Limitations of a convolution method for modeling geometric uncertainties in radiation therapy. II. The effect of a finite number of fractions

Convolution methods can be used to model the effect of geometric uncertainties on the planned dose distribution in radiation therapy. This requires several assumptions, including that the patient is treated with an infinite number of fractions, each delivering an infinitesimally small dose. The error resulting from this assumption has not been thoroughly quantified. This is investigated by comparing dose distributions calculated using the Convolution method with the result of Stochastic simulations of the treatment. Additionally, the dose calculated using the conventional Static method, a Corrected Convolution method, and a Direct Simulation are compared to the Stochastic result. This analysis is performed for single beam, parallel opposed pair, and four-field box techniques in a cubic water phantom. Treatment plans for a simple and a complex idealized anatomy were similarly analyzed. The average maximum error using the Static method for a 30 fraction simulation for the three techniques in phantoms was 23%, 11% for Convolution, 10% for Corrected Convolution, and 10% for Direct Simulation. In the two anatomical examples, the mean error in tumor control probability for Static and Convolution methods was 7% and 2%, respectively, of the result with no uncertainty, and 35% and 9%, respectively, for normal tissue complication probabilities. Convolution provides superior estimates of the delivered dose when compared to the Static method. In the range of fractions used clinically, considerable dosimetric variations will exist solely because of the random nature of the geometric uncertainties. However, the effect of finite fractionation appears to have a greater impact on the dose distribution than plan evaluation parameters.     

Dosimetric characteristics of the Leipzig surface applicators used in the high dose rate brachy radiotherapy

The nucletron Leipzig applicator is designed for (HDR) 192Ir brachy radiotherapy of surface lesions. The dosimetric characteristics of this applicator were investigated using simulation method based on Monte Carlo N-particle (MCNP) code and phantom measurements. The simulation method was validated by comparing calculated dose rate distributions of nucletron microSelectron HDR 192Ir source against published data. Radiochromic films and metal-oxide-semiconductor field-effect transistor (MOSFET) detectors were used for phantom measurements. The double exposure technique, correcting the nonuniform film sensitivity, was applied in the film dosimetry. The linear fit of multiple readings with different irradiation times performed for each MOSFET detector measurement was used to obtain the dose rate of each measurement and to correct the source transit-time error. The film and MOSFET measurements have uncertainties of 3%-7% and 3%-5%, respectively. The dose rate distributions of the Leipzig applicator with 30 mm opening calculated by the validated MC method were verified by measurements of film and MOSFET detectors. Calculated two-dimensional planar dose rate distributions show similar patterns as the film measurement. MC calculated dose rate at a reference point defined at depth 5 mm on the applicator's central axis is 7% lower than the film and 3% higher than the MOSFET measurements. The dose rate of a Leipzig applicator with 30 mm opening at reference point is 0.241+/-3% cGy h(-1) U(-1). The MC calculated depth dose rates and profiles were tabulated for clinic use.     

Experimental verification of IMPT treatment plans in an anthropomorphic phantom in the presence of delivery uncertainties

Clinically relevant intensity modulated proton therapy (IMPT) treatment plans were measured in a newly developed anthropomorphic phantom (i) to assess plan accuracy in the presence of high heterogeneity and (ii) to measure plan robustness in the case of treatment uncertainties (range and spatial). The new phantom consists of five different tissue substitute materials simulating different tissue types and was cut into sagittal planes so as to facilitate the verification of co-planar proton fields. GafChromic films were positioned in the different planes of the phantom, and 3D-IMPT and distal edge tracking (DET) plans were delivered to a volume simulating a skull base chordoma. In addition, treatments planned on CTs of the phantom with HU units modified were delivered to simulate systematic range uncertainties (range-error treatments). Finally, plans were delivered with the phantom rotated to simulate spatial errors. Results show excellent agreement between the calculated and the measured dose distribution: >99% and 98% of points with a gamma value <1 (3%/3 mm) for the 3D-IMPT and the DET plan, respectively. For both range and spatial errors, the 3D-IMPT plan was more robust than the DET plan. Both plans were more robust to range than to the spatial uncertainties. Finally, for range error treatments, measured distributions were compared to a model for predicting delivery errors in the treatment planning system. Good agreement has been found between the model and the measurements for both types of IMPT plan.     

