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

Abstract

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+TV_min) 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+TV_min, 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+TV_min 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+TV_min two-step approach.

I. Introduction

Material decomposition from spectral computed tomography (CT) has demonstrated numerous clinical applications such as improved iodine quantification,¹ distinguishing brain hemorrhage from retained iodine,^2,3,4 generating virtual noncontrast scans,⁵, removal of calcified plaques,⁶ and imaging bone marrow edema.^7,8,9,10 Photon-counting CT detectors have demonstrated additional image quality benefits, such as improved spatial resolution and the potential for improved spectral information.^{11,12,13,14,15}

We have previously proposed a constrained One Step Spectral Image Reconstruction (cOSSCIR) algorithm to directly estimate the material decomposition basis images from the photon counts data.^16,17 The cOSSCIR approach differs from ‘two-step’ methods that either first reconstruct spectral images and then perform basis decomposition (i.e., two-step image domain decomposition) or first estimate basis sinograms and then perform image reconstruction (i.e., two-step projection-domain decomposition). By modeling the physics of polyenergetic x-ray transmission, cOSSCIR can avoid beam hardening artifacts, which is also possible using two-step projection-domain decomposition approaches. Like the two-step image domain decomposition, cOSSCIR can reconstruct from spectral measurements that are spatially unregistered.

The stability of inversion of transmission data into basis material sinograms depends on the spectral content of the x-ray measurements and the basis attenuation energy dependence used in decomposition^18,19. Because the basis attenuation functions are nonorthogonal, the condition number of the inversion is greater than one,¹⁸ thus amplifying the noise. This is true even if photon-counting detectors with many energy bins are used, because the nonorthogonality of the basis attenuation functions is not impacted by the number of energy bins. This inherent instability, despite the number of energy-bin measurements, makes the estimation of basis sinograms or basis images sensitive to noise and errors in the photon-counts data. In two-step projection domain decomposition, each ray is decomposed individually. In one-step direct inversion methods, the spectral information from all rays that pass through an image voxel are used to estimate the basis coefficients for that voxel, which may help stabilize the inversion because the output of the decomposition must be consistent with an image. Constraints can also be placed on the basis images to provide further stability. Several one-step direct inversion approaches have been proposed, which differ in their forward models and optimization approaches.^{20,21,22,23,24,25,26} A unique aspect of the cOSSCIR framework is that it performs the nonlinear and nonconvex optimization of the basis images using the polyenergetic x-ray transmission model, while also allowing non-smooth, convex constraints to be placed on the basis images. In our previous work with simulated data, cOSSCIR resulted in CT images free of metal artifacts caused by beam hardening, photon starvation, and noise.²⁷ In this previous study, a two-step projection-domain approach resulted in large noise streaks in rays through metal due to the instability of the material decomposition step. In comparison, the cOSSCIR algorithm demonstrated reduced noise streaks, even at loose regularization constraint settings, demonstrating a more stable inversion.

Our previous work demonstrated experimental feasibility of cOSSCIR for phantom and tissue specimens using a bench-top photon-counting CT prototype with a small field of view and limited spatial resolution.¹⁷ Since this previous experimental work, the optimization algorithm used within the cOSSCIR framework has been improved based on an alternating direction method of multipliers (ADMM) approach.²⁸ The spectral modeling has also been improved to include a pulse-pileup counts correction.²⁹

This paper investigates the performance of the cOSSCIR algorithm for head CT datasets acquired using a photon-counting CT clinical prototype with a first generation silicon detector. In addition to evaluating cOSSCIR for the first time on a large-scale, high resolution CT dataset of human subjects with complex anatomy, this study is also the first experimental study of cOSSCIR combining the ADMM optimization and the spectral modeling approach that includes a pre-processing pulse pileup correction step. The effects of the cOSSCIR constraint are investigated, with basis and virtual monoenergetic images (VMIs) reconstructed by cOSSCIR compared to images reconstructed by a two-step projection-domain approach that used maximum likelihood for basis decomposition and a filtered backprojection (FBP) or total variation (TV) optimization algorithm for reconstruction.

