Correction to: Detecting focal cortical dysplasia lesions from FLAIR-negative images based on cortical thickness.

It was highlighted that the original article [1] contained an error in the Quantitative evaluation of Methods. A bracket was misplaced in the formula. This Correction article shows the incorrect and correct formula.

Detecting focal cortical dysplasia lesions from FLAIR-negative images based on cortical thickness.

BACKGROUND: Focal cortical dysplasia (FCD) is a neuronal migration disorder and is a major cause of drug-resistant epilepsy. However, many focal abnormalities remain undetected during routine visual inspection, and many patients with histologically confirmed FCD have normal fluid-attenuated inversion recovery (FLAIR-negative) images. The aim of this study was to quantitatively evaluate the changes in cortical thickness with magnetic resonance (MR) imaging of patients to identify FCD lesions from FLAIR-negative images. METHODS: We first used the three-dimensional (3D) Laplace method to calculate the cortical thickness for individuals and obtained the cortical thickness mean image and cortical thickness standard deviation (SD) image based on all 32 healthy controls. Then, a cortical thickness extension map was computed by subtracting the cortical thickness mean image from the cortical thickness image of each patient and dividing the result by the cortical thickness SD image. Finally, clusters of voxels larger than three were defined as the FCD lesion area from the cortical thickness extension map. RESULTS: The results showed that three of the four lesions that occurred in non-temporal areas were detected in three patients, but the detection failed in three patients with lesions that occurred in the temporal area. The quantitative analysis of the detected lesions in voxel-wise on images revealed the following: specificity (99.78%), accuracy (99.76%), recall (67.45%), precision (20.42%), Dice coefficient (30.01%), Youden index (67.23%) and area under the curve (AUC) (83.62%). CONCLUSION: Our studies demonstrate an effective method to localize lesions in non-temporal lobe regions. This novel method automatically detected FCD lesions using only FLAIR-negative images from patients and was based only on cortical thickness feature. The method is noninvasive and more effective than a visual analysis for helping doctors make a diagnosis.

Automatic localization and segmentation of focal cortical dysplasia in FLAIR-negative patients using a convolutional neural network.

PURPOSE: Focal cortical dysplasia (FCD) is a common cause of epilepsy; the only treatment is surgery. Therefore, detecting FCD using noninvasive imaging technology can help doctors determine whether surgical intervention is required. Since FCD lesions are small and not obvious, diagnosing FCD through visual evaluations of magnetic resonance imaging (MRI) scans is difficult. The purpose of this study is to detect and segment histologically confirmed FCD lesions in images of normal fluid-attenuated inversion recovery (FLAIR)-negative lesions using convolutional neural network (CNN) technology. METHODS: The technique involves training a six-layer CNN named Net-Pos, which consists of two convolutional layers (CLs); two pooling layers (PLs); and two fully connected (FC) layers, including 60 943 learning parameters. We employed activation maximization (AM) to optimize a series of pattern image blocks (PIBs) that were most similar to a lesion image block by using the trained Net-Pos. We developed an AM and convolutional localization (AMCL) algorithm that employs the mean PIBs combined with convolution to locate and segment FCD lesions in FLAIR-negative patients. Five evaluation indices, namely, recall, specificity, accuracy, precision, and the Dice coefficient, were applied to evaluate the localization and segmentation performance of the algorithm. RESULTS: The PIBs most similar to an FCD lesion image block were identified by the trained Net-Pos as image blocks with brighter central areas and darker surrounding image blocks. The technique was evaluated using 18 FLAIR-negative lesion images from 12 patients. The subject-wise recall of the AMCL algorithm was 83.33% (15/18). The Dice coefficient for the segmentation performance was 52.68. CONCLUSION: We developed a novel algorithm referred to as the AMCL algorithm with mean PIBs to effectively and automatically detect and segment FLAIR-negative FCD lesions. This work is the first study to apply a CNN-based model to detect and segment FCD lesions in images of FLAIR-negative lesions.

A Novel Approach for Position Verification and Dose Calculation through Local MVCT Reconstruction.

