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PubMed1994

A filtered backprojection algorithm for pinhole SPECT with a displaced centre of rotation.

In this paper the importance of correcting a small centre-of-rotation displacement (approximately 1 mm) in single-photon-emission computed tomography (SPECT) using high-resolution pinhole collimation is demonstrated. A filtered backprojection (FBP) algorithm is derived for a pinhole geometry that has a displaced centre-of-rotation. The centre-of-rotation displacement, or mechanical shift (MS), is the displacement of the midplane of the pinhole collimator from the rotation centre. It is characterized by two orthogonal components: the shift eta of the midplane of the pinhole collimator along the direction of the axis of rotation, and the distance tau between the midline of the pinhole collimator and the axis of rotation. This algorithm is fast and corrects the centre-of-rotation displacement directly by incorporating this displacement into the algorithm. This new algorithm is evaluated using both a three-line source and a micro-SPECT cold rod phantom. The results demonstrate that the pinhole FBP with mechanical shift correction is able to correct the 'doughnut'-type artifacts caused by the mechanical shift and restore the expected system resolution.

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PubMed1994

Figures of merit for comparing reconstruction algorithms with a volume-imaging PET scanner.

For volume-imaging PET scanners, no septa are used to maximize the sensitivity by collecting events oblique to the scanner axis. We answer two questions: (i) how does the performance of an image reconstruction algorithm for a volume-imaging PET scanner depend on its general dimensions? and (ii) at what point is a three-dimensional (3D) reconstruction algorithm needed for a volume-imaging scanner, as the axial extent is increased? A 3D reconstruction algorithm will accurately incorporate the oblique events in a reconstruction of the original source distribution. From simulations of an existing volume PET scanner with a maximum axial acceptance angle (+/-alpha) of alpha = 9 degrees, however, we show that the single-slice rebinning algorithm is a good compromise between sensitivity, speed, and accuracy when compared to standard two-dimensional reconstruction (alpha = 1 degrees), and a 3D reconstruction with alpha = 9 degrees. We also show with simulations that a new scanner with alpha = 27 degrees requires 3D reconstruction in order to achieve maximum sensitivity without unacceptable losses in accuracy. Measurements of scanner performance are based on a series of figures of merit that characterize image quality and quantitative accuracy measured from a set of simulated test phantoms.

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PubMed1994

Noise properties of the EM algorithm: I. Theory.

The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. It is a non-linear iterative algorithm that attempts to find the ML estimate of the object that produced a data set. The convergence of the algorithm and other deterministic properties are well established, but relatively little is known about how noise in the data influences noise in the final reconstructed image. In this paper we present a detailed treatment of these statistical properties. The specific application we have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. We show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in the second paper in this series we demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.

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PubMed1995

A comparative study of three fast algorithms to estimate cerebral blood flow and distribution volume using N-isopropyl-p-[123I]iodoamphetamine and two SPECT scans.

To estimate regional cerebral blood flow (rCBF) and distribution volume (Vd) using N-isopropyl-p-[123]iodoamphetamine (IMP) and two SPECT data (early and delayed data) measured at two discrete time points, three fast algorithms are implemented, which are the table look-up method (TLU), the linear least-squares fitting method (LLSF), and the linear search method (LS). A two-compartment model was used as the IMP kinetic model. These algorithms were applied to the early and delayed data measured by the two scan protocols, t1 = 12 min, t2 = 54 min (scan A) and t1 = 12 min, t2 = 85 min (scan B). The estimated rCBF and Vd values using the present three algorithms were in good agreement with those using the nonlinear least-squares fitting technique and dynamic SPECT data. The quantitative value and signal to noise (S/N) ratio of rCBF images by TLU was not changed whether scan A or scan B was used. However, the S/N of the Vd image by TLU using scan B improved over that using scan A. The same tendency was observed in the results of LLSF and LS. The CPU times for the parametric imaging of CBF and Vd images for one slice (64 x 64 matrix) were 0.04, 0.09, and 0.14 s using TLU, LLSF, and LS, respectively. In conclusion, the present three algorithms gave reasonably accurate estimates of rCBF and Vd using two SPECT data, and are well suited for parametric imaging because of their good statistical properties and high computational efficiency.

