Dose calculation and verification of intensity modulation generated by dynamic multileaf collimators Spiridon Papatheodorou,a) Jean-Claude Rosenwald, Sofia Zefkili, Marie-Claude Murillo, Jean Drouard, and Genevie`ve Gaboriaud Service de Physique Me´dicale, Institut Curie, Paris, France
共Received 9 August 1999; accepted for publication 7 March 2000兲 While the development of inverse planning tools for optimizing dose distributions has come to a level of maturity, intensity modulation has not yet been widely implemented in clinical use because of problems related to its practical delivery and a lack of verification tools and quality assurance 共QA兲 procedures. One of the prerequisites is a dose calculation algorithm that achieves good accuracy. The purpose of this work was twofold. A primary-scatter separation dose model has been extended to account for intensity modulation generated by a dynamic multileaf collimator 共MLC兲. Then the calculation procedures have been tested by comparison with carefully carried out experiments. Intensity modulation is being accounted for by means of a 2D 共two-dimensional兲 matrix of correction factors that modifies the spatial fluence distribution, incident to the patient. The dose calculation for the corresponding open field is then affected by those correction factors. They are used in order to weight separately the primary and the scatter component of the dose at a given point. In order to verify that the calculated dose distributions are in good agreement with measurements on our machine, we have designed a set of test intensity distributions and performed measurements with 6 and 20 MV photons on a Varian Clinac 2300C/D linear accelerator equipped with a 40 leaf pair dynamic MLC. Comparison between calculated and measured dose distributions for a number of representative cases shows, in general, good agreement 共within 3% of the normalization in low dose gradient regions and within 3 mm distance-to-dose in high dose gradient regions兲. For absolute dose calculations 共monitor unit calculations兲, comparison between calculation and measurement reveals good agreement 共within 2%兲 for all tested cases 共with the condition that the prescription point is not located on a high dose gradient region兲. © 2000 American Association of Physicists in Medicine. 关S0094-2405共00兲03505-7兴 Key words: dose calculation, intensity modulation, dynamic collimation, treatment planning
I. INTRODUCTION The term ‘‘Intensity modulation’’ has been widely used in the literature in order to express the spatial variation of the fluence within the beam opening. There are a variety of techniques for generating intensity modulated beams. Conventional compensators and transmission blocks, have been used for decades in order to compensate patient irregular surface and tissue heterogeneities.1 Modern methods for delivering intensity modulation such as tomotherapy2 and dynamic multileaf collimation3–8 are also becoming available due to the advances in radiotherapy machine technology. Dynamic collimation is under development by several linear accelerator manufactures. Recent implementations of dynamic collimation feature computer control of the moving leaves and computerized dynamic control of the dose rate. Different techniques used for the delivery of intensity modulation can introduce varying levels of dosimetric deviation from the desired intensity distribution. For example, missing tissue compensators modify the energy spectrum of the beam and increase significantly the scatter coming from the treatment head.9 Similarly, when dynamic collimation is used, the mechanical and dosimetric properties of the collimating system alter the generated intensity. This is mostly due to leaf transmission and leaf penumbra.10 The variation 960
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of the output according to the variation of collimator opening during the irradiation, needs to be accounted for as well.11,12 These phenomena result in a fluence distribution that can be substantially different from the initial, theoretical, intensity modulation and therefore, should be taken into account by the treatment planning system 共TPS兲 for accurate dose calculations. Chui et al. showed that it is possible to predict the dose distribution resulting from a dynamic-collimated beam by convolving the primary photon fluence corresponding to the intensity distribution with the appropriate pencil beam kernels.13 However, this method requires a good knowledge of the relevant kernels and cannot be directly applied when the dose model is not based on photon convolution. This is the case when the dose is calculated from the separation between its primary and scatter components. Since the primary/scatter dose model has been already validated in clinical use in many centers, it is useful to develop the appropriate extensions to account for intensity modulation, and investigate the possibilities and limitations of this technique to achieve acceptable dose calculations. In this study, a method to extend a primary/scatter dose model to account for intensity modulation generated by dynamic or multiple segment collimation is presented. This
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method has been implemented into the ISIS 3D 共ISiS3D is developed at Institut Curie and commercialized by Technologie Diffusion, Paris, France兲 treatment planning system in order to account for the effect of arbitrary intensity modulation generated by the Varian dynamic multileaf collimator 共Varian Associates, Palo Alto, California兲. We consider that this method could be easily adapted to other treatment planning systems based on a similar dose model and to multileaf collimators developed by other manufacturers. A second distinctive part of this work involves the experimental verification of the dose calculation algorithm. A set of test intensity distributions have been designed in order to verify the accuracy of the algorithm. Calculated and measured dose distributions are compared for a number of representative test cases. II. METHODS AND MATERIALS IMRT 共intensity modulation radiation therapy兲 consists of a number of steps, one of which is the calculation of the dose distribution that will result when a particular arrangement of intensity modulated beams is applied to a patient. In a common inverse planning scheme, the desired intensity modulation patterns for each beam are calculated by specific software applications for dose distribution optimization.14–16 Then the desired intensity modulation is converted to a leaf motion plan for practical delivery with computer controlled multileaf collimators. The software modules that perform this task are often called ‘‘leaf sequencing programs,’’ ‘‘segmentation programs’’ or ‘‘interpreters.’’6–8,17 The purpose of such programs is to prepare the best MLC sequence so that the delivered fluence is as close as possible to the desired one. Other objectives, such as the minimization of the total beam-on time or the minimization of the overall treatment time are also considered. In addition, the influence of a series of undesirable phenomena such as the tongue-and-groove effect, the contribution of radiation transmitted through the MLC leaves and through the rounded leaf tips may be dealt with in that stage.18–20 In other words, an effort is being undertaken in order to anticipate for those negative effects, so that the delivered intensity modulation equals to the desired one. This task is not always fully accomplished, because on the one hand, the implemented techniques are not always successful, and on the other hand, because it is just impossible to fully account for some effects 共e.g., it is impossible to obtain zero dose at a point inside the radiation field, due to transmission兲. Therefore, there is the need for a final ‘‘forward planning’’ dose calculation, for which the purpose is to achieve good accuracy taking into account all possible influences. Even if it is not possible to deliver a dose distribution resembling to the desired one, it is important to calculate the actually delivered dose distribution before approval of any IMRT plan. In our case, the final planning calculation is performed after the MLC sequence has been calculated and particularly using the information of the leaf motion plan in order to take into account the mechanical and dosimetric properties of the MLC. This information is taken directly from the computer Medical Physics, Vol. 27, No. 5, May 2000
FIG. 1. Diagram showing the application of the matrix of correction factors for dose calculations with intensity modulation. The intensity modulation modifies the primary radiation reaching calculation point P. It also modifies the amount of photons reaching the scattering element ⌬S and, therefore, modifies the scatter radiation reaching P.
