Detectors for the future of X-ray imaging

Initial Clinical Images From a Second-Generation Prototype Silicon-Based Photon-Counting Computed Tomography System

RATIONALE AND OBJECTIVES

To demonstrate the feasibility and potential of using a second-generation prototype photon-counting computed tomography (CT) system to provide simultaneous high spatial resolution images and high spectral resolution material information across a range of routine imaging tasks using clinical patient exposure levels.

MATERIALS AND METHODS

The photon-counting system employs an innovative silicon-based photon-counting detector to provide a balanced approach to ultra-high-resolution spectral CT imaging. An initial cohort of volunteer subjects was imaged using the prototype photon-counting system. Acquisition technique parameters and radiation dose exposures were guided by routine clinical exposure levels used at the institution. Images were reconstructed in native slice thickness using an early version of a spectral CT reconstruction algorithm Samples of images across a range of clinical tasks were selected and presented for review.

RESULTS

Clinical cases are presented across inner ear, carotid angiography, chest, and musculoskeletal imaging tasks. Initial reconstructed images illustrate ultra-high spatial resolution imaging. The fine detail of small structures and pathologies is clearly visualized, and structural boundaries are well delineated. The prototype system additionally provides concomitant spectral information with high spatial resolution.

CONCLUSION

This initial study demonstrates that routine imaging at clinically appropriate patient exposure levels is feasible using a novel deep-silicon photon-counting detector CT system. Furthermore, a deep-silicon detector may provide a balanced approach to photon-counting CT, providing high spatial resolution imaging with simultaneous high-fidelity spectral information.

Influence of the thickness of a SiO2 reflective layer on the performance of a structured CsI(Tl) scintillation screen based on an oxidized Si micropore array template in X-ray imaging

To obtain better light guidance and optical isolation effects under a limited microcolumn wall thickness, the influence of the thickness of a SiO₂ reflective layer on the performance of a structured CsI(Tl) scintillation screen based on an oxidized Si micropore array template in X-ray imaging was simulated. The results show that the SiO₂ reflective layer should maintain a certain thickness to achieve good light-guide performance. However, if the template is entirely composed of SiO₂, the light isolation performance of the microcolumn wall will be slightly worse. The results provide a basis for optimizing the thickness of SiO₂ reflective layer.

Virtual monoenergetic images from photon-counting spectral computed tomography to assess knee osteoarthritis

Background

Dual-energy computed tomography has shown a great interest for musculoskeletal pathologies. Photon-counting spectral computed tomography (PCSCT) can acquire data in multiple energy bins with the potential to increase contrast, especially for soft tissues. Our objectives were to assess the value of PCSST to characterise cartilage and to extract quantitative measures of subchondral bone integrity.

Methods

Seven excised human knees (3 males and 4 females; 4 normal and 3 with osteoarthritis; age 80.6 ± 14 years, mean ± standard deviation) were scanned using a clinical PCSCT prototype scanner. Tomographic image reconstruction was performed after Compton/photoelectric decomposition. Virtual monoenergetic images were generated from 40 keV to 110 keV every 10 keV (cubic voxel size 250 × 250 × 250 μm³). After selecting an optimal virtual monoenergetic image, we analysed the grey level histograms of different tissues and extracted quantitative measurements on bone cysts.

Results

The optimal monoenergetic images were obtained for 60 keV and 70 keV. Visual inspection revealed that these images provide sufficient spatial resolution and soft-tissue contrast to characterise surfaces, disruption, calcification of cartilage, bone osteophytes, and bone cysts. Analysis of attenuation versus energy revealed different energy fingerprint according to tissues. The volumes and numbers of bone cyst were quantified.

Conclusions

Virtual monoenergetic images may provide direct visualisation of both cartilage and bone details. Thus, unenhanced PCSCT appears to be a new modality for characterising the knee joint with the potential to increase the diagnostic capability of computed tomography for joint diseases and osteoarthritis.

