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Purpose: To create a user friendly, accurate, real time computer simulator to facilitate the design of dual foil scattering systems for electron beams on radiotherapy accelerators. The simulator should allow for a relatively quick, initial design that can be refined and verified with subsequent Monte Carlo(MC) calculations and measurements. Methods: The simulator consists of an analytical algorithm for calculating electron fluence and a graphical user interface (GUI) C++ program. The algorithm predicts electron fluence using Fermi‐Eyges multiple Coulomb scattering theory with a refined Moliere formalism for scattering powers. The simulator also estimates central‐axis x‐ray dose contamination from the dual foil system. Once the geometry of the beamline is specified, the simulator allows the user to continuously vary primary scattering foil material and thickness, secondary scattering foil material and Gaussian shape (thickness and sigma), and beam energy. The beam profile and x‐ray contamination are displayed in real time. Results: The simulator was tuned by comparison of off‐axis electron fluence profiles with those calculated using EGSnrc MC. Over the energy range 7–20 MeV and using present foils on the Elekta radiotherapy accelerator, the simulator profiles agreed to within 2% of MC profiles from within 20 cm of the central axis. The x‐ray contamination predictions matched measured data to within 0.6%. The calculation time was approximately 100 ms using a single processor, which allows for real‐time variation of foil parameters using sliding bars. Conclusions: A real time dual scattering foil system simulator has been developed. The tool has been useful in a project to redesign an electron dual scattering foil system for one of our radiotherapy accelerators. The simulator has also been useful as an instructional tool for our medical physics graduate students.

Purpose: In this work, recent results from experiments and simulations (with EGSnrc) performed by Rosset al. [Med. Phys.35, 4121–4131 (2008)] on electron scattering by foils of different materials and thicknesses are compared to those obtained using several Monte Carlo codes. Methods: Three codes have been used: GEANT (version 3.21), Geant4 (version 9.1, patch03), and PENELOPE (version 2006). In the case of PENELOPE, mixed and fully detailed simulations have been carried out. Results: Transverse dose distributions in air have been obtained in order to compare with measurements. The detailed PENELOPE simulations show excellent agreement with experiment. The calculations performed with GEANT and PENELOPE (mixed) agree with experiment within 3% except for the Be foil. In the case of Geant4, the distributions are 5% narrower compared to the experimental ones, though the agreement is very good for the Be foil. Transverse dose distribution in water obtained with PENELOPE (mixed) is 4% wider than those calculated by Rosset al. using EGSnrc and is 1% narrower than the transverse dose distributions in air, as considered in the experiment. Conclusions: All the codes give a reasonable agreement (within 5%) with the experimental results for all the material and thicknesses studied.

A protondose computational algorithm, performing an analytical superposition of infinitely narrow proton beamlets (ASPB) is introduced. The algorithm uses the standard pencil beam technique of laterally distributing the central axis broad beam doses according to the Moliere scattering theory extended to slablike varying density media. The purpose of this study was to determine the accuracy of our computational tool by comparing it with experimental and Monte Carlo(MC) simulation data as benchmarks. In the tests, parallel wide beams of protons were scattered in water phantoms containing embedded air and bone materials with simple geometrical forms and spatial dimensions of a few centimeters. For homogeneous water and bone phantoms, the protondoses we calculated with the ASPB algorithm were found very comparable to experimental and MC data. For layered bone slab inhomogeneity in water, the comparison between our analytical calculation and the MC simulation showed reasonable agreement, even when the inhomogeneity was placed at the Bragg peak depth. There also was reasonable agreement for the parallelepiped bone block inhomogeneity placed at various depths, except for cases in which the bone was located in the region of the Bragg peak, when discrepancies were as large as more than 10%. When the inhomogeneity was in the form of abutting air-bone slabs, discrepancies of as much as 8% occurred in the lateral dose profiles on the air cavity side of the phantom. Additionally, the analytical depth-dose calculations disagreed with the MC calculations within 3% of the Bragg peak dose, at the entry and midway depths in the phantom. The distal depth-dose 20%–80% fall-off widths and ranges calculated with our algorithm and the MC simulation were generally within 0.1 cm of agreement. The analytical lateral-dose profile calculations showed smaller (by less than 0.1 cm ) 20%–80% penumbra widths and shorter fall-off tails than did those calculated by the MC simulations. Overall, this work validates the usefulness of our ASPB algorithm as a reasonably fast and accurate tool for quality assurance in planning wide beam proton therapy treatment of clinical sites either composed of homogeneous materials or containing laterally extended inhomogeneities that are comparable in density and located away from the Bragg peak depths.

The GafChromic™ film (GCF) MD-55-2, a radiochromic material, was examined for its optical properties through total reflectance and transmittance measurements in visible spectrum (400–700 nm). By using a multilayer model of the film and Kubelka–Munk’s (KM) theory, absorption and scattering coefficients of the film sensitive layer (K and S, respectively) were obtained from measurements of irradiated and nonirradiated slides. This has allowed calculation of the absorbance A KM of the sensitive layer of the GCF. The model easily splits scattering from absorption. Unlike absorption, scattering is essentially insensitive to irradiation dose and decreases slowly as the wavelength increases. The scattering effect is predominant over absorption in the 400–500 nm range, while beyond 600 nm absorption prevails. The A KM absorbance of the sensitive layer was calculated using the K coefficient and compared with the optical densities (OD) measured considering only ballisticphotons (as in a standard spectrophotometer) as well as the optical densities measured collecting all the transmitted photons (as in many densitometers). The values of A KM found were always lower than OD measured by the other methods and they had the best linearity on the whole visible range. These data support the hypothesis that the sensitive layer reacts to irradiation more linearly than that shown by measurements using standard commercial devices. However, in the 600–680 nm range, correction is not very important because absorption is predominant over scattering. When GCF is used for imaging, scattering produces a loss of spatial information. Consequently, it is necessary to collect only ballisticphotons and to correct absorbance by K and S coefficients.

An adaptation of linear acoustic scattering theory is used to describe the comet tail artifact which is observed in the diagnostic ultrasound imaging of small spherical masses. The calculated backscattered pressure wave is shown to correspond well with the appearance of the artifact in a variety of clinical images.

