monte carlo techniques in medical radiation physicskuncic/lectures/phys5020_resources/andr… ·...

Phys. Med. B i d , 1991, Vol. 36, No 7, 861-920. Printed in the UK

Review

Monte Carlo techniques in medical radiation physics

Pedro Andreo

Department of Radiation Physics, Ka~olimlta Institute and University of Stockholm. Box 60211,104 01 Stockholm, Sweden

Received 31 A u y s t 1990, in final form 27 February 1991

Contents

1. Introduction 1.1. Scope of this review

2. The basics of the Monte Carlo practice 2.1. Random numbers and other computational details 2.2. Photon transport 2.3. Electron transport

2.3.1. Condensed history (‘macroscopic’) techniques 2.3.2. Detailed history (‘microscopic’) techniques 2.3.3. Variance reduction in electron transport

3. Macroscopic Monte Carlo codes in the public domain 4. Applications in medical radiation physics

4.1. Nuclear medicine 4.1.1. Detectors 4.1.2. Imaging correction techniques 4.1.3. Absorbed dose calculations

4.2.1. Detection systems 4.2.2. Determination of physical quantities in diagnostic radiology 4.2.3. Radiation protection aspects

4.3.1. Teletherapy sources and dosimetry equipment 4.3.2. In-phantom simulations 4.3.3. Treatment planning applications 4.3.4. Calculations in brachytherapy

4.4. Radiation protection 4.5. Applications based on microscopic Monte Carlo techniques

4.5.1. Electron microscopy 4.5.2.

4.2. Diagnostic radiology

4.3. Radiotherapy physics

Radiation track structure and microdosimetry 5. Inverse Monte Carlo techniques 6. Vectorized and parallel Monte Carlo simulation 7. Conclusions

Acknowledgments References

0031-9155/91/070861+60%03.50 @ 1991 IOP Publishing Ltd

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1. Introduction

Since the review article by Raeside (1976), where the principles of the Monte Carlo method and its first applications in medical physics were described, the number of publications in this field using the simulation of the transport of radiation continues to increase. Following Nahum (1988a), today practically one scientific article per issue is being published in Physics in Medicine and Biology for instance, and a similar rate can be observed in parallel journals. Several hooks with comprehensive reviews, ‘tech- nical descriptions’ or proceedings from Monte Carlo courses, have also been published recently (Morin 1988, Jenkins e l a/ 1988, Kase e t a/ 1990).

At the time of the 1976 review article by Raeside most of the Monte Carlo work had been developed at large research centers using mainframe computer systems. This was the case with most of today’s well known Monte Carlo codes such as ETRAN (Berger and Seltzer 1968), EGS (Ford and Nelson 1978), MCNP (Thompson 1979, based on previous codes by Cashwell e l a/ 1972, 1973), or MORSE (Straker el a/ 1976), some of which will be discussed in this article. The basic references in the field of photon and electron transport simulation were already available a t that time (Cashwell and Everett 1959, Berger 1963, etc).

A number of Monte Carlo codes were written during the 1970s for application to medical physics, mainly radiotherapy physics. The codes developed by Patau (1972), Nahum (1976) and Abou Mandour (1978) belong to this group, all of them developed on mainframe computer systems and with a strong emphasis on the simulation of electron transport. This aspect explains the weight given to data calculated with ETRAN for comparisons, as the other codes mentioned were generally related to either reactor neutron/photon physics (MCNP, MORSE) or high-energy physics (EGS). The availability of minicomputers (mainly DEC/PDP-II) in many medical institutions made possible the development of ‘smaller’ Monte Carlo codes capable of simulating either specific problems in radiotherapy physics with photon beams (Webb and Parker 1978, Webb 1979), the transport of photons in Compton-scatter tissue densitometry (Bat- tista and Bronskill 1978) or the full electromagnetic cascade used to derive quantities for electron dosimetry in water (Andreo 1980, 1981).

The power of many of the mainframe computer systems initially used for Monte Carlo simulations is available today in many departments of hospital physics around the world, sometimes even on the desk of a hospital physicist. At the same time, some of the general purpose computer codes have become widely distributed through institutions like the Radiation Shielding Information Center (RSIC) at Oak Ridge National Laboratory in USA or the Nuclear Energy Agency Data Bank (NEA) at Gif-sur-Yvette (France) in Europe. The result is a broader group of scientists with knowledge of Monte Carlo techniques; as a consequence, the range of applications for the Monte Carlo method continues to increase.

1.1. Scope o f l h i s reuiew

The main purpose of this work is to review the techniques and applications of the Monte Carlo method in medical radiation physics since Raeside’s review article in 1976. Emphasis will be given to applications where photon and/or electron transport in matter is simulated. Although the method has also been applied to neutrons and heavy charged particles in radiotherapy physics (cf Biiche and Przybilla 1981, Sisterson et a/ 1989, Smith e t a/ 1989), or in the simulation of their track structures (cf Paretzke 1987) the considerably smaller number of applications with these particles, and the

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limited space for this review, excludes them from a detailed survey. The use of Monte Carlo techniques in non-radiation medical physics also falls outside the scope of this review.

Some practical aspects of Monte Carlo practice, mainly related to random num- hers and other computational details, will be discussed in connection with common computing facilities available in hospital environments. Basic aspects of electron and photon transport will he reviewed, followed by the presentation of the Monte Carlo codes widely available in the public domain. Applications in different areas of medical radiation physics, such as nuclear medicine, diagnostic x-rays, radiotherapy physics (including dosimetry), and radiation protection, and also microdosimetry and electron microscopy, will he presented. Actual and future trends in the field, like Inverse Monte Carlo methods, vectorization of codes and parallel processors calculations will also be discussed.

2. The basics of the Monte Car lo p r a c t i c e

The general principles of the Monte Carlo method have been discussed by Raeside (1976) and more recently by Turner e t a1 (1985). They have also been introduced in a number of other publications (cf Cashwell and Everett 1959, Shreider 1966, Carter and Cashwell 1975, James 1980, Lund 1981) and their repetition will be avoided here as much as possible. In what follows some emphasis is given to basic and practical ideas of interest in medical physics.

2.1. Random numbers and oiher computational details

Random numbers will be treated first as they play a fundamental role in Monte Carlo calculations. ‘Philosophical’ arguments on sets of ‘truly-random’, ‘pseudo-random’ and ‘quasi-random’ numbers will not he considered here. The basic ideas on random number generators (RNGS) have been discussed in the initial references cited here; updated (and more complete) reviews have been given by James (1980, 1990) and Ehrman (1981). The classic by Knuth (1969) is still mandatory in the field of RNGs and Sowey (1972) has given about 300 references on random number generation and testing covering the period 1927 to 1971.

Although some new trends in RNGs have been described recently, including ‘intelligent’ random number techniques (learning and biasing) such as those used in the MCNP code (Booth 1988), Lehmer’s method is still the most commonly used for RNGs. I t is called the multiplicative-linear-congruential method. Given a modulus M , a multiplier A , and a starting value t o (‘seed’), random numbers ti are generated according to

ti = (AEi-1 + E ) modulo M (1)

where B is a constant. In multiplicative-linear-congruential generators M is usually chosen t o be 2‘, b being the number of bits in the integer representation of data in the computer. Constants A and B are chosen to give a ‘well behaved’ RNG, which is a difficult task. Congruential RNGs generally have a maximum repetition period of length 2-2M when M is a power of 2, or M - 1 if A4 is prime (Ehrman 1981), which is long enough for most Monte Carlo simulations.

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The length of the period of a RNG must he long enough t o avoid repetitions in the sequence of numbers used during the simulation process, as otherwise correlations can he produced. There are simulations, however, where the set of independent numbers needed might exceed the repetition period of the RNG being used. When the generator is ‘well behaved’, even if the sequence of numbers is used more than once, the probability of having more than one particle history starting in the same position of the sequence of random numbers is practically negligible. This means that when the end of the sequence is reached it will be started again during some of the sampling procedures used along the simulation. The final effect will he equivalent to initiating the sequence of random numbers using different ‘seeds’, but it should be investigated for each computer/RNG/application configuration. The risk of repetition can he decreased using RNGs with very long periods, such as the McGill University package ‘Super-Duper’ in IBM assembler (cf James 1980, 1990) or RNG64 due to Biela- jew (1986), both of which produce periods as long as one might expect from a 64-bit machine. The use of such R N G s however needs longer calculation times than the generators previously reported and their choice must he based on a careful evaluation. This author’s experiences of the simulation of ionization chambers within a phantom, where hundreds of millions of histories of “CO photons were simulated using the EGS code, did not show any change in the results or in the extremely slow convergence of the low uncertainty being sought when RNG64 or the standard generator in EGS (see later) was used.

The RNG of congruential type which has probably received more attention in the Monte Carlo literature is the infamous RANDU, distributed by IBM (1968) in computers of their 360 and 370 series, where A = 65539, B = 0 and M = Z31 (32-bit integer, bit zero reserved for the sign, and a period of Z31 - 1). The same RNG has also been implemented by DEC in minicomputer-based operative systems with 16-hit integer arithmetic (RT-11, RSX-IIM, etc), where i t is possible to call for modulus 232 using two registers which allow the use of the modulus to any power by shifting. (For compati- bility between VAX-11 and PDP-I1 FORTRAN, VAX/VMS offers the option of using this generator also (DEC 1984).) The predicted period for DEC’s RANDU is 230 (in 32-hit DEC machines hit 15 is sign hit) and it produces identical output random numbers to IBM’s generator. The large availability of DEC PDP-11 machines in hospital physics environments during the last 15 years has led to a widespread use of this generator (cf Battista and Bronskill 1978, Bond el a/ 1978, Audreo 1980, etc).

A long period is not the only desirable property in a RNG and different tests must be applied to verify the true randomness of the sequence. Knuth (1969) is still the standard source of information on random tests, which Morin el a/ (1988) have performed on 16-bit minicomputers for RNGs used in different applications of interest in medical physics. Even if the generator RANDU passed the majority of tests, the so called ‘Marsaglia effect’ or how ‘random numbers fall mainly in planes’ (Marsaglia 1968), has shown that this generator produces correlated triplets of random numbers (cf Coldwell 1974). The use of techniques where vi-tuple random numbers are used for decision-making or branching is common in transport simulation, which should warn us against the use of such RNGs. An improvement of the properties of RANDU has been obtained by changing the multiplier A from 65539 to 69069 (Marsaglia 1972), which is the standard RNG used in VAX FORTRAN (MTHWANDOM) with M = Z32, and in the SLAC RNI generator. Another RNG commonly used today in medical physics calculations is the SLAC RN6 generator which has A = 663608 941 and B = 0 and is included in the EGS code. Both generators have similar properties regarding period

Monte Carlo techniques in medical radiation physics 865

length and number of planes in 3-space, RN6 being faster, especially with the ‘in- line’ coding included in EGS which avoids subroutine calls (cf Ehrman 1981). Theory predicts a period of about lo9 for these two generators, but a repetition of 2, 3 and 4 consecutive numbers can be found in certain implementations after generating 3.5 x IO7 random numbers approximately. In VAX FORTRAN the RNG generates numbers in the interval 0 5 < < 1 (DEC 1984), which can yield obvious run-time errors when expressions like In(e), which are frequent in particle transport, are used. One can alternatively use In(1 - C) which is as random as f . SLAC RN6 on the other hand generates numbers in the interval 0 5 < 5 I , and therefore a detailed condition to avoid the generation of zeros should be included.

A very interesting review of RNGs has been published recently by James (1990), where it is recommended that ‘old-firshioned generators (like the multiplicative-linear- congruential previously described) should be archived and made available only upon special request for historical purposes or in situations where the user really wants a bad generator’ and Fibonacci and shift register (also known as Tansworthe) generators should be used instead. Furthermore they should be implemented in the form of subroutines returning an array of random numbers rather than a function returning one number each time. Fibonacci RNGs are based on the series with the same name (where each element is the sum of the two preceding elements) but use some arithmetic or logical operation between two numbers which have occurred somewhere earlier in the sequence, not necessarily the last two:

I

where @ is a binary or logical operation and p and q are the lags (henceforth the name of lagged Fibonacci sequences) defined such that p > q. Shift register RNGs are also based on this formula but with M = 2; therefore only single bits are generated, which are collected into words using a shift register. James (1990) has given the FORTRAN code of the generators RANMAR and ACARRY of the lagged Fibonacci type, with periods of Z 1 4 4 ( ~ 2 x respectively (!), which are completely portable, that is they give bit-identical results on all machines, with at least 24-bit mantissas in the floating-point representation. An ‘in-line’ version of RANMAR l i a s

just become available for the EGS Monte Carlo system (Bielajew 1990a). A problem that has not received enough attention in certain Monte Carlo calcu-

lations is that of truncation errors (different from round-off errors, characteristic of computer hardware). It is well known that the use of single or double precision can yield different results for certain Monte Carlo computations. This is the reason why some Monte Carlo codes which are available for different word-length computers are coded in double precision when they are used on 32-bit machines (for instance ITS, cf Halbleib and Mehlhorn 1984). This might be of importance, for instance, during the simulation of electron transport where extremely short pathlenghts are sometimes selected, especially in the proximity of boundaries. In those cases the use of double precision for certain variables should not be excluded without careful verification; the price of an increase in computation time may be well worth paying.

The treatment of uncertainties in Monte Carlo calculations deserve some additional comments. The classification of uncertainties recommended by CIPM (1981) for experimental methods may also be adopted i i i Monte Carlo procedures (Andreo and Fransson 1989, Andreo 199Oa). In such recommendations, t,he classification of uncertainties commonly used, namely ‘random’ and ‘systematic’ uncertainties, have

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866 P Andreo

been replaced by uncertainties of ‘category A’ (objective, evaluated using statistical methods) and ‘category B’ (subjective, estimated by any other method). Uncertain- ties in both categories are specified by standard deviations (or variances) which can he combined using the general propagation law for uncertainties. However, whereas the evaluation of variances in ‘category A’ is achieved using statistical procedures, both in experiments and in Monte Carlo calculations, uncertainties in ‘category B’ (non-statistical methods) are more difficult to evaluate in Monte Carlo calculations as multiple steps are involved (programming; tabulation of data, in many cases with unknown uncertainties; etc). A conservative estimate is to make both categories of uncertainties (A and B) equal and combine them in quadrature; a multiplicative factor can then be used to obtain a n ‘overall’ uncertainty (CIPM 1981).

