An empirical model for megavoltage x-ray output factors

An empirical model of the factors that determine the central axis dose at 10 cm depth in water for 4 MV, 6 MV and 18 MV photon beams is presented. Backscattering from the variable collimators into the dose monitoring ionization chamber can cause a variation of -0.5% to +1.8% in the dose per monitor unit in accelerators with an electron facility. Forward emission towards the isocentre from the beam flattening filter and upper collimators is more dependent on the position of the upper variable collimator blades than the lower blades, so that they are not interchangeable in determining output factors, which can differ by up to 2%. The model includes the product of the monitor backscatter factor, normalized phantom scatter factor, normalized head scatter factor and inverse square law, corrected for the displacement of the virtual x-ray focus from the target. It can predict the dose to -/+0.83% for 4 MV, -/+0.80% for 6 MV photons and -/+0.82% for 18 MV photons. The normalized head scatter factor is a second-order polynomial of the modified equivalent square collimator, whose coefficients do not vary significantly with x-ray energy. The model was tested by comparison with independent measurements of output factor and generally agreed to around 1%.     

Derivation of electron and photon energy spectra from electron beam central axis depth dose curves

A method for deriving the electron and photon energy spectra from electron beam central axis percentage depth dose (PDD) curves has been investigated. The PDD curves of 6, 12 and 20 MeV electron beams obtained from the Monte Carlo full phase space simulations of the Varian linear accelerator treatment head have been used to test the method. We have employed a 'random creep' algorithm to determine the energy spectra of electrons and photons in a clinical electron beam. The fitted electron and photon energy spectra have been compared with the corresponding spectra obtained from the Monte Carlo full phase space simulations. Our fitted energy spectra are in good agreement with the Monte Carlo simulated spectra in terms of peak location, peak width, amplitude and smoothness of the spectrum. In addition, the derived depth dose curves of head-generated photons agree well in both shape and amplitude with those calculated using the full phase space data. The central axis depth dose curves and dose profiles at various depths have been compared using an automated electron beam commissioning procedure. The comparison has demonstrated that our method is capable of deriving the energy spectra for the Varian accelerator electron beams investigated. We have implemented this method in the electron beam commissioning procedure for Monte Carlo electron beam dose calculations.     

A Monte Carlo multiple source model applied to radiosurgery narrow photon beams

Monte Carlo (MC) methods are nowadays often used in the field of radiotherapy. Through successive steps, radiation fields are simulated, producing source Phase Space Data (PSD) that enable a dose calculation with good accuracy. Narrow photon beams used in radiosurgery can also be simulated by MC codes. However, the poor efficiency in simulating these narrow photon beams produces PSD whose quality prevents calculating dose with the required accuracy. To overcome this difficulty, a multiple source model was developed that enhances the quality of the reconstructed PSD, reducing also the time and storage capacities. This multiple source model was based on the full MC simulation, performed with the MC code MCNP4C, of the Siemens Mevatron KD2 (6 MV mode) linear accelerator head and additional collimators. The full simulation allowed the characterization of the particles coming from the accelerator head and from the additional collimators that shape the narrow photon beams used in radiosurgery treatments. Eight relevant photon virtual sources were identified from the full characterization analysis. Spatial and energy distributions were stored in histograms for the virtual sources representing the accelerator head components and the additional collimators. The photon directions were calculated for virtual sources representing the accelerator head components whereas, for the virtual sources representing the additional collimators, they were recorded into histograms. All these histograms were included in the MC code, DPM code and using a sampling procedure that reconstructed the PSDs, dose distributions were calculated in a water phantom divided in 20000 voxels of 1 x 1 x 5 mm3. The model accurately calculates dose distributions in the water phantom for all the additional collimators; for depth dose curves, associated errors at 2sigma were lower than 2.5% until a depth of 202.5 mm for all the additional collimators and for profiles at various depths, deviations between measured and calculated values were less than 2.5% or 1 mm.     

