Detective quantum efficiency of photon-counting x-ray detectors

PURPOSE
Single-photon-counting (SPC) x-ray imaging has the potential to improve image quality and enable novel energy-dependent imaging methods. Similar to conventional detectors, optimizing image SPC quality will require systems that produce the highest possible detective quantum efficiency (DQE). This paper builds on the cascaded-systems analysis (CSA) framework to develop a comprehensive description of the DQE of SPC detectors that implement adaptive binning.


METHODS
The DQE of SPC systems can be described using the CSA approach by propagating the probability density function (PDF) of the number of image-forming quanta through simple quantum processes. New relationships are developed to describe PDF transfer through serial and parallel cascades to accommodate scatter reabsorption. Results are applied to hypothetical silicon and selenium-based flat-panel SPC detectors including the effects of reabsorption of characteristic/scatter photons from photoelectric and Compton interactions, stochastic conversion of x-ray energy to secondary quanta, depth-dependent charge collection, and electronic noise. Results are compared with a Monte Carlo study.


RESULTS
Depth-dependent collection efficiency can result in substantial broadening of photopeaks that in turn may result in reduced DQE at lower x-ray energies (20-45 keV). Double-counting interaction events caused by reabsorption of characteristic/scatter photons may result in falsely inflated image signal-to-noise ratio and potential overestimation of the DQE.


CONCLUSIONS
The CSA approach is extended to describe signal and noise propagation through photoelectric and Compton interactions in SPC detectors, including the effects of escape and reabsorption of emission/scatter photons. High-performance SPC systems can be achieved but only for certain combinations of secondary conversion gain, depth-dependent collection efficiency, electronic noise, and reabsorption characteristics.


INTRODUCTION
Advances in x-ray detector technology have enabled singlephoton-counting (SPC) x-ray detectors with the ability to identify individual photon interactions.][3][4][5][6][7] It is anticipated that the resulting spectral distribution of energy-depositing events will lead to advanced spectroscopic procedures [8][9][10][11][12][13][14] and improve image quality by reducing image noise from random physical processes including Swank and electronic readout noise. 7,15hile photon-counting methods are receiving a great deal of interest, there remain many challenges to overcome.Stateof-the-art systems are capable of count rates that may be adequate for some applications including mammography [16][17][18][19][20] and breast computed tomography, [21][22][23][24] but remain restrictive. 25,268][29][30] This effect is mitigated with techniques that sum charges in neighboring elements surrounding an interaction and assigns them to a single element, such as that implemented in the MEDIPIX3 prototype. 4In the MEDIPIX approach, elements are clustered into larger binned elements, the total charge generated in binned elements is determined, and a count is attributed to the element (within each cluster) with the largest signal.We call these methods "adaptive binning" and some form of adaptive binning will almost certainly be required to achieve highquality images.
It is also true that these systems will produce the best possible images when they are optimized to achieve the best possible detective quantum efficiency (DQE). 31Cascadedsystems analysis [32][33][34][35][36][37][38][39][40][41] (CSA) has been successful in the development of theoretical models of the DQE, important in the development of conventional energy-integrating systems, and was recently extended to describe the zero-frequency DQE of SPC detectors that implement adaptive binning. 42,43n particular, it was shown that DQE(0) can be expressed in terms of the mean photon-counting signal and associated Wiener noise power spectrum (NPS) and that both can be obtained from the probability density function (PDF) of detector signals.For SPC systems, this leads to 43 DQE(0) = αI SPC , where α represents the detector quantum efficiency and I SPC is a noise factor (I SPC ≤ 1) equal to the probability that a true photon count is recorded given an interaction event (truepositive fraction).This latter term has a pleasing symmetry with the Swank-noise factor 38,[44][45][46] for conventional energyintegrating systems with I SPC becoming the "SPC Swank factor" accounting for degradation in image signal-to-noise ratio (SNR) due to stochastic energy deposition, conversion, and charge-collection processes.Equation (1) is a good description of SPC detector performance when there is minimal noise aliasing at zero spatial frequency, effective adaptive binning approaches are implemented, and a threshold is chosen such that false counts due to electronic noise are suppressed.Tanguay et al. 43 described a method of determining I SPC from the PDF of the total number of detected secondary quanta per x-ray interaction and showed this PDF can be determined by propagating the PDF of the number of imageforming quanta at each stage through a serial cascade of quantum processes.While this established the importance and utility of the cascaded-systems approach for describing the DQE of SPC systems, it was restricted to the simplistic case where all photon energy is deposited at the primaryinteraction site, ignoring photoelectric and Compton emission/scatter photons that escape the detector or are reabsorbed at a remote-interaction site.In the case of reabsorption, photon energy is converted to secondary quanta at both primaryinteraction and reabsorption sites, resulting in a complicated energy response function. 47In addition, liberation of secondary quanta (e.g., charges in a photoconductor or optical photons in a scintillator) is a stochastic process and the PDF when energy is deposited at one site differs to that when the same energy is deposited at multiple sites.
The purpose of this paper is to extend the cascaded-systems approach to describe the zero-frequency DQE of SPC systems including the effects of x-ray reabsorption, imperfect charge collection, and additive readout noise.This is accomplished by describing PDF transfer through parallel cascades of quantum processes and developing a relatively simple closed-form expression for the PDF under conditions of importance for SPC imaging.The utility of this approach is demonstrated in an analysis of the zero-frequency DQE of hypothetical siliconand selenium-based flat-panel SPC detectors.

2.A. I SPC and energy-response function R
The SPC noise factor I SPC is equal to the true-positive fraction of counting interaction events. 42,43The detector signal d after adaptive binning and prior to thresholding is equivalent to the signal from a large detector element in which the x-ray photon always interacts near the center and readout noise is equal to the quadrature sum of noise from each element in the binned cluster.For an interacting photon of energy E, d is used to estimate deposited photon energy ε where, for a linear x-ray detector, ε = κ d for some constant κ.It is convenient to characterize the system response in terms of the energy response function R(ε,E) which is equal to the probability density of ε given interacting energy E. Letting p d ( d| E) represent the PDF of d given an interaction yields and the SPC noise factor is then given by 43 where t is a threshold used to separate x-ray interaction events from electronic noise.This result shows that I SPC can be determined from a knowledge of the PDF of (binned) detector signals.Multiple thresholds can be used to determine the spectral distribution of ε.

