On methods of inhomogeneity corrections for photon transport

Eight methods of photon inhomogeneity correction were examined for their photon transport approximations. The methods were categorized according to the different approaches used to model scatter photon dose contribution. They were the ratio of TAR (RTAR) and the modified Batho power law which utilized only the 1-D density information along the primary photon path; the equivalent TAR (ETAR) and the FFT convolution methods which incorporated the 3-D density information of the medium for empirical scatter dose calculation; the differential SAR (DSAR), the delta volume (DV), dose spread array (DSA), and differential pencil beam (DPB) methods which employed explicit 3-D scatter ray-trace calculation. Cobalt-60 measurements in horizontal slab phantoms were used to allow simpler data analysis. RTAR consistently overestimated lung corrections by approximately 10%. The scatter ray-trace approach was not always better as the DSAR calculations were inferior to those using the Batho method. The ray-tracing DV, DPB, and DSA methods agreed with measurements mostly to within 2%, at the expense of long computation time. The nonscatter ray-tracing ETAR and FFT convolution calculations were only slightly inferior in the same geometries. These methods improve on the current 1-D methods and should be seriously considered for fast optimization purposes in practical 3-D treatment planning.     

Dosimetric evaluation in heterogeneous tissue of anterior electron beam irradiation for treatment of retinoblastoma

A dosimetric study of anterior electron beam irradiation for treatment of retinoblastoma was performed to evaluate the influence of tissue heterogeneities on the dose distribution within the eye and the accuracy of the dose calculated by a pencil beam algorithm. Film measurements were made in a variety of polystyrene phantoms and in a removable polystyrene eye incorporated into a tissue substitute phantom constructed from a human skull. Measurements in polystyrene phantoms were used to demonstrate the algorithm's ability to predict the effect of a lens block placed in the beam, as well as the eye's irregular surface shape. The eye phantom was used to measure dose distributions within the eye in both the sagittal and transverse planes in order to test the algorithm's ability to predict the dose distribution when bony heterogeneities are present. Results show (1) that previous treatment planning conclusions based on flat, uniform phantoms for central-axis depth dose are adequate; (2) that a three-dimensional heterogeneity correction is required for accurate dose calculations; and (3) that if only a two-dimensional heterogeneity correction is used in calculating the dose, it is more accurate for the sagittal than the transverse plane.     

Monte Carlo verification of IMRT dose distributions from a commercial treatment planning optimization system

The purpose of this work was to use Monte Carlo simulations to verify the accuracy of the dose distributions from a commercial treatment planning optimization system (Corvus, Nomos Corp., Sewickley, PA) for intensity-modulated radiotherapy (IMRT). A Monte Carlo treatment planning system has been implemented clinically to improve and verify the accuracy of radiotherapy dose calculations. Further modifications to the system were made to compute the dose in a patient for multiple fixed-gantry IMRT fields. The dose distributions in the experimental phantoms and in the patients were calculated and used to verify the optimized treatment plans generated by the Corvus system. The Monte Carlo calculated IMRT dose distributions agreed with the measurements to within 2% of the maximum dose for all the beam energies and field sizes for both the homogeneous and heterogeneous phantoms. The dose distributions predicted by the Corvus system, which employs a finite-size pencil beam (FSPB) algorithm, agreed with the Monte Carlo simulations and measurements to within 4% in a cylindrical water phantom with various hypothetical target shapes. Discrepancies of more than 5% (relative to the prescribed target dose) in the target region and over 20% in the critical structures were found in some IMRT patient calculations. The FSPB algorithm as implemented in the Corvus system is adequate for homogeneous phantoms (such as prostate) but may result in significant under or over-estimation of the dose in some cases involving heterogeneities such as the air-tissue, lung-tissue and tissue-bone interfaces.     

