On the question of 3D seed reconstruction in prostate brachytherapy: the determination of x-ray source and film locations

Inaccuracy in seed placement during permanent prostate implants may lead to significant dosimetric deviations from the intended plan. In two recent publications (Todor et al 2002 Phys. Med. Biol. 47 2031-48, Todor et al 2003 Phys. Med. Biol. 48 1153-71), methodology was described for identifying intraoperatively the positions of seeds already implanted, thus allowing reoptimization of the treatment plan and correcting for such seed misplacement. Seed reconstruction is performed using fluoroscopic images and an important (and non-trivial) component of this approach is the ability to accurately determine the position of the gantry relative to the treatment volume. We describe the methodology for acquiring this information, based on the known geometry of six markers attached to the ultrasound probe. This method does not require the C-arm unit to be isocentric and films can be taken with the gantry set at any arbitrary position. This is significant because the patient positioning on the operating table (in the lithotomy position) restricts the range of angles at which films can be taken to a quite narrow (typically +/- 10 degrees) interval and, as a general rule, the closer the angles the larger the uncertainty in the seed location reconstruction along the direction from the x-ray source to the film.     

Ir-192 HDR transit dose and radial dose function determination using alanine/EPR dosimetry

Source positioning close to the tumour in high dose rate (HDR) brachytherapy is not instantaneous. An increment of dose will be delivered during the movement of the source in the trajectory to its static position. This increment is the transit dose, often not taken into account in brachytherapeutic treatment planning. The transit dose depends on the prescribed dose, number of treatment fractions, velocity and activity of the source. Combining all these factors, the transit dose can be 5% higher than the prescribed absorbed dose value (Sang-Hyun and Muller-Runkel, 1994 Phys. Med. Biol. 39 1181-8, Nath et al 1995 Med. Phys. 22 209-34). However, it cannot exceed this percentage (Nath et al 1995). In this work, we use the alanine-EPR (electron paramagnetic resonance) dosimetric system using analysis of the first derivative of the signal. The transit dose was evaluated for an HDR system and is consistent with that already presented for TLD dosimeters (Bastin et al 1993 Int. J. Radiat. Oncol. Biol. Phys. 26 695-702). Also using the same dosimetric system, the radial dose function, used to evaluate the geometric dose degradation around the source, was determined and its behaviour agrees better with those obtained by Monte Carlo simulations (Nath et al 1995, Williamson and Nath 1991 Med. Phys. 18 434-48, Ballester et al 1997 Med. Phys. 24 1221-8, Ballester et al 2001 Phys. Med. Biol. 46 N79-90) than with TLD measurements (Nath et al 1990 Med. Phys. 17 1032-40).     

Computed tomography with a linear accelerator with radiotherapy applications

An earlier paper [Simpson et al., Med. Phys. 9, 574 (1982)] described a computed tomography (CT) scanner that was constructed by adding a detector array to a 4-MV isocentric linear accelerator. Since the previous article, the detector array has been improved and we now demonstrate better than 3-mm spatial resolution and better than 1% relative electron density discrimination. A series of pictures from volunteer patients is included. Normal anatomy is visualized with bone, muscle, fat, and air being clearly delineated.     

Optimization of a retrospective technique for respiratory-gated high speed micro-CT of free-breathing rodents

The objective of this study was to develop a technique for dynamic respiratory imaging using retrospectively gated high-speed micro-CT imaging of free-breathing mice. Free-breathing C57Bl6 mice were scanned using a dynamic micro-CT scanner, comprising a flat-panel detector mounted on a slip-ring gantry. Projection images were acquired over ten complete gantry rotations in 50 s, while monitoring the respiratory motion in synchrony with projection-image acquisition. Projection images belonging to a selected respiratory phase were retrospectively identified and used for 3D reconstruction. The effect of using fewer gantry rotations--which influences both image quality and the ability to quantify respiratory function--was evaluated. Images reconstructed using unique projections from six or more gantry rotations produced acceptable images for quantitative analysis of lung volume, CT density, functional residual capacity and tidal volume. The functional residual capacity (0.15 +/- 0.03 mL) and tidal volumes (0.08 +/- 0.03 mL) measured in this study agree with previously reported measurements made using prospectively gated micro-CT and at higher resolution (150 microm versus 90 microm voxel spacing). Retrospectively gated micro-CT imaging of free-breathing mice enables quantitative dynamic measurement of morphological and functional parameters in the mouse models of respiratory disease, with scan times as short as 30 s, based on the acquisition of projection images over six gantry rotations.     