A transport algorithm called the proton loss (PL) model is developed for proton pencil beams of therapeutic energies. The PL model takes into account inelastic nuclear reactions, pathlength straggling, and energy-loss straggling and predicts the 3D dose distribution from a proton pencil beam. In proton beams, the multiple scattering and ionizational energy loss processes approach their diffusional limit where scattering and energy loss probability densities become Gaussian. Therefore we chose to derive the PL model from the Fermi-Eyges diffusional multiple scattering theory and the Gaussian theory of energy straggling. We first introduce a generalization of the Fermi-Eyges equation for proton pencil beams, labeled the proton loss (PL) transport equation. This new equation includes terms that model inelastic nuclear reactions as a depth-dependent absorption and pathlength straggling as a quasi-absorption. Then energy straggling is taken into account by using a weighted superposition of a discrete number of elementary pencil beams. These elementary pencil beams have different initial energies and lose energy according to the CSDA, thus they have different ranges of penetration. A final solution for the proton beam transport is obtained as a linear combination of elementary pencil beam solutions with weights defined by the Gaussian evolution of the proton energy spectrum with depth. A numerical comparison of the dose distribution predictions of the PL model with measurements and PTRAN Monte Carlo simulations indicates the model is both computational fast and accurate.

A simulation experiment for three-dimensional (3D) imaging of exogenous fluorescinated antibodies that specifically bind to infiltrating lymphocytes in minor salivary glands was carried out. Small ( ∼1 mm 3 volume) rhodamine targets, which mimic diseased minor salivary glands labeled with fluorescent antibodies to infiltrating lymphocytes in Sjo/gren’s syndrome, were embedded in a highly scattering tissue phantom consisting of a thick Delrin™ disk covered by index matched Delrin™ slabs of various thickness. In this way the variation of fluorescence profiles on the surface of tissue could be examined corresponding to the range of depths of the salivary glands in vivo. Surface images were obtained for different target depths and radial distances from laser excitation to target fluorophore. These images were analyzed and compared to calculations based on random walk theory in turbid media, using previously determined scattering and absorption coefficients of the Delrin™. Excellent agreement between the surface profiles experimentally measured and those predicted by our random walk theory was obtained. Derivation of these theoretical expressions is a necessary step toward devising an inverse algorithm which may have the potential expressions to perform 3D reconstruction of the concentration distribution of fluorescent labels within tissue.

The purpose of this manuscript is to describe a method for designing a broad beam of electrons suitable for total skin electron irradiation (TSEI). A theoretical model of a TSEI beam from a linear accelerator with a dual scattering system has been developed. The model uses Fermi-Eyges theory to predict the planar fluence of the electron beam after it has passed through various materials between the source and the treatment plane, which includes scattering foils, monitor chamber, air, and a plastic diffusing plate. Unique to this model is its accounting for removal of the tails of the electron beam profile as it passes through the primary x-ray jaws. A method for calculating the planar fluence profile for an obliquely incident beam is also described. Off-axis beam profiles and percentage depth doses are measured with ion chambers, film, and thermoluminescent dosimeters (TLD). The measured data show that the theoretical model can accurately predict beam energy and planar fluence of the electron beam at normal and oblique incidence. The agreement at oblique angles is not quite as good but is sufficiently accurate to be of predictive value when deciding on the optimal angles for the clinical TSEI beams. The advantage of our calculational approach for designing a TSEI beam is that many different beam configurations can be tested without having to perform time-consuming measurements. Suboptimal configurations can be quickly dismissed, and the predicted optimal solution should be very close to satisfying the clinical specifications.

In 1951, Yang derived formulas for computing the pathlength distribution of particles traversing foils, considering only the multiple-scattering process. We here improve upon the accuracy of that work, by using our second-order small-angle approximation. We derive the general solution for a broad parallel beam, and find simple formulas for Yang's two special cases: the pathlength distribution of all the particles at a particular point, taken together; and the pathlength distribution at a particular point of only those particles with zero net angular deflection. From the pathlength (or excess pathlength) distribution, residual range and energy distributions can immediately be deduced. All this work assumes relatively small energy loss, and we consider 5 MeV electrons penetrating lead, which provides considerable scattering without major energy loss. The second-order energy distribution is found to differ considerably from the (first-order) Yang energy distribution, and to agree more closely with EGS4 Monte Carlo calculations.

The Fermi pencil beam formula and the higher-order multiple scattering theory due to Jette are shown to result from a perturbative treatment of the linear Boltzmann equation with Fokker-Planck scattering. Using asymptotic one-dimensional solutions for the transverse integrated (spherical) fluence as well as its variance, approximate higher-order pencil beam theories are constructed. These simple and explicit formulae are shown, by comparison with benchmark Monte Carlo results, to be significantly more accurate than the Fermi and Jette equations, particularly at large distances from the beam axis.

A three-dimensional electron beam dose calculation algorithm implemented on a commercial radiotherapy treatment planning system is described. The calculation is based on the M. D. Anderson Hospital (M.D.A.H.) pencil beam model, which uses the Fermi-Eyges theory of thick-target multiple Coulomb scattering. To establish the calculation algorithm's accuracy as well as its limitations, it was systematically and extensively tested and evaluated against a set of benchmark measurements. Various levels of dose and spatial tolerances were used to validate the calculation quantitatively. Results are presented in terms of the percentage of data points meeting a specific tolerance level. The algorithm's ability to accurately simulate commonly used clinical setup geometries, including standard or extended SSDs, blocked fields, irregular surfaces, and heterogeneities, is demonstrated. Regions of disagreement between calculations and measurements are also shown. The clinical implication of such disagreements is addressed, and the algorithmic assumptions involved are discussed.

A three-dimensional electron beam dose calculational algorithm implemented for use in a three-dimensional treatment planning system is described. The 3-D electron beam calculations have been in use clinically for more than seven years. The algorithm uses a pencil beam model based on small angle multiple Coulomb scattering theory. Our implementation allows volumetric CT-based inhomogeneity corrections and provides for irregular field shapes (fields shaped with cerrobend cutouts) and the use of bolus. As part of NCI-funded work evaluating the state of the art in electron beam treatment planning, extensive algorithm verification was undertaken and results of these tests for our implementation are presented.

The x-ray luminescence of Y2O2S:Eu phosphor screens is studied both in transmission and in reflection mode observation. Detailed experimental data concerning the dependence of the absolute efficiency on the tube voltage and the screen's thickness are presented and comparisons to another rare earth phosphor material (Y2O2S:Tb) are given. With the help of theoretical calculations based on the Hamaker-Ludwig model all the experimental data are explained and the intrinsic efficiency and scattering coefficients of the phosphor are estimated.