To evaluate uncertainties of category A (i.e. with statistical methods), some Monte Carlo codes divide the total number of histories to be simulated into a number of batches, and statistical uncertainties are evaluated using the mean values of the variables (tallies) scored in each batch. Typically, 10 batches are used in Monte Carlo calculations as with EGS user-codes (Rogers 1982) or ITS (Halbleib and Mehlhorn 1984), although they treat the re-start of a calculation differently. Bond e t al (1978) have referred to 15-100 hatches for each simulation. A different procedure consists of scoring both the quantity of interest (cumulative tally) and its squared value (cumulative squares of the tally) whenever a particle is transported a given pathlength (cf Shteider 1966). Uncertainties can then be evaluated during the simulation of each history (inside the main loop of the simulation), and the process is equivalent to a.- suming that every history is a batch on its own. This is the method used, for instance, in the codes MCNP (Soran 1980) or MCEF (Andreo 1980,1981). As can be observed in figure 1 for Gaussian distributed numbers, only when a very large number of batches is considered in the first technique, will the statistical evaluation approach the correct estimation given by the second method.

2.2. Photon fransporl

Most codes dealing mainly with photon transport assume that electrons generated through different interactions are absorbed ‘on the spot’ and the simulation process becomes therefore relatively simple. All the physical interactions of photons (or neutrons) are completely simulated following the general techniques described by Raeside (1976) or Turner et a l (1985), for example, without much comput,ational effort. From the exponential attenuation distribution, the appropriate cumulative distribution can be evaluated and the distance s between interactions in a medium ( s t e p length) is determined by

s = -Xln(l- () (3)

A being the mean free path a t the photon energy at the beginning of the step and f a random number, 0 5 E < 1. The type of interaction event ocurring after the step s is sampled from the appropriate relative probabilities pi (ratios of single cross sections to the total cross section), using their cumulative distribution function Pi. Another random number selects the interaction event i(() such that

j - 1 j

CPi = Pj-1 5 E < c p i = Pj (4)

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r? 0 1.00 c U . - f 0 0 In

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(1983, 1986) whereas a weighted sampling from the Thompson distribution has been used by Chen et a/ (1980). The significance of electron binding corrections in the scattering of low-energy photons has been investigated by Beernink et a/ (1983) and Williamson et nl (1984); they have concluded that neglecting these effects results in a significant underestimation of scattering angles which are of importance in Monte Carlo calculations.

The small number of interactions that take place when photons traverse matter has motivated the development of ‘variance reduction’ techniques to decrease uncertainties that can be evaluated by statistical methods (i.e. category A). In such techniques the ‘natural physics’ (and scoring procedures of tallies) is manipulated in a number of different ways so as to increase the relative occurrence of certain events. Forced interactions, importance sampling, Russian roulette, particle spliting, etc are commonly used techniques well described in many of the classic references on the field (cf Cashwell and Everett 1959, Carter and Cashwell 1975, etc) which were also briefly discussed by Raeside (1976). A collision density estimator to increase the efficiency of small-angle scatter calculations in x-ray diagnostic has been developed by Persliden and Alm-Carlsson (1986); and Gardner e l nl (1987) have derived algorithms to force scattered radiation into detectors of different shape. An up-to-date review describing the techniques in more common use today has been given by Bielajew and Rogers

It is interesting to note that some applications have been developed, mainly in the area of radiotherapy physics, which can be considered as an important variance reduction technique. In general they combine Monte Carlo and analytical techniques, yielding results that would otherwise require extremely long CPU-times for a direct simulation. A simple application, where Monte Carlo calculated depth-doses for monoenergetic photons have been used to produce data for bremsstrahlung spectra, has been reported by Andreo and Nahum (1985) as equivalent to the direct Monte Carlo simulation of millions of photons. A more elaborated approach consists in the convolution of spatial absorbed dose distributions from monoenergetic photons to determine dose distributions for radiotherapy treatment planning (see section 4.3.3), which yields results corresponding to the direct simulation of billions of photon histories (cf Ah- nesjo et a/ 1987). The potential of this method has not been yet applied to some of the problems which demand intensive variance reduction techniques, such as the simulation of ionization chambers for radiotherapy dosimetry which are described in section 4.3.1.

(1988).

2.9. Electron transport

For the simulation of the complete electromagnetic cascade the inclusion of electron transport adds a new dimension to the problem. In principle, the direct simulation of all the physical interactions (sometimes referred to as ‘microscopic’ simulation) could be used for electrons, as i t has been described for photons or neutrons. The only difficulty in doing so is to keep track of all generations of electrons and photons produced during successive interactions. Their transport has to be considered in a systematic way until all the particles have been simulated. However, the very large number of interactions that take place during electron slowing down (of the order of IO4 collisions in aluminium, from 0.5 MeV to 1 keV, cf Berger and Wang 1988), makes i t unrealistic to simulate all the physical interactions in the majority of applications. This aspect has motivated the development of the so called ‘condensed history’ or ‘macroscopic’


techniques (Sidei e l a/ 1957, Schneider and Cormack 1959), where interactions are grouped in different ways.

2.3.1. Condensed history (‘macroscopic’) techniques. The principles of the ‘condensed history’ technique will he described following the classical review on Monte Carlo charged particle transport by Berger (1963). Physical interactions of electrons are classified into groups which provide a detailed ‘macroscopic picture’ of the physical process:

SO,Sl, s2,. . . , sn,. . . EO, E l , Ez, , , . , En, , U O , U l , U Z . . . . , U , , . . T O , P I , P Z , ..., r. ,...

where s, is the distance travelled, E, the energy, U, the direction and rn the position of the electron. T h e transition from step n t o n + 1 accounts for many interactions where multiple collision models, like multiple scattering or stopping-power theory, are considered. T h e step size (distance travelled or energy loss between two steps) has t o be chosen in such a way that the total number of steps is kept as small as possible, as computation time will be proportional t o the number of steps. On the other hand, the step size has t o be such that multiple collision models for angular deflections and energy losses are valid, t ha t is short pathlengths and many single interactions per step.

According t o Berger (1963) the ‘condensed history’ technique can be classified in two procedures:

(1) Class I, which groups all the interactions and uses a predetermined set of pathlengths, the random sampling of interactions being performed a t the end of the step. T h e simplest choice is a constant pathlength. The disadvantage is t ha t the angular deflection increases from step t o s tep as the electron energy decreases, which demands further re-sizing in order t o be able to use a multiple scattering theory. Logarithmic spacing is a better selection as angular deflections change little from step t o step. This is chosen so that , on average, the energy is reduced by a constant factor k per step, t ha t is the fraction of energy lost per step is constant.

(2) Class 11, or ‘mixed procedures’, which groups only minor collisions where energy losses or deflections are small, but considers the individual sampling of ninjor events or ‘catastrophic’ collisions, where a large energy loss or deviation occurs (figure 2). Further details of the technique can be found in Berger (1963) and Andreo (1985). An extension of Class I1 procedures has been introduced by Andreo and Brahme (1984) t o account for energy losses and deflections below the threshold of catastrophic collisions, using restricted energy-loss straggling and restricted multiple scattering during the classification between major and minor events.

In particular, the initial s ta te of knock-on electrons and bremsstrahlung photons is unambiguosly defined, angular deviations can be treated more accurately and the correlation between energy loss and angular deflection is always conserved. On the other hand, Class I schemes can include complete energy-loss straggling which is inherently independent of the electron transport cut-off (Tc) and threshold energies of knock-on electrons (7’8). T h e dependence on Ta is an important limitation of Class I1 schemes, as the sampling of electron energy-loss straggling is limited to the interval [Ta, 0.5To] ( [ E , TO]

Class II schemes have some advantages over Class I procedures.

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Figure 2. Energy-distance diagram of the Class I1 procedure used to describe the electron (and photon) tracks ahown in the upper part of the figure. Electron 'ca1astrophic'collirionJ occur at positions 1 (bremsstrahlung interaction), 2 and 3 (knodi-on inelastic collisions), and 5 (elastic collision, no energy loss). The photon is scattered by a Compton interaction at position 4. All secondary electrons (bottom of lower figure) are followed down to the transport cut-off of the simulation, E,.,, where they are absorbed ( S t a r s ) . The broken descendent line from Eo corresponds to a calculation of the electron energy loss based on the continuous-slowing-d~~,, approximation (CSDA) .

for positrons) which might become critical during the treatment of low-energy electrons (below 100 keV). To overcome this limitation, restricted-Class II procedures (Andreo and Brahme 1984) perform an additional sampling below Ts which makes energy straggling independent of the threshold energy of knock-on electron production. The technique need to be developed further to include, for example, binding effects in inelastic collisions.

The consideration of 'catastrophic' collisions in a Class I1 scheme is analogous to the transport of photons, where all the interactions are considered individually. For electrons a 'catastrophic' mean free path may he defined, Acat, obtained from the addition of single events, such as inelastic, A,,,,, elastic, A.,, and bremsstralung interactions, Abrem. The distance (step length) between major events is again given by equation (3), i.e. s = -ACatIn(l -<) (cf Andreo 1985). In fact by changing the thresh- olds of catastrophic events it would be possible to perform a continuous transition from full grouping (macroscopic) to singleinteraction (microscopic) simulation.

2.9.2. Detailed history ('microscopic 7 techniques. Most existing charged particle Monte Carlo codes are based on the multiple collision models for scattering and energy


loss previously mentioned. The majority of applications used in radiotherapy physics (the main area of medical physics where high-energy electron transport has become of importance) are based on these procedures. There are, however, important limitations with such models, for instance when geometrical regions (or distance to boundaries) are very small, when dealing with very low-energy transfers (smaller than or close to binding energies) or when the transport of low-energy electrons, below, say, 100 keV has to be simulated. In these situations either the number of collisions is not enough to consider a process as multiple, or the physics behind the theory itself is violated. In many of these cases the condensed histories approach has t o be abandoned and the simulation of the transport has to be performed using single-interaction models. This is the case with the majority of the Monte Carlo applications simulating track-length structures, microdosimetry or electron microscopy.

As already mentioned, the microscopic Monte Carlo technique in electron transport is identical to the approach used in photon (or neutron) simulation. In this sense, the techniqne itself is simpler than the macroscopic model as it does not depend on additional parameters governing the grouping and distance between collisions. However, depending on the electron energy, the physics involved might become considerably more complicated. There are still a number of poorly known cross sections at very low energy and, in certain instances, the simulation in condensed media must rely on experimental data extrapolated from gas-phase data, sometimes using Fano plots (UT against In") t o minimize energy dependence.

The complexity of the techniques used in microscopic modelling varies considerably, although a common approach is to neglect bremsstrahlung interactions. Simple models used in early applications, in electron microscopy for example, have been based on the simulation of all the elastic scattering events, calculating the step length between consecutive collisions with the elastic-mean free path. Energy losses were determined from the Bethe theory of stopping power and, in some cases, an approximation to account for energy-loss straggling has been included. An improvement to this model has been to take inelastic collisions into account, the mean free path being based on U,, + aine,, which is the basic techniqne commonly used today. From this point on, the main difference between existing microscopic Monte Carlo applications lies mainly in the different theories, models or experimental data used to account for energy-loss single-processes contributing to (such as electronic ionization and excitation), and refinements to treat atomicshell structure (according to Paretzke (1987) molecular changes become important a t residual energies below, say, 10 eV). Other differences like the accuracy of elastic cross sections (from screened Rutherford cross sections to Mott partial wave expansions), or the generation and transport of secondary and higher-order generation electrons can also be mentioned. Detailed reviews on the different approaches used in electron microscopy and radiation track structure (and microdosimetry) have been given by Kyser (1981) and Paretzke (1987) respectively. An up-to-date review of cross sections at low-energies for different materials (mainly gases and liquid water) has been performed by Grosswendt (1988a).

2.9.9. Variance reduclion in eleclron transport. Variance reduction techniques developed for application to photon transport are, i n principle, also applicable to electrons, but in general the simulations do not involve events which are particularly rare. Apart from correlated sampling techniques, where different particles are simulated using the same sequence of random numbers, the techniques in common use are mostly based on a careful study of the physics of the problem. 'Running-parameters' used in the

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simulation, such as transport cut-off and threshold energies for secondary electron and photon production become extremely important. In Class I1 schemes for instance, the threshold for the production of knock-on electrons Ts can be set equal to the energy cut-off T, for the transport (provided that energy-loss straggling is properly taken into account), as i t is time consuming t o create a particle which will not be simulated.

Other techniques are directly related to the geometry of the problem. In general T, should be chosen such that an electron which has been slowed down to such cut-off has a small probability of crossing to a different region. More details are involved in the so called ‘range rejection’ or ‘electron trapping’ techniques, where the history of an electron without the energy to escape from a certain region, or alternatively to reach another region, will be disregarded; this requires the determination of the electron range and comparison with the distance to the closest boundary (figure 3(a)). Less time consuming is the inclusion of regions where the transport cut-off T,, is larger than in the volume of interest, which can be considered as the simplest technique. A ‘virtual envelope’ surrounding such a volume of interest can be considered, whose dimensions are chosen such that a n electron outside the envelope cannot reach the volume of interest, and the additional transport cut-off T,, is chosen on this basis (figure 3 ( b ) ) .

Figure 3. Variance reduction techniques in electron transport. (a) Range vejection or elrctron trapping technique: an electron et (z,y,z)o is requested to travel down to ( z , y , z , )~ . Even if the energy To is larger than the electron transport cut-off, T,, in most c a e s the history can be disregarded as the CSDA range 70 is shorter than the distance to the closest boundary d. An energy trapping Tt..p can he selected to check the condition TO < d only when To < Tt,,,. (b) Virtual envelope technique: a region E is defined having a thickness t and the same electron trqnsport cut-otl Tc as the volume of interest V . Electrons produced outside the region E U V will be able to reach V only if their CSDA range is longer than t . A cut-otl T,, (larger than T,) can be set outside E U V such that the CSDA range associated with the electron energy T,, js smaller than t. Radiation yield sliould be investigated prior to the selection ofTtrap or Tc, .

Similar region-by-region analysis can be performed to ‘relax’ the requirements on the fraction of energy loss, maximum allowable travelling distance or any other run-time parameter used in general to achieve a more detailed traneport through the volumes of interest. In all cases i t is important to realize that ignoring an electron history involves excluding the probability of bremsstrahlung emission, which could otherwise result in energy deposition in different regions. Therefore the radiation yield


at the different cut-off energies in the simulated medium should always be investigated and kept reasonably low.

