Material decomposition with a prototype photon-counting detector CT system: expanding a stoichiometric dual-energy CT method via energy bin optimization and K-edge imaging

Objective. Computed tomography (CT) has advanced since its inception, with breakthroughs such as dual-energy CT (DECT), which extracts additional information by acquiring two sets of data at different energies. As high-flux photon-counting detectors (PCDs) become available, PCD-CT is also becoming a reality. PCD-CT can acquire multi-energy data sets in a single scan by spectrally binning the incident x-ray beam. With this, K-edge imaging becomes possible, allowing high atomic number (high-Z) contrast materials to be distinguished and quantified. In this study, we demonstrated that DECT methods can be converted to PCD-CT systems by extending the method of Bourque et al (2014). We optimized the energy bins of the PCD for this purpose and expanded the capabilities by employing K-edge subtraction imaging to separate a high-atomic number contrast material. Approach. The method decomposes materials into their effective atomic number (Z eff) and electron density relative to water (ρ e ). The model was calibrated and evaluated using tissue-equivalent materials from the RMI Gammex electron density phantom with known ρ e values and elemental compositions. Theoretical Z eff values were found for the appropriate energy ranges using the elemental composition of the materials. Z eff varied slightly with energy but was considered a systematic error. An ex vivo bovine tissue sample was decomposed to evaluate the model further and was injected with gold chloride to demonstrate the separation of a K-edge contrast agent. Main results. The mean root mean squared percent errors on the extracted Z eff and ρ e for PCD-CT were 0.76% and 0.72%, respectively and 1.77% and 1.98% for DECT. The tissue types in the ex vivo bovine tissue sample were also correctly identified after decomposition. Additionally, gold chloride was separated from the ex vivo tissue sample with K-edge imaging. Significance. PCD-CT offers the ability to employ DECT material decomposition methods, along with providing additional capabilities such as K-edge imaging.


Introduction
Computed Tomography (CT) is a widely used medical imaging technology, and since its inception in the early 1970s (Hounsfield 1973), it has evolved dramatically.One of the most significant breakthroughs in CT technology has been the development of dual-and multi-energy CT.Dual-energy CT (DECT) and multi-energy CT (MECT) refer to techniques in which two or more sets of CT data are acquired at different energy levels.This allows for the extraction of additional information, which is not possible with conventional, single-energy CT (SECT), which only consists of data acquired at one effective energy using an energy-integrating detector (EID).EIDs generate signal by first converting x-rays to light photons within a scintillating crystal, which are then integrated for the final output.
Dual-energy CT for material decomposition was first hypothesized (Alvarez and MacOvski 1976) and demonstrated (Rutherford et al 1976) in the 1970s, shortly after SECT was first implemented.Unfortunately, the use of DECT was limited clinically due to technological constraints.However, with the advancement of CT technology, DECT has become more widely available and is used in a wide range of clinical applications (Van Elmpt et al 2016, Goo andGoo 2017, Albrecht et al 2019).DECT also utilizes EIDs, but captures two data acquisitions at different effective energies.With the two sets of linear attenuation information given by the two sets of energy data, two subsequent sets of data can be decomposed or extracted, usually referred to as the basis functions in the decomposition formula.For example, as shown in the seminal DECT paper by Alvarez and MacOvski (1976), the photoelectric and Compton scattering components of the linear attenuation coefficient can be determined with DECT material decomposition.Though there are many decomposition algorithms and basis functions, one of the most well-studied sets of basis functions is the effective atomic number (Z eff ) and electron density relative to that of water (ρ e ), first demonstrated by Rutherford et al (1976).Since then, many algorithms have been developed to extract these values, including using monochromatic x-rays (Torikoshi et al 2003, Tsunoo et al 2004).The decomposition of data into Z eff and ρ e has a number of uses, including increased diagnostic capabilities by distinguishing between materials with similar attenuation curves.However, one of the most well-studied uses is for its use in improving material segmentation and identification for radiation therapy treatment planning (Bazalova et al 2008a, 2008b, Saito 2012, Landry et al 2013, Tsukihara et al 2013), especially in the fields of low dose rate brachytherapy (Landry et al 2010, 2011, Mashouf et al 2014) and proton therapy (Hünemohr et al 2013, 2014a, 2014b, Bourque et al 2014, Saito and Sagara 2017).Now, with the advent of high-flux photon-counting detector (PCD) technology (Fischer and Helmich 2000, Lindner et al 2001, Iwanczyk et al 2009) which offers energy windowing, the ability to distinguish the energy of incident x-rays and bin each one into a set of discrete energy ranges, MECT is becoming a clinical possibility as well.In fact, clinical PCD-CT scanners are currently available (Rajendran et al 2022), with several others close behind (Si-Mohamed et al 2023, da Silva et al 2019, Zhan et al 2023).PCD-CT typically offers up to eight energy bins (Willemink et al 2018), which significantly increases the options for material decomposition.Additionally, each energy range is captured simultaneously, mitigating any issues with co-registration between the different energy data sets.In DECT and MECT, co-registration refers to matching the multiple energy data sets to one another so that the various structures of the subject match up in the same location in each set.Any mechanism which shifts the subject so that it appears in a slightly different location in each data set will lead to decomposition errors.
PCD-CT systems offer benefits over the current clinical CT systems, even when excluding PCD-CT's energy discrimination abilities.Current clinical systems utilize EIDs, over which PCDs have several potential benefits.The first benefit is higher spatial resolution.EIDs first convert x-rays to light photons, which requires septa to be placed between individual pixels within the detector crystal to mitigate the spread of light photons outside of the pixel the x-ray was incident upon.PCDs are direct conversion detectors, requiring no septa within the detector crystal.The exclusion of the septa in PCDs allows for a higher possible inherent spatial resolution than EIDs, since EIDs suffer from increased dose inefficiency as the pixel size is reduced, as the the septa take up more area (Taguchi 2017, Willemink et al 2018, Leng et al 2019, Wang and Pelc 2021).PCDs also can reduce electronic noise because they function by counting individual photons rather than integrating the detected light signal from individual x-rays (Willemink et al 2018, Leng et al 2019, Flohr et al 2020).PCDs determine each count by measuring whether the charge detection peak is higher than a minimum threshold.Signal below the threshold is discarded as noise, while signal spikes above the threshold are counted as x-ray detections.EIDs integrate all signal, including electronic noise, though the noise has largely been mitigated at clinical acquisition parameters (Duan et al 2013).However, PCD-CT systems have still demonstrated lower image noise compared with stateof-the-art EID-CT systems (Gutjahr et al 2016, Hsieh et al 2021, Rajagopal et al 2021).PCDs' counting behavior also inherently allows for uniform energy weighting, as each x-ray is counted individually without regard to how far over the threshold the detection spike is, other than to bin the x-ray in the appropriate energy range (Taguchi 2017, Willemink et al 2018, Leng et al 2019, Flohr et al 2020).However, as EIDs integrate the signal generated from the detected x-rays, higher energy photons are weighted more highly as they produce a proportionally higher signal.The inherent uniform energy weighting of PCDs allows lower-energy x-rays to contribute equally to the detected signal, which offers better CT image contrast as most soft tissues have larger differences in attenuation at lower energies.Additionally, energy weighting can be applied to PCD data to further improve contrast (Giersch et al 2004, Shikhaliev 2009b, Nguyen et al 2021).
However, PCDs do offer challenges as well; the two main drawbacks being charge-sharing and pulse pileup.Charge-sharing occurs when an x-ray is incident near the border of two or more pixels, and the resulting charge cloud is partially collected by multiple pixels.Thus, the full charge is split, and multiple pixels record a count at a lower energy than the pixel with the incident x-ray.This degrades both the spatial and energy resolution.Pulse pileup occurs when two x-rays are nearly simultaneously incident on a pixel.The resulting signal pulses are summed up and recorded as a single event at a higher energy than either of the two incoming x-rays alone.Pulse pileup degrades the spatial resolution as well as increases the noise within the image (Willemink et al 2018).A trade-off exists in the physical detector itself: smaller pixels will lead to more charge-sharing (Iniewski et al 2007), but less pileup.Thus, a balance must be struck between the two.Methods are being developed to compensate for both effects (Taguchi et al 2022) including coincidence counters which are implemented within the detector architecture to monitor and compensate for charge-sharing events (Iniewski et al 2019a, Taguchi andHsieh 2023).
Much focus has also been devoted to material extraction of high-atomic number (high-Z) contrast agents via K-edge imaging (Roessl and Proksa 2007, Schlomka et al 2008, Cormode et al 2017, Dunning et al 2020, Richtsmeier et al 2020).K-edge imaging can refer to methods that utilize the K-edge discontinuities in the attenuation curves of high-Z materials to extract a third (or higher) material map (Roessl and Proksa 2007), on top of the two material maps possible with DECT, or that utilize K-edge subtraction to separate high-Z materials from conventional CT images, which was first demonstrated using monoenergetic x-ray beams (Dilmanian et al 1997, Zhong et al 1997, Elleaume et al 2002) and later using photon-counting detectors and polyenergetic beams (Dunning et al 2020, Richtsmeier et al 2020).Interest has also been shown in using PCD-CT for material decomposition in a variety of fashions.For example, there are methods that extend DECT methods of Z eff and ρ e decomposition, such as those for predicting properties like tissue stopping power ratios for particle therapies (Jacobsen et al 2020, Kruis 2022).
In this study, we demonstrate that DECT methods can be adapted to PCD-CT systems by applying the method outlined by (Bourque et al 2014) on a bench-top PCD-CT setup, with the added benefit of acquiring the data with a single scan.We optimized the ideal energy ranges with which to perform the decomposition by adjusting the energy bins of the PCD through an extensive set of threshold options.We also expanded the material decomposition capabilities of this method by employing K-edge subtraction imaging to separate a highatomic number contrast material, i.e., gold.From the perspective of a dual-energy measurement, unless the peak energy of the lower energy scan could be set to the K-edge energy of the contrast material, at least three energy bins would be needed to produce Z eff and ρ e images, along with a separate K-edge contrast image.Gold is currently under investigation as a CT contrast agent in the form of gold nanoparticles (Alivov et al 2014, Cormode et al 2017, Mahan and Doiron 2018).The decomposition method was calibrated and evaluated using a number of tissue equivalent materials with known Z eff and ρ e as well as tested on a ex vivo bovine tissue sample which were compared to tabulated tissue values.In addition, the adapted method was compared against the same method used as it was originally intended, with DECT data.

