Assessment of multi-energy inter-pixel coincidence counters for photon counting detectors at the presence of charge sharing and pulse pileup: a simulation study

Abstract

Purpose:

Spectral distortion due to charge sharing (CS) and pulse pileup (PP) in photon counting detectors (PCDs) degrades the quality of PCD data. We recently proposed multi-energy inter-pixel coincidence counters (MEICC) that provided spectral cross-talk information related to CS. When PP was absent, the normalized Cramér–Rao lower bounds (nCRLBs) of 225-μm pixel PCDs with MEICC was comparable to those of 450-μm pixel PCD without MEICC. The aim of this study was to assess the performance of PCDs with MEICC at the presence of both CS and PP using computer simulations.

Methods:

An in-house Monte Carlo program was modified to incorporate the following four temporal elements: (1) A pulse shape with a pulse duration of 20 ns, (2) delays of up to 10 ns in anode arrival times when photons were incident on pixel boundaries, (3) offsets proportional to a vertical separation between the primary and secondary charge clouds at the rate of ±4 ns per ±100 μm, and (4) a stochastic fluctuation of anode arrival times for all of the charge clouds with a standard deviation of 2 ns. We assessed the performance of five PCDs, (a)–(f), for three spectral tasks, (A)–(C): (a) The conventional PCD, (b) a PCD with MEICC, (c) a PCD with one coincidence counter (1CC), (d) a PCD with a 3×3 analog charge summing scheme (ACS), and (e) a PCD with a 3×3 digital count summing scheme (DCS); (A) conventional CT imaging with water (i.e., linear attenuation coefficient maps), (B) water–bone material decomposition, and (C) K-edge imaging with tungsten. The tube current was changed from 1 mA to 1,000 mA and the nCRLB was assessed.

Results:

The recorded count rate curves were fitted by the non-paralyzable detection model with the effective deadtime parameter. The best fit was achieved by 25.8 ns for the conventional PCD, 18.6 ns for MEICC and 1CC, 140.5 ns for ACS, and 209.0 ns for DCS. The nCRLBs were strongly dependent on count rates. MEICC provided the best nCRLBs for all of the imaging tasks over the count rate range investigated except for a few conditions such as K-edge imaging at 1 mA. PP decreased the merit of MEICC over the conventional PCD in addressing CS. Nonetheless, MEICC consistently provided better nCRLBs than the conventional PCD did. The nCRLBs of MEICC were in the range of 49–58% of those of the conventional PCD for K-edge imaging, 45–76% for water–bone material decomposition, and 81–88% for the conventional CT imaging (i.e., linear attenuation coefficient maps). ACS provided better nCRLBs than the conventional PCD did only when the effect of PP was minor (e.g., when the counting efficiency of the conventional PCD was higher than 0.95 with the tube current of up to 100 mA).

Conclusion:

Besides a few cases, MEICC provides the best nCRLBs for all of the tasks at all of the count rates. ACS and DCS provides better nCRLBs than the conventional PCD does only when count rates are very low.

1. Introduction

Photon counting detector (PCD)-based x-ray computed tomography has great potential in many clinical applications,^1–3 and prototype systems have shown excellent performances in phantom and clinical studies.^1,4–6 One of the challenges in PCDs, however, is spectral distortion due to charge sharing (CS) and pulse pileup (PP).^2,7 While non-spectral PCDs with no energy resolution can replace the current energy-integrating detectors, spectral photon counting detectors provide many more clinical merits such as monoenergetic CT images, dual-energy applications, material-specific imaging, and K-edge imaging. Thus, it is critical to address the effect of the spectral distortion. A post-acquisition compensation method for spectral distortion such as Refs. 8,9 may be able to eliminate the bias when the problem is well-conditioned and accurate models for detectors and statistics are used; however, the output of such methods will have higher noise than the output obtained from a (hypothetical) PCD with no CS and no PP.¹⁰

We will use a loose definition of CS in this paper, which includes all of the events that split the incident energy into multiple PCD pixels. When a photon is incident onto a PCD between two pixels, an electronic charge cloud generated by the photon may be split between the two pixels, which will record one count each at lower energies than the original. When a photon interacts with a PCD via the photoelectric effect, a fluorescent x-ray may be emitted and re-absorbed, splitting the energy into two charge clouds. Multiple Compton scattering interactions produce multiple charge clouds. These charge clouds may be detected by more than one PCD pixel. The loose definition of CS includes all of these phenomena that are related to the detection process but independent of the speed of PCDs.

In contrast, PP is attributed to a finite speed of signal formation and recording at PCDs. At high count rates, multiple photons may arrive at PCDs with small time intervals between them, which may result in multiple pulses being piled-up. Slow PCDs cannot resolve the integrated pulse and may produce only one count at a higher energy than the incident photons. PP is negligible at low count rates but is a dominant cause of spectral distortion at high count rates. Incident count rates onto PCDs for clinical x-ray computed tomography may be very high, which is expected to go up to the order of 10⁸ counts per second (cps) per mm²; therefore, it is critical to address PP.

It is challenging to address both CS and PP by simply optimizing design parameters of PCDs. Most of PCDs designed for computed tomography incorporate pulse height analyzers with multiple energy thresholds, which simply count every rising pulse crossing the corresponding energy threshold. It is difficult to implement complex processing because incident count rates onto PCDs may be very high. Major design parameters for pulse height analyzers include the pixel size and the pulse shaping time. Regarding the first parameter, the pixel size, a smaller pixel will have less PP because a smaller pixel area decreases the incident count rates per pixel, but it will have severer CS, which may distort recorded spectra. In contrast, a larger pixel may have less CS at the expense of more adverse effect from PP. Regarding the second parameter, the pulse shaping time, a shorter pulse shaping time makes PCDs faster, thus, is effective in mitigating PP; however, it may make CS worse for the following reason. When a fluorescent x-ray is emitted and absorbed by the same PCD pixel, two charge clouds are generated at two different locations, for example, separated by ~100 μm on average for cadmium telluride PCD.² The two electron clouds will arrive at the anode quasi-coincidentally with the time separation of ~4 ns,² generating two pulses nearly piled-up. The pulse shaping time needs to be sufficiently long in order to integrate the two signals and count as one event. A PCD with a very short pulse shaping time such as 2 ns would end-up resolving the two pulses, recording two counts at lower energies than the incident photon energy, and creating CS problems. As a result, many PCD designs aim at balancing the effects of CS and PP by selecting a moderate pixel size and a reasonably short pulse shaping time.

