Calculating dose and output factors for wedged photon radiotherapy fields using a convolution/superposition method

To account for clinical divergent and polychromatic photon beams, we have developed kernel tilting and kernel hardening correction methods for convolution dose calculation algorithms. The new correction methods were validated by Monte Carlo simulation. The accuracy and computation time of the our kernel tilting and kernel hardening correction methods were also compared to the existing approaches including terma divergence correction, dose divergence correction methods, and the effective mean kernel method with no kernel hardening correction. Treatment fields of 10 x 10-40 x 40 cm2 (field size at source to axis distance (SAD)) with source to source distances (SSDs) of 60, 80, and 100 cm, and photon energies of 6, 10, and 18 MV have been studied. Our results showed that based on the relative dose errors at a depth of 15 cm along the central axis, the terma divergence correction may be used for fields smaller than 10 x 10 cm2 with a SSD larger than 80 cm; the dose divergence correction with an additional kernel hardening correction can reduce dose error and may be more applicable than the terma divergence correction. For both these methods, the dose error increased linearly with the depth in the phantom; the 90% isodose lines at the depth of 15 cm were shifted by about 2%-5% of the field width due to significant underestimation of the penumbra dose. The kernel hardening effect was less prominent than the kernel tilting effect for clinical photon beams. The dose error by using nonhardening corrected kernel is less than 2.0% at a depth of 15 cm along the central axis, yet it increased with a smaller field size and lower photon energy. The kernel hardening correction could be more important to compute dose in the fields with beam modifiers such as wedges when beam hardening is more significant. The kernel tilting correction and kernel hardening correction increased computation time by about 3 times, and 0.5-1 times, respectively. This can be justified by more accurate dose calculations for the majority of clinical treatments.     

Quantification of mean energy and photon contamination for accurate dosimetry of high-energy electron beams

The scientific background of the standard procedure for determination of the mean electron energy at the phantom surface (E0) from the half-value depth (R50) has been studied. The influence of energy, angular spread and range straggling on the shape of the depth dose distribution and the R50 and Rp ranges is described using the simple Gaussian range straggling model. The relation between the R50 and Rp ranges is derived in terms of the variance of the range straggling distribution. By describing the mean energy imparted by the electrons both as a surface integral over the incident energy fluence and as a volume integral over the associated absorbed dose distribution, the relation between E0 and different range concepts, such as R50 and the maximum dose and the surface dose related mean energy deposition ranges, Rm and R0, is analysed. In particular the influence of multiple electron scatter and phantom generated bremsstrahlung on R50 is derived. A simple analytical expression is derived for the ratio of the incident electron energy to the half-value depth. Also, an analytical expression is derived for the maximum energy deposition in monoenergetic plane-parallel electron beams in water for energies between 2 and 50 MeV. Simple linear relations describing the relative absorbed dose and mass ionization at the depth of the practical range deposited by the bremsstrahlung photons generated in the phantom are derived as a function of the incident electron energy. With these relations and a measurement of the extrapolated photon background at Rp, the treatment head generated bremsstrahlung distribution can be determined. The identification of this photon contamination allows an accurate calculation of the absorbed dose in electron beams with a high bremsstrahlung contamination by accounting for the difference in stopping power ratios between a clean electron beam and the photon contamination. The absorbed dose determined using ionization chambers in heavily photon contaminated (10%) electron beams may be too low--by as much as 1.5%--without correction.     

Limiting values of backscatter factors for low-energy x-ray beams

Models for the calculation of upper and lower limiting values to the backscatter factor (BSF) are presented. The upper limit is obtained from Monte Carlo simulations of infinite parallel beams incident on semi-infinite phantoms with the dose contributions from all orders of photon scatter considered. The lower limits are calculated using an analytical photon transport model which considers only the primary dose and the scatter dose from photons that have undergone single scattering interactions in the phantom. The limiting values can be used to evaluate measured and modelled BSF values for x-ray beams with photons of < or = 150 keV. A parametrization of the limiting values in terms of photon energy and irradiation field size is presented so that results determined for monoenergetic beams can be extended to polyenergetic spectra. The utility of the limits is illustrated by comparisons made with BSFs from the literature.     