Bench-top photon-counting CT system
The bench-top PCD-CT system used in this work to evaluate the material decomposition method described above consisted of an x-ray tube (MXR 160/22, Comet Technologies, San Jose, CA, USA) and a PCD (Redlen Technologies, Saanichton, BC, Canada) with a rotating sample stage (Newport Corporation, Irving, CA, USA) mounted between the two (figure 1).The source to isocenter distance was set to 322 mm with a source to detector distance of 578 mm.The PCD has two 8 × 95 mm 2 modules which together create an 8 × 190 mm 2 detector active area.This yielded a 106 mm field of view with 4.5 mm Z-coverage at isocenter.The detector crystal is 2-mm thick cadmium zinc telluride (CZT) which is connected to an Application Specific Integrated Circuit (ASIC).The detector offers a pixel pitch of 330 μm and six energy thresholds which can be set by the user.The six thresholds define five specific energy ranges into which incident x-rays are binned.A sixth bin is in place to record for events over the highest threshold setting, and a seventh bin counts every event above the noise threshold of 24 keV.The PCD operates at up to 650 Mcps mm −2 without polarization and is linear up 6 Mcps mm −2 (deviation from output count rate vs input count rate linearity is less than 1%) (Iniewski et al 2019a(Iniewski et al , 2019b)).The maximum count rate in this study was 12.6Mcps mm −2 with a deviation from linearity of less than 2%, causing negligible errors in our estimates (Iniewski 2016, Richtsmeier et al 2020, 2022, Rodesch et al 2023).The ASIC includes a coincidence circuit, which accounts for some amount of charge-sharing by monitoring pixels immediately adjacent to incident pixel after an x-ray detection.If any charge less than that collected in the incident pixel is measured in the adjacent pixels during a specified time window, those counts in both the incident and adjacent pixel are considered to be charge-shared.The ASIC can operate in two modes; one mode discards all charge-shared counts, and the other mode includes all counts, regardless of charge sharing.We utilized the total counts mode as there was little difference between the two modes at the low flux rates used in this study.Other non-idealities within the PCD system do still exist, such as differences in ASIC dead times and inconsistencies in gain between pixels and variations in the CZT crystal itself.Calibration was conducted on the detector to minimize the inconsistencies in pixel dead time to less than 100 ps.Energy calibration was also performed on a per pixel basis using either an Americium-241 or a Cobalt-57 source.Instabilities between pixels in the time domain were tested for using both 1-second and 1-minute stability tests.If a detector failed any of the above tests it was not utilized in this study.