PCDs that handle CS include an event-based real-time CS correction^11–24 or rejection^25–27 and a reading-based post-acquisition correction/compensation.^28–31 The analog version of event-based real-time CS correction^11,15,19 is called analog charge summing scheme (ACS) and the digital version^13,14,19 is called digital count summing scheme (DCS) in this paper. Both of them function implicitly as follows. For each event, they use a sophisticated inter-pixel communication to recognize a coincidence event, identify the primary incident pixel and the corresponding energy window, and add a count to the associated counter. ACS reconstructs the incident energy accurately; however, it needs to hold onto analog charges until each event is processed and the processing takes time, which, in turn, degrades the detector’s counting capability by a factor of 5–10.^11,15,19 In contrast, DCS digitizes each event into a count in an energy window up-front and the coincidence process keeps and processes digital counts. The accuracy of energy reconstruction may be limited when energy window widths are large; however, DCS is claimed to be much faster than ACS.^19,26

The reading-based post-acquisition correction/compensation that are known are multi-energy inter-pixel coincidence counters (MEICC)^30,31 and one coincidence counter^28,29 (1CC). Both MEICC and 1CC have two sets of counters, one set being N_C primary counters to count the number of up-crossing events within each of N_C energy windows and the other set being coincidence counters to record the number of CS events between one pixel and its neighboring pixels. MEICC has N_C×N_C coincidence counters to capture the “spectral information” of CS events between each pair of energy windows, whereas 1CC has one coincidence counter to record the “intensity” of CS events. When the PP was absent, the performance of 225-μm pixel PCDs with MEICC was comparable to that of 450-μm pixel PCD without MEICC.³¹

It is expected that when PP is present, the counting capability of PCDs with MEICC will be as high as the conventional PCDs because the coincidence counters of MEICC will not interfere with the primary counting process.³⁰ Its ability to tackle CS will remain effective as long as the coincidence counters provide useful information about CS, and will reach the lower limit at the conventional PCDs when the coincidence counters become totally meaningless. In contrast, the CS correction schemes in both ACS and DCS are intricately integrated with the primary counting process, and thus, their abilities to both count photons and reconstruct energies may be degraded at high count rates. No side-by-side comparison has been performed between ACS, DCS, and MEICC, and such an assessment at various count rates is of interest.

Therefore, the purpose of this study was to assess the performance of the conventional PCD, MEICC, 1CC, ACS, and DCS at the presence of both CS and PP using computer simulations. The paper is structured as follows. In Section 2, we outline specific implementations of the five PCDs and simulation methods. We present the results in Section 3, discuss relevant issues in Section 4, and conclude the paper in Section 5.

2. Methods

We outline specific implementations of the five PCDs in Sec. 2.A, the Monte Carlo (MC) simulation methods in Sec. 2.B, and the assessment schemes for the spectral responses in Sec. 2.C, the counting capabilities in Sec. 2.D, and the imaging task-specific performances in Sec. 2.E.

2.A. Five PCDs

In this section, we outline our implementation of the five PCDs assessed in this study.

Conventional PCD:

The conventional PCD had N_C counters for N_C energy windows. It counted the number of up-crossing events for each energy thresholds and the counting process became active as soon as the pulse fell under the corresponding threshold energy. After data were read-out, counts within energy windows were computed by subtracting the output of adjacent thresholds.

MEICC:

MEICC consisted of two parts (Fig. 1), the primary counting part with N_C counters (the components outside the red dashed curve in Fig. 1) and the coincidence processing part with N_C×N_C coincidence counters (the components inside the red dashed curve in Fig. 1). The primary counting part used a direct windowing scheme as follows (see also Fig. 2). An up-crossing of energy threshold j changed the status of the window j to 1 (s_Wj →1) and started a timer for the window, t_Wj. When the timer t_Wj reached a pre-set time T_W, either of the following process took place. If the upper threshold’s status s_Wj+1 was also 1, it reset the status to 0 and did nothing (as the pulse peak was above this energy window). If the status s_Wj+1 was 0 (thus, the pulse peak belonged to this window), it added a count to the primary counter C_j, sent the counting signal to eight neighboring pixels, reset the status s_Wj, set the status s_MAj to 1, and started a timer for the coincidence process for window j, t_MAj, and waited for the next counting event.

Figure 1.

The basic architecture of a PCD pixel with MEICC with the number of thresholds N_C=3 as an example with subscripts L, M, H refer to low, middle, and high energy window, respectively. The neighboring pixels are treated as one area: one or more counts at any of the neighbor pixels will produce the same input to the AND logics. C_X and CC_XY are counters and coincidence counters, respectively, for a pixel-of-interest (POI), and C_X and CC_XY with underlines are those for one of neighboring pixels. Notice that a counter is associated with a two-sided energy window (not a one-sided threshold window) in this study. An underline indicates that it belongs to neighboring pixels. When a count is added coincidently to counter C_X of the POI and counter C_Y of one of neighboring pixels, a count is also added to CC_XY of the POI. In this case, a count is also added to a coincidence counter of the corresponding neighbor pixel, CC_XY, as the circuitry for MEICC in each pixel processes coincidences with its own neighboring pixels independently and in parallel. See Fig. 2 for the timing chart. This figure is from Ref.³⁰.

Figure 2.

A timing chart for the primary counting processing and coincidence processing for MEICC. In this example, pixel i adds a count at the primary counter of window 3, C₃, pixel i′ adds a count at window 1, C₁, and the counting signals are sent to each other (by the red arrow and the magenta arrow, respectively). The coincidence processing is performed at both of the pixels independently and one coincidence count each is added to a coincidence counter CC₃₁ at pixel i and CC₁₃ at pixel i′. Refer to the text in Sec. 2.A for more explanation.

The coincidence processing part functioned as follows (see Fig. 2). When a counting signal for energy window J came from one of eight neighbors, and if the status of the coincidence process s_MBJ was 0, it changed the status to 1 and started a timer t_MBJ. When either t_MAj or t_MBJ reached a preset time T_M, if both s_MAj and s_MBJ were 1, it added a count to the corresponding coincidence counter, CC_jJ, reset the status s_MAj and s_MBJ at the next time tick, and waited for the next coincidence event. See also Ref.³⁰ for more explanation on the coincidence counters.

1CC:

Similar to MEICC, 1CC consisted of the two parts. The primary counting part was the direct windowing with N_C counters as outlined above. It sent an up-crossing signal of the lowest energy threshold to eight neighboring pixels, set the status s_1A to 1, and started a timer for the coincidence process, t_1A, and waited for the next counting event.

The coincidence processing part with one coincidence counter functioned as follows. When an up-crossing signal (for the lowest energy threshold) came from one of eight neighbors, and if the status of the coincidence process s_1B was 0, it changed the status to 1 and started a timer t_1B. When either t_1A or t_1B reached a preset time T₁, if both s_1A and s_1B were 1, it added a count to the coincidence counter, CC, reset the status s_1A and s_1B at the next time tick, and waited for the next coincidence event.

ACS:

We used a straightforward 3×3-pixel ACS scheme, which functioned as follows (see also Figs. 3A–3D). An up-crossing event of the lowest threshold of a pixel i (pixel 5 in Fig. 3A) triggered the following process if its counting status, s_Ai, was 0. Up-crossing events were ignored if s_Ai = 1. It changed the status of all of the 3×3 pixels centering at the pixel i to 1 and started a timer t_Ai. If pixel i′ was already in the counting state (s_Ai′ = 1), pixel i′ and the 2×2 super-pixels that included pixel i′ were excluded from this event. For each of the 3×3 pixels included, it started recording the largest energy of the pulse train above the lowest threshold energy (Fig. 3B). When the timer t_Ai reaches a preset time T_A, it computed a sum of the energies for each of four 2×2 super-pixels (Fig. 3C). It then selected the winner super-pixel with the largest summed energy, selected the winner pixel with the largest deposited energy (Fig. 3D), and added a count to an energy window that corresponded to the energy of the winner super-pixel. Finally, it reset the status of the counting process of all of the 3×3 pixels included and waited for the next event. In an example presented in Fig. 3, the incident energy of 80 keV was correctly reconstructed and one count was correctly added to the energy window 3 of pixel 2. A tie between super-pixels did not affect the final winners; and a tie in deposited energies did not occur because they were continuous analog signals.

Figure 3.

