Experimental study to optimize configurations of PCD Spectral CT

Abstract

BACKGROUND:

High dose efficiency of photon counting detector based spectral CT (PCD-SCT) and its value in some clinical diagnosis have been well acknowledged. However, it has not been widely adopted in practical use for medical diagnosis and security inspection.

OBJECTIVE:

To evaluate the influence on PCD-SCT from multiple aspects including the number of energy channels, k-edge materials, energy thresholding, basis functions in spectral information decomposition, and the combined optimal setting for these parameters and configurations.

METHODS:

Basis material decomposition after spatial reconstruction is applied for PCD-SCT. A “one-step” synthesis method, merging decomposition with synthesis, is proposed to obtain virtual monochromatic images. An I-RMSE is computed using the bias part of I-RMSE to describe the difference of a synthesized signal from ground truth and the standard deviation part of I-RMSE to express the noise level. In addition, virtual monochromatic images commonly used in the medical area are also synthesized. Both numerical simulations and practical experiments are conducted for validation.

RESULTS:

Results indicated that the I-RMSE for matters significantly reduced with an increased number of energy channels compared with dual-energy channel. The maximum reduction is 6% for triple-, 18% for quadruple-and 24% for quintuple-energy, respectively. However, the improvement is not linear, and also slows down after the number of energy channels reaches a certain number. Contrast agents of high concentration can introduce up to 50% error to surrounding matters. Moreover, different energy partitions influence the total error, which demonstrates the necessity of energy threshold optimization. Last, the optimal basis-material combination varies according to targeted imaging matters and the interested monochromatic energies.

CONCLUSIONS:

Gain from more energy channels could be significant with the increase of energy channel number. Introduction of contrast agents in scanned objects will increase overall error in spectral CT imaging. Energy thresholding optimization is beneficial for information recovery. Moreover, the choice of basis materials could also be important to obtain low noise results. With these studies of the effect from various configurations for PCD-SCT, one may optimize the configuration of PCD-SCT accordingly.

Keywords

Spectral CT photon counting detector (PCD)multi-energy material decomposition virtual monochromatic

1 Introduction

Spectral CT (SCT) is competitive in material classification and discrimination since it includes spectrum information. Photon Counting Detectors (PCDs) brings new space in the improvement of image quality and dose efficiency for SCT [16, 17]. Compared with Energy Integrated Detectors (EIDs), PCD can exclude most of the electronic noise by setting proper thresholds and provide energy selective information. Le [10] verified that radiation dose can be decreased to about 50% when PCD is adopted instead of flat panel EIDs. Multi-channel energy sensitive data can be obtained in one acquisition by PCD simultaneously, which benefits to dual-or multi-material decomposition [9 , 26]. In these days, PCD commercially available could be of 2 [19, 20], 5 [1, 12] and 8 [13] energy channels. High energy resolution of a PCD also allows us to decompose contrast agents by k-edge imaging [3 , 25]. These advantages gain a lot of attentions and driving researchers to explore SCT imaging with PCD. Over the past decade, much effort has been devoted to the development of PCD spectral CT scanners. Currently, various prototype systems or laboratory benchtops have been developed, but it has not been widely adopted in clinical use and neither widely available commercially. This might be attributed to the difficulty and non-conclusive opinions on the optimal implementation of PCD-SCT, as well as thorough and robust validation methods [34]. Compared with a traditional CT, many more configurations and parameters must be considered when designing a PCD-SCT system and imaging protocols.

A lot of researches have been done on these aspects. Giersch [6] implements SNR optimization by weighting each energy channel. Rigie et al. [22] apply Hotelling observer to signal detection tasks for spectral CTs to optimize energy thresholds of PCD. Wang et al. [7, 18] determine the optimal thresholds allocated on both sides of k-edge point targeting on maximizing the signal difference to noise ratio in k-edge imaging. Zheng et al. [35] maximize signal-difference-to-noise-ration (SDNR²) to optimize energy thresholds. Huang et al. [8] also propose a threshold optimization method for PCD-SCT. Tkaczyk [29] evaluates the image quality entitlement of PCD over dual kVp imaging by optimal energy weighting and optimal mono-energy imaging under the criterion of CNR² to dose ratio. The optimal monochromatic energy level for virtual monochromatic spectral imaging is also one of the interests to the spectral imaging community [32]. Besides those works, many studies focused on energy information decomposition and reconstruction methods to improve the imaging quality for a spectral CT [2, 30].

