Physical density estimations of single- and dual-energy CT using material-based forward projection algorithm: a simulation study

Abstract

Objectives:

This study aims to evaluate the accuracy of physical density prediction in single-energy CT (SECT) and dual-energy CT (DECT) by adapting a fully simulation-based method using a material-based forward projection algorithm (MBFPA).

Methods:

We used biological tissues referenced in ICRU Report 44 and tissue substitutes to prepare three different types of phantoms for calibrating the Hounsfield unit (HU)-to-density curves. Sinograms were first virtually generated by the MBFPA with four representative energy spectra (i.e. 80 kVp, 100 kVp, 120 kVp, and 6 MVp) and then reconstructed to form realistic CT images by adding statistical noise. The HU-to-density curves in each spectrum and their pairwise combinations were derived from the CT images. The accuracy of these curves was validated using the ICRP110 human phantoms.

Results:

The relative mean square errors (RMSEs) of the physical density by the HU-to-density curves calibrated with kV SECT nearly presented no phantom size dependence. The kV–kV DECT calibrated curves were also comparable with those from the kV SECT. The phantom size effect became notable when the MV X-ray beams were employed for both SECT and DECT due to beam-hardening effects. The RMSEs were decreased using the biological tissue phantom.

Conclusion:

Simulation-based density prediction can be useful in the theoretical analysis of SECT and DECT calibrations. The results of this study indicated that the accuracy of SECT calibration is comparable with that of DECT using biological tissues. The size and shape of the calibration phantom could affect the accuracy, especially for MV CT calibrations.

Advances in knowledge:

The present study is based on a full simulation environment, which accommodates various situations such as SECT, kV–kV DECT, and even kV–MV DECT. In this paper, we presented the advances pertaining to the accuracy of the physical density prediction when applied to SECT and DECT in the MV X-ray energy range. To the best of our knowledge, this study is the first to validate the physical density estimation both in SECT and DECT using human-type phantoms.

Introduction

To establish the relationship between the CT number (in Hounsfield units, HU) of a given voxel and the physical (or electron) density relative to water is one of the crucial processes that control the variance of patient dose calculations in radiotherapy treatment planning.^1–3 The HU-to-density conversion is typically determined by calibration curves, which are experimentally obtained from tissue-substitutes with known densities in a calibration phantom. Several studies have investigated the sensitivity of dose calculation for photon and particle beams relative to the accuracy of this conversion.^4–7 A major concern of this approach is that the elemental composition of these substitutes may differ from that of biological tissues, and consequently, the adopted calibration curves may not be sufficiently accurate. One way to overcome this problem is the state-of-the-art stoichiometric calibration method introduced by Schneider et al⁸, in which the specific parameters of a CT scanner are determined by the measurement of a few tissue-substitutes with known materials. Recently, this method has been re-examined in the context of single-energy CT (SECT) calibration for proton therapy treatment planning, and its accuracy was found to depend on the tissue-substitutes used for calibration.⁹ This dependency implies that the improvement of dual energy CT (DECT) over SECT should also be reassessed, which may depend on the use of tissue substitutes as determined in a number of experiments for predicting relative stopping powers in proton therapy.¹⁰

In this study, the accuracy of physical density prediction using SECT and DECT was evaluated, which is the first step in treatment planning. To reduce the uncertainty of the calibration, the HU-to-density conversion using biological tissues from ICRU Publication 44 was proposed.^11–15 The approach in this study differs from the stoichiometric calibration⁸ by assuming that the X-ray spectra are known for the evaluation of the attenuation coefficients. In this case, CT values were reconstructed from the sinograms generated by the material-based forward projection algorithm (MBFPA), which has been utilised in model-based material decomposition,^16,17 when the material composition of the object was determined. These CT values were then used to perform the calibration to obtain the HU-to-density look-up-table (LUT). Using this LUT, arbitrary CT images can be converted to density images. Namely, this approach is a full simulation of the clinical process because it is based on the modelling of the entire CT system, including the incident X-ray energy spectrum.

