Treatment plan comparison using equivalent uniform biologically effective dose (EUBED).

With the continuing improvement in computer speed, dose distributions can be calculated quickly with confidence. However, the resulting biological effect is known with much less certainty, despite its critical importance when assessing treatment plans. To assess plans accurately, biologically based methods of ranking plans are necessary. Many authors have suggested the use of dose volume histograms with reduction schemes and Niemierko has recently introduced another method based on the cell kill occurring in the tumour. This study presents an investigation into this value and suggests a use in prescribing dose. Equivalent uniform dose (EUD) can obviously be used for assessing treatment plans, although in its current form it is not adequate for assessing normal tissues; however, it can also be used to adjust the prescription dose ensuring all plans deliver the same EUD to the tumour. Once this is performed, plans can more easily be assessed on the effects to the normal tissues. In calculating the EUD another concept is introduced--the equivalent uniform biologically effective dose (EUBED). This value considers the distribution of dose and dose per fraction when comparing plans. Reduced dose per fraction at the edge of the target volume will exacerbate the effect of reduced dose on cell kill. Two methods are suggested for calculating the necessary prescription dose: one using an iterative method and one using the gradient of the EUBED function. A comparison was made for a series of stereotactic cases using different collimator sizes. Interestingly, using this method, although the maximum doses were different, the dose volume histograms (DVHs) for the brainstem were similar in all cases.

Modeling late effects in hypofractionated stereotactic radiotherapy.

PURPOSE: To investigate the effect of increasing fraction size on cell survival in late responding normal tissues. The hypothesis is that total dose can be reduced for constant tumor cell kill and there will be consequent advantage for some surrounding normal tissue cells. Also, the volume of normal tissue that can potentially be damaged by increasing fraction size is minimized by a high degree of dose conformation achievable in stereotactic radiotherapy (SRT). METHODS AND MATERIALS: The linear quadratic (LQ) model has been used to calculate the allowed reduction in total dose with increased fraction size, using tumor alpha/beta ratios of 5 Gy and 10 Gy. Effect on normal tissue is calculated using an alpha/beta ratio of 3 Gy. Maximum dose is normalized to 100% and the effect on normal tissue at different isodose levels assessed. A new quantity, the standard percentage dose, is proposed in order to describe a dose distribution in terms of an isodose distribution for a standard fraction size. Integral biologically effective dose (IBED) in the brainstem is calculated, where the variation with isocenter position and fraction size is considered. RESULTS: The decreasing total dose resulting from increasing the dose per fraction is found to reduce late normal tissue effect for low isodose levels. The threshold isodose level at which there is an advantage corresponds to the ratio of normal tissue to tumor alpha/beta ratios. Brainstem IBED for a higher dose per fraction increases relative to that for a low dose per fraction, when a larger volume of brainstem is covered by high isodose levels. CONCLUSION: Hypofractionation may be biologically sound when a small volume of normal tissue is covered by high isodose levels. There is a calculated advantage in using larger fractions in terms of cell survival at low isodose levels.

Possibilities for tailoring dose distributions through the manipulation of electron beam characteristics.

The influence of the properties of an electron beam on resulting dose distributions, and the potential benefits for dose conformity and optimizing dose distribution characteristics by electron beam manipulation, are theoretically examined. A simulated annealing routine is used to weight electron pencil beams of discrete energies incident at discrete locations and angles on one side of a phantom. The resulting optimal electron phase space provides a dose distribution which most closely approaches a desired distribution on the basis of a physical comparison. For simple desired distributions, intuitive results are obtained such as the benefits of energy modulation for distributing dose with depth, of angular and spatial modulation for overcoming disequilibrium effects and their combination in boosting surface doses. For a complex desired dose distribution, the optimization routine instigates a complex interplay of energy, angular and spatial modulation in attempting to achieve dose conformity. A significant result shows that, for a suitably selected beam energy, angular modulation can compensate for the variation in the depth of the distal edge of a superficial target. The effects of varying just energy for normally incident electrons are compared with those of varying the distribution of incidence angle (for monoenergetic electrons) and the combination of both, indicating the relative merits of the manipulation of available degrees of freedom.

Super-Monte Carlo: a 3-D electron beam dose calculation algorithm.