Effect of tissue heterogeneity on an in vivo range verification technique for proton therapy

It was proposed recently that time-resolved dose measurements during proton therapy treatment by passively scattered beams may be used for in vivo range verification. The method was shown to work accurately in a water tank. In this paper, we further evaluated the potential of the method for more clinically relevant situations where proton beams must pass through regions with significant tissue heterogeneities. Specifically, we considered prostate treatment where the use of anterior or anterior- oblique fields was recently proposed in order to reduce rectal dose by taking advantage of the sharp distal fall-off of the Bragg peak. These beam portals pass through various parts of pubic bone and potential air cavities in the bladder and bowels. Using blocks of materials with densities equivalent to bone, air, etc, arranged in the water tank in relevant configurations, we tested the robustness of the method against range shifting and range mixing. In the former, the beam range is changed uniformly by changes in tissue density in the beam path, while in the latter, variations in tissue heterogeneities across the beam cross section causes the mixing of beam energies downstream, as often occurs when the beam travels along the interface of materials with significantly different densities. We demonstrated that in the region of interest, the method can measure water-equivalent path length with accuracy better than ±0.5?mm for pure range shifting and still reasonable accuracy for range mixing between close beam energies. In situations with range mixing between significantly different beam energies, the dose rate profiles may be simulated for verifying the beam range. We also found that the above performances can be obtained with very small amount of dose (<0.5 cGy), if silicon diodes are used as detectors. This makes the method suitable for in vivo range verification prior to each treatment delivery.     

Experimental verification of convolution/superposition photon dose calculations for radiotherapy treatment planning

This work describes an experimental verification of the two-photon dose calculation engines available on the Helax-TMS (version 6.1) commercial radiotherapy treatment planning system. The performance of the pencil beam convolution and the collapsed cone superposition algorithms was examined for 4, 6, 15 MV beams, under a range of clinically relevant irradiation geometries. Comparisons against measurements were carried out in terms of absolute dose, thus assessment of the accuracy of monitor unit (MU) calculations was also carried out. Results show that both algorithms agree with measurement to acceptable tolerance levels in most cases in homogeneous water-equivalent media irradiated under full scatter conditions. The collapsed cone algorithm slightly overestimates the penumbra width and this is mainly due to discretization effects of the fluence matrix. The accuracy of this algorithm strongly depends on the resolution of the patient density matrix. It is recommended that the density matrix voxel size used for dose calculations is less than 5 x 5 x 5 mm3. The dose in media irradiated under missing tissue geometry, or in the presence of low or high-density heterogeneities, is modelled best with the collapsed cone algorithm. This is of particular clinical interest in treatment planning of the breast and of the thorax. For these treatment sites, a retrospective study of treatment plans indicated in certain cases significant overestimation of the dose to the planning target volume when using the pencil beam convolution model.     

Influence of detector size in photon beam profile measurements

Correction is necessary to account for the detector size in clinical dosimetry of photon and electron beams. This correction is due to the absorbed dose gradient present in a finite-size detector. Further corrections are necessary when the detector and phantom materials are not the same. These corrections are due to the perturbation in the charged-particle fluence. Generally these corrections are applied to measurements along the central axis of the beam. Cross beam profile measurements, however, are not usually corrected for detector size. The ionization profile is also usually assumed to be equivalent to the absorbed dose profile. We have corrected the ionization chamber size effect by two approaches: extrapolation of measurements to zero detector size and deconvolution of measurements using a simple model for the detector response function. We have measured absorbed dose profiles to water using a small water-equivalent plastic scintillation detector. Film profile measurements were also studied. The ionization profile corrected for detector size and absorbed dose profile were not equal, probably due to loss of charged-particle equilibrium in the beam edges. For ionization chamber measurements, knowledge of the charged-particle spectrum is needed to convert ionization to absorbed dose to water. This is not necessary for relative absorbed dose measurements under charged-particle equilibrium. Film has been shown to be a straightforward and reliable method for cross beam profile measurements.     

Evaluation of film and thermoluminescent dosimetry of high-energy electron beams in heterogeneous phantoms.