A technique for the fast calculation of three-dimensional photon dose distributions using the superposition model

Techniques for reducing computation time in 3D photon dose calculations are addressed with specific emphasis given to the convolution/superposition approach. A single polyenergetic superposition model calculating absorbed dose per incident photon fluence (Gy cm2) was developed in terms of TERMA and a total energy deposition kernel (a total point spread function). A novel approach was devised for reducing calculation time. The method, named the CF method, was based on the use of a conventional, fast model (here a modified power-law method was used) for the generation of 3D dose distributions on a fine dose matrix. Superposition calculations were carried out on a coarse matrix and calculation speed was increased simply by reducing the number of calculations. A set of correction factors was derived on the coarse grid from the ratio of the dose values from superposition to those from the conventional algorithm. These were interpolated onto the fine matrix and used to modify the dose calculation from the conventional algorithm. The method was tested in a worst-case example where large dose gradients were present and in a clinically relevant irradiation geometry. It is shown that the time required for the generation of a 3D matrix with superposition can be reduced by at least a factor of 100 with no significant loss in accuracy.     

Theoretical and experimental determination of phantom scatter factors for photon fields with different radial energy variation

The output factor used for monitor unit determination in radiotherapy can be divided into two factors: the head scatter factor and the phantom scatter factor. Theoretical and experimental phantom scatter factors have been compared for different beam qualities between 4 MV and 50 MV and field sizes from 5 cm x 5 cm to 30 cm x 30 cm. The theoretical data were obtained through a convolution method based on Monte Carlo calculated energy spectra and dose kernels. The calculations have been performed both for accelerators with a rather large energy variation within the field and for accelerators with a constant energy distribution in the field. Deviations between theoretical and experimental data were found to be less than 1%.     

A 10 MV x-ray zero-area tissue-maximum ratio expression constructed by taking into account depth and off-axis beam quality change

A 10 MV x-ray zero-area TMR (tissue-maximum ratio) expression was constructed using an attenuation coefficient expression which is a function of depth (z) and off-axis distance (r). The zero-area TMR expression can take into account depth and off-axis beam quality change. The construction is based on a method of convolving one-dimension forward and backward primary dose-spread functions with the primary water collision kerma. Additional information required to construct the zero-area TMR expression for a given fanline at an off-axis distance of r is the zero-area TMR value at z = 0 [Ds(r)] and the v(r) and w(r) values in the backward primary dose-spread function. It was found that the depth of peak zero-area TMR [zpeak(r)] decreases with increasing r and that for large depths, the zero-area TMR expression changes significantly as a function of r. Obtaining an accurate expression for zero-area TMR is important for accurate construction of the three-dimensional (3D) forward and backward primary dose-spread functions needed for a 3D convolution method.     

A dual source photon beam model used in convolution/superposition dose calculations for clinical megavoltage x-ray beams

A realistic photon beam model based on Monte Carlo simulation of clinical linear accelerators was implemented in a convolution/superposition dose calculation algorithm. A primary and an extra-focal sources were used in this beam model to represent the direct photons from the target and the scattered photons from other head structures, respectively. The effect of the finite size of the extra-focal source was modeled by a convolution of the source fluence distribution with the collimator aperture function. Relative photon output in air (Sc) and in phantom (Scp) were computed using the convolution method with this new photon beam model. Our results showed that in a 10 MV photon beam, the Sc, Sp (phantom scatter factor), and Scp factors increased by 11%, 10%, and 22%, respectively, as the field size changed from 3 x 3 cm2 to 40 x 40 cm2. The variation of the Sc factor was contributed mostly by an increase of the extra-focal radiation with field size. The radiation backscattered into the monitor chamber inside the accelerator head affected the Sc by about 2% in the same field range. The output factors in elongated fields, asymmetric fields, and blocked fields were also investigated in this study. Our results showed that if the effect of the backscattered radiation was taken into account, output factors in these treatment fields can be predicted accurately by our convolution algorithm using the dual source photon beam model.     

Equivalent squares of irregular photon fields

The tables of equivalent fields published by the British Journal of Radiology (BJR) are intended for calculation of depth-dose functions in rectangular photon fields. We have investigated the validity of the equivalent-field concept for fields of arbitrary shape over a range of photon energies, field sizes and depths. We show that the empirical scatter-radius function (Day function) used to generate the equivalent-field tables is a good approximation to the average over energy of normalized scatter-air ratios extracted from BJR beam data for depths up to 10 cm. However, this function tends to diverge from the data as depth increases. Accuracy can be improved by making the Day function depend on depth. Equivalent squares, determined by sector integration of the original or modified Day functions, are suitable for megavoltage photon-beam dose calculations at central-axis and off-axis points in irregular as well as rectangular fields.     