The aim of this study was threefold. First, the feasibility of the proposed approach using MBFPA for physical density prediction was presented, which is significantly relevant to estimate proton stopping powers for treatment planning. To achieve this, the HU-to-density curves were simulated using three types of phantoms, where phantom (size/shape and composition) dependence in the calibration was also analysed. Second, the advantage of DECT over SECT for calibrations based on the proposed approach was quantitatively evaluated. Furthermore, the issue of whether a large energy gap in DECT, such as in the kilovoltage–megavoltage (kV–MV) range, could improve the accuracy of estimating the physical density was reassessed.

Methods and Materials

The schematic workflow is shown in Figure 1. The method starts with the preparation of three types of two-dimensional material-based digital calibration phantoms. Next, virtual projections (or sinograms) were produced using MBFPA, and CT images were sequentially reconstructed with the sinograms. Using the reconstructed images, the HU-to-density LUTs were calculated for each phantom. Finally, these LUTs were validated by predicting the physical density distributions of the ICRP110 human phantom. Four different X-ray energy sources (i.e. 80 kVp, 100 kVp, 120 kVp, 6 MVp) were employed. Thus, four SECTs and their six pairwise combinations for DECTs were considered.

Figure 1.

Workflow of the current study for physical density evaluation. BT(A), TS, and BT(B) phantoms were used for the calibration, and the ICRU human phantoms were used for the validation (for more details, see the main text). BT and TS refer to biological tissues and tissue substitutes, respectively.

Digital phantoms

Three types of phantoms were employed for the calibrations in this study, including the biological tissue (BT) phantoms in two different sizes and shapes, as well as the Gammex phantom (Gammex Inc., Middleton, WI). The former two are referred to as the BT(A) and BT(B) phantoms, respectively. The latter is known to have better tissue substitutes (TS) than the other tissue-mimicking phantoms,⁹ and is hereon referred to as the TS phantom. The shape of BT(A) is similar to the Shepp-Logan phantom,^18,19 whereas those of TS and BT(B) mimicked that of the Gammex phantom. The composed materials of BT(A) were taken from the ICRU publication 44.^13–15 Details of the elemental compositions and mass fractions are listed in Table 1 of Ref.¹² Following Hünemohr et al.’s study,¹² the mass fractions of six major elements (H, C, N, O, P, Ca) were normalised to provide a sum of 100%. From the 71 materials listed in Table 1 of Ref,¹²3 materials were randomly selected to construct one BT(A) phantom; a total of 54 phantoms were generated. Meanwhile, the TS phantom was composed of 1 base material (solid water) and 12 insertions (LN-300, LN-450, AP6, BR12, water, SR2, LV1, IB3, B200, CB2-30%CaCO₃, CB2-50%CaCO₃, and SB3), which were generated with weight fractions and mass densities of 8 elements (H, C, N, O, Mg, Si, P, and Ca) provided in Ref.⁹ The BT(B) phantoms have the same shape as the TS phantom but consist of 12 standard human biological tissues^13–15 and water instead of tissue-substitutes. By using different biological tissues, six phantoms were created. The width (major axis) and height (minor axis) of each BT(A) phantom were 260 mm and 197 mm, respectively, with a pixel scale of 1 mm, while the TS and BT(B) phantoms had a radius of 115 mm. Each of the insertions in the latter two had a radius of 13 mm. All the phantoms were placed in the air to obtain “complete” phantoms (images), for which the sizes were 300 × 300 mm² for the BT(A) and 512 × 512 mm² for the TS and BT(B). These three types of phantoms were considered for the following reasons: (1) to analyse the size and shape dependence by comparing the calibration results from the BT(A) and BT(B) phantoms, and (2) to analyse the material dependence by comparing the TS and BT(B) phantoms. Finally, 30 images were prepared from the ICRP110 human phantoms for validation, which consisted of five head, lung, and pelvis phantom slices for both, females and males, which were all 512 × 512 mm² in size with a 1 mm scale; six elements (H, C, N, O, P, and Ca) were considered. The specifics of all the phantoms used in this study are summarised in Table 1. As a representative case, the materials and elemental compositions of the TS phantom are listed in Table 2. The phantoms in the calibration are shown in Figure 2, and the selected ICRP110 human phantoms for a female are shown in Figure 3.