Traditional positioning verification using cone-beam computed tomography (CBCT) may incur errors due to potential misalignments between the isocenter of CBCT and the treatment beams in radiotherapy. This study introduces an innovative method for verifying patient positioning in radiotherapy. Initially, the transmission images from an electronic portal imaging device (EPID) are acquired from 10 distinct angles. Utilizing the ART-TV algorithm, a sparse reconstruction of local megavoltage computed tomography (MVCT) is performed. Subsequently, this MVCT is aligned with the planning CT via a three-dimensional mutual information registration technique, pinpointing any patient-positioning discrepancies and facilitating corrective adjustments to the treatment setup. Notably, this approach employs the same radiation source as used in treatment to obtain three-dimensional images, thereby circumventing errors stemming from misalignment between the isocenter of CBCT and the accelerator. The registration process requires only 10 EPID images, and the dose absorbed during this process is included in the total dose calculation. The results show that our method's reconstructed MVCT images fulfill the requirements for registration, and the registration algorithm accurately detects positioning errors, thus allowing for adjustments in the patient's treatment position and precise calculation of the absorbed dose.

The value of convolutional neural networks-based deep learning model in differential diagnosis of space-occupying brain diseases.

Objectives: It is still a challenge to differentiate space-occupying brain lesions such as tumefactive demyelinating lesions (TDLs), tumefactive primary angiitis of the central nervous system (TPACNS), primary central nervous system lymphoma (PCNSL), and brain gliomas. Convolutional neural networks (CNNs) have been used to analyze complex medical data and have proven transformative for image-based applications. It can quickly acquire diseases' radiographic features and correct doctors' diagnostic bias to improve diagnostic efficiency and accuracy. The study aimed to assess the value of CNN-based deep learning model in the differential diagnosis of space-occupying brain diseases on MRI. Methods: We retrospectively analyzed clinical and MRI data from 480 patients with TDLs (n = 116), TPACNS (n = 64), PCNSL (n = 150), and brain gliomas (n = 150). The patients were randomly assigned to training (n = 240), testing (n = 73), calibration (n = 96), and validation (n = 71) groups. And a CNN-implemented deep learning model guided by clinical experts was developed to identify the contrast-enhanced T1-weighted sequence lesions of these four diseases. We utilized accuracy, sensitivity, specificity, and area under the curve (AUC) to evaluate the performance of the CNN model. The model's performance was then compared to the neuroradiologists' diagnosis. Results: The CNN model had a total accuracy of 87% which was higher than senior neuroradiologists (74%), and the AUC of TDLs, PCNSL, TPACNS and gliomas were 0.92, 0.92, 0.89 and 0.88, respectively. Conclusion: The CNN model can accurately identify specific radiographic features of TDLs, TPACNS, PCNSL, and gliomas. It has the potential to be an effective auxiliary diagnostic tool in the clinic, assisting inexperienced clinicians in reducing diagnostic bias and improving diagnostic efficiency.

Super-resolution reconstruction for parallel-beam SPECT based on deep learning and transfer learning: a preliminary simulation study.

Background: Single-photon emission computed tomography (SPECT) is widely used in the early diagnosis of major diseases such as cardiovascular disease and cancer. High-resolution (HR) imaging requires HR projection data, which typically comes with high costs. This study aimed to obtain HR SPECT images based on a deep learning algorithm using low-resolution (LR) detectors. Methods: A super-resolution (SR) reconstruction network based on deep learning and transfer learning for parallel-beam SPECT was proposed. LR SPECT sinograms were converted into HR sinograms. Training data were designed and generated, including digital phantoms (128×128 pixels), HR sinograms (128×128 pixels), and LR sinograms (128×64 pixels). A series of random phantoms was first used for pre-training, and then the extended cardiac-torso (XCAT) phantom was used to fine-tune the parameters. The effectiveness of the method was evaluated using an unknown cardiac phantom. To simulate a wide range of noise levels, the total count levels of the projection were normalized to 1e7 (100%), 1e6 (10%), and 1e5 (5%). Finally, the training sets for different count levels were generated. Transfer learning was employed to accelerate the training. Results: The proposed network was validated on the simulation data sets using different Poisson noise levels. The quantitative values of the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) indicators of the reconstructed images were improved compared to those recorded using the benchmark methods. Using the proposed method, an image resolution comparable to that of images reconstructed from the HR projection could be achieved. Conclusions: Based on deep learning and transfer learning, an SR reconstruction network in the projection domain of the parallel-beam SPECT was developed. The simulation results under a wide range of noise levels evidenced the potential of the proposed network to improve SPECT resolution for LR detector scanners.