An electron beam dose calculation algorithm has been developed which is based on a superposition of pregenerated Monte Carlo electron track kernels. Electrons are transported through media of varying density and atomic number using electron tracks produced in water. The perturbation of the electron fluence due to each material encountered by the electrons is explicitly accounted for by considering the effect of (i) varying stopping power, (ii) scattering power, and (iii) radiation yield. For each step of every electron track, these parameters affect the step length, the step direction, and for energy deposited in that step respectively. Dose distributions in both homogeneous water and nonwaterlike phantoms, and heterogeneous phantoms show consistent agreement with "standard" Monte Carlo results. For the same statistical uncertainty in broad beam geometries, this new calculation method uses a factor of 9 less computation time than a full Monte Carlo simulation.

Some characteristics of tumour control probability for heterogeneous tumours.

Some mathematical characteristics of tumour control probability (TCP) have been examined in the light of potential intra-tumour variations in clonogenic cell characteristics. In particular, an explicit expression for the relationship between the dose distribution and the probability of local clonogen eradication is obtained which maximizes the TCP using both general and specific TCP models, for the case of fixed energy deposition into the tumour volume. Characteristics of this expression are considered for the cases of varying clonogen cell density (uniform radiosensitivity) and uniform cell density (uniform radiosensitivity), yielding the 'uniform TCP' and 'uniform dose' results respectively for TCP maximization. These situations are examined graphically to highlight the possible consequences of neglecting intra-tumour heterogeneity in dose prescription.

Modelling clinical accelerator beams: a review.

Radiotherapy dose calculation algorithms currently under development (and some in use) require knowledge of the characteristics of particles in linear accelerator-generated radiation beams. A range of techniques have been developed which allow determination of those characteristics requiring minimal interaction with the physical beam. Such techniques may be analytical in nature, making use of analytical transport results and cross sections, or they may be numerical, employing the increasing utility of Monte Carlo techniques. These techniques provide us with an extensive description of clinical beams and the ability to refine beam production and collimation systems. This review details several published analytical and numerical approaches to megavoltage photon and electron beam modelling, the characteristics that they provide, and their relative accuracy and utility.

A review of electron beam dose calculation algorithms.

The advantage of accurate dose calculation in radiotherapy is the availability of better quality information with which to prescribe treatments, and also increased confidence when optimisation procedures are applied in the planning process. Due to the continual increase in computation speed through improvements in technology, a number of advanced electron beam dose calculation algorithms have recently been developed which incorporate physically rigorous modelling of the scattering and interaction processes involved in electron transport. These algorithms are significantly more accurate than those employed by commercially available radiotherapy planning systems. The advantages and disadvantages of the 3D Pencil beam method, the Pencil Beam Redefinition Method, the Multi-ray model, theoretical perturbative methods, the Phase Space Evolution model, Monte Carlo techniques, the Superposition/Convolution method, the Macro-Monte Carlo algorithm, the Super-Monte Carlo method and the Voxel-based Monte Carlo method are discussed in this review.

The energy and angular characteristics of the applicator scattered component of an electron beam.

Using a previously developed Monte Carlo based model for determining the characteristics of particles scattered from electron beam applicators (cones), the properties of such particles are investigated for the 6 MeV and 12 MeV beams of a Siemens Mevatron KD-2 linear accelerator. Properties examined are energy and angular spectra averaged across the full beam, mean energy and angle and particle number with radial distance from isocentre, and mean energy with angle for particles scattered from each of the three trimmers of the applicator considered. The particles examined are electrons scattered from applicator trimmers, and photons generated in the applicator materials by primary beam electrons. It is found that the properties observed are characteristic of the effective source that each applicator trimmer represents.

Calculating the angular standard deviation of electron beams using Fermi-Eyges theory.

Knowledge of the angular distribution of an electron beam at the applicator face is a necessary parameter in defining a beam when the Hogstrom pencil beam method of dose calculation is used. The angular spread can be found experimentally using penumbra widths measured at various distances from the applicator face. Using knowledge of the geometry and composition of the scattering foils of the linear accelerator, the angular standard deviation was calculated theoretically using Fermi-Eyges theory. The obtained angular spread values agree with experimentally derived values to within experimental error for electron energies from 6 to 21 MeV. The Fermi-Eyges calculation is fast, and can be used as a quick check to validate experimental angular spread values.

Superposition dose calculation incorporating Monte Carlo generated electron track kernels.