The sensitivity of a technique for the measurement of trabecular bone mineral concentration has been examined theoretically and experimentally. The technique is based on coherent gamma ray scattering and corrections for attenuation are obtained from transmitted photons rather than Compton scatteredphotons. For an incident photon energy of 60 keV, the minimum detectable bone mineral difference is practically independent of scattering angle while for an incident energy of 100 or 122 keV the scattering angle must be less than 70° to optimize the minimum detectable difference.

In this fourth article in a series on the calculation of electron dose using multiple‐scattering theory, we deal with localized inhomogeneities by solving the Fermi equation for scattering power which is an arbitrary function of position. In fact, we go further, by solving the second‐order multiple‐scattering equation which supersedes the (first‐order) Fermi equation, again for scattering power which is an arbitrary function of position. Thus, we are no longer restricted to a horizontally layered medium, as is the case with the Fermi–Eyges theory. Our general solution is in the form of a perturbation series which evidently converges rapidly enough that only its first two or three terms need be taken for accurate dose calculation. Regarding the energy directly deposited by the primary electrons, the formulas developed in this article give very good agreement with Monte Carlo calculations for the thick half‐slab configuration, as will be seen in the next article in this series. Moreover, our first‐rank, second‐order formulas, when expressed in Fourier‐transformed space, are simple enough to be implemented in a treatment planning system providing full three‐dimensional electron dose calculation for arbitrary configurations of inhomogeneities.

An experiment was performed at the Lawrence Berkeley Laboratory BEVALAC to measure the multiple Coulomb scattering of 650-MeV/A uranium nuclei in 0.19 radiation lengths of a Cu target. Differential distributions in the projected multiple scattering angle were measured in the vertical and horizontal planes using silicon position-sensitive detectors to determine particle trajectories before and after target scattering. The results were compared with the multiple Coulomb scattering theories of Fermi and Molière, and with a modification of the Fermi theory, using a Monte Carlo simulation. These theories were in excellent agreement with experiment at the 2 sigma level. The best quantitative agreement is obtained with the Gaussian distribution predicted by the modified Fermi theory.

A two-source model for the calculation of relative output factors (ROF) for clinical applications of electron beams has been developed. The model consists of (1) an effective extended source above the final field-defining aperture (cutout) plane and (2) a source due to scattering from the aperture. Calculations are based on Fermi–Eyges theory and a pencil beam algorithm with parameters determined independently for each major scattering component. The model predicts a modified inverse square law for determining the dose rate for the electron beams. It also generalizes the “square-root method” and “one-dimensional method” that are often used clinically for ROF calculations. A computer program based on the model has been developed to calculate ROF for irregular fields. The predictions of ROF values have been compared with measurements on a Varian CLINAC 2100C/D accelerator for different cutout size, energies, applicators, and SSDs for square fields, rectangular fields, circular fields, and irregular fields. The agreement between prediction and measurement of the ROF for these wide range of conditions is generally within 1% for energies from 6 to 20 MeV. This two-source model can be used for clinical applications and it requires a minimal set of measured input data.

The dose parameters for the beta-particle emitting 90Sr/90Y source for intravascular brachytherapy (IVBT) have been calculated by different investigators. At a distant distance from the source, noticeable differences are seen in these parameters calculated using different Monte Carlo codes. The purpose of this work is to quantify as well as to understand these differences. We have compared a series of calculations using an EGS4, an EGSnrc, and the MCNP Monte Carlo codes. Data calculated and compared include the depth dose curve for a broad parallel beam of electrons, and radial dose distributions for point electron sources (monoenergetic or polyenergetic) and for a real 90Sr/90Y source. For the 90Sr/90Y source, the doses at the reference position (2 mm radial distance) calculated by the three code agree within 2%. However, the differences between the dose calculated by the three codes can be over 20% in the radial distance range interested in IVBT. The difference increases with radial distance from source, and reaches 30% at the tail of dose curve. These differences may be partially attributed to the different multiple scattering theories and Monte Carlo models for electron transport adopted in these three codes. Doses calculated by the EGSnrc code are more accurate than those by the EGS4. The two calculations agree within 5% for radial distance

Pencil beam kernels describing scattered photon fluence behind homogeneous water slabs at various air gap distances were generated using the EGS Monte Carlo code. Photon scatter fluence was scored in separate bins based on the particle's history: singly scattered, multiply scattered, and bremsstrahlung and positron annihilation photons. Simultaneously, the mean energy and mean angle with respect to the incident photon pencil beam were tallied. Kernels were generated for incident photon pencil beams exhibiting monoenergetic spectra of 2.0 and 10.0 MeV, and polyenergetic spectra representative of 6 and 24 MV beams. Reciprocity was used to generate scatter fractions on the central axis for various field sizes, phantom thicknesses, and air gaps. The scatter kernels were further characterized by full width at half-maximum estimates. Modulation transfer functions were calculated, providing theoretical estimates of the limit of performance of portal imaging systems due to the intrinsic scattering of photon radiation through the patient.

Purpose: Spot size, i.e. the lateral extent of the proton pencil beams used to compose a patient treatment field, is an important characteristic of a proton therapy scanning nozzle. Among other things, the spot size determines the lateral penumbra that may be achieved through pencil beam weight optimization. In this paper we describe measurements of spot size made on a synchrotron based proton therapy scanning system located at the University of Texas MD Anderson Cancer Center, in Houston. Method and Materials: Spots size is measured using three different techniques: pinpoint chamber, EBT film, and Lanex screen with CCDcamera. Spot size is measured by fitting a Gaussian distribution to the measured data. The Lanex CCD system consists of a Lanex screen, enclosed in a light tight box, which is being observed with a CCDcamera by way of a Mylar mirror. (See Figure 1.) Software was written in order to allow for an efficient acquisition of data under the condition of a pulsed proton beam. Results: In all three techniques spot size was measured at isocenter, in air. All three systems were found to give the same result within the margins of uncertainty for each measurement technique. Spot size was found to strongly depend on energy. Furthermore the spot proved not to be circularly symmetric but rather to have a larger extent in the cross plane direction. See Figure 2. We also found a non‐Gaussian component to the beam spot, consistent with Molier's theory of multiple Coulomb scattering. (See Figure 3.) Conclusion: We are able to characterize the spot size of a proton therapy scanning nozzle using three different techniques. The three different techniques differ in signal to noise ratio and in the time it takes to take the measurement.