Visualization of blurring process due to analog components in a digital radiography system using a simple method

To obtain useful medical X-ray images for diagnosis, it is necessary to accurately evaluate the characteristics and fully understand the features of the system to determine the imaging target and conditions. This study describes the performance of a digital radiography (DR) system in detail and proposes a method for characterizing the blurring process due to the presence of analog components in an imaging system. Our method does not involve any specialized technique, such as a simulation that requires long computing time on a high-performance computer. The method is based on the concept of the modulation transfer function (MTF). The functions that corresponded to the MTF of the analog components of the system when the Fourier transform was performed were examined. Indirect conversion type flat-panel detectors (FPD) in the general radiography energy range and direct conversion type FPD in the mammography were used for verification. A Gaussian function and Lorentz function were synthesized for modelling the blurring process of the indirect type FPDs. The spreads and shapes of the modeled functions depended on the systems. For the direct conversion type FPD, a combination of a Lorentz function with a narrow width and an impulse function characterized the results. Furthermore, based on the structure of each detector and the signal transfer process, the visualized results were considered reliable. The results of these experiments will aid in the detailed understanding of the image quality characteristics of the DR system by the feasible approach.

BriXS, a new X-ray inverse Compton source for medical applications

MariX is a research infrastructure conceived for multi-disciplinary studies, based on a cutting-edge system of combined electron accelerators at the forefront of the world-wide scenario of X-ray sources. The generation of X-rays over a large photon energy range will be enabled by two unique X-ray sources: a Free Electron Laser and an inverse Compton source, called BriXS (Bright compact X-ray Source). The X-ray beam provided by BriXS is expected to have an average energy tunable in the range 20-180 keV and intensities between 10¹¹ and 10¹³ photon/s within a relative bandwidth ΔE/E=1-10%. These characteristics, together with a very small source size (~20 μm) and a good transverse coherence, will enable a wide range of applications in the bio-medical field. An additional unique feature of BriXS will be the possibility to make a quick switch of the X-ray energy between two values for dual-energy and K-edge subtraction imaging. In this paper, the expected characteristics of BriXS will be presented, with a particular focus on the features of interest to its possible medical applications.

Development of ZnO-based nanorod arrays as scintillator layer for ultrafast and high-spatial-resolution X-ray imaging system

Under excitation of high-energy and low-flux density of X-ray beam, a 1-μm system spatial resolution was initially achieved by using an 18-μm thickness ZnO nanorod array as the scintillator layer in X-ray imaging beamline at Shanghai Synchrotron Radiation Facility. The decay time measurements indicated the ultraviolet and visible emissions of the arrays were subnanosecond and nanosecond, respectively. Through hydrogen annealing treatment, the ultraviolet luminescence was intensively enhanced and the visible luminescence was remarkably suppressed simultaneously. In conclusion, it can be determined that the ZnO-based nanorod arrays are the decent candidates for applications in ultrafast and high-spatial-resolution X-ray imaging systems.

Measurement of effective detective quantum efficiency for a photon counting scanning mammography system and comparison with two flat panel full-field digital mammography systems

Effective detective quantum efficiency (eDQE) describes the resolution and noise properties of an imaging system along with scatter and primary transmission, all measured under clinically appropriate conditions. Effective dose efficiency (eDE) is the eDQE normalised to mean glandular dose and has been proposed as a useful metric for the optimisation of clinical imaging systems. The aim of this study was to develop a methodology for measuring eDQE and eDE on a Philips microdose mammography (MDM) L30 photon counting scanning system, and to compare performance with two conventional flat panel systems. A custom made lead-blocker was manufactured to enable the accurate determination of dose measurements, and modulation transfer functions were determined free-in-air at heights of 2, 4 and 6 cm above the breast support platform. eDQE were calculated for a Philips MDM L30, Hologic Dimensions and Siemens Inspiration digital mammography system for 2, 4 and 6 cm thick poly(methyl methacrylate) (PMMA). The beam qualities (target/filter and kilovoltage) assessed were those selected by the automatic exposure control, and anti-scatter grids were used where available. Measurements of eDQE demonstrate significant differences in performance between the slit- and scan-directions for the photon counting imaging system. MTF has been shown to be the limiting factor in the scan-direction, which results in a rapid fall in eDQE at mid-to-high spatial frequencies. A comparison with two flat panel mammography systems demonstrates that this may limit image quality for small details, such as micro-calcifications, which correlates with a more conventional image quality assessment with the CDMAM phantom. eDE has shown the scanning photon counting system offers superior performance for low spatial frequencies, which will be important for the detection of large low contrast masses. Both eDQE and eDE are proposed as useful metrics that should enable optimisation of the Philips MDM L30.

