On the conditioning of basis materials and its impact on multimaterial decomposition-based spectral imaging in photon-counting CT

Image-domain material decomposition for dual-energy CT using unsupervised learning with data-fidelity loss

BACKGROUND

Dual-energy computed tomography (DECT) and material decomposition play vital roles in quantitative medical imaging. However, the decomposition process may suffer from significant noise amplification, leading to severely degraded image signal-to-noise ratios (SNRs). While existing iterative algorithms perform noise suppression using different image priors, these heuristic image priors cannot accurately represent the features of the target image manifold. Although deep learning-based decomposition methods have been reported, these methods are in the supervised-learning framework requiring paired data for training, which is not readily available in clinical settings.

PURPOSE

This work aims to develop an unsupervised-learning framework with data-measurement consistency for image-domain material decomposition in DECT.

METHODS

The proposed framework combines iterative decomposition and deep learning-based image prior in a generative adversarial network (GAN) architecture. In the generator module, a data-fidelity loss is introduced to enforce the measurement consistency in material decomposition. In the discriminator module, the discriminator is trained to differentiate the low-noise material-specific images from the high-noise images. In this scheme, paired images of DECT and ground-truth material-specific images are not required for the model training. Once trained, the generator can perform image-domain material decomposition with noise suppression in a single step.

RESULTS

In the simulation studies of head and lung digital phantoms, the proposed method reduced the standard deviation (SD) in decomposed images by 97% and 91% from the values in direct inversion results. It also generated decomposed images with structural similarity index measures (SSIMs) greater than 0.95 against the ground truth. In the clinical head and lung patient studies, the proposed method suppressed the SD by 95% and 93% compared to the decomposed images of matrix inversion.

CONCLUSIONS

Since the invention of DECT, noise amplification during material decomposition has been one of the biggest challenges, impeding its quantitative use in clinical practice. The proposed method performs accurate material decomposition with efficient noise suppression. Furthermore, the proposed method is within an unsupervised-learning framework, which does not require paired data for model training and resolves the issue of lack of ground-truth data in clinical scenarios.

Accurate Reconstruction of Multiple Basis Images Directly From Dual Energy CT Data

OBJECTIVE

We develop optimization-based algorithms to accurately reconstruct multiple ( 2) basis images directly from dual-energy (DE) data in CT.

METHODS

In medical and industrial CT imaging, some basis materials such as bone, metals, and contrast agents of interest are confined often spatially within regions in the image. Exploiting this observation, we develop an optimization-based algorithm to reconstruct, directly from DE data, basis-region images from which multiple ( 2) basis images and virtual monochromatic images (VMIs) can be obtained over the entire image array.

RESULTS

We conduct experimental studies using simulated and real DE data in CT, and evaluate basis images and VMIs obtained in terms of visual inspection and quantitative metrics. The study results reveal that the algorithm developed can accurately and robustly reconstruct multiple ( 2) basis images directly from DE data.

CONCLUSIONS

The developed algorithm can yield accurate multiple ( 2) basis images, VMIs, and physical quantities of interest from DE data in CT.

SIGNIFICANCE

The work may provide insights into the development of practical procedures for reconstructing multiple basis images, VMIs, and physical quantities from DE data in applications. The work can be extended to reconstruct multiple basis images in multi-spectral or photon-counting CT.

An Image-Based Prior Knowledge-Free Approach for a Multi-Material Decomposition in Photon-Counting Computed Tomography

Photon-counting CT systems generally allow for acquiring multiple spectral datasets and thus for decomposing CT images into multiple materials. We introduce a prior knowledge-free deterministic material decomposition approach for quantifying three material concentrations on a commercial photon-counting CT system based on a single CT scan. We acquired two phantom measurement series: one to calibrate and one to test the algorithm. For evaluation, we used an anthropomorphic abdominal phantom with inserts of either aqueous iodine solution, aqueous tungsten solution, or water. Material CT numbers were predicted based on a polynomial in the following parameters: Water-equivalent object diameter, object center-to-isocenter distance, voxel-to-isocenter distance, voxel-to-object center distance, and X-ray tube current. The material decomposition was performed as a generalized least-squares estimation. The algorithm provided material maps of iodine, tungsten, and water with average estimation errors of 4% in the contrast agent maps and 1% in the water map with respect to the material concentrations in the inserts. The contrast-to-noise ratio in the iodine and tungsten map was 36% and 16% compared to the noise-minimal threshold image. We were able to decompose four spectral images into iodine, tungsten, and water.

