Fast Fourier Transform Convolution Research Articles

Method to enhance time series rolling fault prediction by deep fast Fourier convolution

Deep learning realizes the adaptive extraction of features in remaining useful life (RUL), and most of the methods combined with traditional signal processing stay in the preprocessing stage. A fast Fourier convolutional gated recurrent unit (FFCGRU) method is proposed, which incorporates fast Fourier transform (FFT) into a convolutional neural network for adaptive feature extraction, enhancing fault prediction. First, a plug-and-play FFT convolution block is proposed. This method combines signal processing with the adaptive deep learning process and achieves a larger receptive field through a smaller convolution kernel. Secondly, an adaptive pooling layer is designed before the feature map is input into the gated recurrent unit. The feature maps of different channels are aggregated into one feature value to reduce the redundant information of the signal feature map. Comparative studies indicate that the FFCGRU method utilizes a small number of parameters even with large convolutional kernels and exhibits robustness.

Siamese Trackers Based on Deep Features for Visual Tracking

Visual object tracking poses challenges due to deformation of target object appearance, fast motion, brightness change, blocking due to obstacles, etc. In this paper, a Siamese network that is configured using a convolutional neural network is proposed to improve tracking accuracy and robustness. Object tracking accuracy is dependent on features that can well represent objects. Thus, we designed a convolutional neural network structure that can preserve feature information that is produced in the previous layer to extract spatial and semantic information. Features are extracted from the target object and search area using a Siamese network, and the extracted feature map is input into the region proposal network, where fast Fourier-transform convolution is applied. The feature map produces a probability score for the presence of an object region and an object in a region, where the similarities are high to search the target. The network was trained with a video dataset called ImageNet Large Scale Visual Recognition Challenge. In the experiment, quantitative and qualitative evaluations were conducted using the object-tracking benchmark dataset. The evaluation results indicated competitive results for some video attributes through various experiments. By conducting experiments, the proposed method achieved competitive results for some video attributes, with a success metric of 0.632 and a precision metric of 0.856 as quantitative values.

Continuous learning deraining network based on residual FFT convolution and contextual transformer module

Single image deraining work has attracted a lot of attention. So an effective single image deraining network model, RFCTNet, is designed. Specifically, the residual Fast Fourier Transform convolution module is used as a baseline for the deraining network. The advantage is its ability to capture both long and short-term information interactions and to integrate high and low-frequency background information for delivery. And a novel Transformer module, the Contextual Transformer module, is introduced. Contextual knowledge of the rain streaks neighbourhood is learned to improve the recovery of textures and information in the image background based on focusing on the rain pattern features. Because of the overlap between rain streaks, removing them in one stage is difficult, so a multi-stage progressive rain removal method is used. Considering the catastrophic forgetting problem of deep learning, a continuous learning method, PIGWM, is adopted, which is c and enables the network model to memorize of the previous data set. Also, the rain removal method RFCTNet has been proven through extensive experiments to outperform other advanced methods' evaluation metrics on both synthetic and real datasets. The source code can be found on https://github.com/SUTwu/RFCTNet

A Monte Carlo evaluation of dose distribution of commercial treatment planning systems in heterogeneous media.

Calculations from a treatment planning system (TPS) in heterogeneous regions may present significant inaccuracies due to loss of electronic equilibrium. The purpose of this study is to evaluate and quantify the differences of dose distributions computed by some of the newest dose calculation algorithms, including collapsed cone convolution (CCC), fast Fourier transform (FFT) convolution, and superposition convolution, in heterogeneity of the lung. A 6-MV Siemens Primus linear accelerator was simulated by MCNPX Monte Carlo (MC) code, and the results of percentage depth dose (PDD) and dose profile values were compared with measured data. The ISOgray TPS was used and PDDs of CCC, FFT, and superposition convolution algorithms were compared with the results obtained by MCNPX code. CT2MCNP software was used to convert the computed tomography images of the lung tissue to MC input files, and dose distributions from the three algorithms were compared to MC method. For PDD curves in buildup region, the maximum underdosage of ISOgray TPS was at the surface (19%) and comes in closer agreement when depth increases (average 7.08%). Dose differences (DD) between different algorithms and MC were typically 4.81% (range: 1.95% to 7.30%), -1.55% (range: -5.14% to 5.26%) and 4.96% (range: 2.00% to 7.4%) in the lung for the CCC, FFT, and superposition algorithms, respectively. The difference between monitor units and maximum dose calculated using the three algorithms were 0.5% and 1.61%, respectively. The maximum DD of 7% was observed between MC and TPS results. Significant differences were found when the calculation algorithms were compared with MC method in lung tissue, and this difference is not negligible. It is recommended to use of MC-based TPS for the treatment fields including lung tissue.