Modelling 6 MV photon beams of a stereotactic radiosurgery system for Monte Carlo treatment planning

The goal of this work is to build a multiple source model to represent the 6 MV photon beams from a Cyberknife stereotactic radiosurgery system for Monte Carlo treatment planning dose calculations. To achieve this goal, the 6 MV photon beams have been characterized and modelled using the EGS4/BEAM Monte Carlo system. A dual source model has been used to reconstruct the particle phase space at a plane immediately above the secondary collimator. The proposed model consists of two circular planar sources for the primary photons and the scattered photons, respectively. The dose contribution of the contaminant electrons was found to be in the order of 10(-3) of the total maximum dose and therefore has been omitted in the source model. Various comparisons have been made to verify the dual source model against the full phase space simulated using the EGS4/BEAM system. The agreement in percent depth dose (PDD) curves and dose profiles between the phase space and the source model was generally within 2%/1 mm for various collimators (5 to 60 mm in diameter) at 80 to 100 cm source-to-surface distances (SSD). Excellent agreement (within 1%/1 mm) was also found between the dose distributions in heterogeneous lung and bone geometry calculated using the original phase space and those calculated using the source model. These results demonstrated the accuracy of the dual source model for Monte Carlo treatment planning dose calculations for the Cyberknife system.     

Electron spectra derived from depth dose distributions

The technique of extracting electron energy spectra from measured distributions of dose along the central axis of clinical electron beams is explored in detail. Clinical spectra measured with this simple spectroscopy tool are shown to be sufficient in accuracy and resolution for use in Monte Carlo treatment planning. A set of monoenergetic depth dose curves of appropriate energy spacing, precalculated with Monte Carlo for a simple beam model, are unfolded from the measured depth dose curve. The beam model is comprised of a point electron and photon source placed in vacuum with a source-to-surface distance of 100 cm. Systematic error introduced by this model affects the calculated depth dose curve by no more than 2%/2 mm. The component of the dose due to treatment head bremsstrahlung, subtracted prior to unfolding, is estimated from the thin-target Schiff spectrum within 0.3% of the maximum total dose (from electrons and photons) on the beam axis. Optimal unfolding parameters are chosen, based on physical principles. Unfolding is done with the public-domain code FERDO. Comparisons were made to previously published spectra measured with magnetic spectroscopy and to spectra we calculated with Monte Carlo treatment head simulation. The approach gives smooth spectra with an average resolution for the 27 beams studied of 16+/-3% of the mean peak energy. The mean peak energy of the magnetic spectrometer spectra was calculated within 2% for the AECL T20 scanning beam accelerators, 3% for the Philips SL25 scattering foil based machine. The number of low energy electrons in Monte Carlo spectra is estimated by unfolding with an accuracy of 2%, relative to the total number of electrons in the beam. Central axis depth dose curves calculated from unfolded spectra are within 0.5%/0.5 mm of measured and simulated depth dose curves, except near the practical range, where 1%/1 mm errors are evident.     

Collimator-produced energy degradation of megavoltage x rays

Monte Carlo computer calculations have been made to assess the effects of small collimators (i.e., output field sizes less than or equal to 1 cm) on megavoltage x-ray beams. Such collimators are often used in experiments to measure beam energy, half value layers, and other beam parameters. Our calculations demonstrate, however, that such procedures can introduce significant changes in these same parameters. Under certain conditions, more than 40% of the transmitted photons exiting the collimator do so with degraded energy. Optimum collimator design, however, can greatly reduce these effects. In particular, it is shown that the thickness of material required to effect a good collimator is not only dependent upon the photon energy, but also upon the size of the collimator hole.     

Perturbation of radiotherapy beams by radiographic film: measurements and Monte Carlo simulations.

Radiographic film is an established practical tool used in the measurement of the dose distribution for radiotherapy purposes. The accuracy and reproducibility of film optical density as an indicator of dose has been associated with several factors including photon energy, processing conditions and film plane orientation. Few studies have investigated the factors causing variability in film dosimetry, due to the difficulty of separating the individual contributions. The effect that a sheet of radiographic film in a water phantom has on its response to a 6 MV photon and a cobalt-60 teletherapy beam, when orientated perpendicular and parallel to the beam central axis, is reported. Monte Carlo generated spectra were used to calculate collision kerma (Kcoll) for water and film elements. Measured and calculated results indicate a potential over-response at 25 cm depth of the order of 14 +/- 2.4% and 18 +/- 6.0% respectively for 6 MV photons and 15 +/- 3.4% and 32 +/- 4.5% respectively for a cobalt beam. For film exposed parallel as compared to perpendicular to the central axis of the beam, the calculated results suggest an explanation in terms of the predominantly forward directed secondary electrons for the measured difference in film response at depth. It is proposed that the difference in response of the parallel as compared to perpendicular exposed film be due to the predominantly 'upstream' photon interactions giving rise to energy deposition in film. The simulations indicate that the variation with depth of relative energy imparted in film and water elements correlates with the observed variation in film response with depth.     

