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A method for photon beam Monte Carlo multileaf collimator particle transport

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2002 Phys. Med. Biol. 47 3225 (http://iopscience.iop.org/0031-9155/47/17/312) View the table of contents for this issue, or go to the journal homepage for more

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INSTITUTE OF PHYSICS PUBLISHING

PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 47 (2002) 3225–3249

PII: S0031-9155(02)36442-X

A method for photon beam Monte Carlo multileaf collimator particle transport Jeffrey V Siebers, Paul J Keall, Jong Oh Kim and Radhe Mohan Department of Radiation Oncology, Medical College of Virginia Hospitals, Virginia Commonwealth University, Richmond, VA, USA E-mail: [email protected]

Received 1 May 2002 Published 21 August 2002 Online at stacks.iop.org/PMB/47/3225 Abstract Monte Carlo (MC) algorithms are recognized as the most accurate methodology for patient dose assessment. For intensity-modulated radiation therapy (IMRT) delivered with dynamic multileaf collimators (DMLCs), accurate dose calculation, even with MC, is challenging. Accurate IMRT MC dose calculations require inclusion of the moving MLC in the MC simulation. Due to its complex geometry, full transport through the MLC can be time consuming. The aim of this work was to develop an MLC model for photon beam MC IMRT dose computations. The basis of the MC MLC model is that the complex MLC geometry can be separated into simple geometric regions, each of which readily lends itself to simplified radiation transport. For photons, only attenuation and first Compton scatter interactions are considered. The amount of attenuation material an individual particle encounters while traversing the entire MLC is determined by adding the individual amounts from each of the simplified geometric regions. Compton scatter is sampled based upon the total thickness traversed. Pair production and electron interactions (scattering and bremsstrahlung) within the MLC are ignored. The MLC model was tested for 6 MV and 18 MV photon beams by comparing it with measurements and MC simulations that incorporate the full physics and geometry for fields blocked by the MLC and with measurements for fields with the maximum possible tongueand-groove and tongue-or-groove effects, for static test cases and for sliding windows of various widths. The MLC model predicts the field size dependence of the MLC leakage radiation within 0.1% of the open-field dose. The entrance dose and beam hardening behind a closed MLC are predicted within ±1% or 1 mm. Dose undulations due to differences in inter- and intra-leaf leakage are also correctly predicted. The MC MLC model predicts leaf-edge tongue-andgroove dose effect within ±1% or 1 mm for 95% of the points compared at 6 MV and 88% of the points compared at 18 MV. The dose through a static leaf tip is also predicted generally within ±1% or 1 mm. Tests with sliding windows of various widths confirm the accuracy of the MLC model for dynamic delivery 0031-9155/02/173225+25$30.00

© 2002 IOP Publishing Ltd Printed in the UK

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and indicate that accounting for a slight leaf position error (0.008 cm for our MLC) will improve the accuracy of the model. The MLC model developed is applicable to both dynamic MLC and segmental MLC IMRT beam delivery and will be useful for patient IMRT dose calculations, pre-treatment verification of IMRT delivery and IMRT portal dose transmission dosimetry.

1. Introduction The purpose of this paper is to describe an accurate transport-based multileaf collimator (MLC) particle transport model to be used with Monte Carlo (MC) dose computations that is accurate, fast and efficient, and therefore applicable for iterative intensity-modulated radiation therapy (IMRT) dose calculations. For current IMRT systems, application of a patient’s beams to a phantom often results in measured dose distributions that differ from those predicted by the treatment-planning system. (Wang et al 1996, Tsai et al 1998, Siebers et al 2000c, Keall et al 2001). Apart from actual beam delivery errors, the cause of the dose error is partly due to inadequate patient dose calculation algorithms and partly to the inadequate accounting of the beam delivery system scattering and leakage characteristics into such dose calculation algorithms. Although many attempts have been made to incorporate the delivery device effects into the intensity matrices during conversion from optimized to deliverable beam intensity distributions, discrepancies between measured and calculated dose distributions remain (Convery and Rosenbloom 1992, Spirou and Chui 1994, Stein et al 1994, Svensson et al 1994, Spirou et al 1996, van Santvoort and Heijmen 1996, Convery and Webb 1997, Webb et al 1997, Dirkx et al 1998, LoSasso et al 1998, Balog et al 1999, Pasma et al 1999, Ma et al 2000b). The properties of radiation passing through moving MLCs have been studied extensively (Wang et al 1996, LoSasso et al 1998, Arnfield et al 2000). Detailed MC simulations of simple static fields found that in addition to the MLC being a broad source of scattered photons, it is also a source of electrons that contribute to the surface dose (Kim et al 2001). Additionally, the photon energy spectrum transmitted through the MLC is hardened, altering the depth–dose characteristics. This beam hardening is also observed for dynamic IMRT fields (Fix et al 2001a). Given these facts, it is no wonder that approximating the MLC as an intensity matrix for IMRT dose calculations results in discrepancies with respect to measurements. In general, it is recognized that MC algorithms are the most accurate dose computation methodologies. The strength of MC stems from the fact that it can realistically model radiation transport and interaction processes through the accelerator head, beam modifiers and the patient geometry. For IMRT delivered with segmental MLC (SMLC) or dynamic MLC (DMLC) delivery, accurate dose calculation, even with MC, is challenging. Several investigators have used MC for IMRT dose calculations. Specifically, Jeraj and Keall (1999) used MC-generated beamlets or bixels for IMRT optimization. Their method ignored practical beam delivery. Laub et al (2000), Pawlicki and Ma (2001) and Ma et al (1999, 2000a) used fluence or intensity matrices to approximate the effects of IMRT delivery during MC dose computation. The intensity matrices were used to modify the statistical weights of the incident particles to achieve intensity modulation. The scattering and transmission characteristics of the delivery device were approximately incorporated into the intensity matrix, but beam hardening and electron contamination were ignored. While intensity matrix-based MC can match in-phantom measurements at a single depth, without accounting for beam hardening, at other depths, the dose distribution will differ. Although the