The theory of electron penetration as predicted by the Fokker–Planck equation is first reviewed within a restricted context that considers the multiple scattering and transport of charged particles. We then broaden the context and show that range straggling effects also fit successfully into this framework, which completes an electron model initiated by Yang. We introduce those effects with a superposition of Fokker–Planck solutions, i.e., by using an incident beam that contains a spectrum of initial energies, or equivalently, a set of csda ranges. Straggling effects appear to be a beam property in this approach but are returned to the material when we use it. All the information needed to construct the spectrum is obtained from a measurement of the electron rest charge distribution in polystyrene. To illustrate the correctness of this procedure, we consider the case of a 20 MeV electron beam incident on water. We predict the absorbed dose distribution as a function of depth and also measure it with an ionization chamber in a water tank. We find nearly perfect agreement between calculation and experiment in this case where all the results derive and apply to a clinically operational machine.

Characteristics of dual-foil scattered electron beams shaped with a multileaf collimator (MLC) (instead of an applicator system) were studied. The electron beams, with energies between 10 and 25 MeV, were produced by a racetrack microtron using a dual-foil scattering system. For a range of field sizes, depth dose curves, profiles, penumbra width, angular spread in air, and effective and virtual source positions were compared. Measurements were made when the MLC alone provided collimation and when an applicator provided collimation. Identical penumbra widths were obtained at a source-to-surface distance of 85 cm for the MLC and 110 cm for the applicator. The MLC-shaped beams had characteristics similar to other machines which use trimmers or applicators to collimate scanned or scattered electron beams. Values of the effective source position and the angular spread parameter for the MLC beams were similar to those of the dual-foil scattered beams of the Varian Clinac 2100 CD and the scanned beams of the Sagittaire linear accelerators. A model, based on Fermi-Eyges multiple scattering theory, was adapted and applied successfully to predict penumbra width as a function of collimator-surface distance.

Previous experimental and theoretical investigations of the utility of time-resolved methods as a means of optical imaging through the human breast have indicated that a spatial resolution of approximately 1 cm is achievable by isolating the shortest path length photons which propagate through the tissue. Studies have also shown that resolution may be improved further by extrapolating the measured distribution using an appropriate model of photon transport. The experiments described here were performed in order to observe the relationship between achievable spatial resolution and the thickness of the medium. For a given time gate, an improvement in the spatial resolution was observed as the object thickness was reduced. Overall, the results indicate that a breast compression of about 1 cm may improve the limiting spatial resolution by as much as 7 mm. Less encouraging is the implication that temporal extrapolation over several orders of magnitude in intensity is required to achieve a comparable improvement in spatial resolution.

The Method of Moments is generalized to predict the dose deposited by a prescribed source of electrons in a homogeneous medium. The essence of this method is (i) to determine, directly from the linear Boltzmann equation, the exact mean fluence, mean spatial displacements, and mean-squared spatial displacements, as functions of energy; and (ii) to represent the fluence and dose distributions accurately using this information. Unlike the Fermi-Eyges theory, the Method of Moments is not limited to small-angle scattering and small angle of flight, nor does it require that all electrons at any specified depth z have one specified energy E(z). The sole approximation in the present application is that for each electron energy E, the scalar fluence is represented as a spatial Gaussian, whose moments agree with those of the linear Boltzmann solution. Numerical comparisons with Monte Carlo calculations show that the Method of Moments yields expressions for the depth-dose curve, radial dose profiles, and fluence that are significantly more accurate than those provided by the Fermi-Eyges theory.

Purpose: Phosphor materials provide challenges to both fundamental research and breakthrough development of technologies in research areas. In recent years, with the development of science and technology in the field of materials, a number of physical or chemical synthesis methods have been developed and successfully used for the preparation of submicrometer-sized phosphors. The present paper provides a rigorous analysis of light diffusion capabilities of phosphor materials in submicrometer-scale investigating the effect of light wavelength. Methods: Mie scattering theory and Monte Carlo simulation techniques were used for the optical diffusion performance providing numerical calculations. The Monte Carlo model included: (i) phosphor layers composed of different thickness (200, 500, 1000μm) and (ii) different light wavelength values (420, 545, 610 nm) corresponding to different types of activators, such as Ce, Tb, and Eu activators, respectively. Results: Based on Mie calculations, it was found that for low values of refractive index (e.g., 1.4) and for particle radius from 250 up to 500 nm no significant variations occurred on optical parameters. Monte Carlo simulations showed that the resolution increases as light wavelength decreases, respectively, however, this increase is more obvious at lower thickness values (i.e., at 200μm). In particular, as light wavelength decreases from 610 down to 545 and 420 nm, the resolution increases 4.4% and 13.9%, respectively (at 200 μm layer thickness). In addition, as layer thickness increases from 200 up to 500 μm the resolution decreases 50.2% while an increase up to 1000 μm causes a decrease of 70.2% (at 420 nm light wavelength). Conclusions: The goal of the author's study was to investigate the optical diffusion characteristics of submicrometer phosphor materials using Mie scattering theory and Monte Carlo simulation. The present investigation indicated that a key parameter on resolution improvement was the amount of light loss which depends on the choice of activator and affects the lateral spreading.

Purpose: To develop an analytical model for fast and accurate scatter estimation in kV cone‐beam CT images. Methods: With a point source of 100 keV x‐rays we used the Klein‐Nishina (KN) formula for Compton scattering differential cross‐section and integrated the point scatter along an interaction line of tissue‐equivalent medium to obtain a beam‐scatter‐kernel (BSK). We limited our derivation to the case of small imaging field and small scattering angles. The Taylor series of the integrand was approximated by terms up to the 4th order. The integral was then simplified to an analytical form. To calculate attenuation in the medium for scattered photons, we assumed all the scattered photons from a BSK originate from the beam center and have the same energy as the primary photons. We then performed robust calculations using the point scatter theorem for a cubic water phantom (10×10×10 cm3) and treated the results as theoretical expectations for inter‐comparison. Results: Compared to the point‐kernel method, BSK calculations show a +10% systematic difference for either a perpendicular or a tilted beam for the absolute value of scatter fluence. However, the largest difference on the relative scatter distribution is as low as 0.4% and 2.5% for perpendicular and tilted beams, respectively. The largest relative difference for the whole phantom is about ‐ 1.8%. In computation time, our integrated scatter model is faster by over 2 orders than the point kernel method. Conclusion: Our kV scatter model shows a very high computational efficiency in scatter estimation. The expected behavior of scattering distribution is observed and well estimated. We are validating more thoroughly for small imaging fields (∼10×10cm2) and the model is expected to have great potential for imaging small objects and with interior reconstruction algorithms. Future work will extend the frame for multiple photon energies and tissue non‐homogeneity.

