Xiangyang Tang

Characterization of imaging performance in differential phase contrast CT compared with the conventional CT—Noise power spectrum NPS(k)

PURPOSEnThe differential phase contrast CT is emerging as a new technology to improve the contrast sensitivity of the conventional CT. Via system analysis, modeling, and computer simulation, the authors study the noise power spectrum (NPS)-an imaging performance indicator-of the differential phase contrast CT and compare it with that of the conventional CT.nnnMETHODSnThe differential phase contrast CT is implemented with x-ray tube and gratings. The x-ray propagation and data acquisition are modeled and simulated with Fourier analysis and Fresnel analysis. To avoid any interference caused by scatter and beam hardening, a monochromatic x-ray source (30 keV) is assumed, which irradiates the object to be imaged by 360° so that no weighting scheme is needed. A 20-fold up-sampling is assumed to simulate x-ray beams propagation through the gratingsG1 and G2 with periods 8 and 4 μm, respectively, while the intergrating distance is 193.6 mm (1/16 of the Tabolt distance). The dimension of the detector cell for data acquisition ranges from 32u2009×u200932 to 128u2009×u2009128 μm2 , while the field of view in data acquisition is 40.96u2009×u200940.96 mm2 . A uniform water phantom with a diameter 37.68 mm is employed to study the NPS, with its complex refraction coefficient nu2009=u20091u2009-u2009δu2009+u2009iβu2009=u20091u2009-u20092.5604u2009×u200910-7 u2009+u2009i1.2353u2009×u200910-10 . The x-ray flux ranges from 106 to 108 photon/cm2 ·projection and observes the Poisson distribution, which is consistent with that of micro-CT in preclinical applications. The image matrix of reconstructed water phantom is 1280u2009×u20091280, and a total of 180 regions at 128u2009×u2009128 matrix are used for NPS calculation via 2D Fourier Transform in which adequate zero padding is applied to avoid aliasing.nnnRESULTSnThe preliminary data show that the differential phase contrast CT manifests its NPS with a 1/|k| trait, while the distribution of the conventional CTs NPS observes |k|. This accounts for the significant difference in their noise granularity and the differential phase contrast CTs substantial advantage in noise over the conventional CT, particularly, in the situations where the detector cell size for data acquisition is smaller than 100 μm.nnnCONCLUSIONSnThe differential phase contrast CT detects the projection of refractive coefficients derivative and uses the Hilbert filter for image reconstruction, which leads to the radical difference in its NPS and the advantage in noise in comparison to that of the conventional CT.

Characterization of imaging performance in differential phase contrast CT compared with the conventional CT: Spectrum of noise equivalent quanta NEQ(k)