Evaluation of the accuracy of various dose calculation algorithms of a commercial treatment planning system in the presence of hip prosthesis and comparison with Monte Carlo.

High atomic number elements are commonly used in a hip prosthesis which can cause uncertainty in accurate dose calculations in radiation therapy. The aim of this study is to assess the accuracy of the three various algorithms of ISOgray treatment planning system in the presence of hip prosthesis by Monte Carlo (MC). A MC model of Siemens PRIMUS linear accelerator has been built and verified by the measured data of the different algorithms of ISOgray treatment planning systems (TPS) in 6 and 15 MV photon beam energies. Two types of hip prosthesis have been used: stainless steel and titanium. The accuracy of mentioned dose calculation algorithms in the presence of hip prosthesis was evaluated. There were 24.78%, 27.68%, and 27.72% errors in fast Fourier transform (FFT) Convolution, collapsed cone (CC), and superposition in 6 MV photon beam and 26.45%, 30.45%, and 28.63% in 15 MV photon beam for titanium type, respectively. However, there were 32.84%, 35.89%, and 35.57% in 6 MV photon beam and 38.81%, 47.31%, and 39.91% errors in 15 MV photon beam in steel type, respectively. In addition, the ISOgray TPS algorithms are not able to predict the dose enhancement and reduction at the proximal and distal prosthesis interfaces, respectively. Hip prosthesis creates a considerable disturbance in dose distribution which cannot be predicted accurately by the FFT convolution, CC, and superposition algorithms. It is recommended to use of MC-based TPS for the treatment fields including the hip prosthesis.

Accurate pairwise convolutions of non-negative vectors via FFT

A novel method is presented for fast convolution of a pair of probability mass functions defined on a finite lattice with guaranteed accuracy of all computed values. This method, called aFFT-C (accurate FFT convolution), utilizes the Fast Fourier Transform (FFT) for the gain in speed, but relying on a rigorous analysis of the propagation of roundoff error, it can detect and circumvent the accumulation of this numerical error that is otherwise inherent to the Fourier transform. In the worst case scenario aFFT-C’s execution time is on par with the accurate naive convolution but in a typical application it is comparable with a direct FFT-based convolution (FFT-C). The properties of the proposed algorithm are demonstrated both theoretically and empirically.

SU‐E‐T‐91: Accuracy of Dose Calculation Algorithms for Patients Undergoing Stereotactic Ablative Radiotherapy

Purpose:The purpose of this study was to investigate and quantify the variation in dose distributions in small field lung cancer radiotherapy using seven different dose calculation algorithms.Methods:The study was performed in 21 lung cancer patients who underwent Stereotactic Ablative Body Radiotherapy (SABR). Two different methods (i) Same dose coverage to the target volume (named as same dose method) (ii) Same monitor units in all algorithms (named as same monitor units) were used for studying the performance of seven different dose calculation algorithms in XiO and Eclipse treatment planning systems. The seven dose calculation algorithms include Superposition, Fast superposition, Fast Fourier Transform ( FFT) Convolution, Clarkson, Anisotropic Analytic Algorithm (AAA), Acurous XB and pencil beam (PB) algorithms. Prior to this, a phantom study was performed to assess the accuracy of these algorithms. Superposition algorithm was used as a reference algorithm in this study. The treatment plans were compared using different dosimetric parameters including conformity, heterogeneity and dose fall off index. In addition to this, the dose to critical structures like lungs, heart, oesophagus and spinal cord were also studied. Statistical analysis was performed using Prism software.Results:The mean±stdev with conformity index for Superposition, Fast superposition, Clarkson and FFT convolution algorithms were 1.29±0.13, 1.31±0.16, 2.2±0.7 and 2.17±0.59 respectively whereas for AAA, pencil beam and Acurous XB were 1.4±0.27, 1.66±0.27 and 1.35±0.24 respectively.Conclusion:Our study showed significant variations among the seven different algorithms. Superposition and AcurosXB algorithms showed similar values for most of the dosimetric parameters. Clarkson, FFT convolution and pencil beam algorithms showed large differences as compared to superposition algorithms. Based on our study, we recommend Superposition and AcurosXB algorithms as the first choice of algorithms in lung cancer radiotherapy involving small fields. However, further investigation by Monte Carlo simulation is required to confirm our results.