Two replica blotting methods for fast immunological analysis of common proteins in two-dimensional electrophoresis

Knowledge of the photon spectrum of a radiotherapy beam is often needed for three-dimensional (3-D) dose calculations using Monte Carlo methods and/or algorithms employing energy deposition kernels. Direct measurement of the x-ray energy fluence spectrum is not feasible for the high-energy photon beams used clinically. In this paper, the spectrum is extracted from basic beam data that are readily obtained for a clinical beam. We describe the photon spectrum using just two parameters. One parameter, which determines the high-energy part of the spectrum, is obtained using the measured dose in the buildup region for a small field, where electron contamination of the beam can be neglected. The other parameter is extracted from the photon beam attenuation in water. The results compare favorably to spectra generated from Monte Carlo simulations.     

Use of CEA TVS film for measuring high energy photon beam dose distributions

CEA TVS film is a therapy verification film that has been recently introduced in the North American market. This film features linear characteristic curves for photon energies from 137Cs to 18 MV as reported by Cheng and Das [Med. Phys. 23, 1225 (1996)]. In Saskatoon, TVS film was investigated for its application in the measurement of dose distributions with 4 and 18 MV linacs and a 60Co unit. The TVS film jacket has a layer of conductive material that has a minimal effect on the film's response. Film sensitivity generally increases for exposures normal to the incident beam as compared with parallel exposures, but was highly dependent on beam energy and depth of measurement. Fractional depth doses obtained in the parallel orientation agreed well with ion chamber measurements for the linac beams at depths beyond Dmax; ion chamber measurements differed by a maximum of 1.6% and 2.6% for the 4 and 18 MV beams, respectively. In the buildup region, an increase in film response was found when compared to the ion chamber measurements for both linac beams. With the 60Co beam, the TVS film showed an increase in sensitivity with depth as the proportion of scattered soft x rays increases; the maximum difference between ion chamber and film fractional depth doses was 7.8%. The TVS film demonstrates a substantial improvement over Kodak X-Omat V film for measuring depth doses in the parallel orientation, for all beams considered. Generally, the results confirm TVS film as an accurate and practical dosimeter for the measurement of dose distributions in high energy photon beams.     

Direct determination of kerma for a d(48.5) + Be therapy beam

An experimental determination of the neutron kerma ratio between muscle tissue and A-150 plastic was performed at the newly commissioned d(48.5)+ Be therapy facility in Detroit. Low-pressure proportional counters with separate walls made from A-150 plastic, graphite, zirconium oxide and zirconium served to measure ionization yield spectra. The absorbed dose in the wall of each counter was determined and rendered the A-150 and carbon kerma directly, whilst that for oxygen was deduced from differences between the matched metal oxide and metal pair. This enabled the evaluation of an effective kerma ratio as a function of radiation field size and hydrogenous filtration. Although filtration was observed to harden the beam, the application of a single kerma ratio for the various irradiation conditions investigated was found to be appropriate. A neutron kerma ratio of 0.90+/-0.03 was assessed for the Detroit facility, which is lower at the 1sigma level than the 0.95 currently recommended in the dosimetry protocol for high-energy neutron beams.     

Added copper filtration in digital paediatric double-contrast colon examinations: effects on radiation dose and image quality

Paediatric double-contrast barium enema examinations are usually performed at high tube voltage, 102-105 kV. The aim of this study was to investigate how much the effective dose to the child could be reduced by increasing the X-ray energy further by adding copper filter in the beam, and if this dose reduction could be achieved without endangering image quality. Organ doses to an anthropomorphic phantom simulating a 1-year-old child was measured using thermoluminescence dosimetry for assessment of the effective dose and this value was compared with the energy imparted which was obtained from kerma-area product measurements. To verify that the image quality achieved with this added filtration was still diagnostically acceptable, the study included 15 patient examinations. Since the increased X-ray energy will most probably affect low-contrast objects, image quality was also evaluated with two different phantoms containing low-contrast objects. Effective dose for a complete examination can be decreased 44 % and energy imparted 77 % when a 0.3-mm copper filter is inserted in the beam at tube voltage 102 kV. The patient study did not show any significant deterioration of image quality, whereas phantom measurements of contrast-detail resolution and signal-to-noise ratio was marginally impaired by the added copper filtration. This technique is now in clinical practice for paediatric colon examinations.     