files 共DMLC files兲 that are used for controlling the MLC motion during the dynamic irradiation. The final planning dose calculation is based only on the DMLC files and can be performed independently of the origin of those files. A. Dose calculation algorithm
The algorithm used for dose calculation at a point with the ISiS3D treatment planning system is based on the separation of primary-scatter radiation.21–23 Figure 1 illustrates the principle of dose calculation for an intensity modulated photon beam. A beam of finite size is irradiating a water equivalent object. The dose is to be calculated at point P. The largest part of the energy deposited at P is due to photons coming along the direct path 共Source-P兲 and interacting at P. A smaller part of the deposited dose is due to photons that will come along the path Source-S, interact in the sector ⌬S and contribute to scatter at point P. Intensity modulation significantly modifies the spatial fluence distribution, incident to the patient. So the incident photon fluence differs from that of the corresponding open beam in what it is further modulated by the dynamic collimator. The dose, D M , at a point P with intensity modulation is calculated as follows: D M 共 x P ,y P ,z P 兲 ⫽D P,o 共 x P ,y P ,z P 兲 •FC 共 x P ,y P 兲 ⫹
兺i 兺j 关 D S,o共 r i , j ,z s 兲
•FC 共 r i , j 兲兴 .
共1兲
The first term describes the primary component. D P,o (x P ,y P ,z P ) is the dose due to the primary photons as calculated for the corresponding open beam. FC(x P ,y P ) is
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the correction factor applied to the open beam primary component of the dose to account for intensity modulation. The second term describes the energy deposited at point P due to photons that are scattered from every scattering element of the irradiated volume. In this term D S,o (r i , j ,z s ) is the dose contribution to P, due to photons scattered in the element ⌬S(r i , j ,z s ),, as calculated for the corresponding open beam. FC(r i , j ) is the correction factor applied to every scatter component of the dose to account for intensity modulation. The summation is restricted to the field area, and is performed over the range of the angles and along the radii of each sector as shown in Fig. 1. In order to implement this technique, a 2D matrix of correction factors in the plane perpendicular to the beam axis is precalculated for each beam. The dose calculation for the corresponding open field is then adjusted by these correction factors. The correction factors are used in order to weigh both the primary and the scatter component of the dose at a given point, as shown in Eq. 共1兲. The correction factor that is applied in the primary component of the dose is found on the fanline that links the point with the source. The correction factor that is applied on the scatter component of the dose is found on the fanline that links the center of each scattering sector with the source. The correction factors are calculated as the ratio of the incident fluence generated by the dynamic motion of the MLC leaves to the incident fluence of an outline segment corresponding to the total field area: FC 共 x,y 兲 ⫽
⌽ M 共 x,y 兲 . ⌽ O 共 x,y 兲
冕
t⫽T tot
t⫽0
N sgm
F 共 x,y 兲 •dt⬵
FIG. 2. Schematic representation of the beam geometry with the assumption of an extended source situated on a scattering plane inside the treatment head. Collimator jaws as well as other elements in the treatment head are omitted. The extended source, seen be a point P situated on the isocenter plane, may be partially obscured by the collimating system. Assuming an extended source, the resulting fluence profile is not any more a step function. It can be approximated by analytical functions such as a combination of exponential functions. Note that some of the elements of this diagram are depicted disproportionally for clarity purposes.
共2兲
For the calculation of the correction factors, the continuous motion of the collimator leaves is approximated by a sequence of static MLC defined irregular fields, often called ‘‘segments.’’ The superposition of a large number of MLC defined segments represented in the following equation by the sum is an effective approximation of the incident fluence generated by the dynamic motion of the leaves 共represented in the following equation by the integral兲: ⌽ M 共 x,y 兲 ⫽
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兺
i⫽1
F i 共 x,y 兲 •⌬Ti.
