skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: General Formula for Fan-Beam Computed Tomography


In this Letter, we derive a general reconstruction formula for fan-beam computed tomography (CT) utilizing the two-dimensional Dirac function, which is useful in CT imaging for moving objects.

;  [1];  [2]
  1. CT Lab, Radiology Department, University of Iowa, Iowa City, Iowa 52242 (United States)
  2. Applied Science Lab, GE Healthcare Technologies, Milwaukee, Wisconsin 53201 (United States)
Publication Date:
OSTI Identifier:
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review Letters; Journal Volume: 95; Journal Issue: 25; Other Information: DOI: 10.1103/PhysRevLett.95.258102; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States

Citation Formats

Wei Yuchuan, Wang Ge, and Hsieh Jiang. General Formula for Fan-Beam Computed Tomography. United States: N. p., 2005. Web. doi:10.1103/PhysRevLett.95.258102.
Wei Yuchuan, Wang Ge, & Hsieh Jiang. General Formula for Fan-Beam Computed Tomography. United States. doi:10.1103/PhysRevLett.95.258102.
Wei Yuchuan, Wang Ge, and Hsieh Jiang. Fri . "General Formula for Fan-Beam Computed Tomography". United States. doi:10.1103/PhysRevLett.95.258102.
title = {General Formula for Fan-Beam Computed Tomography},
author = {Wei Yuchuan and Wang Ge and Hsieh Jiang},
abstractNote = {In this Letter, we derive a general reconstruction formula for fan-beam computed tomography (CT) utilizing the two-dimensional Dirac function, which is useful in CT imaging for moving objects.},
doi = {10.1103/PhysRevLett.95.258102},
journal = {Physical Review Letters},
number = 25,
volume = 95,
place = {United States},
year = {Fri Dec 16 00:00:00 EST 2005},
month = {Fri Dec 16 00:00:00 EST 2005}
  • The objective of this work is to increase system sensitivity in cardiac single-photon emission-computed tomography (SPECT) studies without increasing patient imaging time. For imaging the heart, convergent collimation offers the potential of increased sensitivity over that of parallel-hole collimation. However, if a cone-beam collimated gamma camera is rotated in a planar orbit, the projection data obtained are not complete. Two cone-beam collimators and one fan-beam collimator are used with a three-detector SPECT system. The combined cone-beam/fan-beam collimation provides a complete set of data for image reconstruction. The imaging geometry is evaluated using data acquired from phantom and patient studies. Formore » the Jaszazck cardiac torso phantom experiment, the combined cone-beam/fan-beam collimation provided 1.7 times greater sensitivity than standard parallel-hole collimation (low-energy high-resolution collimators). Also, phantom and patient comparison studies showed improved image quality. The combined cone-beam/fan-beam imaging geometry with appropriate weighting of the two data sets provides improved system sensitivity while measuring sufficient data for artifact free cardiac images.« less
  • Purpose: To evaluate a novel four-dimensional (4D) image-guided radiotherapy (IGRT) technique in stereotactic body RT for liver tumors. Methods and Materials: For 11 patients with 13 intrahepatic tumors, a respiratory-correlated 4D computed tomography (CT) scan was acquired at treatment planning. The target was defined using CT series reconstructed at end-inhalation and end-exhalation. The liver was delineated on these two CT series and served as a reference for image guidance. A cone-beam CT scan was acquired after patient positioning; the blurred diaphragm dome was interpreted as a probability density function showing the motion range of the liver. Manual contour matching ofmore » the liver structures from the planning 4D CT scan with the cone-beam CT scan was performed. Inter- and intrafractional uncertainties of target position and motion range were evaluated, and interobserver variability of the 4D-IGRT technique was tested. Results: The workflow of 4D-IGRT was successfully practiced in all patients. The absolute error in the liver position and error in relation to the bony anatomy was 8 {+-} 4 mm and 5 {+-} 2 mm (three-dimensional vector), respectively. Margins of 4-6 mm were calculated for compensation of the intrafractional drifts of the liver. The motion range of the diaphragm dome was reproducible within 5 mm for 11 of 13 lesions, and the interobserver variability of the 4D-IGRT technique was small (standard deviation, 1.5 mm). In 4 patients, the position of the intrahepatic lesion was directly verified using a mobile in-room CT scanner after application of intravenous contrast. Conclusion: The results of our study have shown that 4D image guidance using liver contour matching between respiratory-correlated CT and cone-beam CT scans increased the accuracy compared with stereotactic positioning and compared with IGRT without consideration of breathing motion.« less