Dose-related cardiovascular effects of amrinone and epinephrine in reversing bupivacaine-induced cardiovascular depression

Effective dose, an indicator of the stochastic effect of radiation, has been widely used in dose evaluation in the environment. Though conversion factors have been used to obtain E from the air kerma or air absorbed dose, the variation of the conversion factors due to the change of exposure conditions has not been sufficiently investigated. This report documents an investigation of the variation of the effective dose per air kerma for environmental gamma rays depending on the exposure conditions using anthropomorphic phantoms and Monte Carlo calculations, taking into account the precise angular and energy distributions of the environmental gamma rays incident on the human body. As causes of the variation, posture of human bodies, biases of environmental source distributions, and body size were considered. The variation of effective dose in a prone position compared with that in a standing position was found to be within 30%. The bias of environmental sources causes the effective dose per air kerma to vary by 20% at maximum, but in some cases for low-energy gamma rays the variation was found to be up to 40% due to the change in the energy spectrum. The effective dose for a new born infant was estimated to be higher than that for an adult by a maximum of 80-90% for low-energy gamma rays from anthropogenic sources because of a lower shielding effect of the smaller body. The variation of the effective dose equivalent shows a similar tendency to the effective dose. Consequently, this study made it possible to estimate the uncertainties of effective dose and effective dose equivalent evaluated from air kerma or absorbed dose in air using the standard available conversion factors.     

Flow Depletion of Semipervious Streams Due to Unsteady Pumping Discharge

Analytical expressions for rate and volume of flow depletion of semipervious streams due to sinusoidal variation in pumping rate are obtained. An analytical but approximate method is developed for obtaining the rate and volume of stream flow depletion due to arbitrary unsteady pumping discharge. The method uses the ramp kernel and convolution. The use of ramp kernels permits linear interpolation between two consecutive discretized discharge values. The analytical equations for the ramp kernels for the rate and volume of stream flow depletion are derived. The proposed method is applicable for homogeneous and isotropic aquifers that are hydraulically connected to streams.     

Aquifer Response to Sinusoidal or Arbitrary Stage of Semipervious Stream

Analytical expressions for the aquifer responses, viz., groundwater head, rate of flow and cumulative volume of flow, to a generalized sinusoidal stage of semipervious streams considering the stream boundary resistance, are derived. The analytical aquifer responses to a linear stream stage and to a typical analytical flood wave that was used by Cooper and Rorabaugh, are also derived. For a zero-stream resistance, the aquifer responses converge to those for a fully penetrating stream. Also, two analytical methods, a “ramp kernel method” and a “Fourier series method,” for obtaining the aquifer responses to an arbitrary temporal stage of sempervious stream, are developed. The analytical expressions of the ramp kernels for different aquifer responses are developed. The ramp kernel method is found superior to the conventional convolution that uses numerical integration or pulse kernels for obtaining the convolution integral. In the Fourier series method, the aquifer responses to sinusoidal stage are used along with Fourier series. The results obtained using both methods are in close agreement. The new methods are also applicable to fully penetrating streams by assigning a zero value to the stream resistance.     

No association between c-myc amplification and TP53 mutation in sarcoma tumorigenesis

Accuracy of dose delivery at low monitor unit setting is studied for a dual photon energy linear accelerator. Dose delivered per MU is found to be constant for both the photon beams for MU settings above 30. For lower MUs there is definite deviation from the calibrated value and the error is found to be increasing as fewer MUs are set for dose delivery. This dose/MU ratio at low MU setting is found to be dose-rate dependent, showing an increasing trend with pulse repetition frequency (PRF). Also, the dosimetric ratio is observed to be mode dependent; its value for an 18 MV beam is almost double that observed in the case of a 6 MV beam at very low MU setting. The magnitude of this error should be determined for each energy so that appropriate corrections can be applied if very low MUs are to be used.     

Endothelin in migraine patients

AIM: Improvement of the dose homogeneity in radiation treatment of the intact breast using 3D-planning and dose volume histograms. PATIENTS AND METHOD: 3D-planning, including the calculation of dose volume histograms of the planning target volume, was performed on 15 patients, who underwent radiation therapy with tangential photon beams. A standard plan and 2 modified or optimized plans were evaluated. Different dosimetric parameters like maximum dose, mean dose, standard deviation and the fractional volume which receives doses from 95 to 105% of the reference dose were compared and correlated with breast size. RESULTS: With increasing breast size standard planning leads to increased overdosage, both in magnitude and volume. Individual optimization by modifying weights and wedges gives no improvement in dose homogeneity, whereas a photon energy of 10 MV results in a more homogeneous dose distribution. The drawback of the higher energy is the increased underdosage of the skin. CONCLUSION: Using the standard geometry of tangential fields the dose homogeneity cannot be improved significantly by 3D-planning, compared to our standard technique.     