共3兲
In the integral above, F(x,y) is a momentary fluence distribution corresponding to the irregular field shape formed by the leaves at a given time t. The integration is over the total beam-on time, T tot 共expressed in monitor units兲. This, corresponds to the sum of N sgm elementary fluence distributions weighted by the beam-on time ⌬T i corresponding to each segment. In order to calculate the elementary fluence distribution corresponding to each segment, we have used a ‘‘semiempirical’’ model of an extended source situated in a plane inside the treatment head.22 Depending on the calculation point, the extended source can be partially obscured by the collimating system. Using this model, it is possible to describe the penumbra region of each line segment of the irregular field, that Medical Physics, Vol. 27, No. 5, May 2000
depends on the geometrical and physical properties of the collimator as well as on the energy and the quality of the photon beam. Figure 2 shows the problem in one dimension. The fluence profile is represented by a mathematical function which includes the scattering from the treatment head. We have used an empirical function that has been already used for the open beam dose calculations.23,24 This is an exponential function that involves few arbitrary constants that can be derived from measurements of the leaf penumbra. Therefore, a fluence profile along a leaf pair in the in-plane direction is influenced by the leaf-tip penumbra 共Fig. 2兲. The dose profile is obtained from the integration through the collimator opening of an extended effective source, where the relative surface activity decreases exponentially from center to periphery according to the following equation: S 共 r 兲 ⫽ 共  2 /2 兲 •e ⫺  •r . In this equation, S(r) represents the relative strength of the source per unit area as a function of the distance r from the center. The beta constant 共also called collimation constant兲 defines the strength fall-off of the source. For large beta values, the source will tend to be a point source and the resulting profile will be a step function. Typical values of collimating constants are 5.8 cm⫺1 for 6 MV photons and 4.1 cm⫺1 for 20 MV photons. Those values are extracted empirically by fitting calculated dose profiles to measured ones.
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Electron transport is not explicitly taken into account. However, it can be taken into account indirectly by including it in the ‘‘scatter’’ component after specific adjustment of values for small radii.25 By assigning different beta values for the inner and outer part of the profile, one can model nonsymmetrical profiles. Similarly, it is possible to adapt the influence of the leaf tip 共in-plane or leaf motion direction兲 or the leaf side 共cross-plane direction兲 by assigning different beta constants for each direction. The importance of the later is that it allows to take into account the influence of the ensemble of the leaf positions and not only the influence of the leaf pair that corresponds to the calculation point. LoSasso et al.20 reported that on Varian MLCs, there is a considerable contribution of radiation transmitted through the rounded leaf ends. This means that the width of the effective gap between opposing leaves is larger than the geometric gap between leaf tips. In many sliding window cases, this effective widening can account for a significant fraction of the variable gap between leaves. Whether this problem has been taken into account during the calculation of the MLC sequence 共for the generation of optimal leaf motion plans兲 or not, it has to be reconsidered for the final forward dose calculation. Our technique takes into account the effect of rounded tips by shifting the leaf positions outwards by a given offset. This is the opposite of what happens in the leaf sequencing program, where in order to anticipate for the effect of rounded tips, one has to shift the leaves inwards 共whenever this is allowed, i.e., whenever opposite leaves do not collide兲. The fluence generated by a given leaf sequence is calculated as follows. First, the leaf positions at a given time are converted to an irregular field shape. Then different penumbra constants are assigned for leaf-tip and leaf-side. In order to take into account the effect of the ensemble of the leaf positions at each point, the irregular field is subdivided into triangles, then each triangle into at most 10 degree sectors. Finally the contribution of each sector to the fluence at the calculation point is computed. The calculation is being repeated for all points in the correction matrix. This results into a 2D elementary fluence F(x,y) distribution for the given segment. We repeat the calculation for all segments. The superposition of F(x,y) corresponding to each segment, weighted by the corresponding beam-on time ⌬T, yields ⌽ M (x,y) according to Eq. 共3兲. ⌽ M (x,y) is used for the generation of the correction factors. In our treatment planning system we have also used ⌽ M (x,y) in order to graphically represent the intensity modulation related to each beam for verification purposes. Finally, for the generation of the correction factors, ⌽ M (x,y) has to be divided by the fluence distribution of an ‘‘outline segment’’ which is a static-collimated field corresponding to the extreme positions traversed by the jaws or leaves during the entire sequence of dynamic motion: ⌽ O 共 x,y 兲 ⫽F O 共 x,y 兲 •T tot .
