Tiny changes in tomographic system matrices can cause large changes in reconstruction quality.

This paper studies the impact of tiny changes in region-of-interest (ROI) tomography system matrices on the variance of the reconstructed ROI. In small-scale and medium-scale examples, the variance in the reconstructed ROI was estimated for different system matrices. The results revealed a striking and counterintuitive phenomenon: a tiny change in the system matrix can dramatically affect the variance of the ROI estimate. In one of our examples, a decrease of 0.1% in one element out of hundreds of thousands of the system matrix resulted in a systematic reduction of the variance inside the ROI, and by a factor of 5 to 10 for some pixels. Our results agree with a recently proven theorem about the ability of additional measurements to reduce the variance in ROI tomography.

Centers and centroids of the cone-beam projection of a ball.

In geometric calibration of cone-beam (CB) scanners, point-like marker objects such as small balls are imaged to obtain positioning information from which the unknown geometric parameters are extracted. The procedure is sensitive to errors in the positioning information, and one source of error is a small bias which can occur in estimating the detector locations of the CB projections of the centers of the balls. We call these detector locations the center projections. In general, the CB projection of a ball of uniform density onto a flat detector forms an ellipse. Inside the ellipse lie the center projection M, the ellipse center C and the centroid G of the intensity values inside the ellipse. The center projection is invariably estimated from C or G which are much easier to extract directly from the data. In this work, we quantify the errors incurred in using C or G to estimate M. We prove mathematically that the points C, G, M and O are always distinct and lie on the major axis of the ellipse, where O is the detector origin, defined as the orthogonal projection of the cone vertex onto the detector. (The ellipse can only degenerate to a circle if the ball is along the direct line of sight to O, and in this case all four points coincide.) The points always lie in the same order: O, M, G, C which establishes that the centroid has less geometric bias than the ellipse center for estimating M. However, our numerical studies indicate that the centroid bias is only 20% less than the ellipse center bias so the benefit in using centroid estimates is not substantial. For the purposes of quantifying the bias in practice, we show that the ellipse center bias ||CM|| can be conveniently estimated by eA/(π ƒ(≈) where A is the area of the elliptical projection, e is the eccentricity of the ellipse and ƒ(≈) is an estimate of the focal length of the system. Finally, we discuss how these results are affected by physical factors such as beam hardening, and indicate extensions to balls of non-uniform density.

Direct determination of geometric alignment parameters for cone-beam scanners.

This paper describes a comprehensive method for determining the geometric alignment parameters for cone-beam scanners (often called calibrating the scanners or performing geometric calibration). The method is applicable to x-ray scanners using area detectors, or to SPECT systems using pinholes or cone-beam converging collimators. Images of an alignment test object (calibration phantom) fixed in the field of view of the scanner are processed to determine the nine geometric parameters for each view. The parameter values are found directly using formulae applied to the projected positions of the test object marker points onto the detector. Each view is treated independently, and no restrictions are made on the position of the cone vertex, or on the position or orientation of the detector. The proposed test object consists of 14 small point-like objects arranged with four points on each of three orthogonal lines, and two points on a diagonal line. This test object is shown to provide unique solutions for all possible scanner geometries, even when partial measurement information is lost by points superimposing in the calibration scan. For the many situations where the cone vertex stays reasonably close to a central plane (for circular, planar, or near-planar trajectories), a simpler version of the test object is appropriate. The simpler object consists of six points, two per orthogonal line, but with some restrictions on the positioning of the test object. This paper focuses on the principles and mathematical justifications for the method. Numerical simulations of the calibration process and reconstructions using estimated parameters are also presented to validate the method and to provide evidence of the robustness of the technique.

In vivo prompt gamma neutron activation analysis for the screening of boron-10 distribution in a rabbit knee: a simulation study.