CT acquisitions
For all CT acquisitions, the beam was collimated to 172.5 mm horizontal and 17.25 mm vertical coverage at isocenter.The sample stage was rotated at 2 deg/s for a total of 180 s over which 720 projection images were acquired for 0.25 s each.
PCD-CT data sets were acquired at a tube voltage of 120 kVp, with a tube current of 2 mA, the small focal spot (∼0.4 mm), and 6 mm Al filtration in order to replicate the filtration of a clinical head protocol.
DECT data sets were acquired at 80 kVp and 120 kVp.DECT scans were still acquired with the PCD as our lab does not have access to an EID capable of CT acquisitions.The 120 kVp acquisitions were acquired at the same time as the PCD-CT scans (at 2 mA with 6 mm Al), but only utilized the bin which counts every x-ray above 24 keV.The 80 kVp scans were obtained with a tube current of 2.5 mA, the small focal spot, and 3 mm Al filtration.The parameters for the 80 kVp scans were chosen to replicate the same dose as the 120 kVp scans, calculated in SpekPy (Poludniowski et al 2021).For the 80 kVp scans, the sixth energy bin was used, containing energies from 24-80 keV.

Image reconstruction and normalization
The projection images from each CT acquisition, along with flat field and water phantom projection first had dead pixels corrected for using nearest neighborʼs interpolation.The flat field and water phantom projection were acquired with the same parameters as the CT projections.The projection images were then corrected for air in each bin using the equation: in which p is the corrected sinogram image, I is the number of counts in the projection image, and I 0 is the number of counts in the flat field scan.n refers to the energy bin number; bins 1-5 are the individual energy bins and bin 6 is the total counts bin.Additionally, an in-house ring artifact correction method was applied to the sinograms using the water phantom scan (Richtsmeier et al 2022).The water phantom was chosen to cover the full field-of-view (FOV) of the detector, leaving no air showing at the edges of the detector after the scan.The water phantom data was corrected for air using equation (1).Then, each row of the corrected water phantom data was filtered with a median filter to smooth out any large spikes in the data, as the water phantom should appear a smooth curve when a single row is plotted.Each median-smoothed row was fit with an 8-degree polynomial and the ratio between the polynomial fit and the air-corrected data was found.This created a gainmatrix for each pixel which was multiplied to each projection image to correct for small pixel response variations.Finally, all of the gain-corrected projections were summed together to create a single summed projection.Each row of the summed projection was smoothed with a median filter and linearly interpolated to find the 'ideal' summed projection.The difference between the 'ideal' and the original summed projection was found and any pixels still displaying large deviations were corrected for using nearest-neighbor interpolation.
CT images for each energy bin were reconstructed separately using the Feldkamp-Davis-Kress (FDK) algorithm (Feldkamp et al 1984) with a Hamming filter, as implemented in the TIGRE package (Biguri et al 2016) in Python.Images were reconstructed with 24 slices, each containing 512 × 512 pixels.The reconstructed image slices were 105 × 105 mm 2 with a slice thickness of 0.184 mm.The images were then converted to CT number (HU) using the equation: where μ is the un-normalized image signal and μ w is the mean signal within the water vials.For the ex vivo acquisitions, the water vial signal from the calibration insert scans was used for μ w .

Effective atomic number
For mixtures composed of multiple elements, an effective atomic number (Z eff ) can be defined which in theory takes into account the atomic numbers of each of the consituent elements.Here, Z eff was calculated based on the method detailed in Bourque et al (2014).Briefly, for elements, we can define CT number in the following manner: where, f (Z) is the electronic cross section relative to water; therefore, CT number is a function of ρ e and σ e .We seek to define Z eff for mixtures based on the electronic cross section so that CT data can be used to tissue characterization (Bourque et al 2014).Thus, we parameterize the electronic cross section ( ˆ( ) ) such that it is a bijective function over the Z domain that covers human tissues.For a given photon spectrum, the parametric electronic cross section at Z eff would be the electronic cross section for the mixture averaged over the spectrum (σ e,mix ).This gives s s = - Thus, Z eff can be defined for a material at a specific energy.

Material decomposition methodology
The material decomposition method used in this study is an adaptation of the DECT stoichiometric calibration method detailed by Bourque et al (2014), which is based on the method by Schneider et al (1996).In the method laid out in Bourque et al (2014), a DECT scan of a set of calibration materials with known Z eff and ρ e values is performed.The average CT numbers in both of the energy scans are calculated for each of the various calibration materials and paired with their known Z eff and ρ e values.A pair of calibration coefficient vectors are then estimated via least-squares which define the relationship between the CT numbers and the physical material properties, Z eff and ρ e .In subsequent scans under the same conditions, the calibration coefficients can then be used to determine Z eff and ρ e for unknown materials.In this study, the method remains the same, but the information from two PCD energy bins collected from a single CT acquisition are used in place of two different energy scans from a DECT acquisition.The following is a short explanation of the method as detailed in Bourque et al (2014).
CT numbers in both the low and high data sets can be converted to reduced CT numbers, u r , in which r is defined as either L or H for either the low and high energy ranges, respectively.The reduced CT number is calculated as follows and is the ratio of the average linear attenuation coefficient of the material (μ r ) to that of water (μ w,r ) within that energy range: Jackson and Hawkes (1981) stated that some have argued, to some degree of accuracy, that the linear attenuation coefficient can be parameterized into the photoelectric, Compton scattering, and Rayleigh scattering cross sections: e e e ph ray KN 3.62 1.86 where n e is the electron density, σ e is the electronic cross section, the k values are coefficients associated with the three cross sections, and the values z and ẑ are given by the power law additivity rule (Mayneord 1937, Spiers 1946): = and ẑ z 1.86 = .γ i are the fractional weights of the elements in the mixture.Dividing equation ( 7) by μ w leads to the reduced CT number or relative linear attenuation coefficient (u), and n e becomes the relative electron density, ρ e .However, White (1977) and Jackson and Hawkes (1981) state that such a model is not sufficient as the power law additivity rule varies with photon energy and with samples containing elements with a large range of atomic numbers.Jackson and Hawkes (1981) also state that empirical formula have been explored by polynomial or exponential functions that are valid over a specific range of Z and energy.Bourque et al (2014) develop such a model.They show that the electronic cross section (σ e ) can be parameterized for any Z: a m are coefficients which can be estimated through a least-squares fit of the electronic cross section from the NIST database (Berger et al 1998).The behavior of the electronic cross section with respect to Z can be seen in (Bourque et al 2014).Thus, for u, a relationship can be defined for one of the two energy scans: Here, b m,r are the first sets of coefficients which are estimated through calibration with a set of known materials.One set of coefficients corresponds to the low energy data set and the other corresponds to the high energy data set.It should be noted that the polynomial expansion in equation (10) (Kirby et al 2003, Midgley 2004) offers more degrees of freedom compared to equation (7).Additionally, for integer values of Z eff in equation (10), corresponding to elements, the b m values would be equal to the a m values from equation (9) divided by the linear attenuation coefficient for water at the effective energy of the spectra.However, when looking at mixtures, the b m values for said mixtures differ from the elemental a m values.
Next, we define a variable (Γ) called the dual-energy index (Johnson 2011), which incorporates both CT numbers for a specific material or voxel, which is independent of ρ e .

