Material separation in x-ray CT with energy resolved photon-counting detectors

Energy-dependent x-ray attenuation

In x-ray CT, the linear attenuation coefficient μ(E) is determined by Compton scattering and photoelectric absorption cross sections. As a result, μ(E) can be approximated by a weighted combination of two basis functions²⁴

where ρ is the material’s mass density, f_Compton(E) approximates the energy dependence of Compton scattering, and g_PE(E) approximates the energy dependence of photoelectric absorption. Both interactions have been extensively studied and tabulated data or empirical formulas are available.^{24, 25}

The constants a₁ and a₂ only depend on the composition of the material.²⁴ Together with ρ, they describe the dependence of μ(E) on f_Compton(E) and g_PE(E) and thus, the dependence of μ(E) on photon energy, E.

The photoelectric basis function has element-specific discontinuities at atomic energy levels. As a result, materials containing elements with atomic edges in the energy range of interest exhibit energy dependence different from those without edges (Fig. 1). In diagnostic imaging applications, the most significant discontinuities occur at the K-shell binding energies. For a material having a K-edge in the energy range of interest, Eq. 1 needs to be modified to

where the additional basis function g_PE-Kshell(E) describes the element-specific photoelectric absorption that results due to its K energy level. When multiple K-edges are present, each adds a unique corresponding basis function.

The contrast between the attenuation coefficients of two materials results from differences in density, composition, or both. For example, muscle and fat have very similar chemical compositions, while muscle has a higher density (1.06 g∕cm³ vs 0.9 g∕cm³). Thus, the observed linear attenuation coefficient of muscle is higher, due largely to its greater density. On the other hand, the contrast between water and iodine solution is due primarily to the differences in their compositions. Given their similar composition, the linear attenuation of muscle and fat at two different energies is expected to change by a similar ratio. That is, similarity in material composition results in similarity in energy dependence of the attenuation coefficient. However, this energy dependence will be different for materials with different compositions (e.g., water and iodine solution).

In single-energy CT, the linear attenuation coefficient is measured using a single-energy spectrum and the energy information is not retained in the detector. Thus, it is not possible to obtain information about the energy dependence of the attenuation coefficient. However, with spectral CT techniques, such as dual-energy or multi-threshold energy resolved detectors, the attenuation of the object can be measured at multiple energies, allowing the use of this energy dependence information to separate different materials based on their chemical compositions.

Material separation using energy resolved detectors

In the absence of K-edges, a₁ and a₂ can be uniquely determined by measuring the x-ray absorption at two different energies, e.g., using two different energy spectra.²⁷ The measurements for the different spectra contain information about the spectral dependence of the x-ray absorption in the object and can be used to extract information about the object’s composition. For example, in dual-energy CT, a dual-energy index⁴ (DEI) is calculated to quantify the degree of spectral differentiation of different materials provided by images acquired at two tube voltages

where x_low is the CT number measured (in Hounsfield units or HU) at the lower voltage and x_high is that measured at the higher voltage. However, with the inclusion of one or more K-edges in the energy range of interest, the attenuation is a combination of three or more basis components (Eq. 2), and dual-energy CT does not directly provide the requisite number of measurements to fully exploit the energy dependence of the attenuation coefficients (e.g., simultaneously separate soft tissue, bone, and iodine).

One solution is the use of energy resolved photon-counting detectors. For such a detector with n energy thresholds (and therefore n energy windows), this is equivalent to obtaining n nonoverlapping measurements simultaneously with a single x-ray exposure. When n≥3 and properly chosen energy windows are used, there is potentially enough information to allow separation of materials with K-edges (e.g., contrast agents) in the energy range of interest. For example, Refs. ^{13, 23} used an energy resolved detector to investigate iodine-based and gadolinium-based contrast agents and, iodine-based and gold nanoparticle contrast agents, respectively, by basis decomposition analysis.

Alternatively, in this study we applied classification methodology to distinguish soft-tissue-like, bone-like materials, and contrast agents in reconstructed images. The method can be viewed as an extension of the low-vs-high-kVp plots used in dual-energy CT (e.g., Fig. 3 in Ref. 28) to multi-energy CT.