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use of an intensity matrix lends itself to clever schemes to accelerate the Monte Carlo dose calculation, (Laub et al 2000), the inaccuracies introduced in the conversion from desired beam intensity profiles into physical beam delivery (MLC leaf sequences) may exist when this method is used, thereby compromising the accuracy possible with Monte Carlo. Deng et al (2001) used MC dose calculations to study the impact of MLC tongue-andgroove effects on IMRT dose distributions by using an intensity matrix that ignored tongueand-groove effects and another that included tongue-and-groove effects using Chen’s method (Chen et al 2000). X-ray transmission through the MLC was computed assuming a pointsource approximation and a single effective attenuation coefficient. The resultant intensities were convolved with a source-spread function to approximate the effects of the finite source size and scattered photons and electrons. Note that since this method uses a single effective attenuation coefficient, it ignores the beam hardening effect of the MLC. For single fields, Deng et al concluded that tongue-and-groove effects could be up to 10% of the maximum dose, yet for two multiple field test cases in which patient motion was modelled, the effect was less than 1.6%. Deng et al (2001) acknowledged that, in general, neglecting tongue-and-groove effects may lead to dose errors of 5% of the maximum dose or more, particularly for cases with few fields and/or small setup uncertainty. Other investigators have utilized standard MC codes with some simplifying assumptions regarding the MLC geometry and/or the beam delivery for IMRT dose calculation (Fix et al 2001a, 2001b, Liu et al 2001). Fix et al (2001a, 2001b) used GEANT to model the Varian 80-leaf MLC for dynamic and segmental IMRT. The leaf edges parallel to leaf motion were modelled with the tongue-and-groove, but the rounded leaf tips were simplified and modelled as planes focused at the source. An effective leaf tip offset was used to approximate the rounded leaf tip radiation transmission (Arnfield et al 2000), and dynamic MLC delivery was approximated by a series of static MLC segments. These approximations can lead to errors in the dose computation for complex IMRT fields. On the other hand, Liu et al (2001) created a component module for the EGS4 (Nelson et al 1985) user code BEAM (Rogers et al 1995) code called DMLCQ. DMLCQ is an enhanced version of the MLCQ component module (Palmans et al 2000) that allows dynamic motion of the MLC during Monte Carlo simulation. In the DMLCQ module, individual MLC leaves of the Varian 80leaf MLC are modelled with straight edges in the direction parallel to the leaf motion and are focused to the radiation point source. As pointed out by Liu et al, the techniques used to modify the MLCQ module can also be applied to the VARMLC module that includes the proper leaf edges for the 80 leaf MLC. (Kapur et al 2000) However, without simplifying assumptions, use of the standard MC codes for dynamic MLC simulation is computer resource intensive. We previously developed an efficient probability-based MLC model for dynamic IMRT (Siebers et al 2000a, 2000c, Keall et al 2001) that attenuated photons by the probability that they interacted in the MLC and generated first Compton scatter from the MLC. Although the model correctly predicted the beam hardening produced by the MLC during dynamic IMRT delivery and included the rounded MLC leaf tips, it presumed that the neighbouring leaf was always in the same position as the leaf through which the particle was being transported; thus, it neglected tongue-and-groove and other leaf-edge effects. This paper describes a transport-based MLC model that is accurate, fast and efficient, and consequently applicable for iterative IMRT dose calculations. The transport-based model predicts beam hardening and leaf-edge effects (tongue-and-groove) and includes first Compton scatter. By determining the average transmission for each particle through the MLC, the efficiency is comparable to that for open-field dose computations. The techniques described in this paper apply equally to SMLC and DMLC IMRT deliveries.

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2. The MLC model 2.1. Introduction The basis of the MLC model developed is that the complex geometry of the MLC can be broken into simple geometrical regions that facilitate simplified radiation transport. The amount of attenuating material an individual particle encounters while traversing the entire MLC is determined by adding the individual amounts from each of the simplified geometric regions. Since some approximations are made in the transport through a single region, the greater the number of geometric regions used in the model, the smaller the approximations, and the closer the model will be to a complete model of the MLC. With as few as two regions in the beam direction, the most important geometric properties for radiation transmitted through the MLC are considered: leaf-edge or tongue-and-groove effects. While the model is applicable to individual static field delivery, it was developed primarily for use with IMRT beam delivery. To efficiently sample particles for IMRT beam delivery (either SMLC or DMLC), for each incident particle, the probability of interacting in the MLC is evaluated multiple times to determine the average interaction probability. The MLC model was integrated into the MCV MC system (Siebers et al 2000b). Thus, it reads particles from a BEAM (Rogers et al 1995) format phase space file that contains the position, direction and energy of particles (photons and electrons) exiting the treatment head that are incident upon the MLC. After transporting the particles through the MLC, the model writes the exiting particles to another phase space file, which is later used as input to an MC dose calculation algorithm. The input/output routines are separate from the transport modules, making the code readily adaptable for other MC code systems. As implemented in this paper, the MLC model is described for the Varian1 Millennium 120 leaf MLC. Only minor modification of the input is required to apply the model to other Varian MLCs or MLCs from other manufacturers. 2.2. General flow The general flow for incorporating the MLC model into MC dose calculation is shown in figure 1. The positions of the MLC leaves are read from the MLC leaf-sequence file that is normally sent from the treatment-planning system to the treatment machine for use in treatment. This file specifies the projection of the MLC light field to 100-cm SAD at specified fractional monitor units of the total beam delivery (see the appendix for further discussion of the leaf-sequence file). The sequence of sub-fields in the file can result in static beam delivery, SMLC IMRT beam delivery, or DMLC IMRT beam delivery. The positions specified in the leaf-sequence file are translated into physical leaf coordinates using the same method as the treatment machine. That is, they are translated to physical leaf tip positions at the MLC plane using a table-look-up and demagnification from the treatment machine mlctable.txt file. The table look-up accounts for the rounded leaf tip light projection to the isocentre plane while the demagnification projects the field to the MLC location. It is important to note that when the A (right) and B (left) leaves of an MLC pair are specified to be at the same position (that is the leaf pair is specified to be closed), instead of using the positions directly from the table look-up, which could result in a physical gap between the leaves but no light field gap, both leaves are positioned at the mean location from the table-look-up before the demagnification is applied. This results in the leaves physically touching. If this were not the case, at distances 1

Varian Medical Systems, Palo Alto, CA 94304-1129.

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Read MLC leaf sequence Translate to MLC plane coordinates Read particle from incident phase space Transport particle to MLC entrance location Transport particle through leaf† Write particle to exiting phase space

Yes

More incident particles?