  • Purpose: To evaluate 2 deformable image registration (DIR) algorithms for the purpose of contour mapping to support image-guided adaptive radiation therapy with 4-dimensional cone-beam CT (4DCBCT). Methods and Materials: One planning 4D fan-beam CT (4DFBCT) and 7 weekly 4DCBCT scans were acquired for 10 locally advanced non-small cell lung cancer patients. The gross tumor volume was delineated by a physician in all 4D images. End-of-inspiration phase planning 4DFBCT was registered to the corresponding phase in weekly 4DCBCT images for day-to-day registrations. For phase-to-phase registration, the end-of-inspiration phase from each 4D image was registered to the end-of-expiration phase. Two DIR algorithms—smallmore » deformation inverse consistent linear elastic (SICLE) and Insight Toolkit diffeomorphic demons (DEMONS)—were evaluated. Physician-delineated contours were compared with the warped contours by using the Dice similarity coefficient (DSC), average symmetric distance, and false-positive and false-negative indices. The DIR results are compared with rigid registration of tumor. Results: For day-to-day registrations, the mean DSC was 0.75 ± 0.09 with SICLE, 0.70 ± 0.12 with DEMONS, 0.66 ± 0.12 with rigid-tumor registration, and 0.60 ± 0.14 with rigid-bone registration. Results were comparable to intraobserver variability calculated from phase-to-phase registrations as well as measured interobserver variation for 1 patient. SICLE and DEMONS, when compared with rigid-bone (4.1 mm) and rigid-tumor (3.6 mm) registration, respectively reduced the average symmetric distance to 2.6 and 3.3 mm. On average, SICLE and DEMONS increased the DSC to 0.80 and 0.79, respectively, compared with rigid-tumor (0.78) registrations for 4DCBCT phase-to-phase registrations. Conclusions: Deformable image registration achieved comparable accuracy to reported interobserver delineation variability and higher accuracy than rigid-tumor registration. Deformable image registration performance varied with the algorithm and the patient.« less
  • Purpose: To demonstrate that fiducial tracking during pretreatment Cone-Beam CT (CBCT) can accurately measure tumor motion and that this method should be used to validate 4-dimensional CT (4DCT) margins before each treatment fraction. Methods and Materials: For 31 patients with abdominal tumors and implanted fiducial markers, tumor motion was measured daily with CBCT and fluoroscopy for 202 treatment fractions. Fiducial tracking and maximum-likelihood algorithms extracted 3-dimensional fiducial trajectories from CBCT projections. The daily internal margin (IM) (ie, range of fiducial motion) was calculated for CBCT and fluoroscopy as the 5th-95th percentiles of displacement in each cardinal direction. The planning IMmore » from simulation 4DCT (IM{sub 4DCT}) was considered adequate when within ±1.2 mm (anterior–posterior, left–right) and ±3 mm (superior–inferior) of the daily measured IM. We validated CBCT fiducial tracking as an accurate predictive measure of intrafraction motion by comparing the daily measured IM{sub CBCT} with the daily IM measured by pretreatment fluoroscopy (IM{sub pre-fluoro}); these were compared with pre- and posttreatment fluoroscopy (IM{sub fluoro}) to identify those patients who could benefit from imaging during treatment. Results: Four-dimensional CT could not accurately predict intrafractional tumor motion for ≥80% of fractions in 94% (IM{sub CBCT}), 97% (IM{sub pre-fluoro}), and 100% (IM{sub fluoro}) of patients. The IM{sub CBCT} was significantly closer to IM{sub pre-fluoro} than IM{sub 4DCT} (P<.01). For patients with median treatment time t < 7.5 minutes, IM{sub CBCT} was in agreement with IM{sub fluoro} for 93% of fractions (superior–inferior), compared with 63% for the t > 7.5 minutes group, demonstrating the need for patient-specific intratreatment imaging. Conclusions: Tumor motion determined from 4DCT simulation does not accurately predict the daily motion observed on CBCT or fluoroscopy. Cone-beam CT could replace fluoroscopy for pretreatment verification of simulation IM{sub 4DCT}, reducing patient setup time and imaging dose. Patients with treatment time t > 7.5 minutes could benefit from the addition of intratreatment imaging.« less
  • In deriving algorithms to reconstruct single photon emission computed tomography (SPECT) projection data, it is important that the algorithm compensates for photon attenuation in order to obtain quantitative reconstruction results. A convolution backprojection algorithm was derived by Tretiak and Metz to reconstruct two-dimensional (2-D) transaxial slices from uniformly attenuated parallel-beam projections. Using transformation of coordinates, this algorithm can be modified to obtain a formulation useful to reconstruct uniformly attenuated fan-beam projections. Unlike that for parallel-beam projections, this formulation does not produce a filtered backprojection reconstruction algorithm but instead has a formulation that is an inverse integral operator with a spatiallymore » varying kernel. This algorithm thus requires more computation time than does the filtered backprojection reconstruction algorithm for the uniformly attenuated parallel-beam case. However, the fan-beam reconstructions demonstrate the same image quality as that of parallel-beam reconstructions.« less