共4兲
In this equation F O is the elementary fluence corresponding to the outline segment and T tot is the total beam-on time. Medical Physics, Vol. 27, No. 5, May 2000
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Note that the calculated correction factors account only for the effect of the intensity modulation. Any other corrections such as patient heterogeneity corrections are included in the calculation of the corresponding open beam. B. Experimental verification
The implementation of intensity modulation on our system involved another distinctive step: Verify that the calculated doses are in good agreement with measurements on our machine. Dose measurement with dynamic collimation requires that the dose at a measurement point be integrated during the entire exposure as the moving leaves sweep the field to form the dose distribution. This major difference from static fields, where dose can be measured using a single moving probe, has driven many investigators to use radiographic films which can measure multiple points within the field simultaneously.13,20,26 However, considering the accuracy in dose measurements needed for the validation of a new dose calculation algorithm, we have decided to complete our measurements using other detectors as well. Dose profiles were measured using a multidetector linear array and an ion chamber was used for spot measurements at several points of interest 共e.g., normalization points兲. A set of test intensity distributions was created for evaluating our technique 共Fig. 3兲. These tests were designed to determine which properties have been correctly modeled. The leaf trajectories of the 20th leaf pair for the test cases of Fig. 3 are shown in Fig. 4. This figure shows the leaf positions as a function of time expressed in relative monitor units 共1.0 corresponds to the total beam-on time兲. Intensity modulation along each leaf has been defined using a commercial spreadsheet program and then converted into a 2D fluence map. This 2D map constituted the input for our home-made leaf sequencing program6 that calculates the leaf trajectories, for practical delivery of the desired intensity, using the sliding window technique.7 In this program, it is possible to apply leaf synchronization in order to prevent tongue-andgroove effects. Furthermore, an optimal leaf sequence can be calculated at that stage, anticipating whenever it is possible for the effects of leaf transmission and rounded leaf tips. However, the forward dose calculation algorithm must be able to carry out accurate dose calculations for any leaf motion pattern, optimized or not. For that reason, in the verification process, we have preferred using unoptimized leaf motion plans as an input. The resulting DMLC files 共the files that actually drive the MLC leaves兲 are the starting point for the comparisons of this study. The ISiS3D treatment planning system has been used for dose calculations into a virtual uniform phantom with unity density and a fine dose calculation grid. Measurements have been repeated for both 6 and 20 MV photon energies at several measurement depths in a flat polystyrene phantom. A correction has been applied to the measurement depth taking into account the polystyrene density. The dimensions of both virtual and real phantom were sufficiently larger than the field size. Intensity modulated fields have been delivered using a commercial 40 leaf-pair dynamic collimator, mounted
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FIG. 3. A set of test intensity distributions used to verify experimentally the dose model. Those theoretical intensity maps are depicted on a plane perpendicular to the beam axis and correspond to 10⫻10 cm2 fields. Different shades of gray correspond to different dose levels with black corresponding to 100%. The test intensity distributions have been practically delivered with a sliding window dynamic collimation technique. The 10 central leaf pairs participating in the dynamic collimation swept the fields from left to right 共x direction兲. Leaf trajectories for the 20th leaf pair, which sweeps the field area close to the central axis, are shown in Fig. 4.
on a Varian Clinac 2300C/D linear accelerator. Collimator leaves are 1 cm large when projected to the isocenter plane. Two-dimensional dose distributions have been measured by radiographic films 共Agfa Structurix兲 that have been sandwiched between polystyrene sheets and placed at the desired depth. The dose delivered to the film was adjusted so that the maximum dose integrated by the film remained within the films’ linear response. The films where processed manually with regular controls of the film processor temperature. A computer controlled densitometer 共Wellho¨fer 2D densitometer兲 with a spatial resolution of 0.5 mm has been used for reading the films’ optical density. We have used film density-to-dose conversion curves measured using films placed in the phantom in the same orientation used to expose the intensity modulated fields. The density-to-dose conversion curve was determined using static 10 cm⫻10 cm fields for a series of exposures up to 80 cGy. A linear array of diode detectors, the SNC Profiler 1170 共Sun Nuclear Corporation, Melbourne, Florida兲, has been used for the direct measurement of beam profiles. The ProMedical Physics, Vol. 27, No. 5, May 2000
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filer consists of a linear array of 46 p-type diodes, spaced in 0.5 cm increments. The Profiler spatial resolution can be doubled by performing two acquisitions per beam profile with the diodes of the second acquisition shifted by the midspace between diodes. The array is sandwiched between two built-in acrylic plates, with a total buildup thickness of 0.9 ⫾0.1 g/cm2. The bottom plate is 2.8 g/cm2 共2.3 cm兲. A detailed description of the system is given by Zhu et al.27 Measurement depth has been adjusted by adding polystyrene plates over the profiler surface. In that case, the measurement depth was defined using the water equivalent depth of the total buildup material. The sensitivity of individual diodes depends on some intrinsic properties of the diode such as the diode active area and diffusion length. The relative sensitivity from detector to detector may vary by as much as 15%. Therefore, a careful calibration of the array has been performed. Calibration has been repeated for each available photon energy, due to the possible energy dependence in diode sensitivity. The sensitivity of individual diodes, also changes with total dose accumulated in the diode, and should be carefully monitored. Finally, point doses were measured using a cylindrical 0.03 cm3 ion chamber 共IC3 Wellho¨ffer Dosimetrie, Schwartzenbruck, Germany兲, coupled to a Keithley electrometer. The dose at each measurement point was integrated during the entire dynamic collimation exposure. Polystyrene phantoms 共40⫻30 cm2 rectangular area兲 of various thickness were used to vary the depth of measurement point. In order to experimentally assess the effect of rounded leaves and transmission a series of measurements, with sliding windows of various nominal gaps, have been performed. The DMLC files for those test cases have been prepared assuming a constant dose rate of 300 MU/min and a maximum mechanical leaf speed of 2 cm/s 共0.4 cm/MU兲. Measurements have been performed using an ion chamber, in a flat polystyrene phantom at the depth of maximum dose for 20 MV photons. In this work, the coordinate system (x,y) is defined on a plane perpendicular to the collimator rotation axis. The inplane direction 共x axis兲 is the direction of leaf motion on our machine. It coincides with the x b axis as defined in IEC Report 1217.28 The cross-plane direction 共y direction兲 coincides with the y b axis of the beam limiting device as defined in IEC Report 1217. III. RESULTS AND DISCUSSION After the integration of intensity modulation into the ISiS3D treatment planning system, we have compared calculated dose distributions with measurements on our machine. Only a few representative results are presented in this work.31 A. Relative dose calculations
The measured and calculated isodose distributions for an arbitrary, irregularly shaped, intensity modulated beam are shown in Fig. 5. This is a clinical intensity modulated field for treatment of a patient with prostate cancer. Maximum
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FIG. 4. Leaf trajectories of the 20th leaf pair for the test cases of Fig. 3. The figure shows leaf positions as a function of time expressed in relative monitor units 共1.0 corresponds to the total beam-on time兲.