Quantitative contrast-enhanced spectral mammography based on photon-counting detectors: A feasibility study

PURPOSE

To investigate the feasibility of accurate quantification of iodine mass thickness in contrast-enhanced spectral mammography.

MATERIALS AND METHODS

A computer simulation model was developed to evaluate the performance of a photon-counting spectral mammography system in the application of contrast-enhanced spectral mammography. A figure-of-merit (FOM), which was defined as the decomposed iodine signal-to-noise ratio (SNR) with respect to the square root of the mean glandular dose (MGD), was chosen to optimize the imaging parameters, in terms of beam energy, splitting energy, and prefiltrations for breasts of various thicknesses and densities. Experimental phantom studies were also performed using a beam energy of 40 kVp and a splitting energy of 34 keV with 3 mm Al prefiltration. A two-step calibration method was investigated to quantify the iodine mass thickness, and was validated using phantoms composed of a mixture of glandular and adipose materials, for various breast thicknesses and densities. Finally, the traditional dual-energy log-weighted subtraction method was also studied as a comparison. The measured iodine signal from both methods was compared to the known value to characterize the quantification accuracy and precision.

RESULTS

The optimal imaging parameters, which lead to the highest FOM, were found at a beam energy between 42 and 46 kVp with a splitting energy at 34 keV. The optimal tube voltage decreased as the breast thickness or the Al prefiltration increased. The proposed quantification method was able to measure iodine mass thickness on phantoms of various thicknesses and densities with high accuracy. The root-mean-square (RMS) error for cm-scale lesion phantoms was estimated to be 0.20 mg/cm² . The precision of the technique, characterized by the standard deviation of the measurements, was estimated to be 0.18 mg/cm² . The traditional weighted subtraction method also predicted a linear correlation between the measured signal and the known iodine mass thickness. However, the correlation slope and offset values were strongly dependent on the total breast thickness and density.

CONCLUSION

The results of this study suggest that iodine mass thickness for cm-scale lesions can be accurately quantified with contrast-enhanced spectral mammography. The quantitative information can potentially improve the differential power for malignancy.

Efficient, Non-Iterative Estimator for Imaging Contrast Agents With Spectral X-Ray Detectors

An estimator to image contrast agents and body materials with x-ray spectral measurements is described. The estimator is usable with the three or more basis functions that are required to represent the attenuation coefficient of high atomic number materials. The estimator variance is equal to the Cramèr-Rao lower bound (CRLB) and it is unbiased. Its parameters are computed from measurements of a calibration phantom with the clinical x-ray system and it is non-iterative. The estimator is compared with an iterative maximum likelihood estimator. The estimator first computes a linearized maximum likelihood estimate of the line integrals of the basis set coefficients. Corrections for errors in the initial estimates are computed by interpolation with calibration phantom data. The final estimate is the initial estimate plus the correction. The performance of the estimator is measured using a Monte Carlo simulation. Random photon counting with pulse height analysis data are generated. The mean squared errors of the estimates are compared to the CRLB. The random data are also processed with an iterative maximum likelihood estimator. Previous implementations of iterative estimators required advanced physics instruments not usually available in clinical institutions. The estimator mean squared error is essentially equal to the CRLB. The estimator outputs are close to those of the iterative estimator but the computation time is approximately 180 times shorter. The estimator is efficient and has advantages over alternate approaches such as iterative estimators.

