Modelling pencil-beam divergence with the electron pencil-beam redefinition algorithm.

The electron pencil-beam redefinition algorithm (PBRA), which is used to calculate electron beam dose distributions, assumes that the virtual source of each pencil beam is identical to that of the broad beam incident on the patient. In the present work, a virtual source specific for each pencil beam is modelled by including the source distance as a pencil-beam parameter to be redefined with depth. To incorporate a variable pencil-beam source distance parameter, the transport equation was reformulated to explicitly model divergence resulting in the algorithm divPBRA. Allowing the virtual source position to vary with individual pencil beams is expected to better model the effects of heterogeneities on local electron fluence divergence (or convergence). Selected experiments from a measured data set developed at The University of Texas M D Anderson Cancer Center were used to evaluate the accuracy of the dose calculated using divPBRA. Results of the calculation showed that the theory accurately predicted the virtual source position in regions of side-scatter equilibrium and predicted reasonable virtual source positions in regions lacking side-scatter equilibrium (i.e. penumbra and in the vicinity and shadow of internal heterogeneities). Results of the evaluation showed the dose accuracy of divPBRA to be marginally better to that of PBRA, except in regions of extremely sharp dose perturbations, where the divPBRA calculations were significantly greater than the measured data. Dose calculations using divPBRA took 45% longer than those using PBRA. Therefore, we concluded that divPBRA offers no significant advantage over PBRA for the purposes of clinical treatment planning. However, the results were promising and divPBRA might prove useful if further modelling were to include large-angle scattering, low-energy delta rays and brehmsstrahlung.     

A two-dimensional pencil-beam algorithm for calculation of arc electron dose distributions

A two-dimensional pencil-beam algorithm is presented for the calculation of arc electron dose distributions in any plane that is perpendicular to the axis of rotation. The dose distributions are calculated by modelling the arced beam as a single broad beam defined by the irradiated surface of the patient. The algorithm is two-dimensional in that the anatomical cross section of the patient and the skin collimators are assumed identical in parallel planes outside the plane of calculation. The broad beam is modelled as a collection of strip beams, each strip beam being characterised by its planar fluence, mean projected angular direction and a root-mean-square spread about the mean direction. Using these parameters, the dose distribution is calculated using pencil-beam theory. Examples of strip-beam parameters and resulting dose distributions for patient geometries are presented. Features of the algorithm, which include (1) incorporation of pencil-beam theory for the calculation of dose in heterogeneous tissue, (2) run times of only about twice that of comparable-sized fixed electron fields and (3) the input requirement of only a single depth dose and four off-axis dose profiles of measured data, make the algorithm practical for clinical use.     

Effect of dimensionality of heterogeneity corrections on the implementation of a three-dimensional electron pencil-beam algorithm

Electron beam dose distributions were calculated on a three-dimensional grid using three pencil-beam algorithms, each taking into account irregularities in field shape. The algorithms differ in that patient anatomy in either one, two, or three dimensions is used in the calculation of dose to a point. Algorithms were optimized for speed by such techniques as precalculation and storage of several quantities, reordering of pencil-beam and grid-point loops, selection of cut-off values for some calculated quantities, and invoking error function symmetries. Execution times for optimized versions of each of the algorithms as implemented on a three-dimensional treatment planning system were comparable for both the one- and two-dimensional heterogeneity correction requires an additional calculational loop over fan lines. Execution times for the three-dimensional heterogeneity correction were approximately a factor of four longer than those for the two-dimensional correction. For certain geometries, three-dimensional heterogeneity corrections were necessary to calculate dose distributions accurately, in spite of the additional cost in calculation times.     