Super-resolution acquisition and reconstruction for cone-beam SPECT with low-resolution detector.

BACKGROUND AND OBJECTIVE: Single-photon emission computed tomography (SPECT) imaging, which provides information that reflects the human body's metabolic processes, has unique application value in disease diagnosis and efficacy evaluation. The imaging resolution of SPECT can be improved by exploiting high-performance detector hardware, but this exploitation generates high research and development costs. In addition, the inherent hardware structure of SPECT requires the use of a collimator, which limits the resolution in SPECT. The objective of this study is to propose a novel super-resolution (SR) reconstruction algorithm with two acquisition methods for cone-beam SPECT with low-resolution (LR) detector. METHODS: A SR algorithm with two acquisition methods is proposed for cone-beam SPECT imaging in the projection domain. At each sampling angle, multi LR projections can be obtained by regularly moving the LR detector. For the two proposed acquisition methods, we develop a new SR reconstruction algorithm. Using our SR algorithm, a SR projection with the corresponding sampling angle can be obtained from multi LR projections via multi-iterations, and then, the SR SPECT image can be reconstructed. The peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), signal-to-noise ratio (SNR) and contrast recovery coefficient (CRC) are used to evaluate the final reconstruction quality. RESULTS: The simulation results obtained under clean and noisy conditions verify the effectiveness of our SR algorithm. Three different phantoms are verified separately. 16 LR projections are obtained at each sampling angle, each with 32 × 32 bins. The high-resolution (HR) projection has 128 × 128 bins. The reconstruction result of the SR algorithm obtains an evaluation value that is almost the same as that of the HR reconstruction result. Our results indicate that the resolution of the resulting SPECT image is almost four times higher. CONCLUSIONS: The authors develop a SR reconstruction algorithm with two acquisition methods for the cone-beam SPECT system. The simulation results obtained in clean and noisy environments prove that the SR algorithm has potential value, but it needs to be further tested on real equipment.

A feasibility study for in vivo treatment verification of IMRT using Monte Carlo dose calculation and deep learning-based modelling of EPID detector response.

BACKGROUND: This paper describes the development of a predicted electronic portal imaging device (EPID) transmission image (TI) using Monte Carlo (MC) and deep learning (DL). The measured and predicted TI were compared for two-dimensional in vivo radiotherapy treatment verification. METHODS: The plan CT was pre-processed and combined with solid water and then imported into PRIMO. The MC method was used to calculate the dose distribution of the combined CT. The U-net neural network-based deep learning model was trained to predict EPID TI based on the dose distribution of solid water calculated by PRIMO. The predicted TI was compared with the measured TI for two-dimensional in vivo treatment verification. RESULTS: The EPID TI of 1500 IMRT fields were acquired, among which 1200, 150, and 150 fields were used as the training set, the validation set, and the test set, respectively. A comparison of the predicted and measured TI was carried out using global gamma analyses of 3%/3 mm and 2%/2 mm (5% threshold) to validate the model's accuracy. The gamma pass rates were greater than 96.7% and 92.3%, and the mean gamma values were 0.21 and 0.32, respectively. CONCLUSIONS: Our method facilitates the modelling process more easily and increases the calculation accuracy when using the MC algorithm to simulate the EPID response, and has potential to be used for in vivo treatment verification in the clinic.

Applications of artificial intelligence in nuclear medicine image generation.