The superposition/convolution method and the transport of pregenerated Monte Carlo electron track data have been combined into the Super-Monte Carlo (SMC) method, an accurate 3-D x-ray dose calculation algorithm. The primary dose (dose due to electrons ejected by primary photons) is calculated by transporting pregenerated (in water) Monte Carlo electron tracks from each primary photon interaction site, weighted by the terma for that site. The length of each electron step is scaled by the inverse of the density of the medium at the beginning of the step. Because the density scaling of the electron tracks is performed for each individual transport step, the limitations of the macroscopic scaling of kernels (in the superposition algorithm) are overcome. This time-consuming step-by-step transport is only performed for the primary dose calculation, where current superposition methods are most lacking. The scattered dose (dose due to electrons set in motion by scattered photons) is calculated by superposition. In both a water-lung-water phantom and a two lung-block phantom, SMC dose distributions are more consistent with Monte Carlo generated dose distributions than are superposition dose distributions, especially for small fields and high energies-for an 18-MV, 5 X 5-cm(2) beam, the central axis dose discrepancy from Monte Carlo is reduced from 4.5% using superposition to 1.5% using SMC. The computation time for this technique is approximately 2 h (depending on the simulation history), 20 times slower than superposition, but 15 times faster than a full Monte Carlo simulation (on our platform).

A comparison of dosimetry techniques in stereotactic radiosurgery.

Accurate dosimetry of small-field photon beams used in stereotactic radiosurgery (SRS) can be made difficult because of the presence of lateral electronic disequilibrium and steep dose gradients. In the published literature, data acquisition for radiosurgery is mainly based on diode and film dosimetry, and sometimes on small ionization chamber or thermolominescence dosimetry. These techniques generally do not provide the required precision because of their energy dependence and/or poor resolution. In this work PTW diamond detectors and Monte Carlo (EGS4) techniques have been added to the above tools to measure and calculate SRS treatment planning requirements. The validity of the EGS4 generated data has been confirmed by comparing results to those obtained with an ionization chamber, where the field size is large enough for electronic equilibrium to be established at the central axis. Using EGS4 calculations, the beam characteristics under the experimental conditions have also been quantified. It was shown that diamond detectors are potentially ideal for SRS and yield more accurate results than the above traditional modes of dosimetry.

Accounting for the variation in collision kerma-to-terma ratio in polyenergetic photon beam convolution.

In photon beam convolution, the distribution of energy deposition about a primary photon interaction site due to charged particles set in motion at that site is represented by the primary kernel. Energy deposited due to scattered photons, bremsstrahlung, and annihilation photons is represented by the scatter kernel. As the energy deposited in each kernel voxel is normalized to the energy imparted at the interaction site, it is known as a fractional energy distribution. In terma-based convolution, where kernels are normalized to total energy imparted at the interaction site and are convolved with the terma in the dose calculation process, the sum of fractional energies contained in the primary kernel is equal to the ratio of collision kerma (Kc) to terma (T) corresponding to the energy spectrum used to generate the kernel. Since the ratio of collision kerma to terma increases with depth as the beam hardens, the integral fractional energy in a primary kernel formed for the spectrum at the surface is less than the ratio Kc/T at depth. This causes primary dose to be increasingly underestimated with depth and scatter dose to be increasingly overestimated. Single polyenergetic convolution (using polyenergetic primary and scatter kernels formed using a polyenergetic primary photon spectrum) is thus not as rigorous as if a separate convolution is performed for each energy component. The ratio of true primary dose to single polyenergetic primary dose increases almost linearly with depth and is almost equal to the Kc/T ratio. Primary and scatter dose are calculated correctly if a single polyenergetic convolution is performed in terms of Kc (for primary) and T-Kc (for scatter), where the kernels are weighted sums of monoenergetic kernels normalized to Kc and T-Kc. With this method, it is ensured that total primary energy deposited due to primary photon interactions in a unit mass at a point is equal to Kc at that point.

A model for electron-beam applicator scatter.

Applicators (or cones), used in conjunction with patient specific cutouts in electron-beam radiotherapy, may interact with the primary electron beam to produce a secondary beam component (applicator scatter). This component affects machine output as well as the shape of resulting dose distributions. A model has been developed to simulate this scatter component for applicators consisting of trimming plates of arbitrary shape. This model involves sampling established kernels of scatter from edge elements of appropriate materials, obtained through Monte Carlo simulations. The result of the model is a phase space (position, direction, energy, charge, weighting) of applicator scattered particles which can be incorporated into a further Monte Carlo simulation, or as input into another advanced treatment planning algorithm. This model is evaluated by comparison of measured profiles and applicator scatter component depth dose curves with Monte Carlo simulations using simulated phase-space data as input. Results are very consistent and reveal information on the angular and spatial variation characteristics of this beam component. The results obtained verify the developed model as an accurate predictor of the characteristics of applicator scattered particles.

