Resolution Single Photon Emission Computed Tomography Research Articles

CZT camera systems may provide better risk stratification for low-risk patients

BackgroundThe photon sensitivity and spatial resolution of single-photon emission-computed tomography (SPECT) has been significantly improved by solid-state camera systems using cadmium zinc telluride (CZT) detectors. While the diagnostic accuracy of these systems is well established, there is little evidence directly comparing the prognostic utility to conventional NaI cameras. Methods and ResultsRetrospective analysis of patients undergoing SPECT between 2008 and 2012. Visual SPECT assessment was performed utilizing the 17-segment model to determine summed stress scores (SSS). We identified 12,830 consecutive patients, mean age 63.2 ± 13.7 and 56.1% male, 5072 of whom underwent CZT and 7758 NaI imaging. During a median follow-up duration of 7.0 years (IQR 5.5-8.2), a total of 2788 (21.7%) patients died. Compared to SSS 0, minimal perfusion abnormality (SSS 1-3) was associated with increased all-cause mortality with CZT camera (adjusted HR 1.32, P = .017) and NaI camera (adjusted HR 1.29, P = .001, interaction P = .803). Increasing stress abnormality was associated with a similar increase in risk with CZT or NaI imaging (interaction P > .500). In a propensity matched analysis, patients with normal perfusion stress perfusion assessed with a CZT was associated with decreased mortality compared to normal perfusion assessed by a NaI camera system (hazard ratio .88, 95% CI .78-.99, P = .040). ConclusionsIncreasing stress perfusion abnormality was associated with similar increase in all-cause mortality with CZT or NaI cameras. CZT and NaI camera systems provide similar risk stratification, however, normal myocardial perfusion may be associated with a more benign prognosis when assessed with a CZT camera system.

Simulation study of a novel target oriented SPECT design using a variable pinhole collimator.

In the past decade, demands for organ specific (target oriented) single-photon emission computed tomography (SPECT) is increasing, and several groups have conducted studies on developing clinical dedicated SPECT with pinhole collimator to improve the spatial resolution. However, acceptance angle of the collimator cannot be adjusted to fit the different ROIs of target objects because the shape of pinhole could not be changed, and the magnifying factor cannot be maximized as the collimator-to-detector distance is fixed. Furthermore, those dedicated pinhole SPECTs are typically made for a single purpose and therefore possess a drawback in that it cannot be utilized for any other purpose. In this study, we propose a novel SPECT system using variable pinhole collimator (VP SPECT) whose parameters are flexible. The proposed variable pinhole collimator is modeled on conventional pinhole by piling several tungsten layers of different apertures. Depending on the combination of the holes in each layer, a variety of hole diameters and acceptance angles of the pinhole can be made. In addition, VP SPECT system allows attaching the collimator to the object as close as possible to maximize the sensitivity and adjust the distance of the pinhole from the scintillation detector to optimize the system resolution for each rotation angle, automatically. For quantitative measurement, we compared the sensitivity and spatial resolution of VP SPECT with those of conventional pinhole SPECT. To determine the possibility of the clinical and preclinical use of proposed system, a digital mouse whole-body (MOBY) phantom is used for simulating the live mouse model. The result of simulation using ultra-micro hot spot phantom shows that the sensitivity of the proposed VP SPECT system is about 297% of that of the conventional system. While hot rods of diameter 0.6mm can be distinguished in the image with the proposed VP SPECT system, 1.2-mm hot rods are barely discernible in the conventional pinhole SPECT image. According to the result of MOBY phantom simulation, heart walls separated by 3mm were not distinguished in conventional pinhole SPECT images, but were clearly discerned in VP SPECT images. In this study, we designed a novel pinhole collimator for SPECT and presented preliminary results of target oriented imaging with a simulation study. Currently, we are pursuing strategies to realize the proposed system, with the goal to apply the technology into a high-sensitivity and high-resolution preclinical SPECT. Should VP SPECT be applied to the clinical setting, we anticipate a high-sensitivity, high-resolution system for applications such as heart dedicated SPECT or related fields.

Dynamic Positron Emission Tomography Imaging of Renal Clearable Gold Nanoparticles.