CT calibration for two-dimensional scaling of proton pencil beams

For proton dose calculations in heterogeneous media, it was shown in a previous work that the conventional pencil beam approach based on pathlength scaling does not properly account for scattering effects in nonwater media (Szymanowski and Oelfke 2002 Phys. Med. Biol. 47 3313-30). A two-dimensional scaling method was therefore introduced, which is able to predict with high accuracy the propagation of proton pencil beams both along the depth and the lateral directions in inhomogeneous media. In order to integrate this improved pencil beam algorithm in a CT based treatment planning system, two CT calibration curves are needed. The first one relates the Hounsfield numbers to the relative stopping powers, as for the conventional pencil beam approach. The second curve is to relate the Hounsfield numbers to the material-specific lateral scaling factors. The purpose of this work is to provide the CT calibration curves needed for the integration of the pencil beam algorithm featuring the two-dimensional scaling method. Similarly to as suggested by Schneider et al (1996 Phys. Med. Biol. 41 111-24) for the calibration curve in terms of stopping powers, we follow a stoichiometric procedure to get the calibration curve in terms of material-specific lateral scaling factors. The calibration curves for a CT scanner of the type Siemens Somatom Plus 4 are obtained from the analytical calculation of the CT Hounsfield numbers, relative stopping powers and material-specific lateral scaling factors for human biological tissues.     

An EPID response calculation algorithm using spatial beam characteristics of primary, head scattered and MLC transmitted radiation

We have developed an independent algorithm for the prediction of electronic portal imaging device (EPID) response. The algorithm uses a set of images [open beam, closed multileaf collimator (MLC), various fence and modified sweeping gap patterns] to separately characterize the primary and head-scatter contributions to EPID response. It also characterizes the relevant dosimetric properties of the MLC: Transmission, dosimetric gap, MLC scatter [P. Zygmansky et al., J. Appl. Clin. Med. Phys. 8(4) (2007)], inter-leaf leakage, and tongue and groove [F. Lorenz et al., Phys. Med. Biol. 52, 5985-5999 (2007)]. The primary radiation is modeled with a single Gaussian distribution defined at the target position, while the head-scatter radiation is modeled with a triple Gaussian distribution defined downstream of the target. The distances between the target and the head-scatter source, jaws, and MLC are model parameters. The scatter associated with the EPID is implicit in the model. Open beam images are predicted to within 1% of the maximum value across the image. Other MLC test patterns and intensity-modulated radiation therapy fluences are predicted to within 1.5% of the maximum value. The presented method was applied to the Varian aS500 EPID but is designed to work with any planar detector with sufficient spatial resolution.     

A systematic review of the precision and accuracy of dose measurements in photon radiotherapy using polymer and Fricke MRI gel dosimetry

The purpose of this work is to undertake a critical appraisal of the evidence in the published literature concerning the basic parameters of accuracy and precision associated with the use of Fricke and polymer gels (in conjunction with MR imaging) as radiation dosimeters in photon radiotherapy, condensing and analysing the body of published information (to the end of April 2002). A systematic review was undertaken addressing specific issues of precision and accuracy asking defined questions of the published literature. Accuracy and precision in relation to gel dosimetry were defined. Information was obtained from published, peer-reviewed journals. A defined search strategy utilizing MeSH headings and keywords, with extensive use of cross-referencing, identified 115 references dealing with gel dosimetry. Exclusion criteria were used to select only data from publications which would give unequivocal evidence. For accuracy, results had to be compared with an ionization chamber as gold standard and all gel samples had to be manufactured in the same batch. For precision, in addition to gels being from the same batch, samples must all have been irradiated at the same time and scanned simultaneously (or within a short time frame). Many results were found demonstrating 'dose mapping' examples using gels. However, there were very few publications containing firm evidence of precision and accuracy. There was no evidence which fulfilled our criteria about accuracy or precision using Fricke gels. For polymer gels only one paper was found for accuracy (4% (Low et al 1999 Med. Phys. 26 1542-51)) and precision (1.7% (Baldock et al 1998 Phys. Med. Biol. 43695-702)); however, both were carried out at only one dose level. If the exclusion criteria were relaxed to include accuracy results comparing gel to a non gold standard dosimeter (e.g. TLD), results give a median accuracy of 10% (range 8-23.5%) for polymer gel (Cosgrove et al 2000 Phys. Med. Biol. 45 1195-210, De Deene et al 1998 Radiother: Oncol. 48 283-91, Farajollahi et al 2000 Br. J. Radiol. 72 1085-92, McJury et al 1999b Phys. Med. Biol. 44 2431-44, Murphy et al 2000b Phys. Med. Biol. 45 835-45, Oldham et al 2001 Med. Phys. 28 1436-45) and 5% for Fricke gel (Chan and Ayyangar 1995b Med. Phys. 22 1171-5). Evidence also points to accuracy worsening at lower dose levels for both gels. The precision data should be viewed with caution as repeated MR measurements were not performed with the same samples. The only precision data for Fricke gels was 1.5% (Johansson Back et al 1998 Phys. Med. Biol. 43 261-76), but for zero dose. In conclusion, despite the amount of published data, sparse research has been undertaken which provides clear evidence of the accuracy and precision for both gels. That which has been published has used higher doses than would be routine in radiotherapy. The basic radiation dosimeter qualities of accuracy and precision have yet to be fully quantified for polymer and Fricke gels at clinically relevant dose levels.     