Film and thermoluminescent dosimetry (TLD) are investigated in heterogeneous phantoms irradiated by high-energy electron beams. Both film and TLD are practical dosimeters for multiple and moving beam radiotherapy. The accuracy and precision of these dosimeters for radiation dose measurements in homogeneous water-equivalent phantoms has been discussed in the literature. However, film and TLD are often used for dose measurements in heterogeneous phantoms. In those situations perturbations are produced which are related to the density and atomic number of the phantom material and the physical size and orientation of the dosimeter. In our experiments the relative dose measurements in homogeneous phantoms were the same regardless of dosimeter or dosimeter orientation. However, significant differences were observed between the dose measurements within the inhomogeneity. These differences were influenced by the type and orientation of the dosimeter in addition to the properties of the heterogeneity. These differences could be reproduced with Monte Carlo calculations and modeling of the experimental conditions.     

The development and investigation of a prototype three-dimensional compensator for whole brain radiation therapy

Whole brain radiation therapy (WBRT) is the standard treatment for patients with brain metastases, and is often used in conjunction with stereotactic radiotherapy for patients with a limited number of brain metastases, as well as prophylactic cranial irradiation. The use of open fields (conventionally used for WBRT) leads to higher doses to the brain periphery if dose is prescribed to the brain center at the largest lateral radius. These dose variations potentially compromise treatment efficacy and translate to increased side effects. The goal of this research was to design and construct a 3D 'brain wedge' to compensate dose heterogeneities in WBRT. Radiation transport theory was invoked to calculate the desired shape of a wedge to achieve a uniform dose distribution at the sagittal plane for an ellipsoid irradiated medium. The calculations yielded a smooth 3D wedge design to account for the missing tissue at the peripheral areas of the brain. A wedge was machined based on the calculation results. Three ellipsoid phantoms, spanning the mean and +/- two standard deviations from the mean cranial dimensions were constructed, representing 95% of the adult population. Film was placed at the sagittal plane for each of the three phantoms and irradiated with 6 MV photons, with the wedge in place. Sagittal plane isodose plots for the three phantoms demonstrated the feasibility of this wedge to create a homogeneous distribution with similar results observed for the three phantom sizes, indicating that a single wedge may be sufficient to cover 95% of the adult population. The sagittal dose is a reasonable estimate of the off-axis dose for whole brain radiation therapy. Comparing the dose with and without the wedge the average minimum dose was higher (90% versus 86%), the maximum dose was lower (107% versus 113%) and the dose variation was lower (one standard deviation 2.7% versus 4.6%). In summary, a simple and effective 3D wedge for whole brain radiotherapy has been developed. The wedge gives a more uniform dose distribution than commonly used techniques. Further development and shape optimization may be necessary prior to clinical implementation.     

Evaluation of the first commercial Monte Carlo dose calculation engine for electron beam treatment planning

The purpose of this study is to perform a clinical evaluation of the first commercial (MDS Nordion, now Nucletron) treatment planning system for electron beams incorporating Monte Carlo dose calculation module. This software implements Kawrakow's VMC++ voxel-based Monte Carlo calculation algorithm. The accuracy of the dose distribution calculations is evaluated by direct comparisons with extensive sets of measured data in homogeneous and heterogeneous phantoms at different source-to-surface distances (SSDs) and gantry angles. We also verify the accuracy of the Monte Carlo module for monitor unit calculations in comparison with independent hand calculations for homogeneous water phantom at two different SSDs. All electron beams in the range 6-20 MeV are from a Siemens KD-2 linear accelerator. We used 10,000 or 50,000 histories/cm2 in our Monte Carlo calculations, which led to about 2.5% and 1% relative standard error of the mean of the calculated dose. The dose calculation time depends on the number of histories, the number of voxels used to map the patient anatomy, the field size, and the beam energy. The typical run time of the Monte Carlo calculations (10,000 histories/cm2) is 1.02 min on a 2.2 GHz Pentium 4 Xeon computer for a 9 MeV beam, 10 x 10 cm2 field size, incident on the phantom 15 x 15 x 10 cm3 consisting of 31 CT slices and voxels size of 3 x 3 x 3 mm3 (total of 486,720 voxels). We find good agreement (discrepancies smaller than 5%) for most of the tested dose distributions. We also find excellent agreement (discrepancies of 2.5% or less) for the monitor unit calculations relative to the independent manual calculations. The accuracy of monitor unit calculations does not depend on the SSD used, which allows the use of one virtual machine for each beam energy for all arbitrary SSDs. In some cases the test results are found to be sensitive to the voxel size applied such that bigger systematic errors (>5%) occur when large voxel sizes interfere with the extensions of heterogeneities or dose gradients because of differences between the experimental and calculated geometries. Therefore, user control over voxelization is important for high accuracy electron dose calculations.     