Experimental evaluation of validity of simplified Monte Carlo method in proton dose calculations

It is important for proton therapy to calculate dose distributions accurately in treatment planning. Dose calculations in the body for treatment planning are converted to dose distributions in water, and the converted calculations are then generally evaluated by the dose measurements in water. In this paper, proton dose calculations were realized for a phantom simulating a clinical heterogeneity. Both dose calculations in the phantom calculated by two dose calculation methods, the range-modulated pencil beam algorithm (RMPBA) and the simplified Monte Carlo (SMC) method, and dose calculations converted to dose distributions in water by the same two methods were verified experimentally through comparison with measured distributions, respectively. For the RMPBA, though the converted calculations in water agreed moderately well with the measured ones, the calculated results in the actual phantom produced large errors. This meant that dose calculations in treatment planning should be evaluated by the dose measurements not in water but in the body with heterogeneity. On the other hand, the results calculated in the phantom, even by the less rigorous SMC method, reproduced the experimental ones well. This finding showed that actual dose distributions in the body should be predicted by the SMC method.     

The effects of target size and tissue density on the minimum margin required for random errors

The minimum margins required to compensate for random geometric uncertainties in the delivery of radiotherapy treatment were determined for a spherical Clinical Target Volume, using an analytic model for the cumulative dose. Margins were calculated such that the minimum dose in the target would be no less than 95% of the prescribed dose for 90% of the patients. The dose distribution model incorporated two Gaussians, and could accurately represent realistic dose profiles for various target sizes in lung and water. It was found that variations in target size and tissue density lead to significant changes in the minimum margin required for random errors. The random error margin increased with tissue density, and decreased with target size. The required margins were similar for dose distributions of spherical and cylindrical symmetry. Significant dose outside the spherical high dose region, as could result from multiple incident beams, lead to an increased margin for the larger targets. We could confirm that the previously proposed margin of 0.7 times the standard deviation of the random errors is safe for standard deviations up to 5 mm, except for very small targets in dense material.     

The effect of set-up uncertainties,contour changes,and tissue inhomogeneities on target dose-volume histograms

Understanding set-up uncertainty effects on dose distributions is an important clinical problem but difficult to model accurately due to their dependence on tissue inhomogeneities and changes in the surface contour (i.e., variant effects). The aims are: (1) to evaluate and quantify the invariant and variant effects of set-up uncertainties, contour changes and tissue inhomogeneities on target dose-volume histograms (DVHs); (2) to propose a method to interpolate (variant) DVHs. We present a lung cancer patient to estimate the significance of set-up uncertainties, contour changes and tissue inhomogeneities in target DVHs. Differential DVHs are calculated for 15 displacement errors (with respect to the isocenter) using (1) an invariant shift of the dose distribution at the isocenter, (2) a full variant calculation, and (3) a B-spline interpolation applied to sparsely sampled variant DVHs. The collapsed cone algorithm was used for all dose calculations. Dosimetric differences are quantified with the root mean square (RMS) deviation and the equivalent uniform dose (EUD). To determine set-up uncertainty effects, weighted mean EUDs, assuming normally distributed displacement errors, are used. The maximum absolute difference and RMS deviation in the integral DVHs' relative dose between (1) the invariant and calculated curves are 65.2% and 5.8% and (2) the interpolated and calculated curves are 16.9% and 2.5%. Similarly, the maximum absolute difference and RMS deviation in mean EUD as a function of the set-up uncertainty's standard deviation between (1) the invariant and calculated curves are 0.02 and 0.01 Gy; and (2) the interpolated and calculated curves are 0.01 and 0.006 Gy. Since a "worst-case" example is selected, we conclude that, in the majority of clinical cases, the variant effects of contour changes, tissue inhomogeneities and set-up uncertainties on EUD are negligible. Interpolation is a valid, efficient method to approximate DVHs.     

Compensating for heterogeneities in proton radiation therapy

Our method for predicting, and compensating for, the effects of surface irregularities and tissue heterogeneities in proton radiation therapy was evaluated by comparing the predicted and measured dose distributions. Two heterogeneity configurations in a D-shaped water-filled phantom were handled in exactly the same way as patients. Target volumes were designated on thin-section CT scans, a single en face portal was defined, compensating boli were designed and made, and the dose distribution behind the phantom measured and compared with that intended. The compensation was accurate to within 1 mm for the phantom with a single air heterogeneity and to within 2.5 mm for the phantom with multiple bone and air heterogeneities. The bolus and phantom were misaligned by 3 mm and the dramatic change in the dose distribution demonstrated the need to address the problems of patient motion and imperfect immobilisation through compensator design. A philosophy of 'expanding' the bolus is described, and dose distributions measured with the 'expanded' boli indicate that target volume treatment can be assured within prespecified repositioning and motion uncertainties. The uncertainty in the alignment of bolus and heterogeneities leads to corresponding uncertainty in the penetration of the protons. Ranges within which they will stop are calculated and shown to encompass adequately the measured distributions in both the aligned and misaligned cases.     