( )
Γ is independent of ρ e , and we can assume Z eff is independent of the photon spectrum over the range of energies used (Bourque et al 2014).Thus, we can say Γ ≡ Γ(Z) within the domain

Î
, where both Γ and Z are bijective.Z min and Z max refer to the lowest effective atomic numbers of the calibration materials.So, we can then define Z eff as a polynomial sum of the Γ of order K.
Here, c k is the second set of coefficients which are found through a least squares solution given known Z eff and Γ values.
Once the b m,r and c k coefficients have been found, Γ can be defined per voxel for subsequent scans and Z eff can be estimated using equation (12) and ρ e found using: The only other modification made to the method from Bourque et al (2014) occurs if the CT number of a voxel in either of the bins is below −900 HU, that voxel is automatically assigned to air with a Z eff value of 7.66 and ρ e value of 0.0011.Excluding air, the material with the lowest CT number is the LN-300 Lung insert, which averages approximately -710 HU over multiple imaging modalities, with the lowest CT number being −805 HU (Mehta et al 2023).Alternatively, they demonstrated air had a maximum CT number of −960 HU.A threshold of −900 HU straddles the difference between the two, erring towards considering values as not air.Our system does suffer from ring artifacts which occur in regions of air as well, so utilizing a slightly higher threshold mitigates those artifacts from propagating into the Z eff and ρ e maps.

Tissue equivalent materials
Twelve inserts from an RMI electron density phantom (Gammex, Middleton, WI, USA) were used for the stoichiometric calibration for both PCD-CT and DECT, as well as for method evaluation.The insert information can be seen in table 1 including their ρ e and elemental compositions.The inserts' relative electron densities were provided with the phantom, and the elemental compositions were requested and obtained from the manufacturer.However, the phantom model was older than the manufacturer's records so a number of relative electron densities of the materials from the requested data did not match with the original data sheet provided with the phantom.The elemental composition of these materials was then found in other publications.Specifically, the elemental compositions of LN-450 Lung and IB Inner Bone were obtained from Higuchi and Haga (2023) and the two CaCO 3 cylinders were obtained from Landry et al (2013).The materials covered most tissue types, with soft-tissue like materials (Z eff < 8) and materials mimicking various bone densities (Z eff > 10).The tissue-equivalent inserts were only calibrated for ρ e , while Z eff was calculated based on the parametrization of the electronic cross section (equation ( 5)) as Z eff varies with photon energy.Z min and Z max , which define the Z domain over which decomposition is valid were [6.338, 13.582] for 13.562] for DECT.
In practice, the electronic cross sections for each of the elements that appear in the tissue-equivalent material compositions were obtained from the XCOM photon cross section database (Berger et al 1998).Beam spectra were generated in SpekPy (Poludniowski et al 2021), previously validated for our x-ray source (Clements et al 2022).Separately, each element's electronic cross section was averaged over the photon spectrum or energy bin spectrum in question, weighted by the relative number of photons of each energy.The effective energy of each energy bin was also calculated by taking the mean energy weighted by the photon spectrum within the bin.Then, a composite electronic cross section over all photon energies was then defined for each tissue-equivalent insert by calculating a weighted sum of the electronic cross sections for the elements that made up the insert, with the percent compositions as the weights for each element (table 1), before taking the weighted average over the same photon spectrum.The theoretical Z eff of each insert was then estimated based on a cubic spline interpolation between the energy weighted electronic cross sections of each of the elements and their atomic number.As the theoretical Z eff varies with energy, the average of the two Z eff values from the two energy ranges was used in the calibration and evaluation of the model, and the reason for the variation between PCD-CT and DECT.Bourque et al (2014) showed that theoretical Z eff varies weakly with the photon spectrum and can be considered as a systematic error.Table 1 shows the variation between the theoretical Z eff values as the distance from the average (the value following the± ).Additionally, given the small variation in Z eff values, we also consider variation to be a systematic error and that Z eff is independent over the energy range of 24-120 keV.