The use of an extra-focal electron source to model collimator-scattered electrons using the pencil-beam redefinition algorithm

Currently, the pencil-beam redefinition algorithm (PBRA) utilizes a single electron source to model clinical electron beams. In the single-source model, the electrons appear to originate from a virtual source located near the scattering foils. Although this approach may be acceptable for most treatment machines, previous studies have shown dose differences as high as 8% relative to the given dose for small fields for some machines such as the Varian Clinac 1800. In such machines collimation-scattered electrons originating from the photon jaws and the applicator give rise to extra-focal electron sources. In this study, we examined the impact of modeling an additional electron source to better account for the collimator-scattered electrons. The desired dose calculation accuracy in water throughout the dose distribution is 3% or better relative to the given dose. We present here a methodology for determining the electron-source parameters for the dual-source model using a minimal set of data, that is, two central-axis depth-dose curves and two off-axis profiles. A Varian Clinac 1800 accelerator was modeled for beam energies of 20 and 9 MeV and applicator sizes of 15 x 15 and 6 x 6 cm2. The improvement in the accuracy of PBRA-calculated dose, evaluated using measured two-dimensional dose distributions in water, was characterized using the figure of merit, FA3%, which represents the fractional area containing dose differences greater than 3%. For the 15 x 15 cm2 field the evaluation was restricted to the penumbral region, and for the 6 x 6 cm2 field the central region of the beam was included as it was impacted by the penumbra. The greatest improvement in dose accuracy was for the 6 x 6 cm2 applicator. At 9 MeV, FA3% decreased from 15% to 0% at 100 cm SSD and from 34% to 4% at 110 cm SSD. At 20 MeV, FA3% decreased from 17% to 2% at 100 cm SSD and from 41% to 10% at 110 cm SSD. In the penumbra of the 15 x 15 cm2 applicator, the improvement was less, but still significant. At 9 MeV, FA3% changed from 11% to 1% at 100 cm SSD and from 10% to 12% at 110 cm SSD. At 20 MeV, FA3% decreased from 12% to 8% at 100 cm SSD and from 14% to 5% at 110 cm SSD. Results demonstrate that use of a dual-source beam model can provide significantly improved accuracy in the PBRA-calculated dose distribution that was not achievable with a single-source beam model when modeling the Varian Clinac 1800 electron beams. Time of PBRA dose calculation was approximately doubled; however, dual-source beam modeling of newer accelerators (e.g., the Varian Clinac 2100) may not be necessary because of less impact of collimator-scattered electrons on dosimetry.     

Dosimetric evaluation of a two-dimensional, arc electron, pencil-beam algorithm in water and PMMA.

The accuracy of dose calculations from a pencil-beam algorithm developed specifically for arc electron beam therapy was evaluated at 10 and 15 MeV. Mid-arc depth-doses were measured for 0 degrees and 90 degrees arcs using 12 and 15 cm radius cylindrical water phantoms. Calculated depth-doses for the 90 degrees arced beams in the build-up region were as much as 3% less than measured values; the maximum dose was similar in magnitude but at a greater depth; and the therapeutic depth, R80, was 2-4 mm deeper. Calculated values of output (dose per monitor unit) at the depth of the maximum calculated dose were compared with measured values; for arcs ranging from 0-90 degrees, 12 and 15 cm radius water phantoms, and collimator widths of 4, 5 and 6 cm, results showed differences as great as 7%. Isodose countours for a 90 degrees arc were also measured in a 15 cm radius PMMA phantom. At the depth of maximum dose the algorithm predicted doses in the penumbral regions, both with and without collimation, which agreed within a few per cent of measured values. The largest discrepancies were 5%, which occurred in the penumbral portion of the depth-dose fall-off region. Differences between measurement and calculation are not believed to be clinically significant and are believed to be primarily due to the fact that the algorithm models neither large-angle scattering nor the effects of range straggling on the pencil-beam dose distribution.     

Electron pencil-beam redefinition algorithm dose calculations in the presence of heterogeneities.

