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Modelling of output factors for conformal megavoltage X-ray beams

Department of Oncology Physics, Clinical Oncology Directorate, Western General Hospital, Crewe Road South, Edinburgh EH4 2XU, UK

	Abstract

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

 
A method is described for calculating the output from conformally shaped megavoltage X-ray beams. The model has been developed for Varian accelerators but is shown to work for accelerators from another manufacturer. The use of dynamic wedging and both static and dynamic multileaf collimated beams are included in the model. For any linear accelerator, the data required are a set of measured output factors for square beams, an in-air profile and a limited number of readily available parameters defining the geometry of the head of the accelerator. The three components of the output, namely primary, head scatter and phantom scatter are modelled and calculated individually for any point in a beam. An optimization procedure is developed that automatically determines the eight parameters required to model an accelerator in order for these calculations to be performed. The performance of the method is demonstrated for shaped beams using asymmetric and multileaf collimation, both with and without wedging, and for a range of beam energies. The model has been incorporated into a computer program that is used clinically.

	Introduction

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

 
Current linear accelerators are almost exclusively installed with multileaf collimation (MLC) as standard with the result that the use of arbitrarily shaped beams conforming to the shape of the planning target volume is widespread. However, many current planning systems still use simplified methods for calculating beam output (dose per monitor unit (MU)) and consequently the result may be inaccurate. The use of intensity-modulated beams generated by dynamic MLC is becoming more common and an independent calculation method for confirming the resulting dose distribution is important. The aim of this work is to produce a model that will deal with the calculation of output in the above situations and that is also simple to configure and implement for any linear accelerator.

Beam output consists of three components:

1. A primary component owing to photons that have not interacted between the accelerator target and the point of measurement;
   
2. Scattered radiation from the head of the treatment machine, comprising first order and multiply scattered photons as well as electrons originating from the treatment head and the air volume between the head and the surface of the phantom or patient;
   
3. Scattered radiation again consisting of first order and multiply scattered photons from interactions within the phantom.
   

These three components must be modelled and calculated individually in order to determine an accurate estimate of the output. Although the model is developed for Varian (Palo Alto, CA) accelerators using dynamic wedges, its use is also demonstrated for equipment from another manufacturer. The model is developed by first considering the output at the beam reference point. It is then extended to any other off axis point within the beam and finally the effect of both dynamic wedges and dynamic MLC is included. The model is developed using a limited amount of data that is readily available for a linear accelerator.

	Definition of parameters used in the computation of output at the reference point

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

 
The reference point in this work is defined as the point of maximum dose on the central axis of the radiation beam, although it is noted that the reference point may be defined at a depth in water and the model could be adapted to deal with the change by use of appropriate input data. Beam sizes are defined at the isocentric distance and the isocentre is placed on the surface of a water equivalent phantom. The definitions in this section refer to the various components of the dose at the reference point. However output is specified as dose per MU and normalization to the monitor chamber signal will be considered in a later section.

I₀: primary dose
I₀ is the dose due to primary photons that have not interacted between the target and the reference point and therefore does not contain any head scattered radiation. The primary dose is independent of collimator opening but is assumed to vary over the beam in a radially symmetric manner. It is not possible to measure I₀ experimentally because of the presence of head scatter.

I_c: effective primary dose
I_c is the dose that consists of both the primary dose I₀ and the dose due to photons and electrons that originate from any scattering material in the head of the treatment machine and the air volume between the head and the surface of the phantom or patient. Its value therefore depends on C, the collimator opening. This parameter is often referred to as the primary dose, where the argument is that all incident radiation is regarded as "primary" regardless of its origin. It can be measured experimentally with a detector positioned in a mini-phantom large enough to provide electron equilibrium [1]. The relationship between the effective primary dose and the primary dose is given by:

where H_c is the additional dose to the reference point resulting from head scatter due to the collimator opening C.

S_c: collimator scatter correction factor
S_c is the ratio of the effective primary dose for a particular collimator opening to that for a reference collimator opening (100 mm x 100 mm) so

where I_ref is the effective primary dose from the reference collimator opening. It follows from Equation 1 that

where H_ref is the additional dose due to head scatter resulting from the reference collimator setting.