Theoretically, in the absence of K-edge absorption, materials with similar compositions (e.g., various soft-tissue-like materials) are expected to exhibit a similar dependence on Compton and photoelectric components of their attenuation coefficients (i.e., similar a₁ and a₂), with differences resulting mainly from differences in density. Consider a graph where the axes are the attenuation coefficient values measured at two energies; for a voxel containing a particular material, the slope of the line from the origin to the point defined by the attenuation coefficient in the voxel obtained at two energies is given by a density-independent constant

Thus, plotting attenuation coefficients in this space results in clusters of points lying along lines that define their material type (i.e., one line for soft-tissue-like materials, one line for bone-like materials, etc.). The same argument can be applied for two energy spectra or two energy windows, as long as no K-edge is present in the energy range investigated.

Using measurements from two energies, two pieces of spectral information are obtained, based on which two types of materials (e.g., soft-tissue-like and bone-like) can be separated. However, materials with K-edges in the energy range of interest demonstrate a different energy dependence from those without, especially around their K-shell binding energies. This is illustrated in Fig. 2, which shows the average linear attenuation coefficients for several materials in different energy ranges. In both plots, the soft-tissue-like materials (i.e., water, polymethyl methacrylate (PMMA), polytetrafluoroethylene (PTFE), and polyoxymethylene (POM)) tend to cluster along a line that passes through the origin. In Fig. 2a, the iodine solution lies far from the line, as the two energy ranges were chosen such that one was above (33–38 keV) and one was below (27–33 keV) the iodine K-edge (33.2 keV). Also from Fig. 2a, we can make the following observations: (1) Bone cannot be separated from soft-tissue-like materials due to their similar energy dependence in the two energy windows plotted in that figure (Fig. 1) and (2) since both ranges are below the gadolinium K-edge (50.2 keV), the attenuation of gadolinium behaves very similar to non-contrast-agent materials (Fig. 1) and, as a result, the gadolinium solution is collinear with and cannot be separated from the other soft-tissue-like materials.

In Fig. 2b, if one range was chosen below (in this case 44–50 keV) the gadolinium K-edge (50.2 keV) and one above (in this case 50–80 keV), a similar significant separation can be observed for gadolinium as for iodine in Fig. 2a. Note that: (1) Bone is separated from soft-tissue-like materials, due to their different energy dependence in the two energy windows plotted (Fig. 1) and (2) unlike gadolinium in Fig. 2a, the attenuation coefficient of the iodine solution is not collinear with the attenuation coefficients of the soft-tissue-like materials, possibly because the iodine K-edge at 33.2 keV results in a different slope at this energy range, as indicated by Eq. 4.

The goal of this paper was to demonstrate, via simulations and physical experiments, that the above behavior can indeed be observed using a microCT system based on an energy resolved photon-counting detector.

Physical experiments

We constructed a table-top microCT system²⁹ based on an energy resolved photon-counting detector, designed and manufactured by Gamma Medica-Ideas (AS), Norway. The detector consisted of a cadmium telluride (CdTe) radiation sensor bonded to readout electronics implemented as a set of ASICs. It had one row of 512 pixels with an effective pixel pitch of 0.4 mm. The readout electronics had six adjustable energy thresholds for each pixel, each with an associated independent 16-bit counter. The useful operating energy range was 25–122 keV. In this study, the energy thresholds were set to 27, 29, 33, 40, 50, and 60 keV to straddle the iodine and gadolinium K-edges and to ensure comparable count levels in other energy windows.

The system used a SourceRay (Bohemia, NY) SB-120-350 x-ray generator system capable of up to 350 μA and 120 kVp with a 75 μm focal spot. The x-rays were collimated by a 1.6 mm wide lead slit to provide fan-beam x-ray irradiation. The x-ray tube had a beryllium exit window and was further filtered using a 0.79-mm thick aluminum filter to virtually eliminate x-rays with energies below the noise floor of the detector.

The detector was positioned 88.8 cm from the x-ray tube focal spot, while the phantom rotational stage was 20.3 cm from the tube focal spot (Figs. 3a, 4). Hence, magnification was 4.3 and the expected spatial resolution at isocenter was approximately 0.09 mm.