Table 1.

Specifics of all the phantoms used in this study

	Calibration			Validation
Name	BT (A)	TS	BT (B)	ICRP110 human
Materials	Biological tissues	Tissue substitutes	Biological tissues	–
Shape	Shepp-Logan	Gammex	Gammex	Head/Lung/Pelvis
Width (mm)	260	330	330	–
Height (mm)	197	330	330	–
Image (mm2)	3002	5122	5122	5122
Slice(s)a	54	1	6	30

BT, biological tissues; TS, tissue substitutes.

^anumber of phantoms (slices) used in calibration and validation.

Table 2.

Weight fraction (in units of %) and physical density $\rho$ (in units of g/cm³) of the TS (tissue substitutes) phantom used in this study

	H	C	N	O	Mg	Si	P	Ca	$\rho$
Lung (LN-300)	8.46	59.38	1.96	18.14	11.19	0.78			0.280
Lung (LN-450)	8.47	59.57	1.97	18.11	11.21	0.58			0.400
Adipose (AP6)	9.06	72.30	2.25	16.27					0.942
Breast (BR12)	8.59	70.11	2.33	17.90				0.95	0.977
Water	11.19			88.81					1.000
Solid water	8.02	67.23	2.41	19.91				2.31	1.018
Brain (SR2)	10.83	72.54	1.69	14.86					1.053
Liver (LV1)	8.06	67.01	2.47	20.01				2.31	1.097
Inner Bone (IB3)	6.67	55.64	1.96	23.52			3.23	8.86	1.143
Bone mineral (B200)	6.65	55.52	1.98	23.64			3.24	8.87	1.154
Bone (CB2-30% CaCO3)	6.68	53.48	2.12	25.61				12.01	1.335
Bone (CB2-50% CaCO3)	4.77	41.63	1.52	32.00				20.02	1.560
Cortical bone (SB3)	3.41	31.41	1.84	36.50				26.81	1.825

TS, tissue substitutes.

Figure 2.

Shape, size, and composition of the BT(A), TS, and BT(B) phantoms in the calibration. BT and TS refer to biological tissues and tissue substitutes, respectively.

Figure 3.

Ground truth of the selected ICRP110 human phantoms for female. The images in the first, second, and third row indicate head, lung, and pelvis, respectively.

Virtual projection and reconstruction

In this study, four different X-ray energies (80 kVp, 100 kVp, 120 kVp, and 6 MVp) were used for SECTs, and their six pairwise combinations for DECTs. This makes it simple to compare the results from SECT, kV-kV DECT, and kV-MV DECT. For diagnostic CT, an X-ray within the MV range is not applicable, whereas for image-guided radiation systems, equipment to produce clinical or non-clinical projection images with MV X-rays exists.^20–22 In this study, a 6 MVp X-ray, which is a typical X-ray energy used in radiation treatment, was selected. The spectra of the X-ray sources were obtained by Monte-Carlo (MC) simulations using the GEANT4 toolkit (v. 10.4) for a linear accelerator with kV imaging capability (Synergy, Elekta, UK). The Livermore low-energy electromagnetic model was employed in the simulation. For kV X-rays, low-energy photons generated from 10¹⁰ accelerated electrons in a nominal peak voltage were decimated by filters composed of aluminium and copper, and the beam shape was formed by lead-cone and cassette collimators. For the MV X-rays, the photons generated from 2.5$\times$10⁹ accelerated electrons at a nominal peak voltage were decimated by a flattening filter, and the beam shape was formed by primary, jaw, and multileaf collimators. In both cases, the energy spectrum was formed by the photons collected on the plane located 70 cm from the sources, where the mean energies and statistical errors were 48.70$\pm$0.08 keV, 53.24$\pm$0.04 keV, 59.28$\pm$0.04 keV, and 1.30$\pm$0.00 MeV for 80 kVp, 100 kVp, 120 kVp, and 6 MVp, respectively. Figure 4 presents the X-ray spectra produced for the kV and MV beams. The spectra for 120 kVp and 6 MVp are almost the same as those shown in Figure 3 in Ref.²³ The spectrum of the 6 MVp X-ray (Figure 5d) exhibits a slight deviation from that shown in Ref,²⁴ which is primarily owing to the averaging over the phase space plane in the present study. The X-ray spectrum has an off-axis dependence, which could be a problem; however, the flattening filter-free version such as tomotherapy (Accuray Inc. Madison, WI, US), which has a low off-axis dependence, can be used in practical applications. Variations in the X-ray spectra may also affect the accuracy of the physical density predictions. However, in this study, we used only one spectrum for each X-ray energy and compared the results obtained for SECT and DECT.