II. Materials and Methods

II.A. Photon-counting head CT acquisitions

Head CT scans were performed at the Karolinska Institute under IRB approval using a clinical photon-counting CT prototype with an edge-on silicon detector. The prototype system and the silicon detector have been described in previous publications.^30,31,32 The silicon detector had eight energy bins, with thresholds selected, as described in previous work,³¹ to be above the electronic noise floor and to detect an equal number of counts after transmission through 6 cm of polyethylene (PE). The photon-counting detector was mounted on a Lightspeed VCT scanner (GE Healthcare, Chicago, Illinois) with a modified x-ray tube with 0.4 mm focal spot size. The system provided a 0.19 mm in-plane resolution at isocenter. Single-slice CT acquisitions were acquired at 120 kV, 200 mA, one second rotation time, with a medium bowtie and 3334 projections per rotation. The PCCT scanner used in this work is an early prototype with a first-generation silicon detector and does not represent the performance of more recent silicon PCCT efforts but is appropriate to specifically evaluate the cOSSCIR algorithm.

II.B. Photon counts measurement model

The cOSSCIR algorithm, spectrum estimation algorithm, and the basis decomposition step of the two-step projection-domain algorithm all require a mathematical model relating the measured photon counts to the imaged object. The following equation models the continuous, polyenergetic transmission of spectral X-ray measurements, given the spectral response of each energy bin window.

$$I_{w,\ell }=\int S_{w,\ell }\left(E\right)\text{exp }\left[-{\int }_{\ell }\mu \left(E,\overrightarrow{r}\left(t\right)\right)dt\right]dE,$$ (1)

where $I_{w,\ell }$ is the transmitted X-ray photon fluence in energy window $w$ along ray $\ell ;t$ is a parameter indicating location along $\ell ;S_{w,\ell }\left(E\right)$ is the spectral response of energy-bin window $w$ along ray $\ell$, which includes the effects of the tube spectrum and the detector response; and $\mu (E,\overrightarrow{r}(t\left)\right)$ is the spatial and spectral distribution of linear X-ray attenuation.

For material decomposition, the X-ray attenuation map is expanded into basis functions, for example the attenuation functions of two materials,

$$\mu \left(E,\overrightarrow{r}\left(t\right)\right)=\sum _m \left(\frac{{\mu }_m\left(E\right)}{{\rho }_m}\right){\rho }_m{f}_m\left(\overrightarrow{r}\left(t\right)\right),$$ (2)

where ${\rho }_m$ is the density of material $m;{\mu }_m\left(E\right)/{\rho }_m$ is the mass attenuation coefficient of material $m$; and $f_m\left(\overrightarrow{r}\right)$ is the spatial fractional density map for material $m$.

The discretized data model used in this work is obtained by combining Eqs. (1) and (2), normalizing the spectral sensitivity and discretizing the energy and spatial variables,

$${\widehat{c}}_{w,\ell }\left(f\right)=N_{w,\ell }\sum _i{s}_{w,\ell ,i}\text{ exp }\left(-\sum _{m,k}{\mu }_{m,i}{X}_{\ell ,k}{f}_{k,m}\right),$$ (3)

where ${\hat{c}}_{w,\ell }\left(f\right)$ is the mean transmitted photon count; $N_{w,\ell }$ is the mean total number of incident photons along ray $\ell$ in energy window $w;s_{w,\ell ,i}$ is the normalized spectral sensitivity ${\sum }_i s_{w,\ell ,i}=1;i$ indexes the sum over energy; $X_{\ell ,k}$ is X-ray projection matrix element indicating the weighting of pixel $k$ to the line integration along ray $\ell$; and $f_{k,m}$ is the pixelized basis material image with $k$ and $m$ indexing pixel and basis-material, respectively.

II.C. Calibration and Estimation of Spectral Response Functions

An estimate of the spectral sensitivity functions for each energy window and ray, $s_{w,\ell ,i}$ in Eq. 1, is needed by both the cOSSCIR and two-step projection-domain decomposition algorithms to respectively estimate the basis images and basis sinograms from the counts measurements. The spectral response functions represent the spectral distribution of photons that are detected by each energy window for measurements through air. The spectral response functions model the tube spectrum, the detector spectral efficiency, and the response of the detector to flux-independent effects, such as Compton scatter within the silicon detector. Flux-dependent pulse pileup effects are also present in the measurement but are not modeled in the forward model of Eq. 3.

