Theoretical modeling and performance analysis of a direct‐indirect dual‐layer flat‐panel detector for contrast‐enhanced breast imaging

Abstract

Background

Contrast‐enhanced (CE) breast imaging highlights tumors with neo‐angiogenesis in dual‐energy (DE) images after iodinated contrast injection and has shown promise in improving diagnostic performance. Conventionally, DE images are generated by weighted subtraction of sequentially acquired low‐energy (LE) and high‐energy (HE) images, which are prone to motion artifacts caused by patient movement between exposures. A direct‐indirect dual‐layer flat‐panel detector (DI‐DLFPD) based imaging system, incorporating a silver (Ag) K‐edge filter at the x‐ray source, has recently been proposed to acquire LE and HE images simultaneously to eliminate motion artifacts. The DI‐DLFPD uses a direct amorphous selenium (a‐Se) front‐layer (FL) detector for LE acquisition, and an indirect cesium iodide (CsI) back‐layer (BL) detector for HE acquisition.

Purpose

To develop a theoretical model to optimize a DI‐DLFPD based imaging system for maximizing image quality and lesion detectability in CE breast imaging.

Methods

Established models for the x‐ray spectrum and a‐Se detector response are adopted, while the model for the CsI detector is extended to incorporate depth‐independent optical blur variability. These optical properties are empirically characterized from measured spatial‐frequency‐dependent detector performance of CsI samples with varying thicknesses. The x‐ray spectrum, a‐Se FL, and CsI BL models are then cascaded to construct a comprehensive model of the DI‐DLFPD based imaging system. Measurements from two fabricated DI‐DLFPD prototypes with different CsI BL thicknesses are used to validate the model calculations. For CE breast imaging, modulation transfer function matching and weighted subtraction are incorporated. The imaging task involves detecting a cylindrical iodinated lesion (2 mm diameter and height, 1 mg/mL iodine concentration) embedded in 4 and 6 cm thick breasts with 50% glandularity. The impact of component thicknesses on energy separation, absolute iodine contrast, and detectability index (d′) is systematically evaluated.

Results

Incorporation of depth‐independent optical blur variability improved the accuracy of predicting spatial‐frequency‐dependent CsI detector performance across different thicknesses. DI‐DLFPD model calculations agreed well with prototype measurements, confirming the validity of the developed model. Energy separation increased with both Ag K‐edge filter and a‐Se FL thickness. Iodine contrast was primarily determined by the relative position of the BL HE average to the iodine K‐edge, rather than by energy separation alone. The main limiting factor for d′ was noise in the BL HE and DE images. The optimal configuration comprises a 100 µm Ag filter, 200 µm a‐Se FL, and 400 µm CsI BL. It provided d′ greater than 3 for the iodinated object in both breasts at typical clinical dose levels.

Conclusions

The developed dual‐layer detector model accurately predicts the imaging performance of DI‐DLFPD systems comprising a‐Se and CsI detectors. The model reveals optimal system design parameters for a CE breast imaging task; the optimized configuration demonstrates performance comparable to current clinical mammography systems, but additionally offers spectral breast imaging that is free from motion artifacts. The modeling framework developed in this work may be extended to other CsI‐based detectors and multilayer systems for a wide range of imaging applications.

1. INTRODUCTION

Contrast‐enhanced (CE) breast imaging exploits iodine contrast agents and dual‐energy (DE) x‐ray acquisition to highlight breast lesions exhibiting neo‐angiogenesis, thereby improving tumor conspicuity and enhancing both screening and diagnostic performance. ¹ , ² , ³ , ⁴ CE breast imaging is typically implemented using a dual‐shot acquisition procedure, in which low‐energy (LE) and high‐energy (HE) images are acquired sequentially. The temporal interval between exposures often leads to motion artifacts in the weighted subtraction DE images due to patient movement. ⁵ , ⁶ , ⁷ , ⁸ , ⁹

A promising solution to this limitation is to use a dual‐layer (DL) detector, which enables simultaneous acquisition of LE and HE images with a single exposure. ¹⁰ This approach has attracted attention in spectral imaging applications. ¹¹ , ¹² , ¹³ , ¹⁴ , ¹⁵ , ¹⁶ For CE breast imaging, a dedicated system based on a direct‐indirect DL flat‐panel detector (DI‐DLFPD) has been proposed, as shown in Figure 1(a). ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ It employs a silver (Ag) filter at the x‐ray tube port to shape the incident spectrum into two peaks, straddling the K‐edge of iodine (Figure 1(b)). The front layer (FL) utilizes a direct amorphous selenium (a‐Se) detector to predominately capture LE x‐rays and produce a LE image, while the back layer (BL) employs an indirect cesium iodide (CsI) scintillator to absorb essentially all x‐rays transmitted through the FL and produce a HE image simultaneously (Figure 1(c)). This design has been shown to remove background tissue effectively and enhance the visibility of iodinated lesions in CE breast imaging. ¹⁹ , ²⁰ , ²¹

FIGURE 1.

(a) Diagram of the DI‐DLFPD based CE imaging system; (b) An example of energy spectrum of the incident x‐ray beam; (c) An example of energy spectra of x‐rays absorbed by the a‐Se FL and CsI BL.