3. Macroscopic Monte Carlo codes in the publ ic domain

Some of the Monte Carlo codes developed in large research centres are used today for research in medical physics. This has been possible with the availability of considerable computing power in hospital physics departments, together with a well supported distribution of the codes (through the RSIC a t ORNL or the European NEA Data Bank, for instance). Table 1 lists the Monte Carlo codes and systems cited by their name throughout this review.

Two such codes, ETRAN and EGS, or results computed with them, have become widespread and a brief discussion of their different approaches in dealing with the simulation of radiation transport will be given here. Readers wishing to expand the summary given here are referred to the so called ‘Erice book’ (Jenkins et al 1988) where the two codes are treated in depth. Other generally available Monte Carlo codes like MORSE (cf Palta 1981) or OGRE (cf Nilsson and Bralime 1981) have also been used in medical physics calculations, but their use has been less extensive than the two codes which will be described.

The ETRAN Monte Carlo model, an acronym for electron-transport (Berger and Seltzer 1968), was originally developed at the National Bureau of Standards, USA, to simulate the transport of electrons and photons involving energies up to a few MeV, being extended later on for calculations at higher energies. The extension of the (ETRAN-based) ITS system of codes to the multi-GeV region, has been developed recently (Miller 1989). Since the late 1960s, ETRAN has been used in calculations related to the dosimetry of therapeutic beams, mainly electrons (Berger and Seltzer 1969, 1982; Berger el ol 1975). It has provided most of the Monte Carlo results included in the ICRU Report 35 on electron dosimetry (ICRU 1984a) and most of today’s electron dosimetry procedures are based on data computed with this code (cf Andreo 1988a). ETRAN is a Class I code, which emphasizes the physics of electron transport, its main characteristics being the accurate treatment of electron multiple scattering (using the theory of Goudsmit and Sauuderson) and bremsstrahlung interactions (including cross sections differential in energy and angle). To take into account low-energy transport, ETRAN includes characteristic x-rays from the K-shell and Auger electrons after the emission of a photoelectron, but neglects coherent scattering and binding corrections to incoherent scattering (cf Seltzer 1988) which have been included in an updated version of the ITS system.

Discrepancies between calculations with ETRAN and other Monte Carlo codes (and experiments) have been reported in the literature (Andreo 1980, 1981; Andreo and Brahme 1981, 1984). Such discrepancies have been of importance in the determination of quantities of interest in electron dosimetry, snch as the variation of the mean energy of primary electrons with depth, where the values calculated with ETRAN were consistently higher than those obtained with the code MCEF (Andreo and Brahme 1981). It was suggested by these authors that the disagreements were caused by the different treatment of energy-loss straggling in the respective Monte Carlo codes, as ETRAN (Class I) performs a sampling from the Landau/Blunck-Leisegang distribution whereas MCEF (Class 11) combines restricted stopping-power with a sampling from the M ~ l l e r distribution. Discrepancies in electron depth-dose distributions, compared with experiments and other Monte Carlo calculations, have also heen shown

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to be noticeable, especially in the build-up region and the slope of the fall-off section (Andreo and Brahme 1984). Rogers and Bielajew (1986) have demonstrated that ETRAN underestimated the mean collision energy loss as a result of underestimating the number of large energy-loss events, which confirmed the findings of Andreo and Brahme (1981). According to Seltzer (1988) the energy-loss underestimation was due to the incorrect energy-loss sampling procedure from the energy straggling distrihu- tion used in ETRAN. RSIC (1988) has reported the implementation of the corrected straggling algorithm in the ITS system of Monte Carlo codes, primarily based on the ETRAN code (Halbleib and Mehlhorn 1984).

The ETRAN model provides sophisticated electron transport techniques hut does not include the treatment of any geometries other than those of infinite media or plane-parallel slabs of different materials (Seltzer and Berger 1987). The family of ITS c o d a provides geometrical packages of increasing complexity based on the ETRAN system (Halbleib 1988). ITS consists of three main codes, TIGER, CYLTRAN and AC- CEPT, which allow the user to simulate electron and photon transport down to 1 keV in plane-parallel slabs, cylindrical geometry or any combination of the geometrical bodies included in its default combinatorial package, respectively. The ITS system also consists of special versions of these three codes (called ‘P-codes’) which include ionization of all shells and atomic relaxation from the K , L, M and N shells. Upon installation of the system, the use of the codes is quite straightforward, programming is not needed and a simulation is based on a set of few order-independent input statements (source, transport cut-offs and geometry, etc). At the time of writing, the ITS system is being distributed as version 2.1. Unfortunately, the electron stopping-power data included in this version is based on the Sternheimer and Peierls (1971) approximation t o determine the correction due to the density effect in stopping-powers. As is well known this correction yields stopping powers which differ from the actual recommended values (ICRU 1984b). The cross section generator XGEN is being updated to account for recent changes in physical data in a forthcoming version 3, which includes ICRU-37 electron stopping-powers, numerical bremsstrahlung cross sections differential in photon energy, coherent scattering with binding corrections, and binding corrections to incoherent scattering (Halbleib e t a / 1990). A developing version of the new cross section data generator has been used by Andreo (199Ob) to compute electron depth-dose distributions which have been compared with calculations with MCEF and EGS. Very good agreement among the three codes but large discrepancies with the existing ETRAN results used today in electron dosimetry (figure 4) was found.

The EGS (Electron Gamma Shower) Monte Carlo system w a s originally developed at SLAC to simulate high-energy electromagnetic cascades. Version 3 of the system (Ford and Nelson 1978) has been in use at high-energy accelerator centres (SLAC, CERN, etc) for many years, although applications in medical physics were infrequent (cf Nelson and Jenkins 1980). I ts high-energy origin, where forward scattering dom- inates, is the main reason for some of the ‘weaknesses’ of the physics in EGS compared with ETRAN: multiple electron scattering according to the theory of hloliere, bremsstrahlung differential in energy only and a simple treatment of low-energy electrons and photons. According to the classification giveir here, EGS is a Class 11 code where the production of knock-on electrons and bremsstrahlung are treated individually. As a consequence, one of the requirements to run the system is to define tlireshold energies for such events and precompute data (using the code PEGS) for each threshold, which in general will vary for different types of calculation. The entire EGS system is written in MORTRAN, a FORTRAN ‘pre-processor’ that includes powerful macro ca-


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Figure 4. Comparison of depth-dose data for plene-parallel monoenergetic electron beams in water ca l~ lated with different Monte Carlo codes: ITS system (full lines), E G S I system (chain), MCEF (broken) and previous ETRAN (long-chain w i t h diamond). Eleclrori incident energies of IO, 20 and 50 MeV (adapted from Andreo (1990b)).

pabilities. However, MORTRAN has debugging characteristics which are difficult to get acquainted with.

During recent years the low-energy electron transport in EGS has been improved, mainly by the inclusion of user-defined restrictions in step-sizes (maximum energy loss and distance travelled per step, following Berger (1963)), yielding a more accurate system (cf Rogers 1984). A new version has been released, EGS4 (Nelson el a / 1985), which has added electron transport dawn to 1 keV and Rayleigh scattering t o the original physics in the code. Further improvements to the system have been developed by Bielajew and Rogers (1986a, 1987) including an angular distribution for the emission of photoelectrons and a powerful algorithm called PRESTA. The latter includes a pathlength correction to account for the detour due to multiple scattering of calculated straight paths, lateral correlation acording to Berger (1963), and a boundary crossing algorithm which allows very large step sizes except near medium boundaries (step sizes vary depending on the distance to the boundary). PRESTA is based on the Moliere multiple-scattering theory, but whereas Moliere considered his theory to be valid for a number of collisions WO > 20, the algorithm considers instead WO > e (base of natural logarithms), which is the mathematical (instead of physical) limit of the theory. For many simulations PRESTA has simplified some of the cumber- some problems previously involved in user-defined restrictions in step-sizes, and has overcome default limitations in the EGS system mainly regarding multiple scattering. I t still requires further improvements to deal properly with step-size dependence and

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backscatter; in addition, the problem of minimum step-sizes when different energy cut-offs (Tc and Ts in different regions for instance) are used in a simulation is a non- negligible limitation. The strong influence of recent developments in radiation physics has led to new improvements in the code, such as the implementation of ICRU-37 stopping-powers (Duane el a1 1989). The sampling of bremsstrahlung photon angles bas also been incorporated in connection with the simulation of targets for therapeutic accelerators (Bielajew el a/ 1989).

One of the important characteristics of the EGS system is the high degree of flexibility in its operation; at the same time this demands a great deal of programming effort. It is important to note that the EGS system is not a stand-alone Monte Carlo code, but a package that the user must link to his/ber own main code and to :user- written routines, describing the tracking of the transport through the geometry of the problem and the scoring algorithms to extract the relevant quantities from the transport. In a simple way it could be compared with a mathematical or plotting library, as the user ‘connects’ with the EGS system through COMMON blocks, macros and calls to two user-callable subroutines, HATCH and SHOWER. The resulting Monte Carlo code (user-code according to the terminology used in the EGS manual) is therefore largely programmer-dependent and subject to individual errors. To avoid ambiguities regarding the reliability of results which in general are only specified as ‘calculated with EGS’, a proper name should be given to the user-code when referring to calculations using the EGS system. Some of the user-codes developed at NRCC are now widely available through the EGS distribution package.

I t has to be emphasized tha t the coding of the simulation geometry is the most challenging programming aspect in EGS and a very important component of any other Monte Carlo code. In general i t is necessary to determine the intersection of the track of a particle with a given surface. Mathematical tools for different types of surfaces are given as subroutines in the EGS package (cf Nelson and Jenkins 1988), which the user must connect with his own routines. Most of the EGS user-codes in medical physics calculations are, however, based on simple cylindrical geometries. This is also the case for calculations with other Monte Carlo codes. More sophisticated approaches are available, the combinatorial geometry package, CG (Guber el Q / 1967) being by far the most commonly used due to its simplicity for the user. The combinatorial geometry describes three-dimensional complex bodies using Boolean algebra with a few elementary bodies such as parallelepipeds (RPP), spheres (SFH), cylinders (RCC), wedges (WED), etc, which can be oriented arbitrarily in the space. Although the use of the combinatorial geometry is possible in the EGS system, its implementation bas not received much attention. The ITS code ACCEPT (as well as the MORSECG code, cf Straker e l a/ 1976) is, on the other hand, fully based on this mathematical package which does not require more than a few lines to construct any spatial geometry (cf Nelson and Jenkins 1988, Halbleib 1988). A different geometry technique has been implemented in MCNP (Thompson e t a1 1980), where instead of pre-defined geometrical bodies, the user defines geometrical cells using first- and second-degree surfaces (and some special fourth-degree, like elliptical tori) which are combined by using Boolean operators.

It may be of interest to know the differences between the two main Monte Carlo codes, EGS and ITS, that can handle most geometries, energies down to 1 keV, and any material composed of elements with atomic numbers Z from 1 to 100. Rogers and Bielajew (1988, 199Oa) have compared some aspects of the two codes from the EGS- developet point of view, but opinions based on independent experiences with both


codes will he given here. The accurate physics included in the actual ETRAN and ITS systems, mainly regarding multiple scattering (cf Berger and Wang (1988) for a detailed comparison between the theories included in the two codes), bremsstrahlung, and, in general, the transport ofelectrons at low-energies (below, say, 100 keV), makes them superior in all the problems where such phenomena are of importance. For instance, below 20 keV or so, Berger (1973) h a s shown that a modified screening parameter from Moliere’s theory (a constant between l and 2) should be used, but this is not considered in calculations using EGS user-codes. In particular, the treatment of energy-loss straggling in low-energy electron transport might become imprecise in EGS (and in any other Class I1 code without restricted energy straggling, cf Andreo and Brahme (1984)) as the energy of secondary electrons is not sampled below the threshold for the production of knock-on electrons. EGS also has limitations in simulating low-energy photons (energy less than 50 keV or so) as it neglects binding corrections in coherent and incoherent scattering. Positrons are also simulated differently in the two codes. Whereas ETRAN treats them as electrons (except for the emission of annihilation photons when the posit,ron slows to rest), EGS takes into account electron-positron differences in collision energy losses, inelastic scattering and in-flight annihilation. The superiority of EGS in this topic is, however, limited as i t ignores differences in radiative energy losses (and therefore in radiation yield, cf ICRU 1984h), bremstrahlung cross sect,ions and multiple scattering; furthermore, the lack of spin dependence in Moliere’s theory does not allow an implemcntation of true positron multiple scattering, which is intrinsically possible in the more accurate theory of Goudsmit-Saunderson included in ETRAN or ITS.

Important limitations to the two codes (and to any other code based on the condensed history technique) appear a t the time of treating electron scattering during boundary crossings, where the pathlength of the electron has to be truncated. This might result in segment lengths where the use of a theory for electron multiple scattering is no longer justified. ETRAN and the ITS codes determine the direction at the time of crossing by sampling from a Gaussian approximation to the Goudsmidt-Saunderson distribution (Berger and Seltzer 1968, Seltzer 1988), where the mean-square angle is assumed to he proportional to the fraction of the step to the boundary; the electron is forced to cross the boundary througli the rejection of any sampled angle which would direct i t hack into the initial material (Lockwood el a1 1976). The EGS system, on the contrary, forces the validity of Moliere’s theory down to I collision approximately, which imposes a restriction for the minimum step length allowable t,i.. The occurrence of a step smaller than tmin switches-off the sampling from the Moliere distribution and no deflection of the initial direction of the electron occurs. The PRESTA algorithm in EGS4 optimizes the distance to a boundary to avoid as much as possible a violation of t,i. (which is restricted to e collisions in this case), but the sampling is also switched-off when an unavoidable very short electron step occurs. Divergent results a t high atomic numbers produced by these two approximations for the simulation of backscattering experiments with “CO y-rays can be observed in figure 5 , together with discrepancies even at low atomic numbers.

The way in which the two Monte Carlo systems classify boundaries is also different. For most EGS user-codes all scoring regions are separated by in~erfaces, eveu if they are made of the same material; this also requires boundary clieckiiig calculations in the simulation of homogeneous media. In ETRAN and ITS interface boundaries are only considered to separate different materials and not to define scoring regions, where no special action is taken if an electron track crosses them. In general this will make

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Figure 5. A benchmark of cYLTRAN (ITS 1 . 1 t X G E N 3 ) and D O S a Z I o ( E G S q P R E S T A + P E o S ~ ) simulating experiments with 6oCo y-rays that involve interface effects (based on a large plane-parallel ionization chamber with interchangeable front and back walls, cf Nilsson et a1 (1988)). Experiments (crosses, broken line) correspond to a fixed aluminium front wall and varying Zs in the back walls, cavity hei&t 1 mm; results are normalized to t h e c a e AI-AI front-back walls. The simulations use a realistic input “CO y-spectrum (Rogers cl nl 1988), transport electrons and photons down to 1 keV, force photons to interact in the chamber, and perform electron range rejection lor energies below 0.5% radiation yield. Both EGS (open symbols) and ITS (closed symbols) include the latest enhancements in cross section data described in the text. The R A N M A R random number generator is implemented in EGS (James 1990, Bielajew 1990a). Uncertainties are less than 1% in all calculations (one standard deviation, category-A).