光子射野剂量计算参数研究

目的：介绍医用加速器常规光子射线的机器数据测量方法及剂量计算模型中基本参数的计算过程。以百分深度剂量与散射因子为基础数据,根据原散射线模型通过测量数据推导出原射线组织最大剂量比、散射最大剂量比、原射线在水中线性衰减系数、能量注量等,为进一步还原射野在水模体中的剂量分布提供方法与理论。方法：用Blue Phantom三维水箱在医科达Synergy加速器上测量6MV光子线的百分深度剂量、离轴比剂量、总散射因子、准直器散射因子,先从测量的百分深度剂量曲线中按照原散射模型剥离出原射线百分深度剂量,然后在Matlab软件中拟合处理测量的散射因子数据,外推出零野的模体散射因子,从而按照给定公式计算出组织最大剂量比、散射最大剂量比。按照离轴比剂量,利用平方反比规律推出最大开野在模体表面的能量注量。结果：计算出准直器散射因子、总散射因子的拟合公式,外推零野模体散射因子（s。）、根据原射线的百分深度剂量曲线计算出原射线在水中线性衰减系数,组织最大剂量比（TMR）、散射最大剂量比（SMR）、以及射野能量注量分布（Fluence Matrix）。结论：这些基本参数是剂量计算建模的关键,也是进一步研究各种剂量计算模型的基础。     

Monte Carlo study of a Cyberknife stereotactic radiosurgery system

This study investigated small-field dosimetry for a Cyberknife stereotactic radiosurgery system using Monte Carlo simulations. The EGSnrc/BEAMnrc Monte Carlo code was used to simulate the Cyberknife treatment head, and the DOSXYZnrc code was implemented to calculate central axis depth-dose curves, off-axis dose profiles, and relative output factors for various circular collimator sizes of 5 to 60 mm. Water-to-air stopping power ratios necessary for clinical reference dosimetry of the Cyberknife system were also evaluated by Monte Carlo simulations. Additionally, a beam quality conversion factor, kQ, for the Cyberknife system was evaluated for cylindrical ion chambers with different wall material. The accuracy of the simulated beam was validated by agreement within 2% between the Monte Carlo calculated and measured central axis depth-dose curves and off-axis dose profiles. The calculated output factors were compared with those measured by a diode detector and an ion chamber in water. The diode output factors agreed within 1% with the calculated values down to a 10 mm collimator. The output factors with the ion chamber decreased rapidly for collimators below 20 mm. These results were confirmed by the comparison to those from Monte Carlo methods with voxel sizes and materials corresponding to both detectors. It was demonstrated that the discrepancy in the 5 and 7.5 mm collimators for the diode detector is due to the water non-equivalence of the silicon material, and the dose fall-off for the ion chamber is due to its large active volume against collimators below 20 mm. The calculated stopping power ratios of the 60 mm collimator from the Cyberknife system (without a flattening filter) agreed within 0.2% with those of a 10 X 10 cm2 field from a conventional linear accelerator with a heavy flattening filter and the incident electron energy, 6 MeV. The difference in the stopping power ratios between 5 and 60 mm collimators was within 0.5% at a 10 cm depth in water. Furthermore, kQ values for the Cyberknife system were in agreement within 0.3% with those of the conventional 6 MV-linear accelerator for the cylindrical ion chambers with different wall material.     

Peripheral dose measurements for 6 and 18 MV photon beams on a linear accelerator with multileaf collimator