Energy bin optimization
The PCD of the bench-top system offers six energy thresholds, which enabled a large number of potential energy range options for the low and high energy bins to be used in the decomposition.In order to find the energy ranges that offered the best results, fourteen sets of energy thresholds were chosen to encompass all possible energy ranges of 10 keV and larger between 35 and 120 keV, as seen in table 2. The minimum width of an energy range was 10 keV, as the energy resolution of the detector is ∼8 keV.The tissue-equivalent materials were broken into three groups, as shown in figures 2(a)-(c), as the FOV of the system could not accommodate all of the cylinders in a single scan.The model was still optimized on all twelve materials and water.This resulted in 42 total scans, three at each set of energy thresholds.To minimize the number of CT acquisitions necessary, the summation of counts in adjacent bins was used to obtain larger energy ranges.Each of the scans was reconstructed and normalized according to section 2.3, with the water normalization value being taken as the mean value within the water vial of each scan.An exhaustive search for best set of energy ranges was conducted by pairing each energy range with every other possible range; the starting energy of the higher energy range was restricted to be at least 5 keV above the ending energy of the lower range.For each pair, the CT volume was split in half and the model calibrated on half of the slices and evaluated on the other half.Additionally, at each pair of energy range, the model was tested at K and M values ranging between 2 and 7. Some combinations of energy ranges and K and M values did not converge on values of c and b, or produced negative mean values when evaluated, both were discarded.The performance of each pair of energy ranges, at each value of K and M, was evaluated by computing the average root mean squared percent error (RMSPE) of the tissue-equivalent materials excluding LN-300 Lung for both Z eff and ρ e .The RMSPE of Z eff and ρ e for LN-300 Lung was calculated separately, as due to its in-homogeneity it appears as an outlier, and skews the results of the average RMSPE.The equation for RMSPE can be seen below: where ŷi is the measured value of either Z eff or ρ e , y i is the corresponding theoretical value, and n is the number of data points in the Z eff or ρ e sample.The data points for each tissue-equivalent materials were calculated as the mean values within a ∼22 mm diameter ROI within each of the slices of the CT scan.A weighted sum of the four values was computed with weights of 0.4, 0.35, 0.1, and 0.15 for the average Z eff RMSPE, the average ρ e RMSPE, the Z eff RMSPE of LN-300 Lung, and the ρ e RMSPE of LN-300 Lung, respectively.The weights were chosen in order to minimize each of the four RMSPE values while limiting one of the four from growing too large.
For DECT, all values of K and M between 2 and 7 were used to calibrate and evaluate the model.The values which minimized the sum of the average RMSPE for Z eff and ρ e were used for further analysis.2.8.Model performance analysis After optimization, the tissue-equivalent cylinders were scanned at the energy thresholds dictated by the results of the optimization.A water phantom, as well as the inserts (figures 2(d)-(g)) were scanned to acquire a data set to be used to calibrate the model.The inserts were again separated into three groups as they were unable to all fit on the rotation stage simultaneously.The entire group of twelve inserts, plus water, was used to calibrate the model.After model calibration, the calibration coefficients were tested by predicting elemental linear attenuation coefficients.Additionally, the CT number predicted by the model for each of the tissue equivalent materials was compared to the average CT number found experimentally.The residual HU was found by subtracting the experimental CT number from the predicted value.An additional set of the same scans was taken to obtain a data set to be used to evaluate the model, using all of the inserts and the water phantom.The values of K and M used in the model were also defined during the optimization.Once the Z eff and ρ e maps of the evaluation data set were extracted, the accuracy of the Z eff and ρ e values for each material was evaluated and compared to the calculated values.Mean Z eff and ρ e values were calculated from a 21 mm ROI within each cylinder over all slices without significant ring artifacts, and the RMSPE of each cylinder was evaluated using the mean value within the cylinder's ROI from each slice as data points, referred to as the ROI points.To compute the absolute error relative to water, the difference between the theoretical values and the ROI points was divided by the corresponding Z eff or ρ e of water.The mean and standard deviation of the results were then computed.The total mean of Z eff and ρ e for each material was also evaluated as the mean of all voxels within the ROIs from each of the slices, with the variation of Z eff and ρ e computed as the standard deviation of the voxels defined by the ROIs in the same slices.Additionally, to evaluate the noise between PCD-CT and DECT, the standard deviation of the voxels within the water insert were evaluated in the low and high energy CT images and the resulting Z eff and ρ e maps for each method.The model was also tested by scanning a 10 cm high-density polyethylene phantom containing an approximately 10 mm diameter piece of pure aluminium to extract the atomic number and relative electron density.

Ex vivo bovine tissue samples
An ex vivo bovine tissue sample and was stored at 4 °C in a refrigerator until it was scanned with PCD-CT and DECT.The same acquisition parameters as defined by the results of the optimization of the tissue-equivalent inserts were used in both cases.The time between the samples removal from the refrigerator and the end of scanning was approximately two hours.The bovine sample consisted of fat, muscle, and bone and was approximately 75 mm in height and 45 and 60 mm in length and width, respectively (figure 2(i)).One acquisition was acquired with the bovine sample alone for both PCD-CT and DECT and another acquisition was conducted for PCD-CT after the same sample was injected with 0.3 ml of 50 mg/ml AuCl 3 (GG3CS-25.4-100Lot AUY03-7077, Nanopartz Inc, Loveland, CO) into the muscle and fat portion to mimic a contrast-enhanced CT scan.

Ex vivo tissue data analysis
The Z eff and ρ e maps for the short rib scans were extracted after the stoichiometric calibration was computed on the values from all of the tissue-equivalent material cylinders.The Z eff and ρ e values of adipose, muscle, spongiosa (or inner bone), and cortical bone were compared to reference values.The reference material values were calculated from the mass densities and elemental compositions of human tissues referenced from ICRP50 as tabulated in (Bourque et al 2014).The measured ex vivo tissue sample's Z eff and ρ e values found after material decomposition were calculated by placing ROIs in the conventional CT image within areas containing only the specific material.The mean and standard deviation of the voxels defined by the ROIs in all slices without significant ring artifacts were then calculated and compared to the reference value.

K-edge subtraction
Gold K-edge images were reconstructed using the K-edge decomposition method described by (Zhang et al 2020).The K-edge subtraction algorithm used the two energy ranges of the sinogram surrounding the K-edge of gold (80.7 keV), as one of the energy thresholds was placed placed at 81keV.Once reconstructed, the signal of a reference 50 mg/ml AuCl 3 gold sample and water were used to normalize K-edge image signal linearly between 0 and 50 mg/ml of Au.As the injected gold in the short rib was 50 mg/ml, signal would not exceed that level.

Energy bin optimization
We found that the energy ranges of 35-50 keV and 65-95 keV gave the best balance of minimizing the average RMSPE of Z eff and ρ e for all materials (excluding LN-300 Lung) and the RMSPE of LN-300 Lung itself.K and M values of 5 also gave us the best results for PCD-CT.For DECT, we obtained the best results given K and M values of 2 and 3, respectively.The RMSPE values for the energy ranges of 35-50 keV and 65-95 keV can be seen in table 3 with all combinations of K and M between 2 and 7.The full list of results for PCD-CT and DECT can be found in the Supplementary Materials.The effective energy for the 80 kVp spectrum with 3 mm Al was 44.9 keV, for the 120 kVp spectrum with 6 mm Al was 58.9 keV, for the 35-50 keV bin with 6 mm Al was 42.7 keV, and for the 65-95 keV bin with 6 mm Al was 76.4 keV.