2.B. Mean number of photon counts
The number of counts reported by a SPC system is given by the RV c.For fast readouts and detectors that implement adaptive binning to sum charges in neighboring elements surrounding an interaction and assign them to a single element, the mean c is given by 43 where qo [mm −2 ] describes the mean number of incident quanta, λ = qo a t a(λ ≪ 1) describes the number incident during integration time a t , I SPC is given by Eq. ( 3), and ζ represents the probability of observing a false count due to electronic readout noise.Unlike conventional energy-integrating detectors, it is seen readout noise may result in an increase in the mean SPC image signal.Suppression of false noise counts requires a threshold high enough to keep the false-positive fraction low and the second term in square brackets small.

2.C. Determining R from the PDF of image quanta
Equations ( 3) and (4) show that I SPC and c are determined by the energy response function R and in turn by the PDF of d.9][50] Here, we describe a method of obtaining R from the PDF of d that includes the effects of x-ray reabsorption and stochastic conversion and collection processes.We start by summarizing simpler PDFtransfer relationships and then extend to include a description of x-ray reabsorption.

2.C.1. Elementary processes
We refer to five simple processes that are used frequently as "elementary" processes, consisting of (1) quantum gain, (2) quantum selection, (3) quantum scatter, (4) general gain, (5) general filter, and (6) sampling.Quantum gain 32,33 is the process of replacing the jth point in a distribution of points with gj (a random variable with integer-only sample values) points at the same location, such as liberation of secondary quanta (optical photons or e-h pairs).Quantum selection 33 is a special case where a random selection of input points is passed to the output (quantum gain by factor 1 or 0), such as the selection of interacting quanta from a distribution of incident quanta.Quantum scatter 32,33 is the random relocation of individual points in the image plane according to a specified PDF, such as charge carrier diffusion in a semiconductor.General gain is the scaling of an input signal by a gain RV that may have a continuum of values (e.g., amplifier gain) and general filter corresponds to any deterministic image-blurring process that can be described as a convolution operation. 34ampling describes the process of evaluating the detector signal at discrete locations corresponding to the center of detector elements and can be represented by multiplication with a series of Dirac delta functions. 51Temporal lag effects can be incorporated 52 and these processes combined in serial and parallel cascades 53,54 to create comprehensive models of signal and noise (and thereby DQE) of complex quantumbased imaging systems.

2.C.2. PDF following a cascade of elementary processes
Letting Ñi represent the total number of image quanta (points) after the ith stage in a serial cascade of quantum processes gives the PDF of Ñi as 43 where p N i (N i |N i−1 ) represents the conditional PDF of Ñi given Ñi−1 .Recursive application yields the PDF after n processes F. 1. Schematic representation of PDF transfer in quantum processes.Each PDF consists of scaled Dirac δ-functions describing probabilities of integer numbers of quanta.
where p N 0 (N 0 ) is the PDF of the number of incident (x-ray) quanta, as illustrated in Fig. 1, with each process characterized by a conditional PDF.Since the number of quanta must be an integer, both p N i (N i ) and p N i (N i |N i−1 ) are generalized functions consisting of a sequence of scaled δ-functions at integer values of N.
Gain stages are characterized by an integer gain RV g having PDF p g and the total number of quanta in the output comes from replacing each quantum in the input with g quanta.Since p g is assumed identical for input quanta having the same energy, for monoenergetic photon fluence, the conditional PDF is expressed as p g convolved with itself N i−1 − 1 times, 43,55 expressed as a function of N i−1 , where and pr g represents the probability mass function (PMF) of g with the summation over all possible values of g.Extension to polyenergetic fluence would require additional averaging over the incident photon spectrum.For quantum scatter, the number of quanta is unchanged and the conditional PDF is a unity operator.
Three special cases of quantum gain are illustrated.For each, pr g must be normalized over the domain of integer values.
2.C.2.a.Poisson gain.For Poisson gain with mean ḡ, for integer k.Combining with Eqs. ( 7) and ( 8) yields a Poisson distribution with mean ḡ N i−1 , 42,55 2.C.2.b.Deterministic gain.For the case of deterministic gain, where δ k ḡ represents the Kronecker delta and is equal to 1 when k = ḡ and 0 otherwise.The translation property of the Dirac δ-function yields 1 or 0 only with probabilities α and 1 − α, respectively, and pr g is given by a Bernoulli distribution resulting in a Binomial distribution 42,55 These results and others, summarized in Appendix B, may be used in combination with Eq. ( 6) to provide a complete description of the PDF for a single serial cascade of energydepositing events.This result is generalized to include scatterreabsorption events using a parallel-cascades approach in Sec.2.C.3.A list of selected variables is included in Appendix A.

