Computation of mean and variance of the radiotherapy dose for PCA-modeled random shape and position variations of the target.

Radiotherapy dose delivery in the tumor and surrounding healthy tissues is affected by movements and deformations of the corresponding organs between fractions. The random variations may be characterized by non-rigid, anisotropic principal component analysis (PCA) modes. In this article new dynamic dose deposition matrices, based on established PCA modes, are introduced as a tool to evaluate the mean and the variance of the dose at each target point resulting from any given set of fluence profiles. The method is tested for a simple cubic geometry and for a prostate case. The movements spread out the distributions of the mean dose and cause the variance of the dose to be highest near the edges of the beams. The non-rigidity and anisotropy of the movements are reflected in both quantities. The dynamic dose deposition matrices facilitate the inclusion of the mean and the variance of the dose in the existing fluence-profile optimizer for radiotherapy planning, to ensure robust plans with respect to the movements.

A population-based model to describe geometrical uncertainties in radiotherapy: applied to prostate cases.

Local motions and deformations of organs between treatment fractions introduce geometrical uncertainties into radiotherapy. These uncertainties are generally taken into account in the treatment planning by enlarging the radiation target by a margin around the clinical target volume. However, a practical method to fully include these uncertainties is still lacking. This paper proposes a model based on the principal component analysis to describe the patient-specific local probability distributions of voxel motions so that the average values and variances of the dose distribution can be calculated and fully used later in inverse treatment planning. As usually only a very limited number of data for new patients is available; in this paper the analysis is extended to use population data. A basic assumption (which is justified retrospectively in this paper) is that general movements and deformations of a specific organ are similar despite variations in the shapes of the organ over the population. A proof of principle of the method for deformations of the prostate and the seminal vesicles is presented.

Conversion of measured percentage depth dose to tissue maximum ratio values in stereotactic radiotherapy.

For many treatment planning systems tissue maximum ratios (TMR) are required as input. These tissue maximum ratios can be measured with a 3D computer-controlled water phantom; however, a TMR measurement option is not always available on such a system. Alternatively TMR values can be measured 'manually' by lowering the detector and raising the water phantom with the same distance, but this makes TMR measurements time consuming. Therefore we have derived TMR values from percentage depth dose (PDD) curves. Existing conversion methods express TMR values in terms of PDD, phantom scatter factor (Sp), and inverse square law. For stereotactic treatments circular fields ranging from 5-50 mm (19 cones) are used with the treatment planning system XKnife (Radionics). The calculation of TMR curves for this range is not possible with existing methods. This is because PDD curves of field sizes smaller than 5 mm (smallest cone size) are needed, but these cones are not provided. Besides, for field sizes smaller than 40 mm, the phantom scatter factor is difficult to determine and will introduce significant errors. To overcome these uncertainties, an alternative method has been developed to obtain TMR values from PDD data, where absolute doses are expressed in terms of PDD, total scatter factor and inverse square law. For each depth, the dose as a function of field size is fitted to a double exponential function. Then the TMR is calculated by taking the ratio of this function at the depth of interest and the reference depth, for the correct field size. For all 19 cones the total scatter factor and PDDs have been measured with a shielded diode in water for a 6 MV photon beam. Calculated TMR curves are compared with TMR values measured with a diode. The agreement is within 2%. Therefore this relatively simple conversion method meets the required accuracy for daily dose calculation in stereotactic radiotherapy. In principle this method could also be applied for other small field sizes such as those formed with a mini multileaf collimator.

A model to determine the initial phase space of a clinical electron beam from measured beam data.

Advanced electron beam dose calculation models for radiation oncology require as input an initial phase space (IPS) that describes a clinical electron beam. The IPS is a distribution in position, energy and direction of electrons and photons in a plane in front of the patient. A method is presented to derive the IPS of a clinical electron beam from a limited set of measured beam data. The electron beam is modelled by a sum of four beam components: a main diverging beam, applicator edge scatter, applicator transmission and a second diverging beam. The two diverging beam components are described by weighted sums of monoenergetic diverging electron and photon beams. The weight factors of these monoenergetic beams are determined by the method of simulated annealing such that a best fit is obtained with depth-dose curves measured for several field sizes at two source-surface distances. The resulting IPSs are applied by the phase-space evolution electron beam dose calculation model to calculate absolute 3D dose distributions. The accuracy of the calculated results is in general within 1.5% or 1.5 mm; worst cases show differences of up to 3% or 3 mm. The method presented here to describe clinical electron beams yields accurate results, requires only a limited set of measurements and might be considered as an alternative to the use of Monte Carlo methods to generate full initial phase spaces.

