One-dimensional transport equation models for sound energy propagation in long spaces: theory.

In this paper, a three-dimensional transport equation model is developed to describe the sound energy propagation in a long space. Then this model is reduced to a one-dimensional model by approximating the solution using the method of weighted residuals. The one-dimensional transport equation model directly describes the sound energy propagation in the "long" dimension and deals with the sound energy in the "short" dimensions by prescribed functions. Also, the one-dimensional model consists of a coupled set of N transport equations. Only N=1 and N=2 are discussed in this paper. For larger N, although the accuracy could be improved, the calculation time is expected to significantly increase, which diminishes the advantage of the model in terms of its computational efficiency.

Adaptive diffusion smoothing: a diffusion-based method to reduce IMRT field complexity.

Inverse-planned intensity modulated radiation therapy (IMRT) is often able to achieve complex treatment planning goals that are unattainable with forward three-dimensional (3D) conformal planning. However, the common use of IMRT has introduced several new challenges. The potentially high degree of modulation in IMRT beams risks the loss of some advantages of 3D planning, such as excellent target coverage and high delivery efficiency. Previous attempts to reduce beam complexity by smoothing often result in plan degradation because the smoothing algorithm cannot distinguish between areas of desirable and undesirable modulation. The purpose of this work is to introduce and evaluate adaptive diffusion smoothing (ADS), a novel procedure designed to preferentially reduce IMRT beam complexity. In this method, a discrete diffusion equation is used to smooth IMRT beams using diffusion coefficients, automatically defined for each beamlet, that dictate the degree of smoothing allowed for each beamlet. This yields a method that can distinguish between areas of desirable and undesirable modulation. The ADS method has been incorporated into our optimization system as a weighted cost function penalty, with two diffusion coefficient definitions designed to promote: (1) uniform smoothing everywhere or (2) smoothing based on cost function gradients with respect to the plan beamlet intensities. The ADS method (with both coefficient types) has been tested in a phantom and in two clinical examples (prostate and head/neck). Both types of diffusion coefficients produce plans with reduced modulation and minimal dosimetric impact, but the cost function gradient-based coefficients show more potential for reducing beam modulation without affecting dosimetric plan quality. In summary, adaptive diffusion smoothing is a promising tool for ensuring that only the necessary amount of beam modulation is used, promoting more efficient and accurate IMRT planning, QA, and delivery.

An anatomically realistic lung model for Monte Carlo-based dose calculations.

Treatment planning for disease sites with large variations of electron density in neighboring tissues requires an accurate description of the geometry. This self-evident statement is especially true for the lung, a highly complex organ having structures with a wide range of sizes that range from about 10(-4) to 1 cm. In treatment planning, the lung is commonly modeled by a voxelized geometry obtained using computed tomography (CT) data at various resolutions. The simplest such model, which is often used for QA and validation work, is the atomic mix or mean density model, in which the entire lung is homogenized and given a mean (volume-averaged) density. The purpose of this paper is (i) to describe a new heterogeneous random lung model, which is based on morphological data of the human lung, and (ii) use this model to assess the differences in dose calculations between an actual lung (as represented by our model) and a mean density (homogenized) lung. Eventually, we plan to use the random lung model to assess the accuracy of CT-based treatment plans of the lung. For this paper, we have used Monte Carlo methods to make accurate comparisons between dose calculations for the random lung model and the mean density model. For four realizations of the random lung model, we used a single photon beam, with two different energies (6 and 18 MV) and four field sizes (1 x 1, 5 x 5, 10 x 10, and 20 x 20 cm2). We found a maximum difference of 34% of D(max) with the 1 x 1, 18 MV beam along the central axis (CAX). A "shadow" region distal to the lung, with dose reduction up to 7% of D(max), exists for the same realization. The dose perturbations decrease for larger field sizes, but the magnitude of the differences in the shadow region is nearly independent of the field size. We also observe that, compared to the mean density model, the random structures inside the heterogeneous lung can alter the shape of the isodose lines, leading to a broadening or shrinking of the penumbra region. For small field sizes, the mean lung doses significantly depend on the structures' relative locations to the beam. In addition to these comparisons between the random lung and mean density models, we also provide a preliminary comparison between dose calculations for the random lung model and a voxelized version of this model at 0.4 x 0.4 x 0.4 cm3 resolution. Overall, this study is relevant to treatment planning for lung tumors, especially in situations where small field sizes are used. Our results show that for such situations, the mean density model of the lung is inadequate, and a more accurate CT model of the lung is required. Future work with our model will involve patient motion, setup errors, and recommendations for the resolution of CT models.

Reduction of IMRT beam complexity through the use of beam modulation penalties in the objective function.

Inverse planned intensity modulated radiation therapy (IMRT) has become commonplace in treatment centers across the world. Due to the implications of beam complexity on treatment planning, delivery, and quality assurance, several methods have been proposed to reduce the complexity. These methods include beamlet intensity restrictions, smoothing procedures, and direct aperture optimization. Many of these methods typically sacrifice target coverage and/or normal tissue sparing in return for increased beam smoothness and delivery efficiency. In the present work, we penalize beam modulation in the inverse planning cost function to reduce beam complexity and increase delivery efficiency, while maintaining dosimetric quality. Three modulation penalties were tested: two that penalized deviation from Savitzky-Golay filtered versions of the optimized beams, and one that penalized the plan intensity map variation (a measure of overall beam modulation). The modulation penalties were applied at varying weights in a weighted sum objective (or cost) function to investigate their ability to reduce beam complexity while preserving IMRT plan quality. The behavior of the penalties was characterized on a CT phantom, and then clinical optimization comparisons were performed in the brain, prostate, and head/neck. Comparisons were made between (i) plans with a baseline cost function (ii) plans with a baseline cost function employing maximum beamlet intensity limits, and (iii) plans with each of the modulation penalties added to the baseline cost function. Plan analysis was based upon dose-volume histograms, relevant dose metrics, beam modulation, and monitor units required for step and shoot delivery. Each of the techniques yielded improvements over a baseline cost function in terms of MU reduction. In most cases, this was achieved with minimal change to the plan DVHs and metrics. In all cases, an acceptable plan was reached with each of the methods while reducing MU substantially. Each individual method has merit as a tool for reducing IMRT beam complexity and could be easily applied in the clinic to improve overall inverse plan quality. However, the penalty based upon the plan intensity map variation consistently produced the most delivery-efficient plans with the fewest computations.