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Background Purpose Methods Results Conclusion The first step in computed tomography (CT) reconstruction is to estimate attenuation pathlength. Usually, this is done with a logarithm transformation, which is the direct solution to the Beer‐Lambert Law. At low signals, however, the logarithm estimator is biased. Bias arises both from the curvature of the logarithm and from the possibility of detecting zero counts, so a data substitution strategy may be employed to avoid the singularity of the logarithm. Recent progress has been made by Li et al. [IEEE Trans Med Img 42:6, 2023] to modify the logarithm estimator to eliminate curvature bias, but the optimal strategy for mitigating bias from the singularity remains unknown.The purpose of this study was to use numerical techniques to construct unbiased attenuation pathlength estimators that are alternatives to the logarithm estimator, and to study the uniqueness and optimality of possible solutions, assuming a photon counting detector.Formally, an attenuation pathlength estimator is a mapping from integer detector counts to real pathlength values. We constrain our focus to only the small signal inputs that are problematic for the logarithm estimator, which we define as inputs of <100 counts, and we consider estimators that use only a single input and that are not informed by adjacent measurements (e.g., adaptive smoothing). The set of all possible pathlength estimators can then be represented as points in a 100‐dimensional vector space. Within this vector space, we use optimization to select the estimator that (1) minimizes mean squared error and (2) is unbiased. We define “unbiased” as satisfying the numerical condition that the maximum bias be less than 0.001 across a continuum of 1000 object thicknesses that span the desired operating range. Because the objective function is convex and the constraints are affine, optimization is tractable and guaranteed to converge to the global minimum. We further examine the nullspace of the constraint matrix to understand the uniqueness of possible solutions, and we compare the results to the Cramér‐Rao bound of the variance.We first show that an unbiased attenuation pathlength estimator does not exist if very low mean detector signals (equivalently, very thick objects) are permitted. It is necessary to select a minimum mean detector signal for which unbiased behavior is desired. If we select two counts, the optimal estimator is similar to Li's estimator. If we select one count, the optimal estimator becomes non‐monotonic. The oscillations cause the unbiased estimator to be noise amplifying. The nullspace of the constraint matrix is high‐dimensional, so that unbiased solutions are not unique. The Cramér‐Rao bound of the variance matches well with the expected I−0.5${{I}^{ - 0.5}}$ scaling law and cannot be attained.If arbitrarily thick objects are permitted, an unbiased attenuation pathlength estimator does not exist. If the maximum thickness is restricted, an unbiased estimator exists but is not unique. An optimal estimator can be selected that minimizes variance, but a bias‐variance tradeoff exists where a larger domain of unbiased behavior requires increased variance. [ABSTRACT FROM AUTHOR]

Background: Silicon (Si) is a possible sensor material for photon counting detectors (PCDs). A major drawback of Si is that roughly two‐thirds of x‐ray interactions in the diagnostic energy range are Compton scattering. Because Compton scattering is an energy‐insensitive process, it is commonly assumed that Compton events retain little spectral information. Purpose: To quantify how much information can be recovered from Compton scattering events in models of Si PCDs. Methods: We built a simplified model of Si interactions including two interaction mechanisms: photoelectric effect and Compton scattering. We considered three different binning options that represent strategies for handling Compton events: in Compton censoring, all events under 38 keV (the maximum energy possible from Compton scattering for a 120 keV incident photon) were discarded; in Compton counting, all events between 1 and 38 keV were placed into a single bin; in Compton binning, all events were placed into energy bins of uniform width. These were compared to the ideal detector, which always recorded the correct energy (i.e., 100% photoelectric effect). Every photon was assumed to interact once and only once with Si, and the energy bin width was 5 keV. In the primary analysis, the Si detector was irradiated with a 120 kV spectrum