We have recently proposed a calibration method to estimate both the spectral response functions and a nonlinear correction to the photon counts measurements to account for pileup. A brief summary is provided here, while details and a quantitative evaluation of this method can be found in our previous publication.³³

The spectrum estimation problem involves estimating spectral response functions, $S_{w,\ell }\left(E\right)$ in the continuous model or normalized sensitivity function, $s_{w,\ell ,i}$, in the discrete model, from measurements through materials of known composition and thicknesses. Because spectrum estimation is an ill-conditioned problem, the proposed method represents the spectral response with a low-dimensional model. Approximate initial guesses of the tube spectrum and detector response model are also incorporated, as expressed in

$$S_w\left(E,\beta \right)=D_w\left(E\right)R\left(E\right)\text{ exp }\left(-\sum _{j=1}^{N_{\beta }}{\beta }_{w,j}{\left(E/E_{\text{max}}\right)}^j\right),$$ (4)

where $D_w\left(E\right)$ is the detector response model for window $w;R\left(E\right)$ is the X-ray source spectrum model; and the exponential factor with a polynomial function is the low-order representation of the spectrum that enables modifying the initial spectral guesses to better match the calibration transmission measurements. The ${\beta }_{w,j}$ coefficients are the unknown parameters optimized using calibration measurements.

Incorporating the low-order spectrum representation into the discretized counts model for normalized transmission measurements through materials of known thickness, yields the following relationship:

$$T_{w\ell }^{\text{trans}}\left(\beta \right)=\sum _i s_{wi}\left(\beta \right)\text{exp}\left[-\sum _m {\mu }_{mi}{t}_{m\ell }\right]\Delta E,$$ (5)

where $T_{w\ell }$ is the mean fluence transmission fraction (i.e., counts measured through the object normalized by counts measured through air for each ray measurement), which we also refer to as the transmission fraction; $s_{wi}$ is the normalized spectral response discretized over the energy, as expressed in Eq.6; ray index $\ell$ now indicates discrete thickness combinations of the calibration materials indexed by $m;{\mu }_{mi}$ is the energy discretized linear attenuation coefficient ${\mu }_m\left(E\right)$ of the $m^{\mathit{\text{th}}}$ calibration material; $t_{m\ell }$ is the thickness of calibration material $m$ along ray $\ell$; and $\Delta E$ is the energy discretization width.

$$s_w(E,\beta )=\frac{S_w(E,\beta )}{\int S_w(E,\beta )dE}$$ (6)

$$s_{wi}\left(\beta \right)=s_w\left(E_i,\beta \right).$$

To correct for pulse-pileup effects, we assume a low-order polynomial relationship between the transmission fraction incident on the detector and the measured transmission fraction:

$$F_w(T,\alpha )=T+\sum _{k=0}^{N_{\alpha }} {\alpha }_{w,k}{T}^k$$ (7)

$$T_{w\ell }^{\text{meas}}\left(\alpha ,\beta \right)=F_w\left(T_{w\ell }^{\text{trans}}\left(\beta \right),\alpha \right).$$ (8)

To estimate the spectral model parameters, calibration measurements were acquired on the prototype clinical PCCT scanner through 23 combinations of PE (0 cm to 23 cm) and polyvinyl chloride (PVC) (0 cm to 6.5 cm). The flux-independent spectral response correction was modeled as a 10^th degree polynomial, while the nonlinear counts model was modeled with a 4^th degree polynomial. The 14 total $\alpha$ and $\beta$ parameters of the nonlinear correction and spectral response models were optimized by minimizing the transmission Poisson likelihood between the transmission fraction measured through the known calibration materials and transmission fraction estimated by Eqs 5–8 assuming ground truth attenuation values from the NIST XCOM database³⁴, where the optimization was subject to Tikhonov regularization using the identity matrix. The resulting estimated spectra are those optimized to best match the transmission fraction, but may not be accurate representations of the true spectra because multiple spectral shapes can provide similar transmission fractions.

After estimation, the nonlinear counts model of Eq. 7 was inverted by first using $F_w(T,\alpha )$ in Eq. 7 and the estimated $\alpha$ values to calculate a vector of modeled measured transmission fractions for incident transmission fractions ranging between zero and one in increments of 10⁻⁴. Then, for each measured transmission fraction for each ray in each energy-bin sinogram, the corrected incident transmission fraction was interpolated from the vectors of modeled measured transmission and corresponding incident transmission. The corrected transmission fraction sinograms were converted back into counts by multiplying with the air scan measurement. The estimated spectral response functions and the pulse-pileup-corrected counts were used for subsequent reconstruction by both cOSSCIR and the comparison two-step method.