The electron pencil-beam redefinition algorithm (PBRA) is currently being refined and evaluated for clinical use. The purpose of this work was to evaluate the accuracy of PBRA-calculated dose in the presence of heterogeneities and to benchmark PBRA dose accuracy for future improvements to the algorithm. The PBRA was evaluated using a measured electron beam dose algorithm verification data set developed at The University of Texas M. D. Anderson Cancer Center. The data set consists of measurements made using 9 and 20 MeV beams in a water phantom with air gaps, internal air and bone heterogeneities, and irregular surfaces. Refinements to the PBRA have enhanced the speed of the dose calculations by a factor of approximately 7 compared to speeds previously reported in published data; a 20 MeV, 15 x 15 cm2 field electron-beam dose distribution took approximately 10 minutes to calculate. The PBRA showed better than 4% accuracy in most experiments. However, experiments involving the low-energy (9 MeV) electron beam and irregular surfaces showed dose differences as great as 22%, in albeit a small fractional region. The geometries used in this study, particularly those in the irregular surface experiments, were extreme in the sense that they are not seen clinically. A more appropriate clinical evaluation in the future will involve comparisons to Monte Carlo generated patient dose distributions using actual computed tomography scan data. The present data also serve as a benchmark against which future enhancements to the PBRA can be evaluated.     

Modeling of beam customization devices in the pencil-beam splitting algorithm for heavy charged particle radiotherapy

A broad-beam-delivery system for radiotherapy with protons or ions often employs multiple collimators and a range-compensating filter, which offer complex and potentially useful beam customization. It is however difficult for conventional pencil-beam algorithms to deal with fine structures of these devices due to beam-size growth during transport. This study aims to avoid the difficulty with a novel computational model. The pencil beams are initially defined at the range-compensating filter with angular-acceptance correction for upstream collimation followed by stopping and scattering. They are individually transported with possible splitting near the aperture edge of a downstream collimator to form a sharp field edge. The dose distribution for a carbon-ion beam was calculated and compared with existing experimental data. The penumbra sizes of various collimator edges agreed between them to a submillimeter level. This beam-customization model will be used in the greater framework of the pencil-beam splitting algorithm for accurate and efficient patient dose calculation.     

Golden beam data for proton pencil-beam scanning

Proton, as well as other ion, beams applied by electro-magnetic deflection in pencil-beam scanning (PBS) are minimally perturbed and thus can be quantified a priori by their fundamental interactions in a medium. This a priori quantification permits an optimal reduction of characterizing measurements on a particular PBS delivery system. The combination of a priori quantification and measurements will then suffice to fully describe the physical interactions necessary for treatment planning purposes. We consider, for proton beams, these interactions and derive a 'Golden' beam data set. The Golden beam data set quantifies the pristine Bragg peak depth-dose distribution in terms of primary, multiple Coulomb scatter, and secondary, nuclear scatter, components. The set reduces the required measurements on a PBS delivery system to the measurement of energy spread and initial phase space as a function of energy. The depth doses are described in absolute units of Gy(RBE) mm2 Gp?1, where Gp equals 10? (giga) protons, thus providing a direct mapping from treatment planning parameters to integrated beam current. We used these Golden beam data on our PBS delivery systems and demonstrated that they yield absolute dosimetry well within clinical tolerance.     

A three-dimensional reconstruction algorithm for an inverse-geometry volumetric CT system

An inverse-geometry volumetric computed tomography (IGCT) system has been proposed capable of rapidly acquiring sufficient data to reconstruct a thick volume in one circular scan. The system uses a large-area scanned source opposite a smaller detector. The source and detector have the same extent in the axial, or slice, direction, thus providing sufficient volumetric sampling and avoiding cone-beam artifacts. This paper describes a reconstruction algorithm for the IGCT system. The algorithm first rebins the acquired data into two-dimensional (2D) parallel-ray projections at multiple tilt and azimuthal angles, followed by a 3D filtered backprojection. The rebinning step is performed by gridding the data onto a Cartesian grid in a 4D projection space. We present a new method for correcting the gridding error caused by the finite and asymmetric sampling in the neighborhood of each output grid point in the projection space. The reconstruction algorithm was implemented and tested on simulated IGCT data. Results show that the gridding correction reduces the gridding errors to below one Hounsfield unit. With this correction, the reconstruction algorithm does not introduce significant artifacts or blurring when compared to images reconstructed from simulated 2D parallel-ray projections. We also present an investigation of the noise behavior of the method which verifies that the proposed reconstruction algorithm utilizes cross-plane rays as efficiently as in-plane rays and can provide noise comparable to an in-plane parallel-ray geometry for the same number of photons. Simulations of a resolution test pattern and the modulation transfer function demonstrate that the IGCT system, using the proposed algorithm, is capable of 0.4 mm isotropic resolution. The successful implementation of the reconstruction algorithm is an important step in establishing feasibility of the IGCT system.     

