Principle of simultaneous XFCT and CSCT

To realize simultaneous XFCT and CSCT, the X-ray fluorescence and Compton scattering signals must be distinguishable, which can be achieved using linearly polarized X-rays. For linearly polarized X-rays, the differential Compton scattering cross-section is described by the Klein-Nishina formula: $\begin{array}{l}\frac{\text{d}{\sigma }_{\text{KN,LP}}}{\text{d}\text{Ω}}\left(E_0\right)=\frac{1}{2}{r}_{\text{e}}^2{\epsilon }^2\left(\epsilon +{\epsilon }^{-1}-2{{\displaystyle \sin }}^2\theta {{\displaystyle \cos }}^2\varphi \right)\mathrm{,}\hfill \end{array}$ (1) where r_e is the classical electron radius, θ and ϕ are the polar and azimuthal angles, respectively, and $\begin{array}{l}\epsilon =\frac{E_{\text{f}}}{E_{\text{0}}}=\frac{1}{1+\frac{E_{\text{0}}}{m_{\text{e}}{c}^2}\left(1-\cos \theta \right)}\mathrm{,}\hfill \end{array}$ (2) where E_f and E₀ are the Compton-scattered and incident photon energies, respectively, and m_ec² is the rest energy of an electron. Obviously, the differential Compton scattering cross-section is ϕ-dependent; therefore, it can be increased or decreased by adjusting the polarization direction of the incident X-rays.

For XFCT and CSCT, the signal detector is typically placed perpendicular to the incident X-ray beam, for example, θ = 90°. In this direction, the differential Compton scattering cross-section reaches its minimum and maximum values at $\varphi {=0}^{\circ }{/180}^{\circ }$ and $\varphi {=90}^{\circ }{/270}^{\circ }$, respectively, as shown in Fig. 1. If the signal detector is arranged in the x-axis direction, a single Compton scattering background can be completely suppressed using incident X-rays with horizontal polarization (x-axis direction), and the X-ray fluorescence projection P_XFCT used for XFCT reconstruction can be obtained. Furthermore, when the incident X-ray polarization changes to the vertical direction (y-axis), the Compton scattering projection P_CS,VP that is intermingled with the X-ray fluorescence background is acquired. Usually, the energy difference between an X-ray fluorescence photon and a Compton-scattered photon is not distinct; therefore, they cannot be distinguished via photon energy. However, if the two projections are taken at the same photon flux and scan time, the Compton scattering projection P_CSCT used for CSCT reconstruction can be obtained by subtracting the Compton scattering projection P_CS,HP obtained in the horizontal polarization case from the Compton scattering projection P_CS,VP, i.e., $P_{\text{CSCT}}=P_{\text{CS,VP}}-P_{\text{CS,HP}}$ (P_CS,VP and P_CS,HP are acquired at the same energy region). Thus, XFCT and CSCT can be realized simultaneously.

Fig. 1Full-screen View

(Color online) Differential Compton scattering cross-section distribution at (a) the full space and (b) θ = 90° plane. The incident x-ray energy is 83 keV, and ${\sigma }_{\text{T}}$ is the total Thomson scattering cross-section

Monte Carlo simulations

To demonstrate the feasibility of the simultaneous XFCT and CSCT, MC simulations were performed using the Geant4 toolkit [33]. The image layout is shown in Fig. 2. The X-ray source was modeled based on a Thomson scattering light source that can provide quasi-monochromatic, continuously energy-tunable, and polarization-controllable X-rays. To satisfy the field-of-view (FOV) requirements of a small-animal-sized imaging object (～ 5 cm) and quasi-monochromatic spectral conditions, a large source-to-sample distance is required for Thomson scattering light sources [34, 32]. To simplify the unnecessary X-ray transport before the phantom, a quasi-parallel X-ray beam (2D) was adopted in the MC simulations, which was incident on the phantom along the +z axis. The X-ray spectrum has a Gaussian distribution with a peak energy of E₀ = 83 keV and an RMS bandwidth of 1.5%, which can be easily achieved using existing technologies [35, 36]. The incident X-rays are linearly polarized, and their polarization is tunable between the horizontal (x-axis) and vertical directions (y-axis).