Optical imaging has been the primary imaging modality for nearly all of the renal clearable nanoparticles since 2007. Due to the tissue depth penetration limitation, providing accurate organ kinetics non-invasively has long been a huge challenge. Although a more quantitative imaging technique has been developed by labeling nanoparticles with single-photon emission computed tomography (SPECT) isotopes, the low temporal resolution of SPECT still limits its potential for visualizing the rapid dynamic process of renal clearable nanoparticles in vivo. The dynamic positron emission tomography (PET) imaging of renal clearable gold (Au) nanoparticles by labeling them with copper-64 ((64) Cu) to form (64) Cu-NOTA-Au-GSH is reported. Systematic nanoparticle synthesis and characterizations are performed to demonstrate the efficient renal clearance of as-prepared nanoparticles. A rapid renal clearance of (64) Cu-NOTA-Au-GSH is observed (>75%ID at 24 h post-injection) with its elimination half-life calculated to be less than 6 min, over 130 times shorter than previously reported similar nanoparticles. Dynamic PET imaging not only addresses the current challenges in accurately and non-invasively acquiring the organ kinetics, but also potentially provides a highly useful tool for studying renal clearance mechanism of other ultra-small nanoparticles, as well as the diagnosis of kidney diseases in the near future.

Spatial resolution is dependent on image content for SPECT with iterative reconstruction incorporating distance dependent resolution (DDR) correction.

The aim of this study is to determine the dependence of single photon emission computed tomography (SPECT) spatial resolution on the content of images for iterative reconstruction with distance dependent resolution (DDR) correction. An experiment was performed using a perturbation technique to measure change in resolution of line sources in simple and complex images with iterative reconstruction with increasing iteration. Projections of the line sources were reconstructed alone and again after the addition of projections of a uniform flood or a complex phantom. An alternative experiment used images of a realistic brain phantom and evaluated an effective spatial resolution by matching the images to the digital version of the phantom convolved with 3D Gaussian kernels. The experiments were performed using ordered subset expectation maximisation iterative reconstruction with and without the use of DDR correction. The results show a significant difference in reconstructed resolution between images of line sources depending on the content of the added image. The full width at half maximum of images of a line source reconstructed using DDR correction increased by 20-30% when the added image was complex. Without DDR this difference was much smaller and disappeared with increasing iteration. Reported SPECT resolution should be taken as indicative only with regard to clinical imaging if the measurement is made using a point or line source alone and an iterative reconstruction algorithm is used.

Analytical cone-beam SPECT reconstruction algorithm with non-uniform attenuation for general non-circular orbit

In single photon emission computed tomography (SPECT), due to the attenuation of gamma photons, the analytical reconstruction is complicated, where attenuation should be compensated to obtain quantitative results. We know that the resolution of SPECT is low. The cone-beam SPECT reconstruction can improve the photon density and spatial resolution of the reconstructed image. In practice, to minimize the effect of distance-dependent resolution variation (DDRV), the detector should be set as close as possible to the patient. Therefore it would be more efficacious for the orbit of the detector to be elliptical or another shape. In this paper, based on the Novikov's reconstruction formula and our Ray-driven Technology, we present an analytical cone-beam SPECT reconstruction algorithm for general non-circular orbit. The simulation results demonstrate the accuracy and robustness of our method.

WE-G-500-09: Robotic Multi-Pinhole Scenarios for SPECT Molecular and Functional Imaging Onboard and in Other Applications

Purpose: Single Photon Emission Computed Tomography (SPECT) is one of the two predominant modes of nuclear emission tomography, the other being Positron Emission Tomography (PET). An important recent advance has been region-of-interest (ROI) SPECT for cardiac imaging. For SPECT imaging onboard radiation therapy machines, we have proposed a robotic, multi-pinhole SPECT approach. Robotic maneuvering facilitates navigation of the SPECT system about patient, patient table, and gantry, and it also enables more flexible positioning of the pinholes. Even more pinhole-positioning flexibility can be provided by enabling more degrees of freedom (DOFs) in the motion of pinholes relative to one another. This will likely increase imaging performance, but also system cost. The purpose of this study is to investigate imaging performance as a function of DOFs in relative pinhole motion. This has bearing for molecular and functional imaging onboard radiation therapy machines, and also for other applications of SPECT ROI imaging. Methods: A 49-pinhole system which we have previously presented is considered for imaging ROIs smaller than, equal in size to, and larger than the ROI for which the system was optimized. This system has one internal DOF which enables relative motion among the seven 7-pinhole pods that comprise the system. Also considered is a second system in which 7 additional DOFs enable relative motion of the pinholes within each pod. Imaging performance of these two 49-pinhole systems is compared over a range of ROI sizes. Results: The additional DOFs can improve proximity to smaller ROIs and thereby improve SPECT sensitivity and resolution for those ROIs. Conclusion: SPECT ROI imaging can be advanced by a multi-pinhole approach that enables greater relative motion of pinholes. This work was supported by PHS/NIH/NCI grant 5R21CA156390.