In this work, we study impact factors for PCD-SCT from four aspects. Though it has been acknowledged that more than two energy channels could be beneficiary, it is not clear how much gain can be achieved by increasing the number of energy channels (NEC). Hence, we firstly study the image quality in case of different NEC and threshold configuration in this work. Secondly, contrast agents with k-edges are important subjects in PCD-SCT. We look into the effect from the inclusion of contrast agents accompanied with k-edge thresholding. Thirdly, with multiple energy channels, energy thresholds are critical to PCD-SCT image quality. We will demonstrate their influence with simulation experiments. Finally, we examine the effect of basis function choices in the spectral information decomposition for optimal SCT reconstructions. Since virtual monochromatic images are very useful and commonly interested in medical applications [15 , 28], we targeted on virtual monochromatic image reconstruction in this study.

This paper is organized as follows. Section I introduces the background of proposed method. Section II describes the basic theories and methods for spectral CT imaging. Section III presents performance evaluation with experimental studies and result analysis. Section IV discusses the key factors affecting the image quality of a spectral CT with a PCD.

2 Theories and methods

2.1 The physical model for spectral CT imaging

Data acquisition in spectral CT imaging with a PCD is normally modeled as:

$I_{i} (l) = I_{0} \int_{E_{i}}^{E_{i + 1}} {dE}^{'} \int_{0}^{\infty} S (E) e^{- \int_{l} μ (\vec{r}, E) d \vec{r}} h (E^{'}; E) dE, i = 1, \dots, N_{E}$ (1)

with I₀ denoting the incident photons and S (E) the normalized energy spectrum of the source, $μ (\vec{r}, E)$ the spatial distribution of energy-dependent linear attenuation coefficient of a scanned object at location $\vec{r}$ , and l the X-ray path from the X-ray source to a detector bin, h (E′; E) the detector response. We define energy channel to be [E_i, E_i+1) with N_E being the NEC. In post-processing method, ${\tilde{μ}}_{i} (\vec{r})$ will be reconstructed for each energy channel according to:

$\int_{l} {\tilde{μ}}_{i} (\vec{r}) d \vec{r} \approx ln \int_{E_{i}}^{E_{i + 1}} {dE}^{'} \int_{0}^{\infty} S_{in} (E) h (E^{'}; E) dE \equiv p_{i} (l)$ (2)

Here, the quantity p_i (l) is generally referred as projection data. For energy information decomposition, we can assume [31]:

$μ (\vec{r}, E) = \sum_{τ = 1}^{Γ} a_{τ} (\vec{r}) φ_{τ} (E)$ (3)

where basis functions φ_τ (E) represents the energy dependent components in $μ (\vec{r}, E)$ , and $a_{τ} (\vec{r})$ the spatial distribution of the coefficient corresponding to τ^th basis function.

The decomposition model of Equation (3) can be extended to each energy channel in case of τ = 2 so that we have: ${\begin{matrix} {\tilde{μ}}_{1} (\vec{r}) = {〈 φ_{1} (E) 〉}_{[E_{1}, E_{2})} a_{1} (\vec{r}) + {〈 φ_{2} (E) 〉}_{[E_{1}, E_{2})} a_{2} (\vec{r}) \\ {\tilde{μ}}_{2} (\vec{r}) = {〈 φ_{1} (E) 〉}_{[E_{2}, E_{3})} a_{1} (\vec{r}) + {〈 φ_{2} (E) 〉}_{[E_{2}, E_{3})} a_{2} (\vec{r}) \\ \dots \dots \dots \\ {\tilde{μ}}_{N_{E}} (\vec{r}) = {〈 φ_{1} (E) 〉}_{[E_{N_{E}}, E_{N_{E} + 1})} a_{1} (\vec{r}) + {〈 φ_{2} (E) 〉}_{[E_{N_{E}}, E_{N_{E} + 1})} a_{2} (\vec{r}) \end{matrix}$ (4)