Boron neutron capture synovectomy (BNCS) is under development as a potential treatment modality for rheumatoid arthritis (RA). RA is characterized by the inflammation of the synovium (the membrane lining articular joints), which leads to pain and a restricted range of motion. BNCS is a two-part procedure involving the injection of a boronated compound directly into the diseased joint followed by irradiation with a low-energy neutron beam. The neutron capture reactions taking place in the synovium deliver a local, high-linear energy transfer (LET) dose aimed at destroying the inflamed synovial membrane. For successful treatment via BNCS, a boron-labeled compound exhibiting both high synovial uptake and long retention time is necessary. Currently, the in vivo uptake behavior of potentially useful boronated compounds is evaluated in the knee joints of rabbits in which arthritis has been induced. This strategy involves the sacrifice and dissection of a large number of animals. An in vivo (10)B screening approach is therefore under investigation with the goal of significantly reducing the number of animals needed for compound evaluation via dissection studies. The 'in vivo prompt gamma neutron activation analysis' (IVPGNAA) approach uses a narrow neutron beam to irradiate the knee from several angular positions following the intra-articular injection of a boronated compound whose uptake characteristics are unknown. A high-purity germanium detector collects the 478 keV gamma photons produced by the (10)B capture reactions. The (10)B distribution in the knee is then reconstructed by solving a system of simultaneous equations using a weighted least squares algorithm. To study the practical feasibility of IVPGNAA, simulation data were generated with the Monte Carlo N-particle transport code. The boron-containing region of a rabbit knee was partitioned into 8 compartments, and the (10)B prompt gamma signals were tallied from 16 angular positions. Results demonstrate that for this level of spatial resolution, an estimate of (10)B distribution inside the joint can be obtained to within 10% uncertainty, under ideal conditions. Variations of the anatomic dimensions among individual rabbit knees and potential knee positioning errors will result in an uncertainty of over 20%. IVPGNAA thus provides sufficient resolution and quantification regarding the in vivo uptake characteristics of boronated pharmaceuticals to serve as a useful means of screening new compounds of potential use in BNCS.

Even more inversion formulas for the 2D Radon Transform of functions of compact and convex support.

In 2004, Clackdoyle and Noo published a class of inversion formulas for the 2D Radon Transform which depends on the known radius of support of the unknown function. In this work, we extend this class of inversion formulas from functions of circular support to functions with any compact and convex support. We point out the potential benefits of these new inversion formulas in the context of reconstruction from truncated projections. A preliminary implementation of these new inversion formulas is also presented.

Algebraic and analytic reconstruction methods for dynamic tomography.

In this work, we discuss algebraic and analytic approaches for dynamic tomography. We present a framework of dynamic tomography for both algebraic and analytic approaches. We finally present numerical experiments.

The effect of acquisition interval and spatial resolution on dynamic cardiac imaging with a stationary SPECT camera.

The current SPECT scanning paradigm that acquires images by slow rotation of multiple detectors in body-contoured orbits around the patient is not suited to the rapid collection of tomographically complete data. During rapid image acquisition, mechanical and patient safety constraints limit the detector orbit to circular paths at increased distances from the patient, resulting in decreased spatial resolution. We consider a novel dynamic rotating slant-hole (DyRoSH) SPECT camera that can collect full tomographic data every 2 s, employing three stationary detectors mounted with slant-hole collimators that rotate at 30 rpm. Because the detectors are stationary, they can be placed much closer to the patient than is possible with conventional SPECT systems. We propose that the decoupling of the detector position from the mechanics of rapid image acquisition offers an additional degree of freedom which can be used to improve accuracy in measured kinetic parameter estimates. With simulations and list-mode reconstructions, we consider the effects of different acquisition intervals on dynamic cardiac imaging, comparing a conventional three detector SPECT system with the proposed DyRoSH SPECT system. Kinetic parameters of a two-compartment model of myocardial perfusion for technetium-99m-teboroxime were estimated. When compared to a conventional SPECT scanner for the same acquisition periods, the proposed DyRoSH system shows equivalent or reduced bias or standard deviation values for the kinetic parameter estimates. The DyRoSH camera with a 2 s acquisition period does not show any improvement compared to a DyRoSH camera with a 10 s acquisition period.

Normalizing projection images: a study of image normalizing procedures for single particle three-dimensional electron microscopy.

In the process of three-dimensional reconstruction of single particle biological macromolecules several hundreds, or thousands, of projection images are taken from tens or hundreds of independently digitized micrographs. These different micrographs show differences in the background grey level and particle contrast and, therefore, have to be normalized by scaling their pixel values before entering the reconstruction process. In this work several normalization procedures are studied using a statistical comparison framework. We finally show that the use of the different normalization methods affects the reconstruction quality, providing guidance on the choice of normalization procedures.

Fast fully 3-D image reconstruction in PET using planograms.

We present a method of performing fast and accurate three-dimensional (3-D) backprojection using only Fourier transform operations for line-integral data acquired by planar detector arrays in positron emission tomography. This approach is a 3-D extension of the two-dimensional (2-D) linogram technique of Edholm. By using a special choice of parameters to index a line of response (LOR) for a pair of planar detectors, rather than the conventional parameters used to index a LOR for a circular tomograph, all the LORs passing through a point in the field of view (FOV) lie on a 2-D plane in the four-dimensional (4-D) data space. Thus, backprojection of all the LORs passing through a point in the FOV corresponds to integration of a 2-D plane through the 4-D "planogram." The key step is that the integration along a set of parallel 2-D planes through the planogram, that is, backprojection of a plane of points, can be replaced by a 2-D section through the origin of the 4-D Fourier transform of the data. Backprojection can be performed as a sequence of Fourier transform operations, for faster implementation. In addition, we derive the central-section theorem for planogram format data, and also derive a reconstruction filter for both backprojection-filtering and filtered-backprojection reconstruction algorithms. With software-based Fourier transform calculations we provide preliminary comparisons of planogram backprojection to standard 3-D backprojection and demonstrate a reduction in computation time by a factor of approximately 15.