Purpose: Pencil beam algorithms are commonly used for proton therapy dose calculations. Szymanowski and Oelfke [“Two-dimensional pencil beam scaling: An improved proton dose algorithm for heterogeneous media,” Phys. Med. Biol. 47, 3313–3330 (2002) doi: 10.1088/0031-9155/47/18/304.] developed a two-dimensional (2D) scaling algorithm which accurately models the radial pencil beam width as a function of depth in heterogeneous slab geometries using a scaled expression for the radial kernel width in water as a function of depth and kinetic energy. However, an assumption made in the derivation of the technique limits its range of validity to cases where the input expression for the radial kernel width in water is derived from a local scattering power model. The goal of this work is to derive a generalized form of 2D pencil beam scaling that is independent of the scattering power model and appropriate for use with any expression for the radial kernel width in water as a function of depth. Methods: Using Fermi-Eyges transport theory, the authors derive an expression for the radial pencil beam width in heterogeneous slab geometries which is independent of the proton scattering power and related quantities. The authors then perform test calculations in homogeneous and heterogeneous slab phantoms using both the original 2D scaling model and the new model with expressions for the radial kernel width in water computed from both local and nonlocal scattering power models, as well as a nonlocal parameterization of Moliere scattering theory. In addition to kernel width calculations, dose calculations are also performed for a narrow Gaussian proton beam. Results: Pencil beam width calculations indicate that both 2D scaling formalisms perform well when the radial kernel width in water is derived from a local scattering power model. Computing the radial kernel width from a nonlocal scattering model results in the local 2D scaling formula under-predicting the pencil beam width by as much as 1.4 mm (21%) at the depth of the Bragg peak for a 220 MeV proton beam in homogeneous water. This translates into a 32% dose discrepancy for a 5 mm Gaussian proton beam. Similar trends were observed for calculations made in heterogeneous slab phantoms where it was also noted that errors tend to increase with greater beam penetration. The generalized 2D scaling model performs well in all situations, with a maximum dose error of 0.3% at the Bragg peak in a heterogeneous phantom containing 3 cm of hard bone. Conclusions: The authors have derived a generalized form of 2D pencil beam scaling which is independent of the proton scattering power model and robust to the functional form of the radial kernel width in water used for the calculations. Sample calculations made with this model show excellent agreement with expected values in both homogeneous water and heterogeneous phantoms.

The development of treatment planning methods in radiation therapy requires dose calculation methods that are both accurate and general enough to provide a dose per unit monitor setting for a broad variety of fields and beam modifiers. The purpose of this work was to develop models for calculation of scatter and transmission for photon beam attenuators such as compensating filters, wedges, and block trays. The attenuation of the beam is calculated using a spectrum of the beam, and a correction factor based on attenuation measurements. Small angle coherent scatter and electron binding effects on scattering cross sections are considered by use of a correction factor. Quality changes in beam penetrability and energy fluence to dose conversion are modeled by use of the calculated primary beam spectrum after passage through the attenuator. The beam spectra are derived by the depth dose effective method, i.e., by minimizing the difference between measured and calculated depth dose distributions, where the calculated distributions are derived by superposing data from a database for monoenergetic photons. The attenuator scatter is integrated over the area viewed from the calculation point of view using first scatter theory. Calculations are simplified by replacing the energy and angular-dependent cross-section formulas with the forward scatter constant r2(0) and a set of parametrized correction functions. The set of corrections include functions for the Compton energy loss, scatter attenuation, and secondary bremsstrahlung production. The effect of charged particle contamination is bypassed by avoiding use of dmax for absolute dose calibrations. The results of the model are compared with scatter measurements in air for copper and lead filters and with dose to a water phantom for lead filters for 4 and 18 MV. For attenuated beams, downstream of the buildup region, the calculated results agree with measurements on the 1.5% level. The accuracy was slightly less in situations where the scatter component is very large, as for very large fields with very short filter to detector distances. The implementation of the model into treatment planning systems is discussed.

Fermi-Eyges electron-scattering theory has been incorporated into the primary dose calculation for external x-ray beam radiotherapy using the convolution method. Incorporating scattering theory into the convolution technique accounts for the density distribution between the interaction and deposition sites, whereas conventional convolution methods only consider the average density between these two points. As the lateral spread of electrons ejected from an interaction site depends on the density distribution, the energy deposition (and hence dose distribution) is predicted more accurately if scattering is accounted for. This new method gives depth dose curves which show better agreement with Monte Carlo calculations in a (slab inhomogeneity) lung phantom than a conventional convolution method, especially at high energies and small field sizes where lateral electronic disequilibrium exists at the central axis. For a 5 x 5-cm2 18-MV beam incident on the lung phantom, a reduction in the maximum error between the convolution and Monte Carlo depth dose curves from 5% to 2.5% is obtained when scattering theory is used in the primary dose calculation. Incorporating scattering theory into the convolution calculation increases the computation time of the primary dose by a factor of 3.

The purpose of this study is to develop a highly accurate and fast method for calculating electron beam dose distributions in Modulated Electron Radiation Therapy (MERT). An algorithm has been developed for creating phase-space files at the exit of a linear accelerator for any arbitrary intensity and energy electron beam without the need of full Monte Carlo simulations. The model assigns each particle to one of the 3 following sources: primary, secondary collimator and electron collimator scatter. The primary component is derived by fast MC transport in air. The scatter components are derived by the use of MC pre-calculated leaf kernels. Each kernel includes the fluence distribution, energy distribution and scatter probability of generating an electron from a leaf. The original position is sampled from tunable Gaussian or uniform distributions. The direction is estimated by geometrical means. According to the projection of the direction a particle is rejected if it is expected to suffer a leaf-hit. A leaf-hit counter is used to calculate the output of scatter particles based on the pre-calculated scatter probabilities. To account for multiple coulomb scattering in air a MC-corrected version of the Fermi-Eyges scattering theory was implemented. Depth and profile dose distributions were derived for the largest and smallest square field sizes, as well as for irregular and off-axis fields. The model agreed with full MC dose distributions within 3 % in all cases. Output at the depth of maximum dose exhibited discrepancies less than 2.6 % in all cases. The model was 16-22 times faster in generating a phase-space file than a full MC simulation with the BEAMnrc code.

