling, which describe complex sttures as set-theoretic combinations of simple objects, areited in their ease of use and place unrealistic constraints


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the geometric relations between objects such as exclucommon boundaries. An approach to volume-based smodeling has been developed which is based upon topolcally consistent definitions of boundary, interior and exterof a region.250 From these definitions, union, intersection adifference routines have been developed that allow involuand deeply nested structures to be described as set-theocombinations of ellipsoids, elliptic cylinders, prisms, conand planes that accommodate shared boundaries.

An octree-based method~OCT! which describes an objecby using several sizes of cubic regions was proposedOgawa251 to increase the calculation speed in photon traport since the number of voxels is much smaller than thathe VB approach. The ‘‘octree string’’ is generated fromset of serial cross-sections automatically. The same audeveloped a modeling method called the maximum rectgular region~MRR! method.246 In this approach, a MRR fora given voxel is selected within a homogeneous, irregulashaped region from a set of cross-sections. The searcperformed by checking the six sides of a box~MRR! includ-ing the voxel of interest. With the MRR representation of tobject, high speed calculation of photon transport canaccomplished because an object can be described by mof fewer regions than in the VB or the OCT representatmethods. Figure 14 illustrates the calculation time requifor the VB OCT and MRR approaches for different imamatrix sizes.

B. Anthropommorphic phantoms

Modeling of imaging and other medical applicationsbest done with phantom models that match the gross pareters of an individual patient. Computerized anthropomphic phantoms can either be defined by mathematical~ana-lytical! functions, or digital volume arrays. The mathematicspecifications for phantoms that are available assume a

FIG. 14. Calculation times for VB, OCT, and MRR representation ofobject ~reprinted with permission from Ref. 246!.

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cific age, height and weight. People, however, exhibit ariety of shapes and sizes. In the first MIRD pamphlets, seral organs including the skeletal system, were represeschematically using geometric forms~cylinders, cones andellipsoids!.199 The representation of internal organs with thmathematical phantom is very crude since the simple eqtions can only capture the most general description oforgan’s position and geometry.202 A version of this phantomhas been updated to include female organs.252 The most stud-ied phantom is defined as the reference man weigh70 kg.253

Mathematical phantoms are still evolving and are beconstantly improved. The heterogeneity of the body has btaken into account by including soft tissues, bone and luwith different compositions and densities. For certain orgsuch as the stomach and the bladder, a distinction shoulmade between the organ contents and the organ wall. Avised head and brain models was developed by Bouche254

Unlike previous head models, the neck and head are treas two separate compartments. The neck is representedcircular cylinder. It is topped by a cylindrical head region cin the back by a cone, so that its bottom base coincides wthe top of the neck. Two vertical planes on the back joincone to the cylinder of the head. The top of the headdefined by a half ellipsoid. The trunk region of the SnydeFisher phantom without its internal organs is incorporainto the model. Based on the atlas of sectional humanatomy, a 3D computer model of a human torso, includfour cavities of the heart, two lobes of the lung and the bosurface and a 3D model of the myocardium wdeveloped.255 The torso model, with more than 10000 suface triangles, depicts the structures and appropriate protions of the internal organs, especially of the heart.

The Mathematical CArdiac Torso~MCAT! phantom is ananthropomorphic phantom, developed at the UniversityNorth Carolina at Chapel Hill, that has been used in emisscomputed tomography imaging research.256 Using math-ematical formulas, the size, shape and configurations ofmajor thoracic structures and organs such as the heart, lbreasts and rib cage are realistically modeled for imagpurposes. Though anatomically less realistic than phantderived from CT or MR images of patients, the MCAT phatom has the advantage that it can be easily modified to silate a wide variety of patient anatomies. In addition, tMCAT phantom simulates a dynamic, beating heart incluing changes in myocardial wall thickness, changes in chber volumes, apical movement and heart rotation duringcardiac cycle. The phantom consists of two physical moda 3D distribution of attenuation coefficients and a 3D disbution of radionuclide uptake for the various thoracic orgaThe 3D attenuation coefficient phantom classifies all thoratissues into one of 5 types: muscle~vasculature and othesoft tissues!, lung, fat ~such as in the breasts!, trabecularbone and cortical bone. The MCAT phantom has becomvaluable tool in imaging studies where reasonably realisbut anatomically variable patient data needs to be simulaThe graphic in Fig. 15 illustrates the MCAT phantom. T