A new pencil beam model for photon dose calculations in heterogeneous media

The pencil beam method is commonly used for dose calculations in intensity-modulated radiation therapy (IMRT). In this study, we have proposed a novel pencil model for calculating photon dose distributions in heterogeneous media. To avoid any oblique kernel-related bias and reduce computation time, dose distributions were computed in a spherical coordinate system based on the pencil kernels of different distances from source to surface (DSS). We employed two different dose calculation methods: the superposition method and the fast Fourier transform convolution (FFTC) method. In order to render the superposition method more accurate, we scaled the depth-directed component by moving the position of the entry point and altering the DSS value for a given beamlet. The lateral components were thus directly corrected by the density scaling method along the spherical shell without taking the densities from the previous layers into account. Significant computation time could be saved by performing the FFTC calculations on each spherical shell, disregarding density changes in the lateral direction. The proposed methods were tested on several phantoms, including lung- and bone-type heterogeneities. We compared them with Monte Carlo (MC) simulation for several field sizes with 6 MV photon beams. Our results revealed mean absolute deviations <1% for the proposed superposition method. Compared to the AAA algorithm, this method improved dose calculation accuracy by at least 0.3% in heterogeneous phantoms. The FFTC method was approximately 40 times faster than the superposition method. However, compared with MC, mean absolute deviations were <3% for the FFTC method.

FFT-Based Block Processing in Speech Enhancement: Potential Artifacts and Solutions

Most speech enhancement applications perform frequency shaping by means of multiplication in the frequency domain. Operating in the frequency domain is equivalent to convolution in the time domain. In these speech enhancement algorithms, the updating of frequency response alone cannot ensure the fulfillment of the conditions required for multiplication in frequency to correspond to linear convolution instead of circular convolution. As a result, artifacts and distortions may be present in the output of a standard fast Fourier transform (FFT)-based algorithm. Typical methods to deal with these artifacts involve overlapping and windowing. However, even using these strategies, artifacts may be perceptually noticeable under certain signal-to-noise ratio (SNR) conditions and/or when a high sampling frequency is employed. This paper analyzes the efficiency of the standard methods, explains the source of these distortions, provides a perceptual evidence of these artifacts, and proposes two alternative methods to perform artifact-free and distortion-free FFT convolution. These methods are based on the extension of the impulse response and the splitting of the impulse response in two impulse responses, operations that are performed in the frequency-domain. Computational costs and performance of the proposed techniques are also discussed.

Influence of dose calculation algorithms on the predicted dose distributions and NTCP values for NSCLC patients