A Monte Carlo method to evaluate the impact of positioning errors on detector response and quality correction factors in nonstandard beams

During experimental procedures, an adequate evaluation of all sources of uncertainty is necessary to obtain an overall uncertainty budget. In specific radiation dosimetry applications where a single detector is used, common methods to evaluate uncertainties caused by setup positioning errors are not applicable when the dose gradient is not known a priori. This study describes a method to compute these uncertainties using the Monte Carlo method. A mathematical formalism is developed to calculate unbiased estimates of the uncertainties. The method is implemented in egs_chamber, an EGSnrc-based code that allows for the efficient calculation of detector doses and dose ratios. The correct implementation of the method into the egs_chamber code is validated with an extensive series of tests. The accuracy of the developed mathematical formalism is verified by comparing egs_chamber simulation results to the theoretical expectation in an ideal situation where the uncertainty can be computed analytically. Three examples of uncertainties are considered for realistic models of an Exradin A12 ionization chamber and a PTW 60012 diode, and results are computed for parameters representing nearly realistic positioning error probability distributions. Results of practical examples show that uncertainties caused by positioning errors can be significant during IMRT reference dosimetry as well as small field output factor measurements. The method described in this paper is of interest in the study of single-detector response uncertainties during nonstandard beam measurements, both in the scope of daily routine as well as when developing new dosimetry protocols. It is pointed out that such uncertainties should be considered in new protocols devoted to single-detector measurements in regions with unpredictable dose gradients. The method is available within the egs_chamber code in the latest official release of the EGSnrc system.     

Fast Fourier transform convolution calculations of x-ray isodose distributions in homogeneous media

Convolution concepts were implemented using the discrete fast Fourier transform (FFT) to model the three-dimensional dose distribution due to x-rays produced by a medical linear accelerator. Convolution kernels were employed that had been calculated by Mackie using the EGS4 Monte Carlo code. The EGS4 code was also used to estimate initially the spectrum by simulating the production, filtering, and flattening of the beam in the collimator of the linear accelerator. The continuous bremsstrahlung spectrum was modeled using five discrete energies. The more subtle field-size effects of collimator scattering on the spectrum were obtained by calculating corrections to the spectral components using a least-squares search technique. Dose distributions were obtained using FFT convolutions of the kernels for each energy with the spectrally weighted fluence distributions for that energy. The dose distributions were compared with isodose distributions measured in a water phantom. The agreement was generally found to be better than 1% on the central axis. The calculation time for a single three-dimensional beam was approximately 20 min using a VAX/750 without an array processor. Methods were explored to reduce the calculation time using similar hardware, and estimates were made of how to reduce the calculation time using a more sophisticated computer system.     

Accuracy of the phase space evolution dose calculation model for clinical 25 MeV electron beams

The phase space evolution (PSE) model is a dose calculation model for electron beams in radiation oncology developed with the aim of a higher accuracy than the commonly used pencil beam (PB) models and with shorter calculation times than needed for Monte Carlo (MC) calculations. In this paper the accuracy of the PSE model has been investigated for 25 MeV electron beams of a MM50 racetrack microtron (Scanditronix Medical AB, Sweden) and compared with the results of a PB model. Measurements have been performed for tests like non-standard SSD, irregularly shaped fields, oblique incidence and in phantoms with heterogeneities of air, bone and lung. MC calculations have been performed as well, to reveal possible errors in the measurements and/or possible inaccuracies in the interaction data used for the bone and lung substitute materials. Results show a good agreement between PSE calculated dose distributions and measurements. For all points the differences--in absolute dose--were generally well within 3% and 3 mm. However, the PSE model was found to be less accurate in large regions of low-density material and errors of up to 6% were found for the lung phantom. Results of the PB model show larger deviations, with differences of up to 6% and 6 mm and of up to 10% for the lung phantom; at shortened SSDs the dose was overestimated by up to 6%. The agreement between MC calculations and measurement was good. For the bone and the lung phantom maximum deviations of 4% and 3% were found, caused by uncertainties about the actual interaction data. In conclusion, using the phase space evolution model, absolute 3D dose distributions of 25 MeV electron beams can be calculated with sufficient accuracy in most cases. The accuracy is significantly better than for a pencil beam model. In regions of lung tissue, a Monte Carlo model yields more accurate results than the current implementation of the PSE model.     