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PubMed1996

A pencil beam algorithm for proton dose calculations.

The sharp lateral penumbra and the rapid fall-off of dose at the end of range of a proton beam are among the major advantages of proton radiation therapy. These beam characteristics depend on the position and characteristics of upstream beam-modifying devices such as apertures and compensating boluses. The extent of separation, if any, between these beam-modifying devices and the patient is particularly critical in this respect. We have developed a pencil beam algorithm for proton dose calculations which takes accurate account of the effects of materials upstream of the patient and of the air gap between them and the patient. The model includes a new approach to picking the locations of the pencil beams so as to more accurately model the penumbra and to more effectively account for the multiple-scattering effects of the media around the point of interest. We also present a faster broad-beam version of the algorithm which gives a reasonably accurate penumbra. Predictions of the algorithm and results from experiments performed in a large-field proton beam are presented. In general the algorithm agrees well with the measurements.

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PubMed1998

Algorithms for calculating detector efficiency normalization coefficients for true coincidences in 3D PET.

Accurate normalization of lines of response in 3D PET is a prerequisite for quantitative reconstruction. Most current methods are component based, calculating a series of geometric and intrinsic detector efficiency factors. We have reviewed the theory behind several existing algorithms for calculating detector efficiency factors in 2D and 3D PET, and have extended them to create a range of new algorithms. Three of the algorithms described are 'fully 3D' in that they make use of data from all detector rings for the calculation of the efficiencies of any one line of response. We have assessed the performance of the new and existing methods using simulated and real data, and have demonstrated that the fully 3D algorithms allow the rapid acquisition of crystal efficiency normalization data using low-activity sources. Such methods enable the use of scatter-free scanning line sources or the use of very short acquisitions of cylindrical sources for routine normalization.

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PubMed1998

Dose accuracy check of the 3D electron beam algorithm in a treatment planning system.

The accuracy of the recently implemented three-dimensional electron beam dose calculating algorithm in CADPLAN version 2.62 manufactured by Varian Dosetek was investigated. The algorithm uses a generalized Gaussian pencil beam model and the dose distributions are calculated as the sum of three weighted Gaussians. To use the calculating program in an optimum way, one needs to know the dose calculation accuracy of the algorithm as well as its limitations. This investigation includes comparisons of measured relative dose distributions with calculated dose distributions and also comparisons of measured and calculated monitor units. The geometries tested were quadratic fields, irregularly shaped fields, oblique fields, irregularly shaped phantom surfaces and internal heterogeneities and were most often irradiated with 8 and 20 MeV electrons. The results indicate that the algorithm is well suited for clinical three-dimensional dose planning. Some deviations occurred but they were most often within the limits of international criteria of acceptability.

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PubMed1999

A spline-regularized minimal residual algorithm for iterative attenuation correction in SPECT.

In SPECT, regularization is necessary to avoid divergence of the iterative algorithms used for non uniform attenuation compensation. In this paper, we propose a spline-based regularization method for the minimal residual algorithm. First, the acquisition noise is filtered using a statistical model involving spline smoothing so that the filtered projections belong to a Sobolev space with specific continuity and derivability properties. Then, during the iterative reconstruction procedure, the continuity of the inverse Radon transform between Sobolev spaces is used to design a spline-regularized filtered backprojection method, by which the known regularity properties of the projections determine those of the corresponding reconstructed slices. This ensures that the activity distributions estimated at each iteration present regularity properties, which avoids computational noise amplification, thus stabilizing the iterative process. Analytical and Monte Carlo simulations are used to show that the proposed spline-regularized minimal residual algorithm converges to a satisfactory stable solution in terms of restored activity and homogeneity, using at most 25 iterations, whereas the non regularized version of the algorithm diverges. Choosing the number of iterations is therefore no longer a critical issue for this reconstruction procedure.

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PubMed2001

An algorithm for stereotactic localization by computed tomography or magnetic resonance imaging.