PURPOSEnDifferential phase contrast CT (DPC-CT) is emerging as a new technology to improve the contrast sensitivity of conventional attenuation-based CT. The noise equivalent quanta as a function over spatial frequency, i.e., the spectrum of noise equivalent quanta NEQ(k), is a decisive indicator of the signal and noise transfer properties of an imaging system. In this work, we derive the functional form of NEQ(k) in DPC-CT. Via system modeling, analysis, and computer simulation, we evaluate and verify the derived NEQ(k) and compare it with that of the conventional attenuation-based CT.nnnMETHODSnThe DPC-CT is implemented with x-ray tube and gratings. The x-ray propagation and data acquisition are modeled and simulated through Fresnel and Fourier analysis. A monochromatic x-ray source (30 keV) is assumed to exclude any system imperfection and interference caused by scatter and beam hardening, while a 360° full scan is carried out in data acquisition to avoid any weighting scheme that may disrupt noise randomness. Adequate upsampling is implemented to simulate the x-ray beams propagation through the gratings G(1) and G(2) with periods 8 and 4 μm, respectively, while the intergrating distance is 193.6 mm (1∕16 of the Talbot distance). The dimensions of the detector cell for data acquisition are 32 × 32, 64 × 64, 96 × 96, and 128 × 128 μm(2), respectively, corresponding to a 40.96 × 40.96 mm(2) field of view in data acquisition. An air phantom is employed to obtain the noise power spectrum NPS(k), spectrum of noise equivalent quanta NEQ(k), and detective quantum efficiency DQE(k). A cylindrical water phantom at 5.1 mm diameter and complex refraction coefficient n = 1 - δ + iβ = 1 -2.5604 × 10(-7) + i1.2353 × 10(-10) is placed in air to measure the edge transfer function, line spread function and then modulation transfer function MTF(k), of both DPC-CT and the conventional attenuation-based CT. The x-ray flux is set at 5 × 10(6) photon∕cm(2) per projection and observes the Poisson distribution, which is consistent with that of a micro-CT for preclinical applications. Approximately 360 regions, each at 128 × 128 matrix, are used to calculate the NPS(k) via 2D Fourier transform, in which adequate zero padding is carried out to avoid aliasing in noise.nnnRESULTSnThe preliminary data show that the DPC-CT possesses a signal transfer property [MTF(k)] comparable to that of the conventional attenuation-based CT. Meanwhile, though there exists a radical difference in their noise power spectrum NPS(k) (trait 1∕|k| in DPC-CT but |k| in the conventional attenuation-based CT) the NEQ(k) and DQE(k) of DPC-CT and the conventional attenuation-based CT are in principle identical.nnnCONCLUSIONSnUnder the framework of ideal observer study, the joint signal and noise transfer property NEQ(k) and detective quantum efficiency DQE(k) of DPC-CT are essentially the same as those of the conventional attenuation-based CT. The findings reported in this paper may provide insightful guidelines on the research, development, and performance optimization of DPC-CT for extensive preclinical and clinical applications in the future.PURPOSEnDifferential phase contrast CT (DPC-CT) is emerging as a new technology to improve the contrast sensitivity of conventional attenuation-based CT. The noise equivalent quanta as a function over spatial frequency, i.e., the spectrum of noise equivalent quanta NEQ(k), is a decisive indicator of the signal and noise transfer properties of an imaging system. In this work, we derive the functional form of NEQ(k) in DPC-CT. Via system modeling, analysis, and computer simulation, we evaluate and verify the derived NEQ(k) and compare it with that of the conventional attenuation-based CT.nnnMETHODSnThe DPC-CT is implemented with x-ray tube and gratings. The x-ray propagation and data acquisition are modeled and simulated through Fresnel and Fourier analysis. A monochromatic x-ray source (30 keV) is assumed to exclude any system imperfection and interference caused by scatter and beam hardening, while a 360° full scan is carried out in data acquisition to avoid any weighting scheme that may disrupt noise randomness. Adequate upsampling is implemented to simulate the x-ray beams propagation through the gratingsG1 and G2 with periods 8 and 4 μm, respectively, while the intergrating distance is 193.6 mm (1/16 of the Talbot distance). The dimensions of the detector cell for data acquisition are 32 × 32, 64 × 64, 96 × 96, and 128 × 128 μm2 , respectively, corresponding to a 40.96 × 40.96 mm2 field of view in data acquisition. An air phantom is employed to obtain the noise power spectrum NPS(k), spectrum of noise equivalent quanta NEQ(k), and detective quantum efficiency DQE(k). A cylindrical water phantom at 5.1 mm diameter and complex refraction coefficient n = 1 - δ + iβ = 1 -2.5604 × 10-7 + i1.2353 × 10-10 is placed in air to measure the edge transfer function, line spread function and then modulation transfer function MTF(k), of both DPC-CT and the conventional attenuation-based CT. The x-ray flux is set at 5 × 106 photon/cm2 per projection and observes the Poisson distribution, which is consistent with that of a micro-CT for preclinical applications. Approximately 360 regions, each at 128 × 128 matrix, are used to calculate the NPS(k) via 2D Fourier transform, in which adequate zero padding is carried out to avoid aliasing in noise.nnnRESULTSnThe preliminary data show that the DPC-CT possesses a signal transfer property [MTF(k)] comparable to that of the conventional attenuation-based CT. Meanwhile, though there exists a radical difference in their noise power spectrum NPS(k) (trait 1/|k| in DPC-CT but |k| in the conventional attenuation-based CT) the NEQ(k) and DQE(k) of DPC-CT and the conventional attenuation-based CT are in principle identical.nnnCONCLUSIONSnUnder the framework of ideal observer study, the joint signal and noise transfer property NEQ(k) and detective quantum efficiency DQE(k) of DPC-CT are essentially the same as those of the conventional attenuation-based CT. The findings reported in this paper may provide insightful guidelines on the research, development, and performance optimization of DPC-CT for extensive preclinical and clinical applications in the future.