Figure 4.

Simulated X-ray spectra of 80 kVp, 100 kVp, 120 kVp, and 6 MVp. The mean energies were 48.70 keV, 53.24 keV, 59.28 keV, and 1.30 MeV, respectively.

Figure 5.

Schematic of geometry applied in sinogram production. IDD and SDD indicate the isocenter-to-detector distance and the source-to-detector distance, respectively. A total of 609 detectors were used, and the size of each one was 0.15 cm.

Using the spectra and the digital calibration phantoms indicated above, virtual projections using MBFPA were then simulated; further details can be found in the Supplementary Material 1. The geometry of the simulated CTs with relevant factors is shown in Figure 5. The source and detectors were rotated 360° in 0.45° increments (in total 800 projections). There were 609 detectors, which were aligned in equal intervals (0.15 cm). Furthermore, random noise is inevitable in CT imaging, which is also a main source of artefacts. A Gaussian distributed random noise was added to the sinograms (detectors) for the simulations in this study, for which the standard deviation was determined to produce a signal-to-noise ratio of ~20 in the reconstructed images of a homogeneous water phantom (which has the same size and shape as the TS phantom), which is consistent with that in clinical practice.

HU-to-density relation for SECT calibrations

For the calibrations, the average CT number (HU) of each material from the reconstructed images along with the corresponding physical density (known for the calibration phantoms) were used to form the LUTs. A fourth-order polynomial was applied to relate the physical density $\rho$ with HU in Hounsfield units as follows:

$${\displaystyle \rho \left(HU\right)=c_0+c_{1}HU+c_{2}H{U}^2+c_{3}H{U}^3+c_{4}H{U}^4}$$ (1)

where $c_i(i=0,\cdots ,4)$ are the parameters; note, $c_0$ can be expressed with $c_i(i=1,\cdots ,4)$ because $\rho (-1000)$ = 0. Therefore, the last four can be determined from a least-squares fit to the LUT of the SECT scans. The values of the parameters may differ depending on the X-ray energies and calibration phantoms and should be treated accordingly.

HU-to-density relation for DECT calibration

The $\Delta HU-{\rho }_e$ conversion method^25,26 was applied for the DECT calibration, however, the electron density ${\rho }_e$ was replaced with the physical density $\rho$. This is appropriate because $\rho$ is proportional to ${\rho }_e$ . The dual energy subtracted quantity $\Delta HU$ was defined as follows:

$${\displaystyle \mathrm{\Delta }HU\equiv \left(1+\alpha \right)H{U}_H-\alpha H{U}_L}$$ (2)

where ${HU}_H$ and ${HU}_L$ denote the high-energy and low-energy CT numbers in Hounsfield units, respectively. Further, $\mathrm{\alpha }$ is a weighting factor for the subtraction, which is regarded as material-independent. Similar to a previous study,²⁵ the relation between $\Delta HU$ and $\rho$ was assumed to be linear for materials with low effective atomic numbers as follows:

$${\displaystyle \rho \left(\mathrm{\Delta }HU\right)=a\frac{\mathrm{\Delta }HU}{1000}+b}$$ (3)

where $a$, $b$, and $\mathrm{\alpha }$ can be determined by a least squares fit to $({HU}_H,{HU}_L)-\rho$ data obtained from DECT scans of materials with a known density $\rho$ in calibration phantoms, which is similar to the SECT calibration.