Stereotactic localization of an intracranial lesion by computed tomography or magnetic resonance imaging requires the use of a head frame that is fixed to the skull of the patient. To such head frames are attached either N-shaped or V-shaped localization rods. Because of patient positioning, the transverse imaging slices may not be parallel to the frame base; a coordinate transformation algorithm that takes this possibility into consideration is crucial. Here we propose such an algorithm for a head frame with V-shaped localization rods. Our algorithm determines the transformation matrix between the image coordinate system of a transverse image and the frame coordinate system. The determining procedure has three steps: (a) calculation of the oblique angles of a transverse image relative to the head frame and calculation of the image magnification factor; (b) determination of the coordinates of four central markers in both coordinate systems; and (c) determination of the 3 x 3 transformation matrix by using the coordinates of the four markers. This algorithm is robust in principle and is useful for improving the accuracy of localization.

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PubMed2001

A multiobjective gradient-based dose optimization algorithm for external beam conformal radiotherapy.

A multiobjective gradient-based algorithm has been developed for the purpose of dose distribution optimization in external beam conformal radiotherapy. This algorithm is based on the concept of gathering the values of all objectives into a single value. The weighting factors of the composite objective values are varied in different steps, allowing the reconstruction of the trade-off surfaces (three or more objectives) or curves (two objectives) which define the boundary between the feasible and non-feasible domain regions. The analysis of these curves allows the decision-maker to select the solution that best fits the clinical goals. In contrast to all the other algorithms, our method provides not a single solution but a sample of solutions representing all possible clinical importance factors (weights) for the objectives used. The application of this algorithm to two test cases shows that a correct selection for the importance factors to multiply the individual objectives in the global objective value is not trivial and that the location and shape of the boundary region between the feasible and non-feasible solution regions are case dependent. Provided that the individual objective functions are analytically differentiable and that the number of objectives is the range of two to three, the computation times are acceptable for clinical use. Furthermore, the optimization for a unique combination of importance factors within the aggregate objective function is performed in less than 1 min.

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PubMed2002

A unified model for the speed of sound in cranial bone based on genetic algorithm optimization.

The density and structure of bone is highly heterogeneous, causing wide variations in the reported speed of sound for ultrasound propagation. Current research on the propagation of high intensity focused ultrasound through an intact human skull for non-invasive therapeutic action on brain tissue requires a detailed model for the acoustic velocity in cranial bone. Such models have been difficult to derive empirically due to the aforementioned heterogeneity of bone itself. We propose a single unified model for the speed of sound in cranial bone based upon the apparent density of bone by CT scan. This model is based upon the coupling of empirical measurement, theoretical acoustic simulation and genetic algorithm optimization. The phase distortion caused by the presence of skull in an acoustic path is empirically measured. The ability of a theoretical acoustic simulation coupled with a particular speed-of-sound model to predict this phase distortion is compared against the empirical data, thus providing the fitness function needed to perform genetic algorithm optimization. By performing genetic algorithm optimization over an initial population of candidate speed-of-sound models, an ultimate single unified model for the speed of sound in both the cortical and trabecular regions of cranial bone is produced. The final model produced by genetic algorithm optimization has a nonlinear dependency of speed of sound upon local bone density. This model is shown by statistical significance to be a suitable model of the speed of sound in bone. Furthermore, using a skull that was not part of the optimization process, this model is also tested against a published homogeneous speed-of-sound model and shown to return an improved prediction of transcranial ultrasound propagation.

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PubMed2002

A comparison of MCNP4C electron transport with ITS 3.0 and experiment at incident energies between 100 keV and 20 MeV: influence of voxel size, substeps and energy indexing algorithm.

