Imaging Technologies

In theory, as X-rays pass through an object, not only do their intensity attenuate, but their phase also shifts. This can be explained well by the refractive index formula n=1 - δ + iβ, where β represents the absorption factor and δ represents the phase-shift factor. Both XACT and DECT extract the intensity attenuation of X-rays, whereas XPCT focuses on extracting the phase-shift of the X-rays. Considering that the emission energy spectrum of the X-ray is Ω (E) and the spatial position of the object under test is (x, y, z), where the direction of is from the source to the center of the detector, the absorption projection quantity P (s, θ) of the detector in different directions θ can be expressed as follows: (1)where Dirac represents the Dirac function, s represents the projection distance, l is the propagation distance of the X-rays within the object, l_front indicates the position at which the X-rays enter the object, and l_end represents the position at which the X-rays exit the object. Equation 1 is a modified form of the Lambert–Beer law. By collecting multiple projections at different angles and subsequently employing filtered back-projection (FBP) reconstruction algorithms, the distribution of absorption coefficients within an object can be calculated and reconstructed, yielding either slice or three-dimensional images.

DECT employs a widely used basis material decomposition model that leveraging the distinct absorption coefficient μ characteristics of objects under varying X-ray energies. Through mathematical modeling, the imaging data were decomposed on different bases. In this process, the linear attenuation coefficient μ (x, y, z) is transformed into μ_L (x, y, z) + μ_H (x, y, z), where the subscripts L and H represent the conditions of low and high X-ray tube voltages, respectively. Correspondingly, the projection quantity P that describes DECT-based absorption projection formula is transformed as follows: (2)where b₁ and b₂ represent decomposition coefficients. Given the projection data from the dual-energy spectra, b₁ and b₂ can be obtained by solving the system in Eq. 2. Using calcium and water as the basis for decomposition, the relative electron density ρ_e and effective atomic number Z_eff of the object can be represented by Eqs. 3 and 4. (3) (4)where ρ_e,calcium and ρ_e,water are the electron densities of calcium and water, respectively; and Z_calcium and Z_water are the effective atomic numbers of calcium and water, respectively. Equations 3 and 4 reflect the absorption contrast of the material at different energies.

Unlike the first two techniques, XPCT extracts phase-shift information from X-rays. The double-mask EI method requires the illumination curve (IC) to be read by moving the two masks laterally relative to each other. IC acquisition required at least three exposures. Absorption, refraction, and scattering information can be obtained in the presence and absence of an object, which means that the imaging process requires the positional alignment of the two optics and multiple exposures, and the accuracy and stability requirements of the system are challenging for clinical conditions. The SMSS EI method utilizes two adjacent pixels to detect the offset of beamlets in a single exposure, thereby quantitatively extracting refraction information, as shown in Eq. 5 [28]. (5)where W is the distribution of individual beamlets on the detector related to the experimental conditions, Z_od is the distance from the object to the detector, and I₁ and I₂ are the intensity values of two neighboring pixels. SMSS EI achieves concise and efficient single-exposure imaging with a certain loss of spatial resolution, which is significantly less difficult to implement and reduces the time and dose required for imaging. This loss of resolution can be compensated for by two dither steps, which are equivalent to a two-exposure imaging method. The quantum projection quantity of the differential phase is expressed as (6)where k represents a constant that enables the reconstruction of phase information δ. However, the term is a vector that cannot be used directly in conventional FBP algorithms. This issue was encountered in the earliest attempts at ABI to reconstruct phase information. Zhu et al. [29, 30] multiplied the term by for each projection, thus transforming the term into an invariant. In addition to the indirect method, the direct integration of to obtain δ is another reconstruction method. Another direct method uses a Hilbert filter to directly resolve the δ signal using . A more detailed reconstruction algorithm for the phase information can be found in Zhang’s paper [31].

Simulation Model

We used GEANT4 version 11.1.0, with Ubuntu 20.04.02, for all the simulations. The GEANT4 simulations were run on a desktop machine with an Intel Core i7-13700KF 5.40 GHz CPU processor. The simulations involve the physical principles of electromagnetism and X-ray refraction, adhering to the Beer–Lambert law (considering photon-matter interactions such as the photoelectric effect and Compton effect, but excluding electron pair effects as photon energies are below 1.02 MeV in our setup) and Snell’s law (refracting X-rays when transitioning between media based on their refractive indexes). G4ElectromagneticPhysics is used to make X-ray follow the absorption process, and G4MaterialPropertiesTable is used to define the RINDEX of an object to make X-ray follow the refraction process. The emitted particles employ the G4GeneralParticleSource module using the command /gps/hist/point to weigh the energy of each emitted photon based on the corresponding energy spectrum intensity.