Validation

The minimum ${\chi }^2$ value of the fitting curves for the physical density were evaluated for all three calibration phantoms. A total of 30 virtual images based on the ICRP110 human phantoms (shown in Table 1) were independently prepared via virtual projections and image reconstructions. The physical density distribution converted by the LUTs of each energy spectrum and calibration phantom were compared with the ground truth. Statistical analysis was performed to determine the differences in the RMSE among the chosen energies (for SECT), their combinations (for DECT), or the chosen calibration phantoms. In particular, the following differences were assessed: (1) between SECT and DECT, (2) between TS and BT(B), and (3) among the energies with BT(A) and BT(B). For the statistical analysis, a Student's t-test was employed for the first two cases, while Tukey’s range test was employed for the last.

Results

Generated sinograms and reconstructed images

Figure 6 presents the representative sinograms generated from the MBFPA of the TS phantom with four energies, and Figure 7 presents the corresponding reconstructed images.

Figure 6.

Sinograms of the TS phantom with 80 kVp, 100 kVp, 120 kVp, and 6 MVp. The display value ranges are 0–0.01 for kV CT and 0.1–0.3 for MV CT. The horizontal dimension of each sinogram corresponds to 609 detectors, while the vertical dimension corresponds to 800 projections. TS, tissue substitutes.

Figure 7.

Reconstructed images of the TS phantom with 80 kVp, 100 kVp, 120 kVp, and 6 MVp. The display value ranges of the attenuation coefficients (in units of cm⁻¹) are 0–0.4 for kV CT and 0–0.1 for MV CT. TS, tissue substitutes

SECT calibration results

A data set of CT values and physical densities in the calibration phantoms can be derived from the above reconstructed images. The data points and fitted curves for all three phantoms and four energies are shown in Figure 8, and the corresponding residuals are shown in Figure 9. The three calibration phantoms present similar behaviours, which is reasonable because similar or same materials were used. However, the BT(A) phantoms present a relatively different behaviour compared to the TS and BT(B) phantoms in the case of 6 MVp because the size of the former is smaller.

Figure 8.

HU-to-density relations of three calibration phantoms with 80 kVp, 100 kVp, 120 kVp, and 6 MVp in SECT. HU, Hounsfield unit; SECT, single energy CT.

Figure 9.

Residuals corresponding to the SECT fittings in Figure 8. SECT, single energy CT.

DECT calibration results

The data of $\mathrm{\Delta }HU$ and $\rho$, as well as the fitted results, are shown in Figure 10. The corresponding residuals are shown in Figure 11. It was found that the TS and BT(B) phantoms have similar behaviours in all six DECT calibrations, which are different from the BT(A) phantoms. However, in the case of kV SECTs, all three curves present similar behaviours, as shown in Figure 9. This is because a fourth-order polynomial was applied for the SECT fitting, while a linear relationship was assumed for the DECT calibration. Therefore, the difference in the data points is suppressed by the higher order terms in the SECT situation. However, for the DECT calibration, such differences become notable. In addition, the fact that the size/shape of BT(A) and BT(B) phantoms are different also implies that the physical density estimation in the $\mathrm{\Delta }HU$ approach depends on (the size/shape of) the calibration phantoms as well. In particular, this dependence appears stronger in kV–MV DECT calibrations.

Figure 10.

