The effect of statistical uncertainty on inverse treatment planning based on Monte Carlo dose calculation

The effect of the statistical uncertainty, or noise, in inverse treatment planning for intensity modulated radiotherapy (IMRT) based on Monte Carlo dose calculation was studied. Sets of Monte Carlo beamlets were calculated to give uncertainties at Dmax ranging from 0.2% to 4% for a lung tumour plan. The weights of these beamlets were optimized using a previously described procedure based on a simulated annealing optimization algorithm. Several different objective functions were used. It was determined that the use of Monte Carlo dose calculation in inverse treatment planning introduces two errors in the calculated plan. In addition to the statistical error due to the statistical uncertainty of the Monte Carlo calculation, a noise convergence error also appears. For the statistical error it was determined that apparently successfully optimized plans with a noisy dose calculation (3% 1sigma at Dmax), which satisfied the required uniformity of the dose within the tumour, showed as much as 7% underdose when recalculated with a noise-free dose calculation. The statistical error is larger towards the tumour and is only weakly dependent on the choice of objective function. The noise convergence error appears because the optimum weights are determined using a noisy calculation, which is different from the optimum weights determined for a noise-free calculation. Unlike the statistical error, the noise convergence error is generally larger outside the tumour, is case dependent and strongly depends on the required objectives.     

Comparison of IMRT optimization based on a pencil beam and a superposition algorithm

To investigate the role of sophisticated dose calculation methods for treatment planning, we compared conventional pencil beam optimized 6 and 15 MV intensity-modulated treatment plans with optimizations based on the superposition technique. Five lung and five head and neck IMRT cases with spatial resolutions of bixels and dose voxels usually employed in clinical practice were considered for tumor volumes between 15 and 500 cm3. We investigated the systematic error of the pencil beam algorithm and the pencil beam induced error to the optimal solution of bixel weights. For the lung cases, the pencil beam overestimated the mean dose deposited inside the planning target volume (PTV) by about 8%, for small lung tumors even up to 20.6%. In the head and neck cases only a slight overestimation in mean PTV dose of 1.5% was observed. The optimization with the superposition method substantially improved the dose coverage of the considered radiation targets. Additionally, for the head and neck cases, the brainstem was significantly spared by about 4% mean PTV dose through the use of the superposition technique. Our studies showed that, in target regions with intricate tissue inhomogeneities, superposition or Monte Carlo techniques have to be used for the optimization and the final dose calculation of intensity-modulated treatment plans.     

Dose calculations for external photon beams in radiotherapy

Dose calculation methods for photon beams are reviewed in the context of radiation therapy treatment planning. Following introductory summaries on photon beam characteristics and clinical requirements on dose calculations, calculation methods are described in order of increasing explicitness of particle transport. The simplest are dose ratio factorizations limited to point dose estimates useful for checking other more general, but also more complex, approaches. Some methods incorporate detailed modelling of scatter dose through differentiation of measured data combined with various integration techniques. State-of-the-art methods based on point or pencil kernels, which are derived through Monte Carlo simulations, to characterize secondary particle transport are presented in some detail. Explicit particle transport methods, such as Monte Carlo, are briefly summarized. The extensive literature on beam characterization and handling of treatment head scatter is reviewed in the context of providing phase space data for kernel based and/or direct Monte Carlo dose calculations. Finally, a brief overview of inverse methods for optimization and dose reconstruction is provided.     