S_cp: total scatter correction factor
The total dose to the reference point contains both head scatter and phantom scatter. S_cp is defined as the ratio of the total dose to the reference point for a collimator opening C, to that for the reference collimator opening. It can be measured with the detector positioned in a phantom sufficiently large to give full phantom scatter. If P_c is the additional dose to the reference point resulting from phantom scatter due to a collimator opening C, then

where P_ref is the additional dose due to phantom scatter resulting from the reference collimator setting.

Using Equation 1, then

Let S_cp for the reference collimator setting be normalized to 1.0, so that

and therefore

S_p: phantom scatter correction factor
S_p is defined as the dose to the reference point for any beam size divided by the dose to the reference point for the reference beam size, when the reference beam is produced without changing the collimator setting. The aim is therefore, to keep head scatter constant and that is achieved by defining the reference beam size by blocking the beam on the surface of the phantom [2]. If the beam is smaller than the reference beam, then the reference beam is blocked to the size of the beam to be measured. Conceptually, S_p is then given by

where in the denominator, the phantom scatter for the reference beam has to be scaled by the ratio of the effective primary dose in the two situations.

Substituting Equation 1 for I_c gives

which, when using Equation 6, reduces to

Equations 3, 7 and 10 show the well known relationship

and using these three equations to model and predict I₀, H_c and P_c from measured values of S_c, S_p and S_cp for a range of collimator openings, can allow S_cp to be predicted for any other collimator opening C.

	Beam model

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

 
H_c: head scatter
Several different approaches have been investigated to model head scatter [3–8]. These include an analytical method, Monte Carlo techniques and measured output factors in-air. The latter approach is used in this work for, as already stated, it allows the development of a model from a limited amount of experimental data that is readily available for any linear accelerator.

The photon scatter arising from the head of the treatment machine consists of both first order and multiple scatter from any component within the treatment head and its variation is observed by measurement of the effective primary dose. Electron contamination is included as part of H_c and it is not necessary to model it separately. Any measurement performed is normalized to the monitor signal from the linear accelerator and that signal can vary with collimator setting as well as the signal at the reference point. Head scatter can be split into two components:

1. An extra-focal component due to photons scattered from the primary collimator and the beam flattening filter. Extrafocal radiation has a constant effect on the monitor chamber signal but the amount reaching the reference point will vary with the collimator setting.
   
2. A component due to radiation scattered from the secondary collimation system. This will produce backscattered radiation into the monitor chamber and forward scatter to the reference point and both have been modelled [8–10]. The latter may also include the effect of contamination electrons but is ignored here, as S_c values measured at a depth in a mini-phantom beyond the maximum electron range are not significantly different to those measured at the reference depth [11].
   

H: extra-focal scatter
It has been shown by several workers [5, 12, 13] that the beam flattening filter is the main source of extrafocal scattered radiation. Several different empirical models have been developed and all are based on the concept of an extended scatter source located on a plane in the treatment head above the collimating system. The area on this plane that can be observed from any calculation point can easily be determined by projection from the point through the collimator system onto the plane. The physical dimensions of the treatment head that are required to perform the calculation are the distances from the target to the beam-flattening filter, to the upper surfaces of both sets of collimators and the MLC. As the majority of the scatter results from the beam-flattening filter, it is logical to locate the scatter plane at the level of the filter and this has been discussed in detail [6]. In practice, the position of the scatter plane is not critical and only results in different parameter values for the head scatter model.

The validity of the approach when dealing with an MLC as a tertiary collimator as occurs in the design of Varian accelerators has been questioned [9] and it was claimed that increased scatter from the MLC has to be modelled separately. In beam set-up on Varian accelerators, the collimators are automatically driven to a static position to conform closely to the beam defined by the MLC. Consideration of the geometry of the head then shows that the collimator setting defines the area on the scatter plane seen by the calculation point whereas the MLC defines the irradiated area on the phantom. In the work described [9], the collimator setting was fixed to produce a 200 mm x 200 mm beam or larger and may explain their conclusions. In this work, the same approach has been used to model S_c for both MLC and collimators and the difference in the physical distance of these from the scatter plane accounts for the changes in the observed values of S_c (and hence S_cp) for the same beam size set by either system.