∆HU to density relations of the three calibration phantoms with six pairwise combinations of 80 kVp, 100 kVp, 120 kVp, and 6 MVp in DECT. DECT, dual energy CT; HU, Hounsfield unit.

Figure 11.

Residuals corresponding to the DECT fittings in Figure 10. DECT, dual energy CT; HU, Hounsfield unit.

Validation using the ICRP110 human phantom

The validation results using the ICRP110 human phantom are shown in Table 3, where the average RMSEs are listed for all the anatomies. No significant difference in the RMSEs was found between SECT and DECT (p-values: 0.26, 0.16, and 0.68 for BT(A), TS, and BT(B), respectively). Conversely, the predicted results from the calibrations using BT(B) phantom significantly improved compared to that using TS phantom in density calibration (p-values:<0.01), although the mean RSME difference was less than 0.0093. For the energy dependence of the physical density estimation, the RMSEs of the 6 MVp SECT were observed to be significantly larger than those of the others for BT(B), whereas no significant difference was observed in the use of BT(A), mainly caused by the lung area having the large size. In addition, the RMSEs of the 6 MVp SECT from BT(A) phantoms are relatively better than those from BT(B) phantoms. More specifically, the smaller the calibration phantom, the more accurate the validation predictions are. However, such behaviours are not apparent in the kV SECT results.

Table 3.

Predicted average RMSEs (in units of g/cm³) of ICRP110 phantom slices from BT(A), TS, and BT(B) phantoms with SECT and DECT

		SECT predictions				DECT predictions
Calibration	Anatomy	80 kVp	100 kVp	120 kVp	6 MVp	80 kVp-100 kVp	80 kVp-120 kVp	100 kVp-120 kVp	80 kVp-6 MVp	100 kVp-6 MVp	120 kVp-6 MVp
BT(A)	Head	0.1230	0.1236	0.1197	0.1270	0.1264	0.1180	0.1128	0.1155	0.1160	0.1156
	Lung	0.1507	0.1429	0.1356	0.0978	0.1548	0.1394	0.1516	0.0950	0.0963	0.0978
	Pelvis	0.0793	0.0756	0.0739	0.0750	0.0793	0.0781	0.0785	0.0759	0.0754	0.0750
TS	Head	0.1362	0.1424	0.1395	0.2289	0.1758	0.1542	0.1395	0.1981	0.1967	0.1934
	Lung	0.1589	0.1522	0.1445	0.1237	0.1873	0.1595	0.1728	0.1144	0.1141	0.1135
	Pelvis	0.0759	0.0762	0.0770	0.1267	0.0951	0.0917	0.0903	0.1121	0.1108	0.1089
BT(B)	Head	0.1294	0.1375	0.1368	0.2217	0.1644	0.1454	0.1327	0.1771	0.1749	0.1702
	Lung	0.1484	0.1456	0.1418	0.1172	0.1809	0.1558	0.1687	0.1046	0.1048	0.1050
	Pelvis	0.0727	0.0745	0.0757	0.1144	0.0876	0.0850	0.0842	0.0934	0.0918	0.0896

BT, biological tissues; DECT, dual energy CT; SECT, single energy CT;TS, tissue substitutes.

Discussions

This study presented the evaluation of both SECT and DECT calibrations based on the MBFPA and demonstrated that the accuracy of SECT calibration is comparable with that of DECT. In addition, the use of the biological tissue phantom in the calibration was found to improve the physical density prediction compared with that of the Gammex phantom composed of tissue substitutes.

Similar results have already been implicitly obtained by Goma et al who performed SECT calibration using the stoichiometric method, which consists of characterising the CT scanner directly through CT scans of tissue substitutes, and subsequently predicting the CT numbers of biological tissues using this characterisation.⁹ The main difference between the stoichiometric calibration and the present MBFPA approach is that the former requires real CT images acquired from CT scanners, whereas the latter does not. In this study, the CT scanners were modelled, characterised by X-ray energy spectra directly, by which various simulations, with not only the kV-range X-rays but also MV-range X-rays, could be performed. Furthermore, the density results cannot be validated for real patients in the stoichiometric calibration framework. However, using the proposed MBFPA-based calibration approach, the validation is now possible by preparing, e.g. the reconstructed CT data sets using the ICRP110 human phantom, which could be considered as real patients to a certain extent. This study not only supported the results of Goma et al⁹ but also newly presented that the tissue substitutes differ from the biological tissues in physical density calibration.