Signal to noise ratio of energy selective x-ray photon counting systems with pileup

PURPOSE

To derive fundamental limits on the effect of pulse pileup and quantum noise in photon counting detectors on the signal to noise ratio (SNR) and noise variance of energy selective x-ray imaging systems.

METHODS

An idealized model of the response of counting detectors to pulse pileup is used. The model assumes a nonparalyzable response and delta function pulse shape. The model is used to derive analytical formulas for the noise and energy spectrum of the recorded photons with pulse pileup. These formulas are first verified with a Monte Carlo simulation. They are then used with a method introduced in a previous paper [R. E. Alvarez, "Near optimal energy selective x-ray imaging system performance with simple detectors," Med. Phys. 37, 822-841 (2010)] to compare the signal to noise ratio with pileup to the ideal SNR with perfect energy resolution. Detectors studied include photon counting detectors with pulse height analysis (PHA), detectors that simultaneously measure the number of photons and the integrated energy (NQ detector), and conventional energy integrating and photon counting detectors. The increase in the A-vector variance with dead time is also computed and compared to the Monte Carlo results. A formula for the covariance of the NQ detector is developed. The validity of the constant covariance approximation to the Cramèr-Rao lower bound (CRLB) for larger counts is tested.

RESULTS

The SNR becomes smaller than the conventional energy integrating detector (Q) SNR for 0.52, 0.65, and 0.78 expected number photons per dead time for counting (N), two, and four bin PHA detectors, respectively. The NQ detector SNR is always larger than the N and Q SNR but only marginally so for larger dead times. Its noise variance increases by a factor of approximately 3 and 5 for the A1 and A2 components as the dead time parameter increases from 0 to 0.8 photons per dead time. With four bin PHA data, the increase in variance is approximately 2 and 4 times. The constant covariance approximation to the CRLB is valid for larger counts such as those used in medical imaging.

CONCLUSIONS

The SNR decreases rapidly as dead time increases. This decrease places stringent limits on allowable dead times with the high count rates required for medical imaging systems. The probability distribution of the idealized data with pileup is shown to be accurately described as a multivariate normal for expected counts greater than those typically utilized in medical imaging systems. The constant covariance approximation to the CRLB is also shown to be valid in this case. A new formula for the covariance of the NQ detector with pileup is derived and validated.

Energy Window Optimization for X-Ray K-Edge Tomographic Imaging

GOAL

In K-edge tomographic imaging with photon counting detectors, the energy window width of photon counting detectors significantly affects the signal-to-noise ratio (SNR) of measured intensity data and the contrast-to-noise ratio (CNR) of reconstructed images. In this paper, we present an optimization method to determine an optimal window width around a K-edge for optimal SNR and CNR.

METHODS

An objective function is designed to describe SNR of the projection data based on the Poisson distribution of detected X-ray photons. Then, a univariate optimization method is applied to obtain an X-ray energy window width.

RESULTS

Numerical simulations are performed to evaluate the proposed method, and the results show that the optimal energy window width obtained from the proposed method produces not only optimal SNR data in the projection domain but also optimal CNR values in the image domain.

CONCLUSION

The proposed method in the projection domain can determine an optimal energy window width for X-ray photon counting imaging, and achieve optimality in both projection and image domains.

SIGNIFICANCE

Our study provides a practical way to determine the optimal energy window width of photon counting detectors, which helps improve contrast resolution for X-ray K-edge tomographic imaging.

Dimensionality and noise in energy selective x-ray imaging

PURPOSE

To develop and test a method to quantify the effect of dimensionality on the noise in energy selective x-ray imaging.