ITS faster than EGS for calculations in homogeneous media when a similar pathlength division is used.

The EGS system, on the other hand, is a flexible and open code, and with some training i t is relatively easy to program modifications or extract information on transport related quantities. This is particularly useful to implement variance reduction techniques in a user-code. ETRAN and the ITS codes are ‘black-boxes’ where programming is not needed but changes are very difficult to perform. Several key parameters are needed to run EGS codes successfully, and it sometimes require careful comparisons and analysis of the different options available. This might even take longer than the simulation of the problem itself. In ITS codes, input data according to the user manual is the main requirement to run a code; the lack of flexibility of ITS to perform very special calculations, compared with EGS, is compensated by the use of thoroughly checked and verified codes. Finally, the size of computer RAM memory and disk space needed to run and store the ITS system can be three times that for EGS due to the large amount of tabulated da ta and length of FORTRAN source code files.

The so called ‘All-Particle Method’ approach (Cullen e l Q I 1988), presently under development a t the Lawrence Livermore laboratory, might become of special interest in the near future. The system will include the transport of neutrons, photons, electrons and light charged particles as well as the coupling between all species of particles. The code will include a sophisticated geometry package and the latest nuclear and atomic database presently under development a t Livermore (cf Cullen and Perkins

Monte Carlo techniques in rnedzcal radiotion physics 881

1989, Perkins and Cullen 1989). The main goal of the all-particle method in electron transport will be to implement a stratified model where macroscopic and microscopic Monte Carlo techniques will be switched alternately depending on the energy of the electron; in this way each technique will be used in the energy region where it works more efficiently.

4. Applications in medical radiation physics

Applications of the Monte Carlo method in medical radiation physics will be dealt with in this section. The continuously increasing number of these applications during recent years contrasts with the very few references available a t the time of the review by Raeside (1976).

Five main groups will be considered here, namely nuclear medicine, x-ray diag- nostics, radiotherapy physics and dosimetry, radiation protection calculations and transport simulation using microscopic Monte Carlo codes. This classification relies on subjective criteria, as in most cases the boundaries of each group are not strict and overlapping is unavoidable. This is particularly so in applications dealing with radiation protection aspects, both in diagnostic radiology and nuclear medicine, which will be discussed within these groups. A few radiation protection calculations will be treated separately. The same is true for absorbed dose calculations using radioactive sources, which will be treated as sealed or unsealed sources, the first type being included in therapy and the later in nuclear medicine. The simulation of the response of detectors used t o measure spectra of low-energy photons will be included both in nuclear medicine and to a lesser extent in x-ray diagnostic applications.

4.1. Nuclear medicine

As pointed out by Raeside (1976), nuclear medicine is the area where most of the early Monte Carlo calculations in the field were performed. A characteristic shared by most of the existing applications, both old and new ones, is that the codes are mainly based on the simulation of photons only, that is any electrons generated are considered to he absorbed at the location where they are produced. Such an approach is justified by the low energy of the photons being simulated and therefore by the short ranges of the electrons generated and their negligible bremsstrahlung production. Calculations regarding absorbed dose distributions for beta sources clearly fall outside this general characteristic. Three main subgroups of applications will be considered, namely detectors, image reconstruction and dosimetry.

4.1.1. Defectors. The Monte Carlo simulation of detector responses and efficiencies is probably one of the areas in nuclear medicine, and in nuclear physics in general, which has received most attention. The detailed techniques described by Zerby (1963) can be referred to as an early guide for this kind of simulation. There are studies of detectors which include high energies (and therefore incorporate the transport of electrons), but they generally also include energies low enough to be of interest for nuclear medicine. This is the case with the tabulations of the response of 3 in x 3 in NaI detectors between 100 keV and 20 MeV by Berger and Seltzer (1972), and the simulations of planar and cylindrical Ge(Li) detectors between 100 keV and 12 MeV due to Gross- wendt and Waibel(1975). Rogers (1982) has reported simulations for incident photons above 300 keV using cylindrical detectors of different materials which in some cases

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include the effect of the detector housing. Calculations below 300 keV including the same effect have been performed by Saito and Moriuchi (1984) for NaI(TI) detectors with different shapes and volumes.

The design of positron emission tomography (PET) systems using the Monte Carlo method has received considerable attention and a large number of applications have been developed. Derenzo (1981) has simulated arrays of detectors of different materials (NaI(TI), CsF, Ge, plastic detectors, etc) and sizes, as well as other parameters of influence such as the effect of inter-crystal septa. He concluded that narrow bismuth germanate (BGO) had the best detection efficiency. The optimization of the optical coupling between BGO crystals a n d photomultiplier tubes w a s undertaken by Derenzo and Riles (1982) simulating the reflection and scattering along the detection system. The Monte Carlo method has been also used in the design of multi-ring PET cameras used for three-dimensional imaging by Dahlbom e l a / (1989), who have investigated the influence of the inter-plane septa for different source geometries, concluding that the removal of the septa can increase the efficiency of the system by a considerable amount (almost a factor of six). The simulation of a large multiplane PET camera using the EGS system has been reported by Del Guerra el a1 (1983).

The effect of the collimation in a Compton-scatter tissue densitometry scanner has been studied by Battista and Bronskill (1978) in a detailed publication. Monte Carlo techniques have also been employed to determine the geometric response of collima tors in scintillation gamma-cameras and single photon emission tomography (SPECT) detection systems. Metz el a/ (1980) have simulated circular and triangular collimator hole shapes at several collimator-source distances to verify analytical expressions for the transfer function of the collimation system of gamma-cameras. The development of other computer codes to simulate collimators and analyse their performances has been reported by Ljungberg and Strand (1989); recently Gantet el a/ (1990) have used a personal computer t o simulate specific collimation equipment for scintillation cameras and SPECT.

4.1.2. Imoging correciion techniques. Monte Carlo calculations have been found to be powerful tools to quantify and correct for photon scattering, which usually produces blurring of the image and loss of contrast in nuclear medicine imaging procedures. A full simulation of a point-source-phantom-detector assembly allows the determination of the scatter component of the entire system (figure 6) which can be unfolded from the total image to yield an image whithout scattering contribution.

Multiple scattering in Compton-scatter tissue densitometry h a s been analysed by Battista and Bronskill (1978), who concluded that the contamination from high order scattering contains a Considerable fraction of photons having energies identical to those of single scattered photons. The effect could be minimized, but not be elimi- nated, by improving the detector resolution. Speller and IIorrocks (1988) have studied multiple scatter effects a t lower energies, including incident diagnostic x-ray spectra, and obtained correction factors for clinical use in tissue densitometry.

In SPECT, the original Monte Carlo simulat,ions of Beck el al (1982) have provided a method for evaluating scatter correction methods and analysing the contribution of different orders of scattering to the acquired images. Calculations were strongly based on variance reduction techniques, photons being restricted within a given solid angle, and forced to interact with the medium and to be detected. This Monte Carlo SPECT research programme has been fruitfully continued for several years by the group at Duke University, USA. Floyd el a / (1984) have simulated the energy and spatial

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Figure 6. Schematic diagram of the principle of scintillation camera detection and its Monte Carlo simulation including phantom and collimation effects. (a) For an ideal detector, only photons per- pendicular to the collimator will be detected (full lines); in practice, primaryphotonr within a certain solid angle will be recorded (dotted line); due to the finite energy resolution of the system, photons scatteredin the phantom willalso be detected (brokenlines). ( a ) The energy pulse-height distribution due to primary and scattered photons can be measured or calculated (full line); it can be separated into diKeerent contributions (total scattering or dilferent orders of photon scattering) using a Monte Carlo simulation (from Ljungherg c l nf 1990, with permission).

distributions obtained from a line source. They found that second-order scattered photons overlap with non-scattered photons in the photopeak and exhibit an exponential spatial projection; higher order events appear outside the peak and can be excluded using a narrow window. Floyd el a/ (1985a) have used Monte Carlo techniques to evaluate and justify the empirical development of the two windows subtraction technique proposed by Jaszczak et a/ (1984); they showed that the underlying assumptions were valid. Following existing analytical treatments, a deconvolution algorithm for scatter correction has been developed by Floyd el a/ (1985b); Compton scatter is modelled as a convolution of the non-scattered projection with an exponential function describing the scatter component, whose parameters are obtained by fitting Monte carlo simulations of line sources in water. The search for unified reconstruction algorithms led this group to the development of SPECT Inverse Monte Carlo reconstruction techniques

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that are discussed in section 5 . Simple solutions for scatter correction have also been proposed by Floyd el a1 (1988), who presented an analytical polynomial function to relate the scatter fraction to different variables of SPECT systems; the function is a fit to Monte Carlo calculations of the dependence of scattering with depth, energy and detecting window.

A recent comparison between the dual-window and convolution correction techniques based on Monte Carlo simulations has been presented by Ljungberg e t al(1990), who found tha t both techniques underestimate the contribution of scattered photons, and that the depth-dependence of scatter parameters is not taken into account in the existing algorithms. Ljungherg (1990) has developed a depth-dependent scatter correction technique based on Monte Carlo calculations of scatter functions at different lateral positions and depths within a phantom showing enhancement in the image contrast.

Compton scattering effects on PET profiles in water have been simulated by Logan and Bernstein (1983) for comparison with convolution using an exponential function for scatter compensation. Bruno e t a1 (1984) have reported good agreement between Monte Carlo simulations and scattering experiments on a positron tomograph, whereas Bendriem el a1 (1987) have used simulated data to evaluate scatter profiles and scatter fraction and to derive a scatter deconvolution filter.

4.1.3. Absorbed dose calculations. The determination of absorbed dose in different organs due to internal irradiation has traditionally been based on tabulations given by the Medical Internal Radiation Dose (MIRD) Committee of the Society of Nuclear Medicine in USA through their so-called ‘MIRD Pamphlets’. Some of these Pam- phlets have made extensive use of Monte Carlo calculations to derive specific absorbed fractions for photons and electrons, that is the fraction of the emitted energy per unit mass of the medium, which is absorbed a t a distance from the source, and they were already in use a t the time of the 1976 review article by Raeside. During the last decade most of the absorbed dose Monte Carlo calculations involving y-emitter radionuclides have been performed with brachytherapy sources and in consequence will be treated in section 4.3.4.

Interest in dose calculations with @-emitters has been revived with the application of labelled monoclonal antibodies to radioimmunotherapy. The term ‘kernel’ to denote the spatial distribution of absorbed dose around point-isotropic sources of electrons was used originally by Berger (1973). He has calculated kernels for monoenergetic electrons with energies between 10 MeV and 0.5 keV using a combination of macroscopic (full grouping of interactions) and microscopic (single interactions) Monte Carlo techniques for electron energies above and below 20 keV respectively. Details of the microscopic code together with valuable discussions on the limitations of Bethe stopping-power (restricted) and Moliere’s theory of multiple scattering at low energy have been given by Berger (1972). Another microscopic Monte Carlo code developed at ORNL (OREC, cf section 4.5.2) has been used by Turner el a1 (1988) in applications related to P-ray dosimetry in tissue-equivalent materials, finding good agreement with experimental dose distributions.

Based on the Monte Carlo results of Berger (1973), Prestwich el a l (1989) have evaluated kernels for certain nuclides of interest in radioimmunotherapy by weighting the data for monoenergetic electrons over @-ray spectra. An EGS user-code has been used recently by Simpkin and Mackie (1990) to compute p dosekernels for monoenergetic electrons (3 MeV-50 keV) and @-emitters. They have compared their results with

Monle Carlo techn.iques in medical radiation physics 885

the previously described calculations by Berger, finding systematic differences at high energies (justified in terms of the incorrect sampling of energy-loss straggling in the old version of ETRAN). The agreement was better a t low energies even though the 10 keV energy cut-off used in the EGS user-code should make the treatment of energy-loss straggling inaccurate. Good agreement w a s obtained with results for 1 MeV obtained with an updated version of the ETRAN code.

4.2. Diagnostic radiology

The main goal of the application of Monte Carlo techniques to diagnostic radiology is the optimization of diagnostic procedures to improve the image-quality/patient-dose ratio. Many publications in the field have dealt exclusively with radiation protection aspects of different diagnostic techniques, but the amount of scientific work investigat- ing physical quantities or characteristics of the detection systems bas been increasing since the early 1980s. A very interesting and general aspect of these applications has been the improvement of sampling techniques from certain photon interactions (mainly coherent scattering, see section 2.2). Electron transport has been systematically excluded from these simulations for reasons similar to those given for nuclear medicine applications (short electron ranges and negligible bremsstrahlung).

The extensive work in the field done by Chan’s and Doi’s team (University of Chicago) deserves special mention. It covers different aspects of the three snbgroups considered here and has been described in detail in a review by Chau and Doi (1988), which is recommended to those readers wishing to expand this summary.

4.2.1. Detection sysfems. The use of the Monte Carlo method to investigate basic components of the detection system in diagnostic radiology started in the early 1970s. DePalma and Gasper (1972) have simulated photographic emulsion layers to obtain modulation transfer functions (MTF) of different emulsions. Morlotti (1975) has performed simulations to obtain MTF and x-ray efficiency for fluorescent screens to investigate the performance of different phosphors. A Mont,e Carlo code which includes binding corrections to coherent and incoherent scattering (cf Chan and Doi 1983b) has been used by Chan and Doi (1983a, 1984a) to investigate x-ray energy absorption in different screen phosphors, including energy, angular dependence and quantum noise.

As in the preceding section, the effect of image contrast degradation due to scattered radiation has been one of the basic items where Monte Carlo studies have proven to be useful. The performance of anti-scatter grids has been investigated by Kalender (1981a) computing the scatter transmission of different grids. Chan and Doi (1982) have evaluated anti-scatter grids using the contrast improvement factor and the Bucky factor as benefit and cost indicators. This study was complemented with further simulations and experimental verifications of a number of clinical systems by Chan el ai (1985a). The use of an air gap technique as opposed to a grid to reduce scattered radiation from small fields has been proposed by Persliden (1986), who has investigated the influence of air gaps for different detectors in various conditions. Applications of the Monte Carlo technique to the study of anti-scatter grids in mammography have been developed by Dance and Day (1984) where the effect of divergent beams has been included. Chan e l a l (1985b) have conducted a comprehensive study on the application of ultra-high-strip-density grids to mammography, to improve the image contrast, where the effect of various grid parameters has been simulated. A tissue-equivalent

.