Experimental assessment of resolution improvement of a zoom-in PET

We have proposed a zoom-in positron emission tomography (PET) system that incorporates a high-resolution detector into an existing PET scanner to obtain high-resolution images of a region of interest. Previously we have shown by computer simulations that the high-resolution detector can improve the overall system performance in terms of spatial resolution and lesion detectability. In this study, we assessed the resolution improvement in a real system by incorporating a high-resolution detector into our existing microPET II scanner. The high-resolution detector consists of a 14 × 28 array of 0.5 × 0.5 × 10 mm3 lutetium oxyorthosilicate scintillator elements and is placed near the center of the microPET II scanner. It is coupled to two 64-channel photomultiplier tubes (PMTs) via tapered optical fiber bundles. The PMT signals were read out by the electronics in the microPET II scanner. A 15 μCi Na-22 point source was positioned at various locations above the high-resolution detector. Images were reconstructed using the data measured by the microPET II scanner alone and the microPET II data combined with the high-resolution detector data. Profiles taken through the reconstructed point sources show substantial reduction in full-width-at-half-maximum along the direction parallel to the face of the high-resolution detector.     

A note on computing the derivative at a constant direction

The derivative at constant direction is frequently used in inversion of cone-beam data. Several algorithms for computing the derivative have been proposed in the literature. The best algorithm to date has been proposed recently by Noo et al (2007 Phys. Med. Biol. 52 5393-414). In this note we propose a new, simple and efficient formula for computing the derivative. Numerical experiments with helical CT show that our formula and the one of Noo et al provide fairly similar spatial resolution and noise stability, even though the new formula is more efficient and easier to implement.     

Removal and effects of scatter-glare in cone-beam CT with an amorphous-silicon flat-panel detector

Scatter in a detector and its housing can result in image degradation. Typically, such scatter leads to a low-spatial frequency 'glare' superimposed on the primary signal. We infer the glare-spread function (GSF) of an amorphous-silicon flat-panel detector via an edge-spread technique. We demonstrate that this spread (referred to as 'scatter-glare' herein) causes a low-spatial frequency drop in the associated modulation-transfer function. This results in a compression of the range of reconstructed CT (computed tomography) numbers and is an impediment to accurate CT-number calibration. We show that it can also lead to visual artefacts. This explains previously unresolved CT-number discrepancies in an earlier work (Poludniowski et al 2009 Phys. Med. Biol. 54 3847). We demonstrate that after deconvolving the GSF from the projection images, in conjunction with a correction for phantom-scatter, the CT-number discrepancies disappear. We show results for an in-house-built phantom with inserts of tissue-equivalent materials and for a patient scan. We conclude that where scatter-glare has not been accounted for, the calibration of cone-beam CT numbers to material density will be compromised. The scatter-glare measurement method we propose is simple and requires no special equipment. The deconvolution process is also straightforward and relatively quick (60 ms per projection on a desktop PC).     

A synchrotron radiation microtomography system for the analysis of trabecular bone samples.