II.C.1. Constrained One-Step Spectral CT Image Reconstruction (cOSSCIR)

The cOSSCIR algorithm directly inverts the forward model of Eq. 3 to optimize the basis material images $f$ that best match the measured counts data $c$ after the nonlinear counts correction. The cOSSCIR optimization problem is defined by the constrained minimization

$$f^{\star }=\underset{f}{\text{arg min}}{D}_{\text{TPL}}\left(c,\widehat{c}\left(f\right)\right)\text{ such that GTV}\left(f\right)\le \gamma ,$$ (9)

where $D_{\text{TPL}}(c,\hat{c}(f\left)\right)$ is the shifted, negative logarithm of the transmission Poisson Likelihood (TPL)

$$D_{\text{TPL}}\left(c,\widehat{c}\left(f\right)\right)=\sum _{w,\ell }\left[{\widehat{c}}_{w,\ell }\left(f\right)-c_{w,\ell }-c_{w,\ell }\text{ log}\frac{{\widehat{c}}_{w,\ell }\left(f\right)}{c_{w,\ell }}\right],$$ (10)

and the generalized total variation (GTV) is defined as

$$\text{GTV}\left(f\right)=\sum _k\sqrt{\sum _m\parallel {\left(D{f}_m\right)}_k{\parallel }_2^2};$$ (11)

$D$ is a numerical gradient operator. Parameter $\gamma$ is the constraint value for the image GTV. The GTV regularizer sums the absolute magnitude of the gradient image over pixel while performing a quadratic summation over basis material. This form of the GTV was designed to simplify the algorithm, as it requires only one parameter selection. The balance between the regularization of two basis materials is implicit in the algorithm. Because both basis images have values that are close to either zero or one, the contribution of the two basis images to the GTV depends largely on the number of non-zero pixels in the gradient images. The cOSSCIR algorithm could be easily modified to place separate TV constraints on the two basis images or to use a form of GTV with a weighting factor between the two basis images.

We have previously proposed an optimization framework of non-convex alternating direction method of multipliers (ADMM) to optimize large-scale, non-convex functions subject to non-differentiable constraints, such as $D_{\text{TPL}}$.²⁸ More details on this optimization framework and its application to cOSSCIR can be found in previous publications.²⁸

The cOSSCIR algorithm was applied to the head PCCT data for reconstructing bone and adipose basis images. The linear attenuation coefficients for the two basis materials were taken from the NIST XCOM database.³⁴ Basis images were reconstructed onto a 1024 × 1024 pixel grid representing a 34-cm field of view (0.33 mm pixel dimensions). The basis images were initialized with zeros. The resulting bone and adipose basis images were linearly combined to form a 60 keV virtual monoenergetic image.

II.D. Two-step image reconstruction

Whereas cOSSCIR inverts the forward model of Eq. 3 to estimate the basis images from the photon counts measurement, two-step projection-domain material decomposition uses a slightly modified model to estimate the basis material sinograms from the photon counts measurements. The forward model for the two-step approach can be derived from Eq. 3 as equations

$${\widehat{c}}_{w,\ell }\left(z\right)=N_{w,\ell }\sum _i{s}_{w,\ell ,i}\text{ exp }\left(-\sum _m{\mu }_{m,i}{z}_{\ell ,m}\right),$$ (12)

$$z_{\ell ,m}\left(f\right)=\sum _k X_{\ell ,k}{f}_{k,m},$$ (13)

where $z_m$ are the basis material sinograms.

The decomposition step obtains basis material sinograms $z$ from the photon count data $c$. Any CT reconstruction algorithm can then be used to reconstruct the basis images $f$ from the basis sinograms $z$.

In this work maximum likelihood expectation (MLE) algorithm was used to invert Eq. 12 to optimize bone and adipose sinograms from the measured photon counts after nonlinear correction.³⁵ The optimization problem for estimating basis sinograms can be expressed as

$$z^{\star }=\underset{z}{\text{arg min}}{D}_{\text{TPL}}\left(c,\widehat{c}\left(z\right)\right).$$ (14)

It is important to note that both cOSSCIR and the comparison two-step decomposition method used the same forward model (i.e., nonlinear counts correction, spectral response functions, basis attenuation coefficient functions).