Accuracy of patient dose calculation for lung IMRT: A comparison of Monte Carlo, convolution/superposition, and pencil beam computations

The accuracy of dose computation within the lungs depends strongly on the performance of the calculation algorithm in regions of electronic disequilibrium that arise near tissue inhomogeneities with large density variations. There is a lack of data evaluating the performance of highly developed analytical dose calculation algorithms compared to Monte Carlo computations in a clinical setting. We compared full Monte Carlo calculations (performed by our Monte Carlo dose engine MCDE) with two different commercial convolution/superposition (CS) implementations (Pinnacle-CS and Helax-TMS's collapsed cone model Helax-CC) and one pencil beam algorithm (Helax-TMS's pencil beam model Helax-PB) for 10 intensity modulated radiation therapy (IMRT) lung cancer patients. Treatment plans were created for two photon beam qualities (6 and 18 MV). For each dose calculation algorithm, patient, and beam quality, the following set of clinically relevant dose-volume values was reported: (i) minimal, median, and maximal dose (Dmin, D50, and Dmax) for the gross tumor and planning target volumes (GTV and PTV); (ii) the volume of the lungs (excluding the GTV) receiving at least 20 and 30 Gy (V20 and V30) and the mean lung dose; (iii) the 33rd percentile dose (D33) and Dmax delivered to the heart and the expanded esophagus; and (iv) Dmax for the expanded spinal cord. Statistical analysis was performed by means of one-way analysis of variance for repeated measurements and Tukey pairwise comparison of means. Pinnacle-CS showed an excellent agreement with MCDE within the target structures, whereas the best correspondence for the organs at risk (OARs) was found between Helax-CC and MCDE. Results from Helax-PB were unsatisfying for both targets and OARs. Additionally, individual patient results were analyzed. Within the target structures, deviations above 5% were found in one patient for the comparison of MCDE and Helax-CC, while all differences between MCDE and Pinnacle-CS were below 5%. For both Pinnacle-CS and Helax-CC, deviations from MCDE above 5% were found within the OARs: within the lungs for two (6 MV) and six (18 MV) patients for Pinnacle-CS, and within other OARs for two patients for Helax-CC (for Dmax of the heart and D33 of the expanded esophagus) but only for 6 MV. For one patient, all four algorithms were used to recompute the dose after replacing all computed tomography voxels within the patient's skin contour by water. This made all differences above 5% between MCDE and the other dose calculation algorithms disappear. Thus, the observed deviations mainly arose from differences in particle transport modeling within the lungs, and the commissioning of the algorithms was adequately performed (or the commissioning was less important for this type of treatment). In conclusion, not one pair of the dose calculation algorithms we investigated could provide results that were consistent within 5% for all 10 patients for the set of clinically relevant dose-volume indices studied. As the results from both CS algorithms differed significantly, care should be taken when evaluating treatment plans as the choice of dose calculation algorithm may influence clinical results. Full Monte Carlo provides a great benchmarking tool for evaluating the performance of other algorithms for patient dose computations.     

A finite-size pencil beam model for photon dose calculations in three dimensions.

A three-dimensional dose computation model employing a finite-size, diverging, pencil beam has been developed and is demonstrated for Cobalt-60 gamma rays. The square cross-section pencil beam is simulated in a semi-infinite water phantom by convolving the pencil beam photon fluence with the Monte Carlo point dose kernel for Cobalt-60. This finite-size pencil beam is calculated one time and becomes a new data base with which to build larger beams by two-dimensional superposition. The pencil beam fluence profile, angle correction for beam divergence, the Mayneord inverse square correction, radial and angular sampling rates, error propagation, and computation time have been investigated and are reported. Radial and angular sampling rates have a great effect on accuracy and their appropriate selection is important. Percent depth doses calculated by finite-size pencil beam superposition are within 1% of values calculated by full convolution and the agreement with values from the literature is within 6%. The latter disagreement is shown to be due to a low-energy photon component which is not modeled in other calculations. Computation time measurements show the pencil beam method to be faster than full convolution and one implementation of the differential-scatter-air-ratio (dSAR) method.     