Optimization of multileaf collimator settings for radiotherapy treatment planning.

Multileaf collimators have become available in many radiotherapy treatment centres. The cross section of a beam can be shaped to a projection of the target area by moving the leaves of the multileaf collimator into the beam. In this paper, a method is described to optimize the positions of the individual leaves automatically, once the beam directions and weights have been chosen. The individual positions of the leaves are optimized using the variable metric method. Changes in dose resulting from small leaf movements are computed efficiently using a special method. The optimization method was tested on a treatment plan for a phantom patient. It was found that the unnecessary edges of the beams were trimmed efficiently.

Constrained treatment planning using sequential beam selection.

In this paper an algorithm is described for automated treatment plan generation. The algorithm aims at delivery of the prescribed dose to the target volume without violation of constraints for target, organs at risk and the surrounding normal tissue. Pre-calculated dose distributions for all candidate orientations are used as input. Treatment beams are selected in a sequential way. A score function designed for beam selection is used for the simultaneous selection of beam orientations and weights. In order to determine the optimum choice for the orientation and the corresponding weight of each new beam, the score function is first redefined to account for the dose distribution of the previously selected beams. Addition of more beams to the plan is stopped when the target dose is reached or when no additional dose can be delivered without violating a constraint. In the latter case the score function is modified by importance factor changes to enforce better sparing of the organ with the limiting constraint and the algorithm is run again.

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.

Numerical calculation of energy deposition by high-energy electron beams: III-B. Improvements to the 6D phase space evolution model.

The phase space evolution model of Huizenga and Storchi, Morawska-Kaczynska and Huizenga and Janssen et al has been modified to (i) allow application on currently available computer equipment with limited memory (128 Megabytes) and (ii) allow 3D dose calculations based on 3D computer tomographic patient data. This is a further development aimed at the use of the phase space evolution model in radiotherapy electrons beam treatment planning. The first modification regards the application of depth evolution of the phase space state combined with an alternative method to transport back-scattered electrons. This depth evolution method requires of the order of 15 times less computer memory than the energy evolution method. Results of previous and new electron transport methods are compared and show that the new electron transport method for back-scattered electrons hardly affects the accuracy of the calculated dose distributions. The second modification regards the simulation of electron transport through tissues with varying densities by applying distributed electron transport through similarly composed media with a limited number of fixed densities. Results of non-distributed and distributed electron transport are compared and show that the distributed electron transport method hardly affects the accuracy of the calculated dose distributions. It is also shown that the results of the new dose distribution calculations are still in good agreement with and require significantly less computation time than results obtained with the EGS4 Monte Carlo method.

Automatic calculation of three-dimensional margins around treatment volumes in radiotherapy planning.

Following the publication of ICRU Report 50, the concepts of GTV (gross tumour volume). CTV (clinical target volume) and PTV (planning target volume) are being used in radiotherapy planning with increasing frequency. In 3D planning, the GTV (or CTV) is normally outlined by the clinician in CT or MRI slices. The PTV is determined by adding margins to these volumes. Since manual drawing of an accurate 3D margin in a set of 2D slices is extremely time consuming, software has been developed to automate this step in the planning. The target volume is represented in a 3D matrix grid with voxel values one inside and zero outside the target volume. It is expanded by centering an ellipsoid at every matrix element within the volume. The shape of the ellipsoid reflects the size of the margins in the three main orthogonal directions. Finally, the PTV contours are determined from the 50% iso-value lines of the expanded volume. The software tool has been in clinical use since the end of 1994 and has mostly been applied to the planning of prostate irradiations. The accuracy is better than can be achieved manually and the workload has been reduced considerably (from 4 h manually to approximately 1 min automatically).

Phase space evolution distribution functions for high energy electron beams.

The phase space evolution (PSE) model is a 3D electron beam dose calculation model for radiation oncology. The PSE model is based upon the transport of electrons with a specific energy and direction over short distances (typically 0.3-1 cm). The result of the transport of these electrons is described by an energy and direction distribution of the electrons, which is stored in a database. The database is used by the PSE model at the time of the actual electron transport simulation. A good agreement between dose distributions calculated by the PSE model and EGS4 Monte Carlo code for mono-energetic, mono-directional electron beams was found. The differences in point dose are within 1-2% of the maximum dose. These differences can be caused by errors in the database used, or by assumptions made in the PSE model. The aim of this paper is to get more insight into the possible errors introduced by the database. Results show that the data in the database are in good agreement with EGS4 calculated data. Also the influence of the database on a PSE calculated dose distribution has been investigated. The differences between a PSE calculated dose distribution and an EGS4 calculated dose distribution can be reduced to < 0.5% if the database is replaced by a database partly created by EGS4. This shows that small errors in the database have a distinct effect on the dose distribution, and that this dose distribution can be calculated accurately by the PSE model if the right database is used.