Filtered backprojection and a total variation (TV) constrained least-squares optimization²⁹ were used for the second step of reconstructing basis images from the basis sinograms. FBP reconstructions used a ramp filtered apodized by a Hanning window with 9 mm⁻¹ frequency cutoff. The TV least-squares optimization problem is expressed as

$$f^{\star }=\underset{f}{\text{arg min}}\parallel z^{\star }-z\left(f\right){\parallel }_2^2\text{ such that TV}\left(f\right)\le \alpha ,$$ (15)

where each basis image, $f_m$, is optimized individually while subject to a TV constraint.

We represent these two variants of the two-step approaches as MLE+FBP and MLE+TV_min. Images were reconstructed onto the same pixel grid as the cOSSCIR reconstructions. A 60 keV virtual monoenergetic image was generated for additional evaluation.

II.E. Quantitative Evaluation

To validate convergence of the cOSSCIR and two-step reconstruction algorithms under ideal (i.e., inverse crime) conditions, the VMI reconstructed by the two-step approach (MLE+FBP) was thresholded into bone and adipose basis images. These ground truth basis images were then input to the forward model of Eq. 3 to generate noiseless photon counts data. The three investigated algorithms, cOSSCIR, MLE+FBP, and MLE+TV_min were then applied to the simulated counts data, assuming the same forward model as the simulator, for reconstructing bone and adipose basis images. The reconstructed bone and adipose basis images were compared to the ground truth phantom images.

For the experimental head CT reconstructions, the GTV of the bone and adipose basis images reconstructed by the MLE+FBP two-step decomposition method was calculated with Eq. 11 and used as a reference to set the range of evaluated cOSSCIR GTV constraint settings. cOSSCIR reconstruction was performed with the GTV constraint set to 5%, 10%, 20%, 35%, 50% and 100% of the FBP images. The TV of each basis image reconstructed by cOSSCIR at each GTV constraint was calculated and used as the constraint setting for the TV_min reconstruction step of the two-step decomposition. Thus, cOSSCIR and the two-step MLE+TV_min basis images were reconstructed to have the same TV values.

A region of interest (ROI) of size 20 × 20 pixels was extracted from a region of bone and a region of soft-tissue. The mean and standard deviation of the basis image and VMI values within these ROIs were calculated and compared between the COSSCIR and two-step algorithms.

III. Results

The three studied algorithms, cOSSCIR, MLE+FBP, MLE+TV_min converged to the correct basis images in the inverse crime simulation study, verifying that the algorithms were properly implemented.

Figure 1 demonstrates the performance of the spectrum estimation and nonlinear correction applied to the calibration data for one representative detector pixel and energy window. The plot compares the measured transmission fraction to that estimated with Eq. 5 assuming only the initial guess spectral models and the estimated spectral models optimized with and without the nonlinear counts correction. The results demonstrate good agreement between the forward model and measured data when using the estimated spectral model and nonlinear counts correction.

Figure 1:

The transmission fraction measured through the calibration materials is compared to the transmission fraction estimated using Eq. 5 assuming the initial spectral response model, the spectral response optimized for correction of flux-independent effects, and the spectral response optimized jointly with a nonlinear counts correction to account for both flux-independent and pulse-pileup spectral effects.

Figure 2 displays the bone basis image, adipose basis image and the 60-keV VMI reconstructed by the two-step approach consisting of MLE+FBP. The GTV of the bone and adipose basis images reconstructed by the MLE+FBP approach served as the reference GTV constraint from which the range of cOSSCIR GTV settings were derived. The bone and adipose ROIs used for quantitative analysis are also depicted in Figure 2.

Figure 2:

(left) Bone basis, (center) adipose basis, and (right) 60-keV virtual monoenergetic image reconstructed by the two-step approach of MLE+FBP. The display windows are −0.3 to 1.3 for the basis images and −1000 HU to 1000 HU for the VMI. The locations of the bone and adipose ROIs used for quantitative analysis are displayed on the VMI.

Images reconstructed by cOSSCIR are compared to those reconstructed by the two-step approach of MLE+TV_min for the bone basis images (Figure 3), adipose basis images (Figure 4), and 60-keV VMIs (Figure 5). The cOSSCIR reconstructions are displayed at GTV settings equal to 5%, 10%, 20%, and 50% of the GTV value of the basis images reconstructed by MLE+FBP. The MLE+TV_min images were reconstructed so that the TV of the basis images matched those of the cOSSCIR images.