An iterative three-dimensional electron density imaging algorithm using uncollimated compton scattered x rays from a polyenergetic primary pencil beam

X-ray film-screen mammography is currently the gold standard for detecting breast cancer. However, one disadvantage is that it projects a three-dimensional (3D) object onto a two-dimensional (2D) image, reducing contrast between small lesions and layers of normal tissue. Another limitation is its reduced sensitivity in women with mammographically dense breasts. Computed tomography (CT) produces a 3D image yet has had a limited role in mammography due to its relatively high dose, low resolution, and low contrast. As a first step towards implementing quantitative 3D mammography, which may improve the ability to detect and specify breast tumors, we have developed an analytical technique that can use Compton scatter to obtain 3D information of an object from a single projection. Imaging material with a pencil beam of polychromatic x rays produces a characteristic scattered photon spectrum at each point on the detector plane. A comparable distribution may be calculated using a known incident x-ray energy spectrum, beam shape, and an initial estimate of the object's 3D mass attenuation and electron density. Our iterative minimization algorithm changes the initially arbitrary electron density voxel matrix to reduce regular differences between the analytically predicted and experimentally measured spectra at each point on the detector plane. The simulated electron density converges to that of the object as the differences are minimized. The reconstruction algorithm has been validated using simulated data produced by the EGSnrc Monte Carlo code system. We applied the imaging algorithm to a cylindrically symmetric breast tissue phantom containing multiple inhomogeneities. A preliminary ROC analysis scores greater than 0.96, which indicate that under the described simplifying conditions, this approach shows promise in identifying and localizing inhomogeneities which simulate 0.5 mm calcifications with an image voxel resolution of 0.25 cm and at a dose comparable to mammography.     

A second-order finite element algorithm for solving the three-dimensional EEG forward problem

A finite element algorithm has been developed to solve the electroencephalogram (EEG) forward problem. A new computationally efficient approach to calculate the stiffness matrix of second-order tetrahedral elements has been developed for second-order tetrahedral finite element models. The present algorithm has been evaluated by means of computer simulations, by comparing with analytic solutions in a multi-spheres concentric head model. The developed finite element method (FEM) algorithm has also been applied to address questions of interest in the EEG forward problem. The present simulation study indicates that the second-order FEM provides substantially enhanced numerical accuracy and computational efficiency, as compared with the first-order FEM for comparable numbers of tetrahedral elements. The anisotropic conductivity distribution of the head tissue can be taken into account in the present FEM algorithm. The effects of dipole eccentricity, size of finite elements and local mesh refinement on solution accuracy are also addressed in the present simulation study.     

A novel method of three-dimensional localization based on a neural network algorithm

In order to track the three-dimensional position of a non-invasive detecting microcapsule in the alimentary tract, a novel localization method has been developed. Three coils were respectively energized by square wave signals to generate an electromagnetic field. A 3-axis magnetic sensor was sealed in the microcapsule to measure the electromagnetic field strength. Based on the principle of magnetic dipole a corresponding localization model was established. By resolving the localization equations by means of the neural network algorithm, the spatial position of the microcapsule can be obtained. The experiment shows that the localization principle is correct and has high precision. Compared with other methods, this novel method needs no special equipment and after integration the system can be made into a portable type.     