Here, we use 〈φ_τ (E) 〉 _{[E_i,E_{i +1})} to denote the aggregate value of φ_τ (E) for i^th energy channel. Equation (4) can be expressed in matrix-vector format for each pixel at location $\vec{r}$ : $\tilde{μ} (\vec{r}) = \tilde{Φ} (φ, E) a (\vec{r}), \forall \vec{r}$ (5)

where the i^th element of $\tilde{μ} (\vec{r}) \in ℝ^{N_{E} \times 1}$ matches the linear attenuation coefficient reconstructed from the i^th energy channel, the matrix $\tilde{Φ} \in ℝ^{N_{E} \times Γ}$ is with elements ${[\tilde{Φ}]}_{i, τ} \equiv {〈 φ_{τ} (E) 〉}_{[E_{i}, E_{i + 1})}$ . Here, we use $E \in ℝ^{(N_{E} + 1) \times 1}$ to denote the energy thresholds, i.e. the energy channel setting for an SCT. Obviously, $\tilde{Φ}$ is determined by the choices of basis functions φ and energy channel setting E. Decomposition coefficients $a (\vec{r}) \in ℝ^{Γ \times 1}$ represent the component fraction of τ^th basis function for the material at the location $\vec{r}$ . If $\tilde{Φ}$ is reversible, volume fraction $a (\vec{r})$ can be obtained directly by inversing Equation (5). Otherwise it can be calculated by least square fitting. Therefore, the decomposed coefficient images can be calculated as: $\begin{array}{l} a (\vec{r}) = {\begin{matrix} {\tilde{Φ}}^{- 1} (ϕ, E) \tilde{μ} (\vec{r}), if \tilde{Φ} is i n v e r t i b l e \\ {({\tilde{Φ}}^{T} (ϕ, E) \tilde{Φ} (ϕ, E))}^{- 1} {\tilde{Φ}}^{T} (ϕ, E) \tilde{μ} (\vec{r}), else \end{matrix}, \forall \vec{r} \end{array}$ (6)

Since the least square fitting is a general situation, we use this from in this work.

2.2 “One-step” synthesis method

In literature, different information would be synthesized from decomposition coefficients of component fraction depending on different tasks. One type is to obtain images of virtual monochromatic attenuation for scanned objects which are often used in medical applications. It enables us to choose different contrast demonstration and provides accurate Hounsfield unit. Another type is to obtain images of effective atomic number and electron density. It is more commonly used in public security applications. Because virtual monochromatic attenuation is linearly related to decomposition coefficients so that its properties are rather stable and easy to link to system configurations. Also, it is very convenient to transit virtual monochromatic attenuation to other physical quantities. Therefore, in this work, we evaluate spectral CT images based on virtual monochromatic attenuation images.

Virtual monochromatic images can be easily synthesized from decomposed coefficient maps which obtained from projection-or image-domain decomposition [33]. After component fraction $a (\vec{r})$ is obtained via Equation (6), it is straight forward to synthesis virtual monochromatic attenuation images according to Equation (3) at an arbitrary energy E for each pixel at location $\vec{r}$ . For notation convenience, we also write it in a discretized format: $μ^{mono} (\vec{r}, E) = Φ (E) \times a (\vec{r}), \forall E, \vec{r}$ (7)

Different from $\tilde{Φ}$ , elements in $Φ$ (E) is directly [φ₁ (E) φ₂ (E)] for a certain E. Combining Equation (6) and (7)gives us a “one-step” synthesis equation: $μ^{mono} (\vec{r}, E) = Φ (E) [{({\tilde{Φ}}^{T} (φ, E) \tilde{Φ} (φ, E))}^{- 1} {\tilde{Φ}}^{T} (φ, E) \tilde{μ} (\vec{r})] \forall E, \vec{r}$ (8)

Monochromatic images can be obtained naturally from reconstructed attenuating maps according to Equation (8) with pre-determined basis-function matrix $\tilde{Φ}$ and vector $Φ$ (E).