The event processing schemes for ACS (A–D) and DCS (E–H). In this example, the PCD had four energy thresholds at 20, 45, 70, and 95 keV, which made the effective energies for four windows 32.5, 57.5, 82.5, and 107.5 keV. An incident photon carrying 80 keV was incident on the boundary of pixels 2 and 5. For ACS, the largest energy recorded above the lowest threshold 20 keV during the event processing time T_A was 38 keV at pixel 2 and 42 keV at pixel 5 (B). The 2×2 super-pixel energy was 80 keV for both k=1 and 2, 42 keV for both k=3 and 4 (C). The winner super-pixel was k=1, the winner pixel was pixel 5, and the winner window was 3 that corresponded to the energy of super-pixel k=1, 80 keV (D). For DCS, the energy window that gained a count during the event processing time T_D was window 2 at both pixel 2 and pixel 5 (F). The 2×2 super-pixel energy was 65 keV (=32.5+32.5 keV) for both k=1 and 2, 32.5 keV for both k=3 and 4 (G). The winner super-pixel was k=1, the winner pixel was pixel 2 (instead of pixel 5), and the winner window was window 2 (instead of window 3) that corresponded to the energy of super-pixel k=1, 65 keV (H). A random number was used for a tie-breaker. See text in Sec. 2.A for more explanation.

DCS:

We used a straightforward 3×3-pixel DCS scheme, which functioned as follows (see also Figs. 3E–3H). Up-crossing the lowest threshold of the pixel i (pixel 5 in Fig. 3E) triggered the following process if its counting status, s_Di, was 0. Up-crossing events were ignored if s_Di = 1. It changed the status of all of the 3×3 pixels centering at the pixel i to 1 and started a timer t_Di. If pixel i′ was already in the counting state (s_Di′ = 1), pixel i′ and the 2×2 super-pixels that included pixel i′ were excluded from this event. For each of the 3×3 pixels included, it started recording the energy window j that gained a count with the direct windowing scheme outlined before (Fig. 3F). When the timer t_Di reached a preset time T_D, it selected the winner super-pixel which had the largest sum of the effective energies of the windows with a count, and selected the winner pixel with a count at the highest energy window among the 2×2 pixels (Fig. 3H). It then determined the winner energy window that corresponded to the sum of the effective energies, added a count, reset the status for all of the included pixels, and waited for the next event. Here, the effective energy of the window is defined as the mean of the two boundary energies. A random number was used for a tie-breaker. In an example presented in Fig. 3, the winner pixel was pixel 2 instead of pixel 5, and a count was added to energy window 2 as the sum of the effective energies was 65 keV (=32.5+32.5), whereas the incident energy of 80 keV belonged to energy window 3.

The implementation methods outlined above are the most straightforward in our opinion and we chose them to study generic characteristics of the five PCDs. Our implementation of 1CC was different from Hsieh’s,²⁹ and it was chosen to assess the difference between 1CC and MEICC only due to the number of coincidence counters. Other implementations will be discussed in Discussion.

2.B. Monte Carlo (MC) simulator

We used the Monte Carlo simulation program developed for and validated by the previous studies,^30,31 which cascaded the following processes: (1) photon generations with randomized energies and time intervals for a Poisson distribution, (2) CS based on a randomized incident location and interaction and detection processes, (3) pulse train generations with electronic noise, (4) comparator detection signal generation with up-crossing events of energy thresholds. Photon generations and the detection processes were performed for each of 7×7 PCD pixels in parallel. The time tick of pulse trains was 1 ns.

The following four temporal elements were added for this study to the MC program and performed between processes (2) and (3) discussed above. The first element was a finite pulse shape. A slightly asymmetric monopolar pulse shape with a pulse duration of 20 ns at the full-width-at-tenth-maximum was created by connecting two normalized Gaussian functions at the peak; the Gaussian parameters (i.e., the standard deviations) were 3.84 ns for the rising part of the pulse and 5.48 ns for the falling part. The second element was a pixel boundary effect. An electric charge cloud generated near the pixel boundaries traveled a longer distance to the anode than the one generated near the pixel center did; therefore, when the two clouds were generated simultaneously, the former cloud arrived at the anode after the latter one did. This location-dependent delay of up to 10 ns at the pixel boundaries (see Fig. 4) was added to the anode arrival times. The third element was a vertical separation of two charge clouds. When a fluorescent x-ray was emitted and absorbed by a PCD pixel, the primary and secondary charge clouds arrived at the anodes at slightly offset times. The offset was proportional to the vertical separation between the two clouds at a rate of ±4 ns per ±100 μm. The fourth element was a stochastic fluctuation of anode arrival times for all of the charge clouds, which was realized by a zero-mean Gaussian distribution with a standard deviation of 2 ns.

Figure 4.

A contour plot of the pixel boundary effect, i.e., the delay of anode arrival times (ns), up to 10 ns toward the boundaries of pixels.

We believe that parameters for these temporal elements were reasonable choices based on our experiences as explained in the following. The peaking time (which is about a half of the pulse duration) of various PCDs currently known are in the range of 8–1,000 ns;³² thus, the pulse duration of 20 ns is at a lower end of the available PCDs. The pixel boundary delay of PCDs may depend on various factors such as the uniformity of the electric field strengths and the variation of travel lengths, which can be summarized as the small pixel effect.³³ We could not find any literature on the delay measurements; and therefore, the maximum delay of 10 ns was determined based on a preliminary study. When the delay parameter was larger than 10 ns, the monoenergetic x-ray responses had unusually higher K-escape peak and fluorescence peaks. We think that detector physicists will design the detector to minimize the pixel boundary effect and a half of the pulse duration simulated in this study might be an upper-end for well-designed PCDs. The effect of vertical separation depends on the electric field strength and the mobility of electron in detector materials; and 4 ns for 100 μm separation seemed reasonable for ~500 V/mm bias.^2,34

The following settings were used unless otherwise specified. We generated a 140 kVp x-ray spectrum using TASMIP/spektr^35,36 operated at various tube current values from 1 mA to 1,000 mA, which corresponded to the incident count rates of 4.1×10⁴–4.1×10⁷ cps/pixel or 9.6×10⁵–9.6×10⁸ cps/mm². A cadmium telluride PCD had 7×7 pixels with a pixel size of (225 μm)², a thickness of 1.6 mm, and four thresholds (N_C=4) set at (20, 45, 70, and 95 keV). A preliminary study was conducted to confirm that they were close to the optimal threshold energies for various count rates and different tasks. A time duration per reading was varied to make the tube current–time product per reading constant at 2×10⁻² mAs (e.g., 200 μs for 100 mA), which made the expected number of incident photons independent of tube current values, which, in turn, resulted in consistent accuracy in the use of multivariate normal distribution for computing the Fisher information matrix (see Sec. 2.E). The electronic noise added to the pulse train had a standard deviation of 2.0 keV. Other parameters can be found in Ref.³⁷.

The time window parameters outlined in Sec. 2.A were set as follows. T_W for direct windowing used for MEICC, 1CC, and DCS was set at 10 ns, which was a half of the pulse duration (20 ns). The default value of T_M, T₁, T_A, and T_D were determined by the assessment of the spectral responses (see Sec. 2.C and Sec. 3.A) and were set at 20 ns. The effect of different values for T_M, T₁, T_A, and T_D was also assessed.