The proposed DI‐DLFPD based breast imaging system requires thorough design optimization before clinical implementation. For imaging tasks such as detecting small masses or microcalcifications, spatial‐frequency‐dependent analysis is preferred over zero‐frequency assessment. ¹⁷ , ¹⁸ This can be achieved using a cascaded linear system model (CLSM), which characterizes the spatial‐frequency‐dependent propagation of signal and noise through each stage of the imaging chain. ²² , ²³ , ²⁴ , ²⁵ , ²⁶ , ²⁷ For a‐Se detectors, CLSM has been well established for both detector‐level modeling and imaging system‐level analysis. ²⁸ , ²⁹ , ³⁰ For CsI, most studies investigated system‐dependent imaging conditions, such as kVp, exposure, fill factor, and additive noise, whereas only a few studies addressed the impact of detector design parameters such as scintillator thickness. ³¹ , ³² , ³³ , ³⁴ , ³⁵ , ³⁶ However, these models showed systematic overestimation on non‐zero spatial‐frequency performance, even when depth‐dependent effects were included. ³⁴ , ³⁵ This overestimation is attributed to the omission of depth‐independent variation in optical blur in scintillator response, arising from structural inhomogeneities within the CsI scintillator. This effect has been shown to contribute substantially to the spatial‐frequency‐dependent degradation of detective quantum efficiency (DQE). ³⁵ , ³⁷ , ³⁸

In this work, we extend the model for CsI to include depth‐independent optical blur variability in scintillator response, whose values and thickness dependence are empirically characterized from measured spatial‐frequency‐dependent performance of CsI detector samples. A comprehensive theoretical model of the DI‐DLFPD imaging system is then constructed by integrating the x‐ray spectrum, a‐Se FL response, and CsI BL response. Model calculations for both FL LE and BL HE images are validated against measurements obtained from two fabricated DI‐DLFPD prototypes. The validated model is subsequently applied to analyze and optimize system performance for CE breast imaging. Specifically, the effects of component thicknesses on energy separation, absolute iodine contrast, and detectability index (d′) are systematically evaluated.

2. METHODS

2.1. Model For CsI

2.1.1. Development of model for CsI

Figure 2(a) shows x‐ray photoelectric interactions within the CsI scintillator, which follow three parallel pathways: (A) complete energy absorption without K‐fluorescence emission; (B) K‐fluorescence emission and escape from the scintillator, resulting in partial energy loss; (C) emission and reabsorption of K‐fluorescence, producing a secondary light burst. ²⁹ , ³⁹ Figure 2(b) illustrates both depth‐dependent and depth‐independent random variations in optical gain and blur. For the same absorbed energy, scintillation photons generated at different depths exhibit varying escape efficiencies and lateral spread depending on the distance of their origin from the readout surface. ³⁷ , ³⁸ , ³⁹ In addition, the inhomogeneous columnar structure of CsI introduces random variations even for x‐ray interactions occurring at the same depth, which further degrades DQE. ³⁵ , ⁴⁰ Figure 2(c) shows the corresponding serial and parallel cascade stages in the imaging chain. Detailed formulations of individual stages can be found in prior literature and are not repeated here, except for the optical conversion and blurring stages, which are modified to account for depth‐independent random variations. ²⁸ , ²⁹ To simplify notation, the spatial‐frequency‐dependent mean signal and noise power spectrum are defined as ${\overline{\mathrm{\Phi }}}_{in}(f$) and $\mathrm{NP}{\mathrm{S}}_{in}(f)$ for the input, and ${\overline{\mathrm{\Phi }}}_{out}(f)$ and $\mathrm{NP}{\mathrm{S}}_{out}(f)$ for the output at each stage.

FIGURE 2.

Diagram showing (a) x‐ray interactions within the CsI scintillator, (b) depth‐dependent and depth‐independent random variations in optical gain and blur, and (c) the corresponding serial and parallel cascade stages in the imaging chain.

For the optical conversion stage, the output signal and noise are given by ²² , ²⁸ , ⁴⁰

$${\overline{\mathrm{\Phi }}}_{out}\left(f\right)=\overline{\mathrm{g}}{\overline{\mathrm{\Phi }}}_{in}\left(f\right)$$ (1)

$$\mathrm{NP}{\mathrm{S}}_{out}\left(f\right)={\overline{\mathrm{g}}}^2\mathrm{NP}{\mathrm{S}}_{in}\left(f\right)+{{\sigma }^2}_{\mathrm{g}}{\overline{\mathrm{\Phi }}}_{in}\left(f\right)$$ (2)

where $\overline{\mathrm{g}}$ is the mean conversion gain, determined by the light generation and escape efficiency. ³⁸ , ⁴¹ Its variance ${{\sigma }^2}_{\mathrm{g}}$ can be expressed as

$${{\sigma }^2}_{\mathrm{g}}=\overline{\mathrm{g}}{\mathrm{F}}_{\mathrm{g}}$$ (3)

where ${\mathrm{F}}_{\mathrm{g}}$ is the “photon Fano factor”, which quantifies intrinsic, depth‐independent variations in the x‐ray‐to‐light conversion. ⁴⁰ , ⁴² For an ideal Poisson process, ${\mathrm{F}}_{\mathrm{g}}$= 1. For Fg > 1, Poisson excess, ${\mathrm{F}}_{\mathrm{g}}$‐1 quantifies how much the variations exceed ideal Poisson noise.