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Some calculations make use of more accurate represetions of individuals based on volumetric scans, such asMRI, and PET. As an improvement to the mathematicalthropomorphic phantoms, a new family of phantoms wconstructed from CT data.257 The human phantoms preseadvantages towards the location and shape of the organparticular, the hard bone and bone marrow. A physical brphantom has also been developed to simulate the actdistributions found in the human brain in the cerebral bloflow and metabolism studies currently employed in PET.258

The phantom utilizes thin layers of Lucite to provide appent relative concentrations of 4, 1 and 0 for gray mattwhite matter and ventricles, respectively, in the brain.clinically realistic source distribution simulating brain imaing was created in digital format.259 Zubal59,260 developed atypical anthropommorphic VB adult phantom by manusegmentation of CT transverse slices of a living human mperformed by medical experts. A computerized 3D volumarray modeling all major internal structures of the body wthen created. Each voxel of the volume contains an innumber designating it as belonging to a given organ or innal structure. These indexes can then be used to assivalue, corresponding to, e.g., density or activity. Two vsions of the phantom exist, representing either the comphuman torso with an isotropic voxel resolution of 1.5 mm,a dedicated head phantom with 0.5 mm voxel size. The dcated brain phantom was created from the high resoluMRI scans of a human volunteer. This volume array repsents a high resolution model of the human anatomyserves as a VB anthropomorphic phantom.

VI. MONTE CARLO COMPUTER CODES

Many Monte Carlo programs have been in use in the fiof nuclear imaging14,16,17,261 and internaldosimetry,50,220,232,234with many of them available in the

FIG. 15. Surface rendered images of the 3D MCAT phantom developeChapel Hill. ~a! Anterior view with outer body surface and ribs removedshow the various organs modeled.~b! Posterior view with the rib cagepresent~reprinted with permission from Ref. 256!.

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TABLE II. Key features of Monte Carlo codes used in nuclear medical imaging.

MC code General description

EGS4 ~Ref. 262! Coupled photons/electrons transport in any matethrough user specified geometries. Simulationimaging systems not specifically included and requiran extensive amount of user programming inMORTRAN.

ITS including TIGER, CYLTRAN,andACCEPT ~Ref. 263!

Coupled photons/electrons transport in any matethrough slabs, cylinders or combinatorial. Simulationimaging systems not specifically included and requiran extensive amount of user programming inFORTRAN.

MCNP ~Ref. 264! Coupled neutrons/photons/electrons transport in amaterial through user generalized geometry. Simulatof imaging systems not specifically included anrequires an extensive amount of user programmingFORTRAN.

GEANT ~Ref. 43! Coupled photons/electrons transport in any matethrough combinatorial geometry. Simulation of imaginsystems not specifically included and requiresextensive amount of user programming inFORTRAN.

SIMSET ~Ref. 21! Photons transport in any material through voxel-basphantoms. Simulation of SPECT and PET imaginsystems included. User modules written in C couldlinked.

SIMIND ~Ref. 15! Photons transport in any material through voxel-basphantoms. Simulation of SPECT imaging systemincluded. User modules written inFORTRAN could belinked.

SIMSPECT~Ref. 265! Coupled photons/electrons transport in any matethrough voxel-based phantoms. Simulation of SPECimaging systems included. User modules writtenFORTRAN/C could be linked.

MCMATV ~Ref. 266! Photons transport in any material through voxel-basphantoms. Simulation of SPECT imaging systemsincluded. User modules written inFORTRAN could belinked.

PETSIM ~Ref. 20! Photons transport in any material through shape-baphantoms. Simulation of PET imaging systems includeUser modules written inFORTRAN could be linked.

EIDOLON ~Ref. 135! Photons transport in any material through shape-basevoxel-based phantoms. Simulation of 3D PET imagisystems included. User modules written in C/ObjectivC could be linked.

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public domain.9,10 Table II ~Refs. 262–266! lists MonteCarlo codes widely used together with a short descriptiontheir key features.