2.C. Spectral responses

The spectral responses of ACS, DCS, and the conventional PCD were assessed using flat-field monoenergetic x-rays at 80 keV incident onto 7×7 PCD pixels. The incident x-rays were at three different count rates that were comparable to those at 1 mA, 10 mA, and 100 mA with a “baseline spectrum” (outlined in Sec. 2.E). Two hundred energy windows with a width of 2 keV were used. The measurements were repeated 100 times and the mean counts were obtained. T_A and T_D values examined for ACS and DCS, respectively, were 5, 10, 20, and 30 ns.

2.D. Counting capabilities

The counting capability of PCDs was assessed by count rate curves (CRCs) using the baseline data with tube current values changing from 1 mA to 1,000 mA, which corresponded to the incident count rates of 4.1×10⁴–4.1×10⁷ cps/pixel or 9.6×10⁵–9.6×10⁸ cps/mm². All of the PCDs had four thresholds (N_C=4) at (20, 45, 70, and 95 keV) and a sum of the outputs of the four windows was computed. For both MEICC and 1CC, two total counts were computed, one for a sum of all of the primary counters outputs and the other for a sum of all of the coincidence counters output. The measurements were repeated 472–1,100 times for each tube current setting and the mean and the standard deviation of the total counts were computed. The default value of T_M, T₁, T_A, and T_D was 20 ns, but other values, i.e., 5, 10, 30, and 100 ns, were also used for T_M and T₁.

2.E. Imaging-task performances

The performance of PCDs was assessed for the following three imaging-tasks: the conventional CT imaging with water thickness estimation, water thickness estimation as a part of water–bone material decomposition, and K-edge imaging with tungsten as a part of water–bone–tungsten material decomposition. Tungsten was chosen because it has the K-edge in the energy range (60–90 keV) where many x-ray photons are present. There are three elements one needs to consider when assessing the performance of PCDs: input signals, a data acquisition scheme, and a data analysis method. We chose a combined use of flat-field signals,^31,38,39 5×5 super-pixel measurements (i.e., a sum of 5×5 pixels’ outputs to synthesize one large pixel), and Cramér–Rao lower bound (CRLB). The previous study³⁸ has shown that this combination takes into account the effects of CS, n-tuple counting, and the absence of CS across the boundary of 5×5 pixels. It has also presented that 5×5 super-pixel would be sufficient to approximate a detective quantum efficiency at zero frequency, DQE(0). Below we outline the process concisely; refer also the Appendix of Ref. 30 on computation of the Fisher information matrix and CRLBs.

In the image science and signal detection theory, the word “task” (e.g., a task-specific assessment) is specifically used for clinical tasks observers will perform using the obtained images, such as an object/lesion detection and characterization (or classification and parameter estimation).³³ In this paper, we use “imaging-tasks” to indicate the dimensionality of the imaging problem, i.e., the number of parameters required to describe the content of a voxel for a 3-dimensional (D) object—the conventional CT imaging (i.e., linear attenuation coefficient maps) is a 1-D problem, water–bone material decomposition is a 2-D problem, and K-edge imaging is a 3-D problem—each of them includes many clinical tasks. When a detector provides unbiased parameters with minimum variances, an imaging system that uses such a detector is expected to benefit a majority of clinical tasks in the corresponding imaging-task (although there always will be exceptions).

We used CRLBs because of the following reason. The output of the primary counters of MEICC and 1CC are nearly identical to that of the counters of the conventional PCD. The output of the coincidence counters of MEICC and 1CC contain the information associated with CS events that can be used by a post-acquisition algorithm to enhance the signals or decrease the noise or both. Development of such algorithms remains an ongoing research; therefore, we decided to use CRLB to assess the amount of the information each detector outputs when the incident spectrum was changed from the baseline spectrum to the target spectrum.

PCDs had four thresholds (N_C=4) at (20, 45, 70, and 95 keV) and tube current values ranged from 1 mA to 1,000 mA, which corresponded to the incident count rates of 4.1×10⁴–4.1×10⁷ cps/pixel or 9.6×10⁵–9.6×10⁸ cps/mm². The baseline data was generated with the baseline spectrum, which was synthesized by attenuating the 140 kVp spectrum by 5 cm of water, incident onto the 7×7 PCD pixels. The target data were synthesized by changing the spectrum by attenuating the baseline spectrum further by a small amount of either water (0.1 cm), bone (5.0×10⁻² cm), or tungsten (2.0×10⁻⁴ cm). The scans were repeated 600–2,200 times for each condition. For each scan, the outputs of the central 5×5 PCD pixels were summed to form a 5×5 super-pixel output. The mean and covariance of the super-pixel PCD data over noise realizations were calculated, and using them, the Fisher information matrix was computed numerically. We assumed that the noise was multivariate normally distributed and that the covariance remained unchanged between the baseline and target data. We believe that the use of the multivariate normal distribution was justified because the mean detected counts was larger than 30 in most cases and PP made the data non-Poisson-distributed:^40–43 Data from different energy windows were negatively correlated and the variance of data was larger or smaller than the mean, depending on energy windows. The number of entries for the mean vector was N_C for the conventional PCD, ACS, and DCS, N_C + 1 for 1CC, and N_C + N_C² for MEICC. Fisher information matrix was 1×1 for the conventional CT imaging, 2×2 for water–bone material decomposition, and 3×3 for K-edge imaging. A further detail of the processing was presented in the Appendix of Refs.³⁰.

The inverse of the Fisher information matrix was computed and the CRLBs of the basis functions were the corresponding diagonal element of the inverse matrix. CRLBs were then normalized by that of the conventional PCD at 1 mA for each spectral task and were called nCRLBs in this study. The coincidence processing parameter values used were 5, 10, 20, 30, and 100 ns for all of T_M, T₁, T_A, and T_D and 20, 50, 100, 150, and 200 ns for ACS. The default value was 20 ns for all of the four PCDs.

Bootstrap resampling was performed 2,000 times for each PCD data for each condition and the above process was performed for each Bootstrap resample, producing 2,000 nCRLB values. The standard deviation of nCRLB was then computed over 2,000 nCRLB values.

3. Results

3.A. Spectral responses

Figures 5A and 5B show spectra of ACS and DCS, respectively, at 1 mA with various T_A and T_D parameter values, respectively. This count rate was extremely low such that practically there was no PP. It can be seen in Fig. 5A that the spectrum with T_A = 5 ns was distorted severely, with a very high and broad K-escape peak at ~60 keV, a fluorescence peak at ~25 keV, and only few counts at the photo-peak at 80 keV. It demonstrated that the coincidence processing window of T_A=5 ns was too narrow to re-combine multiple pulses at offset times due to the pixel boundary effect and a vertical separation between the primary and secondary charge clouds. The spectrum with T_A = 10 ns showed the same effects as T_A = 5 ns albeit at a reduced magnitude; the spectra with T_A = 20 ns and 30 ns appeared identical and free from these issues, with a sharp photo-peak, a low K-escape peak, and no counts below 40 keV. The DCS spectra with various T_D parameters had the same results (Fig. 5B) as ACS spectra. Therefore, we concluded that the coincidence processing window width had to be at least 20 ns to overcome the CS with the various timing offsets simulated in this study. The default value for T_A for ACS, T_D for DCS, T_M for MEICC, and T₁ for 1CC was set at 20 ns for the reminder of the study.

Figure 5.