No Done Figure 1. Flow for incorporation of the MLC model into the MC simulations. The MLC model is embedded in the box marked with †.

far from the central axis, the leaves, when specified to be closed, would be physically separated by several mm. The table-look-up method differs from the method discussed by others (Arnfield et al 2000, Keall et al 2001), who applied an apparent leaf offset correction, but similar to Liu et al (2001), who used an empirically determined polynomial to essentially reproduce the data in the mlctable.txt file. The difference between using an effective offset and the actual leaf tip positions is apparent when leaves are positioned far from the central axis. When a 0.1 cm field size (light field) is specified 20 cm from the beam central axis, using the table-look-up or polynomial fit results in the leaf tips being physically 0.37 cm apart. The use of the offset correction results in the leaves being separated only by the offset used. A complete discussion of the differences between the light field and radiation field has been given by Boyer and Li (1997) Once all the physical leaf positions as a function of monitor units are determined, particles are read from the phase space of particles exiting the treatment jaws, the particles are transported though the MLC (details given in the sections below), and surviving exiting particles are written to an exiting phase space file that is used as input for Monte Carlo patient

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Figure 2. The coordinate system and major regions used for the MLC model. In each region, each MLC leaf is broken up into sub-regions of various thicknesses to account for thickness variations across the MLC leaf. (The sub-regions shown are not to scale and are for demonstration only).

dose calculations. The transport of particles through the MLC proceeds until all particles in the incident phase space are completed. 2.3. The MLC model The MLC model divides the MLC leaves into simple geometric regions that are non-reentrant; that is, a particle travelling through a region can enter and exit the region only once. The leaf geometry (number of regions, leaf tip shape and thickness as a function of distance) is obtained from a user created configuration file. For the Varian 120-leaf MLC, a minimum of two regions in the beam (Z) direction are required, an upper region and a lower region, split at approximately the mid-thickness point of the MLC. The MLC leaf-coordinate system and the geometric regions for the central leaves of the 120 leaf MLC are shown in figure 2. The MLC leaves move in the X direction. The Y coordinate is perpendicular to the direction of leaf motion. 2.3.1. Photon transport. First, the photon transport for a static leaf position will be described, and in a later section, the application to IMRT delivery will be discussed. Photon transport through the MLC leaf bank at a given static leaf position proceeds as follows: for each MLC leaf region in the Z direction, the incident particle coordinates are evaluated at the centre, entrance and exit of the MLC region. The Y coordinate at the centre of the region is used to determine which leaf the particle is passing through in the region (the leaf index) and the thickness of the region (Tregion). These are determined using a table-look-up. The leaf index to region thickness table is generated using the known MLC geometry from the machinist drawings provided by the MLC vendor and breaking the leaves into two (upper and lower) regions. A portion of this table for the upper-section of the Varian 120-leaf MLC is given in table 1. Note that thickness variations across the MLC leaf are preserved in this table. The particle X coordinates at the entrance, exit and centre of the leaf are used to determine the thickness of material actually traversed in the region. The method to do this is depicted in figure 3(a). Sample particle A travels through the full region thickness (Tregion), particles B and

A method for photon beam Monte Carlo multileaf collimator particle transport

Particle Xenter Xexit Thickness

Xi Xt

A

B

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C D

A

< Xi < Xt Tregion

B

> Xi < Xt tip routine

C

< Xt > Xt tip routine

D

> Xt > Xt

0

(a) Xg

E

Xt

Particle Xcentre Thickness E

< Xg

Tregion-Tgroove

F

> Xg

Tregion

F

(b) Figure 3. The method used to determine the material thickness that a particle goes through in a given MLC region. (a) The entering (Xenter ) and exiting (Xexit ) particle coordinates with respect to the leaf tip locations (Xi and Xt ) are used to determine the section thickness. Particle A passes through the full thickness of the leaf region, B and C call the tip intersection routine and the thickness for particle D is zero. (b) The coordinate of the particle at the centre of the region (Xcentre ) is checked with respect to the coordinate of the groove in the MLC leaf (Xgroove ) for some leaf regions. The thickness for particle E would be corrected for the groove in the MLC leaf. Table 1. Section of table-look-up for upper section of MLC leaf. Using this table, for a particle with a Y coordinate of 0.05 at the centre of the upper section of the MLC leaf, the leaf index would be 30 and the maximum thickness would be 2.998384 cm. Y -end coordinate of section (cm) −0.007224 0.031576 0.038600 0.163500 0.241600 0.248800 0.389000 0.490400 0.529200 0.536224 0.661124 0.739224 0.746424 0.886624 0.988024 1.026824 1.033848 1.158748 1.236848

Leaf index

Thickness of region Tregion (cm)

31 31 30 30 30 30 29 29 29 28 28 28 28 27 27 27 26 26 26

3.377700 3.374999 0.000000 2.998384 3.117263 0.000000 2.846413 3.377536 3.374809 0.000000 2.998135 3.116956 0.000000 2.846048 3.377034 3.374282 0.000000 2.997587 3.116338

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G

H

Figure 4. Sample particles passing through the MLC leaf tips. The MLC model correctly predicts the thickness for particle G, which passes through only a leaf tip in one section and for particle H that passes through leaves in opposing leaf banks.

C travel part-way through the leaf tip and part-way through the leaf, and particle D does not travel through the leaf at all. Particles such as B and C require using a tip-intersection routine.

The tip intersection routine simply searches for the intersection between the particle trajectory and the leaf tip profile curve (broken up into line segments). This is the most computationally intensive portion of the algorithm, however, it is infrequently invoked since most particles do not intersect the leaf tips. For some regions, the particle can potentially travel through a groove cut into the MLC leaf. Each MLC leaf has a groove cut through it to permit mounting of the leaf. This groove is on the thicker portion of the leaf (in Y ), thus it alternates between the upper and lower leaf regions on adjacent leaves. The groove is accounted for in the photon transport by determining if the particle X coordinate at the region centre is further than the groove distance Xg from the leaf tip (see figure 3(b)). When a particle, such as particle E, passes through the groove, the groove thickness is subtracted from the region thickness traversed to obtain the actual thickness of material traversed by the photon. The total thickness traversed by a photon as it travels through all of the MLC sections is obtained by simply summing the thickness from each region. The total thickness, t, is then used to determine the statistical weight of the photon exiting the MLC, wf , using wf = wi e−µ(E)t/ cos θz ,

where wi is the weight of the incident photon, µ(E) is the attenuation coefficient of the MLC material at energy E and cos θz is the z direction cosine of the photon. Tracks of sample particles traversing a simplified MLC leaf using these assumptions are given in figures 4 and 5. Note that transport from the left leaf bank to the right leaf bank through the leaf tips is accounted for (particles G and H, figure 4). However, cross-leaf

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J

K

L

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M

Figure 5. Sample particles through the MLC geometry. The MLC leaf geometry has been simplified in this figure. The thickness used by the MLC model is exactly correct for particles I, J, and K. For particle L, a slight thickness error is introduced because the particle crosses the gap between adjacent leaves, while the maximum thickness error occurs for particle M, which will have a thickness error of half the thickness of the region.