Peripheral dose (PD) to critical structures outside treatment volume is of clinical importance. The aim of the current study was to estimate PD on a linear accelerator equipped with multileaf collimator (MLC). Dose measurements were carried out using an ionization chamber embedded in a water phantom for 6 and 18 MV photon beams. PD values were acquired for field sizes from 5 x 5 to 20 x 20 cm2 in increments of 5 cm at distances up to 24 cm from the field edge. Dose data were obtained at two collimator orientations where the measurement points are shielded by MLC and jaws. The variation of PD with the source to skin distance (SSD), depth, and lateral displacement of the measurement point was evaluated. To examine the dependence of PD upon the tissue thickness at the entrance point of the beam, scattered dose was measured using thermoluminescent dosemeters placed on three anthropomorphic phantoms simulating 5- and 10-year-old children and an average adult patient. PD from 6 MV photons varied from 0.13% to 6.75% of the central-axis maximum dose depending upon the collimator orientation, extent of irradiated area, and distance from the treatment field. The corresponding dose range from 18 MV x rays was 0.09% to 5.61%. The variation of PD with depth and with lateral displacements up to 80% of the field dimension was very small. The scattered dose from both photon beams increased with the increase of SSD or tissue thickness along beam axis. The presented dosimetric data set allows the estimation of scattered dose outside the primary beam.     

Patient-dependent beam-modifier physics in Monte Carlo photon dose calculations

Model pencil-beam on slab calculations are used as well as a series of detailed calculations of photon and electron output from commercial accelerators to quantify level(s) of physics required for the Monte Carlo transport of photons and electrons in treatment-dependent beam modifiers, such as jaws, wedges, blocks, and multileaf collimators, in photon teletherapy dose calculations. The physics approximations investigated comprise (1) not tracking particles below a given kinetic energy, (2) continuing to track particles, but performing simplified collision physics, particularly in handling secondary particle production, and (3) not tracking particles in specific spatial regions. Figures-of-merit needed to estimate the effects of these approximations are developed, and these estimates are compared with full-physics Monte Carlo calculations of the contribution of the collimating jaws to the on-axis depth-dose curve in a water phantom. These figures of merit are next used to evaluate various approximations used in coupled photon/electron physics in beam modifiers. Approximations for tracking electrons in air are then evaluated. It is found that knowledge of the materials used for beam modifiers, of the energies of the photon beams used, as well as of the length scales typically found in photon teletherapy plans, allows a number of simplifying approximations to be made in the Monte Carlo transport of secondary particles from the accelerator head and beam modifiers to the isocenter plane.     

Design of a fast multileaf collimator for radiobiological optimized IMRT with scanned beams of photons, electrons, and light ions

Intensity modulated radiation therapy is rapidly becoming the treatment of choice for most tumors with respect to minimizing damage to the normal tissues and maximizing tumor control. Today, intensity modulated beams are most commonly delivered using segmental multileaf collimation, although an increasing number of radiation therapy departments are employing dynamic multileaf collimation. The irradiation time using dynamic multileaf collimation depends strongly on the nature of the desired dose distribution, and it is difficult to reduce this time to less than the sum of the irradiation times for all individual peak heights using dynamic leaf collimation [Svensson et al., Phys. Med. Biol. 39, 37-61 (1994)]. Therefore, the intensity modulation will considerably increase the total treatment time. A more cost-effective procedure for rapid intensity modulation is using narrow scanned photon, electron, and light ion beams in combination with fast multileaf collimator penumbra trimming. With this approach, the irradiation time is largely independent of the complexity of the desired intensity distribution and, in the case of photon beams, may even be shorter than with uniform beams. The intensity modulation is achieved primarily by scanning of a narrow elementary photon pencil beam generated by directing a narrow well focused high energy electron beam onto a thin bremsstrahlung target. In the present study, the design of a fast low-weight multileaf collimator that is capable of further sharpening the penumbra at the edge of the elementary scanned beam has been simulated, in order to minimize the dose or radiation response of healthy tissues. In the case of photon beams, such a multileaf collimator can be placed relatively close to the bremsstrahlung target to minimize its size. It can also be flat and thin, i.e., only 15-25 mm thick in the direction of the beam with edges made of tungsten or preferably osmium to optimize the sharpening of the penumbra. The low height of the collimator will minimize edge scatter from glancing incidence. The major portions of the collimator leafs can then be made of steel or even aluminum, so that the total weight of the multileaf collimator will be as low as 10 kg, which may even allow high-speed collimation in real time in synchrony with organ movements. To demonstrate the efficiency of this collimator design in combination with pencil beam scanning, optimal radiobiological treatments of an advanced cervix cancer were simulated. Different geometrical collimator designs were tested for bremsstrahlung, electron, and light ion beams. With a 10 mm half-width elementary scanned photon beam and a steel collimator with tungsten edges, it was possible to make as effective treatments as obtained with intensity modulated beams of full resolution, i.e., here 5 mm resolution in the fluence map. In combination with narrow pencil beam scanning, such a collimator may provide ideal delivery of photons, electrons, or light ions for radiation therapy synchronized to breathing and other organ motions. These high-energy photon and light ion beams may allow three-dimensional in vivo verification of delivery and thereby clinical implementation of the BioArt approach using Biologically Optimized three-dimensional in vivo predictive Assay based adaptive Radiation Therapy [Brahme, Acta Oncol. 42, 123-126 (2003)].     

