Non‐stationary model of oblique x‐ray incidence in amorphous selenium detectors: I. Point spread function

Abstract

Purpose

In previous work, a theoretical model of the point spread function (PSF) for oblique x‐ray incidence in amorphous selenium (a‐Se) detectors was proposed. The purpose of this paper is to develop a complementary model that includes two additional features. First, the incidence angle and the directionality of ray incidence are calculated at each position, assuming a divergent x‐ray beam geometry. This approach allows the non‐stationarity of the PSF to be modeled. Second, this paper develops a framework that is applicable to a digital system, unlike previous work which did not model the presence of a thin‐film transistor (TFT) array.

Methods

At each point on the detector, the incidence angle and the ray incidence direction are determined using ray tracing. Based on these calculations, an existing model for the PSF of the x‐ray converter (Med Phys. 1995;22:365‐374) is generalized to a non‐stationary model. The PSF is convolved with the product of two rectangle functions, which model the sampling of the TFT array. The rectangle functions match the detector element (del) size in two dimensions.

Results

It is shown that the PSF can be calculated in closed form. This solution is used to simulate a digital mammography (DM) system at two x‐ray energies (20 and 40 keV). Based on the divergence of the x‐ray beam, the direction of ray incidence varies with position. Along this direction, the PSF is broader than the reference rect function matching the del size. The broadening is more pronounced with increasing obliquity. At high energy, the PSF deviates more strongly from the reference rect function, indicating that there is more blurring. In addition, the PSF is calculated along the polar angle perpendicular to the ray incidence direction. For this polar angle, the shape of the PSF is dependent upon whether the ray incidence direction is parallel with the sides of the detector. If the ray incidence direction is parallel with either dimension, the PSF is a perfect rectangle function, matching the del size. However, if the ray incidence direction is at an oblique angle relative to the sides of the detector, the PSF is not rectangular. These results illustrate the non‐stationarity of the PSF.

Conclusions

This paper demonstrates that an existing model of the PSF of a‐Se detectors can be generalized to include the effects of non‐stationarity and digitization. The PSF is determined in closed form. This solution offers the advantage of shorter computation time relative to approaches that use numerical methods. This model is a tool for simulating a‐Se detectors in future work, such as in virtual clinical trials with computational phantoms.

1. Introduction

The detectors commonly used in breast x‐ray imaging are either indirect‐ or direct‐converting. In an indirect‐conversion detector such as cesium iodide (CsI), x rays are first converted to visible light, which spreads laterally.^{1, 2, 3, 4} The visible light is then converted to charged particles by photodiodes. This signal is digitized by a thin‐film transistor (TFT) array. By contrast, there is no intermediate conversion of x rays to visible light in a direct‐conversion detector such as amorphous‐selenium (a‐Se).^{5, 6, 7} In these detectors, selenium atoms are ionized by x rays, creating electron‐hole pairs. Based on an applied electric field, the electrons and holes migrate to opposite surfaces of the detector, and an image is formed. Badano et al. demonstrated that, for a fixed detector thickness and energy, a‐Se is characterized by a narrower point response function than CsI at normal incidence⁸; in this paper, we use the term point spread function (PSF).

Due to the low atomic number of selenium (Z = 34), a‐Se detectors have low x‐ray absorption at high energies, and hence are more commonly used in low‐energy applications, such as DM and digital breast tomosynthesis (DBT). At 20 keV, the quantum detection efficiency of 0.200 mm thick selenium is 98.3% at normal incidence, assuming a mass attenuation coefficient⁹ of 4.82 × 10³ mm²/g and a density¹⁰ of 4.20 × 10⁻³ g/mm³.

Badano et al. developed a model of the obliquity effect in a‐Se detectors.⁸ They demonstrated that the PSF is broadened due to oblique incidence. In their work, the lateral spread of signal at each depth of a‐Se was modeled with a Lorentzian function. The parameters were fit to the results of Monte Carlo simulations with mantis. Que and Rowlands offered a different approach for calculating the PSF.¹¹ Rather than calculate a single PSF that models all possible sources of blurring, Que and Rowlands identified multiple x‐ray interactions that could introduce blurring, and modeled the PSF for each interaction separately. In their model of the obliquity effect, the PSF is blurred along the ray incidence direction, and is a delta function along the perpendicular direction.