transport within a given MLC Z-region is ignored (figure 5). For particles I, J and K, this has no impact since no cross-leaf transport within a Z region is observed. For a particle such as L the effect of ignoring cross-leaf transport is minimal since the particle passes through neighbouring leaves. However, for particles such as M that escape near the midpoint of a region and for which the neighbouring leaf is absent, the path length error is maximized since no neighbouring leaf can account for the missing material. The maximum path-length error for a given region is one-half of the region thickness. Introducing additional regions along the Z direction can reduce this impact of this effect. For example, by dividing the upper MLC region into two regions at mid-point where particle M escaped the MLC leaf, the path length error for particle M becomes zero in each sub-region. For primary photons travelling from the bremsstrahlung target position, transport using only two MLC regions will be nearly exact. 2.3.2. Compton scattered photons. The principle photon interaction in tungsten from 0.5 to 5 MeV is the Compton interaction. In the MLC model, Compton photons scattered by the MLC are accounted for using the same method as our earlier MLC model (Keall et al 2001). A semi-infinite slab geometry is presumed for the Compton interaction. The slab thickness is the total thickness of the MLC traversed by the incident photon, t. The interaction probability is evaluated using the ratio of the Compton and total attenuation coefficients (µc/µ). The energy

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and angle of the Compton scattered photon are determined using the methods described in the EGS4 user manuals (Nelson et al 1985). The location of the interaction as the photon traverses thickness t is randomly sampled over [0, t) assuming exponential attenuation. The phase space coordinates of the scattered photon are used to determine the remaining thickness to be traversed by the scattered photon (t " ). The attenuation coefficient (µ" (E)), thickness (t " ), and direction cosine (cosθz" ) are used to attenuate the scattered photon as it exits the material slab. Thus, the weight of a scattered photon, wf" , is wf" = wi (1 − e−µ(E)t/ cos θz )

µc −µ(E " )t " / cos θz" e . µ

Compton scattered electrons, photo-electrons, bremsstrahlung photons and pair production products are ignored in this model. 2.3.3. Electron transport. Geometrically, electron transport through the MLC sections is similar to the photon transport, however, the interest lies with whether the electrons intersect the MLC leaves. If the electron intersects the MLC leaf or leaf tip, it is assigned zero weight. Otherwise, the electron travels through the MLC leaf opening with its weight unchanged. 2.3.4. Variance reduction techniques. To improve the efficiency of the later dose calculation algorithm, Russian Roulette is performed on transmitted and scattered photons using the method described in the MCNP manual. (Briesmeister 1997) That is, a statistical weight cut-off wcut is defined. Particles with a final weight wf above the weight cut-off survive with their statistical weight unchanged. Particles with wf less than wcut survive with a probability of wf /wcut . The weight of surviving particles is equal to wcut . The weight cut-off value used for most calculations is 0.10, since the efficiency of delivering an IMRT field is typically greater than this value. 2.3.5. Application to IMRT. The above sections describe the use of the MLC model for a static radiation field. In IMRT, either many static MLC positions may be specified to create the desired intensity profile (SMLC IMRT) or the MLC leaves may be moving when the beam is on (DMLC IMRT). In either case, the MLC leaf-sequencing file specifies the positions of each MLC leaf at specified fractional monitor units (MUs) of the total MUs. Between these MU indices, each MLC leaf travels linearly to the next position and fractional MU. (For SMLC IMRT, leaf travel occurs when different MLC positions are specified at the same number of fractional MUs, and beam irradiation occurs when the same leaf position occurs at different consecutive fractional MU settings.) For a given arbitrary fractional MU, the positions of the A (right) and B (left) MLC leaves for the segment are determined by linear interpolation of positions specified in the MLC file. As a variance reduction technique in this model, each particle is transported through the MLC at N randomly selected fractional MU positions. The MLC leaf positions may be different at each of these different fractional MUs. At each fractional MU setting k, the entering and exiting X coordinates of the particle in each MLC region are used to determine the total thickness the particle traversed at that fractional MU setting. The average statistical weight of a particle exiting the MLC is the geometric mean of the weights from the individual fractional MU settings. That is, for photons wf =

N wi ! −µ(E)tk / cos θz e . N k=1

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To further reduce computational burden, Compton events are sampled only once for each incident photon rather than at each fractional MU setting. That is, one of the kth thicknesses (tk) is used to determine the Compton scatter event. Since the fractional MUs are chosen randomly, the last fractional MU position is as random as any other and is used for determining the Compton scatter. This method preserves the mean number of scattered photons exiting the MLC. The default value of N is 100, and all of the calculations presented here used this value. 3. Benchmark tests and results To benchmark the MLC model, results obtained with the model were compared with measurements and other MC calculations for a variety of test cases. Validation tests were performed using the 6 MV and 18 MV beams from our Varian 21EX linear accelerator with a Millenium 120 leaf MLC attached. Monte Carlo calculations were performed using the MCV MC system. Integration of the MC code with the treatment-planning system, generation of initial phase space, modelling of beam line devices and dose calculation have been described elsewhere (Siebers 1999, Siebers et al 2000b). 3.1. Blocked field tests Several test fields created with the MLC blocking the entire treatment field were used to establish the ability of the MLC to correctly predict the attenuation and scatter characteristics of the MLC. For these tests, the beam-defining jaws were set to produce a given field size, for example, a 10 × 10 cm2 field at an SAD of 100 cm, but the MLC leaves extended across the radiation field, hence blocking the radiation field. We call such a field an MLC-blocked field. 3.1.1. Leakage as function of field size. Overall predictions of MLC leakage (transmitted + scattered) radiation were performed by comparing measurements with MC simulations for multiple MLC-blocked field sizes. Measurements of the radiation leakage through the MLC at a depth of 5 cm in a water equivalent phantom placed at 95 cm SSD were made using a PTW Farmer-type ionization chamber (internal diameter 6.25 mm, length 24.0 mm). Measurements were taken for open and MLC-blocked field sizes of 5 × 5, 10 × 10, and 15 × 15 cm2. For the MLC-blocked fields, measurements were taken with the A and B leaf banks blocking the field and averaged. The ion chamber was oriented perpendicular to the direction of leaf motion to average out the variable transmission through the leaves due to the thickness profile of the leaves. Measurements were taken with the chamber on the central axis and with the chamber offset by ∼±3 mm to ensure that dose variations due to undulations in the leakage dose profile were properly averaged. At each field size, MLC-blocked field results were normalized to the open-field measurements to obtain the fractional MLC leakage. MC simulations were performed for conditions that reproduced the experimental conditions. The MC calculations were performed using the MLC model described in this paper and using the full MLC geometry for an MCNP simulation. The full MC simulation of the MLC is detailed in the paper by Kim et al (2001). The MCNP simulation incorporates all geometric aspects of the MLC-leaf design and includes full physics treatment of photon and electron interactions. MCNP was used to transport only through the MLC. The EGS4 (Nelson et al 1985) BEAM user code (Rogers et al 1995) was used to transport through the other parts of the accelerator head and DOSXYZ (Ma et al 1995) was used for transport through the phantom. A 0.6 × 0.6 × 1.0 cm3 voxel size was used during the MC transport simulations, however, the central five voxels in the direction orthogonal to the beam were averaged to reduce statistical fluctuations, creating an effective voxel size of

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Figure 6. Measured MLC leakage for MLC-blocked fields as a function of length of field size and MC computations using the MLC model and full radiation transport using MCNP. MLC leakage values are normalized to open field of the same field size. 6 MV results are shown with solid lines, 18 MV results with dashed lines.