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filter to reduce scatter radiation in the peripheral region in mammographic imaging has been investigated by Lam and Chan (1990).

The energy response of detectors commonly used in the measurement of x-ray spectra (Ge, Si(Li) and Nal) has been investigated by Chen e t 0 1 (1980), where methods for correcting the measured spectra were examined. Seltzer (1981) has calculated the response of intrinsic germanium detectors giving the results as a set of formulae valid for any energy (up to 300 keV) and detector size. Characteristic x-rays from the germanium have been included in his calculations, but coherent scattering and binding effects in incoherent scattering were excluded (see the description of ETRAN in section 3). A similar study for 21 commercially available detectors has been undertaken by Chan et al (1984), finding consistent differences with the results of Seltzer (1981), which have been explained in terms of the exclusion of coherent scattering and binding effects. The influence of a tungsten absorption edge filter on the x-ray spectra from diagnostic units has been investigated using experimental and Monte Carlo methods by Yamaguchi el al (1983). Analogous simulations have been performed hy Kulkarni and Supe (1984a) for a mammography unit. The fraction of photoelectrons escap- ing from small silicon detectors has been investigated by Aoki and Koyama (1990), who calculated correction factors for different detector sizes which enable spectral measurements to be made comparable with Ge detectors.

4.2.2. Determination of physical quantities in diagnostic radiology. The computation of quantities related to the scattering of photons has been strongly biased towards determination of scatter-to-primary ratios or scatter fractions both using calculated photon fluences or absorbed doses and, to a lesser extent, towards spatial distributions of scattered radiation. Most calculations are based on the scattering of a monoenergetic photon pencil beam incident on a parallelepiped or slab water phantom.

Kalender (1981b) has calculated scatter fractions for different detectors and various object thicknesses, varying also the field size, detector-object distance and incident energy (monoenergetic photons between 30-150 keV). Chan and Doi (1983b, 1985) have derived quantities such as scatter fraction, energy spectra (including mean energies), angular distributions (including mean exit angle), and spatial distributions of the scattered radiation in water and perspex. Neitzel e t a l ( l985) have calculated pencil beam radial profiles in water and the transition to broad beams, studying the importance of coherent and single incoherent scattering. The small-angle distribution of photons transmitted through water slabs has been computed by Persliden and Alm-Carlsson (1986) using a collision density estimator which combines analytical calculations and Monte Carlo simulations. The ITS/ACCEPT code has been used by Barnea and Dick (1986) to simulate experimental arrangements by other authors (using polystyrene instead of water as in most other calculations) and to make comparisons with some of the results previously mentioned.

4.2.3. In evaluating the potential risks of x-rays in diagnostic radiology, Chan and Doi (1984b) have calculated the spatial distributions of energy deposition from pencil beams in water slabs, together with ‘rad/R’ conversion factors and scatter-to-primary ratios for absorbed doses in a phantom. The conversion factors were used to estimate absorbed doses at different locations in the phantom for a given absorbed dose in the recording system, and to select an optimal energy. The factors necessary to obtain the energy imparted to a phantom from measurements of the collision kerma in air have been determined by Persliden and Ah-Carlsson (1984).

Radiation protection aspects.

.

Monte Carlo techniques in medical radiation physics a87

As an application in computed tomography, a Monte Carlo code has been developed by Beck el al(l983) to estimate absorbed doses in specific regions within a cylindrical phantom exposed to rotating x-ray sources.

Studies on absorbed doses from radiological examinations have been mainly dominated by mammography during the last decade. Doi and Chan (1980) have determined 'rad/R' conversion factors and backscatter factors in mammography taking into account x-ray spectra. Their values were found to be considerably higher than previously reported values based on TLD measurements. Calculations of backscatter factors by Grosswendt (1984) for different field sizes were systematically lower than those of Doi and Chan (1980), although the geometry used in the latter (pencil beams) would be more appropriate for very large field sizes. Dance (1980) has developed a Monte Carlo code for the computation of integral dose per unit incident exposure from monoenergetic photons within the energy interval used in mammography; the results have been extended to x-ray spectra by numerical integration. Various filtrations of spec- t ra have been used to derive absorbed dose in the brea3t from a given dose to the recording system using a detailed geometry (see figure 7). Recent simulations have been performed by Dance (1990) to determine conversion factors to estimate the mean glandular breast dose from measurement,s of the incident air kerma to a PMMA phantom. The differing composition of the breast in different age groups has been taken into account by Kulkarni and Supe (1984b).

F o m l spot O f x - r q t"k

Compressed brmst

corsette ond detector

Figure 7. Geometry used in the Monte Carlo calculalions of integral absorbed dose in ma~mno. graphic radiology examinations by Dance (1980) (wit11 permission).

4.3. Radiotherapy physics

Burliu et al (1973) have indicated a number of situations where Monte Carlo methods could play an important role in medical radiology. Since then, applications to radiotherapy and dosimetry have become the most widespread kind of simulations during the last decade, following developments during the late 1970s. An important aspect

888 P Andreo

of modern applications is understanding the significance of the transport of secondary electrons in the energy range of radiotherapy; this had been underestimated (and omitted) in some of the inspiring early applications.

Most recent developments in the', area are based on one of the two major public codes previously described, EGS and ETRAN or ITS, and their use continues to increase. The number of existing applications b&ed on these or other Monte Carlo codes is today so large that only selected cases can be presented here. Further details and references can he found in recent reviews by Mackie (1990) and Rogers and Bielajew (199Oa) on applications of the method to radiotherapy and dosimetry respectively.

4.9.1. Tele-therapy sources and dosimetry equipment A large number of investigations have considered simplified configurations of external electron and photon radiotherapy sources. This is the case with the calculations of Patau e l a1 (1975) on the energy and angular spread of electron beams emerging from foils, and studies of Berger and Seltzer (1978) on the effect of scattering foils in electron central-axis depth-doses, where the influence of energy and angular spread has been considered. Manfredotti e t a1 (1987) have considered a simple geometry t o simulate an electron collimator and score distributions of quantities at a phantom surface which were used later in three-dimensional dose-planning simulations. Electron spectra after foils and air have been calculated by Andreo el a1 (1989) in relation to ionization chamber electron dosimetry. For photon beams McCall e t a1 (1978) have calculated spectra from different high-energy x-ray targets and flattening filters using the EGS system, and Nilsson and Brahme (1981) have investigated scattered photons from flattening filters and collimators in therapeutic photon beams. Patau et a1 (1978) have pioneered the complete simulation of a photon beam, including the photon generation in a W-Cu target, the transport through a flattening filter and collimator, and attenuation in slabs of different materials; they have investigated the energy and angular spectra of photons and electrons at every location using a simplified geometry.

simulations of 6oCo sources and therapy treatment heads using detailed geometry have become ofgreat importance. Pioneer calculations of Berger have been reported by ICRU (1970) for encapsulated 6oCo sources, where photons scattered by the tungsten sleeve were the most important component of the 6oCo spectrum. Similar calculations have been reported by Nan ef a/ (1987) and Rogers et 01 (1988), including collimators and filters to simulate in detail the treatment head'of a 6oCo unit. Petti et a1 (1983a, 1983b) have investigated the electron contamination in photon beams, for the first time simulating in great detail the treatment head of a clinical accelerator; the energy and angular distributions of both photons and contaminating electrons at the phantom surface, and their origin, ha;e been determined using the EGS code. Similar detailed simulations also based on the EGS system have been performed by Mohan et a1 (1985) for photon beam spectra from different acceleratvrs, and by Udale (1988) for electron beams to investigate surface doses due to electron contamination by the beam defining system (figure 8). Detailed investigations on reflecting electron appli- cators and scattering foils in clinical accelerators had been described previously by Ahou Mandour (1978). It should be pointed out that all these EGS-based calculations of bremsstrahlung angular distributions, or spectra a t different angles, have used the default crude approximation for the photon emission angle 8., = mc2/Eo, Eo being the electron total energy and me2 the electron rest energy. As discussed in section 3, Bielajew et a1 (1989) have implemented the sampling of bremsstrahlung photon angles in the EGS system; they have found important discrepancies in the radial variation


of photon spectra compared with calculations using this approximation which might invalidate some of the results mentioned

Mirror -\

A Phantom -

I bl C o s A a c 0 t

Figure 8 . Detailed and schematic diagrams of the geometry used in the Monte Carlo simulation of the treatment head 01 a therapeutic electron accelerator using the E G S I system. The different c-es shown in the schematic diagram produced a progressive improvement in the agreement between the calculated and experimental depth-dose distributions of a 10 MeV electron beam (from Udale 1988, with permission).

A common aspect of the simplified and detailed simulations of therapy sources, when in-phantom studies are included (such as determination of depth-doses or spectra), is the use of a two-step Monte Carlo procedure. During the first step, the energy and/or angular distributions of the radiation field are determined down to the phantom surface after passing through foils, collimators, air, etc. A second step is used for the in-phantom simulation itself, and these distributions are sampled independently to define the initial state of the incident particles. The lack of correlation during the sampling of broad energy and angular distributions during the second step has been pointed out by Andreo and Fransson (1989) for electron beams, which should be a warning for other applications where the siniulations are divided into two parts,

The simulation of the response of ionization chambers exposed to 6oCo beams in air has received considerable attention and raised controversies, but there are still discrepancies which are not yet fully understood (see for instance figure 5 ) . The importance of accurate electron transport and the production of interface artifacts in regions which are small compared with elec'tron step sizes, has been discussed extensively by Bielajew et a / (1985) and Smyth (19%) (cf also Smytli and McEwan (1986) and Bielajew and Rogers (1986b)). This topic has been reviewed in depth by Nahum (1988b). The small number of interactions within the ion chamber volume

890 P Andreo

requires the use of variance reduction techniques. Even so, computation times needed to achieve low uncertainties (less than 1%) are extremely long, specially when ion chambers are simnlated inside a phantom (cf Nilsson e t al 1988) where hundreds of hours of CPU are often needed. This can be considered the major limitation of a direct Monte Carlo simulation, unless more sophisticated variance reduction techniques are used (which usually requires an in-depth knowledge of the code). The combination of analytical and Monte Carlo methods (see section 2.2) has not yet been used in this area.

Bond et a / (1978) and Nath and Schulz (1981) have simulated the response of ion chambers and determined wall correction factors for “CO y-rays in air which are in common use today in the majority of national and international recommended procedures for dosimetry. McEwan and Smyth (1983, 1984) and Rogers el ol (1985) have found, however, important discrepancies between their results and those of Nath and Schulz (1981) for the simulation of chamber response, attributable to defects in the Monte Carlo code of the latter authors, whereas differences in the calculated correction factors (which are ratios of responses) were within statistical uncertainties. Simulations of ion chambers including the effect of the central electrode at the energy of ‘“CO y-rays have been reported by Smyth and McEwan (1984) and Rogers et a! (1985), both finding good agreement with existing experimental data. The stated uncertainties in both set of calculations were on most occasions larger than the correction itself, which confirms the dificulties in simulating ion chambers previously discussed.

Bielajew (199Ob) has developed an analytical theory to derive correction factors for thick-walled ion chambers exposed to WO beams in air, which has been verified using Monte Carlo calculations. Bielajew has simulated some of the chambers from various national primary standards dosimetry laboratories and found discrepancies with Correction factors being used in the laboratories. Further simulations by Rogers and Bielajew (1990b), to determine wall correction factors for these standard cham- hers, have also shown differences with the values actually being used; fortunately the discrepancies, in practice, compensate for each other. To avoid ‘brute force’ calculations of the chamber response, to obtain correction factors as ratios of responses, direct ratios of quantities that provide an alternative definition of the correction factor were scored instead during the simulation. This results in a drastic improvement in computation time, being equivalent to a correlated sampling technique.

Recent applications of the Monte Carlo method have simnlated gap effects in graphite calorimeters (Boutillon 1989); corrections for such effects had been neglected so far by most standards laboratories but have been found to be non-negligible.

4.3.2. In-phantom simulations. A better understanding of the physics of radiation transport in matter has been achieved with the use of Monte Carlo techniques and the potential of ‘switching on/off’ different interactions. Calculations by Seltzer e t al (1978). Andreo and Brahme (1983), Nahum and Brahme (1985), etc, belong to this group, where second and higher order components of different quantities have been investigated in electron beams. The basic geometry of point-monodirectional beams, known as ‘pencil beams’, has proved to be of great interest as the fundamental component of broad beams and for treatment planning algorithms. Berger and Seltzer (1978, 1982) and Andreo (1980, 1981) have performed extensive calculations of energy deposition for electron ‘pencil beam-infinite beams’ based on the so called reciprocity relationship (cf ICRU 1984a). The topic has been reviewed in detail by An- dreo (1988h). Electron depth-doses and ranges for broad beams have been obtained by

Monte Carlo techniques in medical radiation physics 89 1

numerous authors, and the investigations by Rogers and Bielajew (1986), Grosswendt and Roos (1989) and Andreo (199Ob) can he added to the references for pencil beams previously given.

The calculation of other physical quantities in a phantom, like energy spectra, mean energy, stopping-power ratios, energy-absorption coefficient ratios, etc, has been cultivated extensively. The initial calculations by Berger and Seltzer (1969) of electron spectra and water/air stopping-power ratios (sW,*jr) for monoenergetic electron beams (Berger e l al 1975; see also ICRU 1984a), have been improved by developments by Nahum (1976, 1978) for the evaluation of Spencer-Attix ~ ~ , ~ i ~ including a track-end term. Andreo (199Ob) has recently calculated stopping-power ratios for monoenergetic electrons, both with the EGS system and the MCEF code, finding ex- cellent agreement between the two codes. The effect of energy and angular spread on

has been investigated by Audreo et al (1989). It was found that the method recommended in dosimetry protocols yields sw,&ir values that differ by less than 1% from those obtained from direct Monte Carlo calculations for beams with very large energy and angular spread. These calculations have been extended by Andreo and Fransson (1989) to include low-energy electron contamination using the realistic electron beams simulated by Udale (1988), confirming the results previously mentioned. The depth and radial variation of the mean energy in electron beams, together with the practical implications in electron dosimetry, has been investigated by Andreo and Brahme (1981).