To investigate differences in calculated doses and normal tissue complication probability (NTCP) values between different dose algorithms. Six dose algorithms from four different treatment planning systems were investigated: Eclipse AAA, Oncentra MasterPlan Collapsed Cone and Pencil Beam, Pinnacle Collapsed Cone and XiO Multigrid Superposition, and Fast Fourier Transform Convolution. Twenty NSCLC patients treated in the period 2001-2006 at the same accelerator were included and the accelerator used for treatments were modeled in the different systems. The treatment plans were recalculated with the same number of monitor units and beam arrangements across the dose algorithms. Dose volume histograms of the GTV, PTV, combined lungs (excluding the GTV), and heart were exported and evaluated. NTCP values for heart and lungs were calculated using the relative seriality model and the LKB model, respectively. Furthermore, NTCP for the lungs were calculated from two different model parameter sets. Calculations and evaluations were performed both including and excluding density corrections. There are found statistical significant differences between the calculated dose to heart, lung, and targets across the algorithms. Mean lung dose and V20 are not very sensitive to change between the investigated dose calculation algorithms. However, the different dose levels for the PTV averaged over the patient population are varying up to 11%. The predicted NTCP values for pneumonitis vary between 0.20 and 0.24 or 0.35 and 0.48 across the investigated dose algorithms depending on the chosen model parameter set. The influence of the use of density correction in the dose calculation on the predicted NTCP values depends on the specific dose calculation algorithm and the model parameter set. For fixed values of these, the changes in NTCP can be up to 45%. Calculated NTCP values for pneumonitis are more sensitive to the choice of algorithm than mean lung dose and V20 which are also commonly used for plan evaluation. The NTCP values for heart complication are, in this study, not very sensitive to the choice of algorithm. Dose calculations based on density corrections result in quite different NTCP values than calculations without density corrections. It is therefore important when working with NTCP planning to use NTCP parameter values based on calculations and treatments similar to those for which the NTCP is of interest.

放射線治療計画装置における不均質領域の線量計算精度の評価−Ｍｏｎｔｅ Ｃａｒｌｏ計算との比較−

The purpose of this study was to compare dose distributions from three different RTPS with those from Monte Carlo (MC) calculations and measurements, in heterogeneous phantoms for photon beams. This study used four algorithms for RTPS: AAA (analytical anisotropic algorithm) implemented in the Eclipse (Varian Medical Systems) treatment planning system, CC (collapsed cone) superposition from the Pinnacle (Philips), and MGS (multigrid superposition) and FFT (fast Fourier transform) convolution from XiO (CMS). The dose distributions from these algorithms were compared with those from MC and measurements in a set of heterogeneous phantoms. Eclipse/AAA underestimated the dose inside the lung region for low energies of 4 and 6 MV. This is because Eclipse/AAA do not adequately account for a scaling of the spread of the pencil (lateral electron transport) based on changes in the electron density at low photon energies. The dose distributions from Pinnacle/CC and XiO/MGS almost agree with those of MC and measurements at low photon energies, but increase errors at high energy of 15 MV, especially for a small field of 3x3 cm(2). The FFT convolution extremely overestimated the dose inside the lung slab compared to MC. The dose distributions from the superposition algorithms almost agree with those from MC as well as measured values at 4 and 6 MV. The dose errors for Eclipse/AAA are lager in lung model phantoms for 4 and 6 MV. It is necessary to use the algorithms comparable to superposition for accuracy of dose calculations in heterogeneous regions.

SU‐FF‐T‐324: Comparison of RTPS and Monte Carlo Dose Distributions in Heterogeneous Phantoms for Clinical Photon Beams

Purpose: To compare dose distributions from three different radiotherapy treatment planning systems (RTPS) with those from Monte Carlo (MC) calculations and measurements, in heterogeneous phantoms for photon beams. Methods and Materials: This study used four algorithms for RTPS: AAA (analytical anisotropic algorithm) implemented in the Eclipse (Varian Medical Systems) treatment planning system, CC (collapsed cone) superposition from the Pinnacle (Philips), and MGS (multigrid superposition) and FFT (fast Fourier transform) convolution from XiO (CMS). The dose distributions from these algorithms were compared with those from MC and measurements in a set of heterogeneous phantoms. The phantom inhomogeneities mimic relevant clinical treatment sites, which include lung slab, lung‐bone slab, bone‐lung slab, mediastinum, and tumor geometries. The benchmark comparisons were performed in lung densities of 0.30 g/cm3 and a bone density 1.819 g/cm3 for 3×3, 5×5, and 10×10 cm2 field sizes, 4, 6, 10, and 15 MV photon beams. Results: MC dose distributions agreed much better with measured data than the superposition algorithm in a set of heterogeneous phantoms. The difference in dose distributions between Eclipse‐AAA and MC were up to 15%. Eclipse‐AAA underestimated the dose inside the lung region for low energies of 4 and 6 MV, and in constant, overestimated in the tissue region. The dose distributions from Pinnacle‐CC and XiO‐MGS almost agree with those of MC and measurements but increase errors at15 MV, especially for a small field of 3×3 cm2. The FFT convolution extremely overestimated the dose inside the lung slab compared to MC. Conclusions: The dose distributions from the superposition algorithms almost agree with those from MC as well as measured values at 4 and 6 MV. The dose errors for Eclipse‐AAA are lager in lung model phantoms for 4 and 6 MV. The FFT convolution should not use for dose calculations in heterogeneous regions.