SPECT Imaging With Resolution Recovery

Single-photon emission computed tomography (SPECT) is a method of choice for imaging spatial distributions of radioisotopes. Applications of this method are found in medicine, biomedical research and nuclear industry. This paper deals with improving spatial resolution in SPECT by applying correction for the point-spread function (PSF) in the reconstruction algorithm and optimizing the collimator. Several approaches are considered: the use of a depth-dependent PSF model for a parallel-beam collimator derived from experimental data, the extension of this model to a fan-beam collimator, a triangular approximation of the PSF for reconstruction acceleration, and a method for optimal fan-beam collimator design. An unmatched projector/backprojector ordered subsets expectation maximization (OSEM) algorithm is used for image reconstruction. Experimental results with simulated and physical phantom data of a micro-SPECT system show a significant improvement of spatial resolution with the proposed methods.

Monte carlo study of the effect of collimator thickness on T-99m source response in single photon emission computed tomography

In single photon emission computed tomography (SPECT), the collimator is a crucial element of the imaging chain and controls the noise resolution tradeoff of the collected data. The current study is an evaluation of the effects of different thicknesses of a low-energy high-resolution (LEHR) collimator on tomographic spatial resolution in SPECT. In the present study, the SIMIND Monte Carlo program was used to simulate a SPECT equipped with an LEHR collimator. A point source of 99mTc and an acrylic cylindrical Jaszczak phantom, with cold spheres and rods, and a human anthropomorphic torso phantom (4D-NCAT phantom) were used. Simulated planar images and reconstructed tomographic images were evaluated both qualitatively and quantitatively. According to the tabulated calculated detector parameters, contribution of Compton scattering, photoelectric reactions, and also peak to Compton (P/C) area in the obtained energy spectrums (from scanning of the sources with 11 collimator thicknesses, ranging from 2.400 to 2.410 cm), we concluded the thickness of 2.405 cm as the proper LEHR parallel hole collimator thickness. The image quality analyses by structural similarity index (SSIM) algorithm and also by visual inspection showed suitable quality images obtained with a collimator thickness of 2.405 cm. There was a suitable quality and also performance parameters’ analysis results for the projections and reconstructed images prepared with a 2.405 cm LEHR collimator thickness compared with the other collimator thicknesses.

Quantification and reduction of the collimator-detector response effect in SPECT by applying a system model during iterative image reconstruction

Detector blurring and non-ideal collimation decrease the spatial resolution of the single-photon emission computed tomography (SPECT) images. Iterative reconstruction algorithms such as ordered subsets expectation maximization (OSEM) can incorporate degrading factors during reconstruction. We investigated the quantitative errors associated with poor SPECT resolution and evaluated the importance of two-dimensional (2D) and three-dimensional (3D) resolution recovery by modelling system response during iterative image reconstruction. Different phantoms consisted of the NURBS-based cardiac-torso (NCAT) liver phantom with small tumors, the Zubal brain phantom and the NCAT heart phantom were used in this study. Monte Carlo simulation was used to create SPECT projections. Gaussian functions were used to model collimator detector response (CDR). Modeled CDRs were applied during OSEM. Both noise-free and noisy projections were created. Even with noise-free projections, conventional OSEM algorithm provided limited quantitative accuracy compared to both 2D and 3D resolution recovery. The 3D implementation of resolution recovery, however, yielded superior results compared to its 2D implementation. For the liver phantom, the ability to distinguish small tumors in both transverse and axial planes was improved. For the brain phantom, gray to white matter activity ratio was increased from 3.14 ± 0.04 in simple OSEM to 3.84 ± 0.06 in 3D OSEM. For the NCAT heart phantom, 3D resolution recovery, results in images with thinner wall and higher contrast for different noise levels. There are considerable quantitative errors associated with CDR, especially when the size of the target is comparable with the spatial resolution of the system. Between different reconstruction algorithms, 3D OSEM that consider the 3D nature of CDR, improve both the visual quality and the quantitative accuracy of any SPECT studies.