Novel methods of measuring single scan dose profiles and cumulative dose in CT

Computed tomography dose index (CTDI) is a conventional indicator of the patient dose in CT studies. It is measured as the integration of the longitudinal single scan dose profile (SSDP) by using a 100-mm-long pencil ionization chamber and a single axial scan. However, the assumption that most of the SSDP is contained within the chamber length may not be valid even for thin slices. We have measured the SSDPs for several slice widths on two CT scanners using a PTW diamond detector placed in a 300 mm x 200 mm x 300 mm water-equivalent plastic phantom. One SSDP was also measured using lithium fluoride (LiF) TLDs and an IC-10 small volume ion chamber, verifying the general shape of the SSDP measured using the diamond detector. Standard cylindrical PMMA CT phantoms (140 mm length) were also used to qualitatively study the effects of phantom shape, length, and composition on the measured SSDP. The SSDPs measured with the diamond detector in the water-equivalent phantom were numerically integrated to calculate the relative accumulated dose D(L)(0)calc at the center of various scan lengths L. D(L)(0)calc reached an equilibrium value for L > 300 mm, suggesting the need for phantoms longer than standard CT dose phantoms. We have also measured the absolute accumulated dose using an IC-10 small volume ion chamber, D(L)(0)SV, at three points in the phantom cross section for several beamwidths and scan lengths. For one CT system, these measurements were made in both axial and helical scanning modes. The absolute CTDI100, measured with a 102 mm active length pencil chamber, were within 4% of D(L)(0)SV measured with the small volume ion chamber for L approximately 100 mm suggesting that nonpencil chambers can be successfully used for CT dosimetry. For nominal beam widths ranging from 3 to 20 mm and for L approximately 250 mm, D(L)(0)SV values at the center of the water-equivalent phantom's elliptic cross section were approximately 25%-30% higher than the measured CTDI100. For small beamwidths, the difference in D(L)(0)SV for L approximately 250 mm and L approximately 14 x beamwidth (CTDI14nT) reached up to 50%. Peripheral point doses at 70 mm depth along the major axis of the phantom for L approximately 250 mm were up to 22% higher than for L approximately 100 mm. The differences between CTDI100 and D(L)(0)SV for L approximately 250 mm were in good agreement with the predictions made from the numerical integration of the measured SSDPs. Due to the considerable dose measured beyond the length of standard CT phantoms, CT dosimetry for longer body scan series should be performed in longer phantoms. Measurements could be made as we have shown, using a small volume chamber translating through the beam using multiple scans.     

A finite size pencil beam algorithm for IMRT dose optimization: density corrections

For beamlet-based IMRT optimization, fast and less accurate dose computation algorithms are frequently used, while more accurate algorithms are needed to recompute the final dose for verification. In order to speed up the optimization process and ensure close proximity between dose in optimization and verification, proper consideration of dose gradients and tissue inhomogeneity effects should be ensured at every stage of the optimization. Due to their speed, pencil beam algorithms are often used for precalculation of beamlet dose distributions in IMRT treatment planning systems. However, accounting for tissue heterogeneities with these models requires the use of approximate rescaling methods. Recently, a finite size pencil beam (fsPB) algorithm, based on a simple and small set of data, was proposed which was specifically designed for the purpose of dose pre-computation in beamlet-based IMRT. The present work describes the incorporation of 3D density corrections, based on Monte Carlo simulations in heterogeneous phantoms, into this method improving the algorithm accuracy in inhomogeneous geometries while keeping its original speed and simplicity of commissioning. The algorithm affords the full accuracy of 3D density corrections at every stage of the optimization, hence providing the means for density related fluence modulation like penumbra shaping at field edges.     