基于分水岭算法的交互式三维分割方法

背景：在临床中准确对人体组织进行三维分割能提高临床诊断的准确性,但传统的分水岭算法存在过度分割问题,难以实现人体组织的三维分割。 目的：为准确三维分割人体组织,减少图像中伪极小值点对图像分割的影响,提出了一种基于控制标记符分水岭的交互式三维分割方法。 方法：提取CT序列图像的内部和外部标记符,以此修正梯度图像并进行分割;在此基础上,根据序列图像上下层的相似性,利用人机交互进行组织结构的三维分割。首先在第一张序列图像上手工选取感兴趣区域上的一个点,借助同一组织在连续CT序列图像上面积的重叠关系即可从三维序列图上提取出感兴趣区域。 结果与结论：基于控制标记符的分水岭算法解决了直接应用梯度图像进行分割的过度分割问题,便于进一步分割图像。利用基于分水岭算法的交互式三维分割方法得到的三维分割结果经过三维可视化后可清晰、准确地反映组织的三维特征。     

Incorporating partial shining effects in proton pencil-beam dose calculation

A range modulator wheel (RMW) is an essential component in passively scattered proton therapy. We have observed that a proton beam spot may shine on multiple steps of the RMW. Proton dose calculation algorithms normally do not consider the partial shining effect, and thus overestimate the dose at the proximal shoulder of spread-out Bragg peak (SOBP) compared with the measurement. If the SOBP is adjusted to better fit the plateau region, the entrance dose is likely to be underestimated. In this work, we developed an algorithm that can be used to model this effect and to allow for dose calculations that better fit the measured SOBP. First, a set of apparent modulator weights was calculated without considering partial shining. Next, protons spilled from the accelerator reaching the modulator wheel were simplified as a circular spot of uniform intensity. A weight-splitting process was then performed to generate a set of effective modulator weights with the partial shining effect incorporated. The SOBPs of eight options, which are used to label different combinations of proton-beam energy and scattering devices, were calculated with the generated effective weights. Our algorithm fitted the measured SOBP at the proximal and entrance regions much better than the ones without considering partial shining effect for all SOBPs of the eight options. In a prostate patient, we found that dose calculation without considering partial shining effect underestimated the femoral head and skin dose.     

A fast numerical algorithm for electron mean energy calculation in radiation therapy

A numerical algorithm for calculating the mean energy of radiotherapy electron beams has been developed. This algorithm is fast and accurate which makes it suitable for routine clinical use. First, a Gaussian distribution of the electron energy spectrum is derived from the linear Boltzmann equation. Based on this Gaussian spectrum, and after introducing a correction to the CSDA mean energy, a recursive-iterative algorithm for the mean energy calculation is developed. The multiple-scattering correction is taken into account using Yang's pathlength distribution theory. Numerical results of the present algorithm are compared with the results obtained through Monte Carlo simulation as well as Harder's formula. Good agreement with Monte Carlo simulation is achieved. Also the new algorithm is much more accurate than the commonly-used empirical formula of Harder.     

Pencil-beam redefinition algorithm for electron dose distributions

A pencil-beam redefinition algorithm has been developed for the calculation of electron-beam dose distributions on a three-dimensional grid utilizing 3-D inhomogeneity correction. The concept of redefinition was first used for both fixed and arced electron beams by Hogstrom et al. but was limited to a single redefinition. The success of those works stimulated the development of the pencil-beam redefinition algorithm, the aim of which is to solve the dosimetry problems presented by deep inhomogeneities through development of a model that redefines the pencil beams continuously with depth. This type of algorithm was developed independently by Storchi and Huizenga who termed it the "moments method." Such a pencil beam within the patient is characterized by a complex angular distribution, which is approximated by a Gaussian distribution having the same first three moments as the actual distribution. Three physical quantities required for dose calculation and subsequent radiation transport--namely planar fluence, mean direction, and root-mean-square spread about the mean direction--are obtained from these moments. The primary difference between the moments method and the redefinition algorithm is that the latter subdivides the pencil beams into multiple energy bins. The algorithm then becomes a macroscopic method for transporting the complete phase space of the beam and allows the calculation of physical quantities such as fluence, dose, and energy distribution. Comparison of calculated dose distributions with measured dose distributions for a homogeneous water phantom, and for phantoms with inhomogeneities deep relative to the surface, show agreement superior to that achieved with the pencil-beam algorithm of Hogstrom et al. in the penumbral region and beneath the edges of air and bone inhomogeneities. The accuracy of the redefinition algorithm is within 4% and appears sufficient for clinical use, and the algorithm is structured for further expansion of the physical model if required for site-specific treatment planning problems.     