Calculation of a pencil beam kernel from measured photon beam data

Usually, pencil beam kernels for photon beam calculations are obtained by Monte Carlo calculations. In this paper, we present a method to derive a pencil beam kernel from measured beam data, i.e. central axis depth doses, phantom scatter factors and off-axis ratios. These data are usually available in a radiotherapy planning system. The differences from other similar works are: (a) the central part of the pencil beam is derived from the measured penumbra of large fields and (b) the dependence of the primary photon fluence on the depth caused by beam hardening in the phantom is taken into account. The calculated pencil beam will evidently be influenced by the methods and instruments used for measurement of the basic data set. This is of particular importance for an accurate prediction of the absorbed dose delivered by small fields. Comparisons with measurements show that the accuracy of the calculated dose distributions fits well in a 2% error interval in the open part of the field, and in a 2 mm isodose shift in the penumbra region.     

Detection of IMRT delivery errors using a quantitative 2D dosimetric verification system

We investigated the feasibility of detecting intensity modulated radiotherapy delivery errors automatically using a scalar evaluation of two-dimensional (2D) transverse dose measurement of the complete treatment delivery. Techniques using the gamma index and the normalized agreement test (NAT) index were used to parametrize the agreement between measured and computed dose distributions to seven different scalar metrics. Simulated verifications with delivery errors calculated using a commercially available treatment planning system for 9 prostate and 7 paranasal sinus cases were compared to 433 clinical verifications. The NAT index with 5% and 3 mm criteria that included cold areas outside the planning target volume detected the largest percent of delivery errors. Assuming a false positive rate of 5%, it was able to detect 88% of beam energy changes, 94% of a different patient's plan being delivered, 25% of plans with one beam's collimator rotated by 90 degrees, 81% of rotating one beam's gantry angle by 10 degrees, and 100% of omitting the delivery of one beam. However, no instances of changing one beam's monitor unit setting by 10% or shifting the isocenter by 5 mm were detected. Although the phantom shift could not be detected by the small change it made in the dose distribution, our autopositioning algorithm clearly identified the spatial anomaly. Using tighter 3 %/2 mm criteria or combining dose and distance disagreements in an either/or fashion resulted in poorer delivery error detection. The mean value of the 2D gamma index distribution was less sensitive to delivery errors than the other scalar metrics studied. Although we found that scalar metrics do not have sufficient delivery error detection rates to be used as the sole clinical analysis technique, manually examining 2D dose comparison images would result in a near 100% detection rate while performing an ion chamber measurement alone would only detect 54% of these errors.     

Fast range-corrected proton dose approximation method using prior dose distribution

For robust plan optimization and evaluation purposes, one needs a computationally efficient way to calculate dose distributions and dose-volume histograms (DVHs) under various changes in the variables associated with beam delivery and images. In this study, we report an approximate method for rapid calculation of dose when setup errors and anatomical changes occur during proton therapy. This fast dose approximation method calculates new dose distributions under various circumstances based on the prior knowledge of dose distribution from a reference setting. In order to validate the method, we calculated and compared the dose distributions from our approximation method to the dose distributions calculated from a clinically commissioned treatment planning system which was used as the ground truth. The overall accuracy of the proposed method was tested against varying degrees of setup error and anatomical deformation for selected patient cases. The setup error was simulated by rigid shifts of the patient; while the anatomical deformation was introduced using weekly acquired repeat CT data sets. We evaluated the agreement between the dose approximation method and full dose recalculation using a 3D gamma index and the root-mean-square (RMS) and maximum deviation of the cumulative dose volume histograms (cDVHs). The average passing rate of 3D gamma analysis under 3% dose and 3 mm distance-to-agreement criteria were 96% and 89% for setup errors and severe anatomy changes, respectively. The average of RMS and maximum deviation of the cDVHs under the setup error was 0.5% and 1.5%, respectively for all structures considered. Similarly, the average of RMS and maximum deviations under the weekly anatomical change were 0.6% and 2.7%, respectively. Our results show that the fast dose approximation method was able to account for the density variation of the patient due to the setup and anatomical changes with acceptable accuracy while significantly improving the computation time.     

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.     