Systematic Review on Learning-based Spectral CT

Spectral computed tomography (CT) has recently emerged as an advanced version of medical CT and significantly improves conventional (single-energy) CT. Spectral CT has two main forms: dual-energy computed tomography (DECT) and photon-counting computed tomography (PCCT), which offer image improvement, material decomposition, and feature quantification relative to conventional CT. However, the inherent challenges of spectral CT, evidenced by data and image artifacts, remain a bottleneck for clinical applications. To address these problems, machine learning techniques have been widely applied to spectral CT. In this review, we present the state-of-the-art data-driven techniques for spectral CT.

Prototyping optimization-based image reconstructions from limited-angular-range data in dual-energy CT

Image reconstruction from data collected over full-angular range (FAR) in dual-energy CT (DECT) is well-studied. There exists interest in DECT with advanced scan configurations in which data are collected only over limited-angular ranges (LARs) for meeting unique workflow needs in certain practical imaging applications, and thus in the algorithm development for image reconstruction from such LAR data. The objective of the work is to investigate and prototype image reconstructions in DECT with LAR scans. We investigate and prototype optimization programs with various designs of constraints on the directional-total-variations (DTVs) of virtual monochromatic images and/or basis images, and derive the DTV algorithms to numerically solve the optimization programs for achieving accurate image reconstruction from data collected in a slew of different LAR scans. Using simulated and real data acquired with low- and high-kV spectra over LARs, we conduct quantitative studies to demonstrate and evaluate the optimization programs and their DTV algorithms developed. As the results of the numerical studies reveal, while the DTV algorithms yield images of visual quality and quantitative accuracy comparable to that of the existing algorithms from FAR data, the former reconstruct images with improved visualization, reduced artifacts, and also enhanced quantitative accuracy when applied to LAR data in DECT. Optimization-based, one-step algorithms, including the DTV algorithms demonstrated, can be developed for quantitative image reconstruction from spectral data collected over LARs of extents that are considerably smaller than the FAR in DECT. The theoretical and numerical results obtained can be exploited for prototyping designs of optimization-based reconstructions and LAR scans in DECT, and they may also yield insights into the development of reconstruction procedures in practical DECT applications. The approach and algorithms developed can naturally be applied to investigating image reconstruction from LAR data in multi-spectral and photon-counting CT.

Constrained one-step material decomposition reconstruction of head CT data from a silicon photon-counting prototype

BACKGROUND

Spectral CT material decomposition provides quantitative information but is challenged by the instability of the inversion into basis materials. We have previously proposed the constrained One-Step Spectral CT Image Reconstruction (cOSSCIR) algorithm to stabilize the material decomposition inversion by directly estimating basis material images from spectral CT data. cOSSCIR was previously investigated on phantom data.

PURPOSE

This study investigates the performance of cOSSCIR using head CT datasets acquired on a clinical photon-counting CT (PCCT) prototype. This is the first investigation of cOSSCIR for large-scale, anatomically complex, clinical PCCT data. The cOSSCIR decomposition is preceded by a spectrum estimation and nonlinear counts correction calibration step to address nonideal detector effects.

METHODS

Head CT data were acquired on an early prototype clinical PCCT system using an edge-on silicon detector with eight energy bins. Calibration data of a step wedge phantom were also acquired and used to train a spectral model to account for the source spectrum and detector spectral response, and also to train a nonlinear counts correction model to account for pulse pileup effects. The cOSSCIR algorithm optimized the bone and adipose basis images directly from the photon counts data, while placing a grouped total variation (TV) constraint on the basis images. For comparison, basis images were also reconstructed by a two-step projection-domain approach of Maximum Likelihood Estimation (MLE) for decomposing basis sinograms, followed by filtered backprojection (MLE + FBP) or a TV minimization algorithm (MLE + TVmin ) to reconstruct basis images. We hypothesize that the cOSSCIR approach will provide a more stable inversion into basis images compared to two-step approaches. To investigate this hypothesis, the noise standard deviation in bone and soft-tissue regions of interest (ROIs) in the reconstructed images were compared between cOSSCIR and the two-step methods for a range of regularization constraint settings.

RESULTS

cOSSCIR reduced the noise standard deviation in the basis images by a factor of two to six compared to that of MLE + TVmin , when both algorithms were constrained to produce images with the same TV. The cOSSCIR images demonstrated qualitatively improved spatial resolution and depiction of fine anatomical detail. The MLE + TVmin algorithm resulted in lower noise standard deviation than cOSSCIR for the virtual monoenergetic images (VMIs) at higher energy levels and constraint settings, while the cOSSCIR VMIs resulted in lower noise standard deviation at lower energy levels and overall higher qualitative spatial resolution. There were no statistically significant differences in the mean values within the bone region of images reconstructed by the studied algorithms. There were statistically significant differences in the mean values within the soft-tissue region of the reconstructed images, with cOSSCIR producing mean values closer to the expected values.