A Monte Carlo investigation of electron-beam applicator scatter.

An EGS4 Monte Carlo investigation into applicator scatter in clinical electron beams has been undertaken in order to establish the characteristics of electrons incident on the patient surface which have interacted with collimation systems. The applicator scattered component of an electron beam (including that from irregularly shaped cutouts) should be considered when modeling the electron phase space since it represents a component of the beam incident on the patient with widely varying characteristics to those of the primary beam. Scattering off an edge of applicator material is considered in terms of the types and characteristics of incident primary beam, the resulting interactions in the edge, and the fluence and energy characteristics of the emerging particles. Results indicate that the principal component to consider is scattered electrons due to the electron component of the primary beam, and that the fluence and energy characteristics of this component are dependent upon primary beam energy and the configuration of the applicator apertures.

The angular and energy distribution of the primary electron beam.

The angular distribution for electron beams produced by the Siemens KD-2 linear accelerator has been found by simulating electron transport through the scattering foils and air using two methods: Fermi-Eyges multiple Coulomb scattering calculations, and EGS4 Monte Carlo simulations. Fermi-Eyges theory gives solutions where both the angular and spatial fluence distributions are Gaussian, with the angular standard deviation being invariant with off-axis distance. The EGS4 results show slightly non-Gaussian angular and lateral distributions as a result of the use of Moliére theory rather than Fermi-Eyges multiple scattering theory, as well as the simulation of discrete bremsstrahlung and Møller interactions. However, the results from both methods are very similar. The angular standard deviations obtained by these methods agree very closely with those found experimentally. The similar shape of the Monte Carlo and Fermi-Eyges results indicate that a Gaussian approximation to the incident angular distribution will be adequate for use in treatment planning algorithms. Furthermore, the angular standard deviation may be determined using Fermi-Eyges theory as an alternative to experimental methods. Both Monte Carlo simulations, and Fermi-Eyges theory predict that the mean electron angle is proportional to off axis distance for all useful field sizes. For a 15 MeV electron beam, an effective source position of 99 cm and 98 cm from the nominal 100 SSD plane was obtained from Fermi-Eyges and Monte Carlo results respectively for a 15 MeV beam. The effective source position found experimentally for this energy was 98 cm.(ABSTRACT TRUNCATED AT 250 WORDS)

Dose rate dependence of a PTW diamond detector in the dosimetry of a 6 MV photon beam.

The dose rate dependence and current/voltage characteristics of a PTW Riga diamond detector in the dosimetry of a 6 MV photon beam have been investigated. Diamond detectors are radiosensitive resistors whose conductivity (i) varies almost in proportion to dose rate and (ii) is almost independent of bias voltage for a constant dose rate. At the recommended bias of +100 V, and also at +200 V, the detector is operating with incomplete charge collection due to the electron-hole recombination time being shorter that the maximum time for an electron to be collected by the anode. As dose rate is varied by changing FSD or depth (changing dose per pulse), detector current and dose rate are related by the expression i alpha Ddelta where delta is approximately 0.98. This manifests itself in an overestimate in percentage depth-dose at a depth of 30 cm of approximately 1% when compared to ionization chamber results. A similar sublinearity is seen when pulse repetition frequency is varied, indicating that the dependence is an on average rather than an instantaneous dose rate. The dose rate dependence is attributed to the reduction in recombination time as dose rate increases.

Photon beam convolution using polyenergetic energy deposition kernels.

In photon beam convolution calculations where polyenergetic energy deposition kernels (EDKS) are used, the primary photon energy spectrum should be correctly accounted for in Monte Carlo generation of EDKS. This requires the probability of interaction, determined by the linear attenuation coefficient, mu, to be taken into account when primary photon interactions are forced to occur at the EDK origin. The use of primary and scattered EDKS generated with a fixed photon spectrum can give rise to an error in the dose calculation due to neglecting the effects of beam hardening with depth. The proportion of primary photon energy that is transferred to secondary electrons increases with depth of interaction, due to the increase in the ratio mu ab/mu as the beam hardens. Convolution depth-dose curves calculated using polyenergetic EDKS generated for the primary photon spectra which exist at depths of 0, 20 and 40 cm in water, show a fall-off which is too steep when compared with EGS4 Monte Carlo results. A beam hardening correction factor applied to primary and scattered 0 cm EDKS, based on the ratio of kerma to terma at each depth, gives primary, scattered and total dose in good agreement with Monte Carlo results.