Application of an inverse kernel concept to Monte Carlo based IMRT

Inverse treatment planning by means of pencil beam algorithms can lead to errors in the calculation of dose in areas without secondary electron equilibrium. Monte Carlo (MC) simulations give accurate results in such areas but result in increased computation times. We present a new, so-called inverse kernel concept that offers MC precision in inverse treatment planning with acceptable computation times and memory consumption. Inverse kernels are matrices that describe the dose contribution from all bixels of a beam to a distinct voxel of the patient phantom. The concept is similar to other generalized pencil-beam concepts, except that inverse kernel elements are precalculated using a single MC simulation and stored as binary trees. In this procedure a modified MC code (XVMC) is applied to trace the photon history for each dose deposition. Iterative optimization is then applied in a second step. The inverse process is separated into (i) a slower MC simulation and (ii) a faster iterative optimization, followed by (iii) the segmentation procedure, and (iv) a final MC dose calculation step including a segment weight reoptimization. Inverse kernel optimization, or IKO, with segmentation and reoptimization steps is demonstrated by means of a lung cancer case. To demonstrate the superiority of an inverse MC system over pencil-beam or collapsed-cone based systems, the final result of the IKO is compared to plans where all segments have been calculated by pencil beam or collapsed cone, respectively. Dose-volume histograms and dose-difference histograms show remarkable differences, which can be attributed to systematic errors in both algorithms. IKO is a precise, nonhybrid, inverse MC treatment planning system which suits current clinical needs, as several optimization steps can follow one single MC-simulation step for a distinct beam setup.     

Monte Carlo dose calculations for dynamic IMRT treatments

Dose calculations for intensity modulated radiation therapy (IMRT) face new challenges due to the complex leaf geometry and time dependent nature of the delivery. A fast method of particle transport through a dynamic multileaf collimator (MLC) geometry that accounts for photon attenuation and first-scattered Compton photon production has been incorporated into an existing Monte Carlo code used for patient dose calculations. Dosimetric agreement between calculation and measurement for two photon energies and MLC types is within experimental error for the sliding window tests. For a patient IMRT field, the Monte Carlo calculations are closer to measured dose than similar superposition or pencil beam calculations.     

Intensity modulated irradiation of a thorax phantom: comparisons between measurements, Monte Carlo calculations and pencil beam calculations.

The present study investigates the application of compensators for the intensity modulated irradiation of a thorax phantom. Measurements are compared with Monte Carlo and standard pencil beam algorithm dose calculations. Compensators were manufactured to produce the intensity profiles that were generated from the scientific version of the KonRad IMRT treatment-planning system for a given treatment plan. The comparison of dose distributions calculated with a pencil beam algorithm, with the Monte Carlo code EGS4 and with measurements is presented. By measurements in a water phantom it is demonstrated that the method used to manufacture the compensators reproduces the intensity profiles in a suitable manner. Monte Carlo simulations in a water phantom show that the accelerator head model used for simulations is sufficient. No significant overestimations of dose values inside the target volume by the pencil beam algorithm are found in the thorax phantom. An overestimation of dose values in lung by the pencil beam algorithm is also not found. Expected dose calculation errors of the pencil beam algorithm are suppressed, because the dose to the low density region lung is reduced by the use of a non-coplanar beam arrangement and by intensity modulation.     

Direct aperture optimization for IMRT using Monte Carlo generated beamlets

This work introduces an EGSnrc-based Monte Carlo (MC) beamlet does distribution matrix into a direct aperture optimization (DAO) algorithm for IMRT inverse planning. The technique is referred to as Monte Carlo-direct aperture optimization (MC-DAO). The goal is to assess if the combination of accurate Monte Carlo tissue inhomogeneity modeling and DAO inverse planning will improve the dose accuracy and treatment efficiency for treatment planning. Several authors have shown that the presence of small fields and/or inhomogeneous materials in IMRT treatment fields can cause dose calculation errors for algorithms that are unable to accurately model electronic disequilibrium. This issue may also affect the IMRT optimization process because the dose calculation algorithm may not properly model difficult geometries such as targets close to low-density regions (lung, air etc.). A clinical linear accelerator head is simulated using BEAMnrc (NRC, Canada). A novel in-house algorithm subdivides the resulting phase space into 2.5 X 5.0 mm2 beamlets. Each beamlet is projected onto a patient-specific phantom. The beamlet dose contribution to each voxel in a structure-of-interest is calculated using DOSXYZnrc. The multileaf collimator (MLC) leaf positions are linked to the location of the beamlet does distributions. The MLC shapes are optimized using direct aperture optimization (DAO). A final Monte Carlo calculation with MLC modeling is used to compute the final dose distribution. Monte Carlo simulation can generate accurate beamlet dose distributions for traditionally difficult-to-calculate geometries, particularly for small fields crossing regions of tissue inhomogeneity. The introduction of DAO results in an additional improvement by increasing the treatment delivery efficiency. For the examples presented in this paper the reduction in the total number of monitor units to deliver is approximately 33% compared to fluence-based optimization methods.     