The results of this study indicated that kV–MV DECT is not as outperformed as it was expected to be. This is due to the large beam-hardening effect in MV CT, compared to that in kV CT. This can be inferred from the fact that different sizes of the calibration phantoms provided different calibration curves with MV X-rays. For a more apparent indication, a simulation using monochromatic energy X-rays with 3 MeV was also performed, which does not suffer from beam hardening. In this case, no phantom size dependence in the calibration curves was observed. The magnitude of beam hardening in MV CT could also be observed in the homogeneous water phantom (of the same size as the TS phantom) by extracting the reconstructed CT values in the centre and peripheral regions. Their relative difference was ~17% for MV CT due to the cupping artefacts; however, this value is only ~10% for kV CT. Such ambiguity in the MV CT with a practical spectrum significantly affected the accuracy of the calibration. Hence, DECT with MV X-rays does not improve the accuracy of the physical density estimation as well as the SECT of MV X-rays. Furthermore, the use of MV X-rays passing through a titanium filter could, to a certain extent, reduce the beam-hardening effect.

Note that Yang et al²⁷ assessed the superiority of kV–MV DECT in determining proton stopping power by generating 1 MVp beams from MC calculations. According to the authors, when CT number uncertainties and artefacts such as imaging noise and beam-hardening effects were considered, the kV–MV DECT improved the perfectly of SPR estimation substantially over kV–kV or MV–MV DECT methods. The SPR estimation is directly influenced by the electron density (or physical density), and therefore, Yang et al²⁷’s study implied a substantial decrease in the physical density uncertainties in kV–MV DECT. However, this study apparently supports the contrary. This might indicate that the beam-hardening effects were underestimated in MV CT in Yang et al²⁷’s study because the authors assumed the “average spectra” accurately modelled the CT scanner. Thus, the beam-hardening effect should be carefully treated in MV CT.

The proposed method can be considered an improvement over previous stoichiometric approaches in which the parameters, depending on the X-ray spectrum, which characterise the CT scanner are determined by fitting to the effective linear attenuation coefficients of a given material, whereas the proposed method explicitly deals with the X-ray spectrum. Although the explicit handling of the X-ray spectrum is advantageous in CT calibration, the requirement of the X-ray spectrum imposes limitations for practical applications. That is, the exact energy spectrum of medical CT scanners is unknown, and its direct measurement is difficult because of the high photon flux. Nevertheless, novel methods to estimate X-ray spectra in practical CT scanners have been proposed in recent years.^28–31 Therefore, it is reasonable to assume that X-ray spectra are currently available. In the future, the MBFPA approach and the stoichiometric approach should be compared for both virtual and real CT systems. This comparison will not only help bring our study a step closer to clinical practice, but also help validate the MBFPA approach. In this context, several studies should be also done to improve our present results. First, functions other than the polynomials, such as the two-line segments weighted by a sigmoid function, could be used to increase the accuracy in the SECT fitting. Second, an advanced beam-hardening correction method should be applied in the image reconstruction, in particular for MV CT. We will investigate these effects in the future.

Conclusions

The proposed method using the MBFPA is useful in the theoretical analysis of physical density calibrations. The accuracy of SECT calibration is comparable with that of DECT calibration and is improved with the use of biological tissues. The size and shape of the calibration phantom could affect the accuracy, especially for MV CT, mainly because of the beam-hardening effects. The present method based on the MBFPA can also be applied to various other studies, such as effective atomic number estimations and material decompositions.