CONCLUSIONS

The cOSSCIR algorithm, combined with our previously proposed spectral model estimation and nonlinear counts correction method, successfully estimated bone and adipose basis images from high resolution, large-scale patient data from a clinical PCCT prototype. The cOSSCIR basis images were able to depict fine anatomical details with a factor of two to six reduction in noise standard deviation compared to that of the MLE + TVmin two-step approach.

Mineral quantitative characterization method based on basis material decomposition model by dual-energy computed tomography

BACKGROUND

Dual-energy computed tomography (DECT) can reconstruct electron density ρe and effective atomic number Zeff distribution for material discrimination. Image-domain basis material decomposition (IBMD) method is a widely used DECT method. However, IBMD method cannot be used for mineral identification directly due to limitations of complex basis material determination, beam hardening artifacts, and inherent errors caused by approximate empirical formulas.

OBJECTIVE

This study proposes an improved IBMD (IIBMD) method to overcome the above limitations.

METHODS

In IIBMD method, the composition of basis material is optimized to obtain accurate decomposition coefficients, which enables accurate ρe and Zeff distribution. Moreover, the thickness of basis material is optimized to reduce the effect of beam hardening. Furthermore, two formulas in place of empirical formulas are proposed to calculate ρe and Zeff. Finally, a threshold technique is applied to separate different mineral phases.

RESULTS

Numerical simulations and practical experiments using a photon-counting detector CT system are implemented to verify IIBMD method. Results show that the relative errors of ρe and Zeff for seven common minerals are down to 5%, lower than most of the existing DECT methods for rocks. Reasonable volume fraction results of mineral phases are thus obtained through threshold segmentation.

CONCLUSIONS

This study demonstrates that the proposed IIBMD method has high practical value in mineralogical identification.

Noise correlation and its impact on the performance of multi-material decomposition-based spectral imaging in photon-counting CT

PURPOSE

It has been known that noise correlation plays an important role in the determination of the performance of spectral imaging based on two-material decomposition (2-MD). To further understand the basics of spectral imaging in photon-counting CT toward optimal design and implementation, we study the noise correlation in multi-MD (m-MD) and its impact on the performance of spectral imaging.

METHOD

We derive the equations that characterize the noise and noise correlation in the material-specific (basis) images in m-MD, followed by a simulation study to verify the derived equations and study the noise correlation's impact on the performance of spectral imaging. Using a specially designed digital phantom, the study of noise correlation runs over the cases of two-, three-, and four-MD (2-MD, 3-MD, and 4-MD). Then, the noise correlation's impact on the performance of spectral imaging in photon-counting CT is investigated, using a modified Shepp-Logan phantom.

RESULTS

The results in 2-MD show that, in-line with what has been reported in the literature, the noise correlation coefficient between the material-specific images corresponding to the basis materials approaches -1. The results in m-MD (m ≥ 3) are more complicated and interesting, as the noise correlation coefficients between a pair of the material-specific images alternate between ±1, and so do in the case of 4-MD. The m-MD data show that the noise in virtual monochromatic imaging (a form of spectral imaging) is moderate even though the noises in material-specific (basis) images vary drastically.

CONCLUSIONS

The observation of noise correlation in 3-MD, 4-MD, and beyond (i.e., m-MD) is informative to the community. The relationship between noise correlation and the performance of spectral imaging revealed in this work may help clinical medical physicists understand the fundamentals of spectral imaging based on MD and optimize the performance of spectral imaging in photon-counting CT and other X-ray imaging modalities.

Photon-counting CT via interleaved/gapped spectral channels: Feasibility and imaging performance

PURPOSE

Compared to energy-integration, photon-counting x-ray detection facilitates the spectral channelization (energy binning) in spectral CT and thus offers the opportunity of implementing data acquisition via sophisticated schemes, e.g., gapping or interleaving in spectral channels. In this article, we report our investigation of the performance of material decomposition based spectral imaging in photon-counting CT implemented in such data acquisition schemes, and their comparison with the benchmark scheme and other schemes without spectral gapping or interleaving.