Spectra recorded by ACS, DCS, and the conventional PCD with 2-keV-width energy windows and the monoenergetic x-rays at 80 keV incident onto PCDs. (A) ACS with various T_A’s at 1 mA; (B) DCS with various T_D’s at 1 mA; (C–E) ACS with T_A=20 ns, DCS with T_D=20 ns, and the conventional PCD at 1 mA (C), 10 mA (D), and 100 mA (E).

Figure 5C presents 1 mA spectra recorded by the conventional PCD, ACS, and DCS. The spectrum of the conventional PCD was distorted due to CS with a low photo-peak, a high fluorescence peak, and a large continuum. The ACS accurately corrected for the spectral distortion caused by CS: the ACS spectrum presented a sharp photo-peak, a low K-escape peak, almost no continuum, and no fluorescence peak. The DCS spectra lay between those of ACS and the conventional PCD. If they had four energy windows with thresholds at (20, 45, 70, and 95 keV), the relative counts of the conventional PCD, ACS, and DCS were (0.33, 0.36, 0.31, 0.00), (0.00, 0.28, 0.71, 0.01) and (0.04, 0.42, 0.53, 0.01), respectively.

At 10 mA (Fig. 5D), the spectra of both ACS and DCS had slightly lower photo-peaks than at 1 mA. At 100 mA (Fig. 5E), the spectra of both ACS and DCS were severely distorted, with even lower photo-peaks and more counts for 90–160 keV. These changes were attributed to ‘false coincidence,’ where incidental quasi-coincident photons at neighboring pixels were falsely recognized as CS and their energies were added to produce a count at higher energies. The relative counts of the three PCDs at 100 mA were (0.30, 0.35, 0.32, 0.03) for the conventional PCD, (0.02, 0.13, 0.35, 0.50) for ACS, and (0.01, 0.17, 0.28, 0.53) for DCS. Both ACS and DCS showed the effect of significant spectral distortion, whereas the conventional PCD exhibited very little effect.

3.B. Counting capabilities

At 1 mA, the total counts recorded relative to the conventional PCD were 1.00 for the primary counters of both MEICC and 1CC, 0.78 for ACS, and 0.79 for DCS. Both ACS and DCS had fewer counts than the conventional PCD because they successfully moved counts that came from neighboring pixels via CS back to the original pixels. Figure 6A presents the CRCs of the primary counters. All of the CRCs increased monotonically with increasing the tube current values and appeared to have characteristics of the non-paralyzable detection model.⁴⁴ Using the same method as outlined in Ref.⁴⁵, the effective deadtime was estimated as 25.8 ns for the conventional PCD, 18.6 ns for both MEICC and 1CC, 140.5 ns for ACS, and 209.0 ns for DCS. The changes from the conventional PCD to ACS and DCS were a factor of 5.4 and 8.1, respectively, which must be attributed to the effective pixel area for ACS and DCS being 9 (=3×3) times as large as the original pixel area due to the 3×3-pixel coincidence processing. DCS was slower than ACS and it might be because DCS used the direct windowing scheme and a digital signal processing with tie-breakers, whereas ACS used an up-crossing of the lowest threshold for the event detection and analog signal processing. An inverse of the effective deadtime is called the characteristic count rates, and they were 3.9×10⁷ cps/pixel for the conventional PCD, 5.4×10⁷ cps/pixel for both MEICC and 1CC, 7.1×10⁶ cps/pixel for ACS, and 4.8×10⁶ cps/pixel for DCS. The output counts of MEICC and 1CC at higher count rates were up to 14% higher than that of the conventional PCD’s. The reason for these increases at higher count rates is not clear at this moment, although it must be related to the difference between the direct windowing scheme (used for the primary counters of MEICC and 1CC) and the simple pulse height analyzer (used for the conventional PCD).

Figure 6.

(A–C) Count rate curves (CRCs) of five PCDs (A), coincidence counters of MEICC with various coincidence processing time window parameters (T_M) (B), those of 1CC with various T₁’s (C). In (A), the CRC of 1CC was completely overlapped with that of MEICC. (D) The counting efficiency. The counting efficiency of 1CC were completely overlapped with those of MEICC. “Cv-PCD”=Conventional PCD. “Primary”=primary counters of MEICC and 1CC. Legends for (A,D) are presented in top-right; those for (B,C) in bottom-right. Error bars indicate the standard deviations over multiple noise realizations.

The total counts at 1 mA recorded by the coincidence counters for both MEICC and 1CC relative to the conventional PCD were 0.28, 0.64, 1.04, 1.45, and 1.88 for T_M=5, 10, 20, 30, and 100 ns, respectively, which indicated that larger coincidence processing window had more false coincidence events. Figures 6B and 6C show the CRCs of coincidence counters of MEICC and 1CC, respectively, with T_M=5, 10, 20, 30, and 100 ns. The CRCs of the primary counters of MEICC and 1CC were also presented. All of CRCs with T_M≥10 ns were above the CRCs of the primary counters and larger T_M produced a larger CRC. The latter was because at higher count rates, more photons incidentally arrived at neighboring pixels within a larger coincidence processing window and were falsely treated as CS events. The CRCs of MEICC coincidence counters were above the CRCs of 1CC coincidence counter and we believe that it was attributed to our implementation of MEICC and 1CC. MEICC had a coincidence timer t_MAj and the corresponding status s_MAj for each energy window, which only handled events within the energy window j. In contrast, 1CC had a timer t_1A and the corresponding status s_1A for the lowest energy threshold, which handled the events for all of the windows. Consequently, the status s_1A for 1CC was more frequently in a coincidence processing state than the status s_MAj for MEICC was. Thus, the coincidence counters of MEICC had an ability to record more counts in total.

Figure 6D shows the recorded counts per reading at various tube current conditions divided by those at 1 mA, which we call the counting efficiency in the following. The effective counting capability of both ACS and DCS was significantly lower than that of the other PCDs: The counting efficiency at 100 mA was 0.69 for ACS, 0.65 for DCS, and >0.94 for the other three PCDs. The tube current value that corresponded to the counting efficiency of 0.70 was 96 mA for ACS, 83 mA for DCS, 577 mA for the conventional PCD, and 690 mA for both MEICC and 1CC. The changes were a factor of 6.0 between ACS and the conventional PCD, and a factor of 7.0 between DCS and the conventional PCD. Performing the same analysis for the counting efficiency of 0.90, the changes were a factor of 6.9 between ACS and the conventional PCD, and a factor of 7.8 between DCS and the conventional PCD.

3.C. Imaging task performances

Figures 7A–7B and Table 1 show nCRLBs of the five PCDs for the conventional CT imaging task. At 1 mA, MEICC, ACS, and DCS provided the best nCRLB followed by 1CC and the conventional PCD. The rank order of MEICC, 1CC, and the conventional PCD was consistent with the previous study without PP,³¹ because CS was the dominant cause of spectral distortion at 1 mA. Throughout the count rate range investigated, nCRLBs of MEICC remained low with ≤1.08; nCRLBs of 1CC and the conventional PCD also remained low but increased up to 1.33 and 1.25, respectively. MEICC provided the best nCRLBs among the five PCDs studied over the entire count rate range investigated. In fact, the nCRLBs of MEICC relative to those of the conventional PCD at the corresponding tube current values generally improved from 0.88 at 1 mA to 0.81(=1.08/1.33) at 1,000 mA. In contrast, the nCRLBs of both ACS and DCS were comparable to that of the conventional PCD at 100 mA (when the counting efficiency was 0.95 and 0.69 for the conventional PCD and ACS, respectively) and significantly larger for ≥200 mA.