Fig. 2Full-screen View

(Color online) Imaging layout for simultaneous XFCT and CSCT based on linear polarization X-rays (not to scale)

The phantom was a polytetrafluoroethylene (Teflon) cylinder with a diameter of 5.0 cm. Inside the Teflon cylinder were six cylindrical columns 1.0 cm in diameter, containing aluminum (Al), water (H₂O), and gold (Au)-loaded water solution contrast agents. In the contrast agents, the Au weight fractions were 0.5, 1.0, 2.0, and 4.0 wt%, respectively. For Au, its K-edge EK is located at 80.72 keV [37] and its Kα lines generated by the Livermore library in Geant4 are 69.038 and 67.184 keV for $K_{\alpha 1}$ and $K_{\alpha 2}$, respectively.

For fluorescent and Compton-scattered X-ray photon detection, a photon-counting detector (PCD) was placed perpendicular to the x-axis in the x–z plane, and the distance between the PCD and the center of the phantom was 8.0 cm. The PCD was modelled with a pixel size of 0.5 mm and a sensitive energy range of 20–100 keV; currently, these performances can be easily achieved [38]. For simplicity, the energy resolution and detection efficiency of the PCD were assumed ideal. A parallel-hole collimator made of lead (Pb) was placed at the front of the PCD. The length (x-axis direction) and opening size (z-axis direction) of the collimator are 5.0 cm and 0.5 mm, respectively. To improve the fluorescent and Compton-scattered X-ray photon detection efficiency and to improve the XFCT and CSCT reconstruction spatial resolution, a second PCD combined with a parallel-hole collimator was placed on the opposite side of the phantom, and the parallel-hole collimator was offset from the first one by half of the parallel hole pitch, as shown in Fig. 2. To correct the phantom attenuation in XFCT and CSCT, an ideal transmission detector with a pixel size of 0.2 mm was placed 0.5 m downstream of the phantom to acquire the phantom attenuation data.

In the MC simulations, 360-deg projections were acquired at a rotational step of 1° for the CT scan. To balance statistical error and simulation time, 5×10⁹ photons were used for each projection.

Image reconstruction

Based on the imaging geometry in Fig. 2, the fluorescent and Compton-scattered X-ray photon detection can be divided into three steps.

• Stimulation of fluorescent and Compton-scattered X-ray photons in the phantom

As the incident X-ray travels from point A to point P to stimulate the fluorescent and Compton-scattered x-ray photons, the beam intensity is attenuated by the phantom, $\begin{array}{l}I\left(P\right)=I_0\exp \left[-{\displaystyle {\int }_A^P}\mu \left(E_0\mathrm{,}\overrightarrow{r}\right)\text{d}s\right]\mathrm{,}\hfill \end{array}$ (3) where I(P) and I₀ are the X-ray beam intensities at points A and P, respectively, μ is the linear attenuation coefficient of the phantom, E₀ is the incident X-ray energy, and $\overrightarrow{r}$ is the spatial position vector.

• Emission of fluorescent and Compton-scattered x-ray photons

If Au is located at point P, the Kα fluorescence of Au is emitted isotropically as E₀ is higher than EK. The differential intensity of fluorescent x-ray photons can be written as $\begin{array}{l}\frac{\text{d}{I}_{\text{XRF}}}{\text{d}\text{Ω}}\left(P\right)=\frac{1}{4\pi }I\left(P\right){\mu }_{\text{Au,PE}}^{\text{m}}\left(E_0\right){\rho }_{\text{Au}}\left(P\right){\omega }_{K_{\alpha }}\mathrm{,}\hfill \end{array}$ (4a) where ${\mu }_{\text{Au,PE}}^{\text{m}}\left(E_0\right)$ is the photoelectric mass absorption coefficient of Au at X-ray energy E₀, ρ_Au(P) is the local Au concentration (in mass percent) to be reconstructed in XFCT, and ${\omega }_{K_{\alpha }}$ is the fluorescence yield of Au. Meanwhile, the Compton-scattered x-ray photons are also emitted and their spatial distribution is governed by the differential cross-section Eq. (1). The differential intensity of Compton-scattered x-ray photons can be written as $\begin{array}{l}\frac{\text{d}{I}_{\text{CS}}}{\text{d}\text{Ω}}\left(P\right)=I\left(P\right){\rho }_{\text{e}}\left(P\right)\frac{\text{d}{\sigma }_{\text{KN,LP}}}{\text{d}\text{Ω}}\left(E_0\right)\mathrm{,}\hfill \end{array}$ (4b) where ρ_e(P) is the local electron density to be reconstructed in CSCT.