Structured errors in reconstruction methods for Non-Cartesian MR data

BackgroundReconstruction methods for Non-Cartesian magnetic resonance imaging have often been analyzed using the root mean square error (RMSE). However, RMSE is not able to measure the level of structured error associated with the reconstruction process.MethodsAn index for geometric information loss was presented using the 2D autocorrelation function. The performances of Least Squares Non Uniform Fast Fourier Transform (LS-NUFFT) and gridding reconstruction (GR) methods were compared. The Direct Summation method (DS) was used as reference. For both methods, RMSE and the loss in geometric information were calculated using a digital phantom and a hyperpolarized ¹³C dataset.ResultsThe performance of the geometric information loss index was analyzed in the presence of noise. Comparing to GR, LS-NUFFT obtained a lower RMSE, but its error image appeared more structured. This was observed in both phantom and in vivo experiments.DiscussionThe evaluation of geometric information loss together with the reconstruction error was important for an appropriate performance analysis of the reconstruction methods. The use of geometric information loss was helpful to determine that LS-NUFFT loses relevant information in the reconstruction process, despite the low RMSE.     

A quality assurance phantom for IMRT dose verification

This paper investigates a quality assurance (QA) phantom specially designed to verify the accuracy of dose distributions and monitor units (MU) calculated by clinical treatment planning optimization systems and by the Monte Carlo method for intensity-modulated radiotherapy (IMRT). The QA phantom is a PMMA cylinder of 30 cm diameter and 40 cm length with various bone and lung inserts. A procedure (and formalism) has been developed to measure the absolute dose to water in the PMMA phantom. Another cylindrical phantom of the same dimensions, but made of water, was used to confirm the results obtained with the PMMA phantom. The PMMA phantom was irradiated by 4, 6 and 15 MV photon beams and the dose was measured using an ionization chamber and compared to the results calculated by a commercial inverse planning system (CORVUS, NOMOS, Sewickley, PA) and by the Monte Carlo method. The results show that the dose distributions calculated by both CORVUS and Monte Carlo agreed to within 2% of dose maximum with measured results in the uniform PMMA phantom for both open and intensity-modulated fields. Similar agreement was obtained between Monte Carlo calculations and measured results with the bone and lung heterogeneity inside the PMMA phantom while the CORVUS results were 4% different. The QA phantom has been integrated as a routine QA procedure for the patient's IMRT dose verification at Stanford since 1999.     

Accuracy of cranial coplanar beam therapy using an oblique, stereoscopic x-ray image guidance system

A system for measuring two-dimensional (2D) dose distributions in orthogonal anatomical planes in the cranium was developed and used to evaluate the accuracy of coplanar conformal therapy using ExacTrac image guidance. Dose distributions were measured in the axial, sagittal, and coronal planes using a CIRS (Computerized Imaging Reference Systems, Inc.) anthropomorphic head phantom with a custom internal film cassette. Sections of radiographic Kodak EDR2 film were cut, processed, and digitized using custom templates. Spatial and dosimetric accuracy and precision of the film system were assessed. BrainScan planned a coplanar-beam treatment to conformally irradiate a 2-cm-diameter x 2-cm-long cylindrical planning target volume. Prior to delivery, phantom misalignments were imposed in combinations of +/- 8 mm offsets in each of the principal directions. ExacTrac x-ray correction was applied until the phantom was within an acceptance criteria of 1 mm/1 degrees (first two measurement sets) or 0.4 mm/0.4 degrees (last two measurement sets). Measured dose distributions from film were registered to the treatment plan dose calculations and compared. Alignment errors, displacement between midpoints of planned and measured 70% isodose contours (Deltac), and positional errors of the 80% isodose line were evaluated using 49 2D film measurements (98 profiles). Comparison of common, but independent measurements of Deltac showed that systematic errors in the measurement technique were 0.2 mm or less along all three anatomical axes and that random error averaged [formula: see text] 0.29+/-0.06 mm for the acceptance criteria of 1 mm/1 degrees and 0.15 +/- 0.02 mm for the acceptance criteria of 0.4 mm/0.4 degrees. The latter was consistent with independent estimates that showed the precision of the measurement system was 0.3 mm (2sigma). Values of Deltac were as great as 0.9, 0.3, and 1.0 mm along the P-A, R-L, and I-S axes, respectively. Variations in Deltac along the P-A axis were correlated to misalignments between laser isocenter and radiation isocenter as documented by daily clinical Lutz tests. Based on results of comparisons of measured with calculated positions of the 80% dose lines along the major anatomical axes, a 1.25, 1.0, and 1.0 mm (0.75, 0.5, and 0.25 mm) gross tumor volume (GTV)-planning target volume (PTV) margin to account for delivery error would be appropriate for the P-A, R-L, and I-S axes, respectively, for an acceptance criteria of 1 mm/1 degrees (0.4 mm/0.4 degrees). It typically took 2 (3) ExacTrac x-ray image sets to achieve and verify acceptance criteria of 1 mm/1 degrees (0.4 mm/0.4 degrees). Our results demonstrated a measurement technique using a CIRS anthropomorphic head phantom with a modified film cassette, radiographic film (Kodak EDR2) with a custom film cutting template, and film dosimetry software has been developed and successfully applied to our clinic. It is recommended that a third party offer this service. Our goal of achieving accuracy of delivery of 1 mm or better in each of the three major anatomical axes was almost, but not quite achieved, not because of the accuracy of the image guidance system, but likely due to inaccuracy of laser isocenter and other systematic errors.     