Detecting Small Liver Tumors With $^{111}$In-Pentetreotide SPECT—A Collimator Study Based on Monte Carlo Simulations

In 111In-pentetreotide single-photon emission computed tomography (SPECT), the tumor-to-background-uptake ratio is generally high. The noise is, however, also usually on a high level, and in combination with the low spatial resolution of SPECT, this may lead to difficulties in the detection of small tumors. This is especially the case in regions with a relatively high background activity, such as in the liver, which is a common region for somatostatin-positive metastases. Visually detecting the small tumors is important for a successful treatment of the cancer disease. In this paper, we compare three different parallel-hole collimators for 111In-pentetreotide SPECT regarding contrast as a function of image noise for a phantom simulating small tumors in liver background. The corresponding contrast-to-noise ratios are also presented. All raw-data projections are produced using Monte Carlo simulations. The collimators are of type low-energy general-purpose (LEGP), extended LEGP (ELEGP), and medium-energy general-purpose (MEGP). Reconstructions were performed with OSEM both with and without model-based compensation. Of the investigated collimators, the ELEGP collimator proved to be the most optimal for the smallest tumors, both with and without model-based compensation included in the reconstruction. It is also shown that model-based compensation outperforms the conventional reconstruction technique.

Development and Application of a Multimodal Contrast Agent for SPECT/CT Hybrid Imaging

Hybrid or multimodality imaging is often applied in order to take advantage of the unique and complementary strengths of individual imaging modalities. This hybrid noninvasive imaging approach can provide critical information about anatomical structure in combination with physiological function or targeted molecular signals. While recent advances in software image fusion techniques and hybrid imaging systems have enabled efficient multimodal imaging, accessing the full potential of this technique requires development of a new toolbox of multimodal contrast agents that enhance the imaging process. Toward that goal, we report the development of a hybrid probe for both single photon emission computed tomography (SPECT) and X-ray computed tomography (CT) imaging that facilitates high-sensitivity SPECT and high spatial resolution CT imaging. In this work, we report the synthesis and evaluation of a novel intravascular, multimodal dendrimer-based contrast agent for use in preclinical SPECT/CT hybrid imaging systems. This multimodal agent offers a long intravascular residence time (t(1/2) = 43 min) and sufficient contrast-to-noise for effective serial intravascular and blood pool imaging with both SPECT and CT. The colocalization of the dendritic nuclear and X-ray contrasts offers the potential to facilitate image analysis and quantification by enabling correction for SPECT attenuation and partial volume errors at specified times with the higher resolution anatomic information provided by the circulating CT contrast. This may allow absolute quantification of intramyocardial blood volume and blood flow and may enable the ability to visualize active molecular targeting following clearance from the blood.

Illuminating progression: tracking melanoma metastasis in a gene therapy setting

Illuminating progression: tracking melanoma metastasis in a gene therapy setting

Small-animal SPECT and SPECT/CT: application in cardiovascular research

Preclinical cardiovascular research using noninvasive radionuclide and hybrid imaging systems has been extensively developed in recent years. Single photon emission computed tomography (SPECT) is based on the molecular tracer principle and is an established tool in noninvasive imaging. SPECT uses gamma cameras and collimators to form projection data that are used to estimate (dynamic) 3-D tracer distributions in vivo. Recent developments in multipinhole collimation and advanced image reconstruction have led to sub-millimetre and sub-half-millimetre resolution SPECT in rats and mice, respectively. In this article we review applications of microSPECT in cardiovascular research in which information about the function and pathology of the myocardium, vessels and neurons is obtained. We give examples on how diagnostic tracers, new therapeutic interventions, pre- and postcardiovascular event prognosis, and functional and pathophysiological heart conditions can be explored by microSPECT, using small-animal models of cardiovascular disease.

A pinhole gamma camera with optical depth-of-interaction elimination

The performance of pinhole single photon emission computed tomography (SPECT) depends on the spatial resolution of the gamma-ray detectors used. Pinhole cameras suffer from strong resolution loss due to the varying depth-of-interaction (DOI) of gamma quanta that enter the detector material at an angle. We eliminate DOI effects in a scintillation gamma camera via a dedicated optic fiber bundle that acts as a focusing collimator for light generated in a scintillation crystal. A curved crystal is connected to a concavely shaped fiber-optic bundle such that the fibers connect perpendicularly to the crystal's convex surface and point straight at the pinhole opening. Limiting the fiber numerical apertures can be used to suppress resolution losses due to light spread. Here we demonstrate experimentally that this prototype position-sensitive gamma sensor successfully eliminates DOI effects, and has an intrinsic resolution of better than 280 µm full width at half maximum with an interaction probability of 67% for 140 keV photons. Therefore, the detector has great potential for increasing the resolution of pinhole SPECT.