EGS4. The electron gamma shower~EGS! computer codesystem is a general purpose package for Monte Carlo silation of the coupled transport of electrons and photons inarbitrary geometry for particles with energies from a fekeV up to several TeV.262 The code represents the state-othe-art of radiation transport simulation because it is vflexible, well-documented and extensively tested. Some hreferred to theEGS code as thede factogold standard forclinical radiation dosimetryEGS is written in MORTRAN, aFORTRAN pre-processor with powerful macro capabilitieEGS is a ‘‘class II’’ code that treats knock-on electrons abremsstrahlung photons individually. Such events requpredefined energy thresholds and pre-calculated data forthreshold, determined with the cross-section generPEGS.

ITS. The IntegratedTIGER Series~ITS! of coupled electron/

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photon Monte Carlo transport codes is a powerful tooldetermining state-of-the-art descriptions of the productand transport of the electron/photon cascade in timindependent, multi-material, multi-dimensional envronments.263 ITS is a collection of programs sharing a common source code library that can solve sophisticated ration transport problems. A total of eight codes are in tcollection which can be split into six groups: theTIGER codes~for 1D slab geometries!, the CYLTRAN codes~for 2D cylin-drical geometries!, theACCEPTcodes~for arbitrary 3D geom-etries!, the standard codes~for normal applications!, the Pcodes~for applications where enhanced ionization/relaxatprocedures are needed!, and the M codes~for applicationswhich involve 2- or 3D macroscopic electromagnetic field!.The user selects the appropriate code from the librarysupplies it with any special requirements and the physdescription of the problem to be solved in an input file.

MCNP. MCNP is a general-purpose Monte Carlo code thcan be used for neutron, photon, electron or coupled neut

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photon/electron transport.264 The code treats an arbitrarthree-dimensional configuration of materials in geomecells bounded by first- and second-degree surfacesfourth-degree elliptical tori. For photons, the code takescount of incoherent and coherent scattering, the possibilitfluorescent emission after photoelectric absorption, abstion in pair production with local emission of annihilatioradiation and bremsstrahlung. A continuous slowing domodel is used for electron transport that includes positrok-shell x-rays, and bremsstrahlung but does not includeternal or self-induced fields. Important features that maMCNP very versatile and easy to use include a powerful geral source, criticality source, and surface source; bothometry and output tally plotters; a rich collection of varianreduction techniques; a flexible tally structure; and an extsive collection of cross-section data.

GEANT . The GEANT package was originally designed fohigh energy physics experiments, but has found applicatalso outside this domain in the areas of medical and biolocal sciences, radiation protection and astronautics.43 Themain applications ofGEANT are the transport of particlethrough an experimental setup for the simulation of detecresponse and the graphical representation of the setup athe particle trajectories. The two functions are combinedthe interactive version ofGEANT. This is very useful, sincethe direct observation of what happens to a particle insidedetector makes the debugging easier and may reveal posweakness of the setup.

SIMSET. The simulation system for emission tomograp~SIMSET! is a software application designed to perforMonte Carlo simulation of photon creation and transpthrough heterogeneous attenuators for both SPECTPET.21 The package has been in the public domain sincebeginning of 1996 and includes the photon history gener~PHG!, the object editor, a collimator module and detectiand binning modules. The PHG is the module that generand tracks photons within the FOV of the tomograph besimulated. The code is continuously being improved incluing, for instance, the implementation of incoherent scating, random events detection and simulation of coincideimaging using conventional dual-head gamma cameras.

SIMIND . The SIMIND code simulates a clinical SPECscintillation camera and can easily be modified for almany type of calculation or measurement encounteredSPECT imaging,15 including transmission imaging.168 Theentire code has been written inFORTRAN-90and includes ver-sions that are fully operational on VAX-VMS, most UNIXplatforms and on MS-DOS~Lahey LF90 compiler!. In sum-mary, the code works as follows: photons emitted frosimulated decay in the phantom are followed step by stowards the scintillation camera.SIMIND includes an accuratetreatment of photon interaction in the phantom, a protectlayer and in the crystal of the detector. The simulationback-scattering from light guides and photomultipliers is aincluded. Different types of collimators can be selected.SI-

MIND can take advantage of anthropomorphic voxel-baphantoms developed for simulating realistic imaging sittions. The program has been shared among several gr


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code has been widely used for collimators design111 and toevaluate attenuation and scatter correcttechniques.174,176–179,185–186

SIMSPECT. TheSIMSPECTcode developed at MIT is baseon theMCNP Monte Carlo transport code for photon trackinand interaction algorithms. It has been extensively modifito allow complete collimator and source modeling and dirmanipulation of the geometric and physical parameterscountered in SPECT imaging.18,265 The simulation packageallows full tomographic simulation of data from physicalrealistic nonuniform and asymmetric 3D source objects. Pton transport in the detector crystal, light pipe and PMT’snot simulated. The use of positron emitters in SPECT iming has also been modeled viaSIMSPECT in order to betterunderstand the potential improvements of different collimtors design on 511 keV gamma camera imaging.