2.C.3. PDF following parallel cascades
For the case of multiple energy-depositing paths (Fig. 2), such as energy deposition at primary-interaction and scatterreabsorption sites, the total number of secondary quanta is the sum from all paths. 53For the case of two paths, the PDF of the sum ÑA+B = ÑA + ÑB is given by 55 where p N A,B is the joint PDF describing the probability of observing ÑA and ÑB quanta from paths A and B, respectively.When ÑA and ÑB are independent RVs, p N A,B = p N A p N B , giving Equations ( 15) and (16) give the PDF of a sum of quanta from correlated and uncorrelated paths, respectively.Of particular importance in this application is when the input to each path is a subset of a common input distribution as illustrated in Fig. 2. The process of selecting quanta for each path is a branching process in the cascade model.
2.C.3.a.Joint PDF and branch points.The branching process represents a sequence of trials where the jth trial is a random selection of the jth point to follow paths A and/or B with probabilities ξA and ξB , respectively. 53This F. 2. Illustration of PDF transfer where Ñ0 is separated into parallel cascades A and B and then recombined.could represent selection of incident photons that undergo photoelectric or Compton interactions.Each trial is described by Bernoulli RVs ξj,A and ξj,B having sample values 1 or 0 corresponding to being selected, or not, for each path.Each trial is independent but correlations may exist between ξj,A and ξj,B as described by the joint PDF 55 p ξ j,A ,ξ j,B ξ j,A ,ξ j,B = 1 where P(ξ j,A = k and ξ j,B = l) gives the probability that ξj,A = k and ξj,B = l.We use Eq.(17) to derive an expression for p N A+B (N A+B ) in Eq. ( 15) by first considering the simpler case of one input quantum to the branch point and then generalize to a random number.
One input quantum, N 0 = 1.This case represents the situation of a single photon interaction in an adaptively binned element during one integration period, where there can be only one ( ξ1,A = 1) or zero ( ξ1,A = 0) quanta following path A in Fig. 2, and similarly for path B, Therefore, where p ξ j,A ,ξ j,B (N A ,N B ) represents the joint PDF of ξ1,A and ξ1,B evaluated at N A and N B .
Random number of input quanta, Ñ0 .A more general case is to accommodate a random number of quanta interacting in an adaptively binned element during a single integration period, including multiple counts, giving Assuming the joint PDF for ξj,A and ξj,B is the same for all j, ÑA and ÑB are sums of Ñ0 identically distributed independent RVs.We show in Appendix C that p N A,B given Ñ0 input quanta is then given by where the right-hand side denotes the 2D convolution of showing that the joint PDF of the number of quanta in two random subsets of a common input distribution is fully described by the joint PDF of selection variables ξj,A and ξj,B , and the PDF of Ñ0 .2.C.3.b.Joint PDF following cascades of elementary processes.A more general case involves the joint statistics of the number of quanta in two paths after undergoing serial cascades of elementary processes as illustrated in Fig. 3.The RVs ÑA,i and ÑB,i represent the number of quanta after the ith F. 3. Illustration of PDF transfer through parallel cascades of elementary quantum processes.elementary process of each path.Similar to Sec. 2.C.3.a,we derive an expression for p N A,B by first considering the simple case of one input quantum.
One input quantum, N 0 = 1.In the parallel cascades of elementary processes in Fig. 3, ÑA,i = Ñj,A,i represents the number of secondary quanta for the jth input quantum after the ith process of path A (and similarly for path B).For one input quantum, we have where n A identifies the last process in path A, and similarly for path B. Assuming processes in path A is independent of those in path B yields where p N 1,A (N 1,A | ξ1,A ) represents the PDF of Ñ1,A,n A given ξ1,A , and similarly for path B, where we have dropped explicit dependence on n A and n B .Averaging over ξ1,A and ξ1,B yields where ⟨⟩ ξ A ,ξ B represents averaging over ξj,A and ξj,B , and where p N A (N A |ξ A ) represents the PDF of ÑA given ξj,A for one trial and is obtained from Eq. ( 6) and similarly for p N B (N B |ξ B ).
Random number of input quanta, Ñ0 .Similar to Eq. ( 20), the numbers of quanta from paths A and B are In Appendix C, we show that because each process in path A is independent of those in B and ξj,A is independent of ξi,A for i j (and similarly for ξj,B ), the joint PDF of ÑA and ÑB given Ñ0 is where ⟨p⟩ ξ A ,ξ B is the joint PDF given one input quantum in Eq. (25).Averaging over all possible values of Ñ0 yields This is a general PDF-transfer relationship between p N A,B and p N 0 and shows that the joint statistics of ÑA and ÑB are determined by the joint statistics of ξA and ξB , the elementary processes in paths A and B, and the input PDF p N 0 .
We further consider two important branch points: (i) each input quantum is selected for either path A or B (Bernoulli branch), and (ii) each input quantum is selected for both paths A and B (cascade fork). 53.C.3.c.Bernoulli branch.A Bernoulli branch may, for example, describe separation of photoelectric interactions that produce a characteristic emission from those that do not. 37,38,41,53The joint PDF of ξj,A and ξj,B is (31)   from Eq. ( 17).Combining this with Eq. ( 25) yields where Combining the above expression with Eq. ( 30) yields p N A,B after multiple elementary processes.
One input quantum, N 0 = 1.Combining Eqs. ( 15) and ( 32) yields This describes the expected result that when an input quantum is selected for only one path, the PDF of the total number of output quanta is equal to the weighted combination of each.

2.C.3.d. Cascade fork.
A cascade fork may, for example, describe the situation where a photon that has interacted through the Compton effect deposits energy at the site of primary interaction and at a remote site following reabsorption of a Compton-scatter photon.The joint PDF of selection variables is then given by and therefore Combining the above expression with Eq. ( 30) yields p N A,B after multiple elementary processes.
One input quantum, N 0 = 1.Combining Eqs. ( 15) and ( 35) yields F. 4. Schematic illustration of the parallel CSA model describing PDF transfer of the total number of secondary quanta through photoelectric interactions, incoherent interactions, and additive electronic noise using parallel cascades.The RV Ñ0 is the total number of incident x-ray quanta in one readout and is set to unity.
Equations ( 33) and ( 36) are important results of this work.They are special cases of a branch point where the signal from an interacting photon passes through either only one [Eq.( 33)] or both [Eq.( 36)] of two paths and allow us to use the complex models of Figs. 4 and 5 to describe PDF transfer through photoelectric and Compton interactions.In Sec.2.D, we use these results to determine the energy response function and in turn the zero-frequency DQE of hypothetical seleniumbased and silicon-based SPC detectors.