Calibration verification
Given the best energy ranges of 35-50 keV and 65-95 keV, the energy thresholds for the calibration and evaluation scans were set to 30, 35, 50, 65, 81, and

Method evaluation with known tissue-equivalent materials
The results of the decomposition of Z eff and ρ e from the 12 tissue equivalent cylinders and water can be seen in the resulting images in figure 4 for PCD-CT and figure 5 for DECT.Both figures show a conventional CT image with the CB3%-50%, AP6 Adipose, LN-300 Lung, Solid Water, and BR-12 Breast and the resulting Z eff and ρ e maps.Table 5 shows the RMSPE of each of the cylinders and table 6 shows the error of on the materials relative to water.Figure 6 shows the voxel-wise mean and standard deviation of both Z eff and ρ e each of the various tissueequivalent materials plotted as ρ e versus Z eff , for both PCD-CT and DECT.
Table 7 demonstrates the noise in the low and high energy CT images for PCD-CT and DECT as well as the noise in the Z eff and ρ e maps for each method.

Ex vivo tissue sample
The decomposition images of the ex vivo tissue sample can be seen in figure 7 for PCD-CT and figure 8 for DECT.The conventional CT image and the Z eff and ρ e maps are presented in both figures.The mean and standard deviation of Z eff and ρ e within ROIs of cortical bone, adipose, muscle, and inner bone (spongiosa) are shown for both PCD-CT and DECT in table 8.