A finite size pencil beam for IMRT dose optimization

Dose optimization for intensity modulated radiotherapy (IMRT) using small field elements (beamlets) requires the computation of a large number of very small, often only virtual fields of typically a few mm to 1 cm in size. The primary requirements for a suitable dose computation algorithm are (1) speed and (2) proper consideration of the penumbra of the fields which are composed of these beamlets. Here, a finite size pencil beam (fsPB) algorithm is proposed which was specifically designed for the purpose of beamlet-based IMRT. The algorithm employs an analytical function for the cross-profiles of the beamlets which is based on the assumption of self-consistency, i.e. the requirement that an arbitrary superposition of abutting beamlets should add up to a homogeneous field. The depth dependence is stored in tables derived from Monte Carlo computed dose distributions. It is demonstrated that the algorithm produces accurately the output factors and cross-profiles of typical multi-leaf-shaped segments. Due to the accurate penumbra model, the dose distribution features physically feasible gradients at any stage of the iterative optimization, which eliminates the problem of large discrepancies in normal tissue dose due to misaligned gradients between optimized and recomputed treatment plans.     

Quantifying lateral tissue heterogeneities in hadron therapy

In radiotherapy with scanned particle beams, tissue heterogeneities lateral to the beam direction are problematic in two ways: they pose a challenge to dose calculation algorithms, and they lead to a high sensitivity to setup errors. In order to quantify and avoid these problems, a heterogeneity number H(i) as a method to quantify lateral tissue heterogeneities of single beam spot i is introduced. To evaluate this new concept, two kinds of potential errors were investigated for single beam spots: First, the dose calculation error has been obtained by comparing the dose distribution computed by a simple pencil beam algorithm to more accurate Monte Carlo simulations. The resulting error is clearly correlated with H(i). Second, the analysis of the sensitivity to setup errors of single beam spots also showed a dependence on H(i). From this data it is concluded that H(i) can be used as a criterion to assess the risks of a compromised delivered dose due to lateral tissue heterogeneities. Furthermore, a method how to incorporate this information into the inverse planning process for intensity modulated proton therapy is presented. By suppressing beam spots with a high value of H(i), the unfavorable impact of lateral tissue heterogeneities can be reduced, leading to treatment plans which are more robust to dose calculation errors of the pencil beam algorithm. Additional possibilities to use the information of H(i) are outlined in the discussion.     

Two-dimensional pencil beam scaling: an improved proton dose algorithm for heterogeneous media

New dose delivery techniques with proton beams, such as beam spot scanning or raster scanning, require fast and accurate dose algorithms which can be applied for treatment plan optimization in clinically acceptable timescales. The clinically required accuracy is particularly difficult to achieve for the irradiation of complex, heterogeneous regions of the patient's anatomy. Currently applied fast pencil beam dose calculations based on the standard inhomogeneity correction of pathlength scaling often cannot provide the accuracy required for clinically acceptable dose distributions. This could be achieved with sophisticated Monte Carlo simulations which are still unacceptably time consuming for use as dose engines in optimization calculations. We therefore present a new algorithm for proton dose calculations which aims to resolve the inherent problem between calculation speed and required clinical accuracy. First, a detailed derivation of the new concept, which is based on an additional scaling of the lateral proton fluence is provided. Then, the newly devised two-dimensional (2D) scaling method is tested for various geometries of different phantom materials. These include standard biological tissues such as bone, muscle and fat as well as air. A detailed comparison of the new 2D pencil beam scaling with the current standard pencil beam approach and Monte Carlo simulations, performed with GEANT, is presented. It was found that the new concept proposed allows calculation of absorbed dose with an accuracy almost equal to that achievable with Monte Carlo simulations while requiring only modestly increased calculation times in comparison to the standard pencil beam approach. It is believed that this new proton dose algorithm has the potential to significantly improve the treatment planning outcome for many clinical cases encountered in highly conformal proton therapy.     