2.D. Liberation of secondary quanta in x-ray convertor
Figure 4 illustrates the parallel-cascade model we use to describe energy deposition in an x-ray convertor including the effects of stochastic energy deposition through photoelectric or incoherent scattering, conversion to secondary quanta, and collection of secondary quanta, similar to that described by Yun et al. 41 Figure 4 does not illustrate subsequent thresholding and sampling stages discussed in detail by Tanguay et al. 43 In all cases, we assume flat-panel SPC systems.Our goal is to describe the PDF of the total number of secondary quanta Ñtot collected by detector elements per interacting x-ray photon.We therefore let p N 0 (N 0 ) = δ(N 0 − 1) and assume large adaptively binned pixels such that the probability of reabsorption of characteristic or Compton-scatter x-rays in neighboring elements is negligible.This may be a good approximation for systems that use adaptive binning to sum signals from neighboring pixels to get the total energy deposited for every interacting x-ray photon.
The output from each path is the total number of quanta collected from either photoelectric or incoherent interactions.From Eq. ( 15), the total number of collected secondaries Ñtot = Ñpe + Ñinc is given by where p N pe,inc (N pe ,N inc ) is the joint PDF of Ñpe and Ñinc .The Bernoulli branch in Fig. 4 and Eq. ( 32) gives where ξpe and ξinc represent the relative probabilities of photoelectric absorption and incoherent scattering. 41,47. 5. Schematic illustration of the generalized interaction model.(a) Incident x-ray photon interacts in x-ray convertor at depth z 1 with subsequent production of a scatter photon at polar angle θ and azimuthal angle φ.
(b) CSA model describing events that liberate secondary quanta from (path A) primary-interaction site when no scatter/emission photon is generated; (path B) primary-interaction site when a scatter/emission photon is generated; and (path C) remote reabsorption of scatter/emission photon.
Equation (38) shows that a description of p N tot (N tot ) requires the PDFs of the number of collected secondaries resulting from photoelectric and incoherent interactions.These processes are similar in that each may result in emission and reabsorption of a fluorescent/scatter photon.It is therefore convenient to describe photoelectric and incoherent interactions as two special cases of a "generalized" interaction process.

2.E. Generalized x-ray interaction process
Each shaded box in Fig. 4 is a special case of the generalized interaction process in Fig. 5.When used, subscript t represents the interaction type, photoelectric (pe) or incoherent (inc).As illustrated, an incident photon interacts at depth z1 in the x-ray convertor material and may generate a scatter photon with probability S at scatter angle θ and azimuthal angle φ that may be reabsorbed at depth z2 with probability f .Secondary quanta (electron-hole pairs in a photoconductor) are liberated at both primary-interaction and reabsorption sites unless the scatter photon escapes the detector.This model is based on that described by Yun et al. 41 with the extension to depth-dependent collection efficiency.
The collection efficiencies β for each path in Fig. 5 are functions of interaction depths z1 and z2 , which are themselves RVs.The concept of using gain and/or selection variables that are functions of random variables was introduced by Van Metter and Rabbani 56 who called these input-labeled random processes.We adopt this idea to describe the depthdependent collection efficiency in the top shaded path of Fig. 5(b).However, in the lower shaded box, all processes are functions of z1 and/or θ.In addition, these processes are coupled because they are dependent on the same z1 and θ sample values for each interacting photon.In Appendix C we generalize the previous derivation of the PDF of the total number of image quanta from parallel cascades to include these input-labeled parallel processes.The result is used to calculate the PDF of the number of quanta for the generic interaction model illustrated in Fig. 5.
We let ÑA , ÑB , and ÑC represent the number of secondary quanta from top, middle, and bottom paths of Fig. 5(b), respectively, and Ñt = ÑA + ÑB + ÑC represents the total number of secondary quanta for interaction type t.The first branch point in Fig. 5(b) represents separation (Bernoulli branch) of interacting photons that produce a fluorescent/scatter photon (paths B and C) from those that do not (path A), and Eq.(33)  gives the PDF of Ñt , where S t represents the probability that a scatter/emission photon is generated, p N A (N A | St = 0) represents the PDF of ÑA given no scatter/emission photon, ÑB+C = ÑB + ÑC , and p N B+C (N B+C | St = 1) represents the PDF of ÑB+C given a scatter/emission photon.In Secs.2.E.1-2.E.6, we describe how to calculate p N t (N t ) using the PDF-transfer approach.

2.E.1. PDF of ÑA
The PDF of quanta from path A, p N A (N A | St = 0), is obtained from Eq. ( 27), where p N A,0 (N A,0 | St = 0) = δ(N A,0 −1) (the PDF of the number of quanta selected for path A given that one quantum is selected), and therefore, 2.E.1.a.Conversion to secondary quanta.Conversion to electron-hole (e-h) pairs at the primary-interaction site (path A) is described using the PDF-transfer relationship for a quantum gain stage 43 where is the PDF describing the random number of liberated e-h pairs gA .We assume Poisson gain with mean ḡA = E/w, where E is the incident photon energy, w is the effective energy required to liberate one e-h pair, and PDF given by Eq. (10).
2.E.1.b.Depth-dependent collection of secondary quanta.Application of an electric field across the photoconductor causes liberated e-h pairs to drift in opposite directions, generating a current that charges or discharges a capacitor.In flat-panel configurations, the collected charge may be depth dependent as e-h pairs may recombine or be trapped in the x-ray convertor. 57,58 model depth-dependent charge collection as a binomial selection process (second process in path A) characterized by a Bernoulli RV having sample values 1 or 0 with probabilities that depend on interaction depth z1 .We assume the average fraction of collected charges (collection efficiency) β has a depth dependence given by the Hecht relationship [59][60][61] where L (cm) is the convertor thickness, λ e and λ h (cm) are mean-free drift lengths for electrons and holes, respectively, and we have assumed electrons travel toward the entrance surface and holes toward the exit surface.If charges are collected in the opposite direction, it may be necessary to exchange z and L − z on the right-hand side.Low values of λ e and/or λ h result in fewer collected charges and potentially a secondary quantum sink.
Transfer of the PDF through depth-dependent collection processes (summarized in Table III) has recently been described, 42 giving where B (N A ; N A,1 , β(z 1 )) represents the binomial distribution for N A,1 trials and probability of success β(z 1 ), and p z 1 (z 1 ) represents the exponential PDF of z1 .Combining the previous two results yields where The previous two results show that p N A (N A | S = 0) is equal to the binomial distribution with gA trials and probability of success β( z1 ), averaged over all possible values of gA and z1 .