Purpose: To develop a highly accurate and fast method for calculating electron beamdose distributions in Modulated Electron Radiation Therapy (MERT). Method: An algorithm has been developed for creating phase‐ space files at the exit of a linear accelerator for any arbitrary intensity and energy electron beam without the need of full Monte Carlo simulations. The model assigns each particle to one of the 3 following sources: primary, secondary collimator and electron collimatorscatter. The primary component is derived by fast MC transport in air. The scatter components are derived by the use of MC pre‐calculated leaf kernels. Each kernel includes the fluence distribution, energy distribution and scatter probability of generating an electron from a leaf. The original position is sampled from Gaussian or uniform distributions. The direction is estimated by geometrical means. According to the projection of the direction, a particle is rejected if it is expected to suffer a leaf‐hit. A leaf‐hit counter is used to calculate the output of scatter particles based on the pre‐calculated scatter probabilities. To account for multiple coulomb scattering in air a MC‐corrected version of the Fermi‐Eyges scattering theory was implemented. Results: Depth and profile dose distributions were derived for the largest and smallest square field sizes, as well as for irregular and off‐axis fields. The model agreed with full MCdose distributions within 3% in all cases. Output at the depth of maximum dose exhibited discrepancies less than 2.6% in all cases. The model was 16–22 times faster in generating a phase‐space file than a full MC simulation with the BEAMnrc code. Conclusions: Fast, dynamic electron beam calculations open up the possibility for real time delivery of MERT in the clinic and renew interest in electron beam therapy.

Purpose: To develop an improved nuclear halo dose model of a pencil beam algorithm (PBA) for dose calculation of proton beams in heterogeneous media. Methods: The proton PBA consisted of a central axis term and an off axis term. The central axis term was determined from a central axis depth dose of a proton beam in water and was scaled by the mass stopping power ratio to account for other materials. The off axis term was determined from Fermi‐Eyges scattering theory with material‐dependent scattering powers to calculate the spread of the proton beam in heterogeneous media. The nuclear halo dose arising from large angle and non‐elastic scatter was modeled using two terms: a Gaussian and a Cauchy‐Lorentz distribution. Depth‐dependent widths and amplitudes of each distribution were determined by fitting a simulated 1‐mm × 1‐mm pencil beam in water. The PBA was evaluated in approximately 30 test phantoms containing bone and/or air heterogeneities for 4 energies and 2 field sizes. Agreement between PBA and Monte Carlo simulations of the test conditions was quantified by computing the percentage of points within 2 percent dose difference or 1 mm distance to agreement. Results: With the improved nuclear halo model, PBA calculations showed better than of 97% of dose points within 2%/1 mm of MC distributions for all geometries examined. For phantoms containing laterally infinite heterogeneities, agreement between PBA and MC distributions was 100% at 2%/1mm. For phantoms containing laterally finite heterogeneities, agreement was at least 97%. The points failing were due to the central axis approximation of the PBA in regions not influenced by the nuclear halo model. Conclusions: The halo model developed in this work improves the agreement of the PBA with MC simulations in heterogeneous phantoms, particularly in low‐dose regions that can be important for scanned‐beam proton therapy. “This research was supported by contract W81XWH‐10‐1‐0005 awarded by The U.S. Army Research Acquisition Activity, 820 Chandler Street, Fort Detrick, MD 21702‐5014. This report does not necessarily reflect the position or policy of the Government, and no official endorsement should be inferred.”

Purpose: The main objective of this study was to determine the optical properties(scattering and absorption coefficients)in vivo by interstitial measurements during H&N PDT.Methods and Matierals: We developed a model for quick and accurate determination of tissue optical properties(absorption coefficient ⌈a and transportscattering coefficient ⌈s') in semi‐infinite medium, which is theoretically suitable for determination of optical properties head and neck with ALA‐mediated PDT. We measured and determined the optical properties at the treatment wavelength. Measurements were performed before and after the delivery of the light treatment. The light detection device consists of two parallel light transmitting catheters placed 5mm apart. Diffuse light, from a 2mm cylindrical diffusing tip, is collected by an isotropic detector with a 0.5mm scattering bulb whose position is determined by a computer controlled step motor. The system automatically records and plots the light fluence rate per unit source power as a function of position. Results: The result is fitted with a diffusion equation to extrapolate ⌈a and ⌈s'. A theory based on light source on semi‐infinite medium has been developed to interpret the measured data. Another measurement was done using a fiber‐optic based probe. This probe was placed on the tissue surface, and white light was delivered to the tumor surface by one of its source fibers. The diffusely reflected white light was collected by a spectrograph and analyzed to determine the absorption and scatteringspectra of the tissue. Results from both methods were compared for accuracy. To test the ability of this algorithm to accurately recover the optical properties of the tissue, we made measurements in tissue simulating phantoms. Conclusion: We developed a model for quick and accurate determination of tissue optical properties in semi‐infinite medium, which is theoretically suitable for determination of optical properties head and neck PDT.

Purpose: In PET/MR, creation of an attenuation map from the patient's MR image is critical to quantitative PET but remains challenging because MR signals are not directly related to photon attenuation. The existing methods segment the T1‐weighted MR image and then assign average tissue attenuation coefficients to the segmented classes. Before the segmentation, coregistration between the MR image and CT, initially corrected PET and/or a template of MR images is performed. None of the methods are based on rigorous physical theory. For the first time, we developed a method from quantum scatter theory for estimating the attenuation map using the patient's proton density weighted MR image only. No coregistration is needed. Methods:Photon attenuation in human tissues at the photon energy of 511 keV is dominated by Compton scatter of which the probability depends on Z/A of the tissues. Here Z is atomic number and A atomic mass. Since Z/A is equal or close to 1/2 for the major elements in tissues, except hydrogen (Z/A = 1), the tissue mass attenuation coefficient can be calculated from the amount of hydrogen atoms in unit tissue that is obtained from a proton density weighted MR image. Initial assessment has been done by applying the method to calculating mass attenuation coefficients and then comparing with measured values. Results: The calculated attenuation coefficients agree well with the measured values. The differences between the calculated and measured mass attenuation coefficients for water, dry air, muscle and bone are 0.17%, 0.10%, 1.8% and 0.95%, respectively. The calculated number of hydrogen atoms in 1 gram of water also agrees with the exact value with a deviation of 1.8%. Conclusions: The scatter theory based method produces accurate attenuation coefficients. However, experimental evaluation with phantoms or/and patients must be performed and is currently under way.