2.3 Influence factors on the image quality of spectral CT

The performance of a spectral CT could be affected by various factors including source spectrum, detector response, imaging geometry, energy channel configuration, and reconstruction. In this work, we would like to investigate the influence factors in case of x-ray source and detector are fixed.

For convenience, we define the solution of Equation (1) using an implicit reconstruction operator: ${\tilde{μ}}_{i} (\vec{r}) = R^{- 1} {p_{i} (l)}$ (9)

with an reconstruction operator R^-1 (could be inverse Radon transform). By using the chain rule, we can estimate the signal and noise in $μ^{mono} (\vec{r})$ transmitted from p by: ${\bar{μ}}^{mono} (\vec{r}, E) = M \times [\begin{matrix} \bar{R^{- 1} {p_{1}}} \\ \bar{R^{- 1} {p_{2}}} \\ \dots \\ \bar{R^{- 1} {p_{N_{E}}}} \end{matrix}]$ (10)

$σ^{2} {μ^{mono} (\vec{r}, E)} = M Cov (R^{- 1} {p_{1}}, \dots, R^{- 1} {p_{N_{E}}}) M^{T}$ (11)

with ‘transform vector’ $M \equiv Φ (E) {({\tilde{Φ}}^{T} (φ, E) \tilde{Φ} (φ, E))}^{- 1} {\tilde{Φ}}^{T} (φ, E)$ according to Equation (8). Therefore, the signal and noise of synthesized monochromatic images can be directly calculated from reconstructed images according to Equation (10) and Equation (11).

From Equation (10) and Equation (11), we can see that the image quality of a spectral CT is affected by the bias and noise in p_i, the energy channel setting E, the choices of basis functions φ, and the spatial reconstruction operator R^-1.

In addition, quite a few studies [5 , 21] have been focused on the noise propagation of spatial reconstruction operator R^-1 which is not a specific problem for SCT but a general problem for CT reconstruction. Hence, we use a basic FBP reconstruction for avoiding the probable bias and shift-variant effect from various iterative methods [4]. The typical basis function in medical applications are attenuation coefficient curves of basis materials. Therefore, we use different basis materials to study the effect of basis functions. The energy channel setting includes the configuration of NEC and the energy thresholds. The noise in p_i depends on the x-ray source and detector response that are fixed by hardware, and also depends on the energy channel setting. More energy channels will lead to less photons in each energy channel. Changing energy thresholds will distribute photons differently among energy channels.

3 Experimental studies

We carry out simulations as well as practical experiments to examine the influence from the NEC, the selection of basis materials, the energy threshold setting, and contrast agents. According to Equation (10) and Equation (11), we calculate the bias υ^matter and standard deviation σ^matter of a certain matter using neighboring pixels as: $υ^{matter} (E) = \frac{\sum_{\vec{r} \in N {matter}} {\bar{μ}}^{mono} (\vec{r}, E)}{| N {matter} |} - μ * (E)$ (12) $σ^{matter} (E) = \sqrt{\frac{\sum_{\vec{r} \in N {matter}} σ^{2} {μ^{mono} (\vec{r}, E)}}{| N {matter} |}}$ (13)

where N {matter} is the set of neighborhood pixels of a certain matter, and |N {matter} | its size. μ * (E) is the ground-truth attenuation coefficient of a matter at energy E. We introduce spectrum-weighted integrate root mean square error (I-RMSE) for overall quantitative evaluation of image quality: ${\bar{υ}}^{matter} = \sqrt{\frac{\sum_{k = 1}^{K} S (E_{k}) {(υ^{matter} (E_{k}))}^{2}}{K}}$ (14) ${\bar{σ}}^{matter} = \sqrt{\frac{\sum_{k = 1}^{K} S (E_{k}) {(σ^{matter} (E_{k}))}^{2}}{K}}$ (15) ${I - RMSE}^{matter} = \sqrt{{({\bar{υ}}^{matter})}^{2} + {({\bar{σ}}^{matter})}^{2}}$ (16)

Here, discretized K monochromatic energies E_k, k = 1, ⋯, K are considered for the convenience of computation.