Statistical CT noise reduction with multiscale decomposition and penalized weighted least squares in the projection domain

PURPOSESnThe suppression of noise in x-ray computed tomography (CT) imaging is of clinical relevance for diagnostic image quality and the potential for radiation dose saving. Toward this purpose, statistical noise reduction methods in either the image or projection domain have been proposed, which employ a multiscale decomposition to enhance the performance of noise suppression while maintaining image sharpness. Recognizing the advantages of noise suppression in the projection domain, the authors propose a projection domain multiscale penalized weighted least squares (PWLS) method, in which the angular sampling rate is explicitly taken into consideration to account for the possible variation of interview sampling rate in advanced clinical or preclinical applications.nnnMETHODSnThe projection domain multiscale PWLS method is derived by converting an isotropic diffusion partial differential equation in the image domain into the projection domain, wherein a multiscale decomposition is carried out. With adoption of the Markov random field or soft thresholding objective function, the projection domain multiscale PWLS method deals with noise at each scale. To compensate for the degradation in image sharpness caused by the projection domain multiscale PWLS method, an edge enhancement is carried out following the noise reduction. The performance of the proposed method is experimentally evaluated and verified using the projection data simulated by computer and acquired by a CT scanner.nnnRESULTSnThe preliminary results show that the proposed projection domain multiscale PWLS method outperforms the projection domain single-scale PWLS method and the image domain multiscale anisotropic diffusion method in noise reduction. In addition, the proposed method can preserve image sharpness very well while the occurrence of salt-and-pepper noise and mosaic artifacts can be avoided.nnnCONCLUSIONSnSince the interview sampling rate is taken into account in the projection domain multiscale decomposition, the proposed method is anticipated to be useful in advanced clinical and preclinical applications where the interview sampling rate varies.

The second‐order differential phase contrast and its retrieval for imaging with x‐ray Talbot interferometry

PURPOSEnThe x-ray differential phase contrast imaging implemented with the Talbot interferometry has recently been reported to be capable of providing tomographic images corresponding to attenuation-contrast, phase-contrast, and dark-field contrast, simultaneously, from a single set of projection data. The authors believe that, along with small-angle x-ray scattering, the second-order phase derivative Φ() (s)(x) plays a role in the generation of dark-field contrast. In this paper, the authors derive the analytic formulae to characterize the contribution made by the second-order phase derivative to the dark-field contrast (namely, second-order differential phase contrast) and validate them via computer simulation study. By proposing a practical retrieval method, the authors investigate the potential of second-order differential phase contrast imaging for extensive applications.nnnMETHODSnThe theoretical derivation starts at assuming that the refractive index decrement of an object can be decomposed into δ = δ(s) + δ(f), where δ(f) corresponds to the objects fine structures and manifests itself in the dark-field contrast via small-angle scattering. Based on the paraxial Fresnel-Kirchhoff theory, the analytic formulae to characterize the contribution made by δ(s), which corresponds to the objects smooth structures, to the dark-field contrast are derived. Through computer simulation with specially designed numerical phantoms, an x-ray differential phase contrast imaging system implemented with the Talbot interferometry is utilized to evaluate and validate the derived formulae. The same imaging system is also utilized to evaluate and verify the capability of the proposed method to retrieve the second-order differential phase contrast for imaging, as well as its robustness over the dimension of detector cell and the number of steps in grating shifting.nnnRESULTSnBoth analytic formulae and computer simulations show that, in addition to small-angle scattering, the contrast generated by the second-order derivative is magnified substantially by the ratio of detector cell dimension over grating period, which plays a significant role in dark-field imaging implemented with the Talbot interferometry.nnnCONCLUSIONSnThe analytic formulae derived in this work to characterize the second-order differential phase contrast in the dark-field imaging implemented with the Talbot interferometry are of significance, which may initiate more activities in the research and development of x-ray differential phase contrast imaging for extensive preclinical and eventually clinical applications.

Practical interior tomography with radial Hilbert filtering and a priori knowledge in a small round area

PURPOSESnInterior tomography problem can be solved using the so-called differentiated backprojection-projection onto convex sets (DBP-POCS) method, which requires a priori knowledge within a small area interior to the region of interest (ROI) to be imaged. In theory, the small area wherein the a priori knowledge is required can be in any shape, but most of the existing implementations carry out the Hilbert filtering either horizontally or vertically, leading to a vertical or horizontal strip that may be across a large area in the object. In this work, we implement a practical DBP-POCS method with radial Hilbert filtering and thus the small area with the a priori knowledge can be roughly round (e.g., a sinus or ventricles among other anatomic cavities in human or animal body). We also conduct an experimental evaluation to verify the performance of this practical implementation.nnnMETHODSnWe specifically re-derive the reconstruction formula in the DBP-POCS fashion with radial Hilbert filtering to assure that only a small round area with the a priori knowledge be needed (namely radial DBP-POCS method henceforth). The performance of the practical DBP-POCS method with radial Hilbert filtering and a priori knowledge in a small round area is evaluated with projection data of the standard and modified Shepp-Logan phantoms simulated by computer, followed by a verification using real projection data acquired by a computed tomography (CT) scanner.nnnRESULTSnThe preliminary performance study shows that, if a priori knowledge in a small round area is available, the radial DBP-POCS method can solve the interior tomography problem in a more practical way at high accuracy.nnnCONCLUSIONSnIn comparison to the implementations of DBP-POCS method demanding the a priori knowledge in horizontal or vertical strip, the radial DBP-POCS method requires the a priori knowledge within a small round area only. Such a relaxed requirement on the availability of a priori knowledge can be readily met in practice, because a variety of small round areas (e.g., air-filled sinuses or fluid-filled ventricles among other anatomic cavities) exist in human or animal body. Therefore, the radial DBP-POCS method with a priori knowledge in a small round area is more feasible in clinical and preclinical practice.