Convolution-Based Trigonometric Interpolation of Band-Limited Signals

This paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band-limited signal from a finite sequence of samples. The polynomial delivers accurate approximations in the range covered by the sequence, except for a short frame close to the range limits. Besides, its accuracy increases exponentially with the frame width. The method is based on using a band-limited window in order to reduce the truncation error of a convolution series. It is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete Fourier transform (DFT). Specifically, two basic procedures are presented, one based on the fast Fourier transform (FFT), and another based on a recursive update algorithm for the short-time FFT. The paper contains three applications. The first is a variable fractional delay (VFD) filter, which consists of a short-time FFT combined with the evaluation of a trigonometric polynomial. This filter has low complexity and can be implemented using CORDIC rotations. The second is the interpolation of nonuniform Fourier summations, where the proposed method eliminates the need to interpolate any kernel sample. Finally, the third can be viewed as a generalization of the FFT convolution algorithm and makes it possible to interpolate the output of an finite-impulse-response (FIR) filter efficiently.

Variation of dose distribution of stereotactic radiotherapy for small-volume lung tumors under different respiratory conditions

Purpose To clarify the effects of respiratory condition on dose calculation for stereotactic radiotherapy of small lung tumors. Methods and materials Computed tomography (CT) data were obtained for nine tumors (diameter, 2.1–3.6 cm; mean, 2.7 cm) during the stable state, deep expiration, and deep inspiration breath-hold states. Rotational Irradiation with 3 non-coplanar arcs (Rotational Irradiation) and static irradiation with 18 non-coplanar ports (Static Irradiation) using 6-MV photons were evaluated using Fast Fourier Transform (FFT) convolution and Multigrid (MG) superposition algorithms. Dose-volume histograms (DVHs), mean path-length (PL) and mean effective path-length (EPL) were calculated. Results Although the PL was larger for the inspiration state than for the stable state and the expiration state, the EPL was 0.4–0.5 cm smaller in the inspiration state than in the expiration state ( p = 0.01 for Rotational Irradiation; p = 0.03 for Static Irradiation). The isocenter dose obtained by the FFT convolution algorithm was 7–12% higher than that obtained with the MG superposition algorithm. A leftward shift of the DVH obtained by MG superposition was noted for the inspiration state compared with the expiration state. Conclusions The choice of the proper algorithm is important to accounting for changes in respiration state. Differences in isocenter dose were not large among the respiratory states analyzed. EPL was a little shorter for inspiration than for expiration, although there were larger and reverse trends in path length. A leftward shift of the DVH obtained for the inspiration state when MG superposition was used.

Treatment verification in the presence of inhomogeneities using EPID‐based three‐dimensional dose reconstruction