3.0 × 3.0 × 1.0 cm3. Additional MC simulations were performed for a 20 × 20 cm2 field size even though the MLC restrictions prevent the treatment machine from delivering such a field. Results for this test are summarized in figure 6 for the 6 MV and 18 MV beams. The one standard deviation statistical uncertainty in the MC-computed ratios was less than 0.01%. The measurement uncertainty is estimated to be ∼0.3%, one standard deviation. At 6 MV, the MLC model reproduces the measurements and the MCNP simulation within 0.05% of the open field dose for all field sizes. A systematic difference of 0.1% of the open field dose is observed between the 18 MV measurements and MC simulations. (e.g. 1.65% measured and 1.75% calculated for the 5 × 5 cm2 field). This difference is negligible for patient dose calculations and, as pointed out by Kim et al (2001), is likely due to the selection of an MLC density (17.7 g cm−3) such that the 6 MV results matched measurements for the 10 × 10 cm2 field. Selection of a slightly higher MLC density (17.8 g cm−3) would result in a 0.06% difference for both the 6 MV and 18 MV results. 3.1.2. Inter- and intra-leaf leakage differences. To examine how well the MLC model predicts dose undulations perpendicular to the direction of leaf motion for an MLC-blocked field due to differences between inter- and intra-leaf transmission, film measurements with the above conditions (5 cm depth, 95 cm SSD) were made for a 10 × 10 cm2 field. At 18 MV, the film was placed at 10 cm depth, 90 cm SSD. Film measurements used Kodak XV22 film. For each energy, the MLC-blocked film was exposed to 999 MU of beam. Eight calibration films at dose levels of 0–70 MU were exposed in the same session as the other measurements using 10 × 10 cm2 open fields and the same energy beam and depth as the other measurements. All films were processed in the same session. Films were scanned using a Vidar3 VXR 12+ film scanner with the data read into a matrix with resolution 0.1 × 0.1 cm2. Film measurements were compared with MC calculations under the same conditions, but with the Monte Carlo 2 3

Eastman Kodak Company, Rochester, New York. VIDAR Systems Corporation, Herndon, Virginia.

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Figure 7. Measured (solid line) and calculated (dashed line) MLC leakage radiation perpendicular to the direction of MLC leaf motion for a 10 × 10 cm2 MLC-blocked field for 6 MV and 18 MV beams. Dose calculation and measurement for the 6 MV beams occurred at 5 cm depth, 95 cm SSD, and the 18 MV data at 10 cm depth, 90 cm SSD.

calculations having a 2.0 × 0.1 × 0.5 cm3 voxel size. The measured profiles are compared with the MC MLC model for the MLC blocked field in figure 7. Measurement results were averaged over a 2 cm wide band in the X direction centred on X = 0 to match the resolution of the MC calculations in that dimension. The one sigma statistical uncertainty of the MC simulation results is ∼0.3 cGy (∼0.03% of the open-field dose, or 1.8% of the local dose). The calculations reproduce the frequency and the amplitude of the dose undulations for both the 6 MV and 18 MV beams. Note that even the alternating peaks of high and low intensity (present due to the alternating leaf MLC design) are correctly predicted. At 6 MV, the absolute value of the average leakage is correctly predicted, while at 18 MV, the calculated leakage is 1.8 cGy (0.2%) higher than the measured value, consistent with the results in figure 6. 3.1.3. Entrance dose and beam hardening. Previous MC calculations (Kim et al 2001) indicated that for a 10 × 10 cm2 6 MV MLC blocked field, the percent depth dose (%dd) at 10 cm depth is 5% greater than the open-field %dd due to beam hardening. At 18 MV, no beam hardening was observed for the MLC-blocked field. Furthermore, electrons ejected from the MLC were found to contribute ∼18% of the surface dose for a 6 MV beam while for 18 MV the contribution was ∼35%. To investigate how well the MLC model predicts entrance dose and beam hardening produced by the MLC, depth–dose profiles computed using the MLC model for a 10 × 10 cm2 field at 100 cm SSD were compared with full MLC simulation results using MCNP (Kim et al 2001). The voxel size used during transport was 0.4 × 0.4 × 0.2 cm3, and again the central five voxels were combined to create a 2.0 × 2.0 × 0.2 cm3 voxel size for analysis. Depth–dose comparison results are presented in figure 8. For both 6 MV and 18 MV, the statistical precision in the MLC model calculations was 0.2% of the local dose (0.003% of an open-field dose), while for the MCNP results the statistical precision was 0.8% and 0.4% of the local dose for the 6 MV and 18 MV beams, respectively. Panel B of figure 8 shows the dose difference (MCNP-MLC Model/MCNPmax). The 1σ uncertainty in this ratio is also shown on the plot for one 6 MV point. Beyond the depth of maximum dose, the 6 MV

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results are within the statistical uncertainty of this dose ratio and well within the Van Dyk criteria (Van Dyk et al 1993) of ±2%. At 18 MV, the points are within ±2%, however, a slight systematic difference is observed with the MCNP results predicting a slightly lower dose than the MLC model. This is likely due to minor differences in attenuation coefficients used by MCNP and those used by the MLC model. Panel C of figure 8 shows distance-to-agreement (DTA) analysis. In the buildup regions for both the 6 MV and 18 MV beams, the distance to agreement is less than 1 mm. This is well within the typically accepted tolerance of ±2 mm and indicates that the absence of electrons scattered from the MLC in the MLC model does not significantly change the entrance dose. There appears to be no need to add secondary