The mathematical equivalence of consistency conditions in the divergent-beam computed tomography

In this paper, we discuss the mathematical equivalence among four consistency conditions in the divergent-beam computed tomography (CT). The first is the consistency condition derived by Levine et al. by degenerating the Johns equation; the second is the integral invariant derived by Wei et al. using the symmetric group theory; the third is the so-called parallel-fan-beam Hilbert projection equality derived by Hamaker et al.; and the fourth is the fan-beam data consistency condition (FDCC) derived by Chen et al. using the complex analysis theory. Historically, most of these consistency conditions were derived by their corresponding authors using complicated mathematical strategies, which are usually not easy to be precisely understood by researchers with only a general engineering mathematical background. In this paper, we symmetrically re-derive all these consistency conditions using a friendly mathematical language. Based on theoretical derivation, it has been found that all these consistency conditions can be viewed as a necessary condition for the specific solution to Johns equation. From the physical point of view, all these consistency conditions have been essentially expressed as a similar constraint on the projection data acquired with arbitrary two x-ray source points. Numerical simulations have been carried out to experimentally evaluate and verify their merits.

A nanocomposite of Au-AgI core/shell dimer as a dual-modality contrast agent for x-ray computed tomography and photoacoustic imaging

PURPOSEnTo develop a core/shell nanodimer of gold (core) and silver iodine (shell) as a dual-modal contrast-enhancing agent for biomarker targeted x-ray computed tomography (CT) and photoacoustic imaging (PAI) applications.nnnMETHODSnThe gold and silver iodine core/shell nanodimer (Au/AgICSD) was prepared by fusing together components of gold, silver, and iodine. The physicochemical properties of Au/AgICSD were then characterized using different optical and imaging techniques (e.g., HR- transmission electron microscope, scanning transmission electron microscope, x-ray photoelectron spectroscopy, energy-dispersive x-ray spectroscopy, Z-potential, and UV-vis). The CT and PAI contrast-enhancing effects were tested and then compared with a clinically used CT contrast agent and Au nanoparticles. To confer biocompatibility and the capability for efficient biomarker targeting, the surface of the Au/AgICSD nanodimer was modified with the amphiphilic diblock polymer and then functionalized with transferrin for targeting transferrin receptor that is overexpressed in various cancer cells. Cytotoxicity of the prepared Au/AgICSD nanodimer was also tested with both normal and cancer cell lines.nnnRESULTSnThe characterizations of prepared Au/AgI core/shell nanostructure confirmed the formation of Au/AgICSD nanodimers. Au/AgICSD nanodimer is stable in physiological conditions for in vivo applications. Au/AgICSD nanodimer exhibited higher contrast enhancement in both CT and PAI for dual-modality imaging. Moreover, transferrin functionalized Au/AgICSD nanodimer showed specific binding to the tumor cells that have a high level of expression of the transferrin receptor.nnnCONCLUSIONSnThe developed Au/AgICSD nanodimer can be used as a potential biomarker targeted dual-modal contrast agent for both or combined CT and PAI molecular imaging.

Radial differential interior tomography and its image reconstruction with differentiated backprojection and projection onto convex sets.

PURPOSEnInterior tomography has been recognized as one of the most effective approaches in computed tomography (CT) to reduce radiation dose rendered to patients. In this work, the authors propose and evaluate an imaging method of radial differential interior tomography.nnnMETHODSnIn interior tomography, an x-ray beam is collimated to only irradiate the region of interest (ROI) with suspected lesions while the surrounding area∕volume of normal tissues∕organs is spared. In the proposed imaging method of radial differential interior tomography, the outcome is a ROI image that has gone through a radial differential filtering. The image reconstruction algorithm for the radial differential interior tomography is kept in the fashion of differentiated backprojection and projection onto convex sets, but the required a priori knowledge in a small round area becomes zero and may be more readily available in practice.nnnRESULTSnUsing the projection data simulated by computer and acquired by CT scanner, the authors evaluate and verify the performance of the proposed radial differential interior tomography method and its associated image reconstruction algorithm. The preliminary results show that the proposed imaging method can generate an image that is the radial differentiation of a conventional tomographic image and is robust over noise that inevitably exist in practice.nnnCONCLUSIONSnIt is believed that the proposed imaging method may find its utility in advanced clinical applications wherein a ROI-based image processing and analysis is required for lesion visualization, characterization, and diagnosis.