MCMATV . The Monte Carlo Matrix Vectorized~MCMATV !program models photon transport in both homogeneou266

and heterogeneous media.267 The code is designed to modeboth projection data for simulated SPECT studies andcompute photon detection kernels, which can be usedbuild system matrices for use in matrix-based imareconstruction.139 The vectorized code is written inFOR-

TRAN77 and run on a Stellar GS1000 computer for pipelincomputations. It uses an event-based algorithm in whphoton history data are stored in arrays and photon hiscomputations are performed within DO loops. The codeadapted from a history-based Monte Carlo code in whphoton history data are stored in scalar variables and phhistories computed sequentially. Without the use of the vtor processor the event-based code is faster than the hisbased code because of numerical optimization performduring conversion to the event-based algorithm.

PETSIM . A series of programs calledPETSIM have beendeveloped to model the source distribution and its attention characteristics, as well as the collimator and detectorPET.20,44 The different modules are connected by compgamma history files which are stored on a disk or tape. Tstorage of intermediate results on tape reduces simulatime, since most common source geometries need be geated only once. The simulation results include spectranalysis, sensitivity to true coincident events, scattered ccident and single events and the effects of these paramof detector dead-time. The sensitivities in multi-slice systeare presented as matrices of coincident crystal planes.matrix shows the true count sensitivity and the scatter frtion together for each valid combination of planes. This psentation is very useful for assessing the effects of varidegrees of inter-plane collimation. The spatial resolutanalysis includes the effects of positron range, noncolineaof the gamma rays, multiple interaction within the detectoand the effects of quantization into single crystalsmultiple-crystal block detectors. Each of these effects canturned on or off without repeating the simulation. Singcrystals, blocks and crystals with DOI encoding can be spfied, so that the detector geometry can be optimized.

EIDOLON . The Monte Carlo simulator,EIDOLON, was de-

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veloped using modern software engineering techniqmainly for fully 3D PET imaging.22,135The code was writtenin Objective-C, an object-oriented programming languabased on ANSI C. The first version of the program wasveloped using the NextStep development environmentmodular design featuring dynamically loadable program ements or bundles was adopted for software design. The bbuilding block is amodel elementobject class which allowsus elements to be browsed, inspected, adjusted, createddestroyed through a graphical inspector. The user interallows the user to select scanner parameters such as theber of detector rings, detector material and sizes, energycrimination thresholds and detector energy resolution. It aallows us to choose either a complex anthropomorphic phtom or a set of simple 3D shapes, such as parallelepiellipsoid or cylindroid for both the annihilation sources athe scattering media, as well as their respective activity ccentrations and chemical compositions. The user has thesibility to view the reference source image and sinogram dsets as they are generated and are periodically updatedimplementation of the software on a high-performance pallel platform was also reported.38

VII. SCALAR VERSUS VECTORIZED ANDPARALLEL MONTE CARLO SIMULATIONS

Although variance reduction techniques have been deoped to reduce computation time, the main drawback ofMonte Carlo method is that it is extremely time-consuminTo obtain the high statistics (;107 counts! required for im-age reconstruction studies requires us to track hundredmillions of particles. Consequently, a large amount of Ctime ~weeks or even months! may be required to obtain useful simulated data sets. The development of advanced cputers with special capabilities for vectorized or parallel cculations opened a new way for Monte Carlo researchParallel computers are becoming increasingly accessiblmedical physicists.268 This allows research into problemthat may otherwise be computationally prohibitive to be pformed in a fraction of the real time that would be takena serial machine. Historically, however, most programs asoftware libraries have been developed to run on sesingle-processor computers. A modification or adaptationthe code is therefore a prerequisite to run it on a paracomputer. However, it is worth pointing out that amongsimulation techniques of physical processes, the Monte Cmethod is probably the most suitable one for parallel coputing since the results of photon histories are compleindependent from each other. Moreover, computer aidedallelization tools designed to automate as much as possthe process of parallelizing scalar codes are becomavailable.269 Although parallel processing seems to be tideal solution for Monte Carlo simulation, very few invesgations have been reported and only a few papers havepublished on the subject.270