2.E.2. PDF of ÑB + ÑC
In the case that a characteristic/scatter photon is generated, energy may be deposited at primary and secondary absorption sites (paths B and C) as illustrated in Fig. 5.All subsequent processes may be functions of interaction depth and/or emission/scatter angle, and therefore, from Eqs. ( 35) and (C15), p N B+C (N B+C | St = 1) is given by where b = [ z1 , θ] and ⟨⟩b represents an average over all b.In Secs.2.E.2.a-2.E.2.c, we calculate p N B and p N C for fixed b and then average over all possible values of b to get p N B+C .Averaging over b requires the joint PDF of z1 and θ, p z 1 ,θ (z 1 ,θ), described in Appendix C.
2.E.2.a.PDF of ÑB for fixed z1 and θ.There are two processes following the cascade fork in path B of Fig. 5.The first represents conversion to e-h pairs at the primary-interaction site and the second depth-dependent charge collection.For incoherent interactions, energy deposited by the primary interaction is a function of scatter angle θ.Therefore, from Eq. ( 27) Conversion to secondary quanta at primary-interaction site.The first process following the cascade fork in path B represents conversion to e-h pairs at the primary-interaction site.Therefore, similar to Eq. ( 42), where p g B (g B ; θ) represents the PDF of gB for scatter angle θ.Similar to gA , we assume gB is Poisson distributed with mean ḡB = (E − E ′ (θ))/w, where E ′ (θ) is the scatter/emission photon energy for angle θ.
Depth-dependent collection of secondary quanta.For fixed z1 , p N B,1 (N B |N B,1 , z1 ) is given by the binomial distribution, 42,43,62 giving 2.E.2.b.PDF of ÑC for fixed z1 and θ.There are three processes following the cascade fork in path C of Fig. 3. From Eq. ( 27) with p N C,0 (N C,0 | St = 1) = δ(N C,0 − 1), Reabsorption of fluorescent/scatter photon.The first process following the cascade fork in path C represents selection of fluorescent/scatter photons that are reabsorbed in the xray convertor material.Therefore, p N C,1 (N C,1 |N C,0 = 1, θ, z1 ) is equal to the Binomial distribution with one trial and probability of success equal to the reabsorption probability f which is a function of θ and z1 , where where where E s represents fluorescent/scatter photon energy.Combining these expressions yields where Conversion to secondary quanta at reabsorption site.Similar to Eqs. ( 42) and ( 49), where p g C (g C ; θ) represents the PDF of gC for fluorescent/ scatter angle θ, where ḡC = E ′ (θ)/w.Similar to gA and gB , we assume that gC is Poisson distributed.Depth-dependent collection of secondary quanta at reabsorption site.Letting p z 2 (z 2 | z1 , θ) represent the PDF of reabsorption depth z2 given z1 and θ yields 42 Therefore, The first term in the previous expression is a delta function at N C = 0 corresponding to the event that the fluorescent/scatter photon is not reabsorbed.The second term corresponds to the event that fluorescent/scatter photon is reabsorbed.

2.E.3. PDF of Ñt
Combining Eqs. ( 39), (45), and (59) yields the PDF of the total number of quanta for interaction type t using the generic x-ray interaction model illustrated in Fig. 5, The average over z2 in the above expression requires the PDF of z2 given z1 and θ which is derived in Appendix D. Averages over z1 and θ require the joint PDF of z1 and θ, p t z 1 ,θ , which for a given incident photon energy is given by 55 where p t θ depends on the interaction type and has been described in detail by Hajdok et al. 63 and Yun et al. 41,47 for both photoelectric and incoherent interactions, and p z 1 (z 1 ) is given by the exponential PDF µ(E)exp(−µ(E)z 1 ) normalized to unity over the convertor thickness.
The three terms in Eq. ( 62) describe the PDF of image quanta for (i) events that do not result in production of fluorescent/scatter x-rays, (ii) events that result in production of a fluorescent/scatter photon that escapes the detector, and (iii) events that result in production of a fluorescent/scatter photon that is reabsorbed within the detector.In Secs.2.E.4-2.E.6, we use this result to describe the PDFs of the total number of quanta for photoelectric and incoherent interactions.Table I gives mean values and PDFs used for selection and gain variables for each interaction type, where ρ K , ω K , E K , E ′ , w, and f t represent the K-shell participation fraction, K-fluorescent yield, K-fluorescent photon energy, incoherent-scatter energy, average effective energy required to liberate one electron-hole pair, and scatter/emission reabsorption probability, respectively.

2.E.4. Photoelectric interactions
In a photoelectric interaction, path A of Fig. 5 corresponds to events that do not produce a fluorescent photon, and incident energy E is assumed to be absorbed at the primaryinteraction site, liberating gA secondaries (see Table I).Paths B and C describe events that produce a fluorescent photon, T I. Random variables and PDFs defining the type of x-ray interaction used in the generic model shown in Fig. 5.

Photoelectric
Incoherent resulting in gB secondaries emitted locally and gC liberated remotely with probability f pe given by Eq. ( 53) with E s = E K .From Eq. ( 62), the PDF of Ñpe is given by where p pe z 1 ,θ is obtained from Eq. ( 63) with p pe θ (θ) = sin(θ)/ 2. 38,41 The first and third terms in Eq. ( 64) contribute to the photopeak and the second term contributes to the K-escape peak.

2.E.5. Incoherent interactions
In an incoherent event, an incident photon interacts with a loosely bound (free) electron producing a Compton-scatter photon and recoil electron.The energy of the scatter photon E ′ is a function of both incident photon energy and scatter polar angle θ, E ′ = E/(1 + α(1 − cos(θ))), where α = E/m o c 2 represents the incident photon energy in units of electron rest-mass energy (m o c 2 = 511 keV). 64The recoil electron deposits its energy at the primary-interaction site with mean conversion gain ḡB = (E − E ′ )/w.The scatter photon is reabsorbed with probability f inc given by Eq. ( 53).From Eq. ( 62), the PDF of Ñinc is given by where p inc z 1 ,θ is given by Eq. ( 63) with p inc θ (θ) described in detail by Hajdok et al. 63 and Yun et al. 41,47 The first term in Eq. ( 65) represents the distribution of secondaries collected from energy deposition by the recoil electron and the second term contributes to the photopeak.