Three-pinhole collimator to improve axial spatial resolution and sensitivity in pinhole SPECT

The aim of this study was to improve the uniformity of the axial spatial resolution and sensitivity in pinhole single-photon emission computed tomography (SPECT) and to extend the axial field of view, by using a dedicated three-pinhole collimator. A rectangular tungsten plate with three pinhole apertures of 1.5 mm diameter was designed to image a cylindrical field of view of 55 mm diameter and 160 mm axial length using a clinical gamma camera. To evaluate the non-uniformity of spatial resolution and noise, a multiple-disk phantom was built. The phantom was filled with Tc-99m, and data were acquired using a circular orbit and reconstructed with a dedicated iterative reconstruction algorithm. The axial spatial resolution together with the noise was measured in each disk. These measurements were compared to a single-pinhole system using an identical acquisition geometry and reconstruction. At the central slice, a spatial resolution of 2.7 mm was observed for both the three-pinhole and single-pinhole geometries. At 17.5 mm from the central slice, the axial spatial resolution deteriorated to 10.3 mm when using a single pinhole, while the spatial resolution remained 2.7 mm for the three-pinhole system. In the central slice, 19% noise was observed for both geometries. At 31.5 mm from this central slice, the noise remained 19% for the three-pinhole geometry, while it increased to 32% using a single pinhole. The presented three-pinhole collimator improves the uniformity of the axial spatial resolution and sensitivity in pinhole SPECT and consequently extends the axial field of view, a requirement for whole-body small-animal imaging.

Fusion of the MR image to SPECT with possible correction for partial volume effects

Low spatial resolution and the related partial volume effects limit the diagnostic potential of brain single photon emission computed tomography (SPECT) imaging. As a possible remedy for this problem we propose a technique for the fusion of SPECT and MR images, which requires for a given patient the SPECT data and the T1-weighted MR image. Basically, after the reconstruction and coregistration steps, the high-frequency part of the MR, which would be unrecoverable by the set SPECT acquisition system + reconstruction algorithm, is extracted and added to the SPECT image. The tuning of the weight of the MR on the resulting fused image can be performed very quickly, any iterative reconstruction algorithm can be used and, in the case that the SPECT projections are not available, the proposed technique can also be applied directly to the SPECT image, provided that the performance of the scanner is known. The procedure has the potential of increasing the diagnostic value of a SPECT image. Even in the locations of SPECT-MR mismatch it does not significantly affect quantitation over regions of interest (ROIs) whose dimensions are decidedly larger than the SPECT resolution distance. On the other hand, appreciable corrections for partial volume effects are expected in the locations where the contrast in the structural MR matches the corresponding contrast in functional activity.

Improvement of image resolution and quantitative accuracy in clinical Single Photon Emission Computed Tomography

Clinical Single Photon Emission Computed Tomography (SPECT) is a scanning technique which acquires gamma-camera images (‘projections’) over a range of angles around a patient. These projections allow the reconstruction of cross sectional (‘tomographic’) images of the gamma-radiating pharmaceutical distribution in the patient, thus providing interesting information about the functioning of organs and tissues. SPECT images are seriously affected by a variety of image degrading processes. Restrictions on the amount of radio-pharmaceutical that can be administered to a patient cause noise in the projections and the limited spatial resolution of the gamma-camera results in blurring of the projections. In addition to these image degradations, the reconstruction of cross-sections is complicated by Compton scattering of γ-photons in tissue, which causes attenuation of the photon flux received by the gamma-camera and causes improper detection of photons which have been scattered in tissue. This results in some additional blurring and loss of accuracy of the SPECT images in predicting activity concentrations. Tremendous efforts have been made to improve the quantitative accuracy and the spatial resolution of SPECT, and to reduce the noise in the reconstructed images. These efforts have resulted in corrective reconstruction algorithms, which are generally based on incorporation of accurate models of the main image degrading factors. Improvements of the data acquisition hardware can further increase image quality. In this paper, the image formation process of SPECT, including image-degrading factors, is explained. In addition, reconstruction algorithms and hardware modifications are reviewed, and their effects on image quality are illustrated with physical phantom and simulation experiments.