• Detection of the fluorescent and Compton-scattered x-ray photons by the PCD

After passing through the parallel-hole collimator, the fluorescent and Compton-scattered X-ray photons are finally detected by the PCD. Along this path, the beam intensity is attenuated by the phantom from point P to boundary B. Therefore, the beam intensity of fluorescent and Compton-scattered X-ray photons coming from P and detected at detector pixel bin Di can be described as $\begin{array}{c}{I}_{\text{det,u}}\left(P\mathrm{,}{D}_i\right)=\frac{\text{d}{I}_{\text{u}}}{\text{d}\text{Ω}}\left(P\right)\text{Δ}{\text{Ω}}_{\text{c}\mathrm{,}i}\\ \times \exp \left[-{\displaystyle {\int }_P^B}\mu \left(E_{\text{u}}\mathrm{,}\overrightarrow{r}\right)\text{d}s\right],\end{array}$ (5) where the subscript ’u’ indicates XRF or CS for fluorescent or Compton-scattered X-ray photons, respectively, similarly, hereinafter, E_XRF and E_CS are the photon energies of the fluorescent and Compton-scattered X-ray photons, respectively, and $\text{Δ}{\text{Ω}}_{\text{c}\mathrm{,}i}$ is the collecting angle of the parallel-hole collimator corresponding to detector pixel bin Di.

According to the fluorescent and Compton-scattered X-ray photon detection process, the total fluorescent or Compton-scattered beam intensity detected at Di can be described as a volume integral of $I_{\text{det,u}}\left(P\mathrm{,}{D}_i\right)$: $\begin{array}{l}{I}_{\text{u}\mathrm{,}i}={\displaystyle \int }{\displaystyle \int }{\displaystyle {\int }_{V_{P\to D_i}}}{I}_{\text{det,u}}\left(P\mathrm{,}{D}_i\right)\text{d}{V}_P\mathrm{,}\hfill \end{array}$ (6) where the integration domain $V_{P\to D_i}$ is the phantom region subtended by Di toward the parallel-hole collimator. For both XFCT and CSCT, the detected signal intensity I of size $M\times 1$ and the unknown parameters $\rho$ (ρ_Au or ρ_e) of size $N\times 1$ to be reconstructed can be uniformly expressed as a system of linear equations: $\begin{array}{l}I=\mathbb{A}\rho \mathrm{,}\hfill \end{array}$ (7) where $\mathbb{A}={\left[a_{ij}\right]}_{M\times N}$ is a system matrix of size M×N. Similar to other emission tomography techniques (e.g., SPECT and PET), the detected X-ray photon number is relatively low in both XFCT and CSCT, and statistical noise is the major limiting factor for accurate reconstruction. To reduce statistical noise artifacts, the $\rho$ distribution was reconstructed using the commonly used maximum-likelihood expectation maximization (MLEM) algorithm [39] for emission tomography, $\begin{array}{l}{\rho }_j^{\left(k+1\right)}=\frac{{\rho }_j^{\left(k\right)}}{{\displaystyle {\sum }_{i=1}^M{a}_{ij}}}{\displaystyle \sum _{i=1}^M\frac{a_{ij}{I}_i}{{\displaystyle {\sum }_{j^{\prime }=1}^N{a}_{i{j}^{\prime }}}{\rho }_{j^{\prime }}^{\left(k\right)}}}\left(j=\mathrm{1,2,}\dots \mathrm{,}N\right)\mathrm{.}\hfill \end{array}$ (8) To obtain an accurate system matrix $\mathbb{A}$, the attenuation terms $\mu \left(E_0\mathrm{,}\overrightarrow{r}\right)$ in Eq. (3) and $\mu \left(E_{\text{u}}\mathrm{,}\overrightarrow{r}\right)$ in Eq. (5) must be determined, which can be realized using a transmission CT scan of the phantom at the corresponding X-ray energy. For the attenuation CT, statistical noise is not the limiting factor for accurate reconstruction because the signal-to-noise ratio (SNR) in the projections is relatively high. Therefore, the well-known ART-TV iterative algorithm, proven to be effective in our laboratory, was adopted for CT reconstruction at different X-ray energies, and 180 projections taken at a rotational step of 1° were used.