Collapsed cone convolution and analytical anisotropic algorithm dose calculations compared to VMC++ Monte Carlo simulations in clinical cases

The purpose of this work was to study and quantify the differences in dose distributions computed with some of the newest dose calculation algorithms available in commercial planning systems. The study was done for clinical cases originally calculated with pencil beam convolution (PBC) where large density inhomogeneities were present. Three other dose algorithms were used: a pencil beam like algorithm, the anisotropic analytic algorithm (AAA), a convolution superposition algorithm, collapsed cone convolution (CCC), and a Monte Carlo program, voxel Monte Carlo (VMC++). The dose calculation algorithms were compared under static field irradiations at 6 MV and 15 MV using multileaf collimators and hard wedges where necessary. Five clinical cases were studied: three lung and two breast cases. We found that, in terms of accuracy, the CCC algorithm performed better overall than AAA compared to VMC++, but AAA remains an attractive option for routine use in the clinic due to its short computation times. Dose differences between the different algorithms and VMC++ for the median value of the planning target volume (PTV) were typically 0.4% (range: 0.0 to 1.4%) in the lung and -1.3% (range: -2.1 to -0.6%) in the breast for the few cases we analysed. As expected, PTV coverage and dose homogeneity turned out to be more critical in the lung than in the breast cases with respect to the accuracy of the dose calculation. This was observed in the dose volume histograms obtained from the Monte Carlo simulations.     

Dose verification of an IMRT treatment planning system with the BEAM EGS4-based Monte Carlo code

Intensity modulated radiation therapy (IMRT) has been increasingly used in radiotherapy departments during the last several years. A major advantage of IMRT in comparison to traditional three-dimensional conformal radiotherapy is the higher capability in providing dose distributions that conform very tightly to the target even for very complex shapes such as, for instance, concave regions. This results in a significant sparing of adjacent normal tissues. Different types of algorithms are employed in the IMRT dose calculation, from the simple pencil beam method, such as the finite-size pencil beam algorithm, to the more sophisticated algorithms, such as the kernel-based convolution/superposition ones. With the latter ones, electronic disequilibrium and inhomogeneities are better dealt with in comparison to the correction-based models like pencil beam. Nevertheless, even these types of algorithms may have some approximations that can potentially affect the dose results, especially considering that in an IMRT plan small segments or beamlets may be present for which electronic disequilibrium and inhomogeneities effects are of paramount importance. The goal of this work was to determine the accuracy in monitor units (MU) and dose distribution calculation of the algorithm implemented in the commercial treatment planning system PINNACLE3 (P3), for two IMRT plans with 6 MV photon beams. This system is based on a convolution/superposition with the Collapsed Cone approximation algorithm. The "BEAM" Monte Carlo (MC) code was employed as a benchmark in comparing the MU calculation and the dose distribution of P3. The model used to calculate the MU, with the separation of collimator scatter from the phantom scatter, valid for broad beams, was verified for narrow and irregular segments. The attention was focused on the way P3 calculates output factors (OF). A difference of 8% compared to MC was found for a particularly narrow segment analyzed. A dependence of the results on field size was found. For the complete plan, the agreement of dose distribution and MU calculation with MC results (affected by a dose uncertainty less than 0.5%) is very good: the dose difference at isocenter is 2.1% (1 standard deviation) for a "Prostate" site and 2.9% (1 standard deviation) for the "Head and Neck" site.     