For the optical blurring stage, the output signal and noise are given by, ²² , ²³ , ²⁸

$${\overline{\mathrm{\Phi }}}_{out}\left(f\right)={\overline{\mathrm{\Phi }}}_{in}\left(f\right){\overline{MTF}}_{opt}\left(f\right)$$ (4)

$$\mathrm{NP}{\mathrm{S}}_{out}\left(f\right)=\left[\mathrm{NP}{\mathrm{S}}_{in}\left(f\right)-{\overline{\mathrm{\Phi }}}_{in}\left(f\right)\right]{\overline{MT{F}^2}}_{opt}\left(f\right)+{\overline{\mathrm{\Phi }}}_{in}\left(f\right)$$ (5)

where ${\overline{MTF}}_{opt}(f)$ is the mean optical modulation transfer function (MTF), and the expected mean of squared optical MTF is expressed as

$${\overline{MT{F}^2}}_{opt}\left(f\right)={\left[{\overline{MTF}}_{opt}\left(f\right)\right]}^2+{{\sigma }^2}_{MT{F}_{opt}}\left(f\right)$$ (6)

where ${{\sigma }^2}_{MT{F}_{opt}}(f)$ is the intrinsic variance of $MT{F}_{opt}(f)$.

For an absorbed x‐ray fluence of monoenergetic photons with energy $E$ at depth $z$, ${\mathrm{\Phi }}_{in}(E,z)$, the mean output signal and noise after optical conversion and blurring are given by: ²³

$${\overline{\mathrm{\Phi }}}_{out}\left(f,z\right)=\underset{0}{\overset{E_{max}}{\int }}{\overline{\mathrm{\Phi }}}_{in}\left(E,z\right)\overline{\mathrm{g}}\left(E,z\right){\overline{MTF}}_{opt}\left(f,z\right)dE$$ (7)

$$\begin{array}{cc}\hfill \mathrm{NP}{\mathrm{S}}_{out}\left(f,z\right)& =\underset{0}{\overset{E_{max}}{\int }}{\overline{\mathrm{\Phi }}}_{in}\left(E,z\right)\left[\overline{\mathrm{g}}\left(E,z\right)+{\left(\overline{\mathrm{g}}\left(E,z\right)\right)}^2\phantom{\left(1+\frac{{\mathrm{F}}_{\mathrm{g}}}{\overline{\mathrm{g}}\left(E,z\right)}\right)}\right.\hfill \\ & \left.\times \left(1+\frac{{\mathrm{F}}_{\mathrm{g}}}{\overline{\mathrm{g}}\left(E,z\right)}\right){\overline{MT{F}^2}}_{opt}\left(f,z\right)\right]dE\hfill \end{array}$$ (8)

Both depth‐dependent and depth‐independent random variations in optical gain and blur are considered. Signal and noise contributions from all sublayers are subsequently integrated and propagated through deterministic blurring and sampling stages to calculate the overall pre‐sampling MTF, NPS, and DQE for the CsI scintillator. ²⁸ , ³⁴

2.1.2. Determination of optical properties for CsI

Most parameters used in the model for CsI were obtained from prior measurements or established literature values. ³⁴ , ³⁸ , ⁴⁰ , ⁴¹ The optical parameters were based on GPXS‐type CsI samples with a reflective backing and thicknesses ranging from 200 to 600 µm (Hamamatsu Photonics K.K.). ⁴¹ , ⁴³ , ⁴⁴ The depth‐dependent light escape efficiency was characterized in previous measurements and incorporated into the model accordingly. ⁴⁰ For intermediate thicknesses, optical parameters were obtained by linear interpolation of the measured data.

Two parameters were not directly available, ${\overline{MTF}}_{opt}(f)$ for intermediate thicknesses and ${{\sigma }^2}_{MT{F}_{opt}}(f)$ for all thickness values. They were derived using Equations (6) to (8) by fitting the modeled signal and noise to the corresponding measured quantities. The k‐fluorescence effect was incorporated into the model. Figure 3(a) shows the measurement geometry. The CsI‐based FPD had an 85 µm pixel pitch and a 74% fill factor. The x‐ray spectrum was 49 kV (tungsten anode) with 1 mm inherent Al filtration. Additional filtration included a 100 µm Ag foil and a 200 µm a‐Se layer deposited on a 700 µm glass substrate, simulating the incident beam on the BL in the DI‐DLFPD. The scintillator was attached to a carbon fiber plate with holes, enabling coupling to the sensor by compression using thumbscrews. ⁴⁵ Four GPXS CsI samples with thicknesses of 100 µm, 200 µm, 400 µm, and 600 µm were used. Each sample measured 21 × 31 mm². Slanted‐edge and flood images (Figure 3(b)) were acquired and processed to characterize pre‐sampling MTF and NPS, from which DQE was subsequently derived.

FIGURE 3.

(a) Experimental geometry and (b) acquired images for the measurement of the pre‐sampling MTF and NPS of CsI.