For photon beams, Nahum (1976, 1978) w a s the first to calculate stopping-power ratios based on the Monte Carlo method to derive electron slowing-down spectra a t different depths. The technique has been used by Andreo and Nahum (1985) to compute sw+ir values for monoenergetic photon beams, developing a method to weight such data and obtain stopping-power ratios for bremsstrahlung spectra that were verified by Monte Carlo simulations of the complete spectra. Stopping-power ratios correlated consistently with the quality of photon beams have been computed by Andreo and Brahme (1986), who found large discrepancies with existing recommendations of sw,+ir

values based on the nominal accelerating potential of an accelerator (MV). These investigations were performed for a large number of bremsstrahlung spectra (figure 9) and also included the effect on ~ ~ , ~ i ~ of the electron contamination in photon beams and dependence with depth and field size. The results have been used to calculate stopping-power ratios for different materials and derive ion chamber correction factors by Andreo et al (1986); this work giving the complete set of data used today in the majority of dosimetry protocols. A review of Monte Carlo techniques applied to calculations of electron spectra in water produced by electron and photon beams, and further determination of stopping-power ratios, has been performed by Andreo (1988~). Mass energy-absorption coefficient ratios for high-energy photon beams have been computed by Cunningham e l 01 (1986). The same set of bremsstrahlung spectra used by Andreo and Brahme (198G) to calculate stopping-power ratios has been used as input for further calculations of p.,/pratios by Cunningham (cf IAEA 1987), producing a consistent set of data for photon dosimetry.

4.3.3. %atmenl planning applications. Several investigators have repeatedly indicated the non-feasibility of direct treatment planning applications of the Monte Carlo method using then current standard computer technology due to the long computation times involved (cf Nahum 1985, 1988a; Mackie 1989). The main validity of the method has been in verifying algorithms and techniques and, more recently, providing

892 P Andreo

TPR200IlWfor lWmmx1Wmmfleld

Figure 9. Waterlair stopping power ratios as a function of the beam quality for a broad range of photon beams with large variations in spectral shape. The beam quality is specified as the ratio of absorbed doses a t 200 mm and 100 mm depth for a 100 nun x 100 mm field size a t the position of the detector (parallel beam configuration). Triangles, data for monoenergetic photon beams; open squares, for bremsstrahlung thin target spectra; full squares, for thick target spectra (thickness equal to one CSDA range in tungsten); full circles, for accelerator spectra published in the literature; full line, data for realistic clinical spectra used in dosimetry protocols since 1987. The spectra were used as input for Monte Carlo calculations of electron Buence a t different depths in a water phantom (from Andreo 1989).

data for new dose planning procedures. The special case of inhomogeneities and interfaces has received great attention,

and as with ion chamber simulation, the need for accurate simulation of the transport of secondary electrons has been demonstrated (cf Nahum 1988b). Calculations by Webb (1979) using the code of Webb and Parker (1978) ignored electron transport in a 6oCo beam and therefore did not predict correctly the energy deposited close to the interface. Webb and Fox (1980) used the same code to verify inhomogeneity correction algorithms in photon treatment planning. Further Monte Carlo calculations a t interfaces with %o beams have also been reported by Horowitz et al (1986), whereas Seltzer and Berger (1987) have included high-energy photons and electrons as well. The ITS system has been used recently by Das et al (1990) to calculate electron spectra a t interfaces and investigate the validity of stopping-power ratios used to compute absorbed dose from measurements a t interfaces with high-energy photons. For electron beams, treatment planning standard inhomogeneities have been simulated by Shortt e t a/ (1986) and compared wibh experiments, showing that the EGS system can be used to calculate benchmarks against which treatment planning algorithms can be compared.

The use of Monte Carlo techniques in calculations related to electron pencil beam algorithms based on the theory of Fermi-Eyges has been successfully employed by several investigators. Lax e t a / (1983) used Monte-Carlo-calculated pencil beams (from MCEF and ETRAN) to fit the parameters of a multi-Gaussian function, the basic component of a generalized Gaussian algorithm. Bielajew e t al(1987) and Manfredotti e l 01 (1987) used the EGS system to verify different approximations and performance of electron pencil-beam algorithms, whereas Bruinvis el a/ (1989) calculated range straggling functions which have been implemented in a given pencil-beam algorithm.

Monte Carlo techniques an medical radiation physics 893

The traditional technique, used in low-energy photon-beam dose-planning algorithms, of separating dose into primary and scatter components, has been investigated by Mohan and Chui (1985). They have criticized the physical basis of such approach; however the Monte Carlo method has been used by Kijewski et a l (1986) and Rice and Chin (1990), among others, t o develop further improvements in this separation technique.

Calculation methods for photon beams based on the convolution of Monte Carlo calculated kernels, describing the energy deposited by charged particles during photon interactions, have been proposed simultaneously by Mackie et al (1985) and Chui (1985) (cf also Mohan e t a1 1986). Ahnesjo e t a/ (1987) have further developed the method and used the code MCEF to generate a database of kernels of monoenergetic photons (figure IO). The set of data has also been used for the reconstruction of bremsstrahlung spectra from clinical accelerators (Ahnesjo and Andreo 1989) and for the calculation of isotropic pencil beams for inverse Monte Carlo treatment planning (Eklof et a1 1990). Mackie et a l (1988) have computed EGS-based kernels, which have been compared by Mackie (1990) with the MCEF-based data, finding good agreement. The scaling method used in such convolution procedures when inhomogeneities are present has been investigated by Woo and Cunningham (1990).

Recent developments have addressed the possibility of performing direct Monte Carlo calculations in three dimensions using CT images, despite the long computation times involved, and new approaches have been suggested. An important argument for the use of Monte Carlo simulations in the presence of inhomogeneities is that experimental determinations of absorbed dose might be question in transition regions due to the lack of electron equilibrium at high photon arid electron energies, where detector responses are uncertain. The first attempt by Manfredotti et a / (1987), used for comparisons with a pencil beam algorithm, was based on a three-dimensional phantom instead of true CT images, but the final goal of computing three-dimensional dose distributions in small volumes is the same. Calculations in phantoms and actual CT data have been reported by Chui and Mohan (1988), stressing the importance of CT-based Monte Carlo simulations as benchmarks against which other approximate algorithms or dose planning systems could be compared. A very interesting algorithm t o speed-up the direct simulation has been developed by Manfredotti et al (1990), where the very large number of regions existing i n a CT slice is numerically decreased, reducing then the time spent in boundary crossing or checking during the simulation. The algorithm, called UNION and based 011 the EGS system, considers a number of pre- defined volumes of interest and joins regions out,side having tlie same density to form a much smaller number of larger boxes (figure 11). If transport parameters i u such boxes are considered less restrictive than in the volume of interest (see section 2.3.3), further time reduction could be achieved.

4.3.4. Calcvlalions in brachytherapy. With a few exceptions, most of the existing Monte Carlo calculations in brachytherapy refer to dose distributions or specific dose constants in water of lZ5I sources. On some occasions, the simulations have been based on very detailed geometries.

Williamson e t a/ (1983a) have simulated a well detector for the calibration of sources, and found discrepancies with tlie predictions of the Sievert integral model. This topic has been emphasized by Williamson el a / (1983b) using Monte Carlo simulations of encapsulated sources to compare with the Sievert integral, consistently finding overestimations of the exposure rate per unit activity predicted by the integral, Build-

894 P Andrea

Figure 10. Monte Carlo calculated kernels of monoenergetic photon beams in water using the code MCEF (Andreo 1980. 1981). Incident photon energies are 1, 5 , 20 and 50 MeV (left-right, topbottom). The kernels represent the distribution of the energy deposited by charged particles of successive generations produced by photons interacting at a point situated at a depth of 60 mm (1 tic = 10 mm). The importance of the electron transport (bright zone) at high photon energies can be compared with that at low energies where most energy is deposited at the first point of interaction (from Ahnesj6 1990).

up factors for low-energy photons (100-15 keV, including lZ5I) have been computed by Herbold e l ol (1988) using EGS.

Webb and Fox (1979) have computed the dose distribution in water of a number of point isotropic brachytherapy sources commonly used in the UK during the late 1970s, their results being given as radial distributions of the dose-rate. Dale (1982) has included Rayleigh scattering in similar calculations for different tissues and nuclides (including lZ5l and 13'Cs), finding small disagreements with the specific dose constants for water determined by Webb and Fox (1979). Further analysis of his data, emphasizing the results for '"1, has been given by Dale (1983).

Dose distributions in water around commercially available '''I sources (seeds) have been simulated by Burns and Raeside (1987). Scarbrough e t al (1990) used the EGS system and included K-edge photons to calculate similar distributions. Specific dose constants for the same source model have been computed by Williamson (1988) using the detailed geometry of the source and including the emission of K-, L- and M-edge x-rays. Less sophisticated calculations have been performed by Cygler e l ~l (1990) to investigate the effect of gold and silver backings using EGS. Chiu-Tsao e l ol (1986)


Figure 11. Details of an algoritbrn to joiu contiguous regions in a CT slice having tbe same density, what decreaJes the number of boundary crossing and checking during tbe Monte Carlo simulation. (a) CT slices of an anthropomolphic Rando phantom, where a mandible level is selected. (b) CT matrix (160 x 160) of the selected slice having a. spatial resolution of 2.2 nun x 2.2 mm will , CT nmnben replaced by density; both air (oropharinx) and bone (mandible and vertebrae) are left without a grid for sake of clarity. ( c ) Resulting image after the application of the UNION algorithm. which keeps the same inhomogeneities border as tbe original matrix. (d) A large volume of interest is selected having the same resolution as the original matrix; tbe nuniber of geometrical regions is considerable reduced (from Manfredotti e t a/ 1990, with perniieeion).

have used the MORSE code to compute lZ5I and 6oCo dose distributions at distances used for eye plaque therapy and therefore snialler than in the simulations previously cited.

896 P Andrea

4.4. Radiation protection

Specific Monte Carlo applications have been developed in the field of radiation protection which can he classified as an additionalgroup to those considered in nuclear medicine and x-ray diagnostic.

Following the mathematical descriptions of Snyder e l a/ (1969) of anthropomorphic phantoms for the MIRD-5 pamphlet, Cristy (1980) has developed age-dependent hermaphrodite phantoms for the calculation of internal exposures. Extensive work on sex-specific phantoms has been undertaken at Gesellschaft fur Strahlenund Umwelt- forschung (GSF) for the calculation of dose from external photon sources in a variety of situations. The mathematical phantoms Adam and Eva (figure 12) have been described by Kramer et a / (1982), and a number of reports have been published by GSF dealing with organ doses after a number of diagnostic and therapeutic procedures (cf Williams el a / 1984, Drexler et a / 1984, etc).

Figure 12. Perspective superpositions of verticd planes tllrougli the male (Adam) and leitiale (Eva) mathematical phantom used in Monte Carlo calculations of absorbed doses from external photon sources (from Kramer el a1 1082, w i t h permission).

Calculations related to the ICRU sphere, instead of using anthropomorphic phantoms, have been performed by Dimbylow and Francis (1984) among others. An interesting aspect of their photon Monte Carlo code is that it includes electron transport (only under the continuous-slowing-down approximation (CSDA)), which is unusual in radiation protection applications despite the high-energy photon limit usually considered (around 10 MeV). An up-to-date review of ICRU sphere calculations has been given by Grosswendt (1988b).

A few applications in the field have been concerned with the shielding of medical


diagnostic or therapeutic units. Simpkiii (1990) has used the EGS system to compute the transmission of x-ray spectra from computed tomography units through different materials, the results being fitted to a specific analytical model. Nelson and LaRiviere (1984) have combined calculations with the EGS and MORSE codes to evaluate the shielding of medical x-ray accelerators of different energies (up to 25 MeV).

4.5. Applicalions based on microscopic Monte Carlo techniques

The need for Monte Carlo calculations based on the electron’s detailed history (or microscopic) techniques was accentuated in section 2.3.2. Most applications in this area have emphasized different aspects of the physics included in the codes. This can be justified in terms of the complexity of the phenomena governing the transport of low- energy electrons, and the main difference between existing codes and applications lies in the details of how the different interactions are treated. Bremsstrahlung production has been usually neglected due to the low energy of the incident electrons, although some cases have considered initial energies and media where this assumption is not justified. With a few exceptions the energy range of the primary electrons is mainly below 100 keV. The consideration of secondary and higher-order generated electrons is, however, a general difference between the two groups subjectively chosen here to classify microscopic applications.

4.5.1. Eledron microscopy. Traditionally, electron microscopy and related areas have not been included in the field of medical radiation physics. They have been included in this review because the description of the physics included in the Monte Carlo codes will help the reader to understand the more sophisticated approaches used in some applications dealing with radiation track structures and microdosimetry. Most developments in this area are based on existing theoretical cross sections. In what follows, only applications including inelastic processes separated into different contributions will be considered; simple approaches based only on individual elastic collisions and the CSDA to account for energy losses (cf Williamson and Duncan 1986) will not be discussed in this section.

For primary electron energies between 15 and 20 keV, Shimizu el a l (197G) have used a number of elementary excitation processes (conduction-electron, plasmon and L-shell electron excitations) contributing t o inelastic scattering. The approach has provided good agreement with experiments for both energy and angular distributions in aluminium films. A different model using Gryzinski excitation functions for both core (L-shell) and valence electrons has been developed by Adesida el a1 (1980) for materials where not all the inelastic functions are available; they also found good agreement with experimental data for energy and angular distribution, together with backscattering and transmission coefficients, for the same energy range i n aluminium, silicon and PMMA. An interesting aspect of the latter approach has been the use of the screening parameter of Nigam in the Rutherford elastic cross section, and the consideration of angular deflections for each inelastic event. Further details of the physical models used to account for elastic scattering and discrete contribotions to inelastic scattering have been given by Kyser (1981), who has reviewed some of the Monte Carlo techniques used in electron microscopy, microanalysis and microlithography. Applications to magnetic contrast and spatial resolution have also been described in this review.

Electron transmission, backscattering and absorption in solid targets have been investigated by Akkerman and Chernov (1980) for incident energies in the range 10

898 P Andreo

to 30 keV, electron transport being simulated down to 100 eV. In this calculation, electron-hole excitations computed from dielectric theory were added to plasmon excitations and ionization processes. A comparison between individual collision and condensed history techniques has been performed by Akkerman and Gibrekhterman (1985) for electrons below 50 keV in different elements. They have found that the two main errors in macroscopic techniques, namely the neglect of lateral deviation along a path and the incorrect calculation of energy losses (due to neglect of plasma and electron-hole excitations), might compensate for each other as the results appear to be correct in some cases. Approximate atomic number and energy ranges for the validity of condensed history techniques have been suggested in this interesting work.