Radiotherapy X-ray beam inhomogeneity corrections: the problem of lateral electronic disequilibrium in lung.

Accurate dose calculations in lung are important to assess lung and tumour dose in various radiotherapy cancer patients. Those patients of particular relevance are Ca lung and Ca Oesophagus patients because large volumes of lung are irradiated to high doses. In this paper, dosimetry results for megavoltage X-ray beams obtained in a lung phantom are compared with dose computations produced by (1) effective path length, (2) equivalent tissue-air ratio, (3) super-position/convolution and (4) Monte Carlo dose calculation methods. The mid-lung dose error at 10 MV for a 5 x 5 cm field is 10.0%, 6.7%, 1.9% and 0.6% respectively. Tests at the lower energy of 6 MV with a field size of 10 x 10 cm show a mid-lung error of only 2.0% for the equivalent tissue air ratio method. At this energy it appears that central axis dose voids are sufficiently small to enable the routine use of the equivalent tissue air ratio method. At the higher energies tested, 10 and 18 MV, this method is accurate. Superposition and Monte Carlo methods are presented which show good agreement with experimental results in a lung phantom even in regions of lateral electron disequilibrium.

The effect of density on the 10MV photon beam penumbra.

An investigation into the density dependence of the penumbra of the Varian Clinac 18/10 10MV photon beam has been carried out. A water/lung phantom was constructed of polystyrene (r = 1.04 g cm-3) and cork (r = 0.23 g cm-3), in which interfaces exist both parallel and perpendicular to the beam axis. The irradiation of the phantom was also simulated using the EGS4 Monte Carlo system with a cartesian voxel geometry. Experimental (TLD) and Monte Carlo dose profiles are in close agreement, and show a large degree of penumbral broadening in the lung region. This broadening is due primarily to lateral electronic disequilibrium occurring at a larger distance from the geometric beam edge in lung than in water. This disequilibrium can also cause the dose in lung to drop below the dose in water at the same depth and off axis distance, even though the radiological depth is less in lung. Monte Carlo simulations were also performed where the dose is separated into primary and scattered components, for homogeneous media of densities 0.25, 0.50, 0.75 and 1.00 g cm-3. The penumbral width of the primary dose profile was found to be almost constant with depth for a point source of photons (after the initial build-up region), where the lateral distances from the 95-50% and 50-5% dose levels on the dose profile (normalised to the dose at the central axis) are equal in all cases. Also, primary penumbra width was found to be almost inversely proportional to density. The primary penumbra for a unit density material can be fitted accurately by an exponential forming function with empirical determined coefficients. The penumbral shape for the lower densities can then be closely fitted by scaling the coefficients in proportion to density. This scaling method has application in treatment planning, where the predicted primary penumbra shape should take account of inhomogeneities, and is particularly important in matching adjacent fields. When the scattered dose component is added to give the total dose, penumbral width increases because the scattered dose penumbra is wider than that of the primary dose. Also, the inverse proportionality of the penumbra width with density does not hold for the scattered dose. The relative contribution of the scattered dose increases with density. Therefore, the inverse proportionality of penumbra width with density does not hold for the total dose.

Electron contamination in 4 MV and 10 MV radiotherapy x-ray beams.

A thin window parallel-plate ionization chamber was constructed for dose measurement in the build-up region of high energy radiotherapy photon beams. The chamber is an integral part of a perspex block. The entrance window is 12 microns Melinex foil with a thin aluminium surface. Cavity thickness is 1.45 mm. Surface doses for varying field sizes were found to increase almost linearly with the side length of a square field. The surface dose for a 10x10 cm 4 MV photon beam is 12.1% for an open field and this increases to 14.1% with a polycarbonate block tray in the beam. Similarly for a 10 MV photon beam the surface dose is 10.6% for an open field and this increases to 12.4% with a polycarbonate block tray. The difference between the dose for an open field and a field with a polycarbonate block tray inserted becomes more significant for larger field sizes. Electron contamination depth dose curves are determined for a 4 MV and 10 MV photon beam. This is achieved by subtracting a pure photon beam build-up curve generated by an EGS4 Monte Carlo simulation from the experimental build-up curve. The EGS4 curve is a theoretical, electron contamination free curve. The electron contamination curve (of the 10 MV photon beam) has depth dose characteristics similar to that of a broad low energy electron beam.