Recently, the application of artificial intelligence (AI) in medical imaging (including nuclear medicine imaging) has rapidly developed. Most AI applications in nuclear medicine imaging have focused on the diagnosis, treatment monitoring, and correlation analyses with pathology or specific gene mutation. It can also be used for image generation to shorten the time of image acquisition, reduce the dose of injected tracer, and enhance image quality. This work provides an overview of the application of AI in image generation for single-photon emission computed tomography (SPECT) and positron emission tomography (PET) either without or with anatomical information [CT or magnetic resonance imaging (MRI)]. This review focused on four aspects, including imaging physics, image reconstruction, image postprocessing, and internal dosimetry. AI application in generating attenuation map, estimating scatter events, boosting image quality, and predicting internal dose map is summarized and discussed.

Improved detection of focal cortical dysplasia in normal-appearing FLAIR images using a Bayesian classifier.

PURPOSE: Focal cortical dysplasia (FCD) is a malformation of cortical development that often causes pharmacologically intractable epilepsy. However, FCD lesions are frequently characterized by minor structural abnormalities that can easily go unrecognized, making diagnosis difficult. Therefore, many epileptic patients have had pathologically confirmed FCD lesions that appeared normal in pre-surgical fluid-attenuated inversion recovery (FLAIR) magnetic resonance (MR) studies. Such lesions are called "FLAIR-negative." This study aimed to improve the detection of histopathologically verified FCD in a sample of patients without visually appreciable lesions. METHODS: The technique first extracts a series of features from a FLAIR image. Then, three naive Bayesian classifiers with probability (NBCP) are trained based on different numbers of feature maps to classify voxels as lesional or healthy voxels and assign the lesions a probability of correct classification. This method classifies the three-dimensional (3D) images of all patients using leave-one-out cross-validation (LOOCV). Finally, the 3D lesion probability map, including epileptogenic lesions, is obtained by removing false-positive voxel outliers using the morphological method. The performance of the NBCP was assessed for quantitative analysis by specificity, accuracy, recall, precision, and Dice coefficient in subject-wise, lesion-wise, and voxel-wise manners. RESULTS: The best detection results were obtained by using four features: cortical thickness, symmetry, K-means, and modified texture energy. There were eight lesions in seven patients. The subject-wise sensitivity of the proposed method was 85.71% (6/7). Seven out of eight lesions were detected, so the lesion-wise sensitivity was 87.50% (7/8). No significant differences in effectiveness were found between automated lesion detection using four features and lesion detection using manual segmentation, as voxels were quantitatively analyzed in terms of specificity (mean ± SD = 99.64 ± 0.13), accuracy (mean ± SD = 99.62 ± 0.14), recall (mean ± SD = 73.27 ± 26.11), precision (mean ± SD = 11.93 ± 8.16), and Dice coefficient (mean ± SD = 22.82 ± 15.57). CONCLUSION: We developed a novel automatic voxel-based method to improve the detection of FCD FLAIR-negative lesions. To the best of our knowledge, this study is the first to detect FCD lesions that appear normal in pre-surgical 3D high-resolution FLAIR images alone with a limited number of radiomics features. We optimized the algorithm and selected the best prior probability to improve the detection. For non-temporal lobe epilepsy (non-TLE) patients, lesions could be accurately located, although there were still false-positive areas.

Development of an Electronic Portal Imaging Device Dosimetry Method.

Support arm backscatter and off-axis effects of an electronic portal imaging device (EPID) are challenging for radiotherapy quality assurance. Aiming at the issue, we proposed a simple yet effective method with correction matrices to rectify backscatter and off-axis responses for EPID images. First, we measured the square fields with ionization chamber array (ICA) and EPID simultaneously. Second, we calculated the dose-to-pixel value ratio and used it as the correction matrix of the corresponding field. Third, the correction value of the large field was replaced with that of the same point in the small field to generate a correction matrix suitable for different EPID images. Finally, we rectified the EPID image with the correction matrix, and then the processed EPID images were converted into the absolute dose. The calculated dose was compared with the measured dose via ICA. The gamma pass rates of 3%/3 mm and 2%/2 mm (5% threshold) were 99.6% ± 0.94% and 95.48% ± 1.03%, and the average gamma values were 0.28 ± 0.04 and 0.42 ± 0.05, respectively. Experimental results verified our method accurately corrected EPID images and converted pixel values into absolute dose values such that EPID was an efficient radiotherapy dosimetry tool.