Influence of multiple Compton scattering on XFCT

Although single Compton scattering can be greatly suppressed using horizontally polarized X-rays, multiple Compton scattering can still have an impact on XFCT. As shown in Fig. 4(b), the XFCT reconstruction quality, compared with the multiple Compton scattering correction result in Fig. 4(a), deteriorates owing to the influence of multiple Compton scattering. An effective method was developed to correct for this influence. Considering that multiple Compton scattering is almost uniformly distributed in the X-ray spectrum, as shown in Fig. 3(a), the multiple Compton scattering MS_XRF in the X-ray fluorescence detection energy region can be calculated via linear interpolation: $\begin{array}{l}{\text{MS}}_{\text{XRF}}=\frac{1}{2}\left({\text{MS}}_{-}+{\text{MS}}_{+}\right)\mathrm{,}\hfill \end{array}$ (9) where MS_- and MS₊ are the multiple Compton scattering intensities in the low (64.5–67 keV) and high (69.5–72 keV) regions, respectively. The multiple Compton scattering correction was realized by subtracting MS_XRF from the X-ray fluorescence signal detected by the PCD in the horizontal polarization case.

To quantitatively evaluate the contrast improvement caused by multiple Compton scattering correction, the contrast-to-noise ratio (CNR) is calculated as $\begin{array}{l}\text{CNR}=\frac{S_{\text{Au}}-S_{\text{BG}}}{{\sigma }_{\text{BG}}}\mathrm{,}\hfill \end{array}$ (10) where S_Au and S_BG are the mean values of S in the Au and Teflon ROI, respectively, and ${\sigma }_{\text{BG}}$ is the standard deviation in the Teflon ROI. The calculated CNR results for the different materials are shown in Fig. 6. Also depicted in Fig. 6 is the limit of detection (CNR=±5) based on the Rose criterion [40]. It can be seen that the CNR values of the four contrast agents, especially the contrast agent with 0.5% Au concentration, are greatly improved by the multiple Compton scattering correction. Without multiple Compton scattering correction, Al becomes detectable in XFCT (CNR = 5.84), which does not reflect the real situation because this column has no Au. After multiple Compton scattering correction, Al could not be discriminated from the background, as shown in Fig. 4(a). Therefore, it is necessary to correct for the multiple Compton scattering background to improve the CNR in XFCT.

Fig. 6Full-screen View

(Color online) CNR comparison between XFCT, CSCT, and KES. MS: multiple scattering; XF: X-ray fluorescence. The red dot-dashed lines describe the limit of detection (CNR=±5) according to the Rose criterion. The data of H₂O in XFCT, contrast agent with 4.0% Au concentration in CSCT without XF correction, and Al in XFCT with MS correction are not depicted because they cannot be identified by their geometric shape in the corresponding reconstruction images