Numerical calculation of energy deposition by high-energy electron beams: III. Three-dimensional heterogeneous media.

The phase space time evolution model of Huizenga and Storchi and Morawska-Kaczynska and Huizenga has been modified to accommodate calculations of energy deposition by arbitrary electron beams in three-dimensional heterogeneous media. This is a further development aimed at the use of the phase space evolution model in radiotherapy treatment planning. The model presented uses an improved method to control the evolution of the phase space state. This new method results in a faster algorithm, and requires less computer memory. An extra advantage of this method is that it allows the pre-calculation of information, further reducing calculation times. Typical results obtainable with this model are illustrated with the cases of (i) a 20 MeV pencil beam in a water phantom, (ii) a 20 MeV 5 x 5 cm2 beam in a water phantom containing two air cavities, and (iii) a 20 MeV 5 x 5 cm2 beam in a water phantom containing an aluminium region. The results of the dose distribution calculations are in good agreement with and require significantly less computation time than results obtained with Monte Carlo methods.

Numerical calculation of energy deposition by broad high-energy electron beams.

The feasibility is demonstrated of a numerical method to calculate dose deposition by broad high-energy electron beams in homogeneous matter or in heterogeneous matter in which the heterogeneities are arranged in slabs perpendicular to the beam axis. The method is based only on the basic physical interaction processes of high-energy electrons and matter. The method is an extended version of the phase space time evolution method as described by Cordaro and Zucker (1971). The calculated depth-dose curves, energy spectra and angular distributions agree very well with results of the extensive class II Monte Carlo calculations of Andreo and Brahme (1984) and Andreo (1985), but require much less computer time: typically 3 minutes on a VAX 785 with floating point accelerator. This demonstrates the power of a numerical method in comparison with Monte Carlo methods.

Optical diffusion in layered media.

In highly scattering media, light energy fluence rate distributions can be described by diffusion theory. Boundary conditions, appropriate to the diffusion approximation, are derived for surfaces where reflection of diffuse light occurs. Both outer surfaces and interfaces separating media with different indices of refraction can be treated. The diffusion equation together with its boundary conditions is solved using the finite element method. This numerical method allows much freedom of geometry.

The in-air scattering of clinical electron beams as produced by accelerators with scanning beams and diaphragm collimators.

The electron distribution F(x, y, z, theta x, theta y) in air has been evaluated for a clinical electron beam emanating from a scanning beam accelerator in which the collimation of the beam is performed by means of diaphragm collimators. The multiple scattering theory of Fermi turns out to be adequate in describing this electron distribution. In this theory, the only parameter to be determined experimentally is the angular variance at the level of the collimator blocks. Generally, this angular variance features the same energy dependence as the angular scattering power and its value at an arbitrary energy can be derived from measuring the penumbra widths of off-axis profiles in air, at various distances beyond the collimator blocks. Then, the angular variance at the level of a secondary diaphragm collimator can be calculated, as well as off-axis profiles in air at arbitrary distances. In this way, the relative electron distribution at the surface of patients can be calculated easily. This in turn serves adequately as input to the calculation of patient dose distributions in radiation therapy planning.

The use of computed tomography numbers in dose calculations for radiation therapy.

Although corrections for 'beam hardening' and 'scattering' have been implemented in currently available CT scanners, systematic differences exist between a real CT image and an ideal, artefact-free and monochromatic image. The appearance and magnitude of these differences are discussed. Conversion to the ideal image, i.e. conversion from CT number to X-ray attenuation coefficient at diagnostic photon energies, turns out to be possible with an accuracy of 5 per cent. In order to use the CT 'density' information from patients, in clinical photon and electron beam dose calculations, conversions must be made from the X-ray attenuation coefficient at diagnostic energies to relevant high energy radiation interaction properties. These conversions turn out to be possible within an accuracy also of 5 per cent. These limited accuracies cause errors in the photon beam dose calculation of less than 1 per cent of the dose maximum and errors in electron beam dose calculations of less than 2 per cent of the dose maximum.

Evaluation of Cartesian coordinates and radiation doses in points determined with stereo x-ray techniques.

A Fortran computer program STEV (stereo evaluation) is described. The principles of the stereo techniques together with the calculation method of the stereo coordinates are given briefly. The determination of the rectangular coordinates from mean stereo coordinates is described. Radiation doses in anatomical points, during intracavitary and interstitial radiation therapy, are calculated, taking into account a statistical evaluation of the measurement errors.