A method for in vivo treatment verification of IMRT and VMAT based on electronic portal imaging device.

BACKGROUND: Intensity-modulated radiation therapy (IMRT) and volume-modulated arc therapy (VMAT) are rather complex treatment techniques and require patient-specific quality assurance procedures. Electronic portal imaging devices (EPID) are increasingly used in the verification of radiation therapy (RT). This work aims to develop a novel model to predict the EPID transmission image (TI) with fluence maps from the RT plan. The predicted TI is compared with the measured TI for in vivo treatment verification. METHODS: The fluence map was extracted from the RT plan and corrections of penumbra, response, global field output, attenuation, and scatter were applied before the TI was calculated. The parameters used in the model were calculated separately for central axis and off-axis points using a series of EPID measurement data. Our model was evaluated using a CIRS thorax phantom and 20 clinical plans (10 IMRT and 10 VMAT) optimized for head and neck, breast, and rectum treatments. RESULTS: Comparisons of the predicted and measured images were carried out using a global gamma analysis of 3%/2 mm (10% threshold) to validate the accuracy of the model. The gamma pass rates for IMRT and VMAT were greater than 97.2% and 94.5% at 3%/2 mm, respectively. CONCLUSION: We have developed an accurate and straightforward EPID-based quality assurance model that can potentially be used for in vivo treatment verification of the IMRT and VMAT delivery.

A wavelet-based SPECT reconstruction algorithm for nonuniformly attenuated Radon transform.

PURPOSE: Single photon emission computed tomography (SPECT) is a tomography technique that can greatly show information about the metabolic activity in the body and improve the clinical diagnosis. In SPECT, because of photoelectric absorption and Compton scattering, the emitted gamma photons are attenuated inside the body before arriving at the detector. The goal of quantitative SPECT reconstruction is to obtain an accurate image of the radioactivity distribution in the interested area of a human body, so the compensation for nonuniform attenuation and the treatment of Poisson noise are necessary in the quantitative SPECT reconstruction. METHODS: The authors know that the wavelet transform has characteristics of multiresolution and localized nature, and these characteristics can be applied for denoising and localized reconstruction. Based on the explicit inversion formula for the attenuated Radon transform, the authors present a wavelet-based SPECT reconstruction algorithm with compensation for nonuniform attenuation. RESULTS: The wavelet-based SPECT reconstruction algorithm offers the ability for denoising in the reconstruction processing. In simulation, 128 projections were simulated evenly spaced over 360 degrees by a circular orbit, each with 128 bins. Simulation results show that the wavelet-based denoising is effective in SPECT reconstruction. CONCLUSIONS: The authors present a wavelet-based SPECT reconstruction algorithm with compensation for nonuniform attenuation. The reconstruction results from computer simulations show that the wavelet-based SPECT reconstruction algorithm is accurate.

Multiscale penalized weighted least-squares sinogram restoration for low-dose x-ray computed tomography.

In this paper, we propose a novel multiscale penalized weighted least-squares (PWLS) method for restoration of low-dose computed tomography (CT) sinogram. The method utilizes wavelet transform for the multiscale or multiresolution analysis on the sinogram. Specifically, the Mallat-Zhong's wavelet transform is applied to decompose the sinogram to different resolution levels. At each decomposed resolution level, a PWLS criterion is applied to restore the noise-contaminated wavelet coefficients, where the penalty is adaptive to each resolution scale and the weight is updated by an exponential relationship between the data variance and mean at each scale and location. The proposed PWLS method is based on the observations that 1) noise in the CT sinogram after logarithm transform and calibration can be modeled as signal-dependent variables and the sample variance depends on the sample mean by an exponential relationship; and 2) noise reduction can be more effective when it is adaptive to different resolution levels. The effectiveness of the proposed multiscale PWLS method is validated by both computer simulations and experimental studies. The gain by multiscale approach over single scale means is quantified by noise-resolution tradeoff measures.