The theoretical peak performance of a computer is demined by counting the number of floating-point additioand multiplications that can be completed during period


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time, usually the cycle time of the machine.271 Today’s largemachines measure their speed in GFlops (109 operations/s!.Each CPU generally contains 2 multiply and 2 add pipWhen all of these can be employed simultaneously asinstance, in a dot product or a vector update operationFlops/cycle can be attained. The Cray T90 has a cycle tof 2.2 ns and a maximum number of 32 processors, thuspeak performance is 4 operations/1 cycle31 cycle/2.2 ns332 processors558.2 GFlops. Easily portable Monte Caruser codes are generally used for timing benchmark purpon different computers.272 According to van der Steen anDongarra,273 the classification of high-performance compuers is based on the way instructions and data streamsarranged and comprises four main architectural clasThese include the following.

Single Instruction Single Data stream „SISD… ma-chines.These are the conventional systems that containCPU and hence can accommodate one instruction streamis executed serially. Nowadays many large mainframes mhave more than one CPU but each of these execute instion streams that are unrelated. Therefore, such systemsshould be regarded as~multiple! SISD machines acting ondifferent data spaces. Examples of SISD machines areinstance most workstations like those of DEC HewlePackard and Sun Microsystems.

Single Instruction Multiple Data stream „SIMD … ma-chines.Such systems often have a large number of proceing units, ranging from 1,024 to 16,384 that all may execthe same instruction on different data in lock-step. Sosingle instruction manipulates many data items in paralExamples of SIMD machines in this class are the CPP DGamma II and the Alenia Quadrics. Another subclass ofSIMD systems are the vector processors which act on arof similar data rather than on single data items using scially structured CPUs. When data can be manipulatedthese vector units, results can be delivered with a rate of otwo and in special cases of three per clock cycle. So, veprocessors execute on their data in an almost parallel wayonly when executing in vector mode. In this case theyseveral times faster than when executing in conventionallar mode. For practical purposes, vector processors are thfore mostly regarded as SIMD machines. An examplesuch systems is, for instance, the Hitachi S3600.

Multiple Instructions Single Data stream „MISD … ma-chines. Theoretically in these types of machines multipinstructions should act on a single stream of data. As yetpractical machine in this class has been constructed norsuch systems easy to conceive.

Multiple Instructions Multiple Data streams „MIMD …

machines. These machines execute several instructstreams in parallel on different data. The difference withmulti-processor SISD machines mentioned above lies infact that the instructions and data are related becauserepresent different parts of the same task to be executedMIMD systems may run many sub-tasks in parallel in ordto shorten the time-to-solution for the main task to be eecuted. There is a large variety of MIMD systems includithose that behave very differently like a four-processor C

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Y-MP T94 and a thousand processor nCUBE 2S. An imptant distinction between two subclasses of systems shohowever, be made: Shared memory~SM! and distributedmemory~DM! systems.

~i! Shared memory systems.SM systems have multipleCPUs, all of which share the same address space. This mthat the knowledge of where data is stored is of no concerthe user as there is only one memory accessed by all Con an equal basis. Shared memory systems can beSIMD or MIMD. Single-CPU vector processors can be rgarded as an example of the former, while the multi-Cmodels of these machines are examples of the latter.Cray J90 and T90 series belong to this class of compute

~ii ! Distributed memory systems.In this case, each CPUhas its own associated memory. The CPUs are connectesome network and may exchange data between their restive memories when required. In contrast to SM machinthe user must be aware of the location of the data in the lomemories and will have to move or distribute these dexplicitly when needed. Again, DM systems may be eithSIMD or MIMD. The first class of SIMD systems, mentioned above, operate in lock step and all have distribumemories associated to the processors. For the DM-MIsystems again a subdivision is possible: those in whichprocessors are connected in a fixed topology and thoswhich the topology is flexible and may vary from tasktask. The class of DM-MIMD machines is undoubtedly tfastest growing part in the family of high-performance coputers.