Treatment verification is a prerequisite for the verification of complex treatments, checking both the treatment planning process and the actual beam delivery. Pretreatment verification can detect errors introduced by the treatment planning system (TPS) or differences between planned and delivered dose distributions. In a previous paper we described the reconstruction of three-dimensional (3-D) dose distributions in homogeneous phantoms using an in-house developed model based on the beams delivered by the linear accelerator measured with an amorphous silicon electronic portal imaging device (EPID), and a dose calculation engine using the Monte Carlo code XVMC. The aim of the present study is to extend the method to situations in which tissue inhomogeneities are present and to make a comparison with the dose distributions calculated by the TPS. Dose distributions in inhomogeneous phantoms, calculated using the fast-Fourier transform convolution (FFTC) and multigrid superposition (MGS) algorithms present in the TPS, were verified using the EPID-based dose reconstruction method and compared to film and ionization chamber measurements. Differences between dose distributions were evaluated using the gamma-evaluation method (3%/3 mm) and expressed as a mean gamma and the percentage of points with gamma> 1 (P(gamma>1)). For rectangular inhomogeneous phantoms containing a low-density region, the differences between film and reconstructed dose distributions were smaller than 3%. In low-density regions there was an overestimation of the planned dose using the FFTC and MGS algorithms of the TPS up to 20% and 8%, respectively, for a 10 MV photon beam and a 3 x 3 cm2 field. For lower energies and larger fields (6 MV, 5 x 5 cm2), these differences reduced to 6% and 3%, respectively. Dose reconstruction performed in an anthropomorphic thoracic phantom for a 3-D conformal and an IMRT plan, showed good agreement between film data and reconstructed dose values (P(gamma>1) <6%). The algorithms of the TPS underestimated the dose in the low-dose regions outside the treatment field, due to an implementation error of the jaws and multileaf collimator of the linac in the TPS. The FFTC algorithm of the TPS showed differences up to 6% or 6 mm at the interface between lung and breast. Two intensity-modulated radiation therapy head and neck plans, reconstructed in a commercial phantom having a bone-equivalent insert and an air cavity, showed good agreement between film measurement, reconstructed and planned dose distributions using the FFTC and MGS algorithm, except in the bone-equivalent regions where both TPS algorithms underestimated the dose with 4%. Absolute dose verification was performed at the isocenter where both planned and reconstructed dose were within 2% of the measured dose. Reproducibility for the EPID measurements was assessed and found to be of negligible influence on the reconstructed dose distribution. Our 3-D dose verification approach is based on the actual dose measured with an EPID in combination with a Monte Carlo dose engine, and therefore independent of a TPS. Because dose values are reconstructed in 3-D, isodose surfaces and dose-volume histograms can be used to detect dose differences in target volume and normal tissues. Using our method, the combined planning and treatment delivery process is verified, offering an easy to use tool for the verification of complex treatments.

Clinical impact of the detector size effect in 3D-CRT

Objective The detector size artificially increases the measured penumbra of radiotherapy fields. The aim of this work is to determine the influence of the detector size when planning three-dimensional conformal radiation therapy (3D-CRT) treatments. Material and methods Two anatomical sites of interest in 3D-CRT were studied: prostate and hypophysis chordoma. Conventional 3D-CRT treatments for two cases in these locations were planned with a FOCUS 4.0.0 (Computerized Medical Systems, USA) treatment planning system (TPS) equipped with Fast Fourier Transform Convolution (FFTC) and Multigrid Superposition (MGS) algorithms, making use of beams modelled from radiation profiles measured either with a 2.0 mm diameter detector (PFD 3G diode) or with a 5.5 mm diameter detector (PTW-31002 ionisation chamber). These detectors cover up the range of detector sizes commonly used to measure radiation profiles for 3D-CRT. Dose–volume histograms (DVHs), radiobiological indexes, tumor control probability (TCP) and normal tissue complication probability (NTCP) were analysed and compared for planning target volumes (PTVs) and organs at risk (OAR) studied. Results Important differences in DVH were found. OAR received higher dose levels when a 5.5 mm detector was used to measure profiles compared to the case in which a 2.0 mm detector was used. A 2 Gy increment in the mean rectal dose was found when the larger detector was used. In the same way, NTCP of brain stem in hypophysis chordoma treatments was doubled when this detector was used. Conclusion The current use of ionisation chambers of about 5 mm active diameter to get the necessary data to model treatment machines in radiotherapy treatment planning systems (TPS) implies a significant overirradiation of OAR close to the PTV in 3D-CRT treatments due to errors in the measured penumbra of beam profiles. To avoid this overirradiation, the measured profiles should either being acquired with a suitable detector size (2–3 mm active diameter) or being deconvoluted.