filtered by 30 cm of water, with 99.5% of the arriving spectrum above 38 keV so that there was good separation between photoelectric effect and Compton scattering, and the figures of merit were the Cramér–Rao lower bound (CRLB) of the variance of iodine and water basis material decomposition images, as well as the CRLB of virtual monoenergetic images (i.e., linear combinations of material images) that maximize iodine CNR or water CNR. We also constructed a local linear estimator that attains the CRLB. In secondary analyses, we applied other sources of spectral distortion: (1) a nonzero minimum energy threshold; (2) coarser, 10 keV energy bins; and (3) a model of charge sharing. Results: With our chosen spectrum, 67% of the interactions were Compton scattering. Consistent with this, the material decomposition variance for the Compton censoring model, averaged over both basis materials, was 258% greater than the ideal detector. If Compton events carried no spectral information, the Compton counting model would show similar variance. Instead, its basis material variance was 103% greater than the ideal detector, implying that Compton counts indeed carry significant spectral information. The Compton binning model had a basis material variance 60% greater than the ideal detector. The Compton binning model was not affected by a 5 keV minimum energy threshold, but the variance increased from 60% to 107% when charge sharing was included and to 78% with coarser energy bins. For optimized CNR images, the average variance was 149%, 12%, and 10% higher than the ideal detector for the Compton censoring, counting, and binning models, reinforcing the hypothesis that Compton counts are useful for detection tasks and that precise energy assignments are not necessary. Conclusions: Substantial spectral information remains after Compton scattering events in silicon PCDs. [ABSTRACT FROM AUTHOR]

Background: The spatial resolution of energy‐integrating diagnostic CT scanners is limited by interpixel reflectors on the detector, which optically isolate pixels but create dead space. Because the width of the reflector cannot easily be decreased, fill factor diminishes as resolution increases. Purpose: We propose loading (or mixing) a high‐Z element into the reflectors, causing the reflectors to be X‐ray fluorescent. Re‐emitted characteristic X‐rays could be detected in adjacent pixels, increasing the effective fill factor and compensating for fill factor loss with higher‐resolution detectors. The purpose of this work is to understand the physical principles of this approach and to analyze its effectiveness using Monte Carlo simulations. Methods: Detector pixels were modeled using the GEANT4 Monte Carlo package. The width of the reflector was kept constant at 0.1 mm throughout, and we considered pixel pitches between 0.5 and 1 mm. The pixelated scintillator material was gadolinium oxysulfide, 3 mm thick. The baseline reflector material was chosen to be acrylic, and varying concentrations of a high‐Z element were loaded into the material. We assumed that the optical characteristics of pixels were ideal (no absorption within pixels, perfect reflection at boundaries). The detector was irradiated uniformly with 10,000 X‐ray photons to estimate its spectral response. The figure of merit was the variance of the detector signal at zero frequency normalized to that of an ideal single‐bin photon‐counting detector with 100% fill factor. Sensitivity analyses were conducted to understand the effect of varying the high‐Z element concentration and the spectrum. Results: Initial simulations suggested that a k‐edge near 50 keV would be ideal. Gd was therefore selected as the high‐Z material. The relative variances for a conventional energy integrating detector without Gd at 1 mm pixel pitch (81% fill factor) and 0.5 mm pixel pitch (64% fill factor) were 1.38 and 1.74, compared to 1.00 for an ideal photon counting detector, implying a 26% variance penalty for 0.5 mm pitch. When 1 g/cm3 Gd was loaded into the interpixel reflector, the relative variance improved to 1.27 and 1.43, respectively, implying that the variance penalty for including Gd together with 0.5 mm pitch is only 4%. Performance was nearly maximized at 1.0 g/cm3 of Gd, but a concentration of 0.5 g/cm3 of Gd showed most of the benefit. Improvements depend weakly on kV, with lower kV associated with higher improvements. An external anti‐scatter grid was not modeled in our simulations and would reduce the expected benefit, depending greatly on the pitch and dimensionality of the anti‐scatter grid. Conclusions: The losses in fill factor associated with smaller pixel pitch can be reduced if Gd or a similar element could be loaded into the interpixel reflector. These improvements in noise efficiency are yet to be verified experimentally. [ABSTRACT FROM AUTHOR]