3.1 Simulations

We executed our studies on two numerical phantoms shown in Fig. 1. Phantom A is with liquid materials as listed in Table 1:75% Alcohol solution, 8% CH₃COOH solution, Water, 0.9% NaCl solution, 5% Glucose solution, 70% H₂SO₄ solution, Blood, and 10% CaCl₂ solution. All disks in phantom A are in radius of 6 mm. Phantom B is with water of 3 cm in radius as background. Six small disks are of 9, 8, 7, 6, 5, 0.5 mm in radius and with materials: 75% Alcohol solution, Breast, Blood, Bone, Graphite and Al as listed in Table 2.

Fig. 1

Numerical phantoms A and B.

Table 1

Materials in numerical phantom A

Number	Materials
1	75% Alcohol solution
2	8% CH₃COOH solution
3	Water
4	0.9% NaCl solution
5	5% C₆H₁₂O₆ solution
6	70% H₂SO₄ solution
7	Blood
8	10% CaCl₂ solution

Table 2

Materials in numerical phantom B

Number	Materials
1	Water
2	75% Alcohol solution
3	Breast
4	Blood
5	Bone
6	Graphite
7	Fe₃O₄

A 120kVp X-ray source with its spectrum shown in Fig. 2 is modeled in our experiments. Polychromatic CT scan data with Poisson noise are simulated by a homemade toolkit. An ideal PCD with adjustable energy channels are used. In each scan, the iso-center to detector distance is 20 cm and source to iso-center distance is 20 cm. To cover the field of view, 384 detector bins of size 1/ - 30 cm are used. In total, projections at 360 views uniformly distributed over 2π are acquired. Filtered back-projection (FBP) method is used for spatial reconstruction. Without loss of generality, we use basis material decomposition in our spectral CT reconstruction for its ease in change of the basis functions.

Fig. 2

Spectrum of a 120kVp source.

To examine the benefit from multiple energy channels, we experimented on dual-, triple-, quadruple and quintuple-energy imaging. Three kinds of energy channel setting: uniform partition (all energy channel with even photons), k-edge based partition (energy partition with a fixed k-edge point) and average partition (all energy channels with equal width) are studied to examine its influence on image qualities of an SCT. A complete list of energy channel setting in different experiments are in Table 3. Besides, we conducted experiments at full dose (with 1.86e+06 incident photons), 50% dose, 10% dose and 1% dose according to the Zimmerman’s photon setting [36].

Table 3

Configurations of energy channels for simulations

Partition	Energy channels
	Dual	Triple	Quadruple	Quintuple
P1: Uniform	[30, 65]	[30, 60]	[30, 56]	[30, 54]
	[65, 120]	[60, 77]	[56, 64]	[54, 60]
		[77, 120]	[64, 78]	[60, 70]
			[78, 120]	[70, 85]
				[85, 120]
P2: Including K-edge from Gd	[30, 50]	[30, 50]	[30, 50]	[30, 50]
	[50, 120]	[50, 68]	[50, 61]	[50, 60]
		[68, 120]	[61, 78]	[60, 70]
			[78, 120]	[70, 85]
				[85, 120]
P3: Average	[30, 75]	[30, 60]	[30,52]	[30, 48]
	[75, 120]	[60, 90]	[52, 74]	[48, 65]
		[90, 120]	[74, 96]	[65, 84]
			[96, 120]	[84, 102]
				[102, 120]

We use “one-step” synthesis method to obtain virtual monochromatic images. For quantitative analysis, we compute both bias and standard deviation (std) in virtual monochromatic attenuation according to Equation (8–9) for all materials in the phantoms. ROIs are selected within each material region of the phantoms for quantitative metric computation, i.e., there are eight ROIs corresponds to eight materials in phantom A and seven ROIs in phantom B. We present our results with bar charts. In each chart, the height of a bar represents bias and the error bar on its top represents standard deviation (std). This experiment was conducted using uniform energy partition (P1) on both phantoms to investigate the effect of energy channel number.