2.E.6. PDF of detector element signal d
Combining Eqs. ( 38), (64), and (65) yields the PDF of Ñtot = Ñpe + Ñinc , The PDF of d given interacting photon energy E including electronic readout noise is obtained by convolving Eq. ( 37) with the electronic noise PDF, 42,43 where and p e (d) describes the zero-mean distribution of (postbinning) detector readout noise and k is a constant of proportionality converting a number of quanta to the detector-element signal.Equations ( 66) and ( 67) describe the PDF of element signals after adaptive binning prior to thresholding for single-Z detector materials.A description of multi-Z SPC detector materials, such as CdTe and CdZnTe, requires the evaluation of Eq. ( 66) for each atom type and potentially terms that account for emission of characteristic photons from the lower Z atoms (e.g., Cd) following reabsorption of characteristic photons emitted from the higher Z atoms (e.g., Te). 47he response function R, which represents the probability density of estimated photon energy ε given interacting photon energy E, is given by where κ is a constant of proportionality converting a detectorelement signal to estimated photon energy.
To illustrate the utility of this theoretical formalism, we use the previous three expressions with Eqs. ( 1) and ( 3) to calculate the zero-frequency DQE of Si-based SPC detectors, such as the MEDIPIX prototypes, 1,4,29,65,66 and a hypothetical Se-based SPC detector that implements adaptive element binning.While Se is not currently used in photon-counting x-ray imaging, applying the developed formalism to Se is useful in illustrating the potentially degrading effects that escape of characteristic x-rays may have on photon-counting image quality.

2.F. Frequency-dependent DQE
A simple estimate of the DQE frequency dependence comes from recognizing that, assuming ideal adaptive binning, the primary interaction location is known to be somewhere within the detector element of width a at the center of the binned region, resulting in a sinc-shaped MTF.In addition, due to noise aliasing caused by sampling at locations corresponding to the center of detector elements, the Wiener NPS will be uniform 36 with frequency up to the sampling cutoff frequency, resulting in a DQE given by Charge sharing in the detector might enable the primary interaction to be located with subelement resolution, resulting in even better MTF and DQE performance.While these results may be achieved at energies below the detector K-edge energy (e.g., mammography energies), they may not be achieved if the primary-interaction site cannot be identified correctly, such as when more energy is deposited by reabsorption of a scatter photon than by the primary interaction.

MONTE CARLO (MC) COMPARISON
Predictions of this CSA approach were compared with a MC study of the DQE of Si-and Se-based SPC detectors that implement adaptive binning.Simulations were performed in three stages: (1) energy deposition, (2) charge liberation and collection, and (3) adaptive binning, electronic noise, and thresholding.Table II summarizes physical parameters used in combination with the physical models described in Sec.2.E.

3.A. Energy deposition
Energy deposition was simulated using the particle tracking (pTrac) function of the MC user code 5 TM .Energydeposition data were collected in list mode, giving threedimensional (3D) positions and energies of energy-deposition events following each primary photon interaction.Simulations were performed for uniformly distributed photon fluences and 2000 × 2000 10 × 10 µm 2 scoring bins.In all cases, ranges of photoelectrons and Compton recoil electrons were not considered (i.e., site of charge liberation is the same as x-ray interaction).Double (or triple) counting of incident photons caused by the finite range of photoelectrons and Compton recoil electrons was therefore not considered in this analysis.In addition, we assumed count rates low enough that pulse pileup effects could be ignored.A similar MC technique was implemented by Yun et al. 41 for the description of conventional energy-integrating detectors and the reader is referred there for more details.

3.B. Charge liberation and collection
In all cases, we assume that the site of charge collection is the same as charge liberation, i.e., no charge diffusion.This may be a reasonable assumption for larger adaptively binned elements and higher threshold levels where the number of double-counting events caused by partial energy absorption in neighboring elements is reduced, as discussed by Xu et al. 68 and Lundqvist et al., 16 respectively, but represents an optimistic estimate of detector performance for smaller element sizes.
Charge liberation.Liberation of e-h pairs was simulated by sampling a Poisson distribution with mean value E dep /w for each photon interaction, where E dep represents deposited photon energy (obtained from the previous stage).The output of this stage was the 3D position and number of liberated e-h pairs for each energy-deposition event.
Charge collection.We simulated collection of liberated charges by sampling a binomial distribution with number of trials equal to the number of liberated e-h pairs and depthdependent probability of success β(z) given by the Hecht relation [Eq.(43)].We consider two extreme cases of charge collection: asymmetric and symmetric collection of charge carriers.In the case of Se, we consider an asymmetric case and assume the mean-free drift length for electrons (λ e ) is two orders of magnitude lower than that for holes. 61In the case of Si, consider a symmetric case and assume meanfree drift lengths of holes and electrons are of the same order of magnitude.Calculations were performed for each energy-depositing event, giving the 2D position and number of collected e-h pairs for each primary and secondary photon interaction.

3.C. Adaptive binning, electronic noise, and thresholding
We simulated two types of adaptive binning.For the first type, which represents an ideal situation, we summed e-h pairs collected from primary-and secondary-reabsorption sites and then added a sample from a zero-mean Gaussian distribution to simulate electronic noise.When the resulting signal was above the threshold level, a count was attributed to the element with the highest detected charge.
For the second type of adaptive binning, which is similar to that implemented in MEDIPIX Si-based detectors, 4 we added electronic noise (σ pb e ) to each prebinning detector element, clustered elements into larger n × n elements, and summed charges collected from all interactions within each cluster.When the signal from a cluster was above the threshold level, a count was attributed to the element (within the cluster) with the highest detected charge.
Each adaptive-binning approach was performed for prebinning element areas of 100 × 100 µm 2 and a threshold equal to three times the electronic noise level (σ e ) of adaptively binned signals.This resulted in a 2 × 2 cm 2 SPC image from which DQE(0) was determined.

3.D. DQE calculation
We calculated DQE(0) from MC images using the following relationship: where SNR meas and SNR ideal (= α qo a) represent the measured and ideal SPC signal-to-noise ratio, respectively, and α represents the quantum efficiency.