Satellite-Based Energy Balance to Assess Within-Population Variance of Crop Coefficient Curves

Quantifying evapotranspiration (ET) from agricultural fields is important for field water management, water resources planning, and water regulation. Traditionally, ET from agricultural fields has been estimated by multiplying the weather-based reference ET by crop coefficients (Kc) determined according to the crop type and the crop growth stage. Recent development of satellite remote sensing ET models has enabled us to estimate ET and Kc for large populations of fields. This study evaluated the distribution of Kc over space and time for a large number of individual fields by crop type using ET maps created by a satellite based energy balance (EB) model. Variation of Kc curves was found to be substantially larger than that for the normalized difference vegetation index because of the impacts of random wetting events on Kc, especially during initial and development growth stages. Two traditional Kc curves that are widely used in Idaho for crop management and water rights regulation were compared against the satellite-derived Kc curves. Simple adjustment of the traditional Kc curves by shifting dates for emergence, effective full cover, and termination enabled the traditional curves to better fit Kc curves as determined by the EB model. Applicability of the presented techniques in humid regions having higher chances of cloudy dates was discussed.     

On compensator design for photon beam intensity-modulated conformal therapy

Recently the compensator has been shown to be an in expensive and reliable dose delivery device for photon beam intensity-modulated radiation therapy (IMRT). The goal of IMRT compensator design is to produce an optimized primary fluence profile at the patient's surface obtained from the optimization procedure. In this paper some of the problems associated with IMRT compensator design, specifically the beam perturbations caused by the compensator, are discussed. A simple formula is derived to calculate the optimal compensator thickness profile from an optimized primary fluence profile. The change of characteristics of a 6 MV beam caused by the introduction of cerrobend compensators in the beam is investigated using OMEGA Monte Carlo codes. It is found that the compensator significantly changes the energy spectrum and the mean energy of the primary photons at the patient's surface. However, beam hardening does not have as significant an effect on the percent depth dose as it does on the energy spectrum. We conclude that in most situations the beam hardening effect can be ignored during compensator design and dose calculation. The influence of the compensator on the contaminant electron buildup dose is found to be small and independent of the compensator thickness of interest. Therefore, it can be ignored in the compensator design and included as a correction into the final dose distribution. The scattered photons from the compensator are found to have no effect on the surface dose. These photons produce a uniform low fluence distribution at the patient's surface, which is independent of compensator shape. This is also true for very large fields and extremely asymmetric and nonuniform compensator thickness profiles. Compared to the primary photons, the scattered photons have much larger angular spread and similar energy spectrum at the patient's surface. These characteristics allow the compensator thickness profile and the dose distribution to be calculated from the optimized fluence profile of primary photons, without considering the scattered photons.     

The effect of the nasopharyngeal air cavity on x-ray interface doses

We investigated the impact of air cavities in head and neck cancer patients treated by photon beams based on clinical set-ups. The phantom for investigation was constructed with a cubic air cavity of 4 x 4 x 4 cm3 located at the centre of a 30 x 30 x 16 cm3 solid water slab. The cavity cube was used to resemble an extreme case for the nasal cavity. Apart from measuring the dose profiles and central axis percentage depth dose distribution, the dose values in 0.25 x 0.25 x 0.25 cm3 voxels at regions around the air cavity were obtained by Monte Carlo simulations. A mean dose value was taken over the voxels of interest at each depth for evaluation. Single-field results were added to study parallel opposed field effects. For 10 x 10 cm2 parallel opposed fields at 4, 6 and 8 MV, the mean dose at regions near the lateral interfaces of the cavity cube were decreased by 1 to 2% due to the lack of lateral scatter, while the mean dose near the proximal and distal interfaces was increased by 2 to 4% due to the greater transmission through air. Secondary build-up effects at points immediately beyond the air cavity cube are negligible using field sizes greater than 4 x 4 cm2. For most head and neck treatment, the field sizes are usually 6 x 6 cm2 or greater, and most cavity volumes are smaller than our chosen dimensions. Therefore, the influence of closed air cavities on photon interface doses is not significant in clinical treatment set-ups.