METHODS

The Cramèr-Rao lower bound (CRLB), a universal lower limit of the covariance of any unbiased estimator, is used to quantify the noise. It is shown that increasing dimensionality always increases, or at best leaves the same, the variance. An analytic formula for the increase in variance in an energy selective x-ray system is derived. The formula is used to gain insight into the dependence of the increase in variance on the properties of the additional basis functions, the measurement noise covariance, and the source spectrum. The formula is also used with computer simulations to quantify the dependence of the additional variance on these factors. Simulated images of an object with three materials are used to demonstrate the trade-off of increased information with dimensionality and noise. The images are computed from energy selective data with a maximum likelihood estimator.

RESULTS

The increase in variance depends most importantly on the dimension and on the properties of the additional basis functions. With the attenuation coefficients of cortical bone, soft tissue, and adipose tissue as the basis functions, the increase in variance of the bone component from two to three dimensions is 1.4 × 10(3). With the soft tissue component, it is 2.7 × 10(4). If the attenuation coefficient of a high atomic number contrast agent is used as the third basis function, there is only a slight increase in the variance from two to three basis functions, 1.03 and 7.4 for the bone and soft tissue components, respectively. The changes in spectrum shape with beam hardening also have a substantial effect. They increase the variance by a factor of approximately 200 for the bone component and 220 for the soft tissue component as the soft tissue object thickness increases from 1 to 30 cm. Decreasing the energy resolution of the detectors increases the variance of the bone component markedly with three dimension processing, approximately a factor of 25 as the resolution decreases from 100 to 3 bins. The increase with two dimension processing for adipose tissue is a factor of two and with the contrast agent as the third material for two or three dimensions is also a factor of two for both components. The simulated images show that a maximum likelihood estimator can be used to process energy selective x-ray data to produce images with noise close to the CRLB.

CONCLUSIONS

The method presented can be used to compute the effects of the object attenuation coefficients and the x-ray system properties on the relationship of dimensionality and noise in energy selective x-ray imaging systems.

Note: design and construction of a multi-scale, high-resolution, tube-generated x-ray computed-tomography system for three-dimensional (3D) imaging

The design and construction of a high resolution modular x-ray computed tomography (XCT) system is described. The approach for meeting a specified set of performance goals tailored toward experimental versatility is highlighted. The instrument is unique in its detector and x-ray source configuration, both of which enable elevated optimization of spatial and temporal resolution. The process for component selection is provided. The selected components are specified, the custom component design discussed, and the integration of both into a fully functional XCT instrument is outlined. The novelty of this design is a new lab-scale detector and imaging optimization through x-ray source and detector modularity.

A unified statistical framework for material decomposition using multienergy photon counting x-ray detectors

PURPOSE

Material decomposition using multienergy photon counting x-ray detectors (PCXD) has been an active research area over the past few years. Even with some success, the problem of optimal energy selection and three material decomposition including malignant tissue is still on going research topic, and more systematic studies are required. This paper aims to address this in a unified statistical framework in a mammographic environment.

METHODS

A unified statistical framework for energy level optimization and decomposition of three materials is proposed. In particular, an energy level optimization algorithm is derived using the theory of the minimum variance unbiased estimator, and an iterative algorithm is proposed for material composition as well as system parameter estimation under the unified statistical estimation framework. To verify the performance of the proposed algorithm, the authors performed simulation studies as well as real experiments using physical breast phantom and ex vivo breast specimen. Quantitative comparisons using various performance measures were conducted, and qualitative performance evaluations for ex vivo breast specimen were also performed by comparing the ground-truth malignant tissue areas identified by radiologists.

RESULTS

Both simulation and real experiments confirmed that the optimized energy bins by the proposed method allow better material decomposition quality. Moreover, for the specimen thickness estimation errors up to 2 mm, the proposed method provides good reconstruction results in both simulation and real ex vivo breast phantom experiments compared to existing methods.

