### CT phantoms, scan protocols and image reconstruction

Two different CT phantoms were used: a Gammex phantom (467-TOMO; Gammex RMI, Middleton WI, USA) (Fig. 1) and a Catphan phantom (Catphan 504; The Phantom Laboratory, Salem NY, USA) (Fig. 2). The Gammex phantom contains 12 inserts representing tissue-equivalent materials with various known electron densities. Each insert is cylindrical, with a diameter of 30 mm and a length of 70 mm. The Catphan phantom has six homogeneous regions consisting of common materials with known electron densities; each region is a cylinder with a 12.5-mm diameter and a length of 25 mm.

Scans were conducted with a DLCT unit, the IQon Spectral CT (Philips Healthcare, Best, the Netherlands). Similarly to the acquisition protocols applied in clinical routine, Gammex and Catphan phantoms were scanned with an x-ray source voltage of 120 peak kilovoltage (kVp), a collimation width of 0.625 mm, a revolution time of 1.5 s and a spiral pitch factor of 0.983. We made four levels of x-ray tube currents, which were 229, 153, 77 and 56 mA, resulting in x-ray exposures of 350, 234, 117 and 86 mA, respectively, and the corresponding volume CT dose index (CTDI_{vol}) of 30, 20, 10 and 7.5 mGy was recorded in the dose reports.

The spectral data were reconstructed with an iterative reconstruction at vendor-specific level 2 and a standard filter B, according to the settings used in most clinical abdominal examinations. The field of view was 360 mm, and the slice thickness was 0.8 mm. Corresponding VMIs were generated by using vendor-specific spectral software (IntelliSpace Portal v10.1; Philips Healthcare) at 50 and 200 keV. We used these two monoenergetic levels because at 50 keV the photoelectric effect and at 200 keV the Compton effect are the dominant x-ray interactions with matter. Regions of interest were drawn as cylinders with half of the radius and height of the actual insert dimension, and the mean HU values were measured. Regions of interest were synchronised between VMIs, and the measurements were repeated using image-processing software (ImageJ v1.50f; National Institutes of Health, Bethesda, MD, USA) [15].

### Electron density estimation using cross-sectional model

In order to obtain electron density values using the cross-sectional model, the energy-dependent CT numbers HU(*E*) in the corresponding VMI were firstly converted into the mass-attenuation coefficients *μ*(*E*)/*ρ* at the specific energy using the following equation:

$$ \frac{\mu (E)}{\rho }=\frac{\mu_w(E)}{\rho_w}\left(\frac{\mathrm{HU}(E)}{1000}+1\right), $$

(1)

where *μ*_{w}(*E*)/*ρ*_{w} is constant representing mass-attenuation coefficient of water at energy *E*, which can be referenced from the National Institute of Standards and Technology database [16].

In an ideal case of a narrow beam of monoenergetic photons in the range of clinical CT (*E* < 511 keV), the mass-attenuation coefficient *μ*(*E*)/*ρ* can be attributed to three physical interaction mechanisms: photoelectric absorption, incoherent (Compton) scattering and coherent (Rayleigh) scattering. For the energy range used in clinical CT, coherent scattering can often be neglected for standard body tissues, resulting in the well-known two-dimensional parameterisation for the mass-attenuation coefficient [17, 18]:

$$ \frac{\mu (E)}{\rho}\cong {a}_p{f}_p(E)+{a}_c{f}_c(E), $$

(2)

where *a*_{p} and *a*_{c} are characteristic parameters for the different materials in the image. *f*_{p} and *f*_{c} are the energy dependencies of photoelectric absorption and Compton scattering. The photoelectric absorption part is approximated as:

$$ {a}_p{f}_p\cong {\rho}_e{C}_p\frac{Z^m}{E^n}, $$

(3)

where *ρ*_{e} is the absolute electron density (e/cm^{3}), and *Z* is the effective atomic number. *C*_{p} is constant and equals 9.8 × 10^{− 24} [18]. *E* is the energy of the x-ray beam measured in kiloelectron volts. For a numerical fit of the experimental data, *m* is between 3 and 4, and *n* is between 3 and 3.5. In this study, we use *m* = 3.8, *n* = 3.2 [4].