X-ray computed microtomography is particularly well suited for studying trabecular bone architecture, which requires three-dimensional (3-D) images with high spatial resolution. For this purpose, we describe a three-dimensional computed microtomography (microCT) system using synchrotron radiation, developed at ESRF. Since synchrotron radiation provides a monochromatic and high photon flux x-ray beam, it allows high resolution and a high signal-to-noise ratio imaging. The principle of the system is based on truly three-dimensional parallel tomographic acquisition. It uses a two-dimensional (2-D) CCD-based detector to record 2-D radiographs of the transmitted beam through the sample under different angles of view. The 3-D tomographic reconstruction, performed by an exact 3-D filtered backprojection algorithm, yields 3-D images with cubic voxels. The spatial resolution of the detector was experimentally measured. For the application to bone investigation, the voxel size was set to 6.65 microm, and the experimental spatial resolution was found to be 11 microm. The reconstructed linear attenuation coefficient was calibrated from hydroxyapatite phantoms. Image processing tools are being developed to extract structural parameters quantifying trabecular bone architecture from the 3-D microCT images. First results on human trabecular bone samples are presented.     

A geometric calibration method for cone beam CT systems

Cone beam CT systems are being deployed in large numbers for small animal imaging, dental imaging, and other specialty applications. A new high-precision method for cone beam CT system calibration is presented in this paper. It uses multiple projection images acquired from rotating point-like objects (metal ball bearings) and the angle information generated from the rotating gantry system is also used. It is assumed that the whole system has a mechanically stable rotation center and that the detector does not have severe out-of-plane rotation (<2 degrees). Simple geometrical relationships between the orbital paths of individual BBs and five system parameters were derived. Computer simulations were employed to validate the accuracy of this method in the presence of noise. Equal or higher accuracy was achieved compared with previous methods. This method was implemented for the geometrical calibration of both a micro CT scanner and a breast CT scanner. The reconstructed tomographic images demonstrated that the proposed method is robust and easy to implement with high precision.     

Accuracy of device-specific 2D and 3D image distortion correction algorithms for magnetic resonance imaging of the head provided by a manufacturer

For the application of magnetic resonance imaging (MRI) in precision radiotherapy, image distortions must be reduced to a minimum to maintain geometrical accuracy. Recently, two-dimensional (2D) and three-dimensional (3D) algorithms for MRI-device-specific distortion corrections were developed by the manufacturers of MRI devices. A previously developed phantom (Karger C P et al 2003 Phys. Med. Biol. 48 211-21) was used to quantify and assess the size of geometrical image distortions before and after application of the 2D and 3D correction algorithm in the head region. Four different types of MRI devices with different gradient systems were measured. For comparison, measurements were also performed with two computed tomography (CT) devices. Mean distortions of up to 4.6+/-1.4 mm (maximum: 5.8 mm) were found prior to the correction. After the correction, the mean distortions were well below 2.0 mm in most cases. Distortions in the CT images were below or equal to 1.0 mm on average. Generally, the 3D algorithm produced comparable or better results than the 2D algorithm. The remaining distortions after the correction appear to be acceptable for fractionated radiotherapy.     

A megavoltage CT scanner for radiotherapy verification.

We have further developed a system for generating megavoltage CT images immediately prior to the administration of external beam radiotherapy. The detector is based on the scanner of Simpson (Simpson et al 1982)--the major differences being a significant reduction in dose required for image formation, faster image formation and greater convenience of use in the clinical setting. Attention has been paid to the problem of ring artefacts in the images. Specifically, a Fourier-space filter has been applied to the sinogram data. After suitable detector calibration, it has been shown that the device operates close to its theoretical specification of 3 mm spatial resolution and a few percent contrast resolution. Ring artefacts continue to be a major source of image degradation. A number of clinical images have been presented. The next stage of this work is to use the system to make clinical measurements of patient set-up inaccuracies building on our work making such measurements from digital portal images (Evans et al 1992).     