CONCLUSIONS

The proposed statistical framework of PCXD has been successfully applied for the energy optimization and decomposition of three material in a mammographic environment. Experimental results using the physical breast phantom and ex vivo specimen support the practicality of the proposed algorithm.

Quantification of breast density with spectral mammography based on a scanned multi-slit photon-counting detector: a feasibility study

A simple and accurate measurement of breast density is crucial for the understanding of its impact in breast cancer risk models. The feasibility to quantify volumetric breast density with a photon-counting spectral mammography system has been investigated using both computer simulations and physical phantom studies. A computer simulation model involved polyenergetic spectra from a tungsten anode x-ray tube and a Si-based photon-counting detector has been evaluated for breast density quantification. The figure-of-merit (FOM), which was defined as the signal-to-noise ratio of the dual energy image with respect to the square root of mean glandular dose, was chosen to optimize the imaging protocols, in terms of tube voltage and splitting energy. A scanning multi-slit photon-counting spectral mammography system has been employed in the experimental study to quantitatively measure breast density using dual energy decomposition with glandular and adipose equivalent phantoms of uniform thickness. Four different phantom studies were designed to evaluate the accuracy of the technique, each of which addressed one specific variable in the phantom configurations, including thickness, density, area and shape. In addition to the standard calibration fitting function used for dual energy decomposition, a modified fitting function has been proposed, which brought the tube voltages used in the imaging tasks as the third variable in dual energy decomposition. For an average sized 4.5 cm thick breast, the FOM was maximized with a tube voltage of 46 kVp and a splitting energy of 24 keV. To be consistent with the tube voltage used in current clinical screening exam (∼32 kVp), the optimal splitting energy was proposed to be 22 keV, which offered a FOM greater than 90% of the optimal value. In the experimental investigation, the root-mean-square (RMS) error in breast density quantification for all four phantom studies was estimated to be approximately 1.54% using standard calibration function. The results from the modified fitting function, which integrated the tube voltage as a variable in the calibration, indicated a RMS error of approximately 1.35% for all four studies. The results of the current study suggest that photon-counting spectral mammography systems may potentially be implemented for an accurate quantification of volumetric breast density, with an RMS error of less than 2%, using the proposed dual energy imaging technique.

Direct measurement of mammographic X-ray spectra with a digital CdTe detection system

In this work we present a detection system, based on a CdTe detector and an innovative digital pulse processing (DPP) system, for high-rate X-ray spectroscopy in mammography (1-30 keV). The DPP system performs a height and shape analysis of the detector pulses, sampled and digitized by a 14-bit, 100 MHz ADC. We show the results of the characterization of the detection system both at low and high photon counting rates by using monoenergetic X-ray sources and a nonclinical X-ray tube. The detection system exhibits excellent performance up to 830 kcps with an energy resolution of 4.5% FWHM at 22.1 keV. Direct measurements of clinical molybdenum X-ray spectra were carried out by using a pinhole collimator and a custom alignment device. A comparison with the attenuation curves and the half value layer values, obtained from the measured and simulated spectra, from an ionization chamber and from a solid state dosimeter, also shows the accuracy of the measurements. These results make the proposed detection system a very attractive tool for both laboratory research, calibration of dosimeters and advanced quality controls in mammography.

Estimator for photon counting energy selective x-ray imaging with multibin pulse height analysis

PURPOSE

This paper describes a noniterative estimator for the energy dependent information from photon counting detectors with multibin pulse height analysis (PHA). The estimator uses the two function decomposition of the attenuation coefficient [R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733-744 (1976)] and its output is the line integrals of the basis set coefficients. The output noise variance and bias is compared to other noniterative estimators and to the Cramèr-Rao lower bound (CRLB).