Comparison of PENELOPE Monte Carlo dose calculations with Fricke dosimeter and ionization chamber measurements in heterogeneous phantoms (18 MeV electron and 12 MV photon beams)

Different measurements of depth-dose curves and dose profiles were performed in heterogeneous phantoms and compared to dose distributions calculated by a Monte Carlo code. These heterogeneous phantoms consisted of lung and/or bone heterogeneities. Irradiations and simulations were carried out for an 18 MeV electron beam and a 12 MV photon beam. Depth-dose curves were measured with Fricke dosimeters and with plane and cylindrical ionization chambers. Dose profiles were measured with a small cylindrical ionization chamber at different depths. The LINAC was modelled using the PENELOPE code and phase space files were used as input data for the calculations of the dose distributions in every simulation. The detectors (Fricke dosimeters and ionization chambers) were not modelled in the geometry. There is generally a good agreement between the measurements and PENELOPE. Some discrepancies exist, near interfaces, between the ionization chamber and PENELOPE due to the attenuation of the lower energy electrons by the wall of the ionization chamber.     

A convolution method of calculating dose for 15-MV x rays

Arrays were generated using the Monte Carlo method representing the energy absorbed throughout waterlike phantoms from charged particles and scatter radiation set in motion by primary interactions at one location. The resulting "dose spread arrays" were normalized to the collision fraction of the kinetic energy released by the primary photons. These arrays are convolved with the relative primary fluence interacting in a phantom to obtain three-dimensional dose distributions. The method gives good agreement for the 15-MV x-ray dose in electronic disequilibrium situations, such as the buildup region, near beam boundaries, and near low-density heterogeneities.     

Brachytherapy source characterization for improved dose calculations using primary and scatter dose separation

In brachytherapy, tissue heterogeneities, source shielding, and finite patient/phantom extensions affect both the primary and scatter dose distributions. The primary dose is, due to the short range of secondary electrons, dependent only on the distribution of material located on the ray line between the source and dose deposition site. The scatter dose depends on both the direct irradiation pattern and the distribution of material in a large volume surrounding the point of interest, i.e., a much larger volume must be included in calculations to integrate many small dose contributions. It is therefore of interest to consider different methods for the primary and the scatter dose calculation to improve calculation accuracy with limited computer resources. The algorithms in present clinical use ignore these effects causing systematic dose errors in brachytherapy treatment planning. In this work we review a primary and scatter dose separation formalism (PSS) for brachytherapy source characterization to support separate calculation of the primary and scatter dose contributions. We show how the resulting source characterization data can be used to drive more accurate dose calculations using collapsed cone superposition for scatter dose calculations. Two types of source characterization data paths are used: a direct Monte Carlo simulation in water phantoms with subsequent parameterization of the results, and an alternative data path built on processing of AAPM TG43 formatted data to provide similar parameter sets. The latter path is motivated of the large amounts of data already existing in the TG43 format. We demonstrate the PSS methods using both data paths for a clinical 192Ir source. Results are shown for two geometries: a finite but homogeneous water phantom, and a half-slab consisting of water and air. The dose distributions are compared to results from full Monte Carlo simulations and we show significant improvement in scatter dose calculations when the collapsed-cone kernel-superposition algorithm is used compared to traditional table based calculations. The PSS source characterization method uses exponential fit functions derived from one-dimensional transport theory to describe both the primary and scatter dose contributions. We present data for the PSS characterization method to different 192Ir, 137Cs, and 60Cs brachytherapy sources. We also show how TG43 formatted data can be derived from our data to serve traditional treatment planning systems, as to enable for a gradual transfer to algorithms that provides improved modeling of heterogeneities in brachytherapy treatment planning.     