Another trend that has come up in the last few yearsdistributed processing. This takes the DM-MIMD conceone step further: instead of many integrated processorone or several boxes, workstations, mainframes, etc.,connected by Ethernet, for instance and set to work conrently on tasks in the same program. Conceptually, this isdifferent from DM-MIMD computing, but the communication between processors is often orders of magnitude sloMany commercial, and noncommercial packages to readistributed computing are available. Examples of theseParallel Virtual Machine~PVM!,274 and Message Passing Interface~MPI!.275 PVM and MPI have been adopted for instance by HP/Convex, SGI/Cray, IBM and Intel for the trasition stage between distributed computing and MassivParallel Processing~MPP! systems on the clusters of thefavorite processors and they are available on a large amof DM-MIMD systems and even on SM-MIMD systems focompatibility reasons. In addition, there is a tendencycluster SM systems, for instance by HIPPI channels, totain systems with a very high computational power, e.g.,NEC SX-4 and the Convex Exemplar SPP-2000X havestructure, although the latter system could be seen as a mintegrated example~the software environment is much mocomplete and allows SM addressing!. Other interesting re-search systems like the Intel ASCI Option Red systemSandia National Laboratory~with a measured performancof 1.3 TFlops!, the CP-PACS at the University of Tsukub~measured performance of 368 GFlops! and the NumericalWind Tunnel at the National Aerospace Lab. in Japan~230


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GFlops!, are not marketed and only available at the Institumentioned and, therefore, not of much benefit to the supcomputer community at large. It is worth noting that thmarket of parallel and vector machines is highly evasive;rate with which systems are introduced and disappear ais very high and therefore the information provided wprobably be only approximately valid.

Sequential programs make the most effective use ofavailable processing power: they alone guarantee maximuse of the CPU. In parallel programs, communication magement introduces an unavoidable overhead, resultinless efficient use of the overall CPU power. Moreover,cording to Amdahl’s law,276 parallelization efficiency is de-creased by a factor representing the fraction of operatithat must be executed in sequential order. When this fracreaches one we are confronted with a wholly unparallelizacode, and the speed-up is zero no matter how many prosors are used. The efficiency of parallel programs is furthmore reduced by a factor equal to the fraction of procesidle time, which is highly dependent on the software parlelization techniques used by the programmer. Scalar orrial Monte Carlo codes track the history of one particle atime, and the total calculation time is the sum of the timconsumed in each particle history. Many Monte Carlo appcations have characteristics that make them easy to mapcomputers having multiple processors. Some of these palel implementations require little or no inter-processor comunication and are typically easy to code on a parallel coputer. Others require frequent communication asynchronization among processors and in general are mdifficult to write and debug. A common way to parallelizMonte Carlo is to put identical ‘‘clones’’ on the various processors; only the random numbers are different. It is thefore important for the sequences on the different procesto be uncorrelated so each processor does not end up slating the same data.277 That is, given an initial segment othe sequence on one process, and the random numbequences on other processes, we should not be able to prthe next element of the sequence on the first process.example, it should not happen that if we obtain random nubers of large magnitude on one process, then we are mlikely to obtain large numbers on another. In developing aparallel Monte Carlo code, it is important to be able to rproduce runs exactly in order to trace program execution

Since a Monte Carlo particle history is a Markov chathe next interaction or movement of a particle is alwaystermined by the current state of the particle. The historiestwo particles became identical only when the same randnumber sequence is used to sample the next state. To enthat the seed tables on each processor are random and urelated, Mascagni36 described a canonical form for initializing separate cycles of the Fibonacci generators. Therehowever, many approaches to vectorized and paralleldom number generation in the literature.278–280We can dis-tinguish three general approaches to the generation ofdom numbers on parallel computers: centralized, replicaand distributed. In the centralized approach, a sequengenerator is encapsulated in a task from which other ta

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FIG. 16. A comparison between history-based scaprocessing, event-based vector processing and histbased parallel processing. In history-based scalar pcessing, one particle history is tracked at a time.history-based parallel processing, each parti(p1 ,p2 ,...,pm) is assigned to one process which tracits complete history (e1 ,e2 ,...,en). In event-based vec-tor processing, a process treats only part of each parthistory (e1 ,e2 ,...,en) and particles (p1 ,p2 ,...,pm)‘‘flow’’ from process to process.