Exact inversion of the exponential x-ray transform for rotating slant-hole (RSH) SPECT.

The RSH SPECT scanner provides parallel-beam attenuated projections for a fully 3D acquisition geometry. The geometry can be represented by circles on the unit sphere of projection directions, one circle for each position of the detector head. Unlike most other fully 3D geometries this one is particularly challenging because there are no 2D subsets in the data. When no attenuation is present, it is well known that an unmeasured projection can be synthesized if it lies inside one of the measured circles. The main result of this work is that under some assumptions on the attenuation distribution, attenuated projections within a circle can be synthesized from available attenuated projections. One consequence is that RSH SPECT projections can be rebinned into a conventional SPECT geometry for which analytic attenuation correction techniques are available.

Analytic method based on identification of ellipse parameters for scanner calibration in cone-beam tomography.

This paper is about calibration of cone-beam (CB) scanners for both x-ray computed tomography and single-photon emission computed tomography. Scanner calibration refers here to the estimation of a set of parameters which fully describe the geometry of data acquisition. Such parameters are needed for the tomographic reconstruction step. The discussion is limited to the usual case where the cone vertex and planar detector move along a circular path relative to the object. It is also assumed that the detector does not have spatial distortions. We propose a new method which requires a small set of measurements of a simple calibration object consisting of two spherical objects, that can be considered as 'point' objects. This object traces two ellipses on the detector and from the parametric description of these ellipses, the calibration geometry can be determined analytically using explicit formulae. The method is robust and easy to implement. However, it is not fully general as it is assumed that the detector is parallel to the rotation axis of the scanner. Implementation details are given for an experimental x-ray CB scanner.

Attenuation correction in SPECT using consistency conditions for the exponential ray transform.

Using data consistency conditions for the exponential ray transform, a method is derived to correct SPECT data for attenuation effects. No transmission measurements are required, and no operator-defined contours are needed. Furthermore, any 3D parallel-ray geometry can be considered for SPECT data acquisition, even unconventional geometries which do not lead to a set of 2D parallel-beam sinograms. The method is presented for both the 2D parallel-beam geometry and a particular 3D case, called the rotating slant hole geometry. Full details of the algorithms are given. Implementation has been carried out and results are presented in 2D and in 3D using simulated data.

Blind estimation of compartmental model parameters.

Computation of physiologically relevant kinetic parameters from dynamic PET or SPECT imaging requires knowledge of the blood input function. This work is concerned with developing methods to accurately estimate these kinetic parameters blindly; that is, without use of a directly measured blood input function. Instead, only measurements of the output functions--the tissue time-activity curves--are used. The blind estimation method employed here minimizes a set of cross-relation equations, from which the blood term has been factored out, to determine compartmental model parameters. The method was tested with simulated data appropriate for dynamic SPECT cardiac perfusion imaging with 99mTc-teboroxime and for dynamic PET cerebral blood flow imaging with 15O water. The simulations did not model the tomographic process. Noise levels typical of the respective modalities were employed. From three to eight different regions were simulated, each with different time-activity curves. The time-activity curve (24 or 70 time points) for each region was simulated with a compartment model. The simulation used a biexponential blood input function and washin rates between 0.2 and 1.3 min(-1) and washout rates between 0.2 and 1.0 min(-1). The system of equations was solved numerically and included constraints to bound the range of possible solutions. From the cardiac simulations, washin was determined to within a scale factor of the true washin parameters with less than 6% bias and 12% variability. 99mTc-teboroxime washout results had less than 5% bias, but variability ranged from 14% to 43%. The cerebral blood flow washin parameters were determined with less than 5% bias and 4% variability. The washout parameters were determined with less than 4% bias, but had 15-30% variability. Since washin is often the parameter of most use in clinical studies, the blind estimation approach may eliminate the current necessity of measuring the input function when performing certain dynamic studies.

Single-slice rebinning method for helical cone-beam CT.

In this paper, we present reconstruction results from helical cone-beam CT data, obtained using a simple and fast algorithm, which we call the CB-SSRB algorithm. This algorithm combines the single-slice rebinning method of PET imaging with the weighting schemes of spiral CT algorithms. The reconstruction is approximate but can be performed using 2D multislice fan-beam filtered backprojection. The quality of the results is surprisingly good, and far exceeds what one might expect, even when the pitch of the helix is large. In particular, with this algorithm comparable quality is obtained using helical cone-beam data with a normalized pitch of 10 to that obtained using standard spiral CT reconstruction with a normalized pitch of 2.