The extrafocal scatter function can be assumed to be symmetrical with respect to radius r_h and to reduce in intensity from the centre outwards. The concept is mainly derived from the shape of the flattening filter and moving from the centre radially outwards, has an initial sharp reduction in intensity combined with a tail of slowly reducing intensity. A wide variety of empirical functions with respect to radius r_h have been used to model the scatter function as follows: a Gaussian function [7, 14], a series of three Gaussian functions [10], an exponential function [6], a general polynomial in r_h [15], a piecewise linear function [16] and different functions for different values of r_h [8]. Most of these were tested as a starting point for this work and an exponential function was found to give the best result but did not give the accuracy considered necessary. However, through experimentation, it has been found that modifying the exponential function by raising r_h to a power gives an excellent fit for H and the following expression is used:

where h₁, h₂ and h₃ are constants for a machine; r_h is the radial distance from the central axis on the scattering plane; and the summation is over the area projected onto the scatter plane.

The head scatter plane is represented by a two-dimensional matrix of scattering elements located at the level of the beam-flattening filter. A trial and error process was used to determine the size of the elements where the size was reduced until no observable difference occurred in the calculation of H. Square elements are defined for simplicity in calculation and are sufficiently small such that the different resolution in orthogonal directions resulting from the different collimator distances from the target is also not observable in the calculation of H. Each element has dimensions of 1 mm x 1 mm and the geometry of the treatment head on a Varian accelerator leads to approximately 17 scattering elements exposed on the scattering plane for any 10 mm x 10 mm area within the beam portal at the isocentric distance. H is evaluated by summation of the scatter resulting from all elements within the area defined by the projection through the collimator system (including MLC) onto the scattering plane as seen from the point of calculation. The distance r_h from the central axis for each element considered is easily calculated for use in Equation 12.

F: forward scatter from the collimation system
Forward scatter from the collimation system to the reference point has a much smaller effect than extrafocal scatter. It is modelled in a similar manner to that proposed by Olofsson et al [17] and is assumed to be directly proportional to the perimeter L of the beam set by the collimators, so that each collimator is considered as a scatter source. If a MLC is present, the beam perimeter is determined from the leaf positions where each leaf acts as a scatter source. The amount of forward scatter is obviously zero for zero beam size and in the model will increase with increasing beam size up to the largest size available (400 mm x 400 mm). A linear variation with beam size has been assumed and the forward scatter F contributing to the reading at the reference point is modelled by:

where k_f is a normalization constant to be determined in the modelling process and will represent the additional scatter contribution to the reference point for a 400 mm x 400 mm jaw setting; L is the actual beam perimeter as defined by the jaws/MLC; and L₄₀₀ is the beam perimeter for a jaw setting of 400 mm x 400 mm.

M: change in monitor chamber signal owing to backscatter from the collimation system
Phantom scatter has no effect on the monitor signal and the primary component of the beam output has a constant effect, as does the extrafocal component of head scatter. However backscatter from the collimation system will reach the monitor chamber and its magnitude will vary with collimator setting. Beam output is always quoted as dose per monitor unit and the dose contribution from head scatter will be modified owing to the effect of backscatter on the monitor chamber signal. The effect is taken into account in the model by normalizing the head scatter component to the monitor chamber reading.

The primary collimator determines the size of the radiation beam incident on the collimation system and produces a circular beam with a radius of 250 mm at the distance of the isocentre on Varian linear accelerators. For any collimator setting there will be an area within this circular beam that is not irradiated and that is simply the area for the beam when projected to the distance of the isocentre. Backscatter will be a maximum with the collimators closed, decrease as the beam size increases and is assumed to be zero for a circular beam of 250 mm radius. The monitor signal value M is set to 1.0 for a 250 mm radius beam (i.e. representing the constant component of the signal with no backscatter present). M increases in value with decreasing beam size as given by the following equation:

where k_b is a normalization constant to be determined in the modelling process and represents the backscatter contribution to the monitor chamber signal resulting from zero beam area. A is the beam area at the isocentric distance and R is the radius of the circular beam defined by the primary collimator at the isocentric distance.

As the two sets of collimators are at different distances from the monitor chamber, there will be a collimator interchange effect on the value of the backscatter to the monitor chamber. The effect is small enough to be ignored, as backscatter into the monitor chamber is known to have a small effect on beam output [10, 18, 19].