field size is 7.71 cm in x direction and 9 cm in y direction. Only the nine central leaf pairs 共leaf pair 17 to 25兲 participate in the dynamic collimation. Measured data have been obtained by exposing a film to a 6 MV photon beam for a beam-on time of 70 monitor units. The film was placed perpendicularly to the beam axis at the depth of 5 cm. Film optical density has been converted to dose using a measured film calibration curve. Absolute dose on the central axis has been measured using an ion chamber as well. It has been found equal to 31.9 cGy. The maximum integrated dose was 48.4 cGy, so doses integrated by the film remain within the linear response range of the calibration curve. In Fig. 5, the two isodose distributions are normalized to 100% on the central axis. Calculated data are illustrated with gray levels Medical Physics, Vol. 27, No. 5, May 2000
while measured data are represented by solid line contours. The agreement, in general, is quite good with the two sets of isodose curves having similar shapes. In some regions discrepancies may reach to 5% relative to the normalization dose. In high dose gradient regions the maximum distance between measured and calculated isodose lines is in the order of 3 mm. Limitations of film dosimetry, problems in registration of the measured and calculated isodose sets and approximations in the dose calculation model are some of the possible causes for these differences. Considering the problems related to film dosimetry and the difficulties on handling and evaluating two-dimensional dose distributions, we have decided to complete our measurements using ion chambers and a multidetector array.
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FIG. 5. Measured and calculated isodose distributions at the depth of 5 cm from a 6 MV intensity modulated photon beam. Both measured and calculated distributions have been normalized to 100% on the central axis. Calculated data are illustrated with gray levels. Measured data have been obtained by exposing a film for a beam-on time of 70 MU. They are superposed on the calculated data and represented here by solid line contours with the corresponding isodose labels on top of them.
Using the intensity patterns of Fig. 3, dose profiles have been measured and calculated for 6 and 20 MV photons at different depths 共5, 10, and 15 cm兲. All calculations and measurements were made at a source to axis distance 共SAD兲 of 100 cm, for 10⫻10 cm2 intensity modulated fields. Figures 6–9 show dose profiles based upon results of calculations 共solid lines兲, ion chamber 共solid circles兲, and profiler measurements 共dashed lines and circles兲. The theoretical intensity modulation, used for the leaf motion calculation, is represented by the gray solid line. Profiler measurements have been taken with the diode array parallel to the direction of leaf motion, in the leaf center of the 20th leaf pair. The dose profiles along the x axis have been normalized on the central axis regarding the x direction but slightly off-axis 共at the center of the 20th leaf pair兲 regarding the y axis direction. The normalization point expressed in collimator coordinates was, therefore, (x,y)⫽(0 cm, ⫺0.5 cm兲. For the test case of Fig. 3共b兲 共‘‘single step’’兲, the normalization point was at (x,y)⫽(2.5 cm, ⫺0.5 cm兲. For the test case of Fig. 3共c兲 共‘‘steps in y direction’’兲, the diode array was perpendicular to the direction of leaf motion through the central axis. Figure 6 illustrates the comparison between calculation and measurement of the dose profile along the central leaf pair for a dynamic irradiation intended to produce a uniform dose distribution 关uniform pattern, Fig. 3共a兲兴. All leaves participating in the dynamic collimation are instructed to sweep the field with a constant speed and with a constant gap between opposing leaves. This is a simple test that allows to quantify the performance of the dose calculation algorithm as compared with data from open field dosimetry 共20%–80% penumbra, flatness, etc.兲. Conclusions can be drawn for the modeling of the penumbra region 共field edges兲 and the region of the residual dose. If dose calculations for the corresponding open field are accurate enough, there should not be a Medical Physics, Vol. 27, No. 5, May 2000
FIG. 6. Comparison between measured and calculated uniform profiles generated by dynamic collimation 关test of Fig. 3共a兲兴. The calculation– measurement depth is 5 cm. The photon energy is 6 MV in 共a兲 and 20 MV in 共b兲. 关Calculation: Solid line; profiler measurement: Dashed line and open circles; ion chamber measurement: Solid circles; theoretical intensity profile: Grayed line兴.