The use of film dosimetry of the penumbra region to improve the accuracy of intensity modulated radiotherapy

Accurate measurements of the penumbra region are important for the proper modeling of the radiation beam for linear accelerator-based intensity modulated radiation therapy. The usual data collection technique with a standard ionization chamber artificially broadens the measured beam penumbrae due to volume effects. The larger the chamber, the greater is the spurious increase in penumbra width. This leads to inaccuracies in dose calculations of small fields, including small fields or beam segments used in IMRT. This source of error can be rectified by the use of film dosimetry for penumbra measurements because of its high spatial resolution. The accuracy of IMRT calculations with a pencil beam convolution model in a commercial treatment planning system was examined using commissioning data with and without the benefit of film dosimetry of the beam penumbrae. A set of dose-spread kernels of the pencil beam model was calculated based on commissioning data that included beam profiles gathered with a 0.6-cm-i.d. ionization chamber. A second set of dose-spread kernels was calculated using the same commissioning data with the exception of the penumbrae, which were measured with radiographic film. The average decrease in the measured width of the 80%-20% penumbrae of various square fields of size 3-40 cm, at 5 cm depth in water-equivalent plastic was 0.27 cm. Calculations using the pencil beam model after it was re-commissioned using film dosimetry of the penumbrae gave better agreement with measurements of IMRT fields, including superior reproduction of high dose gradient regions and dose extrema. These results show that accurately measuring the beam penumbrae improves the accuracy of the dose distributions predicted by the treatment planning system and thus is important when commissioning beam models used for IMRT.     

Dose calculation models for proton treatment planning using a dynamic beam delivery system: an attempt to include density heterogeneity effects in the analytical dose calculation

The gantry for proton radiotherapy at the Paul Scherrer Institute (PSI) is designed specifically for the spot-scanning technique. Use of this technique to its full potential requires dose calculation algorithms which are capable of precisely simulating each scanned beam individually. Different specialized analytical dose calculations have been developed, which attempt to model the effects of density heterogeneities in the patient's body on the dose. Their accuracy has been evaluated by a comparison with Monte Carlo calculated dose distributions in the case of a simple geometrical density interface parallel to the beam and typical anatomical situations. A specialized ray casting model which takes range dilution effects (broadening of the spectrum of proton ranges) into account has been found to produce results of good accuracy. This algorithm can easily be implemented in the iterative optimization procedure used for the calculation of the optimal contribution of each individual scanned pencil beam. In most cases an elemental pencil beam dose calculation has been found to be most accurate. Due to the long computing time, this model is currently used only after the optimization procedure as an alternative method of calculating the dose.     

Fast Monte Carlo dose calculation for photon beams based on the VMC electron algorithm.

A new Monte Carlo algorithm for 3D photon dose calculation in radiation therapy is presented, which is based on the previously developed Voxel Monte Carlo (VMC) for electron beams. The main result is that this new version of VMC (now called XVMC) is more efficient than EGS4/PRESTA photon dose calculation by a factor of 15-20. Therefore, a standard treatment plan for photons can be calculated by Monte Carlo in about 20 min. on a "normal" personal computer. The improvement is caused mainly by the fast electron transport algorithm and ray tracing technique, and an initial ray tracing method to calculate the number of electrons created in each voxel by the primary photon beam. The model was tested in comparison to calculations by EGS4 using several fictive phantoms. In most cases a good coincidence has been found between both codes. Only within lung substitute dose differences have been observed.     