Background: Photon counting detectors (PCDs) for x‐ray computed tomography (CT) face spectral distortion from pulse pileup and charge sharing. The photon counting scheme used by many PCDs is threshold–subtract (TS) with pulse height analysis (PHA), where each counter counts up‐crossing events when pulses exceed an energy threshold. PCD data are not Poisson‐distributed due to charge sharing and pulse pileup, but the counting statistics have never been studied yet. Purpose: The objectives of this study were (1) to propose a modified photon counting scheme, direct energy binning (DB), that is expected to be robust against pulse pileup; (2) to assess the performance of DB compared to TS; and (3) to evaluate its counting statistics. Methods: With DB scheme, counter k starts a timer upon an up‐crossing event of energy threshold k, and adds a count only if the next higher energy threshold (k+1) was not crossed within a short time window (hence, the pulse peak belongs to the energy bin k). We used Monte Carlo (MC) simulation and assessed count‐rate curves and count‐rate‐dependent spectral imaging task performance for conventional CT imaging as well as water thickness estimation, water–bone material decomposition, and K‐edge imaging with tungsten as the K‐edge material. We also assessed count‐rate‐dependent measurement statistics such as expectation, variance, and covariance of total counts as well as energy bin outputs. The agreement with counting statistics models was also evaluated. Results: The DB scheme improved the count‐rate curve, that is, mean measured counts as a function of input count‐rate, and peaked with 59% higher count‐rate capability than the TS scheme (3.5 × 108 counts per second (cps)/mm2 versus 2.3 × 108 cps/mm2). The Cramér–Rao lower bounds (CRLB) of the variance of basis line integrals estimation for DB was better than those for TS by 2% for the conventional CT imaging, 30% for water–bone material decomposition, and 32% for K‐edge imaging at 1000 mA (at 7.3 × 107 cps/sub‐pixel after charge sharing). When count‐rates were lower, PCD data statistics were dominated by charge sharing: the variance of total counts and lower energy bins was larger than the mean counts; the covariance of bin data was positive and non‐zero. When count‐rates were higher, PCD data statistics were dominated by pulse pileup: the variance of data was lower than the mean; the covariance of bin data was negative. The transition between the two regimes occurred smoothly, and pulse pileup dominated the statistics ≥400 mA (when the count‐rate after charge sharing was 2.9 × 107 cps/sub‐pixel and the probability of count‐loss for DB was 37%). Both DB and TS had good agreement with Yu–Fessler's models of total counts; however, DB had a better agreement with Wang's variance and covariance models for energy bin data than TS did. Conclusions: The proposed DB scheme had several advantages over TS. At low to moderate flux, DB could improve the resilience of PCDs to pulse pileup. Counting statistics deviated from the Poisson distribution due to charge sharing for lower count‐rate conditions and pulse pileup for higher count‐rate conditions. [ABSTRACT FROM AUTHOR]

Photon counting detectors (PCDs) are classically described as being either paralyzable or nonparalyzable. When the PCD is paralyzed, it is no longer sensitive to the detection of additional flux. A recent strategy in PCD design has been to compensate for detector paralysis by embedding specialized paralysis compensation electronics into the application-specific integrated circuit (ASIC). One such compensation mechanism is the pileup trigger, which places an additional energy bin at very high energy that is triggered only during pileup. Another compensation mechanism is the retrigger architecture, which converts a paralyzable PCD into a nonparalyzable PCD. We propose a third mechanism that modifies the retrigger architecture using dedicated secondary counters. We studied the incremental benefit of these three paralysis compensation mechanisms in simulation. We modeled the spectral response using Monte Carlo simulations and then estimated the variance in basis material decomposition of a single pixel using the Cramér-Rao lower bound (CRLB). In the absence of paralysis compensation, noise in basis material images shows sharp increases at moderate flux (near the characteristic count rate) due to contrast inversion and again at high flux. The pileup trigger reduces noise at high flux but does not eliminate contrast inversion. The retrigger architecture eliminates contrast inversion but does not reduce noise at high flux. Our proposed retrigger architecture with dedicated secondary counters reduce noise at both moderate and high flux. [ABSTRACT FROM AUTHOR]