problem in the region within 80% of the field because correction factors have a constant value there. Because both measured and calculated dose profiles are normalized, it is difficult to evaluate the influence of the photons transmitted through the collimator leaves inside the field. However, in the case of the single step pattern of Fig. 3共b兲, one half of the field remains covered by the leaves for half of the beam on time. So the influence of the radiation transmitted through the leaves is much more important and evident even after normalization. Figure 7 shows the corresponding theoretical and measured transverse 共along the x axis兲 intensity profile through the center of the field 共i.e., along the 20th leaf pair兲. The effect of electron lateral transport is evident at the edge of the step. The dose calculation algorithm should be able to predict its variation with depth. Figure 7共a兲 shows the results for a 20 MV photon beam at a depth of 5 cm and Fig. 7共b兲 at a depth of 15 cm. With the sliding window technique, as the leaves sweep across the field, modeling of leaf penumbra becomes very important. In the Varian multileaf collimator, leaves have
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FIG. 7. Comparison between measured and calculated intensity modulated profiles for 20 MV photons 关test of Fig. 3共b兲兴. The calculation–measurement depth is 5 cm in 共a兲 and 15 cm in 共b兲. 关Calculation: Solid line; profiler measurement: Dashed line and open circles; theoretical intensity profile: Grayed line兴.
FIG. 8. Calculated and measured 6 MV dose profiles for stair step intensity distributions. 共a兲 Stair steps along the y direction. 共b兲 Stair steps along the x direction 共leaf motion direction兲. The calculation–measurement depth is 3 cm in 共a兲 and 5 cm in 共b兲. 关Calculation: Solid line; profiler measurement: Dashed line and open circles; ion chamber measurement: Solid circles; theoretical intensity profile: Grayed line兴.
rounded edges in order to maintain a relatively constant penumbra in respect with their off-axis position. However, in the direction perpendicular to the leaf motion, leaf side has a totally different shape 共the characteristic tongue-and-groove arrangement for the minimization of interleaf leakage兲. So beam penumbra corresponding to leaf tip and leaf side may present slight differences. In our model, this can be taken into account by assigning different collimation constants 共beta constants兲 in x or y direction. Leaf side penumbra plays an important role in the case of the pattern described in Fig. 3共c兲 共steps in y direction兲 while leaf tip penumbra plays an important role in the case of the pattern shown in Fig. 3共d兲 共steps in x direction兲. In the first case a beam profile has been measured in the y direction passing through the center of the field. Results are shown in Fig. 8共a兲 for 6 MV photons at the depth of 3 cm. While the theoretical intensity within the width of each leaf is assumed to be uniform, the corresponding dose is affected by the lack of lateral electronic equilibrium and the effects of leaf side penumbra. Similarly in the case of Fig. 3共d兲, a beam profile has been measured in the x
direction 共along the 20th leaf pair兲. Results are shown in Fig. 8共b兲 for 6 MV photons in 5 cm depth. Again effects of leaf penumbra along the x direction and electronic disequilibrium smear the dose profile. The dose calculation algorithm should be able to account for these effects. Finally, Fig. 9 shows the comparison between measured and calculated dose profiles along the 20th leaf pair for the ‘‘roof’’ shape 关Fig. 3共e兲兴 and the intensity well 关Fig. 3共f兲兴 patterns. Figure 9共a兲 shows beam profiles at 5 cm depth for 20 MV photons. The theoretical intensity pattern has a roof shape, comprising ascending and descending slopes. Those kind of slopes can be almost perfectly reproduced by a full dynamic collimating system. However, as described in the Materials and Methods paragraph, a certain discretization is being performed by the dose calculation algorithm during the calculation of the incident fluence 关Eq. 共3兲兴. If the approximation is not accurate enough, the dose calculation algorithm will not predict the generated dose profile and will present a sort of smoothed stair steps instead of a continuous line. In the case of Fig. 9共b兲, beam profiles have been measured and
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80%–100% penumbra region兲. A possible cause is the fact that dose profiles in the penumbra region are calculated based on analytical functions which do not fully account for physical phenomena occurring in the head/collimator assembly. Also, there have been some discrepancies in the region of the residual dose 共outside the field兲 but they are related to limitations of the original dose model for open beam dose calculations, since the correction factors outside the field are always constant and equal to 1. B. Effect of rounded leaf end transmission
FIG. 9. Comparison between calculated and measured dose profiles along the 20th leaf pair for a ‘‘roof’’ shape intensity modulation in 共a兲 and for an intensity well distribution in 共b兲. Both of them have been calculated– measured at a depth of 5 cm. The photon energy is 20 MV for 共a兲 and 6 MV for 共b兲. 关Calculation: Solid line; profiler measurement: Dashed line and open circles; ion chamber measurement: Solid circles; theoretical intensity profile: Grayed line兴.