After evaluating the above three effects, the additional dose reaching the reference point due to head scatter H_c is modelled using the following equation:

P_c: phantom scatter
The phantom scatter component is depth dependent, but for this work has only to be modelled at the depth of maximum dose. There have been many different approaches developed to model phantom scatter, for example the integration of differential scatter–air ratios, but these can be computationally slow. For that reason, a similar approach is used to model phantom scatter as is used for head scatter. The area on the phantom that is irradiated by any beam can easily be determined by projection from the source onto the surface of the phantom through the collimating system. The phantom scatter reaching the reference point is modelled as follows:

where p₁, p₂ and p₃ are again machine specific; r_p is the radial distance from the central axis of the beam; and the summation is over the irradiated area on the phantom.

The phantom scatter plane located at the reference depth is represented by 1 mm x 1 mm scatter elements in an identical manner to the extrafocal scatter plane and the summation takes place over the irradiated area as determined by the collimator system (and MLC).

I₀: primary
Using Equations 15 and 16 to evaluate the head scatter H_ref and the phantom scatter P_ref for the reference collimator setting, allows the primary I₀ to the reference point to be calculated from Equation 6 as follows:

	Determination of machine dependent parameters

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

 
Optimization
An optimization procedure is used to determine the values of the variables h₁, h₂, h₃, p₁, p₂, p_3, k_b and k_f such that the best fit is made to experimental data for S_c, S_p and S_cp values for a machine. S_cp values are readily available for any machine as they are simply the output values used in routine treatment planning. It has been argued [20] that S_p values at the depth of maximum dose (as given by the normalized peak scatter function) are independent of beam energy within experimental uncertainties. Values are taken from that publication for square beam sizes of 40 mm to 400 mm, extrapolated down to a square field size of 20 mm and the same data set used for every treatment machine modelled. S_c values can be measured or simply calculated from S_cp/S_p. A table of S_c, S_p and S_cp values is required as input data for the optimization program. The range should cover all collimator settings used on the machine and could include asymmetric collimator settings. Beam sizes smaller than would normally be measured have been included in the input data because small areas may be irradiated during the use of dynamic MLC for intensity-modulated treatments.

Starting values for all eight variables are estimated and are then adjusted by the algorithm as it is executed. An iterative technique using simulated annealing [21] is employed where, in each iteration, a variable is randomly selected and randomly increased or decreased. The current values of the variables are then used to evaluate H_c and P_c for each collimator setting provided as input data including H_ref and P_ref for the reference setting. For each collimator setting the fitted S_c, S_p and S_cp values are evaluated from Equations 3, 7 and 10, respectively. The primary component I₀ is determined by using Equation 17 as a constraint built into the optimization.

The optimization program minimizes an objective function that is taken as the sum of the squares of the differences between the predicted and measured S_c, S_p and S_cp values. The change to a variable is accepted if the value of the objective function decreases. It is also accepted in a small percentage of cases where the value of the objective function increases to avoid becoming trapped in a local minimum. The percentage accepted decreases as the algorithm progresses. Termination is when no significant improvement in the value of the objective function is apparent and provides the optimum values of all variables, as well as the primary I₀, so that the best least squares fit is obtained to S_c, S_p and S_cp input data.

Results of the optimization
Repeated runs of the optimization algorithm show only small differences in the determination of the parameters required for the model. This is due to the stochastic method employed and these differences produce no observable effect on the S_c, S_p and S_cp values calculated by the model. Table 1 is a typical example showing the fit obtained for a 6 MV beam using square beam sizes from 20 mm to 400 mm. The standard deviation for any individual measurement used as input data (including normalization to the reference collimator setting) is 0.4%. However all data used for input has been smoothed and interpolated for consistency, which increases the standard deviation to an estimated 0.6%. For all accelerators modelled, the maximum deviation between input and fitted data is 0.2% and well within the experimental uncertainty of the input data.

Modification for off-axis calculation points
The above model is based on data at the reference point, but it is possible to modify the components of Equation 7 to determine the output at off-axis calculation points within the beam portal. The output at any point, specified by coordinates (x,y) relative to the central axis, is then given by:

where r=(x²+y²) is the radial distance from the reference point. When r=0 Equation 18 reduces to Equation 7.