The performance of X-ray photon counting detectors (PCDs), especially on spectral tasks, is compromised by charge sharing. Existing mechanisms to compensate for charge sharing, such as charge summing circuitry or larger pixel sizes, increase and aggravate pileup effects. We propose a new mechanism, the coincidence counting bin (CCB), which does not increase pileup and which has implementation similarities to existing energy bins. The CCB is triggered by coincident events in adjacent pixels and provides an estimate of the double counts arising from charge sharing. Unlike charge summing, the CCB does not directly restore corrupted events. Nonetheless, knowledge of the number of coincident counts can be used by the estimator to reduce noise. We simulated a PCD with and without the CCB using Monte Carlo simulations, modeling PCD pixels as instantaneous charge collectors and X-ray energy deposition as producing a Gaussian charge cloud with 75 micron FWHM, independent of energy. With typical operating conditions and at low flux (120 kVp, incident count rate 1% of characteristic count rate, 30 cm object thickness, five energy bins, pixel pitch of 300 microns), the CCB improved dose efficiency of iodine and water basis material decomposition by 70% and 50%, respectively. An improvement of 20% was also seen in an iodine CNR task. These improvements are attenuated as incident flux increases and show moderate dependence on filtration and pixel size. At high flux, the CCB does not provide useful information and is discarded by the estimator. The CCB may be an effective and practical mechanism for charge sharing compensation in PCDs. [ABSTRACT FROM AUTHOR]

Purpose: Charge sharing is a significant problem for CdTe‐based photon counting detectors (PCDs) and can cause high‐energy photons to be misclassified as one or more low‐energy events. Charge sharing is especially problematic in PCDs for CT because the high flux necessitates small pixels, which increase the magnitude of charge sharing. Analog charge summing (ACS) is a powerful solution to reduce spectral distortion arising from charge sharing but may be difficult to implement. We investigate correction of the signal after digitization by the comparator (“digital count summing”), which is only able to correct a subset of charge sharing events but may have implementation advantages. We compare and quantify the relative performance of digital and ACS in simulations. Methods: Transport of photons in CdTe was modeled using Monte Carlo simulations. Energy deposited in the CdTe substrate was converted to electrical charges of a predetermined shape, and all charges within the detector pixel are assumed to be perfectly collected. In ACS, the maximum charge received over any 2 × 2 block of pixels was grouped together prior to digitization. In digital count summing (DCS), the charge was digitized in each pixel, and subsequently, adjacent pixels that detected events grouped their charge to record a single, higher energy event. All simulations were performed at the limit of low flux (no pileup). The default tube voltage was 120 kVp, object thickness was 20 cm of water, pixel pitch was 250 μm, and charge cloud modeled as a Gaussian with σ = 40 μm. Variation of these parameters was examined in a sensitivity analysis. Results: Detectors that used no correction, DCS, and ACS misclassified 51%, 39%, and 15% of incident photons, respectively. For iodine basis material imaging, DCS exhibited 100% greater dose efficiency compared to uncorrected, and ACS exhibited an additional 111% greater dose efficiency compared to digital charge summing. For a nonspectral task, the dose efficiency improvement as estimated by improvement of zero‐frequency detective quantum efficiency, DQE(0) was 10% for DCS compared to uncorrected and 10% for ACS compared to DCS. A sensitivity analysis showed that DCS generally achieved half the benefit of ACS over a range of conditions, although the benefit was markedly less if the charge cloud was instead modeled as a small sphere. Conclusions: Summing of counts after digitization may be a simpler alternative to summing of charge prior to digitization due to the relative complexity of analog circuit design. Over most conditions studied, it provides roughly half the benefit of ACS and may offer certain implementation advantages. [ABSTRACT FROM AUTHOR]