Complex dark‐field contrast and its retrieval in x‐ray phase contrast imaging implemented with Talbot interferometry

PURPOSEnUnder the existing theoretical framework of x-ray phase contrast imaging methods implemented with Talbot interferometry, the dark-field contrast refers to the reduction in interference fringe visibility due to small-angle x-ray scattering of the subpixel microstructures of an object to be imaged. This study investigates how an objects subpixel microstructures can also affect the phase of the intensity oscillations.nnnMETHODSnInstead of assuming that the objects subpixel microstructures distribute in space randomly, the authors theoretical derivation starts by assuming that an objects attenuation projection and phase shift vary at a characteristic size that is not smaller than the period of analyzer grating G₂ and a characteristic length dc. Based on the paraxial Fresnel-Kirchhoff theory, the analytic formulae to characterize the zeroth- and first-order Fourier coefficients of the x-ray irradiance recorded at each detector cell are derived. Then the concept of complex dark-field contrast is introduced to quantify the influence of the objects microstructures on both the interference fringe visibility and the phase of intensity oscillations. A method based on the phase-attenuation duality that holds for soft tissues and high x-ray energies is proposed to retrieve the imaginary part of the complex dark-field contrast for imaging. Through computer simulation study with a specially designed numerical phantom, they evaluate and validate the derived analytic formulae and the proposed retrieval method.nnnRESULTSnBoth theoretical analysis and computer simulation study show that the effect of an objects subpixel microstructures on x-ray phase contrast imaging method implemented with Talbot interferometry can be fully characterized by a complex dark-field contrast. The imaginary part of complex dark-field contrast quantifies the influence of the objects subpixel microstructures on the phase of intensity oscillations. Furthermore, at relatively high energies, for soft tissues it can be retrieved for imaging with a method based on the phase-attenuation duality.nnnCONCLUSIONSnThe analytic formulae derived in this work to characterize the complex dark-field contrast in x-ray phase contrast imaging method implemented with Talbot interferometry are of significance, which may initiate more activities in the research and development of x-ray differential phase contrast imaging for extensive biomedical applications.

Grating-based x-ray differential phase contrast imaging with twin peaks in phase-stepping curves-phase retrieval and dewrapping.

PURPOSEnX-ray differential phase contrast CT implemented with Talbot interferometry employs phase-stepping to extract information of x-ray attenuation, phase shift, and small-angle scattering. Since inaccuracy may exist in the absorption grating G2 due to an imperfect fabrication, the effective period of G2 can be as large as twice the nominal period, leading to a phenomenon of twin peaks that differ remarkably in their heights. In this work, the authors investigate how to retrieve and dewrap the phase signal from the phase-stepping curve (PSC) with the feature of twin peaks for x-ray phase contrast imaging.nnnMETHODSnBased on the paraxial Fresnel-Kirchhoff theory, the analytical formulae to characterize the phenomenon of twin peaks in the PSC are derived. Then an approach to dewrap the retrieved phase signal by jointly using the phases of the first- and second-order Fourier components is proposed. Through an experimental investigation using a prototype x-ray phase contrast imaging system implemented with Talbot interferometry, the authors evaluate and verify the derived analytic formulae and the proposed approach for phase retrieval and dewrapping.nnnRESULTSnAccording to theoretical analysis, the twin-peak phenomenon in PSC is a consequence of combined effects, including the inaccuracy in absorption grating G2, mismatch between phase grating and x-ray source spectrum, and finite size of x-ray tubes focal spot. The proposed approach is experimentally evaluated by scanning a phantom consisting of organic materials and a lab mouse. The preliminary data show that compared to scanning G2 over only one single nominal period and correcting the measured phase signal with an intuitive phase dewrapping method that is being used in the field, stepping G2 over twice its nominal period and dewrapping the measured phase signal with the proposed approach can significantly improve the quality of x-ray differential phase contrast imaging in both radiograph and CT.nnnCONCLUSIONSnUsing the phase retrieval and dewrapping methods proposed to deal with the phenomenon of twin peaks in PSCs and phase wrapping, the performance of grating-based x-ray differential phase contrast radiography and CT can be significantly improved.