The measurements were performed on a new clinical PCCT scanner (Siemens NAEOTOM Alpha, Siemens Healthineers, Forchheim, Germany) at a dose level of CTDI_vol=40 mGy (measured in a head dosimetry phantom with a diameter of 16 cm) with dose modulation turned off, a collimation width of 144 × 0.4mm, and a pitch of 0.55. A calibration phantom (Ø 150 mm) containing twelve Gammex 467 inserts (Gammex, Inc., Middleton, WI, USA) and an evaluation phantom (Ø 130 mm) containing six CIRS 062M inserts (CIRS, Inc., Norfolk, VA, USA), as listed in Table 1 , were scanned at a tube voltage of 120 kVp using a four energy bins mode which are combined to yield two basis images. The four energy thresholds are manufacturer-specified. Subpixels are read out in 2×2 groups for this mode (17). The clinical scanner software does not allow direct access to the reconstructed basis images used in Taasti et al. (16) for RSP estimation. We thus made use of VMIs which are easily accessible. VMIs were reconstructed with the Qr36f kernel at energies ranging from 40 keV to 190 keV in 10 keV increments, resulting in 512×512-pixel images with 0.38 × 0.38mm² pixel spacing and 1 mm slice thickness.

Following the approach adopted by (18), RED (ρ _e ), EAN (Z _eff ), and RSP were converted from PCCT data using the VMIs and the formalism proposed by Saito and Sagara (19) implemented in in-house software. RED, represented by ρ _e , was derived using results of calibrations with mean CT numbers of Gammex inserts based on each pair of high- and low-energy VMIs, denoted by HU_H and HU_L (20):

where a , b , and α were obtained by least-squares fitting of ρ _e of the Gammex inserts to a weighted subtraction of CT numbers from high- and low-energy VMIs.

In order to estimate EAN, denoted by Z _eff , the ratio of EAN of Gammex insert materials to that of water was fitted to the reduced CT number u L = H U L 1000 + 1 and RED (19):

where γ _L and γ ₀ were obtained by least-squares fitting. EAN was evaluated by Mayneord’s equation with an exponent m=3.3 (21, 22):

where ω _i represents the mass weight of the composing element with an atomic number of Z _i and an atomic mass of A _i . The elemental composition of the insert materials are listed in Table 1 .

Mean excitation energy I was calibrated against EAN for soft-tissue and bone-tissue inserts, respectively, with a separation point at EAN=8.8 (21):

where I _w=78 eV is the mean excitation energy of water recommended by ICRU report 90 (23, 24); the mean excitation energy of the studied inserts is defined as (21):

Two sets of C 1 soft / bone and C 0 soft / bone were obtained with least-squares fitting for soft-tissue and bone-tissue inserts, respectively.

In this study, we exploited various pairs of monoenergies from the reconstructed VMIs ranging from 40 keV to 190 keV in 10 keV increments. For VMIs at each low energy E _L, a sequence of high energies ranging from E _L+10 to 190keV was selected, which resulted in 105 different energy pairs. For each high- and low-energy pair, the measured CT numbers were converted to RED, EAN, and mean excitation energy using the obtained fit parameters. RSP of each voxel was calculated from the RED and mean excitation energy by the Bethe-Bloch formula (21):

where m _e is electron mass, c the speed of light, and β the velocity relative to the speed of light.

Mean values of the estimated RED, EAN, and RSP were calculated in a region of interest (ROI) for each insert. For each insert, which has a diameter of 29mm (Gammex) or 30mm (CIRS), a cylindrical ROI covering the central volume of the insert was used, with a margin of 5 pixels (i.e., 1.9mm) excluding voxels at the borders.

In order to evaluate the accuracy of RED, EAN, and RSP, the mean estimated values for the calibration and evaluation inserts were compared to their reference values. The reference RED values were calculated from the mass density ρ of the inserts provided by the manufacturers:

where the denominator is the weighted sum of Z/A for composing elements of water. Equation 3 was used to obtain reference values of EAN. The weighting factors ω _i , atomic numbers Z _i , atomic masses A _i , and mean excitation energies I _i in equations 3, 5, and 7 used in this work are tabulated in Table 1 .

For RSP, the reference values were experimentally assessed through PEAKFINDER (PTW, Freiburg, Germany) water column measurements in a carbon ion beam for all the inserts in this study except the Gammex insert Muscle, as reported in Hudobivnik et al. (25). For RSP accuracy calculations the Muscle insert was thus omitted.