The GEANT4 configurations of the three systems are presented in Fig. 1. Z_so is the distance from the source to the object and Z_sd is the distance from the source to the detector, with values of 80 and 100 cm, respectively. The XACT system is a basic device consisting of an X-ray source, an object, and a detector, where the energy of the X-ray source corresponds only to the low-energy option. The DECT system adds a high-energy X-ray source to the XACT system. The XPCT system utilizes the SMSS EI method by adding a coded-aperture module to the XACT system as a mask.

Fig. 1Full-screen View

(Color online) Schematics of the three systems. (a) depicts XACT system, (b) depicts DECT system, and (c) depicts SMSS EI-based XPCT system

The specific parameters for CT imaging are listed in Table 1. Based on Handschuh et al. [32], X-ray tubes with 40 and 80 kVp voltages were selected for DECT imaging. The X-ray energy spectrum, generated by Spekpy simulation [33], is shown in Fig. 2. XACT and XPCT used a 40 kVp X-ray source, whereas DECT used 40 and 80 kVp sources. Glass filter (1.1 mm) and aluminum filter (0.5 mm) were placed before the 40 and 80 kVp tubes to reduce beamline hardening. Source and detector pixels were 40 and 62 μm, respectively, with the detector having 200 single-row pixels. The system’s magnification factor was 1.25, yielding a reconstructed voxel size of 49.6 μm. Projection data were acquired at 1-degree intervals, obtaining 360 images per rotation. XACT emitted 4×10⁷ photons per angle, whereas DECT emitted 2×10⁷ photons due to dual voltage requirements. In XPCT, the y-direction resolution was halved per Eq. 5; however, linear interpolation achieved 49.6 μm voxel size for comparison. The mask’s projected size doubled the pixel size, with beamlets hitting the pixel midpoints. The mask thickness ensured that the X-rays passed only through the apertures, enabling the absorption of almost all 40 kVp X-rays elsewhere. XPCT photon count was increased tenfold to offset the 90% mask loss (i.e., 4×10⁸).

Fig. 2Full-screen View

The emission spectra of X-ray sources. (a) is the emission spectrum at a tube voltage of 40 kVp, and (b) is the emission spectrum at a tube voltage of 80 kVp (high energy for DECT only)

The simulated soft-tissue phantom is shown in Fig. 3, and the corresponding tissue material information is presented in Table 2. The radius of the phantom was 4 mm, whereas those of the other plugs were 0.6 mm. The phantom height was 2 mm. Tissue material data were obtained from CIRS 062M phantom and International Commission on Radiological Protection. The μ(E) of the phantom was obtained from the database of the National Institute of Standards and Technology and δ was calculated as δ = 4.15×10⁴ ρZ/(AE²) [27].

Fig. 3Full-screen View

(Color online) Schematic view of the soft-tissue phantom model. The soft-tissue phantom consists of nested tubes with soft tissue as the base, with four component material plugs positioned at four locations. The radius of the disk is 4 mm, while the radius of the plugs is 0.6 mm. The height of the phantom is 2 mm

The system setup in GEANT4 corresponds to that shown in Fig. 1. The GEANT4 simulation program for SMSS EI-based XPCT is optimized by Brombal et al. [34, 35]. The energy spectrum is input into the source module, generating photons with the same distribution as in Fig. 2. In the SMSS EI system, photons must pass through the mask and hit the center of two pixels precisely, making mask position calibration essential. The mask must meet requirements in both y and z directions. For y-direction calibration, the normalized intensity position of 0.5 is determined by extracting the single-mask IC curve [36], as shown in Fig. 4(a), ensuring the photon reaches the detection unit’s middle position. For z-direction calibration, the mask is moved until achieving a uniform image. The refractive angle signal at this z-position, shown in Fig. 4(b), confirms z-direction requirements are met. Table 2 and δ containing phantom information are input into the object module. The soft-tissue phantom has five solid parts. We add an object rotation module to adjust positions and rotations during scanning, as the original program lacks direct phantom rotation. Using Python scripting, the program enables simultaneous multitasking and multithreading for efficient simulation.

Fig. 4Full-screen View

(Color online) (a) IC curve for single-mask. (b) is the refractive angle signal extracted by the XPCT after the mask position is well calibrated