3.1.1 Results for phantom A

We used Water and Ca to be the basis materials for the decomposition in this study. Our results show that, bias is small for light materials (low attenuating) so that the gain of increasing NEC is mainly in reducing std in high noise case. Bias in SCT is much bigger for heavy materials (high attenuating) and its decrease as NEC increases is quite obvious. We show the results for the light matter C₆H₁₂O₆ solution and heavy matter H₂SO₄ solution in Fig. 3 as examples for illustration. At full dose, the I-RMSE of C₆H₁₂O₆ decreases about 1% from dual-to quadruple-energy while H₂SO₄ about 14%. With lower dose, bias holds stable and std increases as expected. The reduction in I-RMSE of C₆H₁₂O₆ is up to 9% in quintuple-energy compared with dual-energy at the 10% dose level and the I-RMSE reduction for H₂SO₄ is 13%. Hence, we can see the advantage of increasing NEC in all noise levels. Moreover, I-RMSEs decrease almost linearly from dual-energy to quintuple-energy, and the decrease slows down after four energy channels.

Fig. 3

I-RMSE, bias/std and relative decrease for two materials in case of different NEC and dose. Upper row: I-RMSE results. Mid row: energy dependent bias/std as function of energy. Bottom row: relative decrease of I-RMSE. From left to right: representative light material 5% C₆H₁₂O₆ solution and heavy material 70% H₂SO₄ solution.

3.1.2 Results for phantom B

We did same experiments on phantom B as on phantom A but used Water and Fe as basis materials for decomposition considering the materials in phantom B. Similar to results in the phantom A study, the I-RMSE decreases with increasing NEC. Moreover, variance is also dominant in low-attenuation matters while bias is bigger for heavy materials. We show the results of two representative materials, water and bone in Fig. 4. The I-RMSE of water decreases to 5% in quadruple-energy compared with dual-energy at 10% dose and the I-RMSE of bone also decreases about 17% from dual-to quadruple-energy. We found that all std’s in phantom B decrease with the increasing NEC except for Fe₃O₄ as shown in Fig. 5. We think the slightly increase of std’s in Fe₃O₄ is mainly because of severe metal artefacts. Note that Fe₃O₄ is of high attenuation and more energy channels are still beneficial for Fe₃O₄ perfomance since bias is 1.3 times its std while std merely increases about 6% from dual-to quintuple-energy.

Fig. 4

I-RMSE and relative decrease results for phantom B in case of different NEC and dose. Upper row: I-RMSE results. Bottom row: relative decrease I-RMSE. Left column: typical low attenuating matter Water. Right column: typical high attenuating matter Bone.

Fig. 5

I-RMSE results for Fe₃O₄ in phantom B. Only the stds of Fe₃O₄ slightly increases because of severe metal artefacts. Bias is big for heavy material Fe₃O₄ so that I-RMSEs decrease with increasing NEC.

3.1.3 Study on the influence of k-edge materials

To study the influence from k-edge materials, we substituted 5% C₆H₁₂O₆ solution in phantom A with k-edge materials: 5% Gd solution and 10% Gd solution respectively, and executed the same experiments as in Section 3.1.1 with 10% dose. The combination of Water/Gd was chosen for K-edge material decomposition and Water/Ca for others. Energy partition P2 was used considering the k-edge of Gd. The results are shown in Fig. 6. Both bias and std for k-edge materials decrease with increasing NEC, especially the case with 10% Gd solution results in a 78% reduction of I-RMSE for Gd from dual-to triple-energy. For the rest of materials, I-RMSE gets higher as the concentration of Gd increases. The I-RMSE of NaCl in quadruple energy channel increases about 13% with 5% Gd solution while 50% with 10% Gd solution. Therefore, the introduction of k-edge materials will increase overall I-RMSE in surrounding materials. Higher concentration of contrast agents brings bigger errors.