2.2. Experimental validation of model for the DI‐DLFPD

The refined CsI model (Section 2.1.1) was integrated with established x‐ray spectrum and a‐Se models to predict DI‐DLFPD performance using parameters obtained from literature (Appendix A) and GPXS‐based measurements (Section 2.1.2). ³⁰ , ⁴⁶ For validation, model calculations of both FL LE and BL HE imaging performance were compared with experimental measurements from two fabricated DI‐DLFPD prototypes (Analogic Canada). As shown in Figure 4(a), the prototypes incorporated different CsI thicknesses of 200 and 400 µm. Both had a 200 µm thick a‐Se FL for LE image acquisition, consistent with typical mammography detectors. Pre‐sampling MTF, NPS, and DQE were measured experimentally using the imaging geometry shown in Figure 4(b). The 49 kV tungsten x‐ray spectrum was shaped using a 100 µm Ag filter to produce two energy peaks, one below and one above the K‐edge of iodine.

FIGURE 4.

(a) Cross‐sectional view illustrating the design concept of two DI‐DLFPD prototypes, highlighting the varying CsI BL thicknesses; (b) Experimental setup for measurements.

2.3. Application of the model for lesion detection analysis

The validated model was then applied to predict signal and noise in DE images and to calculate the detectability index (d′) for iodinated objects in DI‐DLFPD based CE digital mammography (CEDM), using the workflow shown in Figure 5. The pixel pitch is 85 µm, and the fill factor for a‐Se and CsI is 1.00 and 0.74, respectively. ²⁸ , ⁴⁵ The modeled x‐ray spectrum, based on TASMICS, ⁴⁶ is incident on the a‐Se FL, with part of the beam transmitted to the CsI BL. The energy separation, defined as the difference in the average energies of the x‐ray spectra absorbed by the FL and BL, was analyzed for a 4 cm breast with 50% glandularity (50‐50 breast). MTF matching and logarithmic transformation were implemented using a deterministic blur process and a gain process ⁴³ in the normalized noise calculation for the HE and blurred LE images, $NNP{S}_{HE}$ and $NNP{S}_{LE}$. Inter‐layer misalignment was assumed to be corrected using projective transformation matrices derived from grid phantom images. Scatter radiation was not considered, as its impact and correction strategies have been extensively discussed in previous studies. ⁴⁴ , ⁴⁷ , ⁴⁸ The weighting factor ($w$) was analytically determined based on LE and HE spectra and used to calculate noise in subtracted DE images ($NNP{S}_{DE}$). Task function ($W_{task}$) was modeled for a cylindrical iodinated object (2 mm diameter and height, 1 mg/mL iodine concentration) by multiplying absolute iodine contrast ($C_{DE}$) in DE images by object function ($O_{lesion}$). ⁴⁹ The detectability index was calculated as: ⁴⁹

$$d^{\prime }=\sqrt{\int \int \frac{MT{F^2}_{DE}}{NNP{S}_{DE}}{W^2}_{task}d{f}_{x}d{f}_y}$$ (9)

$$NNP{S}_{DE}=NNP{S}_{LE}{w}^2+NNP{S}_{HE}$$ (10)

$$W_{task}=C_{DE}{O}_{lesion}$$ (11)

$$\begin{array}{cc}\hfill C_{DE}& =Lesio{n}_{DE}-Bk{g}_{DE}\hfill \\ & =\left[{\mu }_{lesion}^{HE}-{\mu }_{bkg}^{HE}-w\left({\mu }_{lesion}^{LE}-{\mu }_{bkg}^{LE}\right)\right]t\hfill \end{array}$$ (12)

where $Lesio{n}_{DE}$ and $Bk{g}_{DE}$ denote the signals for the iodinated lesion and the normal breast background in the DE image, respectively, calculated from their attenuation coefficients at LE and HE (${\mu }_{lesion}^{HE}$, ${\mu }_{bkg}^{HE}$, ${\mu }_{lesion}^{LE}$, and ${\mu }_{bkg}^{LE}$) and the lesion thickness ($t$). Energy‐dependent x‐ray attenuation coefficients for breast tissues, the Ag filter, and detector materials were obtained from the NIST XCOM database. ⁵⁰

FIGURE 5.

Workflow to model the signal and noise in DE images and d′ of iodinated objects.

3. RESULTS

3.1. Model for CsI

Figure 6(a) and (b) show the measured and modeled pre‐sampling MTF and DQE for GPXS CsI samples with different thicknesses based on the setup and beam conditions described in Section 2.1.2. Good agreement is achieved between model calculations and experimental measurements when the estimated optical MTF and its variance are incorporated into the model. Conversely, omitting optical MTF variability leads to a substantial overestimation of the non‐zero spatial‐frequency DQE, as illustrated for the 600 µm CsI detector in Figure 6(b). This discrepancy underscores the importance of depth‐independent MTF variability in the model. The pre‐sampling MTF drops more rapidly with spatial frequency for thicker CsI samples, following the trend of optical MTF, as shown in Figure 6(c). This trend arises from increased lateral spread of optical photons before they reach the readout surface. It also contributes to the faster drop in DQE at higher spatial frequencies for thicker samples (Figure 6(b)). The higher zero‐frequency DQE observed for thicker CsI samples results from increased x‐ray absorption efficiency. The estimated optical MTF for the 200 µm CsI sample agrees well with previous measurements using the single x‐ray imaging (SXI) method, confirming the validity of the parameter estimation method. ³⁸ Minor discrepancies for the 600 µm sample are likely due to experimental uncertainties in both SXI and measurements in Section 2.1.2. ³⁵ Figure 6(d) shows the estimated variance of optical MTF as a function of CsI thickness at different spatial frequencies. The variance decreases with frequency due to the decreased mean optical MTF at higher frequencies, in agreement with previous SXI observation. ³⁵ At non‐zero frequencies, both the mean optical MTF and its variance decrease with CsI thickness. This consistent relationship enables reliable linear interpolation of these parameters for CsI samples with thicknesses between 100 and 600 µm.