Incorporation of the aperture thickness in proton pencil-beam dose calculations

Field-specific apertures, of sufficient range-absorbing thickness, are used in the majority of proton-therapy treatments today. In current practice, these apertures are modelled as objects of infinitesimal thickness. Such an approximation, however, is not accurate if the aperture edge is close to, or extends over, the beam axis. Practical situations in which this occurs include off-axis patch fields, small apertures, and fields shaped with a multileaf collimator. We develop an extension of the pencil-beam dose model to incorporate the aperture thickness. We derive an exact solution as well as a computationally simpler approximate implementation. The model is validated using measurements of the lateral penumbra. For a set-up with a source size of 2.76 cm, a source-to-axis distance of 227 cm, and a aperture-to-axis distance of 35 cm, the maximum increase in penumbra for a 6 cm thick aperture compared to the thin-aperture model is about 2 mm. The maximum shift in the 95% isodose contour line is larger. The overall effect depends on the aperture thickness, the position of the aperture edge and the intrinsic source size and SAD, but is fairly insensitive to aperture-to-skin distance and depth in patient.     

Application of the electron pencil beam redefinition algorithm to electron arc therapy

This project investigated the potential of summing fixed-beam dose distributions calculated using the pencil-beam redefinition algorithm (PBRA) at small angular steps (1 degree) to model an electron arc therapy beam. The PRBA, previously modified to model skin collimation, was modified further by incorporating two correction factors. One correction factor that is energy, SSD (source-to-surface distance), and field-width dependent constrained the calculated dose output to be the same as the measured dose output for fixed-beam geometries within the range of field widths and SSDs encountered in arc therapy. Another correction factor (single field-width correction factor for each energy) compensated for large-angle scattering not being modeled, allowing a more accurate calculation of dose output at mid arc. The PBRA was commissioned to accurately calculate dose in a water phantom for fixed-beam geometries typical of electron arc therapy. Calculated central-axis depth doses agreed with measured doses to within 2% in the low-dose gradient regions and within 1-mm in the high-dose gradient regions. Off-axis doses agreed to within 2 mm in the high-dose gradient regions and within 3% in the low-dose gradient regions. Arced-beam calculations of dose output and depth dose at mid arc were evaluated by comparing to data measured using two cylindrical water phantoms with radii of 12 and 15 cm at 10 and 15 MeV. Dose output was measured for all combinations of phantom radii of curvature, collimator widths (4, 5, and 6 cm), and arc angles (0 degrees, 20 degrees, 40 degrees, 60 degrees, 80 degrees, and 90 degrees) for both beam energies. Results showed the calculated mid-arc dose output to agree within 2% of measurement for all combinations. For a 90 degree arc angle and 5 x 20 cm2 field size, the calculated mid-arc depth dose in the low-dose gradient region agreed to within 2% of measurement for all depths at 10 MeV and for depths greater than depth of dose maximum R100 at 15 MeV. For depths in the buildup region at 15 MeV the calculations overestimated the measured dose by as much as 3.4%. Mid-arc depth dose in the high-dose gradient region agreed to within 2.2 mm of measured dose. Calculated two-dimensional relative dose distributions in the plane of rotation were compared to dose measurements using film in a cylindrical polystyrene phantom for a 90 degree arc angle and field widths of 4, 5, and 6 cm at 10 and 15 MeV. Results showed that off-axis dose at the ends of arc (without skin collimation) agreed to within 2% in the low-dose gradient region and to within 1.2 mm in the high-dose gradient region. This work showed that the accuracy of the PBRA arced-beam dose model met the criteria specified by Van Dyk et al. [Int. J. Radiat. Oncol. Biol. Phys. 26, 261-273 (1993)] with the exception of the buildup region of the 15 MeV beam. Based on the present results, results of a previous study showing acceptable accuracy in the presence of skin collimation, and results of a previous study showing acceptable accuracy in the presence of internal heterogeneities, it is concluded that the PBRA arced-beam dose model should be adequate for planning electron arc therapy.