Technical Note: On the analytical proton dose evaluation in compounds and mixtures.

PURPOSE: By combining the physical processes occurring due to the interaction of protons with matter, analytical theories published so far have provided acceptable models for calculating depth-dose distributions in homogeneous media. As a well-defined and comprehensive theory, the formula derived by Bortfeld models the dose transferred to the target in terms of the parabolic cylinder function. The model also includes three parameters with values specified for an initial proton energy and for the target material. These parameters are obtainable through the data gathered in nuclear data tables. The analytical studies using this interesting model are therefore restricted to those materials for which the data have been provided in these tables. This study aims to find general solutions for calculation of these parameters for a compound or mixture composed of an arbitrary choice of constituent elements. METHODS: Inspired by formulas dedicated for calculating the range and the probability of undergoing nonelastic nuclear interactions for protons in desired compounds, the analytical methods for finding the three mentioned parameters are investigated. The accuracy of the methods suggested is examined through comparison of the results with those which are calculated using the data taken from nuclear data tables. By employing the calculated parameters using the derived formulas in the Bortfeld model, the dose distribution at depth in a chosen target is calculated. RESULTS: For an arbitrary selection of compounds, the predictions of the analytical depth-dose model using these parameters have been found to closely match the results employing the parameters calculated using the data reported in nuclear data tables. CONCLUSIONS: The formulas presented are general, mathematically easy to handle, and valid for almost every compound or mixture including materials of interest for proton radiotherapy purposes, making the Bortfeld model more practical and advantageous.

Technical Note: On the proton range and nuclear interactions in compounds and mixtures.

PURPOSE: Range and probability of nonelastic nuclear interactions (NNIs) for protons can be found only for a limited number of compounds and mixtures in nuclear data tables, and the proton-related analytical studies are therefore restricted to those materials for which the data are provided in these documents. In this paper, the authors present general solutions for calculating the proton range and probability of NNIs for desired compounds and mixtures. METHODS: Benefiting from the Bragg-Kleeman approximation of mass stopping power, the authors derive a concise formula for calculating the proton range in materials with arbitrary number of constituent elements. Additionally, the authors propose another relation for obtaining the probability of undergoing NNIs which is suggested to be additive. RESULTS: The examination of the formula presented shows that the authors' method can be considered as general solutions for analytical evaluation of the range in compounds and mixtures. The formula proposed for probability of NNIs is valid for almost every compound except for those materials containing H. It is shown that this formula can be modified so that it covers these materials. CONCLUSIONS: The authors present a general analytical method for calculating the range and probability of NNIs for protons which are mathematically easy to handle and valid for desired compounds or mixtures composed of an arbitrary number of constituent elements, including materials of interest for proton radiotherapy purposes.

Inhibition of the epidermal growth factor receptor by erlotinib prevents immortalization of human cervical cells by Human Papillomavirus type 16.

The Human Papillomavirus type-16 (HPV-16) E6 and E7 oncogenes are selectively retained and expressed in cervical carcinomas, and expression of E6 and E7 is sufficient to immortalize human cervical epithelial cells. Expression of the epidermal growth factor receptor (EGFR) is often increased in cervical dysplasia and carcinoma, and HPV oncoproteins stimulate cell growth via the EGFR pathway. We found that erlotinib, a specific inhibitor of EGFR tyrosine kinase activity, prevented immortalization of cultured human cervical epithelial cells by the complete HPV-16 genome or the E6/E7 oncogenes. Erlotinib stimulated apoptosis in cells that expressed HPV-16 E6/E7 proteins and induced senescence in a subpopulation of cells that did not undergo apoptosis. Since immortalization by HPV E6/E7 is an important early event in cervical carcinogenesis, the EGFR is a potential target for chemoprevention or therapy in women who have a high risk for cervical cancer.

Creating a spread-out Bragg peak in proton beams.

The model of Bortfeld and Schlegel (1996 Phys. Med. Biol. 41 1331-9) for determining the weights of proton beams required to create a spread-out Bragg peak (SOBP) gives a significantly tilted SOBP. However, by arbitrarily varying its parameter p, which relates the range of protons to their energy, we have been able to create satisfactory SOBPs. MCNPX Monte Carlo calculations have been carried out to determine p, demonstrating the success of this modification. Optimal values of p are tabulated for various combinations of maximum beam energy E(0) (50, 100, 150, 200 and 250 MeV) and SOBP width χ (15%, 20%, 25%, 30%, 35% and 40%), as well as for a correction factor needed to calculate the SOBP dose. An example shows the application of these results to analyzing the dose deposited by deuterons and alpha particles in broad proton beams.