FIGURE 6.

Experimental measurements and model calculations of (a) pre‐sampling MTF and (b) DQE for GPXS CsI samples of varying thicknesses, (c) estimated optical MTF compared with previous measurements using the single x‐ray imaging (SXI) method, and (d) variance of the optical MTF for GPXS samples.

3.2. Experimental validation of model for the DI‐DLFPD

Figure 7(a) shows the comparison of modeled and measured results of pre‐sampling MTF for FL of both DI‐DLFPD prototypes, which have a nominal thickness of 200 µm a‐Se. The slight discrepancy of measured results between the two prototypes could be attributed to sample thickness variations during fabrication. Figure 7(b) shows the modeled and measured pre‐sampling MTF for the 200 µm CsI BL in the 1st prototype and the 400 µm CsI BL in the 2nd prototype. The model correctly predicts the expected reduction in MTF with the thicker CsI. While the model slightly underestimates the MTF for the first prototype and overestimates it for the second, these differences are likely due to experimental uncertainties and scintillator variation between vendors. Figure 7(c) compares the modeled and measured NNPS for both prototypes. The discrepancies between model and experiment follow similar trends to those observed in the MTF results. Overall, good agreement is achieved, particularly for the a‐Se FL and the 400 µm CsI BL. This validates the effectiveness of the model to accurately predict the impact of layer thickness on the spatial‐frequency‐dependent imaging performance of DI‐DLFPD. Despite the ability to fine‐tune the optical parameters for the subsequent analysis of DI‐DLFPD prototypes, we decided to retain the parameters derived from GPXS samples. This is to ensure consistency and leverage the broader range of CsI thicknesses available in the GPXS samples, which facilitates more accurate interpolation for intermediate thicknesses and supports systematic evaluation of the thickness dependence of imaging performance.

FIGURE 7.

Model calculated and experimentally measured (a) pre‐sampling MTF of 200 µm a‐Se FL in both prototypes, (b) pre‐sampling MTF of the 200 µm CsI BL in the first prototype and the 400 µm CsI BL in the second prototype, and (c) NNPS for both layers in both prototypes.

3.3. Application of the model for lesion detection analysis

3.3.1. Energy separation

Figure 8(a) shows the dependence of energy separation between LE and HE on the a‐Se FL thickness and CsI BL thickness with a 100 µm thick Ag K‐edge filter. Energy separation exhibits minimal dependence on CsI BL thickness, whereas it increases substantially with a‐Se FL thickness. This increase is more pronounced when a thicker Ag filtration is used, as shown in Figure 8(b), where a 400 µm CsI BL is used. Additionally, increasing the Ag filter thickness improves energy separation only when the a‐Se FL is sufficiently thick.

FIGURE 8.

Dependence of energy separation on (a) a‐Se FL thickness and CsI BL thickness, and (b) a‐Se FL thickness and Ag filter thickness.

Figure 9 shows the energy spectra responsible for the trends seen in Figure 8. Shown in Figure 9(a) is the energy distribution of absorbed photons for the CsI BL detector with different thicknesses, which was calculated for a 49 kV tungsten spectrum shaped by a 100 µm Ag K‐edge filter and a 200 µm a‐Se FL. The 400 µm CsI BL absorbs more x‐rays than the 200 µm CsI BL across the entire spectrum. However, this additional absorption has minimal effect on the average energy of the HE images, explaining the weak dependence of energy separation on CsI thickness (seen in Figure 8(a)). Figure 9(b) shows that a thicker a‐Se FL detector absorbs more x‐rays. Consequently, fewer LE x‐rays reach the BL, increasing the average energy for BL HE images and thereby improving energy separation. This improvement is more pronounced with a thicker Ag K‐edge filter (as seen in Figure 8(b)), which produces an incident spectrum with (1) a larger portion of LE x‐rays and (2) the HE peak shifted toward higher energy, as shown in Figure 9(c). These changes in the incident beam also lead to a substantial decrease in the average energy for FL LE images (Figure 9(d)), resulting in improved energy separation even with FL and BL thicknesses unchanged (Figure 8(b)). However, the benefit of thicker Ag filtration disappears when using a thin a‐Se FL (Figure 8(b)), because insufficient stopping power allows many LE x‐rays to transmit to the BL, reducing the spectral distinction between the two layers.

FIGURE 9.