Figure 3:

Bone basis images are displayed resulting from (top) cOSSCIR and (bottom) MLE+TV_min. COSSCIR images were reconstructed with GTV constraint equal to 5%, 10%, 20%, and 50% of the GTV of the MLE+FBP images shown in Figure 2. The MLE+TV_min images were reconstructed to match the TV of the COSSCIR images. The images are windowed to display values between −0.3 and 1.3.

Figure 4:

Adipose basis images are displayed resulting from (top) cOSSCIR and (bottom) MLE+TV_min. cOSSCIR images were reconstructed with GTV constraint equal to 5%, 10%, 20%, and 50% of the GTV of the MLE+FBP images shown in Figure 2. The MLE+TV_min images were reconstructed to match the TV of the cOSSCIR images. The images are windowed to display values between −0.3 and 1.3.

Figure 5:

Virtual monoenergetic images at 60 keV from (top) cOSSCIR and (bottom) MLE+TV_min generated as a linear combination of the bone and adipose basis images displayed in Figures 3 and 4, respectively. The GTV constraint setting increases from left to right. The images are windowed to display values between −500 HU to 500 HU.

Figures 3 and 4 show increasing noise with increasing regularization constraint for both algorithms. Although both algorithms reconstructed images with the same TV value (a measure of variation), the cOSSCIR images show less noise and ring artifacts than the MLE+TV_min images.

The differences between the cOSSCIR and MLE+TV_min images are less noticeable in the VMIs than in the basis images. This is because noise is known to be negatively correlated between the bone and adipose images. The noise effects are therefore reduced in the VMIs, leading to more similar images when comparing cOSSCIR and MLE+TV_min. This cancellation of noise also explains why the VMIs are fairly consistent across the range of studied GTV constraint settings.

The mean and standard deviation of the values in the bone and soft-tissue ROIs are compared in Figure 6 for the bone basis, adipose basis, and VMIs reconstructed by cOSSCIR and MLE+TV_min across the range of studied regularization settings. Table 1 compares the mean and standard deviation of MLE+FBP to that of COSSCIR at 20% of the GTV of the MLE+FBP results and that of MLE+TV_min at TV matched to cOSSCIR. There were no statistically significant differences, as measured by a two-sided T-test, between the mean values in the bone region ROIs reconstructed by the studied algorithms, while statistically significant differences were found for the soft-tissue ROIs. The values in the basis images in the bone and soft tissue regions are expected to be close to either one or zero for most voxels. However, the ground truth values are unknown, as the true patient tissue composition may differ from the NIST models used in this study. There was no statistically significant difference in the mean values in the cOSSCIR ROIs compared between the lowest and highest regularization setting. Standard deviation increased with increasing regularization setting.

Figure 6:

Each plot compares the mean value in an ROI for cOSSCIR and MLE+TV_min, plotted against constraint setting. The cOSSCIR images were reconstructed with GTV constraint set to 5%, 10%, 20%, 35%, 50% and 100% of the GTV of the basis images reconstructed by MLE+FBP. The MLE+TV_min images were reconstructed with the TV constraint of each basis image matched to that of the cOSSCIR reconstructions. The error bars represent the standard deviation of values in the ROIs. The top row displays results from the bone ROI, and the bottom row displays results from the soft-tissue ROI. The plots display the quantitative analysis in the (left column) bone basis images, (center column) adipose basis images, and (right column) 60-keV VMI.

Table 1:

The mean and standard deviation of ROIs in images reconstructed by MLE+FBP, cOSSCIR at 20% of the GTV of the MLE+FBP results, and MLE+TV_min at TV matched to cOSSCIR.

	Bone ROI			Soft-tissue ROI
Image type	Bone	Adipose	60 keV VMI	Bone	Adipose	60 keV VMI
cOSSCIR	0.95±0.05	−0.06±0.14	1809±232	0.02±0.05	0.98±0.15	34±170
MLE+TVmin	0.95±0.32	−0.04±0.77	1817±260	0.05±0.17	0.91±0.45	53±98
MLE+FBP	0.94±0.35	−0.04±0.96	1805±179	0.04±0.27	0.92±0.78	40±100