4.A. PDF of detector element signals, p d (d |E )
Figure 6 illustrates p N tot (N tot ) for Se and Si convertor materials for the model illustrated in Fig. 4 calculated for selected photon energies, convertor thicknesses, and meanfree drift lengths for holes.For systems that implement adaptive element binning, p N tot is related to p d through Eq. ( 67). Figure 6 shows good agreement between theoretical and MC simulations for all conditions considered.
In general, for both Se and Si, low mean-free drift lengths (∼0.1 cm) result in broad and asymmetric photopeaks that are shifted toward lower energies due to depth-dependent collection efficiencies.This effect is more severe for thicker convertor materials, where the collection efficiency has a stronger depth dependence, [59][60][61] and for higher energy photons, where the distribution of interaction depths is more uniform.Although not shown in Fig. 6, electronic noise results in further photopeak broadening.For systems with sufficiently low electronic noise levels and high mean-free drift lengths (i.e., λ h ≫ L), stochastic energy deposition and conversion processes are primary causes of spectral distortion, resulting in finite-width photopeaks, K-escape peaks, F. 6. PDFs of the total number of quanta collected by detector elements for selected photon energies for (left) Se-and (right) Si-based convertor materials for selected mean-free drift lengths for holes (λ h ).Calculations for Se assumed asymmetric charge collection (i.e., λ e ≪ λ h ).Calculations for Si assumed symmetric charge collection (i.e., λ e ∼ λ h ).Curves and symbols represent theoretical and Monte Carlo data, respectively.
Medical Physics, Vol.42, No. 1, January 2015 F. 7. Normalized SPC pixel value co (counts per incident photon) as a function of threshold t for selected Se and Si convertor thicknesses with σ e = e-h pairs and λ = 1/10.Vertical gray lines indicate mean conversion gains for the indicated energies.Calculations for Se assumed asymmetric charge collection (i.e., λ e ≪ λ h ).Calculations for Si assumed symmetric charge collection (i.e., λ e ∼ λ h ). and a distribution of low-energy deposition events with area determined by the Compton cross section.

4.B. Optimal SPC threshold
Figure 7 illustrates the dependence of normalized SPC pixel value ( c/ q0 a) on threshold level for selected Se and Si convertor thicknesses, photon energies, and mean-free drift lengths.All curves in Fig. 7 were calculated assuming σ e = 200 e-h pairs and small λ (=1/10) such that pulse pileup could be ignored.
For both Se and Si, as expected, threshold values lower than approximately 3σ e result in an inflated image signal due to false electronic noise counts.For threshold values greater than 3σ e , a plateau is reached with height approximately equal to the quantum efficiency.Width of the plateau depends on the number of secondaries collected per interacting x-ray photon and, in general, is narrower for lower energy photons than for higher energy photons and for materials with higher w values (such as Se).Decreasing the mean-free drift length of holes and electrons results in fewer collected secondaries which results in a secondary quantum sink and narrowing of the range of acceptable threshold values.

4.C. Zero-frequency DQE of SPC detectors
Figure 8 illustrates the dependence of DQE(0) on incident photon energy for selected Se and Si convertor thicknesses and mean-free drift lengths.In all cases, we have assumed t = 3σ e , λ ≪ 1, and ideal adaptive binning described in Sec.3.C.Good agreement between theoretical and MC results is observed for all conditions considered.
At fluoroscopic and radiographic energies (>40 keV), DQE(0) for Se is approximately equal to the quantum efficiency for all convertor thicknesses, mean-free drift lengths, and electronic noise levels considered.However, for higher levels of electronic noise and mammographic photon energies (<40 keV), DQE(0) for Se is degraded substantially due to loss of energy-deposition events below the electronic noise floor (3σ e ).This effect is caused by a combination of F. 8. DQE(0) as a function of incident photon energy for selected (left) Se and (right) Si convertor thicknesses, mean-free drift lengths, and electronic noise levels.Lines and symbols represent theoretical and Monte Carlo data, respectively.The gray curve indicates the quantum efficiency.
In the case of Si for E ≤ 20 keV, DQE(0) is degraded by high electronic noise and poor collection efficiency.For higher (30-60 keV) energies and thicker convertor materials, there is a slight DQE reduction that is likely due to the combination of lower charge collection and an increase in the number of Compton interactions that together result in more events below the electronic noise floor.Figure 9 illustrates the expected result that as the number of binned elements is increased, image SNR decreases due to an increase in additive noise.In the case of Se, with the exception of the 2 × 2 case for σ pb e 60, there is good agreement between MC and theory.Disagreement for 2 × 2 F. 9. SNR 2 /α qo a as a function of additive noise for selected photon energies incident on Se and Si detectors that implement adaptive binning.Theoretical curves are calculated assuming "ideal" adaptive binning with additive noise σ e equal to n (= 2, 10) times the prebinning electronic noise.

4.D. Effect of adaptive binning on image SNR
elements and σ pb e 60 is likely the result of a characteristic photon escaping from the binned element surrounding the primary interaction and causing an additional count in a neighboring binned element.This "double counting" results in an inflated image signal and a misleading increase in SNR that is not accounted for in the theoretical model.This effect disappears for σ pb e 60 because thresholds t (≥360 e-h pairs) are higher than the signal (≈250 e-h pairs) generated by reabsorbed fluorescent photons.
A similar discrepancy between theory and MC is observed for Si.In the case of Si, this discrepancy may be due to reabsorption of Compton-scatter x-rays.Since Compton-scatter x-rays can retain a large fraction of incident photon energy, increasing the threshold does not reduce the number of double counts.