Oblique incidence for broad monoenergetic proton beams.

PURPOSE: The depth dose of a monoenergetic broad parallel proton beam has been modeled in a number of ways, but evidently not yet for oblique incidence. The purpose of this investigation is to find an accurate analytic formula for this case, which can then be used to model the depth dose of a broad beam with an initial Gaussian angular distribution. METHODS: The Bortfeld model of depth dose in a broad normally incident proton beam has been extended to the case of oblique incidence. This extension uses an empirically determined Gaussian parameter sigma(x) which (roughly) characterizes the off-axis dose of a proton pencil beam. As with Bortfeld's work, the modeling is done in terms of parabolic cylinder functions. To obtain the depth dose for an initial angular distribution, the result is integrated over the angle of incidence, weighted by a Gaussian probability function. The predictions of the theory have been compared to MCNPX Monte Carlo calculations for four phantom materials (water, bone, aluminum, and copper) and for initial proton energies of 50, 100, 150, 200, and 250 MeV. RESULTS: Comparisons of the depth dose predicted by this theory with Monte Carlo calculations have established that with very good accuracy, sigma(x) can be taken to be independent both of the depth and of the angle of incidence. As a function of initial proton range or of initial proton energy, sigma(x) has been found to obey a power law to very high accuracy. Good fits to Monte Carlo calculations have also been found for an initial Gaussian angular distribution. CONCLUSIONS: This investigation is the first step in the accurate modeling of a proton pencil beam with initial Gaussian angular distribution. It provides the longitudinal factor, with its Bragg peak buildup and sharp distal falloff. A transverse factor must still be incorporated into this theory and this will give the lateral penumbra of a collimated proton beam. Also, it will be necessary to model the dose of product particles from nuclear interactions of the proton beam. With the accurate modeling of a pencil beam, it will be possible to accurately take into account the effect of localized tissue inhomogeneities.

Deterministic photon kerma distribution based on the Boltzmann equation for external beam radiation therapy.

A photon transport algorithm for fully three-dimensional radiotherapy treatment planning has been developed based on the discrete ordinates (SN) solution of the Boltzmann equation. The algorithm is characterized by orthogonal adaptive meshes, which place additional points where large gradients occur and a procedure to evaluate the collided flux using the representation of spherical harmonic expansion instead of the summation of the volume-weighted contribution from discrete angles. The Boltzmann equation was solved in the form of SN spatial, energy, and angular discretization with mitigation of ray effects by the first-collision source method. Unlike existing SN codes, which were designed for general purpose for multiparticle transport in areas such as nuclear engineering, our code is optimized for medical radiation transport. To validate the algorithm, several examples were employed to calculate the photon flux distribution. Numerical results show good agreement with the Monte Carlo calculations using EGSnrc.

Three-dimensional electron dose calculation using an improved hybrid pencil beam model.

An improved hybrid-pencil beam model (HPBM) for electron-beam three-dimensional dose calculation has been studied. The model is based on the fact that away from the edges of a large field, the electron distribution function exactly equals that for an infinitely wide electron beam. In the present model, we use the bipartition model to calculate the longitudinal part of the pencil-beam distribution function, and Fermi-Eyges multiple-scattering theory to calculate its transverse part. In order to describe the electron beam characteristics accurately, we introduce a new parameter, which is extracted from measured profile data near the surface of a water phantom, to correct the transverse distribution determined by the Fermi-Eyges theory. Furthermore, we introduce an effective energy spectrum to describe the effect on the collimated electron beam of the accelerator head. The dose distributions calculated with the improved HPBM were compared with the experimental data, and the agreement was within 1% in most of cases. This preliminary study has demonstrated the potential for use of the model in the clinical therapy.

Magnetic fields with photon beams: planar-current-induced magnetic fields.

Strong transverse magnetic fields can produce very large dose enhancements and reductions in localized regions of a patient under irradiation by a photon beam. In this work we consider planar-current-induced magnetic fields ("PCIMFs") generated by arbitrary electric currents in one or two parallel planes, and pose two questions: how much arbitrariness is there in specifying a PCIMF, and how can we solve the "inverse problem" of determining the current distribution which generates a chosen PCIMF? We have completely answered both questions, and have applied the general formulas which we have developed to the case of cylindrical symmetry, giving a concrete example of our method. The present work provides the theoretical tools for designing PCIMFs, but a great deal of systematic research will be required in order to understand and design magnetic fields which produce desired distributions of dose enhancement and dose reduction in photon beams treating patients.