calculated in 5 cm depth for 6 MV photons. The theoretical intensity pattern presents a 50% intensity well of 3⫻3 cm2 inside a 10⫻10 cm2 field. The dose calculation algorithm will accurately predict the width and the depth of the well for the real dose profile only if the effects of leaf transmission, leaf penumbra and electron lateral transport in the medium are adequately modeled. The agreement between measurement and calculation for the examined dose profiles was in general quite good. In low dose gradient regions, deviations were generally smaller than 3% of the normalization dose, which has been our limit of acceptance for intensity modulated fields. In high dose gradient regions and in the penumbra region 共field edges兲, dose agreement has been evaluated in terms of position errors and was found within our 3 mm distance-to-dose 共DTA兲 criterion. Dose points that violate both dose 共3% of the normalization dose兲 and distance-to-dose 共3 mm兲 criteria have been found on the parts of the profiles linking a high dose gradient region with a low dose gradient region 共e.g., 0%–20% and Medical Physics, Vol. 27, No. 5, May 2000
In the test cases presented above, the gap size of the sliding window was relatively large as it can be shown in Fig. 4 where the leaf trajectories for the opposing leaves of the 20th leaf pair are depicted for each test field. However, depending on the gap size of the sliding window, the dose transmitted through the MLC varies significantly and becomes very important for small gap sizes. In order to assess the magnitude of this effect we performed a series of measurements with sliding windows of various nominal gaps. Opposing leaves where instructed to move at constant speed forming a slit that sweeps over the field area. Ideally 共zero transmission, sharp penumbra, no leaf edge effects兲, the integrated dose should be proportional to the size of that slit. However, as the gap size decreases, the fraction of the time a given point remains covered by the leaves increases, so the influence of leaf transmission is greater. Furthermore, as the nominal gap decreases, the additional transmission through the rounded leaf tips accounts for a significant fraction of the dose. With the ion chamber positioned on the collimator central axis, the reading was recorded for a fixed number of monitor units, using the intensity modulated field. Then this was repeated for the open field for the same number of monitor units and the ratio of the two readings was calculated. Table I illustrates a comparison between measured and calculated doses relative to the open beam dose for a series of nominal gaps. The test case of Fig. 6 is also included 共second row of Table I兲. For the dose calculation of column 5, a constant mid-leaf transmission of 2% has been assumed. According to LoSasso et al.,20 the effect of the transmission through the rounded leaf on the dose delivered by dynamic MLCs can be approximated by a 1 mm offset applied to the leaf position. This observation is in good agreement with our results 共column 7; Table I兲, where an effective widening of the nominal gap by 2 mm 共1 mm for each opposed leaf兲 has been applied. Table I shows that the discrepancies between measurements and calculations may be up to 26.6% for a nominal gap of 1 cm if the transmission through the leaves and the leaf rounded tips is not taken into account, but they remain below 2% if these phenomena are properly accounted for in dose calculations. C. Absolute dose calculations
The monitor unit 共MU兲 calculation for an intensity modulated field is based on a correction applied on the number of monitor units calculated for the corresponding open field under equal conditions. This correction factor represents an ef-
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TABLE I. Comparison between calculated and measured doses relative to the open beam dose, obtained for the same number of monitor units and for different slit widths 共gaps兲 being swept at constant speed over the field area. For the dose calculations of column five, a constant mid-leaf transmission of 2% has been used. For the dose calculations of column seven, an effective widening of the nominal gap by 2 mm 共1 mm for each opposed leaf兲 has been assumed. Calculated dose
Test case
Measured dose 共20 MV; d max)
Without any correction
% Difference from meas.
Accounting for transmission only
1 0.806 0.465 0.312 0.197 0.124
1 0.8 0.444 0.286 0.167 0.091
0 ⫺0.7 ⫺4.5 ⫺8.3 ⫺15.2 ⫺26.6
1 0.802 0.452 0.302 0.178 0.107
Open field Fig. 6 test case 8 cm gap 4 cm gap 2 cm gap 1 cm gap
fective transmission through the ‘‘virtual modulator’’ 共similar to the wedge factor for wedges兲 and can be defined as the ratio of dose per MU with intensity modulation to the dose per MU for the corresponding open field. Monitor units calculated by the corresponding open beam must be divided by this effective transmission factor 共ETF兲 to yield the number of monitor units for the corresponding intensity modulated field: MUim⫽MUop /ETF. We expect to have an accuracy in monitor unit calculations comparable with that of the computation of the effective transmission factors. Of course, this is true with the condition that the MU calculations for the open beams are accurate. Otherwise, any error in the calculation of MUop is propagated in the calculation of MUim. Therefore, the best way to evaluate the applied corrections for MU calculations is to compare calculated effective transmission factors with measured ones. Table II illustrates such a comparison for the test cases of Fig. 3 for 6 and 20 MV photon beams. The
% Difference from meas.
Accounting for transmission and rounded leaf tips
0 ⫺0.5 ⫺2.8 ⫺3.2 ⫺9.6 ⫺13.7
1 0.807 0.464 0.313 0.196 0.125
Calculated effective transmission factor
Measured effective transmission factor
Difference 共%兲
6 MV Fig. Fig. Fig. Fig. Fig. Fig. Fig.
3共a兲 共uniform兲 3共b兲 共single step兲 3共c兲 共steps in y direction兲 3共d兲 共steps in x direction兲 3共e兲 共‘roof’ shape兲 3共f兲 共intensity well兲 6 共clinical intensity兲
0.803 0.611 0.650 0.648 0.320 0.793 0.484
0.807 0.653 0.648 0.651 0.322 0.797 0.484
⫺0.5 ⫺6.4 0.3 ⫺0.4 ⫺0.5 ⫺0.5 0
Fig. Fig. Fig. Fig. Fig. Fig.