Heterogeneity of Cerebral Blood Flow: a Fractal Approach

We demonstrate the heterogeneity of regional cerebral blood flow using a fractal approach and single-photon emission computed tomography (SPECT). Tc-99m-labelled ethylcysteine dimer was injected intravenously in 10 healthy controls and in 10 patients with dementia of frontal lobe type. The head was imaged with a gamma camera and transaxial, sagittal and coronal slices were reconstructed. Two hundred fifty-six symmetrical regions of interest (ROIs) were drawn onto each hemisphere of functioning brain matter. Fractal analysis was used to examine the spatial heterogeneity of blood flow as a function of the number of ROIs. Relative dispersion (= coefficient of variation of the regional flows) was fractal-like in healthy subjects and could be characterized by a fractal dimension of 1.17 +/- 0.05 (mean +/- SD) for the left hemisphere and 1.15 +/- 0.04 for the right hemisphere, respectively. The fractal dimension of 1.0 reflects completely homogeneous blood flow and 1.5 indicates a random blood flow distribution. Patients with dementia of frontal lobe type had a significantly lower fractal dimension of 1.04 +/- 0.03 than in healthy controls. Within the limits of spatial resolution of SPECT, the heterogeneity of brain blood flow is well characterized by a fractal dimension. Fractal analysis may help brain scientists to assess age-, sex- and laterality-related anatomic and physiological changes of brain blood flow and possibly to improve precision of diagnostic information available for patient care.

A rotating parallel hole collimator for high resolution imaging of medium energy radionuclides

A rotating parallel hole collimator has been designed and built for high resolution gamma camera imaging of medium energy radionuclides. This cast lead collimator has 2.0 mm thick septa and was designed for imaging 50-100 mCi I-131 radiolabeled monoclonal antibodies being administered intratumorally for brain tumor therapy. Projection data from different collimator rotation positions are summed to create a composite projection dataset with more dense spatial sampling than projection data at a single position. Measured single photon emission computed tomography (SPECT) resolution with I-131 was 5.5-7.5 mm full-width at half-maximum (FWHM) for a 13 cm radius of rotation compared with 9.9-13.9 mm with a conventional medium energy collimator. SPECT resolution for the rotating collimator was only slightly worse with I-131 than with Tc-99m, indicating good attenuation by the septa. Sensitivity was 56 counts/sec/mCi with a tight region of interest (ROI) around the projection of an I-131 line source 12 cm from the collimator surface and 64 counts/sec/mCi with a tight ROI around the projection of an 11 cm diameter I-131 disk source 10 cm from the collimator surface. SPECT studies of 3.7 cm and 6.2 cm diameter shell-core tumor phantoms filled with I-131 were made. The lower activity core and higher activity shell were well resolved in a SPECT study using the rotating hole collimator. SPECT shell to core activity concentration ratios were more accurate with the rotating collimator than with a medium energy collimator. These results show the potential of this rotating collimator for high resolution human tumor studies with I-131 tracers.

Improved estimation of the detector response function for converging beam collimators

Converging beam collimator geometries offer improved tradeoffs between resolution and noise for single photon emission computed tomography (SPECT). The major factor limiting the resolution in SPECT is the collimator-detector response blurring. In order to compensate for this blurring it is useful to be able to calculate the collimator response function. A previous formulation presented a method for calculating the response for parallel and converging beam collimators that assumed that the shape of the holes did not change over the face of the collimator. However, cast collimators are fabricated using pins with a constant cross-section (shape perpendicular to the pin axis). As a result, due to the angulation of the pins, the holes made by these pins have shapes on the front and back faces of the collimator that change with position. This change in hole shape is especially pronounced when the angle between the collimator hole and the collimator normal is large, as is the case for half-fan-beam or short-focal-length collimators. This paper presents a derivation of a modification to the original method that accounts for the change in shape of the collimator holes. The method has been verified by comparing predicted line spread functions to experimentally measured ones for a collimator with a maximum hole angle of with respect to the normal. This formulation is useful for predicting the response of fan-beam collimators in the design process and for use in detector response compensation algorithms.