Influence of X-ray fluorescence on CSCT

Compared with the Compton scattering signal covering a relatively wide energy region, the integrated X-ray fluorescence background was very weak in the vertical polarization case, as shown in Fig. 3(a). However, this weak X-ray fluorescence background can still affect the CSCT reconstruction quality. For comparison, the CSCT of the phantom was reconstructed without subtracting the X-ray fluorescence background acquired for horizontal polarization. The reconstruction results are presented in Fig. 7. Clearly, the contrast agent with 4.0% Au concentration cannot be discriminated from its surrounding background owing to the influence of X-ray fluorescence. To quantitatively analyze the influence of X-ray fluorescence background on CSCT, the CNR was calculated, and the results are shown in Fig. 6. It can be seen that the CNR values of Al, H₂O, and the three contrast agents with lower Au concentrations are improved approximately 1.9-fold on average after the X-ray fluorescence correction. In addition, the CSCT reconstruction results are also influenced by serious artifacts, especially in the central Teflon area, as shown in Figs. 4(c) and 7. Unlike XFCT, in which only contrast agents can emit effective signals, all the materials in the phantom can produce Compton scattering signals in CSCT. For accurate CSCT reconstruction, projections of different materials with a higher SNR are required, and the strong statistical noise of the background material (i.e., Teflon) (while the fluorescence signal is sufficiently high for accurate XFCT reconstruction at the same photon flux) may cause these artifacts. Therefore, effective methods for CSCT reconstruction with a lower SNR should be developed to correct the artifacts in future studies. Owing to the influence of reconstruction artifacts, Al without X-ray fluorescence background correction can hardly be discriminated from the background (CNR～5, see Fig. 6), whereas it can be easily identified (CNR =11.56) after the X-ray fluorescence correction.

Fig. 7Full-screen View

(Color online) CSCT of the phantom without X-ray fluorescence background correction

Comparison with K-edge subtraction imaging

K-edge subtraction (KES) is an effective imaging modality for medical diagnosis that uses the K-absorption edge discontinuity of the contrast agent [41]. For this application, the Thomson-scattering light source has proven to be an excellent tool owing to its quasi-monochromaticity, energy tunability, and high brightness [42-46]. Previous studies using broad-spectrum X-ray tubes demonstrated that the CNR of KES is lower than that of XFCT when the contrast agent concentration is lower than 0.4% [47, 48]. However, this conclusion can be influenced by many factors, including the X-ray spectrum, imaging geometry, phantom type, and reconstruction method. To confirm our prediction, the KES results were analyzed using our imaging layout. Because the attenuation CT of the phantom was reconstructed at X-ray energies above and below the K-edge of Au in our simulation, the KES image of the phantom could be obtained at the same radiation dose level. $\begin{array}{l}\text{Δ}\mu \left(\overrightarrow{r}\right)=\mu \left(E_K^{+}\mathrm{,}\overrightarrow{r}\right)-\mu \left(E_K^{-}\mathrm{,}\overrightarrow{r}\right)\mathrm{,}\hfill \end{array}$ (12) where $E_K^{+}=E_0$ and $E_K^{-}=E_{\text{XRF}}$ are adopted. The KES image of the phantom is shown in Fig. 8 and the CNR is shown in Fig. 6. Compared with CSCT, the four contrast agents and H₂O exhibited quite different contrasts in KES, whereas these materials exhibited a similar contrast in CSCT because of their similar electron density values. Compared with XFCT, the CNR values of the four contrast agents in KES are reduced by 2.7, 2.4, 2.2, and 2.1 times for 0.5%, 1.0%, 2.0%, and 4.0% Au concentrations, respectively. Based on the available CNR data, further extrapolation and interpretation show that the limits of detection for Au concentration (CNR value = 5) are 0.27% and 0.4% for XFCT and KES, respectively, as shown in Fig. 9. However, when the Au concentration was further reduced to 0.19%, the contrast agent could be distinguished from the background in the KES image (CNR value ≤ -5; see the H₂O column in Fig. 8). The anomalous CNR superiority of KES over XFCT at ultralow Au concentrations is attributed to the large energy difference between $E_K^{-}$ and $E_K^{+}$, which causes an attenuation difference between H₂O and Teflon at the two X-ray energies.

Fig. 8Full-screen View

(Color online) KES of the phantom

Fig. 9Full-screen View

(Color online) Relation between CNR and Au concentration for XFCT and KES. The red dot-dashed lines describe the limit of detection (CNR=±5) according to the Rose criterion