A different approach to deal with energy losses from all inelastic collisions has been developed by Liljequist (1985) with a simple generalized oscillator-strength density model. Although the model was applied to conversion electron Mossbauer spectroscopy, the simplicity of the resulting Monte Carlo code has been s t r w e d in successful comparisons with some of the more sophisticated approaches previously referred to.

Applications including higher energies have been developed by Terrisol and Patau (1978); they have calculated energy loss distributions for 1 MeV electrons crossing aluminium foils with thicknesses in the micrometre range; they found good agreement with experimental results obtained with electron microscopes. Their Monte Carlo code is based on calculations of the mean free path using an elastic screened relativistic Mott cross section (taken from Berger et a/ 1970) and inelastic processes consisting of ionization (Gryzinski cross sections) and excitation in the conduction band. Martinez and Balladore (1979) have calculated angular and spatial distributions, and related beam transmission at different angles, for 1-3 MeV electrons scattered by gold and carbon foils with different thicknesses.

4.5.2. Radiaiion irack structure and microdosimetry. Although initial developments in the field were based on theoretical cross sections, the importance of experimental data a t low-energies to overcome limitations of the existing theories was acknowledged in the late 1970s. Many of the experiments available have been performed for gases, and data have been extrapolated to condensed media, or extended to other energies, using Fano plots. Different aspects of the techniques used in this area have been excellently reviewed by Pareztke (1987), which together with the more updated compilation of data given by Groaswendt (1988a), constitute a valuable source of information in this area.

As in many other aspects of Monte Carlo electron transport, pioneer calculations by Berger have greatly contributed to later developments in the field. A Class 11-like procedure has been used by Berger (1970) to calculate energy spectra and spatial distributions of energy deposition from 5 keV electrons in spherical regions in water. The transport has considered all individual elastic scattering events, as well as inelastic collisions producing knock-on electrons with energies greater than 200 eV. Energy losses between collisions have been calculated using a stopping-power restricted to 200 eV. This initial work has discussed some of the limitations of the cross sections used (screened Rutherford and Mdller), mainly those aspects related to the screening correction of the elastic cross section and binding effects in the inelastic collisions. Both limitations have been approximately corrected for in later calculations of the spatial distribution of energy deposited in the atmosphere by Berger el a/ (1970), where limitations of the Moliere screening parameter have been analysed. The resulting code

Monte Carlo fechnipes in medical radiation physics 899

has been used further by Berger (1972) to calculate spatial distributions of absorbed dose in water for different electron energies and emission angles in tracks of heavy charged particles, and to determine microdosimetric event-size spectra from tritium P r a y s in micro-spheres in water and propane. Further detailed calculations of electron kernels in water (cf Berger 1973) together with statistical distributions of energy deposition events and their perturbation in gas-filled cavities surrounded by solid walls have been reported by Berger (1974).

Researchers a t ORNL have undertaken a project to build a Monte Carlo electron transport code (OREC) based on a large number of inelastic cross sections for liquid water, calculated mainly from dielectric response functions and experimental data. Up to 21 separate events, from ionization and excitation to radical formation and dissociative events, have been reported by Hamm el a1 (1976) to obtain mean free paths used during the Monte Carlo procedure to calculate kernels of 1 and 10 keV electrons down to 10 eV in water. The same code bas been used by Hamm el a/ (1978a) to study probability distributions for the number of ionizations and distributions of energy-loss in different size volume elements, together with single-event spectra. Electron slowing-down spectra in liquid water and statistical fluctuations in the ionization yield for primary electrons between 1 keV and 1 MeV followed down to 10 eV have been calculated by Hamm e t a1 (1978b); at low energies, energy spectra and spatial correlation of energy transfer were found to be nearly independent of the initial electron energy in agreement with previous findings of Paretzke (1976) in water vapour using the code MOCA (cf Paretzke et a l 1974).

The importance of low-energy secondary electrons in the mechanisms of energy absorption and track structure of low LET particles has been emphasized by Paretzke (1976) a t GSF, using a Monte Carlo code largely based on experimental cross sections for water vapour (figure 13). A comparative joint study between distributions of different quantities in water in liquid and vapour phases (single collision and slowing-down electron spectra, spatial distributions of energy deposition, etc) has been performed by Turner el a/ (1982) using the codes OREC and MOCA developed at ORNL and GSF respectively.

Berger (1981) has discussed the spatial correlation of ionizations i n water in terms of a restricted ionization yield (defined as the number of ionizations, per 100 eV, which are preceded by another ionization on the track within a distance s or closer) which has been calculated for various separation distances s and electron energies up to 1 MeV. The calculations apply to liquid water, hut experimental (extrapolated) data for water vapour have been used for excitation, dissociative excitation and ionization interactions; elastic scattering bas been based on atomic cross sections for hydrogen and oxygen together with experimental data a t some energies. These results have indicated that most significant spatial correlations occur in regions of the order of nauometres, but not in micrometre regions.

Track structures in liquid water including the production of ions and excitations have been described by Terrisol and Patau (1981) for 30 keV electrons in liquid water, which have been used in the calculation of proximity functions. Turner el a/ (1983) have included the detailed time evolution of chemical species around tracks in water, developing a model to describe the formation and spatial distribution of such species after electrons have been slowed down to 7.4 eV (assumed threshold for electronic excitation of liquid water); the Monte Carlo code has been used to calculate Fricke G-values for 5 and 1 keV electrons. The code has been re-examined by Wright et a1 (1985) regarding details of the pre-chemical and chemical reaction stages, and

900 P Andrea

Figure 13. (adapted from Paretzke 1987, with pemission).

significant improvements have been reported for the calculations of physical and chemical reactions produced in irradiated water containing DNA. In a recent application of this code Bolch el a / (1990) have investigated the production of free ammnonia in aquaeous solutions of glycylglycine, irradiated by x-rays and 6oCo y-rays, whose radiation chemistry is relatively simple; very good agreement with experimental results has shown the result of a unique attempt a t simulating, on a nanometre scale, the events occurring within an irradiated solution.

Calculations for gases have been performed by different groups at Toulouse (University Paul Sabatier) and Braunschweig (Physikalisch-Technische-Bundesanstalt, PTB) which are largely based on experimental data for different gases. The difficulty in collecting data explains why, in general, each independent investigation has dealt with only a few gases. Terrisol and Patau (1974) have simulated the transport of 1 and 0.5 keV electrons down to 20 eV in hydrogen to calculate energy deposition distributions, ranges, and W values (both for ionization and excitation) which were found to be in good agreement with experimental results from other investigators. Their code was based mainly on theoretical cross sections for ionization and excitation. Fur- ther improvements, considering total inelastic cross sections based on experimental data, have been developed by Terrisol e l a / (1976) to calculate similar quantities and event-size spectra in different gases (oxygen, nitrogen, carbon dioxide, methane, air, etc). Terrisol ef a / (1978) have extended these calculations to liquid water. Distri- butions of spatial energy dissipation and ionization, practical ranges, W values and backscattering coefficients in nitrogen and air have been calculated by Grosswendt and Waibel (1978); electron energies between 50 eV and 5 keV were considered in this investigation, which also included the influence of electrostatic fields existing in typical ionization chambers. The treatment of elastic scattering was based on the

Elutic, ionization, and excitation cross sections for electron impact on water vapor

I

Monle Carlo lechniques in medical radiation physics 90 1

developments of Berger el a l (1970), whereas for ionization and excitation processes existing analytical functions were used. Similar quantities have been calculated in methane by Waibel and Grosswendt (1983). Electron slowing down spectra in hydrogen and statistical fluctuations in the ionization yield for primary electrons between 20 eV and 5 keV have been calculated by Grosswendt (1982), and by Grosswendt and Waibel (1985) in methane, carbon dioxide and a gaseous mixture of methane and argon.

5 . Inverse Monte Carlo tecliniques

The term ‘Inverse Monte Carlo’ (IMC), a concept introduced by Dunn (1981), has been used to describe a numerical method for solving a class of inverse problems. I t has been used successfully in nuclear medicine imaging using SPECT (Floyd el al 1985c, 1986a, b) and in the design of compensating filters for photons in radiotherapy accelerators (Dunn el a l 1987). Other applications have been developed which are based also on this technique although different terms, like inverse treatment planning (cf Lind and Brahme 1987) have been used. The basic concepts of IMC will now he introduced following the development of Dunn (1981).

A general formulation can be used to describe the calculation of the expectation value (r(y)) from the outcome r ( z , y ) of a stochastic process which is a function of the random variable I and perhaps another independent variable y:

J-CU

where f(z) is a probability density distribution governing the variable I. A conventional Monte Carlo technique, henceforth called direct, will estimate

(r(y)) a t discrete values of y, given z(x ,y) and f(z). By the law of large numbers (sum of a large number of random variables converges to the expected mean value) i t can be written

where & is a random sample from I(.). i ( y ) is a Monte Carlo estimate of ( ~ ( y ) ) , i.e.

The IMC technique used to solve equation (5) consists of finding f(z) given z(z , y) and z(y) a t discrete values of y, which is equivalent to solving a Fredholm integral equation of the first kind. Two different, procedures have been used to solve this equation using Monte Carlo methods, and both have been called IMC in the literature.

One procedure is based on iterative direct Monte Carlo calculations. In general there will be a set of parameters a of the desired function f , i.e. f(z, a ) ; the approach consists in searching these parameters a unt,il some objective function O(a) is minimized. The method, in gencral, is to choose an initial estimate of ai, perform a direct Monte Carlo simulation sampling from the assumed f(x, a, ) to compute i(yi) , and then iterate the Monte Carlo simulation until the calculated i(yi) matches the known

902 P Andreo

value of r ( y ; ) . The direct simulation is performed for each discrete value z(y) where all the parameters a are iteratively searched. Some approaches of this type simply select a set of parameters a a t random, compute a model (which can be a simple function) and compare with predicted or experimental data to decide if the parameters are acceptable or not (cf Keen and McGreevy 1990).

The second procedure consists in non-iterative Monte Carlo simulations, and is based on inversion techniques as will be seen later; at some stage the method involves a direct (non-iterative) Monte Carlo simulation to obtain the function z(z, y), sometimes referred to as the kernel. As inversion methods are very often based on iterative techniques, this way of solving the final step of the procedure has added confusion to the iterative or non-iterative characteristics of the simulation itself. In the present work, the term non-iterative IMC is then used for an inversion method based on a non-iterative Monte Carlo simulation, and this is independent of the technique used to treat the inversion problem itself.

The non-iterative IMC is strongly related to the variance reduction technique in direct simulation called importance sampling, where the sampling process from f(z) is altered by using random numbers from a modified distribution, f '(z), instead. This introduces a bias which is corrected for with a weighting factor f ( z ) / p ( z ) , so that the estimate F ( y ) is an unbiased estimator of ( ~ ( y ) ) , that is

where ti is a random sample from f '(z). A typical example is when photon interactions are forced to occur within a given distance A , where the conventional sampling of the distance between interactions, given by equation (3), is substituted by the importance sampling described by

s = -Xin[l- [(I - e-'/')] (11)

where the weight factor is (1 - e-'/'). In the non-iterative IMC, given 11 discrete values of y at which z ( z , y) and t (y) are

known, the approach consists in finding a set m(m 5 n) of parameters a o f f , such that

f(z) = aj z E (.j,zj+i)>.i = . . , m

= O otherwise

or

Monte Carlo iechnigues in medico1 radiolion physics 903

Using imporlance sampling, the method involves one direct simulation using an assumed known probability density distribution f ’ ( z ) such that

whexe ti are sampled from f ’ ( z ) . Everything is known in this expression except the a, i.e. it consists of a system of n equations with m unknowns that can in principle be solved, at least numerically. I f f is linear in a, then

which can be solved by straightforward matrix inversion. I f f is nonlinear in a then a nonlinear inversion is needed. The method is then a single-step procedure, that is just one direct simulation is needed and therefore the Monte Carlo process is non- iterative. Depending on the specific problem, the elements of [ A ] computed with one direct simulation may not be adequate to construct all the elements of (21. In this case a different simulation would be required for each element of [ z ] although approximations have been used to avoid this requirement.

An interesting application of the non-iterative IMC method has beeu performed by Floyd el a1 (1986a, b) for image reconstruction from acquired projection profiles in SPECT. Equation (15) was written as

where p j describes the projection data, si the source distribution, and t i j represents the contribution to each projection element j of the scatter from each source element i in a phantom. [ t i j ] is calculated using a single direct simulation, identical to those used in scatter correction techniques, to determine the matrix of probabilities for the phantom-camera setup. The source has been assumed to have rotationally symmetric attenuation, and the simulation performed for only one projection angle; values for other angles were obtained by interpolation on a cylindrical grid. In subsequent steps, given the acquired clinical projections P , the inversion of equation (16) determines the imageS. The reconstruction discussed here yielded 11 520 (64 x 180) equations in 4096 (64 x 64) unknowns, where the use of an array processor reduced the computation time to a few seconds per iteration during the inversion procedure. Further developments have been performed by Floyd e t a1 (1986b) for the improvement of the inversion procedure using a Maximum Likelihood Estimation algorithm.

A similar technique has been used for optimizing of radiation therapy treatment planning (also referred to as inverse treatment planning), cf Lind and Brahme (1987), where equation (15) has been written in the form

In this case dj describes the desired dose distribution in the target volume (see figure 14), k i j is the contribution from the elementary irradiation kernel j at position i, and fi is the spatial distribution of the source (kernel density) which has to be calculated. The kernel It = [ k i j ] is calculated using a single direct Monte Carlo simulation as described in section 4.3.3.

904 P A n d r e o

Figure 14. Illustrationof the use of non-iterative ( IMC) techniques to generate aunifom dose in the target volume (oesophagus) by scanning s multileaf collimator over the palient. Slit photon beam kemels, K , based on direct non-iterative Monte Carlo calculations of energy deposition kernels using the MCEF code ( K i h a n e t a1 1988), are scanned with a density F to generate the required incident beam D. In the IMC procedure D and K are known, and F is determined by inverse techniques (adapted from Brahme e t 0 1 1988).