Measuring the three-dimensional shear from simulation data, with applications to weak gravitational lensing

We have developed a new three-dimensional algorithm, based on the standard P3M method, for computing deflections resulting from weak gravitational lensing. We compare the results of this method with those of the two-dimensional planar approach, and rigorously outline the conditions under which the two approaches are equivalent. Our new algorithm uses a Fast Fourier Transform convolution method for speed, and has a variable softening feature to provide a realistic interpretation of the large-scale structure in a simulation. The output values of the code are compared with those from the Ewald summation method, which we describe and develop in detail. With an optimal choice of the high-frequency filtering in the Fourier convolution, the maximum errors, when using only a single particle, are about 7 per cent, with an rms error less than 2 per cent. For ensembles of particles, used in typical N-body simulations, the rms errors are typically 0.3 per cent. We describe how the output from the algorithm can be used to generate distributions of magnification, source ellipticity, shear and convergence for large-scale structure.

Refinements of the finite-size pencil beam model of three-dimensional photon dose calculation.

Modern three-dimensional (3-D) photon dose calculation algorithms need to be fast and accurate if they are to be practical for treatment optimization. Refinements to a previously proposed finite-size pencil beam (FSPB) method are presented in order to fulfill these needs. Specifically, a fast Fourier transform (FFT) convolution technique is used to speed calculation of the FSPB; the fluence spectrum is modeled, and the effects of finite source size, a Gaussian x-ray source intensity profile and partial transmission through a multileaf collimator (MLC) leaf are approximated. The use of FFT techniques in the calculation of small diverging fields involves approximations that are investigated for a 6 MV beam and shown to introduce errors that vary with energy but do not exceed 0.7% on the central axis. Dose distributions calculated by FSPB superposition are in excellent agreement with those calculated by full field FFT convolution. Two key advances over the original implementation of the FSPB model are demonstrated: the fast calculation of the FSPB facilitates development, and the incorporation of realistic beam parameters enables accurate modeling of clinical beams.

Theoretical developments on fast Fourier transform convolution dose calculations in inhomogeneous media.

A theory is presented on dose calculations in inhomogeneous media that takes advantage of fast Fourier transform (FFT) convolution for practical three-dimensional treatment planning using photon beams. While the initial work of Boyer and Mok [Med. Phys. 13, 503-509 (1986)] provided a theory which is based on first principles, it failed to give satisfactory predictions inside inhomogeneities. Subsequently, Zhu and Boyer [Phys. Med. Biol. 35, 351-368 (1990)] showed that their formulas agreed well with measured data, but these formulas were empirically altered from Boyer and Mok's. In this work, Boyer and Mok's first-order theory is extended to include second-order inhomogeneity effects. A new correction dose formula is derived which corrects the first scattered dose due to the presence of inhomogeneities. This correction dose formula works better than Zhu and Boyer's empirical correction dose formula. Furthermore, the primary dose formula used by Zhu and Boyer, which was empirically modified from Boyer and Mok's, is justified theoretically. Clear statements are made about the assumptions and the approximations that enter into the derivation which in turn uncover the limitations of this FFT convolution dose calculation.

Accuracy requirements of the primary x‐ray spectrum in dose calculations using FFT convolution techniques

The photon beam dose calculation methods using fast Fourier transform (FFT) convolution require that the primary beam energy spectrum is accurately known; however, a quantitative assessment of the required accuracy is not yet provided. In this study, the sensitivity of various parameters such as central axis depth dose and dose under attenuators to the incident photon spectrum was simulated and evaluated for 6- and 18-MV photon beams. The central axis depth-dose data using the modified spectrum were compared with those calculated using the original spectrum. Maximum errors on the central axis between these two sets of depth-dose data were found by changing the weighting of each energy bin of the primary spectrum. The maximum change in percent depth dose is proportional to the change in average incident energy relating to the quality of the x ray. To keep the maximum depth-dose error to less than 1%, the relative change in average energy brought about by changes in single spectral bins should be less than 2%. The accuracy required for each energy bin varies dramatically, dependent on its deviation from the original average energy and its relative weighting. For example, for the 6-MV x-ray spectrum, the energy bins centered around 0.76 MeV (first bin) and 6.8 MeV (fifth bin) need to have accuracies of 5% and 40%, respectively, to obtain a 1% accuracy in percent depth dose.