To investigate the imaging properties, we used a 2.54-cm diameter cylindrical phantom made from PMMA (or acrylic). The phantom (Fig. 3b) had five 4.8-mm diameter holes with centers at equiangular increments on a circle with a radius of 6.25 mm and centered on the axis of the cylinder. These holes could be filled with different solutions or solid rod inserts. We investigated materials including PMMA (or acrylic), PTFE (or Teflon^™), POM (or Delrin^™), polycarbonate, glycol-modified polyethylene terephthalate (PET), bone-equivalent plastic, water, and aqueous solutions with various concentrations of iodine-based (Omnipaque^™350) or gadolinium-based (Magnevist^™) contrast agents. The iodine and gadolinium K-edges are at 33.2 and 50.2 keV, respectively. The compositions of the materials imaged are listed in Table 1.

We used a 60 kV tube voltage and a 100 μA tube current. A low tube current was used in this study to minimize the effect of pulse pileup. Projection data were acquired over 360° with 1° increments. Bad pixels were identified as ones having either anomalously low or high count rates in the projection data. The pixel values in bad pixels were replaced with the average pixel values in the nearest properly operating neighboring pixels. We reconstructed the projections using a standard fan-beam filtered backprojection algorithm.

To exclude pulse pileup, the highest energy threshold was set to an energy equal to the tube voltage. Thus, data from five energy windows were available to perform material separation. A rectangular region of interest (ROI) was drawn over each material in the reconstructed images in each energy window to measure their respective linear attenuation coefficients.

To demonstrate the principles outlined in Sec. 2, we generated scatter plots of the linear attenuation coefficients of materials in various pairs of energy windows, similar to those presented in Fig. 2. Specifically, each axis of the plot represented the attenuation coefficient in one of the five energy windows. The goal was to investigate whether attenuation coefficients for the contrast agents fell along different lines than the other materials, indicating the capability to separate them in the object.

Simulation experiments

To provide a benchmark for performance of the physical experiments in the absence of degrading factors present in the real detector, we simulated an energy resolved photon-counting detector with an analytical CT projector (modified from Ref. 30). The detector specifications and system configurations were identical to those used in the physical experiments.

We measured the 60 kVp spectrum from the x-ray tube on the previously introduced microCT system with a commercial spectrometer (XR-100T-CdTe, Amptek Inc., Bedford, MA) and used it in the simulation. Quantum noise was modeled as a Poisson random variable. Other degrading effects including finite energy resolution, charge sharing, and pulse pileup were not modeled.

For comparison, dual-kVp acquisition with integrating detectors was also simulated using x-ray beams corresponding to tube voltages of 60 and 120 kV. Both the 60 and 120 kVp spectra were measured from the x-ray tube with the spectrometer. Preobject filtration identical to that in Ref. 10 was used to improve the spectral separation. The resulting postfiltering spectra are plotted in Fig. 5. The two spectra were scaled such that they had equal flux and the sum of their total flux equaled that of the 60 kVp spectrum used with the energy resolved photon-counting detector.

The simulated cylindrical phantom (Fig. 6) was 2.54 cm in diameter, with eight 4-mm diameter holes at equiangular increments on a circle with a radius of 6.35 mm and centered on the axis of the cylinder. Its dimension was comparable with the phantom used in the physical experiments except that there were more holes for inserts so that a greater number of materials could be imaged at the same time.

We investigated all of the materials used in the physical experiments, but with more combinations of volume concentrations of the two contrast agent solutions. Similarly, ROIs were drawn for each material and scatter plots were generated for the linear attenuation coefficients between various pairs of energy windows. For dual-kVp acquisition, the linear attenuation coefficients for the lower and higher kVp measurements were plotted (as in a standard low-vs-high-kVp plot used in dual-energy CT).