Advanced single-slice rebinning in cone-beam spiral CT

To achieve higher volume coverage at improved z-resolution in computed tomography (CT), systems with a large number of detector rows are demanded. However, handling an increased number of detector rows, as compared to today's four-slice scanners, requires to accounting for the cone geometry of the beams. Many so-called cone-beam reconstruction algorithms have been proposed during the last decade. None met all the requirements of the medical spiral cone-beam CT in regard to the need for high image quality, low patient dose and low reconstruction times. We therefore propose an approximate cone-beam algorithm which uses virtual reconstruction planes tilted to optimally fit 180 degrees spiral segments, i.e., the advanced single-slice rebinning (ASSR) algorithm. Our algorithm is a modification of the single-slice rebinning algorithm proposed by Noo et al. [Phys. Med. Biol. 44, 561-570 (1999)] since we use tilted reconstruction slices instead of transaxial slices to approximate the spiral path. Theoretical considerations as well as the reconstruction of simulated phantom data in comparison to the gold standard 180 degrees LI (single-slice spiral CT) were carried out. Image artifacts, z-resolution as well as noise levels were evaluated for all simulated scanners. Even for a high number of detector rows the artifact level in the reconstructed images remains comparable to that of 180 degrees LI. Multiplanar reformations of the Defrise phantom show none of the typical cone-beam artifacts usually appearing when going to larger cone angles. Image noise as well as the shape of the respective slice sensitivity profiles are equivalent to the single-slice spiral reconstruction, z-resolution is slightly decreased. The ASSR has the potential to become a practical tool for medical spiral cone-beam CT. Its computational complexity lies in the order of standard single-slice CT and it allows to use available 2D backprojection hardware.     

Computer modeling of the spatial resolution properties of a dedicated breast CT system

Computer simulation methods were used to evaluate the spatial resolution properties of a dedicated cone-beam breast CT system. X-ray projection data of a 70 microm nickel-chromium wire were simulated. The modulation transfer function (MTF) was calculated from the reconstructed axial images at different radial positions from the isocenter to study the spatial dependency of the spatial resolution of the breast CT scanner. The MTF was also calculated in both the radial and azimuthal directions. Subcomponents of the cone beam CT system that affect the MTF were modeled in the computer simulation in a serial manner, including the x-ray focal spot distribution, gantry rotation under the condition of continuous fluoroscopy, detector lag, and detector spatial resolution. Comparison between the computer simulated and physically measured MTF values demonstrates reasonable accuracy in the simulation process, with a small systematic difference (approximately 9.5 +/- 6.4% difference, due to unavoidable uncertainties from physical measurement and system calibration). The intrinsic resolution in the radial direction determined by simulation was about 2.0 mm(-1) uniformly through the field of view. The intrinsic resolution in the azimuthal direction degrades from 2.0 mm(-1) at the isocenter to 1.0 mm(-1) at the periphery with 76.9 mm from the isocenter. The results elucidate the intrinsic spatial resolution properties of the prototype breast CT system, and suggest ways in which spatial resolution can be improved with system modification.     

Fluoroscopic gating without implanted fiducial markers for lung cancer radiotherapy based on support vector machines

Various problems with the current state-of-the-art techniques for gated radiotherapy have prevented this new treatment modality from being widely implemented in clinical routine. These problems are caused mainly by applying various external respiratory surrogates. There might be large uncertainties in deriving the tumor position from external respiratory surrogates. While tracking implanted fiducial markers has sufficient accuracy, this procedure may not be widely accepted due to the risk of pneumothorax. Previously, we have developed a technique to generate gating signals from fluoroscopic images without implanted fiducial markers using template matching methods (Berbeco et al 2005 Phys. Med. Biol. 50 4481-90, Cui et al 2007b Phys. Med. Biol. 52 741-55). In this note, our main contribution is to provide a totally different new view of the gating problem by recasting it as a classification problem. Then, we solve this classification problem by a well-studied powerful classification method called a support vector machine (SVM). Note that the goal of an automated gating tool is to decide when to turn the beam ON or OFF. We treat ON and OFF as the two classes in our classification problem. We create our labeled training data during the patient setup session by utilizing the reference gating signal, manually determined by a radiation oncologist. We then pre-process these labeled training images and build our SVM prediction model. During treatment delivery, fluoroscopic images are continuously acquired, pre-processed and sent as an input to the SVM. Finally, our SVM model will output the predicted labels as gating signals. We test the proposed technique on five sequences of fluoroscopic images from five lung cancer patients against the reference gating signal as ground truth. We compare the performance of the SVM to our previous template matching method (Cui et al 2007b Phys. Med. Biol. 52 741-55). We find that the SVM is slightly more accurate on average (1-3%) than the template matching method, when delivering the target dose. And the average duty cycle is 4-6% longer. Given the very limited patient dataset, we cannot conclude that the SVM is more accurate and efficient than the template matching method. However, our preliminary results show that the SVM is a potentially precise and efficient algorithm for generating gating signals for radiotherapy. This work demonstrates that the gating problem can be considered as a classification problem and solved accordingly.     