An in-air radial profile gives the effective primary variation I_c(x,y) normalized to 1.0 at r=0 for the collimator setting used in the measurement. H_c(x,y) values can be calculated using Equation 15 along the same radius using the same collimator settings as for the measurement of the in-air profile. The primary I₀ at the reference point is known from the modelling process and the head scatter H_c(0,0) from the above calculation. The effective primary I_c(0,0)=I₀+H_c(0,0) is used to re-normalize the in-air profile to give true values of I_c(x,y). The primary variation I₀(r) along a radius can be found by subtracting out the calculated head scatter component H_c(x,y). The calculation should be performed along a specific radius, for although I₀(r) is assumed to be radially independent, I_c(x,y) will show a small variation due to head geometry.

An example of the calculation required to determine I₀(r) is shown in Table 3, where a radial profile normal to the lower collimator is measured for a 400 mm x 400 mm collimator setting on a Varian 600 CD accelerator. The fit parameters for this machine give I₀=0.873 and H₄₀₀(0,0)=0.126 for the 400 mm x 400 mm beam setting. The effective primary I_c(0,0) at the reference point is therefore 0.999. The primary I₀(r) at the calculation point can be determined from this data for use in Equation 18 for points that are off-axis. It should be noted that the head scatter component H₄₀₀(x,y) only shows a small reduction in value of approximately 3% towards the edge of the beam. This is not surprising because the area projected onto the scatter plane does not vary with the position of the off-axis calculation point. Although the projected area moves, it is large for a 400 mm x 400 mm collimator setting and only the "tail" of the head scatter function affects changes in the calculated value.

Inclusion of dynamic wedges and MLC
Varian linear accelerators utilize dynamic wedges where the wedge effect is produced by sweeping one of the upper (Y) collimators across the beam during irradiation. A fraction of the total MU is given before the collimator starts to move. This open beam fraction is large for low angle wedges and small for high angle wedges. Varian accelerators in this centre use an enhanced set of dynamic wedges with a total of 11 wedge angles available in steps of 5° from 10° to 60°. The generation of the movement pattern for the collimator is specified by a segmented treatment table (STT), which describes the relationship between the cumulative MU and collimator position Y necessary to produce the desired wedge angle. A full description of STTs can be found in Varian literature [22]. STTs for open beams and 60° wedged beams are provided by Varian. The STT for any other wedge angle can be calculated from these using a tan relationship as follows:

where STT(0,Y) is for an open field and is constant with respect to collimator position Y.

The output for any wedged beam can be calculated rapidly and with sufficient accuracy from knowledge of the appropriate STT table for the wedge. The beam can be simulated by a summation of a large number of asymmetric beams with the moving collimator travelling from its start position to its stop position at a distance of 5 mm from the static collimator. To achieve the required accuracy in the calculation, the collimator is moved in steps of 1 mm so, for example, 95 asymmetric beams simulate a 100 mm wide wedged beam. For every asymmetric beam used in the simulation, the fraction M to be given of the total monitor units can be interpolated from the STT table. The primary, head scatter and phantom scatter components of output are calculated for each simulating beam. If the calculation point is shielded, the primary component is modified by multiplying by a factor to account for the transmission through the collimation system. The factor was experimentally measured as 0.35% under broad beam conditions and is an average of the energies and head designs investigated as its value is not critical in practice. If the point lies within the penumbral region, the primary component is reduced by an appropriate factor interpolated from a measured penumbral profile for the collimators. The accuracy of the profile is not important as it has been shown to have an insignificant effect on the calculation of the output. The individual components are added, multiplied by the appropriate weighting factor M and summed over all simulating beams to give the output at the calculation point. A wedge factor can be determined by normalizing the result to the output calculated for the same sized open beam. The technique has been shown to agree with measurement to within 0.2% for a variety of accelerators and beam energies.