Dunn el ol (1987) have applied the method for the design of compensating filters in high-energy photon therapeutic beams, where the goal is to produce a desired photon fluence profile Q at certain depth. For simplicity the application considered solid angles subintervals to specify photon directions and filter thicknesses, Q(n,) and t ( n j ) , using M subregions (subindex k) in the phantom and m subregions (subindex j) in the filter. In this case, equation (15) was written in the form

[QIM = I A l ~ , m [ ~ l m + ~ [ B l ~ , m [ P l m (18)

where aj = e-#*; (j = 1 , . . , , m solid angle elements), p is the total attenuation coefficient, pj = ( 1 - aj) and e = C T C ~ ~ ~ ~ ~ ~ / ~ . Elements Ak,j are the direct Monte Carlo determinations of photons that do not interact in a unit thickness filter and Bk,, those that do interact, obtained from one simulation including interactions in the phantom. Equation (18) has to be solved for each t ( O j ) , which i n the case A4 = m can be achieved using simple matrix inversion and in the case M > m requires nonlinear techniques.

In some of the practical cases reported in this section, most efforts have been directed towards the inversion procedure needed to solve equation (15), or equivalently, equations (16-18). It is important to point out that the term non-iterative IMC can be applied only in those cases where the kernel is obtained using Monte Carlo methods, but not when it is obtained by analytical means. The situation can be compared with the use of convolution methods using photon kernels in radiotherapy, where both Monte-Carlebased and analytical kernels have been used.

6. Vector ized and parallel Monte Car lo siniulatioii

Most Monte Carlo applications referred to dnring this work have been developed on traditional or scalar computers where each instruction is processed one at a time (single

Monte Carlo techniques in medical radialion physics 905

instruction, single data stream, SISD). Vector computers where a single instruction is executed simultaneously on an entire list of numbers or vector (single instruction, multiple data streams, SIMD), have been proved to yield much higher performances. The CRAY-1, first installed in the middle 1970s, was the first commercially successful vector processor, initiating the era of supercomputers.

Only a couple of years ago, supercomputer speed was measured in MFLOPS (lo6 floating-point operations per second). This was the unit specified by Nahum (1988a), who has used a benchmarking study by Dongarra (1988) to est,imate the CPU time needed by different computers to calculate a radiotherapy photon-beam dose distribution with an uncertainty of 2% in a phantom divided into 5 mm sided cubes. This would take about 500 h on a VAX-11/780 (0.25 MFLOPs using single precision in the Dongarra study). An estimation of about 600 h to achieve the same uncertainty has been given also by Mackie (1989) for 10 million electron histories between 5 and 25 MeV, which is one order of magnitude larger than that specified by Nahum (1988a) for electron beams. Nahum estimated that a CR.AY supercomputer, rated a t 15 MFLOPS, would need ahout 4.5 h for the photon calculation.

Today’s large machines meamre their speed in GFLOPS (10’ operationsfs), and present performance figures make direct treatment planning on patients perfectly fea- sible. An estimation based on Dongarra’s results for a CRAY Y-MP/832 (8 processors, 6 ns clock cycle) running the LINPACK test, 195 MFLOPS, can be used to verify this statement; one arrives a t a figure of about 50 min for the radiotherapy calculation. The specifications given by Leghart (1990) for t h e CRAYY-MP, the present CRAY C-90 (16 processors, 4 ns clock cycle), and the next gallium arsenide (GaAs) based CRAY-3 (16 processors, 2 us clock cycle), can also be used to estimate peak performancest of 2.7, 8, and 16 GFLOPS respectively. Assuming that the peak-performancefL1NPACK- test ratio remains approximately constant, it can he estimated that the GRAY C-90 and the CRAY-3 would need 16 and 8 min, respectively, to perform the treatment planning calculation using a phantom divided into 5 m m sided cubes and 2% uncertainty.

The CRAY-4 (planned for 1992) will operate at a clock rate of 1 ns and deliver 128 GFLOPS using 64 processors (Bell 1989). Tomorrow’s speed (about 1995), with the full development of massive parallel-processing supercomputers, will he measured in TFLOPS operationsfs), i.e. the power ofmore than 10 million personal computers. Any estimation done with supercomputers today will therefore be obsolete in a few years.

As a consequence of the immense speedup of calculations in different areas of science with the use of vector machines, in combination with highly vectorized programs, Monte Carlo researchers have turned their efforts towards the vectorization of Monte Carlo codes. This means the reorganization of a program so that the maximum number of operations can he executed through vector processing. However, the conventional

t According to Dongsma (1988), the peak performance is a theoretical figure given by computer mandacturelg when describing a system. I1 is obtained by counting the nuniber of floating-point additions and multiplications (full precision) thal can be completed during a period of time, usually the clock cyde time. A GRAY c-90, with a clock cycle of 4 m during which bolh an addition and a multiplication can be completed in each one of its possible 16 processors yields

I

2 operations 1 cycle I cycle 4 ns

16 processors = 8 GFLOPS

906 P Andrea

Monte Carlo approach with a history-based algorithm, where particle histories are independent, does not lend itself to vectorization. This is due to the random nature of the particle simulation, where each step of a history is determined by statistical means, which naturally influences other possible generated particles. As each particle follows its own random walk, an array of particles cannot be operated in common and therefore the vector approach where identical instructions are performed on each vector element (particle) will fail. In addition the computer coding of the simulation process involves many conditional statements which inhibit vector processing.

An alternative to the history-based Monte Carlo coding has been suggested with event-based codes, where a vector of particles is processed for one type of event (free flights, collisions, boundary crossings and termination of the histories). This approach fulfils the requirements of the vectorized concept. Brown and Martin (1984) have analysed in detail the configuration of Monte Carlo codes for vectorization, connecting the scalar and the vectorial approach. A review of vectorized Monte Carlo has been performed by Martin and Brown (1987), where they describe variations of event- based algorithms together with speedup results published by different groups on both neutron and photon transport simulations with vectorized codes.

There are some important aspects in the vectorization and use of vectorized codes which deserve comments. T h e programming effort needed to recode an existing Monte Carlo code is a significant undertaking, together with the benchmarking of the new code against the old scalar code. The vectorized code can only be executed on a vector machine. Even if most of the published comparisons between vectorized and scalar codes show asignificant speedup factor with vector machines (Brown and Martin (1984) have reported speedups of up to 40 times), the real breakthrough for vectorized Monte Carlo in particle transport has riot Iriaterialized. There are large discrepancies in the vectorjscalar performance ratio reported by different authors, but for a well known code like EGS4, Miura (1987) has reported a modest speedup of 8 times with the vectorized EGS4V.

Supercomputer technology has evolved into smaller, mass-produced systems, mini- supercomputers and parallel processor environments. Mini-supercomputers, deliver about a quarter to half the performance of a supercomputer for only one-tenth of the price. With the market now dominated by Convex, DEC and IBM have just entered in the arena with the VAX woo (16 ns clock time, up to 500 MFLOPs with 4 processors) and J-models in the 3090 series (14.5 ns) respectively, and the new IBM Summit generation is expected soon (cf Emmen and Hollenberg 1990). By 1991 almost half of all high-performance mini-supercomputers will use parallel processing technology, compared with 22% in 1986 (Leghart 1990).

Parallel processing utilizes multiple arrays of CPUS operating in union, allowing the computers to process information 10 to 100 times faster than older single-processor systems. Parallel computers are designated as MIMD, multiple instruction streams independently operating on multiple data streams. Several architectures are available, a widely accepted classification being the shared-memory multiprocessor concept (with a common bus connecting all CPUS) and the multicomputer concept where each processor has its own small RAM and communicates with other processors through high-speed serial links (cf Stein 1988, Bell 1989). A drawback of the shared-memory configuration is the limited bus bandwidth, which can be saturated when different processors try to access the memory simultaneously. This is called the von Neumann bottleneck, which does not exists in multicomputer architecture.

Communications are the central problem for parallel computers when a host

Monte Carlo techniques in medical radialion physics 907

processor is connected to an array of perhaps hundreds of parallel processors, each with its own memory. Solutions like hypercube computers have been developed, where the processors are connected together as if they were at the corners of a multidimen- sional cube. The Intel iPSC can have 128 node processors, each based on a 80836 chip; all are connected via Ethernet to a host computer, also 80836-based. The NCUBE fits 8192 processors in such architecture, achieving 27 GFLOPS. A forthcoming Connection Machine will have 65536 processors, etc. New classifications have been suggested for parallel computers: those with less than 33 processors are called moderately parallel; between 33 and 1000 processors are highly parallel; and those with more than 1000 are classified as massively parallel (Leghart 1990).

Special attention has been given to the so called transputer, a small multicom puter building block whose design lends itself to parallel computingeven in a personal computer. The transputer T800 from INMOS is a processor with 2 or 4 Mbyte of RAM. Operating a t 30 MHz, it delivers 2.25 IEEE 32-bit MFLOPS. It incorporates 4 Kbyte of on-chip static memory (not cache memory per se) and four link interfaces used for communication with other transputers at a capacity of 120 Mbit s-’ link I / o , the equivalent of 12 Ethernets (Stein 1988). Plug-in boards have been developed for both the IBM PC and the Macintosh. Connecting several transputers together thus form a multicomputer system where the simulation of different particles can , in theory, run independently on each processori~ thus making a parallel machine out of a personal computer.

Although parallel architecture seems to he the ideal solution for Monte Carlo calculations, software aspects represent again the primary bottleneck to a rapid implementation. Design of true parallel processing operating systems, taking care of the communication between different processors, and the design of optimization language compilers, distributing computational loading among the processors while keeping input and output data in a host computer, are the major obstacles of the implementation. Debugging tools are also difficult to develop for similar reasons. Software has been one of the key problems slowing down the acceptance of parallel processing, and whereas hardware speed has increased at a compound rate of 3% per month, yielding to a thousand-fold improvement over. the past 20 years, software development has increased only at a compound rate of about 5% per year (Leghart 1990).

A more realistic approach in medical physics Monte Carlo calculations can be achieved today by using existing or newly built networks in hospital environments. Figure 15 shows results obtained running the code MCEF on different computers, where depth-dose distributions in water for plane-parallel electron beams of5 MeV have been calculated. Some interesting aspects can be observed. First the Convex C-210 mini- supercomputer, running in scalar mode, is about 16 times faster than a VAX-11/780, microVAX-I1 etc, but i t is only 5 times faster than a micr0VAX-3800 and a VAXstation 3100/38 operating under VMS, or an HP 9000/375 workstation (Motorola68030 at 50 Hz with cache memory) operating under Unix (HP-UX). This shows that under certain circumstances, standard benchmarks (such as LINPACK, which has yielded 12 MFLOPS for the Convex C-210 in the Dongarra’s study) can be misleading and that to gauge a system’s performance accurately, the benchmark program should resemble that for which the system will be used. Second, a Macintosh Ilfx (68030 at 40 Hz, no cache memory) runs the code practically a t the same speed as a microVAX-3600 and a VAXstation 3100/30 under VMS, or the HP 9000/370 workstation under Unix (68030 at 33 Hz with cache memory). This is about the same speed as reported by Ma and Nahum (1990) running EGS4 on one T8oo transputer. In a short time the new

908 P Andreo

VAXSt 3100138

VAXst 3100130

VAX 8200

VAX 780

VAX 750 FPA

microVAX II

mlcroVAX 3800

microVAX 3500

Macintosh iIx

Macintosh llfx

HPs1 90001370

HPsl 90001375

Convex C-210

5 MeV electrons in water full cascade Simulated down Io 275 keV

0 5 10 15 20 Performance relative to a VAX-111780

Figure 15. Comparison of Monte Carlo calculations on different computers using the code MCEF to simulate the transport of 5 MeV electrons in water. CPU times are normalized to that of a VAX.IIIIBO computer. The insert shows an enlargement of the results obtained with computers typically existing at a hospital environment. All VAXes operating under VMS, Convex and HPS under Convex Unix and HP-UX respectively.

Motorola 68040 chip will be available, increasing these performances. A network based on such micro or minicomputers can emulate the performance of a Convex mini-supercomputer a t a small fraction of the cost, thus making it possible to perform today's lengthy Monte Carlo simulations at much higher speed. Similar solutions have been presented by Mackie (1989) a t a slightly higher cost using RI%-based (Reduced Instruction Set Computing chips) DEC stations claiming the speed equivalent of 100 VAX-11/780 computers. Improved performances can be obtained today with the new IBM RS/GOW RISC workstations, that running the LINPACK benchmark at a rate of 7.5 MFLOPS have become the fastest floating-point units available, outside of a mainframe. A final important item in recent computer technology has been the new (RISC) scalar chip Intel 80860. According to Margulis (1989) this 'personal supercomputer' or 'one- chip supercomputer' can operate up to 50 MHz and achieve superscalar performance executing three operations per clock cycle (the 1BM RS/600 also shares such capability). At 40 MHz, the Intel 80860 yields 10 MFLOPS for the LINPACK benchmark, or 33 VAX MIPS (million VAX-like instructions/s) for the Stanford integer benchmark.

7. Conclusions

The large number of applications referred to in this work shows the usefulness, and considerable success, of the Monte Carlo method as an instrument in different areas of medical radiation physics. The different techniques used in the field have contributed to the development of sophisticated calculational tools of great value in radiation


physics. The increasing availability of fast computers and networks at hospitals, and the existence of well documented and supported public codes, will surely enlarge the number of scientific applications and communications in the future.

It is however important to be aware of the limitations of the method. The Monte Carlo method is not a magic black box that provides the correct answer to all kind of simulations. Some shortcomings of the commonly used macroscopic techniques for electron transport, for example, have been discussed in this review. These suggest the combination of macroscopic and microscopic techniques in energy ranges o r geometries where each technique is most efficient. A major issue refers to approximations which are only valid for certain applications, like the neglect of electron transport, but which have been implemented in applications where they are not valid, yielding erroneous resnl ts .

Tests and comparisons of the Monte Carlo code being used with other well known codes and experiments is mandatory; even if the actual simulation cannot b e compared, i t is always possible to select related cases for which results are available. Un- fortunately a non-negligible percentage of the existing codes have not been compared with other codes or experimental results. A recommendation can be given here: ‘Be sceptical of the results of anybody else’s Monte Carlo computer code. B e especially sceptical of your own code. No matter how you word your disclaimer, you wi l l still “carry the can” filled with your own bugs’ (Bielajew 1990a, private communication).

Acknowledgments

Detailed comments and suggestions on t,he manuscript from Anders Brahme, Annette Fransson and Alan E Nahum are gratefully acknowledged. Sincere thanks are due to colleagues Alex Bielajew, Rock Mackie and Dave Rogers for their contributions to the field and discussions during many years; very special thanks to Martin Berger and Alan Najum.

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