Discussion
The ability for the decomposition method to accurately segment materials based on Z eff and ρ e was successfully demonstrated with a set of calibration tissue-equivalent materials along with an ex vivo bovine tissue sample.The   mean RMSPE values for Z eff and ρ e for PCD-CT were 0.76% and 0.72%, respectively, and 1.77% and 1.98% for DECT.One potential explanation for the better performance of the PCD-CT model compared to DECT model can be seen in Supplementary figure 1.Over the entire Z eff range that the PCD-CT or DECT models are calibrated for, the difference in the values for the low and high energy range is larger for PCD-CT, meaning material analysis is easier.For both data sets, however, a number of shortcomings exist that could lead to the the decomposed data not matching the theoretical values.Due to the fact that the phantom was an older model, the manufacturer did not have the composition data for the specific batch of tissue-equivalent inserts.The density and ρ e specifications were included with the phantom, so the predicted ρ e was accurate.The manufacturer did provide an elemental composition table for the current makeup of the various inserts with their density and ρ e .For the materials whose ρ e did not quite match with the manufacturer-provided table, the closest data was found in the literature.As the calculation for the theoretical Z eff values depends significantly on the elemental makeup of the inserts, this introduced uncertainty.However, as seen in Supplementary figure 4, the experimental CT number for the various tissue-equivalent materials was generally within 10 HU of the CT number predicted by the model for all energy bins or spectra, excluding some outliers.This indicates that the composition of the materials in table 1 is close to the real composition for those that were found in the literature.The material that showed the highest consistent residual CT number across both energy bins and both DECT spectra was, in fact, water.Given that the composition of water is well defined, and that the materials whose compositions were found in the literature did not deviate over all energy bins and spectra, the deviation was likely not due entirely from erroneous compositions.In fact, LN-450 Lung, whose density deviated the most from the density found in the literature, still demonstrated similar accuracy when compared with other inserts for the PCD-CT data.
Next, there was also uncertainty in the beam weighting of the electronic cross sections in the calculation of the predicted Z eff values.The spectra used in the calculations were the x-ray tube spectra, but as we were using a PCD, which suffers from effects such as charge-sharing and pulse pileup (Willemink et al 2018), in addition to having a non-finite energy resolution, the number of x-rays in a given bin may not match the actual number of x-rays produced by the x-ray tube in that energy range.For the PCD-CT data, we also restricted the energy ranges to contain energies 35keV and above, which could impact the optimization negatively.This is due to the fact that many soft tissues have larger differences in attenuation in the low energy range, and thus lead to a more accurate decomposition.However, our PCD does suffer from large differences in pixel response at energies below 30 keV, so in order to reduce image noise, the lowest energy threshold in the optimization was restricted to 35 keV.Error could still have been introduced in the DECT data, as it encompassed all energies from 24 keV (the electronic noise threshold) to the peak beam energy.This could be evident in the increased ring artifacts seen in the DECT images (figures 5 and 8) when compared with the PCD-CT images (figures 4 and 7).This introduced the potential for error in CT number for the DECT data, as seen with the higher residual CT numbers for the two DECT spectra (Supplementary figure 4), and increased the noise within the cylinders when compared with the PCD-CT data.
These properties and shortcomings of the PCD were also demonstrated in the prediction of the linear attenuation coefficients of carbon and oxygen (figure 3) at the effective beam energies for both PCD-CT bins and both DECT spectra.Though the model successfully predicted the linear attenuation coefficients of carbon and oxygen (figure 3), closely for PCD-CT, the predictions of the linear attenuation coefficients at the effective beam energies of the 80 kVp beam and the 120 kVp beam were further from the true values.This is likely due to the inclusion of the detections in the PCD below 30 keV, which include more spurious counts and are subject to more noise than other energies.The effective energies were calculated using the beam spectra calculated in Spekpy, which did not take into account the spectra the detector records, which were distorted by chargesharing and pileup.The PCD-CT linear attenuation values were more accurate as the two energy bins suffer less from noise introduced by the detector in the energy range below 30 keV and thus the effective beam energies were more accurate.The performance of the model was also demonstrated in the decomposition of the aluminum (Supplementary figure 2, 3), which was not a calibration material.Although the Z value was reconstructed accurately, beam hardening was seen in the CT images and especially in the Z eff maps.ρ e was underestimated for both PCD-CT and DECT, though it was outside of the calibration values for ρ e , which contributed to the effect.
A potential shortcoming of the bin optimization for PCD-CT was the inclusion of the water vial in the insert scans.As these were cone-beam acquisitions, the water vial was used to normalize the reconstructions to CT number.The placement of the water vial in the same scan as the inserts could have lead to beam hardening artifacts from the bone-equivalent inserts in the water vial within the reconstructed images.Artifacts would affect the normalization and thus the correct calculation of CT number.However, each of the scans was examined qualitatively and the slices chosen in which to calculate the normalization value were free from artifacts.The choice to include the water vial in the other tissue-equivalent insert scans was to reduce to the total number of optimization scans that were necessary.For the calibration and evaluation scans, the separate water phantom was scanned in order to reduce the possibility of beam hardening artifacts.
PCD-CT outperformed DECT overall.Though DECT demonstrated slightly better RMSPE and relative error values for some of tissue-equivalent materials, such as AP6 Adipose and BRN-SR2 Brain, DECT did not offer better results for both Z eff and ρ e for a single insert.Figure 6 demonstrates how the PCD-CT data agreed better with the predicted values than DECT did, with mean values which were closer to the predicted values.The RMSPE and relative errors for Z eff and ρ e were calculated from the mean values in each slice in order to mitigate the effect the ring artifacts had the results.The DECT images also suffered from mis-registration, which is obvious in areas immediately surrounding the cylinders and the ex vivo bovine tissue sample in the Z eff images (figures 5(b) and 8b).Each of the 80 kVp DECT scans was approximately half of a degree out of rotation with the corresponding 120 kVP scan, as shifting the sinogram one projection along the angle space in either the positive or negative resulted in larger rings surrounding the objects in the Z eff images.This was largely mitigated for the calculated Z eff and ρ e values for tissue-equivalent cylinders since the ROIs avoided the edges of the inserts, and excluding the two lung-mimicking materials, the inserts were homogeneous.The results for the two lung materials demonstrated how mis-registration can be a detriment in DECT due to the various pores within the material being misaligned.However, even when excluding the lung samples, PCD-CT demonstrated mean RMSPE values for Z eff and ρ e of 0.78% and 0.39%, respectively, compared with 1.30% and 0.87% for DECT.
For the ex vivo bovine tissue sample, our method was not able to offer results which were as accurate for all of the real tissue types as those that were found for the electron density inserts.Adipose, muscle, and inner bone were accurately determined, with relative errors for Z eff below 2.9% for both PCD-CT and DECT, and errors of 2% or less for ρ e .Cortical bone deviated much more significantly for both methods.There are a few likely sources which contributed to the differences between the predicted and measured values.First, the tissue sample is bovine, which may offer slightly different tissue compositions compared to the measured human tissue compositions from the ICRP.Additionally, the types of bone measured as spongiosa and cortical bone (figures 7 and 8 )in the rib may not actually represent the type of bone that was defined for the ICRP values.The areas adipose and muscle are more obvious in the images and as a result the decomposed values more accurate as well.
Gold chloride was successfully segmented within the bovine tissue sample with K-edge subtraction imaging demonstrating the ability of PCD-CT to extract high-Z contrast materials in images while performing Z eff and ρ e decomposition.K-edge subtraction results in material specific and quantifiable material maps, which can, as demonstrated in this study, be produced from the same set of energy bin data as was used for material decomposition.
One of the major benefits of PCD-CT in regard to material decomposition is that PCD-CT is able to acquire the requisite energy data for material decomposition without the need for a separate DECT system.With PCD-CT, multi-energy scans can be acquired on the same system that acquires conventional CT data.Additionally, more than two sets of energy data can be acquired, which offers the potential to decompose data into more than just two maps (Si-Mohamed et al 2017), as done in this study with Z eff and ρ e .The fact that the PCD-CT spectral data is already co-registered also makes projection-based decomposition methods more easily feasible than many DECT systems where co-registration can be more complicated, as demonstrated in this study.Projectionbased decomposition methods avoid errors that can be introduced by the reconstruction process in image-based methods, as the decomposition is done in the raw data space and reconstructed afterwards (Tatsugami et Liu et al 2022).A large part of this reduction in dose with the dual-source PCD-CT system is due to the improved contrast-to-noise ratio with PCDs compared to EIDs because of the reduction in electronic noise and the lower statistical noise (Danielsson et al 2021).This noise reduction with PCDs enables an equivalent signal-to-noise ratio (SNR) and CNR to be achieved at a lower dose compared with EIDs.In this work we compared a dualsource PCD-CT with a single-source PCD-CT.In addition to lower mean RMSPE values, PCD-CT provided similar noise in the water vial compared to DECT (table 7), with only half the dose.The DECT images suffered from increased ring artifacts which affected the tissue-equivalent material results.The similar level of noise with less dose can be explained by a better spectral separation between low and high energy images with low/high energies of 42.7/76.4keV and 44.9/58.9keV for PCD and DECT, respectively.This spectral separation could be improved for the DECT but would necessitate a high source voltage that is not available in our facility.A fair comparison of the dose efficiency could then be performed.
If we compare other DECT methods decomposing Z eff and ρ e maps to that of our method reproducing (Bourque et al 2014) in PCD-CT, we find better results with the PCD-CT data described here.For reference, with PCD-CT we obtain mean RMSPE values of 0.76% and 0.72% for Z eff and ρ e , respectively, and mean relative errors of 0.84% and 0.46%, respectively.With our DECT scans we obtained mean RMSPE values of 1.77% and 1.98%, respectively, and mean absolute relative errors of 1.69% and 1.07%.(Tsukihara et al 2013) found mean errors on ρ e of 1.01%., using a linear relationship of ρ e and the difference in CT number in a DECT scan.(Landry et al 2013) showed average Z eff RMSPE values of 5.1%, and (Saito and Sagara 2017) demonstrated an RSMPE on average Z eff of 2.9% (Hünemohr et al 2014a) obtained relative errors of 1.7% and 0.4% for Z eff and ρ e , respectively, achieving better results for ρ e than using our method with PCD-CT.And finally, (Bourque et al 2014) found RMSPE values of 3.58% and 0.60% for Z eff and ρ e , respectively, which still outperforms our adaptation in respect to ρ e .More recently, these methods have also been applied experimentally in clinical settings for the purpose of evaluating whether DECT could better predict the stopping power ratio (SPR) for proton therapies.Using a dual-layer DECT (DL-DECT) system, (Landry et al 2019) adapted their previous method to obtain RMSPE values of 1.0% for ρ e and 2.9% for Z eff , and from there obtain a RMSPE of less than 1% for SPR.(Faller et al 2020) also investigated DL-DECT for SPR calculations at the Heidelberg Ion Beam Therapy Center demonstrated similar values to (Landry et al 2019) with reported relative errors on the order of 1% to 2% for both ρ e and Z eff .Given these values, our PCD-CT method offers decomposition results which are comparable to DL-DECT solutions in the clinic.
However, there are currently still a number of drawbacks with PCD-CT when compared to DECT.There is currently only a single PCD-CT scanner that has been cleared by the Food and Drug Administration for clinical scanning (Rajendran et al 2022), though many others are nearing clinical implementation as well (Si-Mohamed et al 2023, Baffour et al 2023, Higashigaito et al 2023, Nehra et al 2023, Zhan et al 2023).However, DECT systems are more readily available in the clinic, in addition to being well integrated into clinical workflows.Ideally, the spectral separation between two energy bins of a PCD would be total but is often not total due to inherent properties of PCDs, as mentioned above.As such, the incident beam spectrum is not accurately captured by the detector and thus spectral distortions are introduced.To mitigate these distortions in the captured spectrum detector energy response models are needed (Tanguay et al 2020) and can be difficult to define (Rodesch et al 2023).As such not every DECT method, if the specific beam spectra are used in the decomposition, may lend itself as easily to PCD-CT translation.However, PCDs are currently the subject of intense interest for clinical CT applications, and many workarounds and corrections for the issues of PCDs, such as charge-sharing and pulse pileup, are being investigated (Hsieh 2020, Taguchi 2020, Hsieh et al 2021, Lewis and Das 2022).As such, PCD-CT has potential as a replacement for both SECT and DECT in clinical use.