Figure 7.

The nCRLBs for the conventional CT imaging (A,B), the water–bone material decomposition (C,D), and K-edge imaging (E,F), plotted over a narrower range (A,C,E) or a wider range (B,D,F) of tube current values. For (B,D,F), markers were placed at slightly horizontally shifted locations for better visibility. Error bars indicate the standard deviations over 2,000 Bootstrap re-samples. “Cv-PCD”=Conventional PCD.

Table 1.

nCRLBs for the conventional CT imaging task

		Tube current (mA)
		1	10	50	100	200	400	600	800	1,000
Incident count rate	(cps/mm2)	9.6×105	9.6×106	4.8×107	9.6×107	1.9×108	3.8×108	5.7×108	7.7×108	9.6×108
	(cps/pixel)	4.1×104	4.1×105	2.1×106	4.1×106	8.2×106	1.6×107	2.5×107	3.3×107	4.1×107
Conventional PCD	Mean	1.00	1.03	1.09	1.01	1.09	1.11	1.18	1.27	1.33
	(St.dev)	(0.06)	(0.05)	(0.03)	(0.03)	(0.03)	(0.03)	(0.04)	(0.04)	(0.04)
MEICC	Mean	0.88	0.92	1.03	0.98	1.06	1.04	1.05	1.04	1.08
	(St.dev)	(0.05)	(0.04)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)
ICC	Mean	0.93	0.96	1.06	1.00	1.07	1.06	1.13	1.19	1.25
	(St.dev)	(0.06)	(0.04)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.04)
ACS	Mean	0.89	0.91	1.03	1.02	1.13	1.59	2.44	4.39	9.63
	(St.dev)	(0.06)	(0.04)	(0.03)	(0.03)	(0.04)	(0.05)	(0.08)	(0.15)	(0.32)
DCS	Mean	0.89	0.91	1.02	1.02	1.20	1.60	2.58	4.87	14.34
	(St.dev)	(0.05)	(0.04)	(0.03)	(0.03)	(0.04)	(0.05)	(0.08)	(0.16)	(0.57)

“Mean”=nCRLBs computed from the raw MC data; “St.dev”=Standard deviation over 2,000 Bootstrap re-samples

Figures 7C–7D and Table 2 present nCRLBs for the water–bone material decomposition task. At 1 mA, nCRLBs of all of MEICC, 1CC, ACS, and DCS were in the range of 0.43–0.50 and significantly better than the conventional PCD at 1.00. At and above 200 mA (when the counting efficiency for the conventional PCD was at 0.89 or lower), however, nCRLBs of both ACS and DCS were higher than those of the conventional PCD and rapidly increased with increasing the count rates. The speed of the degradation was faster with water–bone material decomposition than with the conventional CT imaging. In contrast, MEICC had consistently lower nCRLBs than the conventional PCD and the best nCRLBs among the five PCDs, although the nCRLBs of MEICC relative to those of the conventional PCD at the corresponding tube current values generally increased from 0.45 at 1 mA to 0.76(=0.95/1.24) at 1,000 mA.

Table 2.

nCRLBs for the water–bone material decomposition task

		Tube current (mA)
		1	10	50	100	200	400	600	800	1,000
Incident count rate	(cps/mm2)	9.6×105	9.6×106	4.8×107	9.6×107	1.9×108	3.8×108	5.7×108	7.7×108	9.6×108
	(cps/pixel)	4.1×104	4.1×105	2.1×106	4.1×106	8.2×106	1,6×107	2.5×107	3.3×107	4.1×107
Conventional PCD	Mean	1.00	1.01	0.94	0.96	0.97	0.99	1.07	1.06	1.24
	(St.dev)	(0.07)	(0.05)	(0.04)	(0.04)	(0.04)	(0.04)	(0.05)	(0.05)	(0.06)
MEICC	Mean	0.45	0.50	0.50	0.52	0.55	0.60	0.73	0.74	0.95
	(St.dev)	(0.03)	(0.03)	(0.02)	(0.02)	(0.02)	(0.02)	(0.03)	(0.03)	(0.04)
ICC	Mean	0.50	0.55	0.56	0.65	0.89	0.96	0.98	0.95	1.11
	(St.dev)	(0.03)	(0.03)	(0.02)	(0.03)	(0.04)	(0.04)	(0.04)	(0.04)	(0.05)
ACS	Mean	0.44	0.46	0.49	0.66	1.34	4.07	9.21	13.72	18.12
	(St.dev)	(0.03)	(0.02)	(0.02)	(0.03)	(0.07)	(0.34)	(1.11)	(2.07)	(2.97)
DCS	Mean	0.43	0.46	0.52	0.65	1.11	5.51	24.23	158.39	146.99
	(St.dev)	(0.03)	(0.02)	(0.02)	(0.03)	(0.05)	(0.54)	(6.05)	(>100)	(>100)

“Mean”=nCRLBs computed from the raw MC data; “St.dev”=Standard deviation over 2,000 Bootstrap re-samples

Figures 7E–7F and Table 3 present nCRLBs for the K-edge imaging task. At 1 mA, ACS had the best nCRLB at 0.26, followed by MEICC and DCS both at 0.49, 1CC at 0.98, and the conventional PCD at 1.00. At 50 mA, both MEICC and ACS had comparable nCRLBs; however, nCRLB of ACS became comparable to that of the conventional PCD at 100 mA (at the counting efficiency of 0.69 and 0.95 for ACS and the conventional PCD, respectively) and increased quickly afterwards. In contrast, nCRLBs of MEICC remained low in the range of 49–58% of those of the conventional PCD at the corresponding tube current conditions. The nCRLBs of 1CC were comparable to those of the conventional PCD, and it was consistent with the previous study.³¹

Table 3.

nCRLBs for the K-edge imaging task

		Tube current (mA)
		1	10	50	100	200	400	600	800	1,000
Incident count rate	(cps/mm2)	9.6×105	9.6×106	4.8×107	9.6×107	1.9×108	3.8×108	5.7×108	7.7×108	9.6×108
	(cps/pixel)	4.1×104	4.1×105	2.1×106	4.1×106	8.2×106	1.6×107	2.5×107	3.3×107	4.1×107
Conventional PCD	Mean	1.00	1.04	1.23	1.39	1.70	2.05	3.09	4.60	5.09
	(St.dev)	(0.08)	(0.06)	(0.07)	(0.08)	(0.10)	(0.14)	(0.23)	(0.42)	(0.48)
MEICC	Mean	0.49	0.54	0.68	0.81	1.04	1.42	2.07	2.45	2.96
	(St.dev)	(0.04)	(003)	(0.03)	(0.04)	(0.06)	(0.09)	(0.15)	(0.19)	(0.25)
ICC	Mean	0.98	1.02	1.21	1.34	1.68	2.00	2.93	4.02	4.52
	(St.dev)	(0.08)	(0.06)	(0.06)	(0.07)	(0.10)	(0.14)	(0.22)	(0.35)	(0.42)
ACS	Mean	0.26	0.30	0.65	1.49	7.57	38.69	357.1	307.5	3,623
	(St.dev)	(0.02)	(0.02)	(0.03)	(0.08)	(0.89)	(13.34)	(>100)	(>100)	(>100)
DCS	Mean	0.49	0.64	1.42	2.76	15.77	536.5	1,284	85.68	3,371
	(St.dev)	(0.04)	(0.03)	(0.08)	(0.22)	(2.75)	(>100)	(>100)	(>100)	(>100)

“Mean”=nCRLBs computed from the raw MC data; “St.dev”=Standard deviation over 2,000 Bootstrap re-samples

The nCRLBs of both ACS and DCS increased quickly due to PP and severe effects of false coincidence (see Fig. 5E). As count rates increased, lower energy windows had fewer counts and eventually only the highest energy window had counts. With only one or two effective windows, the estimation problems for the water–bone material decomposition and K-edge imaging became ill-posed and this was the reason why the nCRLBs increased rapidly.