Intensity-modulated beams are generated on Varian accelerators using dynamic MLC and the leaf motion controller software in the Varian planning system (Eclipse version 7.1) uses 320 static MLC beams for simulation by default. It is noted that later versions of this software (version 7.2) use a smaller and variable number of simulating beams. Output is calculated using a similar method to that for dynamic wedges. Eclipse produces a standard MLC file containing the parameters for every static beam including an MU index specifying the fraction of the total MU that should have been delivered up to and including the application of that beam. The MU indices allow the calculation of a weighting factor for every simulating beam. The primary, head scatter and phantom scatter components of the output are determined at the calculation point. The primary component is modified to account for transmission through the MLC if the calculation point is shielded. The transmission factor is taken as 2%, which has been experimentally measured as an average value when considering transmission both through and between the MLC leaves. If the point lies within the MLC penumbral region, the primary component is again reduced by an appropriate factor interpolated from a measured penumbral profile where the factor depends on the distance of the calculation point from the MLC portal. Once again, the accuracy of the penumbral profile is not important and neither is it necessary to attempt to model the tongue and groove effect. The individual components are added, multiplied by the appropriate weighting factor and summed over all simulating beams to give the output.

The model can deal with manual wedges but requires a separate fit to be performed for S_c and S_cp data for every wedge in order to determine the model parameters. This has been performed for motorized wedges on Elekta accelerators but only for output calculation on the beam central axis. Off-axis calculation would require the measurement of orthogonal in-air profiles with the wedge in position. External blocking is simple to deal with once the mounting position of the block tray is known with respect to the geometry of the accelerator head. Neither of these has been investigated in detail because modern linear accelerators either use or are now offering dynamic wedging and all use MLC to provide blocking.

	Results

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

 
Although many different comparisons with experimental measurements have been carried out to show performance, only a sample of these can be illustrated and are given in Tables 4 to 7. The accuracy of the measured data is determined by repeated measurement to have a standard deviation of 0.4% for open beams, increasing to 0.5% for 10° and to 0.7% for 60° wedged beams. Positional errors within larger dose gradients account for the higher standard deviation for wedged beams. As stated above, the model will predict the data used in the fitting process to a maximum error of 0.2% but now has to be tested to determine if it will predict output values to an acceptable accuracy for a variety of other beam geometries. Ideally this accuracy should be in the range 0.5% to 0.7% and therefore within one standard deviation of measurement.

A comparison for wedged asymmetric beams on a Varian 6 MV machine is shown in Table 5. Table 5a gives measured and calculated results for the two extreme wedge angles at four different beam sizes using both Y1 and Y2 as the moving collimator. Table 5b compares wedge outputs at an off-axis point. Five different beam sizes are measured for the same two wedge angles with Y1 as the moving collimator. Wedge outputs are compared at a point 40 mm from the central axis towards the moving Y1 collimator. Overall agreement is again 0.2% on average with a maximum deviation of 0.7%.

View larger version (41K): [in this window] [in a new window]   Figure 1. Dose distributions for an 8 MV intensity-modulated beam in water at the reference depth produced using dynamic MLC on a Varian linear accelerator. The isodose lines are in 0.2 Gy steps from 1.6 Gy to 3.4 Gy. (a) Output calculation program; (b) Eclipse planning system.

	Conclusion

Top Abstract Introduction Definition of parameters used... Beam model Determination of machine... Results Conclusion References

An interactive computer program for the calculation of beam output is shown in Figure 2. The position of the collimators and the output calculation point (shown by + in the graphics window) can be set through the use of sliders or automatically set to the central axis or the centre of the beam. The treatment machine, wedge angle and direction are set from drop down menus and the Varian MLC and BLOCK specification files can be input with their beam portal displayed, again as shown in the graphics window. The output is calculated for the focus–skin distance (FSD) set for the beam and is also shown broken down into its primary, head and phantom scatter components. The output calculation software is in clinical use for static and wedged beams. Calculation for open beams is instantaneous, whereas it takes a few seconds to calculate large wedged beams and dynamic intensity-modulated beams that can require the order of 300 asymmetric beams for simulation.

View larger version (36K): [in this window] [in a new window]   Figure 2. Interactive screen for the output calculation program.

	Acknowledgments

 
Dr D Thwaites and Dr C McKerracher for measurements carried out on Varian accelerators. Dr J Mills for measurements and supplying data for Elekta accelerators.

Received for publication August 23, 2004. Revision received November 22, 2004. Accepted for publication January 5, 2005.

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