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request random numbers. This avoids the problem of geating multiple independent random sequences, but is unlikto provide a good performance. Furthermore, it makes reducibility hard to achieve: the response to a request depeon when it arrives at the generator, and hence the recomputed by a program can vary from one run to the nextthe replicated approach, multiple instances of the sameerator are created~for example, one per task!. Each generatoruses either the same seed or a unique seed, derived, foample, from a task identifier. Clearly, sequences generatethis fashion are not guaranteed to be independent anddeed, can suffer from serious correlation problems. Howethe approach has the advantages of efficiency and easimplementation and should be used when appropriate. Indistributed approach, responsibility for generating a sinsequence is partitioned among many generators, whichthen be parceled out to different tasks. The generators arderived from a single generator; hence, the analysis ofstatistical properties of the distributed generator is simplifiThere are two possibilities for parallelisation of Monte Cacodes.

~i! Particle or history-based parallelization: each partiis assigned to one of the parallel processes andhistories of assigned particles are processed withinprocess.

~ii ! Task- or event-based parallelization: a process tre


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only part of random walk events or tasks, thus pticles ‘‘flow’’ from process to process according ttheir events.

The difference between the different algorithms is illustrain Fig. 16. In a review of vectorized Monte Carlo, Martiand Brown281 described variations of event-based algorithtogether with speed-up results published by different grou

During the last two decades, investigations were carrout to run different Monte Carlo codes on multipletransputer systems,282,283 vector parallel super-computers,266,267,284,285parallel computers38,286,287and a clus-ter of workstations in a local area network using PVM.288

There are large discrepancies in the performance ratioported by different authors. In particular, Miura285 reported aspeed-up of about 8 with the vectorizedESG4code~EGS4V!.A factor varying between 2.9 and 5.1 was also reportedMCMATV 266 depending on whether the detection of scattephotons is modeled or not. A linear decrease in computtime with the number of processors used was also repowith EIDOLON.38

VIII. CONCLUSIONS AND FUTURE PROSPECTS

Nuclear medicine has historically been the field in whimost of the early Monte Carlo calculations in medical phyics were performed. The large number of applications

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ferred to in this paper shows the usefulness and considersuccess of the Monte Carlo method as a tool in differareas of nuclear imaging. The availability of Monte Cacodes in the ‘‘public domain,’’ developed and tested worwide in a variety of applications by many users should pvide confidence for their use as research tools in the diffefields of nuclear imaging. The use of Monte Carlo calcutions to validate scatter and attenuation correction methhas been useful. Furthermore, scatter images can, in pciple, be calculated for any arbitrary source distributioSimulated realistic emission images~including both primaryand scatter! can be corrected for attenuation and scatter,be compared with scatter- and attenuation-free ideal imaThe accuracy in correction methods can thus be evaluatean unbiased way since systematic errors can be controll

The lack of inherent error estimates and relatively slconvergence is a well known limitation of the Monte Cartechnique. In many situations, one can predict the statiserror ~variance! on the estimated parameters from MonCarlo simulations, and hence estimate the number of trneeded to achieve a given error. A few investigations hbeen carried out to quantify the accuracy of Monte Casimulations. However, some authors reported discrepanbetween experimental measurements and results obtawith Monte Carlo programs. For example, it has beenticed that the measured energy spectra for a small psource has a larger magnitude low-energy tail than istained with Monte Carlo simulations.164,165There are severapossible explanations for this effect. The background radtion always present in scanning rooms is a possibility toconsidered. A potential source of those events might‘‘scatter’’ from photomultiplier tubes back into the detectiocrystal.165 This effect could also be caused by the characistics of the scinitillation camera electronics.164 Discrepan-cies also observed in quantification of parameters likescatter fraction could be explained by the sharp enethreshold model used in most codes,70,191i.e., the experimen-tally observed variations in crystal energy response is hataken into account in the analysis of simulated data135

Therefore, it is important that most simulations of complphantom geometries be matched with scatter-free simtions. There are other problems associated with the detecsystem which are too camera specific for Monte Carlo toworthwhile. A typical example for scintillation camerasthe uniformity variation and shift of energy spectra with rtation angle. Faster software implementations might be psible by using post-simulation modules incorporating modfor those effects based on experimental measurements.

Future developments in detector design, together wfaster computer systems, will improve image quality, temral resolution, patient throughout and quantitation. It is epected that Monte Carlo calculations will enter the clinicand scientific arena and will become a method of choicedevelop and to implement patient-specific dosimetry andage correction techniques and to optimize instrumentaand clinical protocols.


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