Energy spectra of x‐ray (a) absorbed by BL layers of varying CsI thicknesses, (b) absorbed by FL layers of varying thicknesses and subsequently by the same CsI BL, (c) incident on the DI‐DLFPD with varying Ag K‐edge filter thicknesses, and (d) absorbed by FL and BL with varying Ag thicknesses. Unless otherwise stated, Ag thickness is 100 µm, a‐Se FL thickness is 200 µm, and CsI BL thickness is 400 µm.

3.3.2. Absolute iodine contrast

Figure 10 shows the impact of thicknesses for a‐Se FL, CsI BL, and Ag K‐edge filter on the absolute contrast of the iodinated object (2 mm diameter and height, 1 mg/mL iodine concentration) embedded in a 4 cm thick 50‐50 breast, calculated according to Equation (12). Absolute iodine contrast generally improves with increasing a‐Se FL thickness and decreasing CsI BL thickness. An optimal Ag filter thickness is observed to be approximately 100 µm, which does not correspond to the configuration that yields the maximum energy separation.

FIGURE 10.

Dependence of absolute contrast of the iodinated object (2 mm diameter and height, 1 mg/mL iodine concentration) on (a) a‐Se FL and CsI BL thicknesses, and (b) a‐Se FL and Ag K‐edge filter thicknesses.

Figure 11(a) shows the mass attenuation coefficients of 50‐50 breast, iodine, a‐Se, and CsI. Absolute contrast of iodine is maximized when the average energy of the HE signal is closest to the iodine K‐edge, rather than when the energy separation is maximized. The detection efficiency of CsI is also enhanced when its absorbed energy average is just above its K‐edges. Figure 11(b) compares the absorbed spectra by FL and BL in two representative scenarios to illustrate this effect. Scenario 1 uses 300 µm Ag, 350 µm a‐Se FL, and 400 µm CsI BL, which yields a larger energy separation (15.04 keV; seen in Figure 8(b)), but lower iodine contrast (2.31 × 10⁻³; seen in Figure 10(b)). This is because the average absorbed photon energy for the BL increases to 42.46 keV, significantly above the K‐edge, as shown in Figure 11(b). This reduces differential attenuation between iodine and breast tissue and reduces CsI detection efficiency for iodine. In contrast, scenario 2 uses 100 µm Ag, 300 µm a‐Se FL, and 100 µm CsI BL, which gives lower energy separation (11.36 keV; seen in Figure 8(a)), but higher iodine contrast (2.48 × 10⁻³; seen in Figure 10(a)). It yields a BL HE spectrum positioned closer to the K‐edge, resulting in higher iodine contrast despite a smaller energy separation.

FIGURE 11.

(a) Mass attenuation coefficients of 50‐50 breast, iodine, a‐Se, and CsI. (b) x‐ray spectra absorbed by the a‐Se FL and CsI BL.

3.3.3. Iodine detectability

Figure 12(a) shows the d′ as a function of a‐Se FL and CsI BL thicknesses for the iodinated object (2 mm diameter and height, 1 mg/mL iodine concentration) embedded in a 4 cm thick 50‐50 breast, using a 100 µm Ag K‐edge filter and 1 mGy mean glandular dose (MGD). With the a‐Se FL thickness of 200 µm, increasing the CsI BL thickness from 100 to 200 µm leads to the largest increase in d′ of 19.0%. Further increases in 100 µm increments result in smaller improvements of 5.9% and 1.9%, respectively. Increase in d′ beyond 400 µm is negligible. Detectability initially increases with a‐Se FL thickness due to improved energy separation and absolute iodine contrast (seen in Figures (8) and (10)), and then declines due to increased noise. While the maximum d′ is achieved at an a‐Se thickness of ∼150 µm, d′ changes very slowly with the increased thickness up to 200 µm, which is the standard a‐Se thickness used in commercial mammography systems. Regarding the dependence on Ag filter thickness, the optimal d′ occurs at ∼ 100 µm, which can provide sufficient x‐ray fluence to achieve the glandular dose used in clinical applications. ¹⁷ These optimal thickness values differ from those for absolute iodine contrast because d′ depends critically on spatial‐frequency‐dependent image noise, especially that in the HE images.

FIGURE 12.

Detectability index (d′) of the iodinated object as a function of (a) a‐Se FL and CsI BL thicknesses with a 100 µm Ag filter, and (b) a‐Se FL and Ag thicknesses with a 400 µm CsI BL. The cylindrical iodinated object (2 mm diameter and height, 1 mg/mL concentration) is embedded in a 4 cm thick 50‐50 breast. MGD = 1 mGy.

Figure 13(a) shows the NNPS in FL LE and BL HE images for varying CsI BL thicknesses using a 100 µm Ag K‐edge filter and 200 µm a‐Se FL. Increasing CsI thickness from 200 to 400 µm reduces BL NNPS at zero frequency by 19.4% due to increased quantum efficiency. Further increase in CsI thickness to 600 µm reduces by only 4.8%. The reduction is more pronounced at higher frequencies due to the reduced MTF of thicker CsI. The BL NNPS remains significantly higher than the FL NNPS because the HE image contains far fewer photons. It is important to note that the NNPS for FL and BL have the same shape, which is due to MTF matching before weighted subtraction. The weighting factor $w$ is always lower than 1, making the NNPS from BL the dominant contributor to DE NNPS. As shown in Figure 13(b), DE NNPS decreases with CsI thickness but plateaus beyond 400 µm, explaining the diminishing improvement in d′ in Figure 12(a) in this thickness range.