Basis images reconstructed by cOSSCIR demonstrated lower noise standard deviation than the MLE+TV_min basis image, even though both algorithms produced images with the same TV. In the bone region, cOSSCIR reduced the noise standard deviation of the basis image values by a factor of 2.8 to 6.1 compared to MLE+TV_min, depending on regularization strength. In the adipose region of the basis images, the noise reduction factor of cOSSCIR ranged from 2.1 to 5.4. In the 60 keV VMIs, the cOSSCIR reconstructions demonstrated lower noise standard deviation for the bone region at low VMI energies, while cOSSCIR demonstrated higher noise levels than MLE+TV_min for higher VMI energies. Figure 7 compares the noise standard deviation of the bone and soft tissue regions of VMIs at energy between 30 and 100 keV, demonstrating that the algorithms have different noise properties at different VMI energies.

Figure 7:

The noise standard deviation in bone and soft tissue ROIs of VMIs are compared between cOSSCIR, MLE+TV_min, and MLE+FBP for VMI energies between 30 keV and 100 keV. Please note the logarithmic scale of the vertical axis.

To compare the visualization of fine anatomical details in the right inner ear region, Figures 8, 9, and 10 display an ROI extracted from the right inner ear region of the bone basis images, adipose basis images, and VMIs, respectively.

Figure 8:

The left inner ear region of the bone basis images, extracted from Figure 3, are displayed resulting from (top) cOSSCIR and (bottom) MLE+TV_min. cOSSCIR images were reconstructed with GTV constraint equal to 5%, 10%, 20%, and 50% of the GTV of the MLE+FBP images shown in Figure 2. The MLE+TV_min images were reconstructed to match the TV of the cOSSCIR images. The images are windowed to display values between −0.3 and 1.3.

Figure 9:

The left inner ear region of the adipose basis images, extracted from Figure 4 are displayed resulting from (top) cOSSCIR and (bottom) MLE+TV_min. cOSSCIR images were reconstructed with GTV constraint equal to 5%, 10%, 20%, and 50% of the GTV of the MLE+FBP images shown in Figure 2. The MLE+TV_min images were reconstructed to match the TV of the cOSSCIR images. The images are windowed to display values between −0.3 and 1.3.

Figure 10:

The left inner ear region of the 60-keV VMI, extracted from Figure 5 are displayed resulting from (top) cOSSCIR and (bottom) MLE+TV_min. The GTV constraint increases from left to right. The images are windowed to display values between −500 HU and 500 HU.

Reconstructed cOSSCIR images of additional head CT acquisitions are available as supplemental material to this article.

IV. Discussion

The cOSSCIR algorithm successfully reconstructed basis material images from the large-scale, high resolution, anatomically complex data acquired by a clinical PCCT prototype. The presented images show that cOSSCIR is able to reconstruct basis images with low noise and fine anatomical detail.

We hypothesized that the cOSSCIR algorithm would provide a more stable inversion into basis material images, due to the decomposition of each image voxel using the spectral data from all rays that pass through the voxel and because of the constraints placed on the basis images. Results demonstrate lower noise in the cOSSCIR basis images compared to those reconstructed by the MLE+TV_min algorithm, with both algorithms producing images at the same TV value. The noise reduction of cOSSCIR ranged from a factor of 2 to 6. Because both the cOSSCIR and MLE+TV_min basis images had the same TV, the increased noise in the MLE+TV_min images signifies reduced spatial resolution compared to cOSSCIR. Spatial resolution is not easily quantified in the anatomically detailed images. Qualitatively, cOSSCIR demonstrated better depiction of fine details, for example in the inner ear regions displayed in Figures 8, 9, and 10.

The MLE+TV_min and MLE+FBP algorithms demonstrated lower noise than cOSSCIR for VMIs in the mid energy range (Figure 7), despite the much higher noise in the basis images. The MLE+FBP algorithm also resulted in lower VMI noise than MLE+TV_min for some cases, despite higher basis image noise. Both of these results suggest that the regularization used by cOSSCIR and MLE+TV_min may reduce the noise correlation across basis images. While the cOSSCIR VMIs resulted in more noise than the two-step images for some VMI energies, the cOSSCIR VMIs depicted more fine anatomical detail, suggesting higher spatial resolution, as can be seen in the inner ear region. All algorithms in our study were designed to optimize the basis images, not the VMIs.