The data read by the detector are imported into MATLAB for processing and imaging, as shown in Fig. 5. Projection images are presented by moving the object in the x-direction to show the projected image with a larger field of view, with CT imaging only imaging a certain height. The simulated profiles in Fig. 5(b), (g), (h) and (q) closely match theoretical profiles. For XACT imaging, 360-degree projections are acquired, and the projection sinogram is obtained using the fanbeam function, followed by ifanbeam for the sliced image. Hamming filters are used in all back-projection (BP) procedures except Fig. 5(w). The XACT image contrast in Fig. 5(d) originates from linear absorption coefficient μ. DECT imaging is based on XACT, with material decomposition performed on low-energy (Fig. 5(k)) and high-energy (Fig. 5(l)) images. Figure 5(k) appears sharper due to decreased X-ray interaction cross-section at higher energies. Using calcium and water as basis materials (μ values: (calcium) 8.31 and 1.47 cm^-1; (water) 0.69 and 0.25 cm^-1 at 40 and 80 kVp, respectively), Eqs. 2, 3 and 4 yield relative electron density Z_eff and effective atomic number images in Fig. 5(m) and (n). The DECT image contrast in Fig. 5(o) combines normalized Z_eff values. XPCT imaging requires complex fanbeam transformation due to non-constant . Three solutions - multiplication, direct integration, and Hilbert transform - yield projection sinograms Fig. 5(r), (s) and (t), with resulting slice images (u), (v) and (w). Figure 5(t) shows center brightness due to direct integration effects. XPCT images Fig. 5(u), (v) and (w) contrast originates from differential phase , phase δ₁ and phase δ₂. Detailed algorithms are reported in reference [31, 35, 37].

Fig. 5Full-screen View

(Color online) Schematic of imaging data processing for XACT, DECT and XPCT with MATLAB. (a), (e), (f) and (p) are the raw projections obtained by GEANT4. (b), (g), (h) and (h) show the red profiles corresponding to the projections and the theoretically calculated curves in orange. (c), (i), (j), (r), (s) and (t) are projection sinograms. (d), (k), (l), (u), (v) and (w) are the sliced images obtained after the ifanbeam operation. Note that the Hamming window filtering is added to all except (w) which is an only BP operation. (m) and (n) are the relative electron density Z_eff image and the effective atomic number Z_eff image of (k) and (l) decomposed with calcium and water as the base material,respectively, which can be combined together to obtain the DECT image (o) with Z_eff and Z_eff as the contrast. The blue and orange arrows show the operation of fanbeam after acquiring multiple projected profiles at low and high energies, respectively. The yellow arrows indicate the operation of FBP after obtaining the projection sinograms. The green arrow indicates a processing operation for dual-energy operation, which recombines the low-energy and high-energy images after material decomposition. The purple arrow indicates the operation of BP only after obtaining the projection sinograms

For a better comparison with the simulation results, we additionally built theoretical CT images of XACT, DECT and XPCT of the phantom with the same voxel based on MATLAB. After modeling in MATLAB, the same process was performed, as shown in Fig. 5. The next step involves the development of methods to evaluate these results.

Evaluation System

In this study, the performance of DECT, XPCT and XACT was assessed through a comprehensive evaluation system based on four aspects: error evaluation metrics, image quality evaluation metrics, contrast and noise evaluation metrics, and data distribution and dispersion evaluation metrics. The error evaluation metrics include the absolute error (AE) and mean absolute error (MAE), which are calculated using Eqs. 7 and 8 respectively, to assess the difference between the measured and theoretical values for different tissues. The image quality evaluation metrics consist of the structural similarity index (SSIM) and peak signal-to-noise ratio (PSNR), where the former compares the similarity between two images, and the latter measures the noise level and distortion of the image, as indicated by Eqs. 9 and 10. The contrast and noise evaluation metrics, represented by the contrast-to-noise ratio (CNR) defined in Eq. 11, assess the clarity and noise level of the reconstructed images. The data distribution and dispersion evaluation metrics were represented by boxplots that visually displayed the distribution and dispersion of the data, including the minimum, lower, median, upper, and maximum values. This method helps visualize the distribution and dispersion of the measurement data, aiding in the identification of outliers and the shape of the data distribution. (7) (8) (9) (10) (11)In Eqs. 7 and 8, O denotes the observed value, where i represents the voxel point, and n is the total voxel count of different tissues. In Eq. 9, A_measured, A_theoretical, , and σ_{measured,theoretical} are the local mean, variance, and covariance of the measured and theoretical images, respectively, and C₁ and C₂ are stability constants. In Eq. 10, Peakval represents the maximum observed value in an image. MSE is the mean squared error between the measured and theoretical images. In Eq. 10, O_signal and O_background represent the intensities of the signal and background, respectively, and σ² is the corresponding variance.

CT Images

Following the reconstruction algorithms described in Sect. 2.2, we obtained slice images for the three techniques as shown in Fig. 6. All the images share identical ROIs, with normalized image intensities. To reveal hidden textures, we use pseudo-color representation, mapping grayscale values to the RGB color range. The colorbar indicates normalized intensity. The lung (inhale) region appears blue, lung (exhale) appears green, adipose appears orange, and water and soft-tissue regions appear in varying shades of red. The similar colors for water and soft tissue reflect their comparable densities. In subsequent analyses, the distinction between water and soft tissue serves as a reference for imaging technology resolution - clearer distinction indicates better discriminatory ability. From the theoretical images in Fig. 6(a) and 6(c), XACT and DECT show similar image contrasts due to their shared principle of intensity attenuation. While the performance of XPCT differs from that of absorption imaging techniques, advantages and disadvantages cannot be determined through simple observations.