METHODS

The estimator first computes an initial estimate from a linearized maximum likelihood estimator. The errors in the initial estimates are determined at a set of points from measurements on a calibration phantom. The errors at these known points are interpolated to create two-dimensional look up tables of corrections to the initial estimates. During image acquisition, the linearized maximum likelihood estimate for each data point is used as an input to the correction look up tables, and the final output is the sum of the estimate and the correction. The performance of the estimator is compared to generalizations of the polynomial and rational polynomial estimators for multibin data. The estimators are compared by the mean square error (MSE) and its components, the bias, and the variance of the output. The variance is also compared to the CRLB. The performance is simulated with two to five bins PHA data. The CRLB at a fixed object thickness is also computed as a function of the number of bins.

RESULTS

For two bin data, all the estimators' variances are equal to the CRLB. With three or more bins, only the proposed estimator achieves the CRLB while the others, which were not optimized for noise performance, have much larger output variance. The bias of the proposed estimator is equal to the polynomial estimator for calibration phantoms with 40 or more steps, that is, 1600 combinations of basis materials, but is larger than the rational polynomial bias. In all cases at the photon counts tested, the MSE is essentially equal to the variance, indicating that the bias errors are negligible compared to the variance.

CONCLUSIONS

The estimator provides a noniterative method to compute the energy dependent information from multibin PHA data that achieves the CRLB over a wide range of operating conditions and has low output bias. The estimator can be calibrated based on the measurements of a calibration phantom; so, it does not require measurements of the x-ray energy spectrum or the detector response functions.

A comparative analysis of OTF, NPS, and DQE in energy integrating and photon counting digital x-ray detectors

PURPOSE

One of the benefits of photon counting (PC) detectors over energy integrating (EI) detectors is the absence of many additive noise sources, such as electronic noise and secondary quantum noise. The purpose of this work is to demonstrate that thresholding voltage gains to detect individual x rays actually generates an unexpected source of white noise in photon counters.

METHODS

To distinguish the two detector types, their point spread function (PSF) is interpreted differently. The PSF of the energy integrating detector is treated as a weighting function for counting x rays, while the PSF of the photon counting detector is interpreted as a probability. Although this model ignores some subtleties of real imaging systems, such as scatter and the energy-dependent amplification of secondary quanta in indirect-converting detectors, it is useful for demonstrating fundamental differences between the two detector types. From first principles, the optical transfer function (OTF) is calculated as the continuous Fourier transform of the PSF, the noise power spectra (NPS) is determined by the discrete space Fourier transform (DSFT) of the autocovariance of signal intensity, and the detective quantum efficiency (DQE) is found from combined knowledge of the OTF and NPS. To illustrate the calculation of the transfer functions, the PSF is modeled as the convolution of a Gaussian with the product of rect functions. The Gaussian reflects the blurring of the x-ray converter, while the rect functions model the sampling of the detector.

RESULTS

The transfer functions are first calculated assuming outside noise sources such as electronic noise and secondary quantum noise are negligible. It is demonstrated that while OTF is the same for two detector types possessing an equivalent PSF, a frequency-independent (i.e., "white") difference in their NPS exists such that NPS(PC) > or = NPS(EI) and hence DQE(PC) < or = DQE(EI). The necessary and sufficient condition for equality is that the PSF is a binary function given as zero or unity everywhere. In analyzing the model detector with Gaussian blurring, the difference in NPS and DQE between the two detector types is found to increase with the blurring of the x-ray converter. Ultimately, the expression for the additive white noise of the photon counter is compared against the expression for electronic noise and secondary quantum noise in an energy integrator. Thus, a method is provided to determine the average secondary quanta that the energy integrator must produce for each x ray to have superior DQE to a photon counter with the same PSF.

CONCLUSIONS

This article develops analytical models of OTF, NPS, and DQE for energy integrating and photon counting digital x-ray detectors. While many subtleties of real imaging systems have not been modeled, this work is illustrative in demonstrating an additive source of white noise in photon counting detectors which has not yet been described in the literature. One benefit of this analysis is a framework for determining the average secondary quanta that an energy integrating detector must produce for each x ray to have superior DQE to competing photon counting technology.