The condensed-history electron transport algorithms in the Monte Carlo code MCNP4C are derived from ITS 3.0, which is a well-validated code for coupled electron-photon simulations. This, combined with its user-friendliness and versatility, makes MCNP4C a promising code for medical physics applications. Such applications, however, require a high degree of accuracy. In this work, MCNP4C electron depth-dose distributions in water are compared with published ITS 3.0 results. The influences of voxel size, substeps and choice of electron energy indexing algorithm are investigated at incident energies between 100 keV and 20 MeV. Furthermore, previously published dose measurements for seven beta emitters are simulated. Since MCNP4C does not allow tally segmentation with the *F8 energy deposition tally, even a homogeneous phantom must be subdivided in cells to calculate the distribution of dose. The repeated interruption of the electron tracks at the cell boundaries significantly affects the electron transport. An electron track length estimator of absorbed dose is described which allows tally segmentation. In combination with the ITS electron energy indexing algorithm, this estimator appears to reproduce ITS 3.0 and experimental results well. If, however, cell boundaries are used instead of segments, or if the MCNP indexing algorithm is applied, the agreement is considerably worse.

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PubMed2003

Multiobjective inverse planning for intensity modulated radiotherapy with constraint-free gradient-based optimization algorithms.

We consider the behaviour of the limited memory L-BFGS algorithm as a representative constraint-free gradient-based algorithm which is used for multiobjective (MO) dose optimization for intensity modulated radiotherapy (IMRT). Using a parameter transformation, the positivity constraint problem of negative beam fluences is entirely eliminated: a feature which to date has not been fully understood by all investigators. We analyse the global convergence properties of L-BFGS by searching for the existence and the influence of possible local minima. With a fast simulated annealing (FSA) algorithm we examine whether the L-BFGS solutions are globally Pareto optimal. The three examples used in our analysis are a brain tumour, a prostate tumour and a test case with a C-shaped PTV. In 1% of the optimizations global convergence is violated. A simple mechanism practically eliminates the influence of this failure and the obtained solutions are globally optimal. A single-objective dose optimization requires less than 4 s for 5400 parameters and 40000 sampling points. The elimination of the problem of negative beam fluences and the high computational speed permit constraint-free gradient-based optimization algorithms to be used for MO dose optimization. In this situation, a representative spectrum of possible solutions is obtained which contains information such as the trade-off between the objectives and range of dose values. Using simple decision making tools the best of all the possible solutions can be chosen. We perform an MO dose optimization for the three examples and compare the spectra of solutions, firstly using recommended critical dose values for the organs at risk and secondly, setting these dose values to zero.

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PubMed2003

Genetic algorithm based deliverable segments optimization for static intensity-modulated radiotherapy.

The static delivery technique (also called step-and-shoot technique) has been widely used in intensity-modulated radiotherapy (IMRT) because of the simple delivery and easy quality assurance. Conventional static IMRT consists of two steps: first to calculate the intensity-modulated beam profiles using an inverse planning algorithm, and then to translate these profiles into a series of uniform segments using a leaf-sequencing tool. In order to simplify the procedure and shorten the treatment time of the static mode, an efficient technique, called genetic algorithm based deliverable segments optimization (GADSO), is developed in our work, which combines these two steps into one. Taking the pre-defined beams and the total number of segments per treatment as input, the number of segments for each beam, the segment shapes and weights are determined automatically. A group of interim modulated beam profiles quickly calculated using a conjugate gradient (CG) method are used to determine the segment number for each beam and to initialize segment shapes. A modified genetic algorithm based on a two-dimensional binary coding scheme is used to optimize the segment shapes, and a CG method is used to optimize the segment weights. The physical characters of a multileaf collimator, such as the leaves interdigitation limitation and leaves maximum over-travel distance, are incorporated into the optimization. The algorithm is applied to some examples and the results demonstrate that GADSO is able to produce highly conformal dose distributions using 20-30 deliverable segments per treatment within a clinically acceptable computation time.

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PubMed2004

An exact Fourier rebinning algorithm for 3D PET imaging using panel detectors.