Fig. 6Full-screen View

(Color online) CT image results. (a), (c) and (e) are theoretical images of XACT, DECT and XPCT, respectively. (b), (d) and (f) are simulated images of XACT, DECT and XPCT, respectively

For simulated images, XACT and DECT show similar characteristics, both struggling with water detection, with DECT showing higher noise due to emitting half the photons as XACT in two emissions. In XPCT, we selected Fig. 5(w) for comparison. The XPCT noise differs from Gaussian noise in absorption images, showing slight ringing artifacts [38], but edges remain clearly visible in all five materials..

For quantitative analysis, we extracted information from each image, as shown by white boxes in Fig. 6(a), with results depicted in Fig. 7. Panels (a), (b) and (c) in Fig. 7 show the intensity profiles of XACT, DECT, and XPCT images, respectively, against soft-tissue background. XPCT demonstrates the best profile smoothing and shows water’s signal intensity, which is not reflected by absorption-contrast imaging. The statistical evaluation process is detailed in the next section.

Fig. 7Full-screen View

(Color online) Profiles of the CT images in Fig. 6. Panels (a), (b) and (c) represent profiles of the XACT, DECT and XPCT images, respectively. The orange lines are from theoretical images and the red lines are from simulated images

Statistical Evaluation

The physical quantities reconstructed from the simulated images and the AE results are compared with the theoretical values in Table 3. Among these, DECT exhibited the largest deviation, which was due to the halving of the dose per emission, leading to relatively poor performance in the simulated images. Conversely, XPCT had the smallest AE, indicating a closer alignment with the theoretical values.

Furthermore, we compare the SSIM and PSNR values of the images shown in Fig. 6, and the results are listed in Table 4. These two metrics are calculated using the simulated and theoretical images. Larger SSIM and PSNR values indicate better-simulated images. Evidently, XPCT is the closest to the theoretical image and exhibits the least noise effect. XACT and DECT do not perform well on image quality assessment metrics because of the effect of Gaussian noise.

Table 5 displays the theoretical and simulated CNR results of the images in Fig. 7. The theoretical CNR shows XACT and DECT have similar resolving power. XACT performs better for tissues with large density differences, while DECT excels for tissues with small density differences. XPCT shows the highest CNR for large density differences, with resolving power between DECT and XACT for small density differences. In simulated results, DECT performs the worst overall, while XACT surpasses XPCT for tissues with large density differences, and XPCT excels for similar-density tissues. The theoretical CNR values are on the order of 10¹⁴, approaching infinity; however, reconstruction algorithm noise prevents infinite values. Simulated CNR values are lower due to noise impact. Comparing both on the same scale reveals XACT maintains excellent resolution for high density differences, XPCT performs the best for smaller differences, while DECT shows weakest performance, primarily due to photon statistical fluctuations proportional to the square root of emitted particles. In the DECT simulated results, increased statistical fluctuation from halving the photon emission affects both low-energy and high-energy slice images. Both projections deviate from true values due to statistical fluctuations. When performing material decomposition using Eq. 2, this deviation amplifies errors in calculating b₁ and b₂ [39]. Boxplots from Fig. 6 are shown in Fig. 8. White lines indicate median position, while white dots show mean position. Box height shows data dispersion, with bottom and top edges at 25th and 75th percentiles. The whiskers extend to the remaining data range. The boxplots show that XPCT has better data distribution than do DECT and XACT, indicating better phase information reconstruction and extraction.

Fig. 8Full-screen View

(Color online) Boxplots calculated from CT simulated results in Fig. 6

In brief, this section provides a quantitative evaluation of DECT and XPCT compared to XACT. Results show XPCT performs relatively better in several performance metrics.

While the SMSS EI-based XPCT method is easier to implement, misalignments in mask position during gantry movement can invalidate calibration, requiring more precise equipment. Radiation safety is crucial in medical applications. Though radiation dose is comparable, exposure time is significantly longer due to reduced X-ray efficiency. Research shows optimizing the mask [40] and reducing exposure through low-dose irradiation [31] can address this. The method requires specific detector capabilities, as pixel crosstalk affects detection sensitivity, making direct-conversion or photon-counting detectors more suitable.

In this study, soft-tissue models with low-density and similar tissues are used owing to limited data on other tissues, and such tissue differentiation remains challenging in clinical diagnosis. XPCT offers a new technique for this, though complex situations require further biological experiments for resolution.