Fig. 6

I-RMSE for phantom A in case of including and excluding k-edge materials. Left column: k-edge matter Gd solution of different concentration. Right column: representative material 0.9% NaCl solution.

3.1.4 Study on the different partitions of energy channels

The partition of energy channels is also of people’s great interest. We repeated our experiment on phantom A at 10% dose in Section 3.1.1 using different partitions of energy channels. Three partitions as listed in Table 3 were used. Our results show that I-RMSE decreases as NEC increases in all energy partition schemes. Gd Partition (P2) is slightly superior to uniform Partition (P1). Average Partition (P3) is an inappropriate choice for this case. Representative results are shown in Fig. 7. We can see that an appropriate energy partition could result in better image quality for scanned materials. Energy threshold optimization is necessary for SCT.

Fig. 7

I-RMSE results for phantom A in case of different partition of energy channels. Left column: typical light matter Blood. Right column: typical heavy matter 10% CaCl₂ solution.

3.1.5 Study on the influence of basis-material combinations for decomposition

We tested six more combinations of basis materials for our experiment in Section 3.1.1 at 10% dose. We can see that the decrease of I-RMSE with the increase of NEC follows similar trend for different basis-material choices (as shown in Fig. 8). The integrated results do not show obvious difference as basis material combination changes.

Fig. 8

I-RMSE results for Water in phantom A in case of different basis material combinations. All combinations have similar I-RMSEs.

For better demonstration, we computed relative error (R-Err) of a certain matter at each monochromatic energy:

$ɛ^{matter} (E) = \sqrt{{(υ^{matter} (E))}^{2} + {(σ^{matter} (E))}^{2}}$ (17)

${R - Err}_{i}^{matter} (E) = \frac{ɛ_{i}^{matter} (E)}{ɛ_{ref}^{matter} (E)}$ (18)

where i denotes i^th basis-material combination and subscription “ref” indicates the reference basis-material group. Here, water/Ca group is chosen as the reference group. The results are plotted in Fig. 9. Different basis material groups result in big difference for μ^mono (E) at low energy (<40 keV in our experiments). The R - Err in Group C/Fe is 1.13 times of that in group water/Ca, whereas it is 11% lower in group C/water than in group water/Ca. The error in μ^mono (E) in light matter using different basis material combinations is similar for higher energy. The error differs more with different basis material combinations for heavy matters. The fluctuation of ${R - Err}_{i}^{matter} (E)$ in H₂SO₄ for C/Al varies from 0.97 to 1.07. Moreover, optimal basis material group may vary for μ^mono (E)at different E. Therefore, basis materials need to be carefully selected based on targeted matters as well as the energy of interest.

Fig. 9

R-Err results for phantom A in case of different basis material combinations. Left column: light material water. Right column: heavy material 70% H₂SO₄ solution.

3.2 Practical experiments

We use two phantoms, as is shown in Fig. 10, in practical experiments for validation. Phantom C is consist of four plastic cylinders of 2 cm in radius: PTFE, PET, PC and POM. Phantom D have three kinds of liquids of 15 mm in radius: 10% CaCl₂ solution, 20% NaCl solution, 5% NaCl solution, and 8% C₆H₁₂O₆ solution. The laboratory SCT system is displayed in Fig. 11. The X-ray source is 100kVp/0.5 mA. The source to iso-center distance is 485 mm and the source to detector distance is 747 mm. The PCD have 1536 pixels of 0.1 mm pitch. Projection data over 2π is collected using the energy partition listed in Table 4. We again use FBP for spatial reconstruction. Alcohol and Al are used as basis materials for decomposition. I-RMSE as well as R-Err of a certain matter in phantoms are calculated for quantitative analysis. Virtual monochromatic images at 50 keV and 70 keV are synthesized for both phantoms and displayed in Fig. 12.

Fig. 10

Practical phantoms C and D.

Fig. 11

Laboratory SCT system.