Purpose: Accurate modeling of multiple Coulomb scattering(MCS) of heavy charged particles is imperative to achieve accurate dose distributions for proton therapy simulations. The Gaussian approximation based on Rossi's theory used in the Monte Carlo N Particle extended (MCNPX) code overestimates the characteristicscattering angle for thin materials. The purpose of this research was to evaluate a new, more accurate MCS algorithm based on Moliere theory that has been implemented in MCNPX.Method and Materials: Simulations of a proton beam with a range in water of 16.1 cm incident on steel spheres of diameters 6.4 mm, 9.5 mm, and 15.9 mm were run and compared to previous simulations completed with the old algorithm and to measured data. Simulated dose was tallied in air at the locations of the film measurements, which were 2 cm, 15 cm, 30 cm, and 63 cm distal to the spheres. Results: When compared to the old MCS algorithm, the new MCS algorithm more accurately modeled the maximum dose enhancement due to scattering 58% of the time and the maximum dose shadow due to scattering 83% of the time. Visual inspection of the dose profiles distal to the steel spheres reveals the consistent improvement in the dose distribution resulting from the new algorithm. Conclusion: The new MCS algorithm appears to be more accurate than the old algorithm. In order to further characterize the algorithm's performance, additional simulations with a simpler geometry and comparisons with measurements are required.

Purpose: The method of obtaining the primary x‐ray spectrum by the Compton scattering correction has been established for relatively high x‐ray tube voltages. However, the influence of Rayleigh scattering can not be neglected in the mammography energy range. Besides, accurate data of Rayleigh and Compton cross sections for the scatterermaterial are required for obtaining the primary x‐ray spectrum by correcting the scattered x‐ray spectrum in mammography. The purpose of this study is to investigate theoretically and experimentally the relative contributions of Rayleigh scattering for PMMA (polymethyl methacrylate) in the mammography energy range. Method and Materials: Fluorescent x‐rays in the energy range of 10 to 25 keV were generated by exciting four metal targets including molybdenum by synchrotron radiation. The fluorescent x‐rays were incident on a sphere of PMMA, and the scattered x‐rays were measured with a CdTe detector at scattering angles of 90, 120, 150, 165 degrees. The scattered fluorescent x‐ray spectrum was separated into the Rayleigh and Compton scattering peaks by using a curve fitting technique with two Gaussian functions. We compared the measured cross section data to those obtained from theoretical values. We also calculated the scattered fluorescent x‐ray spectra using the Monte Carlo simulation and compared them to the measuredspectra.Results: The measured ratio of Rayleigh to total scattering cross section was about 30% at 10 keV. The maximum discrepancy of the measured and theoretically calculated values of the ratio was about 30% for the scattering angles of 90 to 165 degrees. However, at the angle of 120 degrees where the overall error might be minimum, the theoretical and experimental values agreed within 2%. Conclusion: Rayleigh and Compton scattering cross sections should be measured more precisely in order to improve the calculation of the primary mammography x‐ray spectrum from the scattered x‐ray spectrum.

Purpose: Development of an easy to commission electron beammodel using highly accurate fluence benchmarks from Monte Carlo treatment head simulations and measurements. Method and Materials: A simple electron beammodel consists of the key treatment head components: a circular primary source for electrons, a disk representing the secondary scattering foil and a ring representing the monitor chamber sidewall. The primary source accounts for scatter in the exit window and primary foil. The angular distribution of the primary source is chosen to match the lateral dose profiles from measurements with only the primary foil in the treatment head. The energy spectrum is derived from unfolding measured depth dose curves from large fields. The angular distribution of the primary source is broadened analytically using accurate scattering theory to account for the secondary scattering foil. The lateral extent of the beam is restricted to account for the collimation by the sidewall of the monitor chamber. Phase space data generated from the simple electron beammodel is used with the EGSnrc user code BEAMnrc to simulate the secondary collimators and scattering of the electrons in air. Results: Primary foil simulations with the beam model, with no secondary scattering foil or monitor chamber in the beam line, match measured dose profiles of large fields within 1%–2%. Fluence profiles were in comparable agreement with the benchmark. Energy spectra were cleanly unfolded from measured depth dose curves from large fields. Peak energy was within 0.05 MeV of the fluence benchmark. Preliminary results for the full clinical beam with the secondary foil and chamber in place are promising. Conclusions: The influence of the different treatment head components on the beam were accurately determined with the detailed fluence benchmarks, which proved instrumental in development of this beam model. Research sponsored by NIH R01_CA104777.

Purpose: Understanding the limitation of the Monte Carlo codes is essential in order to avoid systematic errors in simulations, and to suggest further improvement of the codes. The purpose of this study was to simulate the treatment head of Elekta (Synergy™) linear accelerator with two different Monte Carlo codes called MCNP4C and BEAMnrc and compare the calculated data of both codes with each other and with experimental measurements. Method and Materials: MCNP4C and BEAMnrc Monte Carlo codes commonly used in medical physics, were compared and evaluated against electron depth dose,beam profile and experimental total scatter factor results obtained using clinical radiotherapybeams. Different physical models and algorithms used in the codes give different depth dose curves, beam profile and total scattering factors. Results: The maximum difference for percentage depths dose curves in the Build‐up region was about 1.1% and it was less than 1mm in the dmax. These differences for the dose profile curves in the flat area was at last 0.9% and for the penumbra was about 1.5mm and in the out field regions which only the scatterradiation reach that area, the maximum difference was about 0.2%. Conclusion: In conclusion, the differences in cross sections used for bremsstralung effect, different multiple scatteringtheory and different electron transport algorithms used by MCNP4C and BEAMnrc will lead to discrepancy in depth dose and beam profile curves, which is negligible for both codes if the simulations were well tuned. The timing study is also shows that BEAMnrc is generally faster than MCNP4C and use of large number of scoring voxels dramatically slows down the MCNP calculation. However, use of a large number of geometry voxels in MCNP only slightly affects the speed of the calculation.

Purpose: Pencil beam approach for proton beams is not accurate in the Bragg peak region because the multiple scatteringtheories are derived for monoenergetic particles and, therefore, can be applied only within CSDA range. We improve the pencil beam model to account for the range straggling in the Bragg peak region. Method and Materials: Depth dose distribution from a monoenergetic proton beam can be calculated by folding a CSDA depth dose distribution with a function which describes range struggling. Performing the integration numerically, we obtain that the dose distribution from a proton beam can be presented as a linear combination of the CSDA depth dose distributions with weights which are defined by range straggling distribution. For particles traveling in each CSDA approximation, we can consistently apply multiple scatteringtheory. The final equation for 3D proton distribution covers entire range of a proton pencil beam. Results: We have obtained an analytical expression for the 3D dose distribution from a proton pencil beam which covers entire penetration range. The equation consistently describes the depth dose distribution and the lateral spread due to range straggling in the Bragg peak region. This approach accounts also for proton loss from the beam at the end of the range. Conclusions: 3D dose distribution from a proton pencil beam can be calculated using a simple analytical formula for entire proton range. This formula is convenient for use in treatment planning with therapeuticproton beams, particularly, in the heterogeneous compositions.