The basis images reconstructed by cOSSCIR and MLE+TV_min demonstrated similar mean values in the bone region, whereas statistically significant differences in the mean were found between cOSSCIR and both two-step approaches for the soft-tissue region. The true values in the bone and soft tissue regions of the basis images depend on the local density and composition of the tissue and are thus unknown. The values are expected to be close to 0 and 1, which was the case for the cOSSCIR algorithm for both the bone and soft tissue regions. The value of the 60-keV VMI in the soft-tissue region, assuming the attenuation coefficient of brain tissue from the NIST XCOM database³⁴ and a value of 0.2 for water in the conversion to Hounsfield units, was expected to be 29 HU. The mean value in the soft-tissue region of the cOSSCIR VMI was 34 HU, compared to 53 HU for MLE+TV_min, which was a statistically significant difference, and 40 HU for MLE+FBP, which was not a statistically significant difference. The mean values in the basis images are expected to be similar for all studied algorithms because they use same forward and spectral models. The cause of the discrepancies in the soft-tissue region of the two-step reconstructions is unknown.

Direct comparison of computation times is difficult because of differences in computer hardware, algorithm implementation, and definition of convergence. The cOSSCIR algorithm generally converged within approximately 1000 iterations, with each iteration consisting of four projection operations and two backprojection iterations. The TV_min algorithm generally converged within approximately 500 iterations, with each iteration consisting of two projection and two backprojection iterations (i.e., one projection and backprojection per iteration for each basis image). The MLE decomposition step of the TV_min also required many iterations per ray, but did not involve projecting through the image array. This step could be computed efficiently with a parallelized implementation. Based on this approximate analysis, the computation time of cOSSCIR is roughly three times longer than the reconstruction step of the MLE+TV_min algorithm.

Ring artifacts were visible in the basis images reconstructed by both cOSSCIR and the MLE+TV_min algorithms. Ring artifacts can be caused by errors in the spectral calibration leading to biases that vary across detector pixels. Because of the ill-posed nature of the spectral calibration and material decomposition (i.e., different combinations of basis materials can lead to similar overall attenuation), these biases are usually anti-correlated across basis images and are reduced or absent in the VMIs. Ring artifacts can also occur when the detector pixel response varies between the time of spectral calibration and CT acquisition. In our previous cOSSCIR work with a different optimization framework, we proposed estimating a spectral scaling factor concurrently with the basis images to correct for detector instability.¹⁷ This approach was found to reduce ring artifacts and could potentially be integrated into the ADMM optimization approach. In the highly regularized cOSSCIR images, the rings may be blurred into artifacts that could be interpreted as anatomical structures or into bands of artifacts. However, choosing a weaker regularization setting can mitigate this issue and still provides high image quality as demonstrated in Figures 3 and 4. Clinical CT images typically undergo some form of ring artifact correction. The images in this paper are presented without any ring artifact correction step. Also, the data used in this study were collected on an early silicon PCCT prototype with a first-generation detector and do not represent the capabilities of newer silicon PCCT systems. Subsequent improvements in manufacturing processes are expected to produce better hardware in comparison to this early prototype, resulting in data with less spectral variation from pixel to pixel.

The cOSSCIR algorithm placed a grouped TV constraint on the basis images during inversion from photon counts to basis images. The comparison MLE+TV_min algorithm placed TV constraints during the inversion from basis sinogram to basis image. One limitation of this work is that the maximum likelihood algorithm used for decomposition into basis sinograms in the MLE+TV_min approach was unregularized. Decomposition algorithms that use regularization or take advantage of noise correlations in the decomposition step have been proposed and may yield improved noise behavior.^36,37,38,39 The MLE algorithm used in this work has been shown to meet the Cramer-Rao lower bound on noise variance for the inversion into basis sinograms,⁴⁰ however additional noise improvements may be possible by taking advantage of noise correlations between energy windows. The TV least-squares reconstruction approach used in the two-step comparison could also be improved with better noise modeling.

Another limitation of this work is that the data were acquired in a single-slice, fan-beam geometry for which scatter effects could be neglected. The cOSSCIR forward model could be easily modified for multi-slice geometries. For larger cone angles, scatter is expected to reduce the material decomposition accuracy and would need to be corrected prior to cOSSCIR reconstruction or incorporated into the forward model.

V. Conclusions

The cOSSCIR algorithm, combined with our previously proposed spectral model estimation and nonlinear counts correction method, successfully reconstructed 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 the MLE+TV_min two-step approach.