Image quality in diagnostic x-ray imaging is ultimately limited by the statistical properties governing how, and where, x-ray energy is deposited in a detector. This in turn depends on the physics of the underlying x-ray interactions. In the diagnostic energy range (10-100 keV), most of the energy deposited in a detector is through photoelectric interactions. We present a theoretical model of the photoelectric effect that specifically addresses the statistical nature of energy absorption by photoelectrons, K and L characteristic x rays, and Auger electrons. A cascaded-systems approach is used that employs a complex structure of parallel cascades to describe signal and noise transfer through the photoelectric effect in terms of the modulation transfer function, Wiener noise power spectrum, and detective quantum efficiency (DQE). The model was evaluated by comparing results with Monte Carlo calculations for x-ray converters based on amorphous selenium (a-Se) and lead (Pb), representing both low and high-Z materials. When electron transport considerations can be neglected, excellent agreement (within 3%) is obtained for each metric over the entire diagnostic energy range in both a-Se and Pb detectors up to 30 cycles/mm, the highest frequency tested. The cascaded model overstates the DQE when the electron range cannot be ignored. This occurs at approximately two cycles/mm in a-Se at an incident photon energy of 80 keV, whereas in Pb, excellent agreement is obtained for the DQE over the entire diagnostic energy range. However, within the context of mammography (20 keV) and micro-computed tomography (40 keV), the effects of electron transport on the DQE are negligible compared to fluorescence reabsorption, which can lead to decreases of up to 30% and 20% in a-Se and Pb, respectively, at 20 keV; and 10% and 5%, respectively, at 40 keV. It is shown that when Swank noise is identified in a Fourier model, the Swank factor must be frequency dependent. This factor decreases quickly with frequency, and in the case of a-Se and Pb, decreases by up to a factor of 3 at five cycles/mm immediately above the K edge. The frequency-dependent Swank factor is also equivalent to what we call the "photoelectric DQE," which describes signal and noise transfer through photoelectric interactions.

The practice of diagnostic x-ray imaging has been transformed with the emergence of digital detector technology. Although digital systems offer many practical advantages over conventional film-based systems, their spatial resolution performance can potentially be a limitation. Detector element size is one important factor limiting resolution in digital systems and manufacturers have addressed the issue by developing smaller elements. However, detector element size can only be reduced so far before which more fundamental effects become important. We present a Monte Carlo study in which the fundamental causes and upper limits of spatial resolution, due to x-ray interactions, were determined for direct conversion amorphous silicon (a-Si), amorphous selenium (a-Se) and lead iodide (PbI 2), and indirect conversion cesium iodide (CsI) detectors. The spatial distribution of absorbed energy (per unit mass), or point spread function (PSF), was scored within each converter material for various incident photon energies (10-150 keV) and converter thicknesses (based on quantum efficiency values between 0.10 and 0.99). The "x-ray interaction" modulation transfer function (MTF) was determined from each PSF, and was used to characterize the energy and thickness dependence of the spatial resolution using the 50% MTF frequency, f 50 , and effective sampling aperture, a ef f. In the diagnostic energy range, f 50 values reach as low as 0.3 cycles/mm in a-Si at 60 keV and above; and 300, 20, and (100,10) cycles/mm at the K-edges of a-Se, CsI, and PbI 2 , respectively. In contrast, a ef f values are similar for each of the materials, ranging from 3 µm at low energies (20 keV) to 20 µm at higher energies (100 keV). Several conclusions can be drawn from the results of these simulations. (i) The main source of MTF degradation in low (a-Si) and high Z (a-Se, CsI, PbI 2) materials is due to reabsorption of Compton scatter x rays and K fluorescent x rays, respectively. (ii) Secondary electrons (e.g. photoelectrons) play an important role in determining the overall shape of the MTF. They not only degrade the high-frequency portion (>5 cy/mm) of the MTF, but also control the magnitude of the low-frequency drop (<5 cy/mm) due to secondary x ray (e.g. K fluorescent x rays) reabsorption. (iii) Current thicknesses of a-Si and a-Se used in practice may be increased to improve quantum efficiency with minimal loss in spatial resolution (from x-ray interactions) for mammography and µ-CT applications. (iv) Although not reached practically, the above a ef f values represent the fundamental spatial resolution limits of the converter materials tested and serve as a fundamental barrier that dictate how small detector elements should be manufactured.

An often neglected assumption related to detector performance metrics such as the modulation transfer function (MTF), noise power spectrum (NPS), and detective quantum efficiency (DQE) is that they only apply to a small region around the centre of an x-ray image. In the periphery of an image, image formation is from obliquely incident x rays. These off-axis x rays will introduce an additional degrading effect on the above detector performance metrics. In our study, we use Monte Carlo simulations to quantify the effects of off-axis radiation on the MTF, NPS, and DQE on common diagnostic x-ray detectors. In our simulations, we vary the incident angle of x rays between 0 • and 12 • , which is a typical range of divergence in diagnostic x-ray imaging. In the case of amorphous selenium, our results show that off-axis incident x rays degrade the MTF above 5 cycles/mm with increasing severity at higher incident angles and x-ray energy, and more importantly has very little effect on the NPS. Hence, the impact is more severe on the DQE due to the MTF squared dependency. For an incident x-ray angle of 12 • (∼13 cm from central axis or chest wall in mammography), the DQE falls to 50% of its initial value at 10 and 7 cycles/mm for x-ray energies of 20 and 40 keV, respectively. This loss of signal-to-noise ratio may be most significant near the skin line in mammography studies.

A frequency-dependent x-ray Swank factor based on the "x-ray interaction" modulation transfer function and normalized noise power spectrum is determined from a Monte Carlo analysis. This factor was calculated in four converter materials: amorphous silicon (a-Si), amorphous selenium (a-Se), cesium iodide (CsI), and lead iodide (PbI2) for incident photon energies between 10 and 150 keV and various converter thicknesses. When scaled by the quantum efficiency, the x-ray Swank factor describes the best possible detective quantum efficiency (DQE) a detector can have. As such, this x-ray interaction DQE provides a target performance benchmark. It is expressed as a function of (Fourier-based) spatial frequency and takes into consideration signal and noise correlations introduced by reabsorption of Compton scatter and photoelectric characteristic emissions. It is shown that the x-ray Swank factor is largely insensitive to converter thickness for quantum efficiency values greater than 0.5. Thus, while most of the tabulated values correspond to thick converters with a quantum efficiency of 0.99, they are appropriate to use for many detectors in current use. A simple expression for the x-ray interaction DQE of digital detectors (including noise aliasing) is derived in terms of the quantum efficiency, x-ray Swank factor, detector element size, and fill factor. Good agreement is shown with DQE curves published by other investigators for each converter material, and the conditions required to achieve this ideal performance are discussed. For high-resolution imaging applications, the x-ray Swank factor indicates: (i) a-Si should only be used at low-energy (e.g., mammography); (ii) a-Se has the most promise for any application below 100 keV; and (iii) while quantum efficiency may be increased at energies just above the K edge in CsI and PbI2, this benefit is offset by a substantial drop in the x-ray Swank factor, particularly at high spatial frequencies.