Purpose: To improve quality and reliability of dose calculation based on pencil beam model, in treatment planning for intensity‐modulated proton therapy (IMPT). Method and Materials: Inverse treatment planning system KonRad (developed at DKFZ Heidelberg, commercially distributed by Siemens) employs pencil beam algorithm for dose calculation. Dose distributions for proton energies of 60 through 250 MeV, and various widths of the beam were precalculated using a Monte Carlo code, based on GEANT4. The results, which included description of depth dose profiles and beam broadening in tissue due to multiple Coulomb scattering, were stored in the pencil beam database. The updated database was used to optimize IMPT treatment plans. Dose calculation, assuming delivery of optimized beam intensity patterns by continuous magnetic scanning, was repeated with Monte Carlo, and the results were then compared to those obtained with KonRad pencil‐beam‐based calculation. Results: Monte Carlo calculation was in good agreement with a set of proton depth dose measurements taken in a water phantom. With the nuclear interactions “turned off”, Monte Carlo description of multiple Coulomb scattering agreed well with the theoretical model. Nuclear interactions contributed substantially to broadening of the dose distribution for higher‐energy (over 150 MeV) proton beams, mainly at relatively shallow depths. Fitting both nuclear and non‐nuclear contributions to a single Gaussian, for the database, produced a negligible systematic error in KonRad dose calculation at shallow depths (less than 0.3% of the peak dose). For sample plans, Monte Carlo and KonRad dose distributions (point doses) agreed to within 5 percent of the target prescription, except in the lung (non‐cancerous) tissue, where disagreement of up to 15% was observed. Conclusion: The use of Monte Carlo data in parametrization of pencil beam kernels improved the precision of proton dose calculation without increasing the calculation time. Supported in part by the NCI grant 5P01‐CA21239‐25.

Purpose: To develop an effective and efficient method for optimizing the beam current modulation (BCM) pattern in commissioning proton beams using passive scattering and range modulation wheels. This modulation pattern is essential in producing beams with properly weighted spread-out Bragg peaks for the desired depth-dose distribution. Method and Materials: The proton beam nozzle houses the spinning modulation wheel (600 rpm) and the various scattering components and translates the modulated beam current to a depth-dose distribution. This translation can be described, for a specific configuration of the nozzle, i.e., with a specific wheel position at a specific time, by a function C(x, t), defined as the dose produced at depth x and time t for a unit beam current at the time t. While the function C(x, t) is difficult to calculate theoretically due to the complex scattering paths in the nozzle and the phantom, it can be easily obtained by measuring the dose as a function of time and by a proper deconvolution taking into account the delays from the measuring circuit. With C(x, t) determined, one can calculate the depth dose distributions for any given BCM. Then the optimal BCM can be found, not by repeated measurements, as was often done before, but by numerical calculations. A window-based software has been developed to measure the time-dose functions and to perform numerical BCM optimizations. Results: The method was applied to optimize the BCM for a number of proton beams. The differences between the calculated and measured depth dose distributions are less than 1%, well within the clinical requirement. Conclusion: We have developed an effective and efficient method for optimizing the beam current modulation for proton therapy beams using passive scattering.

To address the deficiencies of currently available dose calculation algorithms for radiotherapy planning, two dose calculation methods have been devised. The first method incorporates Fermi–Eyges multiple scattering theory into the primary dose calculation of the superposition method for external x-ray beam radiotherapy. The inclusion of scattering theory into the superposition technique accounts for the density distribution between the primary photon interaction and energy deposition sites. The second method developed is the super-Monte Carlo (SMC) method. SMC calculates dose by a superposition of pregenerated Monte Carlo electron track kernels. For x-ray beams, the primary dose is calculated by transporting pregenerated (in water) Monte Carlo electron tracks from each primary photon interaction site. The length of each electron step is scaled by the inverse of the density of the medium at the beginning of the step. The scatter dose is calculated by superposition. SMC can also be applied to electron beam dose calculation. Pregenerated electron tracks are transported through media of varying density and atomic number. The perturbation of the electron fluence due to each material encountered by the electrons is explicitly accounted for. Dose distributions in a variety of phantoms show good agreement with Monte Carlo results. SMC is an accurate, three-dimensional unified photon/electron dose calculation algorithm.

It is shown that predictions can be made of scatter-to-primary ratios for a variety of mammographic configurations. The different configurations can be produced by changes in source-detector distance; source-phantom distance; air gap; photon energy; phantom composition, thickness, and cross-sectional area. A detailed analysis of the effect of a grid is also considered. The predictions are not computationally intensive. Experimental verifications of the predictions for varying phantom cross-sectional dimensions, source-detector distance, and air gaps with and without a grid have been carried out. Variations with phantom thickness were not considered in this paper. Detailed comparisons between experiment and theory indicate that scatter-to-primary ratios were predicted within one standard deviation and the coefficient of variation is within 3% for most of the data points with the worst case coefficient of variation about 6%. Measurements of grid secondary transmission have also been compared to theoretical predictions and agreement is within the coefficient of variation of about +/- 15%.

This article is the first in a series on the calculation of electron dose using multiple-scattering theory. In it we develop a unified theory, which we term Gaussian multiple-scattering theory, starting from a number of contributions already in the literature: the Fermi-Eyges multiple-scattering theory, the Yang path length distribution, the second-order multiple-scattering theory of Jette [Med. Phys. 12, 178 (1985)], and the diffusion theory of Bethe et al. [Proc. Am. Philos. Soc. 78, 573 (1938)]. After examining in detail the ramifications and limitations of Gaussian multiple-scattering theory, we derive basic formulas generalizing the Fermi-Eyges theory, for use in subsequent articles. We also find explicit, accurate expressions for incorporating the scattering power into the theory.

Photon dose calculations are frequently based on the separation of primary and scatter dose components. For small field sizes it is difficult to make this separation from measured data. Such problems lend themselves to solution by calculational methods, which permit rigorous definition of primary and scatter dose. Using Monte Carlo methods, we verify an equation derived from O'Connor's scaling theorem relating scatter-primary ratios at small depths and field sizes to those at larger depths and fields. This equation may prove to be useful when separating primary and scatter dose components, and consequently in dose calculations involving irregular fields or inhomogeneities.