A comparison of step-and-shoot leaf sequencing algorithms that eliminate tongue-and-groove effects

The performances of three recently published leaf sequencing algorithms for step-and-shoot intensity-modulated radiation therapy delivery that eliminates tongue-and-groove underdosage are evaluated. Proofs are given to show that the algorithm of Que et al (2004 Phys. Med. Biol. 49 399-405) generates leaf sequences free of tongue-and-groove underdosage and interdigitation. However, the total beam-on times could be up to n times those of the sequences generated by the algorithms of Kamath et al (2004 Phys. Med. Biol. 49 N7-N19), which are optimal in beam-on time for unidirectional leaf movement under the same constraints, where n is the total number of involved leaf pairs. Using 19 clinical fluence matrices and 100000 randomly generated 15 x 15 matrices, the average monitor units and number of segments of the leaf sequences generated using the algorithm of Que et al are about two to four times those generated by the algorithm of Kamath et al.     

An image-based skeletal dosimetry model for the ICRP reference adult male--internal electron sources

In this study, a comprehensive electron dosimetry model of the adult male skeletal tissues is presented. The model is constructed using the University of Florida adult male hybrid phantom of Lee et al (2010 Phys. Med. Biol. 55 339-63) and the EGSnrc-based Paired Image Radiation Transport code of Shah et al (2005 J. Nucl. Med. 46 344-53). Target tissues include the active bone marrow, associated with radiogenic leukemia, and total shallow marrow, associated with radiogenic bone cancer. Monoenergetic electron emissions are considered over the energy range 1 keV to 10 MeV for the following sources: bone marrow (active and inactive), trabecular bone (surfaces and volumes), and cortical bone (surfaces and volumes). Specific absorbed fractions are computed according to the MIRD schema, and are given as skeletal-averaged values in the paper with site-specific values reported in both tabular and graphical format in an electronic annex available from http://stacks.iop.org/0031-9155/56/2309/mmedia. The distribution of cortical bone and spongiosa at the macroscopic dimensions of the phantom, as well as the distribution of trabecular bone and marrow tissues at the microscopic dimensions of the phantom, is imposed through detailed analyses of whole-body ex vivo CT images (1 mm resolution) and spongiosa-specific ex vivo microCT images (30 μm resolution), respectively, taken from a 40 year male cadaver. The method utilized in this work includes: (1) explicit accounting for changes in marrow self-dose with variations in marrow cellularity, (2) explicit accounting for electron escape from spongiosa, (3) explicit consideration of spongiosa cross-fire from cortical bone, and (4) explicit consideration of the ICRP's change in the surrogate tissue region defining the location of the osteoprogenitor cells (from a 10 μm endosteal layer covering the trabecular and cortical surfaces to a 50 μm shallow marrow layer covering trabecular and medullary cavity surfaces). Skeletal-averaged values of absorbed fraction in the present model are noted to be very compatible with those weighted by the skeletal tissue distributions found in the ICRP Publication 110 adult male and female voxel phantoms, but are in many cases incompatible with values used in current and widely implemented internal dosimetry software.     

A BPF-type algorithm for CT with a curved PI detector

Helical cone-beam CT is used widely nowadays because of its rapid scan speed and efficient utilization of x-ray dose. Recently, an exact reconstruction algorithm for helical cone-beam CT was proposed (Zou and Pan 2004a Phys. Med. Biol. 49 941-59). The algorithm is referred to as a backprojection-filtering (BPF) algorithm. This BPF algorithm for a helical cone-beam CT with a flat-panel detector (FPD-HCBCT) requires minimum data within the Tam-Danielsson window and can naturally address the problem of ROI reconstruction from data truncated in both longitudinal and transversal directions. In practical CT systems, detectors are expensive and always take a very important position in the total cost. Hence, we work on an exact reconstruction algorithm for a CT system with a detector of the smallest size, i.e., a curved PI detector fitting the Tam-Danielsson window. The reconstruction algorithm is derived following the framework of the BPF algorithm. Numerical simulations are done to validate our algorithm in this study.