Simulation experiments

Figure 7 demonstrates the case for an energy resolved photon-counting detector when the choice of energy windows resulted in little or no separation of the materials. For display convenience, only the Omnipaque^™350 and Magnevist^™ solutions (no mixtures) are shown. The energy windows used in this plot were on the same side of the iodine and gadolinium K-edges. Thus, both the Compton and photoelectric dependences were continuous in the energy range considered and the approximation in Eq. 4 held. Note that almost all materials fell on a common line except for PTFE, which lay moderately above the line. This is possibly due to the different composition of PTFE, which contains a relatively large amount of fluorine compared to the other compounds, which contain mainly carbon, hydrogen, and oxygen.

However, with one energy window below and the other above the iodine K-edge (33.2 keV), mixtures containing iodine (e.g., Omnipaque^™350) significantly deviated from the soft tissue common line, as shown in Fig. 8. The concentrations of the two contrast agents in the mixture solutions are indicated by the lines that intersect at the point cluster (the legend is in the right bottom corner). For iodine-containing solutions, the distance to the noniodine-containing line (i.e., soft-tissue-like, bone-like, and gadolinium-only solutions) was approximately proportional to the concentration of iodine present in the sample, while variations in the gadolinium concentration resulted in almost parallel shifts of the clusters. Furthermore, consistent with theoretical prediction (Fig. 2a), this particular energy window combination could not separate bone from soft tissue.

An example of separating gadolinium-containing mixtures is shown in Fig. 9. The two energy windows were above and below the gadolinium K-edge (50.2 keV). The results are largely consistent with those from Fig. 8, with the concentration of gadolinium being proportional to the extent of deviation from the soft tissue common line. The results are also consistent with Fig. 2b, in which bone separated from soft-tissue-like materials in this energy window combination. These simulation experiments demonstrate the potential for using multiple energy windows to simultaneously separate soft tissue, bone, and materials with multiple K-edges.

One way to simultaneously visualize data from more than two (e.g., three) energy windows is to use 3-D “wall plots.” These show the projection of the data on three planes defined by three different energy window axes. With simultaneous data acquisition in multiple (e.g., five in this study) energy windows, it is theoretically feasible to generate 5-D plots including all the spectral information from the acquisitions. However, displaying and understanding scatter plots in a high dimensional space is difficult. The wall plots are a compromise, representing a 3-D scatter plot with projections in three 2-D planes. As suggested by the previous 2-D scatter plots, in some planes the projections were not separable, while in others, significant separation was observed due to the K-edge discontinuity, as shown in Fig. 10 (iodine) and Fig. 11 (gadolinium).

For comparison, a low-vs-high-kVp scatter plot (Fig. 12) was generated using the dual-kVp data. The separation among the contrast agents was smaller compared to Figs. 89. Also, to specify various target materials (e.g., iodine in Fig. 8 and gadolinium in Fig. 9), dual-kVp techniques require choosing suitable filtering materials and multiple acquisitions∕exposures. On the other hand, energy resolved photon-counting only requires the use of appropriate energy threshold settings, which does not require knowledge of the energy spectra and is relatively easy to implement using current detector technology.

For other non-contrast-agent materials, the additional information provided by the multiple energy windows also helped to separate them. To partially quantify the difference in separation, we used the separation angle θ between the vectors of attenuation coefficients. θ is defined as the angle between the vector of linear attenuations coefficients for two materials. For instance, for dual-kVp CT, θ is defined as in Fig. 13, where ${\vec{\mathbf{\mu }}}_{\text{Material}1}=\left({\mu }_{\text{Material}1}^{60\text{kVp}},{\mu }_{\text{Material}1}^{120\text{kVp}}\right)$ and ${\vec{\mathbf{\mu }}}_{\text{Material}2}=\left({\mu }_{\text{Material}2}^{60\text{kVp}},{\mu }_{\text{Material}2}^{120\text{kVp}}\right)$. Thus, $\text{cos}\theta ={\vec{\mathbf{\mu }}}_{\text{Material}1}\cdot {\vec{\mathbf{\mu }}}_{\text{Material}2}∕\left|{\vec{\mathbf{\mu }}}_{\text{Material}1}\right|\left|{\vec{\mathbf{\mu }}}_{\text{Material}2}\right|$. Note that the vectors involved can be of arbitrary dimension.

