2.1

Dynamic Dual-Energy CT Mode

The SME CT diagram is shown in Figure 1a, and the DDE CT diagram is shown in Figure 1b. In SME CT, there are Nk energy thresholds for Nk-energy CT imaging. However, in DDE CT, there are only two energy thresholds. The low-energy threshold is unchanged and fixed, while the high-energy threshold changes among different preset values. When a photon is injected into the detector, it will be counted either in the low-energy bin or in the high-energy bin.

Figure 1.

Diagrams of the SME CT (up) and DDE CT (down) [4].

The high-energy thresholds change randomly for different detector pixels and different scan views. The preset values for the high-energy thresholds are set according to the requirement of multi-energy imaging. Nk-energy imaging requires Nk -1 preset values for the high-energy thresholds. These values are the same as the values of energy thresholds in an SME CT that aims for the same Nk-energy imaging.

2.2

Iterative Dynamic Dual-Energy CT Algorithm

In a DDE CT scan, there are two measurements i_{low, j} and i_{high, j} for the transmitted photons of the j_th ray. Supposing the measurements follow a Poisson distribution, the corresponding random variables of the measurements I_{low, j} and I_{high, j} are equal to:

In the above formula, I_0,k are the incident photons of the k_th-energy bin, which is the same for all X-rays. is the system matrix. N_j is the number of X-rays. N_i is the number of pixels in a reconstruction image. X_k is the accurate, noise-free reconstruction image of the k_th energy bin. [AX_k]_j represents the line integral of the j_th X-ray. N_k is the number of total energy bins. N_th,j is the number of preset values for the high-energy threshold at the j_th ray.

The iterative DDE CT algorithm for calculating the multi-energy projections from the DDE CT data can be concluded to the following two steps in one iteration:

Step 1: Update the multi-energy transmitted photons using the last updated multi-energy CT results and the low- and high-energy measurements i_{low, j} and i_high,,j.

Step 2: Update the multi-energy CT images with the multi-energy transmitted photons .

For step 1, the multi-energy transmitted photons can be calculated according to the following formula:

For step 2, the SIRT [5] algorithm is used for CT reconstruction of all energy bins. The initial values for the multi-energy CT image are set to the values of the mono-energetic CT image, which can be reconstructed from the mono-energetic projections i_low,j+i_high,j.

3.1

The Property of Poisson Random Variable

In this section, we first discuss a property of the Poisson random variable. Considering a Poisson random variable X, the probability density function for the random variable X is:

According to (3), the expectation and variance of the random variable X are equal to:

Consider another Poisson random variable Y, where the expectation of Y is tλ (0 < t < 1). According to (4) and (5), the expectation and variance of Y are:

Consider a new random variable Z. Its definition is:

The probability density function for Z is:

The expectation and variance of Z are:

Random variables Y and Z have the same expectation, while the variance of Z is smaller than the variance of Y. The conclusion for the above derivation is: the variance of a Poisson random variable (Y) is larger than the variance of a random variable (Z), if this variable (Z) is calculated from the Poisson random variable with a larger expectation (X).

3.2

The Noise Model of Dynamic Dual-Energy CT

In this section, we drive the noise model of the DDE CT data and compare it to the SME CT data. Considering an N_k-energy SME CT, the number of transmitted photons in the k_th-energy bin I_k is a Poisson random variable. The number of transmitted photons of all energy bins I_total is also a Poisson random variable, and they satisfy the following relationship:

Therefore, the expectations of these random variables satisfy the following relationship:

If the expectation of the number of total transmitted photons is λ₀, according to (6), the expectations and variances for the number of transmitted photons in different energy bins are:

Now consider the DDE CT data. In DDE CT, the number of transmitted photons of the low-energy bin and the high-energy bin I_low and I_high are Poisson random variables and satisfy the following relationships:

Because the number of transmitted photons measured in the low-energy bin and the high-energy bin in a DDE CT are Poisson random variables, according to (13) and (15), their expectations and variations are equal to:

In DDE CT, the number of transmitted photons of multiple energy bins is not obtained from direct measurement but is calculated from the low- and high-energy bin data:

is still a random variable and represents the number of transmitted photons of the k_th energy bin. According to (9), (10), and (16), the expectation and variance of are:

Comparing the expectation and variance of the number of multi-energy transmitted photons in SME CT and DDE CT, we can obtain the following relationship:

For all k, the following inequalities always satisfy:

Therefore, we obtain the following conclusion: compared to the multi-energy CT transmitted photons I_k that are directly measured from an SME CT, the multi-energy transmitted photons , which are calculated from the DDE CT data, have the same expectation, but the variance is lower.

3.3

Verification

To verify the theory proposed in Section 3B, we simulate a large amount of data. We calculate the variances of the multi-energy CT transmitted photons for these simulation results and verify whether the relationship between the variances of the SME CT and the DDE CT is consistent with the theory.

A 20 cm diameter water cylinder is scanned in the simulation. The number of energy bins is 8. The variance images of the third-energy-bin SME CT transmitted photons and the third-energy-bin DDE CT transmitted photons are calculated from 1000 samples and shown in Figures 2a and 2b, respectively. If the proposed theory is correct, the following formula should be satisfied:

Figure 2.

The variance images and calculation results of the third energy bin. 2a and 2b are variance images of the SME CT and DDE CT, respectively. 2c is the ratio image. 2d is the calculation result. The display windows for 2d is [0.5, 1.5].

k is the number of energy bins. j is the number of X-rays. The ratio image of the third energy bin is shown in Figure 2c. The third-energy bin calculation results are shown in Figure 2d. The mean value of Figure 2d is 1.0022, which is extremely close to 1. This result verifies the correctness of the noise model proposed in Section 3B.

4.1

Calculating Accurate t_k in the Dynamic Dual-Energy CT

Section 3 proves that the noise of projection calculated from DDE CT is lower than that calculated from SME CT. However, there is a premise for this conclusion: the ratios between the number of transmitted photons of a specific energy bin and all energy bins are accurately known, e.g., t_k is known and accurate for any k. In theory, t_k can be accurately calculated only if the noise-free transmitted photons of different energy bins are known. t_k cannot be calculated directly from the original DDE CT data.

The iterative DDE CT algorithm, which is described in Section 2B, is proposed to calculate accurate t_k. When the algorithm converges, (2) will be rewritten as:

is the convergent value for the multi-energy transmitted photons calculated from DDE CT and is an observation of the random variable defined in (17). Therefore, according to the conclusion in Section 3B, has lower statistical noise than i_k,j, which is the transmitted photon measured from the SME CT. In other words, the multi-energy projections calculated from the DDE CT data are less noisy than the SME CT if the iterative DDE algorithm converges.

4.2

Convergence of the Iterative Dynamic Dual-Energy Algorithm

The convergence of the iterative DDE algorithm is verified through simulation. The XCAT thorax phantom [6] is scanned in the simulation. The simulation configurations are shown in Table 1.

Table 3.

Configuration of the simualtion

The noise of the projections is measured in terms of the relative error, which is defined as:

prj_truth is the ground truth of the line integrals of multi-energy CT, which can be obtained from the noise-free SME CT. In the analysis of the convergence, the relative error is calculated for line integrals of three types: the multi-energy line integrals measured in SME CT, the multi-energy line integrals calculated from DDE CT data, and the convergent values for the multi-energy line integrals from DDE CT. If the algorithm converges, the second type of relative error will decrease to the same value as the third type of relative error.

The convergence curve is plotted in Figure 3, which shows that the algorithm converges. This illustrates that the iterative DDE algorithm can accurately calculate t_k. Moreover, the relative errors of the multi-energy line integrals from the DDE CT are much smaller than the relative errors of the multi-energy line integrals from the SME CT, which indicates that a less-noisy multi-energy projection can be calculated from the DDE CT.

Figure 3.

The convergence curve.