First macro Monte Carlo based commercial dose calculation module for electron beam treatment planning--new issues for clinical consideration

The purpose of this study is to present our experience of commissioning, testing and use of the first commercial macro Monte Carlo based dose calculation algorithm for electron beam treatment planning and to investigate new issues regarding dose reporting (dose-to-water versus dose-to-medium) as well as statistical uncertainties for the calculations arising when Monte Carlo based systems are used in patient dose calculations. All phantoms studied were obtained by CT scan. The calculated dose distributions and monitor units were validated against measurements with film and ionization chambers in phantoms containing two-dimensional (2D) and three-dimensional (3D) type low- and high-density inhomogeneities at different source-to-surface distances. Beam energies ranged from 6 to 18 MeV. New required experimental input data for commissioning are presented. The result of validation shows an excellent agreement between calculated and measured dose distributions. The calculated monitor units were within 2% of measured values except in the case of a 6 MeV beam and small cutout fields at extended SSDs (>110 cm). The investigation on the new issue of dose reporting demonstrates the differences up to 4% for lung and 12% for bone when 'dose-to-medium' is calculated and reported instead of 'dose-to-water' as done in a conventional system. The accuracy of the Monte Carlo calculation is shown to be clinically acceptable even for very complex 3D-type inhomogeneities. As Monte Carlo based treatment planning systems begin to enter clinical practice, new issues, such as dose reporting and statistical variations, may be clinically significant. Therefore it is imperative that a consistent approach to dose reporting is used.     

Implementation of pencil kernel and depth penetration algorithms for treatment planning of proton beams

The implementation of two algorithms for calculating dose distributions for radiation therapy treatment planning of intermediate energy proton beams is described. A pencil kernel algorithm and a depth penetration algorithm have been incorporated into a commercial three dimensional treatment planning system (Helax-TMS, Helax AB, Sweden) to allow conformal planning techniques using irregularly shaped fields, proton range modulation, range modification and dose calculation for non-coplanar beams. The pencil kernel algorithm is developed from the Fermi Eyges formalism and Molière multiple-scattering theory with range straggling corrections applied. The depth penetration algorithm is based on the energy loss in the continuous slowing down approximation with simple correction factors applied to the beam penumbra region and has been implemented for fast, interactive treatment planning. Modelling of the effects of air gaps and range modifying device thickness and position are implicit to both algorithms. Measured and calculated dose values are compared for a therapeutic proton beam in both homogeneous and heterogeneous phantoms of varying complexity. Both algorithms model the beam penumbra as a function of depth in a homogeneous phantom with acceptable accuracy. Results show that the pencil kernel algorithm is required for modelling the dose perturbation effects from scattering in heterogeneous media.     

Modelling pencil-beam divergence with the electron pencil-beam redefinition algorithm.

The electron pencil-beam redefinition algorithm (PBRA), which is used to calculate electron beam dose distributions, assumes that the virtual source of each pencil beam is identical to that of the broad beam incident on the patient. In the present work, a virtual source specific for each pencil beam is modelled by including the source distance as a pencil-beam parameter to be redefined with depth. To incorporate a variable pencil-beam source distance parameter, the transport equation was reformulated to explicitly model divergence resulting in the algorithm divPBRA. Allowing the virtual source position to vary with individual pencil beams is expected to better model the effects of heterogeneities on local electron fluence divergence (or convergence). Selected experiments from a measured data set developed at The University of Texas M D Anderson Cancer Center were used to evaluate the accuracy of the dose calculated using divPBRA. Results of the calculation showed that the theory accurately predicted the virtual source position in regions of side-scatter equilibrium and predicted reasonable virtual source positions in regions lacking side-scatter equilibrium (i.e. penumbra and in the vicinity and shadow of internal heterogeneities). Results of the evaluation showed the dose accuracy of divPBRA to be marginally better to that of PBRA, except in regions of extremely sharp dose perturbations, where the divPBRA calculations were significantly greater than the measured data. Dose calculations using divPBRA took 45% longer than those using PBRA. Therefore, we concluded that divPBRA offers no significant advantage over PBRA for the purposes of clinical treatment planning. However, the results were promising and divPBRA might prove useful if further modelling were to include large-angle scattering, low-energy delta rays and brehmsstrahlung.