ACS with a larger T_A had a larger (i.e., worse) nCRLB and the effect of T_A was quite significant (results not presented). T_A=20 ns was sufficient to address CS with the temporal separations in anode arrival times as discussed in Sec. 3.A and larger T_A’s simply degraded the performance. The MEICC with different T_M values had similar effects; however, the differences were very small and MEICC with T_M = 100 ns was still better than the conventional PCD (results not presented).

4. Discussion

Among the five PCDs investigated, MEICC provided the best nCRLBs for all of the imaging tasks over the count rate range investigated except for a few conditions (e.g., K-edge imaging at ≤50 mA, where ACS provided better nCRLBs). PP decreased the merit of MEICC in addressing CS. Nonetheless, MEICC consistently performed better than the conventional PCD did, albeit diminishing merit with increasing count rates. The nCRLBs of MEICC were in the range of 49–58% of those of the conventional PCD for K-edge imaging, 45–76% for water–bone material decomposition, and 81–88% for the conventional CT imaging. Both ACS and DCS had better nCRLBs than the conventional PCD did at lower count rates; however, at moderate or higher count rates (e.g., ≥100 mA for K-edge imaging and ≥200 mA for the other tasks), both ACS and DCS provided much higher nCRLBs than the other three PCDs. At higher count rates, MEICC had a significantly better performance than both ACS and DCS did, which was attributed to the following two reasons. One, the coincidence processing of MEICC did not interfere with the primary counting process; therefore, MEICC had a better counting capability than either ACS or DCS. Two, the inter-pixel communication of MEICC was a one-time one-way communication, whereas that of ACS and DCS was a continuous two-way communication over the coincidence processing time window. With MEICC, the communication was made to spread the counting signal from a pixel that detected an event to its neighboring pixels and these pixels acted independently afterwards. Both ACS and DCS in this study kept the communication open until both the winner pixel and the winner energy window were determined and a count was added. The communication was the key to address CS in real-time and output accurate counts; however, it significantly limited the count rate range within which the energy reconstruction functioned well.

To our knowledge, this is the first simulation study with the four temporal factors to assess the performances of ACS and DCS for specific imaging tasks. Our study showed that both ACS and DCS performed poorly and worse than the conventional PCD did when count rates were higher and counting efficiency were lower than a certain value. The “break-even” points of the counting efficiency for ACS were ~0.69 (at ~100 mA, which corresponds to the counting efficiency of 0.95 for the conventional PCD) for three imaging tasks. As it will be discussed below, different implementations of ACS and DCS may result in different count rate-dependency; however, we expect that these breaking points remain the same as long as other factors such as pixel sizes are not changed.

Our results of 1CC were in good agreement with Hsieh’s report²⁹ despite many differences in experimental settings. At the lowest count rate, Hsieh reported that a relative dose efficiency for iodine material was ~1.7 with 5 bins (which implies an nCRLB of ~0.56=1/1.7), whereas the nCRLB for water–bone material decomposition in our study was 0.50. The merit of 1CC over the conventional PCD in water–iodine material decomposition diminished with increasing count rates in Hsieh’s report (see Fig. 6 of Ref.²⁹) and we observed a similar effect in Fig. 7.

The performances of both ACS and DCS were qualitatively consistent with what has been reported previously in terms of the following aspects: (1) The spectral responses to a monoenergetic input at a very low count rate were excellent (e.g., compare Fig. 5C in this study with Fig. 3 of Ref.¹⁵). (2) The spectral responses were degraded with increasing count rates. The changes from 1 mA to 100 mA (Figs. 5C–5E) appeared very similar to the experimental data showing changes from 5.3×10⁵ cps/mm² to 4.6×10⁷ cps/mm² (Fig. 7 of Ref.¹⁵). (3) The nCRLBs of DCS at a low count rate were between those of ACS and the conventional PCD (Fig. 6 versus Table IV of Ref.¹⁹). (4) CRCs of ACS were lower than that of the conventional PCD due to a limited detector speed (e.g., Fig. 6A versus Fig. 6 of Ref.¹⁵, which showed CRCs of Medipix3 with and without the ACS mode). (5) CRCs of DCS were lower than that of the conventional PCD (e.g., Fig. 6A versus Fig. 10 of Ref. 10 and Fig. 6 of Ref. 11, which showed CRCs of two versions of XCounter with and without DCS mode).

Nevertheless, the ACS and DCS studied in this paper do not necessarily agree with those reported in the literature quantitatively because of differences in various design specifications and experimental settings such as pulse duration time, pixel sizes, pixel thicknesses, electric field strengths, x-ray spectra, and inter-pixel communication schemes. For example, it was reported that with a Medipix3 detector with 110-μm pixels, the count rates where the counting efficiency dropped to 0.90 changed by a factor of 4.3, from 2.8×10⁶ cps/mm² with ACS activated to 1.2×10⁷ cps/mm² without ACS activated.¹⁵ In contrast, the changes from the conventional PCD to ACS in this study were a factor of 6.9 for the counting efficiency of 0.90; and the change in the estimated deadtime was a factor of 5.4 from 25.8 ns to 140.5 ns (see Sec. 3.B). The performance of Medipix3 at higher count rates might be somewhat better than that of the ACS in this study. It was reported that with an XCounter PDT25-DE detector, the change in the characteristic count rate was a factor of 9–10 from DCS on to DCS off. In contrast, we found the change a factor of 8.1 from 4.8×10⁶ cps/pixel for DCS to 3.9×10⁷ cps/pixel for the conventional PCD.

DCS in this study was slower than ACS, despite DCS being expected to be faster than ACS for various reasons.^19,26 It might be because, one, both DCS and ACS used the identical coincidence processing window width of 20 ns in this study, and two, DCS implemented in this study used the direct windowing scheme, whereas ACS used an up-crossing of the lowest threshold for event detection. As discussed before, the changes from the conventional PCD to ACS and DCS were a factor of 5.4 and 8.1, respectively, which were close to the results of Medipix3 and XCounter, respectively. Nonetheless, in actual implementation and circuitry designs, ACS may need a longer processing time than DCS does and DCS can use the minimum time (which was 20 ns in this study) required to address the time-shifted pulses due to the pixel boundary effect and the vertical separations outlined in Sec. 2.B. DCS had larger noise than expected in some cases, for example, K-edge imaging at 1 mA. We suspect that the random numbers used to break ties to determine the winner pixel and winner window might have increased the uncertainty of the estimation and the use of deterministic rules may decrease the noise at the expense of either the accuracy of the energy reconstruction or the incident locations.