Calculated buildup curves for photons with energies up to 60Co

This report investigates the buildup region of the depth-dose curve for broad parallel beams of 60Co and lower energy photons incident on slabs of tissue as defined by the International Commission on Radiation Units and Measurements (ICRU) and other materials. The EGS Monte Carlo system was used to simulate the transport of photons and electrons in the materials. Calculated buildup regions of the depth-dose curves are presented for photons incident on ICRU tissue. For 60Co beams, the primary and scatter dose components are presented, both for the buildup region and at depth in the phantom. Effects of beam radius were investigated and found to have only a small effect in the buildup region of the central axis depth-dose curve. The calculated effects of beam radius are larger at depth and in good agreement with experiment. Effects of scatter and attenuation in a small miniphantom of water were investigated as a function of photon energy and were found to cancel within 0.8%. The mean distance of electron transport was calculated for 60Co beams incident on carbon, water, polystyrene, aluminum, and lead.     

Differential pencil beam dose computation model for photons

Differential pencil beam (DPB) is defined as the dose distribution relative to the position of the first collision, per unit collision density, for a monoenergetic pencil beam of photons in an infinite homogeneous medium of unit density. We have generated DPB dose distribution tables for a number of photon energies in water using the Monte Carlo method. The three-dimensional (3D) nature of the transport of photons and electrons is automatically incorporated in DPB dose distributions. Dose is computed by evaluating 3D integrals of DPB dose. The DPB dose computation model has been applied to calculate dose distributions for 60Co and accelerator beams. Calculations for the latter are performed using energy spectra generated with the Monte Carlo program. To predict dose distributions near the beam boundaries defined by the collimation system as well as blocks, we utilize the angular distribution of incident photons. Inhomogeneities are taken into account by attenuating the primary photon fluence exponentially utilizing the average total linear attenuation coefficient of intervening tissue, by multiplying photon fluence by the linear attenuation coefficient to yield the number of collisions in the scattering volume, and by scaling the path between the scattering volume element and the computation point by an effective density.     

Evaluation of Al2O3:C optically stimulated luminescence (OSL) dosimeters for passive dosimetry of high-energy photon and electron beams in radiotherapy

This article investigates the performance of Al2O3: C optically stimulated luminescence dosimeters (OSLDs) for application in radiotherapy. Central-axis depth dose curves and optically stimulated luminescence (OSL) responses were obtained in a water phantom for 6 and 18 MV photons, and for 6, 9, 12, 16, and 20 MeV electron beams from a Varian 21EX linear accelerator. Single OSL measurements could be repeated with a precision of 0.7% (one standard deviation) and the differences between absorbed doses measured with OSLDs and an ionization chamber were within +/- 1% for photon beams. Similar results were obtained for electron beams in the low-gradient region after correction for a 1.9% photon-to-electron bias. The distance-to-agreement values were of the order of 0.5-1.0 mm for electrons in high dose gradient regions. Additional investigations also demonstrated that the OSL response dependence on dose rate, field size, and irradiation temperature is less than 1% in the conditions of the present study. Regarding the beam energy/quality dependence, the relative response of the OSLD for 18 MV was (0.51 +/- 0.48)% of the response for the 6 MV photon beam. The OSLD response for the electron beams relative to the 6 MV photon beam. The OSLD response for the electron beams relative to the 6 MV photon beam was in average 1.9% higher, but this result requires further confirmation. The relative response did not seem to vary with electron energy at dmax within the experimental uncertainties (0.5% in average) and, therefore, a fixed correction factor of 1.9% eliminated the energy dependence in our experimental conditions.     