The Compton effect can be approximated with electron density and the total Klein-Nishina cross-section [19]:

$$ {a}_c={\rho}_e, $$

(4)

$$ {f}_c\left(\gamma \right)={C}_0\left\{\frac{1+\gamma }{\gamma^2}\left[\frac{2\left(1+\gamma \right)}{1+2\gamma }-\frac{1}{\gamma}\ln \left(1+2\gamma \right)\right]+\frac{1}{2\gamma}\ln \left(1+2\gamma \right)-\frac{\left(1+3\gamma \right)}{{\left(1+2\gamma \right)}^2}\right\}, $$

(5)

$$ \gamma =\frac{E}{510.975\ \mathrm{keV}},\kern0.75em {C}_0=2\pi {r}_0^2, $$

(6)

where *E* is the x-ray energy and has the unit of kiloelectron volts, and *r*_{0} is the classical electron radius, which equals to 2.818 × 10^{− 13} cm.

If we substitute Eqs. (3)–(5) into Eq. (2), we obtain:

$$ \frac{\mu (E)}{\rho}\cong {\rho}_e\left({C}_p\frac{Z^m}{E^n}+{f}_c(E)\right). $$

(7)

The two unknown variables *ρ*_{e} and *Z* in Eq. (7) can be obtained by the acquisition of two VMIs at distinct energy levels and then analytically solving the resulting set of equations. To maximise the difference between two VMIs and thus improve the accuracy of the solution, we use energy levels at 50 keV and 200 keV. Absolute electron densities can then be converted to relative electron density using known water electron density (3.343 × 10^{23} e/cm^{3}).

### Electron density estimation using calibrated conversion function

We used the Gammex phantom to fit a conversion function from HU values measured in two VMIs to relative electron densities. A scan is taken using a relatively high radiation exposure (30 mGy) to acquire almost noise-free calibration images of the phantom at 50 keV and 200 keV. HU values at these two energies are used to fit Saito’s conversion function [14]:

$$ {\rho}_e=a\frac{\left(1+\alpha \right)\mathrm{H}{\mathrm{U}}_{\mathrm{H}}-\alpha \mathrm{H}{\mathrm{U}}_{\mathrm{L}}}{1000}+b, $$

(8)

where *ρ*_{e} is the actual relative electron density taken from the phantom’s data sheet, HU_{H} and HU_{L} are HU values in the VMIs at 50 keV and 200 keV, and *a*, *b*, α are parameters specific to the scanner.

The Gammex phantom consists of twelve materials with known pairs of (*ρ*_{e}, HU_{H}, HU_{L}), and there were three unknown parameters (*a*, *b*, *α*) in the equation; this fitting was computed using MATLAB software (v9.2; MathWorks, Natick MA, USA) and a surface-fitting algorithm. The fitting results were then used to compute relative electron densities of the Catphan phantom containing six materials in varied dose scans.

### Error measurement

To describe measurement errors, we computed the percentage error (%Error) as the ratio of the difference of the estimated value (*ρ*_{e}) to the nominal value of the relative electron density (*ρ*_{n}):

$$ \%\mathrm{Error}=\frac{\rho_e-{\rho}_n\ }{\rho_n}\times 100\%, $$

(9)

and *ρ*_{e} − *ρ*_{n} is noted as absolute error. The overall estimation error is assessed using root mean square error (RMSE) and normalised root mean square error (NRMSE):

$$ \mathrm{RMSE}=\sqrt{\frac{1}{N}\sum {\left({\rho}_e-{\rho}_n\right)}^2}; $$

(10)

$$ \mathrm{NRMSE}=\frac{\mathrm{RMSE}}{\overline{\rho_n}},\kern0.5em \overline{\rho_n}=\left(\frac{1}{N}\sum {\rho}_n\right). $$

(11)

To assess correlations between the estimated values and the nominal values, the Pearson correlation *R* was used. In addition, linear regression analysis was performed as fitting:

$$ {\rho}_e=\beta \bullet {\rho}_n+\epsilon, $$

(12)

where *β* and *ϵ* are regression coefficients (slope and intercept). Coefficient of determination, which describes the goodness of the fit, is noted as *R*^{2}. Analysis of covariance (ANCOVA) was performed for the measurement against the group where \( {\rho}_e^{\prime }={\rho}_n \), which indicates an ideal measurement. Moreover, a paired *t* test was performed for the measurement against the nominal values.

In order to assess the reproducibility of the estimation and the influence of different radiation exposures, Pearson correlation, RMSE, and NRMSE between measurements and nominal values were computed. A paired *t* test was performed for measurements between the highest dose (30 mGy) and lowest dose (7.5 mGy). All error estimation and statistical analyses were performed using MATLAB software.