All of ACS, DCS, and MEICC implemented in this study used a straightforward scheme for each detector and used the identical and short coincidence processing time of 20 ns to study the generic characteristics of these detectors. Different implementations are possible and may produce different quantitative results. For example, ACS and DCS that concerns CS with 4 or fewer neighboring pixels than eight neighbors may have a better counting capability, but one needs to invent a way to reconstruct energy accurately. The ACS implemented by Medipix3 may have a longer coincidence processing time (up to 1 μs) and employ a very complex inter-pixel communication and task-sharing scheme (see Fig. 4 of Ref.¹²). Each pixel computes the energy of one 2×2 super-pixel, determines the winner, and performs instantaneous one-way inter-pixel communication multiple times during the process to share the outcomes and stay consistent between pixels. It would be of interest to implement it and compare the performances when a timing chart similar to Fig. 2 for the scheme becomes available. A DCS with more complex energy reconstruction algorithm may improve the accuracy of the energy recovery. The use of 5×5 pixels instead of 3×3 pixels may further improve the accuracy of the energy recovery for both ACS and DCS. The DCS implemented by Hsieh and Sjolin¹⁹ determined the final winner pixel by rules which were independent of the deposited energy fraction. This implementation will have a better counting capability than our implementation (and possibly better nCRLBs as well, as discussed before), although it will result in a half-pixel shift of object images and have a somewhat limited ability to reconstruct incident photon energies. For MEICC, the use of four neighbors instead of eight neighbors can decrease a probability of false-coincidences, which may improve the performance against PP at higher count rates, although the ability to address CS at lower count rates may be reduced by ~10%.³⁰ A use of fewer number of coincidence counters³⁰ such as 2×N_C (=8) instead of N_C² (=16) and a different configuration for timers and statuses may result in a more energy- and resource-efficient performance.

We wish to comment on the production complexity and the cost for a MEICC ASIC in the future. All of the components used in the MEICC design are currently available. By packing more counters and timers, one may need to carefully consider heat, power consumption, crosstalk between lines, space needed, and transfer of increased volume of data; however, we believe that they are manageable. Once the ASIC is designed, we expect that the complexity and cost for the production will remain comparable to the conventional PCD ASIC and that a majority of photon counting detector system’s cost will come from detector arrays as opposed to ASICs. We remain cautiously optimistic about the cost.

We also wish to comment on complex effects of CS and PP on super-pixel data. When PP is negligible, CS results in a positive correlation between a pair of energy windows at two neighboring pixels as the energy of one photon is split between them. Super-pixel data are then created by adding such neighboring pixels data, resulting in a positive correlation among energy windows of the super-pixel data. (When a CS event produces two pulses at the same energy window at two pixels, it will result in double-counting at the energy window of the super-pixel data. The probability of this double-counting depends on energy window widths, threshold energies, fluorescence x-ray yields and its travel distance, the relative size of detector pixel to charge cloud, etc.) When PP is present, lower and higher energy windows of the same pixel are negatively correlated, as piled-up lower energy pulses will be measured as one count at a higher energy window. Thus, lower and higher energy windows of the super-pixel data are negatively correlated. These two competing effects—a positive correlation by CS and a negative correlation by PP—are present in super-pixel data and a dominant factor changes depending on count rates.

The nCRLBs of the conventional PCD for the conventional CT at higher count rates were higher than those of the water–bone material decomposition at the same tube current; for example, the nCRLBs at 1,000 mA for the two tasks were 1.33 and 1.24, respectively. We found this very surprising, because we expected that the nCRLBs of the conventional CT imaging task would be degraded mainly due to a loss of counts (with no or less effect from spectral distortion), whereas those of more spectrally-demanding task (i.e., water–bone material decomposition) would be degraded further as they would be affected by both a loss of counts and spectral distortion. In fact, in a separate unpublished MC simulation study, the nCRLBs at 1,000 mA were 1.54 and 2.38 for the conventional CT imaging and water–bone material decomposition, respectively, supporting the above expectation. Among several differences between the two studies, that study used single-pixel measurements, while this study used super-pixel measurements. The complex effects of CS and PP on the covariance of super-pixel data outlined in the previous paragraph was absent for single-pixel data and we postulate that that was the main reason for the difference.

The study had several limitations. (1) We used only one base spectrum, one K-edge material, and one pulse duration in this study. Different base spectra and K-edge materials would result in different nCRLB values and a longer/shorter pulse duration would result in more/less effects of PP; however, we think that the general observations would remain the same. For example, the simulation conditions used in a previous study³⁰ were very different from those Hsieh used in Ref.²⁹ in terms of pixel sizes and the spectra; however, the improvement by 1CC over the conventional PCD at the absence of PP was comparable to each other. (2) The number of energy windows N_C was fixed at four. A larger N_C will result in a better nCRLBs with straightforward changes (see Fig. 6 of Ref.³¹). (3) The threshold energies might not be the optimal set for each PCD at each condition, as discussed in Sec. 2.B; however, we have confirmed in a preliminary study that the optimal energies for the conventional PCD did not change significantly over count rates and that the chosen threshold energies were close to the optimal ones at various count rates. Threshold energies had to be set prior to data generation for the MC simulation and it took 2 months to generate data necessary for this study. It was practically impossible to run simulations multiple times with different threshold energies. (4) We used only one pulse shape and one PCD material, and did not include a baseline restoration. A bipolar pulse and/or a pulse with a long tail would result in different effects of PP in terms of the mean counts and the covariance-to-mean ratio of counts among energy windows.⁴⁰ Different materials such as silicon have probabilities of interactions (i.e., more Compton interactions) that are very different from cadmium telluride we studied in this paper. Baseline restoration would suppress dynamic noise floor fluctuations and affect the output of low energy windows. (5) This study was conducted with a sufficiently large number of counts per reading recorded. The condition was chosen to make the CRLB analysis valid: It has been shown that if only a few counts on average per reading were recorded, the asymptotic property of maximum-likelihood would not hold and estimated results might be biased,⁴⁶ which in turn, would violate the foundation of the CRLB analysis: the existence of an unbiased estimator. (6) We did not include cross-sectional images nor a sinogram in this study because we wanted to assess the effect of PP systematically and specifically. Many conditions change from one sinogram data point to another such as the incident spectrum, the count rate, the number of counts per pixel, and those for neighboring pixels. It is our belief that at an early stage of the development, we should study the effect of one factor at a time and build on knowledge and experiences, which are critical to have a good grasp of complex problems and mechanisms. Once the study is completed, the sinogram data would be mere samples from the studied data range. See Ref.^8,9 for such examples. (7) We used one PCD pixel size (225 μm) only in this study; a performance with larger pixel sizes such as 450 μm would also be of interest. Larger pixels are better for CS, but worse for PP, than smaller pixels. We think that it would make more sense to include different pixel sizes when we assess the cross-sectional image quality using sinograms. PCDs with different pixel sizes have different spatial samplings and spectral responses; thus, it would be fair to use the same object with a mixture of different spatial resolution contents, such as high spatial frequency components (e.g., skull edges) and low soft tissue contrast components (e.g., hemorrhage or infarct lesions), to assess the overall performances. We shall leave it to the future work.

5. Conclusion

The MEICC provides the best nCRLBs for all of the imaging tasks over the count rate range investigated except for a few conditions. PP decreases the merit of MEICC in addressing CS; however, MEICC consistently provides better nCRLBs than the conventional PCD does. The nCRLBs of MEICC for K-edge imaging are in the range of 49–58% of those of the conventional PCD, 45–76% for water–bone material decomposition, and 81–88% for the conventional CT imaging (i.e., linear attenuation coefficient maps). In contrast, both ACS and DCS have better nCRLBs than the conventional PCD does only when count rates are very low, e.g., when the counting efficiency for ACS are higher than 0.95.