FIGURE 13.

Modeled NNPS in (a) FL LE and BL HE images, and (b) DE images for varying CsI BL thicknesses, using a 100 µm Ag filter and 200 µm a‐Se FL. MGD = 1 mGy to a 4 cm 50–50 breast.

Figure 14(a) shows the FL and BL NNPS for varying a‐Se FL thickness using a 100 µm Ag filter and 400 µm CsI BL. When a‐Se thickness increases from 200 to 300 µm, FL NNPS decreases by 11.9%, but BL NNPS increases substantially by 46.4% because fewer photons could reach CsI. As shown in Figure 14(b), the resulting DE NNPS increases by 43.4%. The noise increase outweighs the improvements in energy separation and iodine contrast, leading to the decrease in d′ observed in Figure 12(a).

FIGURE 14.

Modeled NNPS in (a) FL LE and BL HE images, and (b) DE images for varying a‐Se FL thicknesses, using a 100 µm Ag filter and 400 µm CsI BL. MGD = 1 mGy to a 4 cm 50–50 breast.

Figure 15 shows d′ for the same iodinated object embedded in a 6 cm thick 50‐50 breast delivered with a 2.25 mGy MGD. The overall pattern is similar to that observed for the 4 cm breast, although the maximum occurs at slightly shifted thicknesses due to increased beam hardening in the thicker breast. The optimized configuration (100 µm Ag filter, 200 µm a‐Se FL, and 400 µm CsI BL) also applies to this 6 cm breast. While a 125 µm Ag provides a 2.8% increase in d′, it cannot provide sufficient x‐ray fluence to achieve the clinically relevant glandular dose. ¹⁷ Decreasing the a‐Se FL thickness to ∼125 µm increases d′ by only 6.1%, but could substantially degrade the image quality for standard LE‐only clinical mammography applications. ⁵¹

FIGURE 15.

Detectability index of the iodinated object as a function of (a) a‐Se FL and CsI BL thicknesses with a 100 µm Ag filter, and (b) a‐Se FL and Ag thicknesses with a 400 µm CsI BL. The cylindrical iodinated object (2 mm diameter and height, 1 mg/mL concentration) is embedded in a 6 cm thick 50‐50 breast. MGD = 2.25 mGy.

4. DISCUSSION

To improve the accuracy of spatial‐frequency‐dependent signal and noise propagation, this study refines the CLSM model for CsI scintillators by incorporating depth‐independent random variations in optical gain and blur alongside established depth‐dependent effects. Optical parameters were derived from measurements of GPXS CsI samples with thicknesses ranging from 100 to 600 µm, with linear interpolation used for intermediate thicknesses. This approach is generalizable to CsI samples from other vendors and provides a practical framework for task‐specific optimization of CsI‐based detector design.

The refined CsI model was integrated with established x‐ray spectrum and a‐Se models to build a complete DI‐DLFPD imaging model. The predicted pre‐sampling MTF and NNPS matched well with measurements from two DI‐DLFPD prototypes with different CsI BL thicknesses, validating the accuracy of the model. Residual discrepancies likely arise from measurement uncertainties and vendor‐dependent CsI sample variations. Overall, the model captures the dependence of spatial‐frequency‐dependent performance on CsI thickness and supports system‐level optimization.

For CEDM, the model incorporates postprocessing steps, including MTF matching, logarithmic transformation, and weighted subtraction, enabling evaluation of energy separation, absolute iodine contrast, and detectability index (d′). Energy separation increases with a‐Se FL and Ag filter thicknesses but shows minimal dependence on CsI BL thickness. This behavior arises because FL preferentially absorbs LE photons, hardening the spectrum incident on BL. This effect is enhanced under stronger Ag filtration, which increases the LE fraction in the incident spectrum and shifts the HE peak to higher energy. In contrast, CsI thickness primarily affects absorbed photon fluence rather than HE average or energy separation. Importantly, the optimal absolute contrast is achieved when HE average is just above the K‐edge of iodine and CsI, rather than when energy separation alone is maximized. Consequently, the optimal combination of Ag filter and a‐Se FL thickness exists; beyond this point, the HE average shifts away from the K‐edge, reducing contrast.

Detectability behaves differently as it is governed by the balance between contrast and noise. The BL NNPS is substantially higher than the FL NNPS due to lower photon fluence in the HE image. In addition, the sub‐unity subtraction weighting factor makes BL noise the dominant contributor to DE noise. For a 4 cm breast, optimal d′ is achieved with a 150 µm a‐Se FL and 100 µm Ag filter, where the improvement in contrast is not outweighed by excessive noise increase. Beyond this point, further FL absorption and Ag filtration elevate BL noise and reduce d′. For a 6 cm breast, the optimal a‐Se FL thickness decreases to 125 µm, while the optimal Ag filter thickness increases to 125 µm. This shift results from increased beam hardening, requiring stronger Ag filtration to maintain sufficient energy separation and iodine contrast, while a thinner a‐Se FL is favored to preserve adequate LE absorption and allow HE transmission to BL. Increasing CsI BL thickness improves quantum efficiency and yields meaningful gain in d′ up to 400 µm. Beyond this point, further absorption improvement is negligible, while MTF degradation limits performance for small lesion detection. When using the 200 µm thick a‐Se common in commercial mammography detectors and a 100 µm thick Ag filter constrained by the tube output, the reduction in d′ is less than 3% and 7% relative to the optimal values for 4 and 6 cm 50‐50 breasts, respectively.