Conclusion
In this study we demonstrated that a PCD-CT bench-top system is able to extract effective atomic number (Z eff ) and relative electron density (ρ e ) using two energy bins from a single acquisition, replicating a DECT stoichiometric material decomposition method.Z eff and ρ e were accurately determined for known tissuemimicking materials with an average RMSPE of 0.76% and 0.72%, respectively, compared to RMSPE of 1.77% and 1.98%, respectively, for DECT.Tissue types within an ex vivo tissue sample demonstrated values within two standard deviations of known human tissue values.In addition, the original method was extended to include material discrimination of high-atomic number contrast agents with K-edge subtraction, demonstrated with gold injected into an ex vivo tissue sample.As such, PCD-CT offers DECT material decomposition capabilities, with additional high-Z contrast agent extraction, along with conventional SECT capacity as well, making it an attractive option for clinical use.
95 keV.The 81 keV threshold was chosen in order to account for K-edge imaging of gold, and the 65-95 keV bin was created by summing the 65-81 and 81-95 keV bins together.The calibration coefficients determined for PCD-CT and DECT can be seen in table4.Using the b m values for both energy ranges for PCD-CT and DECT, the values of the sum b elements between 5 and 15, the approximate Z eff range the model is calibrated for.The results can be seen in Supplementary figure1.The calculated values ranged between ∼0.75-1.0 at Z = 5 and increased to ∼1.4-2.2 at Z = 15.Two examples of the predicted elemental linear attenuation coefficients for oxygen and carbon at the effective energies of the two PCD-CT energy bins and the 80 kVp and 120 kVp spectra can be seen in figure 3. The results of the aluminium extraction can be seen in the Supplementary figures 2 and 3.The comparison between the predicted and experimental CT number for each of the tissue equivalent materials can be seen in Supplementary figure 4. Materials with residual CT numbers over ±10 HU were water in all energy

Figure 3 .
Figure 3. Predicted linear attenuation coefficient at the effective energies of the energy bins for PCD-CT and spectra for DECT.(a) Carbon.(b) Oxygen.

Figure 4 .
Figure 4. PCD-CT images of the second evaluation subset of tissue equivalent materials and water.(a) 35-50 keV CT image with labeled materials and ROIs depicted in red.(b) Effective atomic number image.(c) Relative electron density image.

Figure 5 .
Figure 5. DECT images of the second evaluation subset of tissue equivalent materials and water.(a) 80 kVp CT image with labeled materials and ROIs depicted in red.(b) Effective atomic number image.(c) Relative electron density image.

Figure 6 .
Figure 6.Effective atomic number versus relative electron density for all the tissue equivalent materials and water.Crosses designate the theoretical values, and diamonds show the extracted values, with error bars showing the standard deviation within the ROI for each cylinder.(a) PCD-CT.(b) DECT.

3
.5.K-edge imaging In addition to material decomposition, PCD-CT offers high-Z contrast material extraction and quantification, as demonstrated in figure 9.In the conventional CT image (figure 9(a)) the gold solution is indistinguishable from the inner bone (figure 9(b)).However, gold is extracted in the K-edge subtraction image (figure 9(c)).

Figure 7 .
Figure 7. PCD-CT images of an ex vivo tissue sample.(a) 35-50 keV CT image with the color-coded tissue-specific ROIs.(b) Effective atomic number image.(c) Relative electron density image.

Figure 8 .
Figure 8. DECT images of an ex vivo tissue sample.(a) 80 kVp CT image with the color-coded tissue-specific ROIs.(b) Effective atomic number image.(c) Relative electron density image.

Figure 9 .
Figure 9. K-edge imaging for gold identification in an ex vivo tissue sample.(a) 24-120 keV CT image, gold circled.(b) Profile drawn across the rib, shown as the red line in (a), gold signal circled.(c) K-edge subtraction image of gold.

Table 1 .
Relative electron density (ρ e ), effective atomic number (Z eff ) for PCD-CT and DECT, and elemental composition for the tissue equivalent materials used in this study.

Table 2 .
PCD energy thresholds for the energy range optimization scans.

Table 3 .
The optimization results for K and M values with energy bins of 35-50 keV and 65-95 keV.Mean Z eff and ρ e RMSPE values encompass all tissueequivalent inserts excluding LN-300 Lung.

Table 5 .
RMSPE of individual tissue equivalent materials.

Table 6 .
Absolute error relative to water of individual tissue equivalent materials.

Table 7 .
The noise in the low and high energy CT images and Z eff and ρ e maps for PCD-CT and DECT.

Table 8 .
Effective (Schneider et al 1996)ative electron densities extracted from ex vivo bovine tissue sample.Mean and standard deviation are displayed with the corresponding ICRP human tissue values(Snyder et al 1975), calculated from their elemental compositions(Schneider et al 1996)as detailed in section 2.6.
al 2022).Our lab is currently investigating projection-based methods in addition to image-based methods, as the one detailed in this study.Dual source PCD-CT has been demonstrated to provide lower noise values compared to dual source EID-CT in both conventional (Winkelmann et al 2023, Wrazidlo et al 2023) and spectral mode (Booij et al 2022,