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electrons to the MLC model. Overall, the MLC model predicts depth–dose characteristics and beam hardening for a 10 × 10 cm2 MLC-blocked field in agreement with those for full transport through the MLC with MCNP. 3.2. Leaf-edge effects 3.2.1. Picket fence fields. The impact of the particle-tracking approximations was investigated by creating test fields that maximize the tracking error. The maximum tracking error exists when alternating neighbouring leaves are in and out of the field. Two such ‘picket fence’ test fields were created; one in which the even leaves block the beam and odd leaves are retracted from the beam, and the other in which the odd leaves block the beam and even leaves are retracted. These two cases differ in that in one, the thicker portion of the leaf is towards the target, and in the other, the thicker portion is towards the isocentre. These fields were configured with the jaws set to produce a 10 × 10 cm2 field. Films exposed at 5 cm depth, 95 cm SSD were compared with MC calculations under the same conditions for the 6 MV beam; for the 18 MV beam, the film was exposed at a depth of 10 cm with a 100-cm SSD. The MC statistical uncertainty was ∼0.5% of the local dose for the voxel size of 2.0 × 0.2 × 2.0 cm3. Measurements were averaged over a 2 cm wide band in the X direction to match the resolution of the MC calculations in that direction. The measurements and calculations are compared in figure 9. The 6 MV and 18 MV measurements and calculations are well within measurement and calculation uncertainties. At 6 MV, 95% of points are within ±1% or 1 mm, and 98% of the points are within ±2% or 2 mm. At 18 MV, 88% are within ±1% or 1 mm, and 95% of the points are within ±2% or 2 mm. 3.2.2. Tongue-and-groove effects. The field with the maximum ‘tongue-and-groove’ effect is a combination of the two above fields, with half of the beam delivered with only even MLC leaves blocking the beam and the other half delivered with only odd MLC leaves blocking the beam. Although the Varian 120-leaf MLC does not have a tongue-and-groove

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design, these are called ‘tongue-and-groove’ effects since the effect on particle transport through the stepped leaf edge is similar. The MC calculation used a voxel size of 2.0 × 0.2 × 2.0 cm3, and the statistical uncertainty was ∼0.5% of the local dose. Again, measurements were averaged 2.0 cm in the X direction to match the resolution of the MC calculations. The calculations match the measurements well at both 6 MV and 18 MV as shown in figure 10. For the 6 MV data, 87% of the calculated points are within ±1% or 1 mm of the measured data, and, at 18 MV, 88% of the points fall in this same category. If a ±2% or 2-mm criterion is used, 97% are within the criteria at 6 MV, and 96% are within the criteria

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for 18 MV. In a field with the maximum tongue-and-groove effects, the MLC model matches measurements. 3.2.3. Leaf tip profiles. Radiation transmitted through the leaf tips was verified by comparing MLC model calculations with measurements. The jaws are set to produce a 10 × 10 cm2 field size at 100 cm. One bank of the MLC leaves (the left bank) is set so the leaf tips are on the central axis, and the opposing leaf bank (the right bank) is set to be out of the radiation field. Film is irradiated for 50 MU. For the MC calculations, dose in 0.1 × 2.0 × 2.0 cm3 voxels was scored at 5 cm depth in a water phantom located at 95 cm SSD for each of the MC simulations. The 0.1 cm dimension for the profile is along the X direction, the direction of the leaf tip (see figure 2). MC statistical uncertainty was ∼0.3%. Results are compared in figure 11 for the 6 MV beam. Measurements were averaged over a 2 cm wide band in the Y direction to match the calculation voxel dimension. For the 6 MV beam, 79% of the calculated points in the profile were within ±1% or 1 mm of the measurements. For the 18 MV beam (not shown), 86% of the calculated points were within ±1% or 1 mm. For the ±2% or 2-mm criterion, 96% of the points were within tolerance for both energies. 3.3. Leaf-positioning tests 3.3.1. Sliding windows. A sensitive test for testing the MLC leaf tip contribution to dose in dynamic MLC fields is to produce 10 × 10 cm2 fields by sweeping sliding windows of various widths and to measure and calculate total dose delivered on the central axis for each window width. By varying the window width, different combinations of open-field, leaf tip transmission, and leaf tip leakage contribute to the total dose. Others have used this test to evaluate parameters for MLC trajectory-to-fluence conversion (LoSasso et al 1998, Arnfield et al 2000) and to test an MC MLC model (Keall et al 2001). In this study, window widths ranging from 0.1 cm to 10.0 cm were used. It is important to note that only four control points were used in these files, corresponding to the leaves initially closed on the left-hand side

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Figure 11. Measured (solid line) and calculated (triangles) doses for 6 MV field with leaf tips located on the beam central axis. The per cent difference between the calculated and measured data at each point [100 × (measured–calculated)/(max measured)] is shown to ease the comparison.

of the field, the leaves open at the specified window width on the left-hand side of the field, the leaves open at the specified window width at the right-hand side of the field, and the leaves closed on the right-hand side of the field. This arrangement results in a constant physical window width but a non-constant light-field width as the leaves traverse across the field (see appendix). For these measurements and calculations, the jaws were set to 13 × 10 cm2, and dose was recorded at a depth of 5 cm at 95 cm SSD for 6 MV and 18 MV beams. Dose measurements were performed using the ionization chamber technique described in section 1. The MC simulation used a voxel size of 2.0 × 2.0 × 1.0 cm3. Measurements and calculations were normalized to the output for a 10 × 10 cm2 open field to minimize the effects of day-today beam output variations. Table 2 summarizes results for the sliding window fields. Also included in this table is the output for an MLC-blocked 10 × 10 cm2 field which is useful to show data consistency with results from section 3.1. The 1σ statistical precision values for the MC calculations are given in the table. The uncertainty in repeated measurements is estimated to be ∼0.5% of the local dose. Overall, the MLC model predicts the measured dose within 0.6% of the open-field dose for both the 6 MV and 18 MV beams, however, with respect to the local dose, the difference between the measurement and MLC model prediction increases with decreasing window size, maximizing at ∼7% of the local dose for the 1 mm window width. It is at this window size that error in physical leaf positioning would be most noticeable. The MLC position tolerance set in the leaf-sequence file is 0.2 cm at the isocentre plane. Given the 0.2 cm position tolerance, acceptable window widths during the 1 mm field measurement could have been anywhere in the range from 0 to 3 mm. By extrapolation, the MLC model calculations at 6 MV, a 0 mm window corresponds with a dose of ∼3.2% of the open-field dose, whereas by interpolation, a 3 mm window corresponds to ∼6.2%. The measured value of 4.71% indicates that the actual window width is very close to the desired width. For Varian MLCs, the user can acquire the log files produced by the MLC controller that indicate the MLC leaf positions during the beam delivery. These ‘dynalog’ files record the actual specified and actual MLC leaf positions at a frequency of 20 Hz during dynamic

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Table 2. Measured and computed doses at 5 cm depth in phantom at 95 cm SSD for a field in which the MLC was configured to block the entire radiation field (closed) and for 10 × 10 cm2 fields created by sliding a window of the size indicated (see the appendix for further description of the sliding window fields). The jaws were set to 10 × 10 cm2 for the closed field and 13 × 10 cm2 for the sliding window fields. Doses are given in per cent of open-field dose for a 10 × 10 cm2 field. 6 MV