For each high- and low-energy pair, root-mean-square errors (RMSE) of the estimated RED, EAN, and RSP were evaluated for the calibration and the evaluation phantom, respectively, using the following equations:

Here, value_est is the mean estimated value per voxel in a region of interest (ROI) for each insert, value_ref the reference value obtained either from manufacturer information (RED and EAN) or water column measurements (RSP); N is the number of inserts of each phantom.

Figure 1 shows calibration curves from fitting procedures corresponding to equations 1 ( Figure 1A ), 2 ( Figure 1B ), and 4 ( Figure 1C ) using measured CT numbers of Gammex inserts from VMIs at 60 and 180 keV. RED was well fitted with a coefficient of determination R ^{2 =} 0.9999. For fitting EAN as plotted in Figure 1B , R ² was 0.9934. The outlier is the LN-300 Lung insert with the largest residuals. The calibration for the conversion of EAN to mean excitation energy yielded R ^{2 =} 0.6974 for soft-tissue inserts and R ^{2 =} 0.9804 for bone-tissue inserts.

The fit parameters obtained are listed in Table 2 . Using the acquired fit parameters, the measured CT numbers were converted into RED ( Figure 2B ), EAN ( Figure 2A ), and mean excitation energy ( Figure 2C ) for both the calibration and evaluation phantoms. Figure 2 presents the maps of these quantities, along with RSP ( Figure 2D ) calculated from RED and mean excitation energy, at three different slices of the CIRS (evaluation) phantom.

The residuals of RED, EAN, and RSP evaluated for both calibration and evaluation inserts within the ROIs shown in red in Figure 2A are presented in Figure 3 . As shown in the left-most plot ( Figure 3A ), the estimation of RED demonstrated an accuracy within 0.8% except -1.6% and -4.8% for the two lung inserts. For EAN, an agreement with reference values within 2.8% can be observed for most of the inserts, whereas for Liver (CIRS), LN-300 Lung (Gammex), and BRN-SR2 Brain (Gammex) the errors exceeded 4.4% (see Figure 3B ). For RSP, the majority of the inserts had residuals below 1.0%; two inserts, Liver and Muscle (CIRS), exhibited values slightly above 1.0%. The LN-300 Lung insert had the highest error (-3.9%, see Figure 3C ).

The corresponding RMSEs averaged over phantom inserts are given in Table 3 . As the RSP value of the Gammex Muscle insert was not measured (25), the RMSE calculation of RSP for the Gammex (calibration) phantom excluded the Muscle insert. The two lung-mimicking inserts (LN-450 Lung and LN-300 Lung) showed relatively worse results as also observed in previous studies (18, 25, 26). Therefore, the RMSEs for the Gammex phantom were also calculated excluding lung inserts. Additionally, RMSEs of RSP acquired at a state-of-the-art dual-source DECT scanner (SOMATOM Definition FORCE, Siemens Healthineers, Forchheim, Germany) at 150 kVp with Sn filtration and 90 kVp in previous studies (11, 25) are presented in Table 3 .

Detailed information about measured CT numbers and reference and calculated RED, EAN, and RSP values of the studied inserts are listed in Table 4 .

Figure 4A shows mean and standard deviation of CT numbers at all the monoenergies for four exemplary evaluation (CIRS) inserts with the highest and the lowest density: Cortical Bone, Muscle, Adipose, and Trabecular Bone 200. The corresponding contrast-to-noise ratios (CNRs) are plotted in Figure 4B .

Regardless of the chosen low- and high-energy pair, the RMSE of RED mostly (102 out of 105 pairs) ranges from 0.55% to 0.56% for the CIRS (evaluation) phantom and from 1.51% to 1.54% for the Gammex (calibration) phantom. The few exceptions with a higher RMSE are listed in Table 5 .

Figure 5 presents RMSEs of EAN evaluated for both phantoms using different low- and high-energy pairs of VMIs. RMSE values are displayed on a background of different shades of colors corresponding to the value. Similar patterns can be observed for the two phantoms: energy pairs sharing the same low-energy VMIs had nearly the same RMSE, although a minimum could still be identified. Errors of EAN using 40 keV low-energy VMIs were remarkably high, around 90% and 99% for the Gammex and the CIRS phantom, respectively. The low energy 60 keV had the best EAN accuracy, 50 keV the second. The EAN accuracy decreased with an increasing low energy. The lowest error, 2.90% for the Gammex phantom and 2.32% for the CIRS phantom, obtained with 60 and 180 keV VMIs, is marked by a red box in both plots.

Like EAN, RMSEs of RSP are shown in Figure 6 , and similar trends were observed: For the same low energy, RMSE barely varied despite the different choice of high energy. The RSP accuracy using the optimal energy pair 60 and 180 keV, 1.27% for the Gammex phantom and 0.71% for the CIRS phantom, is marked by a red box in both plots.