For acquisition from energy resolved photon-counting detectors, θ can be similarly computed from the dot product of the attenuation vector $\vec{\mathbf{\mu }}=\left({\mu }^{\text{Window}1},{\mu }^{\text{Window}2},\dots \right)$. Here, $\vec{\mathbf{\mu }}$ consists of measurements from all of the energy windows (five in this study) or any subset of them.

Since the attenuation vectors in this study were measured at multiple pixels in a ROI, an average vector was computed for each material and each energy window∕kVp over multiple pixels. The error bars in the separation angle calculations therefore result directly from noise (and artifacts, if they exist) in the reconstructed images.

The separation angle θ between other non-contrast-agent targets and the PMMA background was computed. For the dual-kVp approach, measurements with both kVps were used, while in the energy resolved photon-counting approach, measurements from all five windows were used. The result is plotted in Fig. 14. For non-contrast-agent materials, energy resolved photon-counting provided improved separation from the background compared to the dual-kVp approach.

For contrast agents, as described previously, energy resolved photon-counting provided an easy way to choose different energy window pairs to specify various target material with a single acquisition. Figure 15a plots the separation angles (with respect to PMMA background) for different concentrations of Omnipaque^™350 solutions. Here, for energy resolved photon-counting, only measurements from 29–33 to 33–40 keV were used (“K-edge-counting”) to separate iodine.

Figure 15b shows similar results for Magnevist^™ solutions: Measurements from 40–50 and 50–60 keV were used to separate gadolinium using energy resolved photon-counting. Consistent with Figs. 8912, as well as theoretical predictions, comparing attenuation coefficients below and above a K-edge for contrast agents provided improved separation with respect to the background and the separation angle θ monotonically increased with increasing contrast agent concentrations.

Physical experiments

We simultaneously imaged three cylindrical phantoms with the microCT system described in Sec. 3A. Two of the cylinders were filled with mixtures of different volume concentrations of the two contrast agents and the other cylinder contained solid inserts of bone-equivalent plastic and the soft-tissue-like materials PTFE, POM, polycarbonate, and water.

In Fig. 16, when one energy window was below the iodine K-edge (33.2 keV) and the other above it, the attenuation coefficient values from voxels in the soft-tissue-like materials (PMMA, PTFE, POM, polycarbonate, and water), bone-equivalent plastic, and gadolinium-only solutions clustered along a common line (dotted), consistent with theoretical calculations (e.g., see Fig. 2a). As the simulation experiments predicted, solutions with high iodine concentrations, both those with and without gadolinium, lay above the soft tissue common line, although they displayed less separation compared to the simulation results (Fig. 19, to be discussed in Sec. 5).

A similar phenomenon was observed (Fig. 17) with one energy window below and the other above the gadolinium K-edge (50.2 keV). There was strong separation for solutions with high gadolinium concentrations, which again confirms that the discontinuity in the attenuation coefficient curve causes this separation. Also, in contrast with iodine (Fig. 16), the result for gadolinium was in very good agreement with the previous simulation experiments (Fig. 19). Bone-equivalent plastic and iodine-only solutions fell below the soft tissue common line, also consistent with the theoretical calculations (Fig. 2b).

These data suggest that the contrast agent and bone could be separated from soft tissue and each other by scatter plots from multi-energy-window measurements and classification methods. We observed that the energy window combination that provided the best separation from soft-tissue-like materials was different for each material. For the two contrast agents used in this study, the largest separation was obtained with one window above and one window below the K-edge of the contrast agent.

Similar to Figs. 1415, the separation angles with respect to the PMMA background were computed. Figure 18 compares the results for the non-contrast-agent materials, while Fig. 19 compares the results for the nonmixture contrast agent solutions. Figure 18 demonstrates consistency between the simulations and physical experiments, although, as would be expected, due to degrading factors such as ring artifacts, finite energy resolution, charge sharing, and pulse pileup, the separation angles are smaller with the real detector than with the simulation for all materials except POM. Figure 19 further supports the previous observation that the degree of separation provided by the K-edge is proportional to the contrast agent concentration. While dual-kVp physical experiments were not available for comparison in this study, the consistency between the observed degree of angular separation between the simulations and physical experiments is indirect evidence of the superior material separation provided by the energy resolved photon-counting device.