DISCUSSION
A theoretical framework is presented for obtaining the energy-response function of photon-counting x-ray detectors.This was made possible by introducing new relationships that describe propagation of the PDF of the total number of imageforming quanta through complicated parallel cascades of image-forming processes for photon-counting x-ray detectors.This is required when there is more than one image-forming process that contributes to an image signal, such as in the case of reabsorption of fluorescent and Compton-scatter photons.Using this approach, the zero-frequency DQE and average count rate of hypothetical silicon-and selenium-based SPC detectors were determined including the effects of escape and reabsorption of fluorescent and Compton-scatter photons, stochastic conversion to secondary quanta, depth-dependent charge collection, and electronic noise.
For photon-counting systems that implement adaptive element binning, the zero-frequency DQE is equal to the quantum efficiency multiplied by a new SPC noise factor, I SPC .This term is equal to the probability of counting a photon given an interaction event, i.e., the true-positive fraction of photon counts, and has the appearance of a "photon-counting" Swank-noise factor.
The CSA model of I SPC based on a generalized depthdependent interaction model incorporating the statistics of liberation and collection of secondary quanta showed that the DQE is degraded by escape of fluorescent and Compton-scatter photons, depth-dependent collection efficiency, and electronic noise.It was demonstrated that for Si-and Se-based SPC systems, there is a narrow range of acceptable thresholds that depends on photon energy, collection efficiency, and electronic noise level.Thresholds that adequately suppress electronic noise without thresholding out interaction events will provide a DQE that is approximately equal to the quantum efficiency.In this case, as expected, the DQE is not compromised by Swank noise or electronic noise.However, in some cases, this condition may not be satisfied, such as at lower mammographic energies, higher levels of electronic noise, or poor collection efficiencies.Under these conditions, our results suggest that Si detectors that liberate more charges per interacting x-ray photon will achieve superior DQE performance.
In all cases, our theoretical models assumed an adaptive binning approach where all interacting photon energy was assumed to be collected in a single large element.Comparison with MC simulations showed that this can be good approximation to systems that cluster elements into n×n larger elements, then sum charge collected from all interactions within each cluster and attribute a count to the element (within a cluster) with the largest detected charge.For example, our theoretical model agreed very well with MC predictions of the DQE of Se-based systems that cluster elements into 200 × 200 µm 2 binned elements and applies a threshold that eliminates double counts caused by reabsorption of fluorescent photons.For sufficiently low electronic noise levels, loss of signal generated by thresholding out reabsorbed fluorescent photons has negligible effect on the DQE.Similar trends and agreement were observed for Si-based systems, although our results suggest that larger (>200×200 µm 2 ) binned elements may be required to adequately suppress double counting reabsorbed Comptonscattered x-rays.Furthermore, our Monte Carlo analysis illustrates that uncorrected double (or triple) counting of interacting photons may result in a misleading increase in image SNR (and therefore DQE) of SPC systems.This effect will be more severe when charge sharing caused by charge diffusion and electron transport, which were not modeled in either theoretical or MC analyses, is considered.We suspect this apparent increase may be due to nonlinear relationships between reabsorption, thresholding, and image noise, although such a relationship was not considered in detail in this study.In the future, it will be essential to quantify such relationships to avoid potentially over estimating the DQE of SPC systems.
In this paper, results have been compared with Monte Carlo simulations only for single-Z detector materials.Multi-Z detector materials have the potential to differ slightly from single-Z models when the characteristic emission from a high-Z atom interacts with a lower Z atom to create a new lower energy characteristic emission.The impact of this is known to be minimal with conventional energy-integrating CsI detectors where the two atoms have very similar atomic numbers, but is less clear with some SPC detectors such as CdZnTe.

CONCLUSIONS
A method of describing the zero-frequency DQE of photon-counting x-ray imaging detectors, including the effects of reabsorption of characteristic and Compton-scatter photons in a convertor material, is described.This approach uses the concepts of cascaded-systems analysis to determine the PDF of the total number of image-forming quanta resulting from parallel image-forming processes, such as energy deposition at primary-interaction and reabsorption sites.The model demonstrated good agreement with a Monte Carlo simulation of Si-and Se-based detectors under a realistic range of parameters.Application of this PDF-transfer approach shows that low conversion gain, depth-dependent collection efficiency, high electronic noise, and escape of characteristic/scatter photons can result in degradation of the photon-counting DQE under some conditions and provides a framework for the development of high-performance photoncounting and energy-resolving x-ray systems.

APPENDIX A: LIST OF VARIABLES
In the following, overhead ˜indicates a random variable, p x represents the PDF of the variable x, and p x, y represents the joint PDF of x and ỹ.Total number of quanta (and associated PDF) for interaction of type t.B (N t ;g, β)

Symbol
Binomial distribution for g trials and probability of success β, evaluated at N t .
B X (N t ; β) Binomial distribution for probability of success β averaged over all possible gain values ( gX ) of path X, evaluated at N t .

Figure 9
Figure 9 illustrates the dependence of SNR 2 /α q0 a on prebinning additive noise level (σ pb e ) for Se-and Si-based systems.Monte Carlo results were calculated assuming 2 × 2 and 10 × 10 binned elements.Theoretical results were calculated assuming ideal adaptive binning described in Sec.3.C with additive noise levels equal to 2σ pb e and 10σ pb e for 2 × 2 and 10 × 10 cases, respectively.In all cases, a threshold of t ≥ 3 × nσ pb e (n = 2, 10) was implemented.A value of SNR 2 /α q0 a = 1 represents the ideal situation where image SNR is not degraded by stochastic deposition, liberation, or collection processes, or electronic noise.Figure9illustrates the expected result that as the number of binned elements is increased, image SNR decreases due to an increase in additive noise.In the case of Se, with the exception of the 2 × 2 case for σ

Medical Physics, Vol. 42, No. 1, January 2015
T II.Material properties for a-Se and Si.
Conversion gain (and associated PDF).pr g (g)PMF of discrete conversion gain variable g.Ñi , p N i (N i )Total number of secondary quanta (and associated PDF) after the ith stage of a serial cascade of quantum processes.pNi(Ni|Ni−1)PDF of the total number of secondary quanta after the ith stage of a serial cascade of quantum processes given N i−1 quanta after the (i − 1)th stage.ÑA+B , p N A+B (N A+B ) Total number of quanta (and associated PDF) from parallel paths A and B.p N A N B (N A ,N B )Joint PDF of the total number of quanta from parallel paths A and B.p N A,B (N A ,N B | Ñ0 )Joint PDF of the total number of quanta from parallel paths A and B given Ñ0 input quanta.ξj,ABernoulliRVdescribingselection of the ith quanta to follow path A of a parallel cascade.pξj,Aξj,B(ξj,A ξ j,B ) Joint PDF of Bernoulli RVs ξj,A and ξj,B .pNA(NA|ξA ) PDF of ÑA given ξA .ÑA,iTotal number of quanta after the ith elementary process of path A .p N A, i ( ÑA,i | ÑA,i−1 ) PDF of ÑA,i given ÑA,i−1 .Ñtot , p N tot (N tot )Total number of collected secondaries per x-ray interaction and associated PDF.p PPU N tot (N tot )PDF of the total number of secondary quanta collected per integration period including the effects of pulse pileup.d, p d (d|E) PDF of adaptively binned element signal d given photon energy E. Ñt , p N t (N t )