Impact of dose-distribution uncertainties on rectal ntcp modeling. I: Uncertainty estimates

A trial of nonescalated conformal versus conventional radiotherapy treatment of prostate cancer has been carried out at the Royal Marsden NHS Trust (RMH) and Institute of Cancer Research (ICR), demonstrating a significant reduction in the rate of rectal bleeding reported for patients treated using the conformal technique. The relationship between planned rectal dose-distributions and incidences of bleeding has been analyzed, showing that the rate of bleeding falls significantly as the extent of the rectal wall receiving a planned dose-level of more than 57 Gy is reduced. Dose-distributions delivered to the rectal wall over the course of radiotherapy treatment inevitably differ from planned distributions, due to sources of uncertainty such as patient setup error, rectal wall movement and variation in the absolute rectal wall surface area. In this paper estimates of the differences between planned and treated rectal dose-distribution parameters are obtained for the RMH/ICR nonescalated conformal technique, working from a distribution of setup errors observed during the RMH/ICR trial, movement data supplied by Lebesque and colleagues derived from repeat CT scans, and estimates of rectal circumference variations extracted from the literature. Setup errors and wall movement are found to cause only limited systematic differences between mean treated and planned rectal dose-distribution parameter values, but introduce considerable uncertainties into the treated values of some dose-distribution parameters: setup errors lead to 22% and 9% relative uncertainties in the highly dosed fraction of the rectal wall and the wall average dose, respectively, with wall movement leading to 21% and 9% relative uncertainties. Estimates obtained from the literature of the uncertainty in the absolute surface area of the distensible rectal wall are of the order of 13%-18%. In a subsequent paper the impact of these uncertainties on analyses of the relationship between incidences of bleeding and planned rectal dose-distributions is explored.     

Accuracy limits of the equivalent field method for irregular photon fields

A mathematical approach is developed to evaluate the accuracy of the equivalent field method using basic clinical photon beam data. This paper presents an analytical calculation of dose errors arising when field equivalencies, calculated at a certain reference depth, are translated to other depths. The phantom scatter summation is expressed as a Riemann-Stieltjes integral and two categories of irregular fields are introduced: uniform and multiform. It is shown that multiform fields produce errors whose magnitudes are nearly twice those corresponding to uniform fields in extreme situations. For uniform field shapes, the maximum, local, relative dose errors, when the equivalencies are calculated at 10 cm depth on the central axis and translated to a depth of 30 cm, are 3.8% and 8.8% for 6 MV and cobalt-60 photon beams, respectively. In terms of maximum dose those errors are within 1-2%. This supports the conclusion that the equivalencies between rectangular fields, which are examples of uniform fields, are applicable to dose ratio functions irrespective of beam energy. However, the magnitude of such errors could be of importance when assessing the exit dose for in vivo monitoring. This work provides a better understanding of the influence of the irregular field shapes on the accuracy of the equivalent field method.     

Calculation of dose in homogeneous phantoms for partially attenuated photon beams

Measured and calculated dose distributions under attenuators, which are of smaller cross-sectional dimensions than the radiation field, are presented. The study was performed on a 4-MV linac at a source-surface distance of 120 cm on the beam central axis in a water phantom for several thickness and cross sections of lead attenuators. Dose correction factors, which are used to multiply the open beam data to get dose distributions under partial attenuators, depend strongly on attenuator parameters and on depths in phantom. A method to calculate dose correction factors for any combination of attenuator parameters and any phantom depth is presented. The calculated dose distributions under partial attenuators agree well with measured data, which indicates that the method can be applied in clinical situations.