MATERIALS AND METHODS

Using a deliberately designed anthropomorphic head phantom that mimics the intracranial soft tissues and bony structures, a simulation study is carried out with the focus on two-material decomposition-based spectral imaging in photon-counting CT, under both ideal and realistic detector spectral responses. The projection data are acquired in four spectral channels, and then are sorted to implement the schemes of gapping ((ch₁ , ch₃ ); (ch₂ , ch₄ ); (ch₁ , ch₄ )) and interleaving ((ch₁ , ch₃ )+(ch₂ , ch₄ ); (ch₁ , ch₄ )+(ch₂ , ch₃ ); ((ch₁ +ch₃ ), (ch₂ +ch₄ )); ((ch₁ +ch₄ ), (ch₂ +ch₃ ))) in spectral channels, in addition to the benchmark scheme ((ch₁ +ch₂ ), (ch₃ +ch₄ )) and other conventional schemes (ch₁ , ch₂ ), (ch₂ , ch₃ ) and (ch₃ , ch₄ ), where 'ch' denotes channel, '+' denote addition, and (·,·) the operation of material decomposition and image reconstruction. Using the contrast-to-noise ratio between targeted regions of interest as the figure of merit, we study the performance of spectral imaging (material specific and virtual monochromatic) associated with these spectral channelization schemes.

RESULTS

Under ideal detector spectral response, the scheme (ch₁ , ch₄ ) outperforms the benchmark scheme ((ch₁ +ch₂ ), (ch₃ +ch₄ )) and others in gapped and/or interleaved spectral channelization in material specific imaging, while the interleaved scheme (ch₁ , ch₄ )+(ch₂ , ch₃ ) performs the best in virtual monochromatic imaging. Notably, only about half x-ray dose is utilized in the scheme (ch₁ , ch₄ ) for image formation. Under realistic detector spectral response, the difference in imaging performance over all spectral channelization schemes diminishes, along with degradation in each scheme's individual performance. The results suggest that (i) different strategy in spectral channelization may have to be exercised in material specific imaging and virtual monochromatic imaging, respectively, and (ii) the spectral distortion in realistic detector's response due to charge-sharing, Compton scattering and fluorescent escaping should be mitigated as much as possible.

CONCLUSION

The spectral channelization schemes and associated imaging performance reported herein are novel and thus informative to the community, which may further the understanding of physical fundamentals and design principles for material decomposition based spectral imaging in photon-counting CT and other x-ray related imaging modalities. This article is protected by copyright. All rights reserved.

Invertibility of multi-energy X-ray transform

PURPOSE

The goal is to provide a sufficient condition for the invertibility of a multi-energy (ME) X-ray transform. The energy-dependent X-ray attenuation profiles can be represented by a set of coefficients using the Alvarez-Macovski (AM) method. An ME X-ray transform is a mapping from N AM coefficients to N noise-free energy-weighted measurements, where N ≥ 2 .

METHODS

We apply a general invertibility theorem to prove the equivalence of global and local invertibility for an ME X-ray transform. We explore the global invertibility through testing whether the Jacobian of the mapping J ( A ) has zero values over the support of the mapping. The Jacobian of an arbitrary ME X-ray transform is an integration over all spectral measurements. A sufficient condition for J ( A ) ≠ 0 for all A is that the integrand of J ( A ) is ≥ 0 (or ≤ 0 ) everywhere. Note that the trivial case of the integrand equals 0 everywhere is ignored. Using symmetry, we simplified the integrand of the Jacobian to three factors that are determined by the total attenuation, the basis functions, and the energy-weighting functions, respectively. The factor related to the total attenuation is always positive; hence, the invertibility of the X-ray transform can be determined by testing the signs of the other two factors. Furthermore, we use the Cramér-Rao lower bound (CRLB) to characterize the noise-induced estimation uncertainty and provide a maximum-likelihood (ML) estimator.

RESULTS

The factor related to the basis functions is always negative when the photoelectric/Compton/Rayleigh basis functions are used and K-edge materials are not considered. The sign of the energy-weighting factor depends on the system source spectra and the detector response functions. For four special types of X-ray detectors, the sign of this factor stays the same over the integration range. Therefore, when these four types of detectors are used for imaging non-K-edge materials, the ME X-ray transform is globally invertible. The same framework can be used to study an arbitrary ME X-ray imaging system, for example, when K-edge materials are present. Furthermore, the ML estimator we presented is an unbiased, efficient estimator and can be used for a wide range of scenes.

CONCLUSIONS

We have provided a framework to study the invertibility of an arbitrary ME X-ray transform and proved the global invertibility for four types of systems.