有无均整器模式下臂架与准直器不同角度条件下的剂量学比较

目的:探讨臂架或准直器角度的改变对均整（FF）与非均整（FFF）两种模式的射线剂量的影响。方法:选用Versa HD直线加速器配备的6 MV/10 MV光子束FF/FFF模式4档能量在设定好九点位置的10 cm×10 cm标准射野内进行实验。首先,借助IMF等中心夹具将Mapcheck2固定于治疗机机头,并用Mapcheck2测量相同臂架与准直器角度条件下4种光子束输出的平面剂量值;其次,用Mapcheck2测量在相同臂架角度、不同准直器角度与相同准直器角度、不同臂架角度两种条件下4种光子束的中心轴剂量值;最后,固定准直器为0°,设立两组臂架对穿射野（0°与180°,90°与270°）。拆除Mapcheck2,采用固体水和FC65-G电离室建立一个测量模体来测量4种光子束在两组等中心对穿野的剂量。运用SPSS统计软件对该实验收集到的数据进行对比分析。结果:在相同臂架与准直器角度条件下,4种光子束辐照9个点的平面剂量之间均存在明显统计学差异（P6 MV FF =0.020, P6 MV FFF=0.017, P10 MV FF =0.030, P10 MV FFF=0.016）;而不同臂架角度或不同准直器角度条件下,4种能量光子束的中心轴点剂量值均无统计学差异。在0°与180°的对穿野,4种能量光子束的输出剂量存在统计学差异（P6 MV FF =0.001, P6 MV FFF=0.002, P10 MV FF =0.003, P10 MV FFF=0.001）,而在90°与270°的对穿野无统计学差异。结论:Versa HD直线加速器拥有优良的机械等中心性能。在实际工作时,臂架和准直器的旋转,均不影响光子束的中心轴剂量的准确输出。在FF模式下,射线能量越高,受治疗床影响越小;FFF模式射线由于射线质软,能量越高,更易受到治疗床的衰减作用,在实际中应引起重视。     

Collimator scatter in modeling radiation beam profiles

In computer dose calculations using scatter-air ratio sector summation algorithms, the primary dose from the target to points away from the central axis of a beam is computed using an exponential intensity model of the source and a transmission parameter for the collimator. This model works well inside the beam and near edges but is inaccurate outside the beam at distances of more than 1-2 cm from beam edges. We have modified the standard beam profile model to include a dose contribution representing photon radiation scattered from the collimators. Collimator edges are treated mathematically as line sources and an adjustable parameter is introduced which represents the activity per unit length of the collimator edges. Dose from the collimator edges is assumed to decrease purely geometrically as the inverse of the square of the distance and no modification is made for tissue attenuation. With these assumptions, the total collimator scatter dose to a point is most accurately computed by a line integral over the edges of the beam outline. This modification fits naturally into the standard scatter-air ratio sector summation computer algorithm but adds significantly to dose computation time. Some approximations eliminate the line integration and lead to a collimator scatter term which is proportional to field perimeter and independent of off-axis distance. The modified dose model was tested by comparing measured dose profiles with computed ones using x-ray beams from Philips (6 and 15 MV) and Varian (4 and 6 MV) accelerators. There was significant improvement in fit compared to the standard beam model for points outside the radiation beam.     

Dosimetric characteristics of dynamic wedged fields: a Monte Carlo study.

We have developed a Monte Carlo (MC) technique using the EGS4/BEAM system to calculate dosimetric characteristics of dynamic wedges (DW) for photon beam radiotherapy. The simulation of DW was accomplished by weighting the history numbers of the electrons, which are incident on the target in accordance with the segmented treatment table. Calculations were performed for DW with wedge angles ranging from 15 degrees to 60 degrees as well as for open fields with different field sizes for both degrees 6 and 18 MV beams. The MC-calculated percentage depth dose (PDD) and beam profiles agreed with the measurements within +/- 2% (of the dose maximum along the beam axis) or +/- 2 mm in high dose gradient region. The DW slightly affects energy spectra of photons and contaminating electrons. These slight changes have no significant effects on PDD as compared to the open field. The MC-calculated dynamic wedge factors agree with the measurements within +/- 2%. The MC method enables us to provide more detailed beam characteristics for DW fields than a measurement method. This beam characteristic includes photon energy spectra, mean energy, spectra of contaminating electrons and effects of moving jaw on off-axis beam quality. These data are potentially important for treatment planning involving dynamic wedges.