An inversion formula for the exponential radon transform in spatial domain with variable focal-length fan-beam collimation geometry.

Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.

An efficient reconstruction method for nonuniform attenuation compensation in nonparallel beam geometries based on Novikov's explicit inversion formula.

This paper investigates an accurate reconstruction method to invert the attenuated Radon transform in nonparallel beam (NPB) geometries. The reconstruction method contains three major steps: 1) performing one-dimensional phase-shift rebinning; 2) implementing nonuniform Hilbert transform; and 3) applying Novikov's explicit inversion formula. The method seems to be adaptive to different settings of fan-beam geometry from very long to very short focal lengths without sacrificing reconstruction accuracy. Compared to the conventional bilinear rebinning technique, the presented method showed a better spatial resolution, as measured by modulation transfer function. Numerical experiments demonstrated its computational efficiency and stability to different levels of Poisson noise. Even with complicated geometries such as varying focal-length and asymmetrical fan-beam collimation, the presented method achieved nearly the same reconstruction quality of parallel-beam geometry. This effort can facilitate quantitative reconstruction of single photon emission computed tomography for cardiac imaging, which may need NPB collimation geometries and require high computational efficiency.

Evaluation of an efficient compensation method for quantitative fan-beam brain SPECT reconstruction.

Fan-beam collimators are designed to improve the system sensitivity and resolution for imaging small objects such as the human brain and breasts in single photon emission computed tomography (SPECT). Many reconstruction algorithms have been studied and applied to this geometry to deal with every kind of degradation factor. This paper presents a new reconstruction approach for SPECT with circular orbit, which demonstrated good performance in terms of both accuracy and efficiency. The new approach compensates for degradation factors including noise, scatter, attenuation, and spatially variant detector response. Its uniform attenuation approximation strategy avoids the additional transmission scan for the brain attenuation map, hence reducing the patient radiation dose and furthermore simplifying the imaging procedure. We evaluate and compare this new approach with the well-established ordered-subset expectation-maximization iterative method, using Monte Carlo simulations. We perform quantitative analysis with regional bias-variance, receiver operating characteristics, and Hotelling trace studies for both methods. The results demonstrate that our reconstruction strategy has comparable performance with a significant reduction of computing time.

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.

Super resolution SPECT reconstruction with non-uniform attenuation.

In this paper, based on Novikov's explicit inversion formula for the attenuated Radon transform, we present a super resolution SPECT reconstruction algorithm with compensation for non-uniform attenuation. Unlike the former methods improving the medical image resolution via super resolution (SR) in the reconstructed image, the proposed method apply the SR algorithm in the low resolution (LR) sinogram, which needs only 1-D shift of the detector, and the PSF is easy to obtain. Simulation results show that our reconstruction algorithm is effective.

A SPECT reconstruction method for extending parallel to non-parallel geometries.

Due to its simplicity, parallel-beam geometry is usually assumed for the development of image reconstruction algorithms. The established reconstruction methodologies are then extended to fan-beam, cone-beam and other non-parallel geometries for practical application. This situation occurs for quantitative SPECT (single photon emission computed tomography) imaging in inverting the attenuated Radon transform. Novikov reported an explicit parallel-beam formula for the inversion of the attenuated Radon transform in 2000. Thereafter, a formula for fan-beam geometry was reported by Bukhgeim and Kazantsev (2002 Preprint N. 99 Sobolev Institute of Mathematics). At the same time, we presented a formula for varying focal-length fan-beam geometry. Sometimes, the reconstruction formula is so implicit that we cannot obtain the explicit reconstruction formula in the non-parallel geometries. In this work, we propose a unified reconstruction framework for extending parallel-beam geometry to any non-parallel geometry using ray-driven techniques. Studies by computer simulations demonstrated the accuracy of the presented unified reconstruction framework for extending parallel-beam to non-parallel geometries in inverting the attenuated Radon transform.