3共a兲 共uniform兲 3共b兲 共single step兲 3共c兲 共steps in y direction兲 3共d兲 共steps in x direction兲 3共e兲 共‘roof’ shape兲 3共f兲 共intensity well兲
0.804 0.613 0.649 0.648 0.330 0.790
0.808 0.648 0.649 0.653 0.325 0.798
⫺0.5 ⫺5.4 0 ⫺0.7 1.6 1.0
20 MV
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0 0.1 ⫺0.2 0.3 ⫺0.5 0.8
clinical example of Fig. 5 is also included for 6 MV photons. We have measured the effective transmission factors on the central axis at 5 cm depth using an ion chamber. For the calculation of effective transmission factors, the algorithm calculates the dose on the prescription point 共in our case on the central axis at 5 cm depth兲 using the corrected incident fluence matrix and then for the corresponding open field at the same point, and forms the ratio of the two. Table II shows a very good agreement 共within 2%兲 between calculated and measured data for all tested cases except for the case of Fig. 3共b兲 共‘‘single step’’兲, for which discrepancies are above acceptable limits for both 6 and 20 MV photons. This is because, in the case of Fig. 3共b兲 the prescription point is located exactly on the limit between a low intensity level and a high intensity level with a difference of 50% between them. As a consequence, the prescription point is found in a region of high dose gradient in the resulting dose distribution. This illustrates the importance of well choosing the prescription point so that it is not situated in a high intensity gradient region. Therefore, in some cases it could be neces-
TABLE II. Comparison between calculated and measured effective transmission factors for monitor unit calculations with intensity modulated beams. The comparison is made for the set of test intensity distributions shown in Fig. 3 and for the clinical intensity modulation of Fig. 6. Data have been obtained for 6 and 20 MV photons. Discrepancies have been expressed as per cent differences relative to the measured data.
Test case
% Difference from meas.
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sary to prescribe the dose on a relevant off-axis point. The effective transmission factor in that case is defined similarly, i.e., the ratio of dose per MU with intensity modulation to the dose per MU for the corresponding open field at the given point. The validity of this calculation has not been verified yet. Furthermore, the accuracy of effective transmission factor calculations for different depths of the prescription point should be examined. Such dependencies deserve a careful analysis, which is not however, the subject of the present paper. IV. CONCLUSIONS A method for calculating dose distributions generated by intensity modulated 共IM兲 photon beams has been presented. The method constitutes an extension of an existing primaryscatter separation dose model. It is computationally efficient because the necessary corrections to account for IM are based on the modulation of the incident fluence and not on the integration of multiple segment dose distributions. Once the correction matrices have been prepared for each beam, the computation time is the same as that of the corresponding open beams. It is clear that the mechanical and dosimetric characteristics of the system used for practical IM delivery should be taken into account by the dose calculation model. In the case of IM delivery with dynamic collimators, our dose calculation model accounts for the transmission through the leaves, the leaf tip and leaf side penumbra effects. These phenomena play a very important role in the case of modern dynamic collimation techniques, such as the sliding window, because they influence the dose distribution not just at the field edges but at any point inside the radiation field. The fact that the radiation source behaves rather as an extended source than as a point source has been modeled using an exponential function. Limitations and approximations existing in the original model for open beam dose calculations remain in the present extension for IM dose calculations. Furthermore, effects such as the tongue-and-groove and the inter leaf leakage are not taken into account for the moment. Many authors18,19 have presented solutions for dealing with the tongue-and-groove problem in an earlier stage of the IM generation and delivery chain, i.e., within the leaf sequencing algorithm that translates the desired IM to leaf trajectories. We acknowledge that it could be desirable to be able to perform precise forward dose calculation even in the interleaf region where those effects generate discrepancies. A dosimetric verification of the calculation procedures has been performed with measurements of 2D dose distributions using films, dose profiles using a multidetector array and spot measurements using ion chambers. Film dosimetry is a powerful method for the verification of intensity modulation delivered by dynamic collimation because it permits to measure 2D dose distributions simultaneously. However, the inconveniences of film dosimetry such as the limited accuracy and the time consuming procedures outline the limitations of the method. A set of test intensity modulation patterns has been deMedical Physics, Vol. 27, No. 5, May 2000
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signed in order to verify dose calculations and form a basis for QA. It permits to draw conclusions by performing only dose profile measurements. Intensity modulated beam profiles measured by the Profiler agree very well with those measured by an ionization chamber. The Profiler proves to be a very useful tool for measuring intensity modulation profiles with the condition that the relative sensitivity of diodes is carefully corrected. Comparisons between calculations and measurements at several depths, for 6 and 20 MV photon beams are satisfactory. Discrepancies were in most of the cases within our acceptance criteria, which have been 3% of the normalization dose in low dose gradient regions and 3 mm in high dose gradient regions. We should note here that, in the case of intensity modulation, the use of specific tools for the registration and the automatic quantitative evaluation of measured and calculated data is mandatory.29,30 For absolute dose calculations 共MU兲 the calculated effective transmission factor was in good agreement 共within 2%兲 with the measured one for all the examined cases except of one, for which the prescription point was on a very high gradient dose region. For that case discrepancies where of the order of 6%. We consider that for clinically relevant cases 共like the one of Fig. 5兲 the prescription point must be chosen outside of a high dose gradient region. Clinical implementation of intensity modulation necessitates, among other things, the development of accurate dose calculation algorithms and the accompanying verification by carefully carried out experiments, both of which prove to be very demanding tasks.
ACKNOWLEDGMENT The authors wish to thank Varian Medical Systems for the interest and support of this work. a兲
Author to whom correspondence should be addressed. Telephone: ⫹33共0兲1 44324490; Fax: ⫹33共0兲 144323509. Electronic mail:
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