Task‐specific analysis was performed using a small, low‐contrast iodinated object (2 mm, 1 mg/mL), representing a challenging detection scenario in CEDM. While absolute contrast and detectability values increase with lesion size and iodine concentration, the trend for optimal design parameters remains relatively unchanged, as they are largely governed by x‐ray spectral shaping and noise propagation. ¹⁸ For high‐spatial‐frequency tasks, such as microcalcification material decomposition, the MTF for CsI becomes more influential, and optimal parameters (such as CsI thickness and optical design) are expected to change accordingly.

Overall, our findings suggest that the configuration of 100 µm Ag filter, 200 µm a‐Se FL, and 400 µm CsI BL allows identical performance for existing clinical imaging applications, while providing optimal DI‐DLFPD based CEDM performance. In DBT mode, for x‐rays incident at the most oblique angle (${25}^{\circ }$), the effective a‐Se FL and CsI BL thickness increase, but d′ remains above 97% of the value at the central angle ($0^{\circ }$), supporting uncompromised CEDBT performance.

5. CONCLUSIONS

We developed a spatial‐frequency‐dependent model for signal and noise propagation in DI‐DLFPD based CEDM systems, enabling system optimization. Building upon established x‐ray spectrum and a‐Se FL models, the model incorporates depth‐independent optical blur variability within the CsI scintillator. This improves the accuracy of DQE prediction, as validated by measurements from a prototype system. The analysis demonstrates that noise in the BL HE images, rather than energy separation or absolute iodine contrast, is the principal factor limiting lesion detectability. Considering practical implementation and compatibility with existing clinical workflows, an optimized system configuration consists of a 100 µm Ag filter, a 200 µm a‐Se FL layer, and a 400 µm CsI BL layer. This design provides reliable detectability for a 2 mm, 1 mg/mL iodinated lesion in both 4 and 6 cm 50–50 breasts at clinical dose levels. The modeling framework and the methodology for determining optical parameters from measurements are broadly applicable and may be used to guide the optimization of CsI‐based detectors or other multilayer imaging systems for a wide range of imaging tasks.

CONFLICT OF INTEREST STATEMENT

The authors have no relevant conflicts of interest to disclose.

APPENDIX A. MODEL ARCHITECTURE AND PARAMETERS USED FOR A‐SE AND CSI

A.1.

Figures A1 and A2 show the flowcharts of the cascaded stages in the a‐Se and CsI imaging chains. The general characteristic parameters used in the model are listed in Table A1. ¹⁹ , ²⁸ , ²⁹ , ³⁰ , ⁴⁰ , ⁴¹ , ⁵² Additional blurring introduced by charge trapping in a 7 µm thick blocking layer between the pixel electrodes and the a‐Se was included. ³⁰ A Swank factor of 0.99 was used for the intrinsic variation in charge conversion gain, while the k‐fluorescence effect was modeled separately. ²⁸ , ²⁹ , ⁵³

FIGURE A1.

Flow diagram showing the cascaded stages involved in the direct‐conversion a‐Se imaging chain.

FIGURE A2.

Flow diagram showing the cascaded stages involved in the indirect‐conversion CsI imaging chain.

TABLE A1.

General characteristic parameters for a‐Se (operated at an electric field of 10 V/µm) and CsI.

Parameter	a‐Se	CsI	Parameter	a‐Se	CsI
Density (g/cm3)	4.27	3.38	K‐shell probability	0.864	0.832 (Cs) 0.838 (I)
W (eV/quanta)	50	17	K‐fluorescence yield	0.596	0.895 (Cs) 0.882 (I)
Pixel electronic noise (e)	1200	1200	Kα production fraction	0.862	0.812 (Cs) 0.814 (I)
K‐edge energy (keV)	12.7	36.0 (Cs) 33.2 (I)	Kα, Kβ energies (keV)	11.2, 12.5	30.7, 35.0 (Cs) 28.6, 32.3 (I)

For CsI, intrinsic depth‐independent variations in x‐ray‐to‐light conversion were characterized by a “photon Fano factor” of 64.3. ⁴⁰ The depth‐dependent light escape efficiency was implemented based on previously measured GPXS CsI data obtained using the single‐x‐ray imaging technique. ⁴⁰ For CsI thicknesses up to 200 µm, the light escape efficiency was treated as constant (0.850). For a 600 µm CsI layer, the escape efficiency varied from 0.819 in the deepest sublayer to 0.944 near the readout surface.

Wu X, Howansky A, Huang H, et al. Theoretical modeling and performance analysis of a direct‐indirect dual‐layer flat‐panel detector for contrast‐enhanced breast imaging. Med Phys. 2026;53:e70588. 10.1002/mp.70588