Abstract We present an exact Fourier-based algorithm for rebinning 3D data generated by a stationary dual-panel PET system to obtain direct slices for subsequent slice-by-slice reconstruction. The algorithm is computationally efficient and can greatly reduce the problem dimensionality and computation complexity of the reconstruction task. By conducting computer simulation studies in which the effects of scatter, randoms, detector response functions and attenuation correction are not considered, we demonstrate that direct slices generated by the proposed algorithm are quantitatively accurate and exhibit substantially better noise characteristics than the original direct slices. However, image artefacts, occurring at axial intensity discontinuities and in regions close to the axial axis, can be observed. With preliminary supporting evidence, we stipulate that these artefacts are due to errors in discrete implementations of partial derivatives of the 3D data. Although general weightings can be used in the proposed rebinning algorithm, in this work we only study the use of uniform weightings and the resulting direct slices exhibit non-uniform noise distributions, with the central slices being less noisy. In future studies, by using general weightings to reduce contributions of the data that are acquired at large oblique angles, we expect that more uniform noise distributions can be achieved, and axial blurring due to parallax errors can be reduced, at the cost of overall image variance.

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PubMed2004

A second-order finite element algorithm for solving the three-dimensional EEG forward problem.

A finite element algorithm has been developed to solve the electroencephalogram (EEG) forward problem. A new computationally efficient approach to calculate the stiffness matrix of second-order tetrahedral elements has been developed for second-order tetrahedral finite element models. The present algorithm has been evaluated by means of computer simulations, by comparing with analytic solutions in a multi-spheres concentric head model. The developed finite element method (FEM) algorithm has also been applied to address questions of interest in the EEG forward problem. The present simulation study indicates that the second-order FEM provides substantially enhanced numerical accuracy and computational efficiency, as compared with the first-order FEM for comparable numbers of tetrahedral elements. The anisotropic conductivity distribution of the head tissue can be taken into account in the present FEM algorithm. The effects of dipole eccentricity, size of finite elements and local mesh refinement on solution accuracy are also addressed in the present simulation study.

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PubMed2004

A simple derivation and analysis of a helical cone beam tomographic algorithm for long object imaging via a novel definition of region of interest.

We derive and analyse a simple algorithm first proposed by Kudo et al (2001 Proc. 2001 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine (Pacific Grove, CA) pp 7-10) for long object imaging from truncated helical cone beam data via a novel definition of region of interest (ROI). Our approach is based on the theory of short object imaging by Kudo et al (1998 Phys. Med. Biol. 43 2885-909). One of the key findings in their work is that filtering of the truncated projection can be divided into two parts: one, finite in the axial direction, results from ramp filtering the data within the Tam window. The other, infinite in the z direction, results from unbounded filtering of ray sums over PI lines only. We show that for an ROI defined by PI lines emanating from the initial and final source positions on a helical segment, the boundary data which would otherwise contaminate the reconstruction of the ROI can be completely excluded. This novel definition of the ROI leads to a simple algorithm for long object imaging. The overscan of the algorithm is analytically calculated and it is the same as that of the zero boundary method. The reconstructed ROI can be divided into two regions: one is minimally contaminated by the portion outside the ROI, while the other is reconstructed free of contamination. We validate the algorithm with a 3D Shepp-Logan phantom and a disc phantom.

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PubMed2004

Prostate segmentation algorithm using dyadic wavelet transform and discrete dynamic contour.

Knowing the location and the volume of the prostate is important for ultrasound-guided prostate brachytherapy, a commonly used prostate cancer treatment method. The prostate boundary must be segmented before a dose plan can be obtained. However, manual segmentation is arduous and time consuming. This paper introduces a semi-automatic segmentation algorithm based on the dyadic wavelet transform (DWT) and the discrete dynamic contour (DDC). A spline interpolation method is used to determine the initial contour based on four user-defined initial points. The DDC model then refines the initial contour based on the approximate coefficients and the wavelet coefficients generated using the DWT. The DDC model is executed under two settings. The coefficients used in these two settings are derived using smoothing functions with different sizes. A selection rule is used to choose the best contour based on the contours produced in these two settings. The accuracy of the final contour produced by the proposed algorithm is evaluated by comparing it with the manual contour outlined by an expert observer. A total of 114 2D TRUS images taken for six different patients scheduled for brachytherapy were segmented using the proposed algorithm. The average difference between the contour segmented using the proposed algorithm and the manually outlined contour is less than 3 pixels.

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