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Calculated

Calculated with offset

Closed 1 mm 2 mm 5 mm 10 mm 20 mm 40 mm 50 mm 70 mm 100 mm

1.64 4.71 5.66 8.39 12.6 19.8 31.0 35.8 43.6 57.4

1.64 ± 0.01 4.38 ± 0.01 5.33 ± 0.01 8.01 ± 0.01 12.16 ± 0.02 19.44 ± 0.03 30.81 ± 0.05 35.52 ± 0.06 43.04 ± 0.07 57.09 ± 0.09

1.66 ± 0.01 4.69 ± 0.01 5.65 ± 0.01 8.29 ± 0.01 12.42 ± 0.01 19.60 ± 0.02 30.98 ± 0.03 35.63 ± 0.03 43.08 ± 0.04 57.14 ± 0.05

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1.74 4.98 5.90 8.58 12.8 20.0 31.2 36.0 43.8 57.5

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1.83 ± 0.01 5.02 ± 0.01 5.94 ± 0.01 8.63 ± 0.01 12.70 ± 0.01 19.89 ± 0.02 31.31 ± 0.03 35.80 ± 0.03 43.26 ± 0.04 57.30 ± 0.05

MLC delivery and have been described in detail by Litzenberg (Litzenberg et al 2002). Using this dynalog file, the leaf positions reported by the MLC controller were converted to a leafsequence file for use with the MC MLC model. The leaf-sequence file for the 1 mm sliding window had 3692 control points in it. Using this input file, the output at 6 MV was 4.44% of that of the 10 × 10 cm2 open field, a slight improvement over using the original leaf-sequence file. However, it was noted that for the first and last segments in the log files, when every leaf pair was closed, each and every opposing MLC leaf pair was reported to cross over by 0.016 cm (physically, at the MLC plane). This is due to overdriving the MLC motors to remove backlash built into the MLC system (Huntzinger 2002). To account for the backlash error in the MLC system, a set of MC simulations was performed for the sliding window test cases assuming that the 0.016 cm is an error in the actual leaf read-out positions, corresponding to a leaf position error of 0.008 cm per leaf. When this leaf position error is accounted for in the MC simulation, the output using the MLC sequence for the 1 mm sliding window from the dynalog files was 4.78%, and when the original input leaf sequence input file was used and this leaf position error was accounted for, the output was 4.66%. The outputs for all sliding window fields, when the leaf position error is accounted for, are listed in table 2. When a 0.008 cm leaf position offset is used to account for the backlash error, the MC MLC model predicts the measured output within 0.5% of the open-field dose and within 1.5% of the local dose in all cases. Three things are indicated by this result. First, for this test case, the leaf positions achieved by the MLC were much closer to the desired values than the 0.2 cm tolerance. Secondly, the read-back of the ‘actual’ MLC positions from the dynalog file are not necessarily the physical leaf positions, possibly due to backlash in the MLC system. Thirdly, for our sliding-window MLC simulations, an offset of 0.008 cm per leaf improves agreement with measured values. 3.3.2. Off–axis field. It is commonly known that the leaf positions in the Varian MLC sequence file specify the leaf positions at the isocentre plane and typically correspond to the light-field projection of the MLC leaf edge, (Boyer and Li 1997) although this can be modified by the user using the mlctable.txt file mentioned in section 2.2 and in the appendix. When

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Figure 12. Measured (solid line) and calculated (dashed line) outputs for a 1 mm sliding window that goes from −20 cm to −6 cm. Jaws were set at −20 cm and −2 cm.

using the default mlctable.txt that uses the light-field projection, MLC leaf tip positions far from the central axis can differ from those that would be obtained using simple back projection by several millimeters. Thus, it is important to check the MLC model far from the central axis to ensure that the leaf tips are correctly positioned. To ensure the MLC model accurately reproduces leaf positions far from the central axis, a 4-point dynamic leaf sequence file was created that specified the leaves being initially closed on the left-hand side of the field at X = −20.0, the leaves open to a 1 mm light-field window on the left-hand side of the field (leading leaf at X = −19.9 cm), the leaves open at a 1 mm light-field window at the right-hand side of the field (leading leaf at X = −6.0), and the leaves closed on the right-hand side of the field. The Y -jaws were set to produce a 10-cm wide field, and the X-jaws were set at −20.0 cm and −2.0 cm. The X-jaw setting was limited to −2.0 by the maximum jaw over travel beyond the beam central axis. Film measurements at 5 cm depth, 95 cm SSD with a 6 MV beam for a 999 MU irradiation were compared with MC calculations that used the MLC model for identical conditions. The voxel size for the MC computations was 0.2 × 2.0 × 1.0 cm3. Results from this test are given in figure 12. Two calculated profiles are given: one that assumes that the MLC leaf position error is zero, and the other presuming that the leaf position error is 0.008 cm. When the MLC position error is presumed to be zero, the calculated profile is ∼7% less than the measured profile, but when the leaf position error is presumed to be 0.008 cm, the calculated profile is ∼2.5% less than the measured profile. Slight changes in the leaf position have a large effect on this profile. 4. Discussion and summary When MC methods are used for patient-dose computations, the accuracy in the final result is limited by the accuracy of all of the components involved in the computation. This includes approximations made in source modelling and in radiation transport through various patientbeam modifiers, including the MLC. This study developed a method for simulating radiation transport through the MLC for use with MC dose calculation algorithms. Rigorous benchmark