Table 4

Configurations of energy channels for practical experiments

Energy channels	Dual	Triple	Quadruple	Quintuple
Partition	[24, 46]	[24, 44]	[24, 36]	[24, 34]
	[46, 100]	[44, 56]	[36, 48]	[34, 44]
		[56, 100]	[48, 60]	[44, 54]
			[60, 100]	[54, 63]
				[63, 100]

Fig. 12

Virtual monochromatic images at 50 keV and 70 keV for (a) Phantom C and (b) Phantom D.

The bias of phantom C, as is displayed in Fig. 13, is rather small which indicates that the synthesized monochromatic images are accurate as expected. However, stds of phantom C increases from triple to quintuple energy channels. Similarly, the bias of phantom D is also small compared with its std. The noise level is the main concern in this case. The std also starts to increase when the NEC is four.

Fig. 13

I-RMSE results for practical phantoms. Left column: I-RMSEs for phantom C. Right column: I-RMSEs for phantom D.

We use another basis material group Water and 10% CaCl₂ solution to decompose phantom C for contrast. The results for representative matter PTFE are shown in Fig. 14. Without loss of generality, we only examine the dual energy case. This suggests that the group Water/10% CaCl₂ solution are better basis materials for PTFE overall. Moreover, the R-Err curves confirm that the error of μ^mono (E) varies rather significantly at different E. A careful choice of basis material combination can improve the monochromatic image quality at a certain energy for targeted matter.

Fig. 14

I-RMSE and R-Err results for PTFE in phantom C in case of different basis material combinations at dual energy.

4 Discussion and conclusion

In this work, we carefully examined the multiple factors influencing the spectral CT imaging. Virtual monochromatic images are used for performance evaluation by the I-RMSE. Our study shows that increasing the number of energy channels will help to reduce the bias in reconstructions in all situations of different phantoms, different noise levels, as well as different energy partitions. The difference in multi-energy I-RMSE from dual-energy case is statistically significant according to our calculation of p-values (≤0.05) for our simulation study. The advantage of multi-energy (>2 energy channels) be justified by two facts. Firstly, narrowing the width of each energy channel will lead to data closer to monochromatic energy case and hence lowers the error in the synthesized virtual monochromatic attenuation coefficients of scanned objects. Secondly, since objects are characterized by nonlinear energy-dependent attenuation coefficients, multiple narrow windows would help us to catch this energy-dependency, i.e., we might be able to extract more accurate information from more energy channels with narrow windows. The noise reduction by increasing NEC in simulation accords with the results published by Roessl [24] except for Fe₃O₄ in phantom B because of severe metal artefacts. These results tell us that an optimal decomposition can well absorb information from multiple energy channels in most cases even though noise-level in each energy channel becomes higher with increasing NEC. However, metal artifacts caused by high attenuating matter is disastrous to post-processing methods and extremely noisy data in multi-energy channels aggravate the artefacts. In addition, the results in practical experiments also indicate that the gain of error increases with the number of energy channels. However, the gain disappears above three channels. We think this is because of the limitation in spectral resolution and energy calibration for a practical PCD. These systematic error could be bigger than the gain of additional energy channels over three. Hence, according to our results, three or four energy channels might be a good choice for practical applications. It would need much higher performance PCD detectors to cash the gain of more energy channels than that.

We also examined the influence of k-edge materials, energy-window partition, and basis-material combination. K-edge materials would increase the error of overall images. Higher concentration of Gd contributes to a big increase of I-RMSE. The experiment with three different energy partitions tells us that optimal energy partition could result in high accuracy and low noise. Moreover, the basis-material choice shall vary according to targeted imaging matters, which could help to improve the quality of virtual monochromatic images at energies of interest.

In summary, we in this work used basic FBP to emphasize the effects from systematical configuration. We will further our study to optimize image reconstruction for practical application of multi-energy spectral CT systems in future work.

Footnotes

Acknowledgments

This work is supported by grants from the National Natural Science Foundation of China (No. 61771279 and No. 11435007), National key research and development program of China (No. 2016YFF0101304), and the National Science Fund for Distinguished Young Scholars (No. 11525521).

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