Investigation of intensity-modulated radiotherapy optimization with gEUD-based objectives by means of simulated annealing

Inverse treatment planning of intensity-modulated radiation therapy (IMRT) is complicated by several sources of error, which can cause deviations of optimized plans from the true optimal solution. These errors include the systematic and convergence error, the local minima error, and the optimizer convergence error. We minimize these errors by developing an inverse IMRT treatment planning system with a Monte Carlo based dose engine and a simulated annealing search engine as well as a deterministic search engine. In addition, different generalized equivalent uniform dose (gEUD)-based and hybrid objective functions were implemented and investigated with simulated annealing. By means of a head-and-neck IMRT case we have analyzed the properties of these gEUD-based objective functions, including its search space and the existence of local optima errors. We found evidence that the use of a previously published investigation of a gEUD-based objective function results in an uncommon search space with a golf hole structure. This special search space structure leads to trapping in local minima, making it extremely difficult to identify the true global minimum, even when using stochastic search engines. Moreover, for the same IMRT case several local optima have been detected by comparing the solutions of 100 different trials using a gradient optimization algorithm with the global optimum computed by simulated annealing. We have demonstrated that the hybrid objective function, which includes dose-based objectives for the target and gEUD-based objectives for normal tissue, results in equally good sparing of the critical structures as for the pure gEUD objective function and lower target dose maxima.     

Comparison of dose calculation algorithms with Monte Carlo methods for photon arcs

The objective of this study is to seek an accurate and efficient method to calculate the dose distribution of a photon arc. The algorithms tested include Monte Carlo, pencil beam kernel (PK), and collapsed cone convolution (CCC). For the Monte Carlo dose calculation, EGS4/DOSXYZ was used. The SRCXYZ source code associated with the DOSXYZ was modified so that the gantry angle of a photon beam would be sampled uniformly within the arc range about an isocenter to simulate a photon arc. Specifically, photon beams (6/18 MV, 4 x 4 and 10 x 10 cm2) described by a phase space file generated by BEAM (MCPHS), or by two point sources with different photon energy spectra (MCDIV) were used. These methods were used to calculate three-dimensional (3-D) distributions in a PMMA phantom, a cylindrical water phantom, and a phantom with lung inhomogeneity. A commercial treatment planning system was also used to calculate dose distributions in these phantoms using equivalent tissue air ratio (ETAR), PK and CCC algorithms for inhomogeneity corrections. Dose distributions for a photon arc in these phantoms were measured using a RK ion chamber and radiographic films. For homogeneous phantoms, the measured results agreed well (approximately 2% error) with predictions by the Monte Carlo simulations (MCPHS and MCDIV) and the treatment planning system for the 180 degrees and 360 degrees photon arcs. For the dose distribution in the phantom with lung inhomogeneity with a 90 degrees photon arc, the Monte Carlo calculations agreed with the measurements within 2%, while the treatment planning system using ETAR, PK and CCC underestimated or overestimated the dose inside the lung inhomogeneity from 6% to 12%.     

Feasibility of a fast inverse dose optimization algorithm for IMRT via matrix inversion without negative beamlet intensities

A fast optimization algorithm is very important for inverse planning of intensity modulated radiation therapy (IMRT), and for adaptive radiotherapy of the future. Conventional numerical search algorithms such as the conjugate gradient search, with positive beam weight constraints, generally require numerous iterations and may produce suboptimal dose results due to trapping in local minima. A direct solution of the inverse problem using conventional quadratic objective functions without positive beam constraints is more efficient but will result in unrealistic negative beam weights. We present here a direct solution of the inverse problem that does not yield unphysical negative beam weights. The objective function for the optimization of a large number of beamlets is reformulated such that the optimization problem is reduced to a linear set of equations. The optimal set of intensities is found through a matrix inversion, and negative beamlet intensities are avoided without the need for externally imposed ad-hoc constraints. The method has been demonstrated with a test phantom and a few clinical radiotherapy cases, using primary dose calculations. We achieve highly conformal primary dose distributions with very rapid optimization times. Typical optimization times for a single anatomical slice (two dimensional) (head and neck) using a LAPACK matrix inversion routine in a single processor desktop computer, are: 0.03 s for 500 beamlets; 0.28 s for 1000 beamlets; 3.1 s for 2000 beamlets; and 12 s for 3000 beamlets. Clinical implementation will require the additional time of a one-time precomputation of scattered radiation for all beamlets, but will not impact the optimization speed. In conclusion, the new method provides a fast and robust technique to find a global minimum that yields excellent results for the inverse planning of IMRT.     