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tests were used to quantify the accuracy of the MLC model. One strength of the MLC model is that is uses the same method as the treatment machine to determine the MLC leaf positions as a function of the number of monitor units delivered. One goal of the MLC model was that it be substantially faster than simulations using a standard MC code with full physics treatment. Using the MLC-blocked field depth–dose test case, in which both MCNP and MLC model calculations were performed, we found that the MLC model was 200 times faster than MCNP for the same problem on the same computer processor. A particle transport rate of 10,000–20,000 photons/second is achieved on 500 MHz Pentium III processors. The MLC model thus has met the goal of being reasonably fast. Since the probability of a given photon interacting in the MLC is evaluated at multiple random fractional monitor units during the beam delivery to determine the average attenuation probability, the model is also efficient. Tests with the MLC blocking the radiation field indicate that the physics included in the MLC model is sufficient. MLC-blocked fields will contribute the maximum amount of electron scatter from the MLC, have the maximum amount of bremsstrahlung interactions from electrons incident upon the MLC, and have the maximum amount of pair production from photon interactions in the MLC. Although the MLC model neglects each of these interactions, it accurately predicts the surface dose, beam hardening, and field size dependence of the leakage radiation from the MLC. This suggests that modelling only photon attenuation and first Compton scatter may be sufficient for other upstream beam modifiers without affecting the final MC dose calculation result, particularly those modifiers located upstream of the MLC location such as the jaws or upper wedges. The tests with sliding windows of various widths indicate the ability of the MLC model to accurately integrate the dose delivery from the closed, leaf tip, and open portions of DMLC beam delivery. This accuracy is obtained because the MLC motion is modelled using the same technique as that used to control the MLC on the accelerator. Because the MLC leaf positions have some position uncertainty due to backlash, sliding a small (∼1 mm or less) window width across the field can be used to determine the magnitude of this position uncertainty. Inclusion of a physical leaf position error to account for backlash in the MLC simulation had a larger effect on the results than using the leaf positions reported in the MLC controller-produced dynalog files, even though the leaf position tolerance specified in the input files used in this study was 0.2 cm. The leaf position error determined in this study (0.008 cm) is of little consequence for typical, clinically used window widths of 0.5 cm and greater. Furthermore, the backlash position error likely depends on the prior history of the leaf motion. However, given the sensitivity of the output to the leaf position error for small sliding windows, use of a ∼1 mm sliding window field in routine MLC quality assurance would be a good way of verifying the stability of the MLC. The ability of the MLC model to correctly predict the dose undulations in MLC-blocked fields, the dose for picket-fence fields, and the dose in instances with the maximum tongueand-groove effect indicate that the geometry simplifications made by the MLC model are inconsequential at the 1% dose level for test cases that most severely test the model. In routine clinical cases where leaves are at least partially synchronized, the effects of these approximations will be even less. Overall, the MLC model developed has been shown to reproduce dose measurements generally within ±1% or 1 mm. The MLC model is fast and efficient, and is applicable to and has been used for static beam, DMLC and SMLC IMRT beam deliveries. This model will be useful for inclusion in MC-based IMRT dose calculations, for pre-treatment verification of IMRT beam delivery, and for calculation of portal imager transmission images for patient treatment verification.

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5. Information The source code for the MLC model described in this study and the MLC leaf sequence files used in this study are available on request from the first author. Acknowledgments This work was supported by grants CA74043 and CA74158 from the National Cancer Institute. The authors would like to thank Varian Oncology Systems for providing information for MC modelling of the accelerator and MLC, and for supporting research efforts at Virginia Commonwealth University. We would like to thank Chris Bartee for assistance with MLC measurements and providing ready answers to our questions about the MLC, Devon Murphy for her meticulous editing of our manuscripts, and Jim Satterthwaite of Phillips Medical Systems for providing us with the EGS4-based Compton scattering routines in C. Appendix A. Control of Varian MLCs Accurate modelling of the motion of the MLC during dynamic delivery is crucial for conversion of leaf motions to ‘deliverable’ intensities for conventional dose calculation and for MC simulations that incorporate the MLC motion. Unfortunately, the literature is not clear in its description of MLC motion for Varian MLCs. This appendix discusses this motion. The discussion will first cover a single static MLC field and then expanded to cover DMLC and SMLC beam deliveries. The discussion will be given in the context of the 120 leaf MLC used in this study, but, according to Varian documentation (2000), will equally apply to the 80 leaf and 52 leaf MLCs. The MLC consists of 120 MLC leaves. These leaves are contained in two leaf banks termed the A- and B-leaf banks; thus there are 60 leaves in each leaf bank. A leaf pair consists of leaves with the same index in A and B leaf banks. For static beam delivery, the position of each MLC leaf is contained in a computer file (a .mlc file). The MLC controller interprets the MLC positions from this file for each leaf pair as follows: For a given leaf pair, the positions of the A and B leaves are read from the file. A table-look-up is performed from a file called mlctable.txt to transform the positions indicated in the file to leaf tip positions. Note that if the specified position of the A leaf is the same as that of the B leaf (that is A leaf = −B leaf, the leaves touching) then the mean position for the A and B leaves is determined from the mlctable.txt look-up, and both leaves are set to this same position. This keeps the MLC leaves touching when the leaf pair is specified to be closed. A demagnification of the A and B leaf positions to the MLC plane is then performed using a demagnification factor specified in the mlctable.txt file. The default mlctable.txt provided by Varian is configured such that the light-field projection of the rounded leaf tip will be at the position indicated in the leaf position file. This has been discussed at length by Boyer and Li (1997). It is possible for the user to modify the mlctable.txt so that some other quantity is projected to the isocentre plane, such as the 50% isodose line, however, this is not recommended. For DMLC and SMLC beam deliveries, the MLC motion is specified in a dynamic leafsequence file (.dml or .dma file). This format of this file is similar to that of the .mlc file except that MLC positions are listed at various control points. At each control point, the fractional number of monitor units (MU) [0,1] is also listed. The leaf positions at these control points are converted to the MLC plane using the same method as for static fields. For dynamic delivery, for MU values between those indicated in the leaf-sequence file, the leaf positions are linearly interpolated. This interpolation occurs on the leaf positions after

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Figure 13. Physical window width (solid line) and the projected light-field width (dashed line) for the 1 mm window width leaf-sequence file indicated in table 3. Note that the light-field width increases in the centre of the field. Table 3. Leaf positions in leaf sequence file for a 1 mm sliding window field. The index is the fractional number of monitor units for the leaves at the indicated positions. A negative in the leaf position indicates that it has crossed the beam central axis. Index

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5.00 5.00 −4.90 −5.00

they are converted to the MLC plane. As an example, consider the .dml file with leaf positions as indicated in table 3. This .dml file with four control points indicates a 10 cm wide field produced with the leaves initially closed, opening to a 1 mm window width at −5.00 cm, sliding this window across to +5.00 cm, then closing the window (this is the 1 mm window file used in this study). Figure 13 shows the physical distance between the leaves and the projected light field for this field. Note, the interpolation method results in a constant physical window width, however, the light field and the radiation field are non-constant, increasing to a maximum at the centre of the field. For the sliding-window field measurements and calculations used in this study (section 3.3), the .dml files contained four control points similar to those in the test case discussed. Since the MC MLC model described in this paper uses the same algorithms as the accelerator (described above) to set the MLC positions, the MC MLC model correctly predicted MLC positions. Note that the sliding-window files are the same as those used by Arnfield et al (2000) and Keall et al (2001). These files do not produce a uniform light-field window width across the field, which makes using these files to determine parameters such as the ‘equivalent leaf shift’ or ‘leaf gap error’ (Arnfield et al 2000) problematic.

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