Purpose: Photon‐counting detectors using CdTe or CZT substrates are promising candidates for future CT systems but suffer from a number of nonidealities, including charge sharing and pulse pileup. By increasing the pixel size of the detector, the system can improve charge sharing characteristics at the expense of increasing pileup. The purpose of this work is to describe these considerations in the optimization of the detector pixel pitch. Methods: The transport of x rays through the CdTe substrate was simulated in a Monte Carlo fashion using GEANT4. Deposited energy was converted into charges distributed as a Gaussian function with size dependent on interaction depth to capture spreading from diffusion and Coulomb repulsion. The charges were then collected in a pixelated fashion. Pulse pileup was incorporated separately with Monte Carlo simulation. The Cramér–Rao lower bound (CRLB) of the measurement variance was numerically estimated for the basis material projections. Noise in these estimates was propagated into CT images. We simulated pixel pitches of 250, 350, and 450 microns and compared the results to a photon counting detector with pileup but otherwise ideal energy response and an ideal dual‐energy system (80/140 kVp with tin filtration). The modeled CdTe thickness was 2 mm, the incident spectrum was 140 kVp and 500 mA, and the effective dead time was 67 ns. Charge summing circuitry was not modeled. We restricted our simulations to objects of uniform thickness and did not consider the potential advantage of smaller pixels at high spatial frequencies. Results: At very high x‐ray flux, pulse pileup dominates and small pixel sizes perform best. At low flux or for thick objects, charge sharing dominates and large pixel sizes perform best. At low flux and depending on the beam hardness, the CRLB of variance in basis material projections tasks can be 32%–55% higher with a 250 micron pixel pitch compared to a 450 micron pixel pitch. However, both are about four times worse in variance than the ideal photon counting detector. The optimal pixel size depends on a number of factors such as x‐ray technique and object size. At high technique (140 kVp/500 mA), the ratio of variance for a 450 micron pixel compared to a 250 micron pixel size is 2126%, 200%, 97%, and 78% when imaging 10, 15, 20, and 25 cm of water, respectively. If 300 mg/cm2 of iodine is also added to the object, the variance ratio is 117%, 91%, 74%, and 72%, respectively. Nonspectral tasks, such as equivalent monoenergetic imaging, are less sensitive to spectral distortion. Conclusions: The detector pixel size is an important design consideration in CdTe detectors. Smaller pixels allow for improved capabilities at high flux but increase charge sharing, which in turn compromises spectral performance. The optimal pixel size will depend on the specific task and on the charge shaping time. [ABSTRACT FROM AUTHOR]

A bowtie is a filter used to shape an x-ray beam and equalize its flux reaching different detector channels. For development of spectral CT with energy discriminating photon-counting (EDPC) detectors, here we propose and evaluate a dynamic bowtie for performance optimization based on a patient model or a scout scan. With a mechanical rotation of a dynamic bowtie and an adaptive adjustment of an x-ray source flux, an x-ray beam intensity profile can be modulated. First, a mathematical model for dynamic bowtie filtering is established for an elliptical section in fan-beam geometry, and the contour of the optimal bowtie is derived. Then, numerical simulation is performed to compare the performance of the dynamic bowtie in the cases of an ideal phantom and a realistic cross-section relative to the counterparts without any bowtie and with a fixed bowtie respectively. Our dynamic bowtie can equalize the expected numbers of photons in the case of an ideal phantom. In practical cases, our dynamic bowtie can effectively reduce the dynamic range of detected signals inside the field of view. Although our design is optimized for an elliptical phantom, the resultant dynamic bowtie can be applied to a real fan-beam scan if the underlying cross-section can be approximated as an ellipse. Furthermore, our design methodology can be applied to specify an optimized dynamic bowtie for any cross-section of a patient, preferably using rapid prototyping technology. [ABSTRACT FROM AUTHOR]