Monte Carlo evaluation of a photon pencil kernel algorithm applied to fast neutron therapy treatment planning

When dedicated software is lacking, treatment planning for fast neutron therapy is sometimes performed using dose calculation algorithms designed for photon beam therapy. In this work Monte Carlo derived neutron pencil kernels in water were parametrized using the photon dose algorithm implemented in the Nucletron TMS (treatment management system) treatment planning system. A rectangular fast-neutron fluence spectrum with energies 0-40 MeV (resembling a polyethylene filtered p(41)+Be spectrum) was used. Central axis depth doses and lateral dose distributions were calculated and compared with the corresponding dose distributions from Monte Carlo calculations for homogeneous water and heterogeneous slab phantoms. All absorbed doses were normalized to the reference dose at 10 cm depth for a field of radius 5.6 cm in a 30 x 40 x 20 cm3 water test phantom. Agreement to within 7% was found in both the lateral and the depth dose distributions. The deviations could be explained as due to differences in size between the test phantom and that used in deriving the pencil kernel (radius 200 cm, thickness 50 cm). In the heterogeneous phantom, the TMS, with a directly applied neutron pencil kernel, and Monte Carlo calculated absorbed doses agree approximately for muscle but show large deviations for media such as adipose or bone. For the latter media, agreement was substantially improved by correcting the absorbed doses calculated in TMS with the neutron kerma factor ratio and the stopping power ratio between tissue and water. The multipurpose Monte Carlo code FLUKA was used both in calculating the pencil kernel and in direct calculations of absorbed dose in the phantom.     

Analysis of photon beam exit dose using photon point kernels

The Monte Carlo method is used to analyse the dose fall-off at the exit surface of a megavoltage photon beam. The convolution/superposition method of dose calculation using Monte-Carlo-generated homogeneous photon kernels is shown to be in error for exit dose calculation. Instead, photon kernels that incorporate modelling of the exit surface were generated, also using Monte Carlo, to analyse the problem, and the calculated dose fall-off using these kernels agrees well with measured data. In addition, the physics underlying the characteristics of the dose fall-off is analysed based on complete Monte Carlo modelling. Practical improvements to the convolution/superposition method are suggested.     

The energy-dependent electron loss model for pencil beam dose kernels

The 'monoenergetic' electron loss model was derived in a previous work to account for pathlength straggling in the Fermi-Eyges pencil beam problem. In this paper, we extend this model to account for energy-loss straggling and secondary knock-on electron transport in order to adequately predict a depth dose curve. To model energy-loss straggling, we use a weighted superposition of a discrete number of monoenergetic pencil beams with different initial energies where electrons travel along the depth-energy characteristics in the continuous slowing down approximation (CSDA). The energy straggling spectrum at depth determines the weighting assigned to each monoenergetic pencil beam. Supplemented by a simple transport model for the secondary knock-on electrons, the 'energy-dependent' electron loss model predicts both lateral and depth dose distributions from the electron pencil beams in good agreement with Monte Carlo calculations and measurements. The calculation of dose distribution from a pencil beam takes 0.2 s on a Pentium III 500 MHz computer. Being computationally fast, the 'energy-dependent' electron loss model can be used for the calculation of 3D energy deposition kernels in dose optimization schemes without using precalculated or measured data.