One limitation of these previous works is that the presence of the TFT array was not included in the modeling assumptions, and hence the PSF models a non‐pixelated system. The purpose of this paper is to develop a complementary model for a digital system, in which the signal in the x‐ray converter is binned into detector elements (dels). For the purpose of this paper, we expand upon the model of the PSF developed by Que and Rowlands.¹¹ The PSF of the digital system is derived by convolving the PSF of the x‐ray converter with the product of two rectangle functions, which match the del size in each direction.^{12, 13, 14} It is shown that this convolution can be calculated in closed form. The advantage of deriving an analytical formula is that computation time is minimized relative to numerical integration methods.

In x‐ray imaging, the shape of the PSF varies with position based on the angle of incidence and the directionality of the incident ray. An additional aim of this paper is to demonstrate how the PSF varies with position. We use this formulation to quantify non‐stationarity in a‐Se detectors.

2. Materials and methods

2.A. Coordinate system

A model of the PSF was previously developed by Que and Rowlands for an arbitrary incidence angle¹¹; however, they did not model the non‐stationarity of the PSF and did not consider the impact of digitization. These effects are now modeled from first principles for a breast imaging system.

Figure 1 shows a diagram of the acquisition geometry. The exit surface of the detector is defined to be the plane z = 0. The origin (O) is the point in this plane at which the ray is normally incident. The x‐ray source is treated as a point source at the coordinate (0, 0, d). Figure 1 also allows for the possibility of transforming to a coordinate system with a different reference point (M); in a breast imaging system, point M is the midpoint of the chest wall side of the detector. This transformation is considered in Section 2.D.

Figure 1.

A diagram of the acquisition geometry is shown. The direction of ray incidence is parallel with vector $\overrightarrow{\text{SB}}$ (or equivalently, vector $\overrightarrow{\text{OC}}$). Also, θ is the incidence angle, and Γ is the azimuthal angle of the ray. The path length of the ray through a‐Se (thickness l) begins at point B and ends at point C. This figure is not to scale.

At each point along the ray, there are x‐ray interactions with selenium atoms (Fig. 2). The electrons and holes that are produced migrate to opposite sides of the x‐ray converter due to the applied electric field. We assume that signal is recorded at the exit surface as opposed to the entrance surface. For this reason, the directionality of the electric field determines whether the signal is produced by electrons or holes. The theoretical model can be applied to either. However, in this paper, the signal is taken to be produced by holes, like the AXS‐2430 detector (Analogic Canada Corporation, Montreal, Quebec, Canada) modeled in Section Results.

Figure 2.

Electron‐hole pairs are produced by the ionization of selenium atoms along the path length of the ray. Due to the applied electric field, the electron and hole migrate to opposite ends of the detector. Signal is formed along the line segment $\overline{\text{QC}}$, which lies along the ray incidence direction (x′). For the purpose of this paper, it is presumed that there is no blurring along the perpendicular direction (y′).

Figure 2 shows how signal is produced along the x′ direction in the exit surface of the x‐ray converter; this is called the ray incidence direction throughout this paper. This direction lies along the angle Γ relative to the x direction (Fig. 1). To derive the PSF, the signal needs to be calculated in this rotated frame using the transformation

$$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right)+\left(\begin{array}{cc}cos\mathrm{\Gamma }& -sin\mathrm{\Gamma }\\ sin\mathrm{\Gamma }& cos\mathrm{\Gamma }\end{array}\right)\left(\begin{array}{c}{v}_1^{\prime }\\ v_2^{\prime }\end{array}\right)$$ (1)

where $v_1^{\prime }$ is position along the ray incidence direction, $v_2^{\prime }$ is position in the perpendicular direction, and ξ ₁ and ξ ₂ are the x and y coordinates of the entrance point (B), respectively. Two trigonometric substitutions can be made based on Fig. 1

$$cos\mathrm{\Gamma }=\frac{{\xi }_1}{r}$$ (2)

$$sin\mathrm{\Gamma }=\frac{{\xi }_2}{r}$$ (3)

where

$$r=\sqrt{{\xi }_1^2+{\xi }_2^2}$$ (4)

giving

$$v_1^{\prime }=\frac{{\xi }_1}{r}x+\frac{{\xi }_2}{r}y-r$$ (5)

$$v_2^{\prime }=\frac{-{\xi }_2}{r}x+\frac{{\xi }_1}{r}y$$ (6)

as the inverse transformation. The derivation that follows presumes that r > 0. There is normal incidence at the position for which r = 0; the PSF at this position is simply the TFT sampling function described in Section 2.B.

2.B. Depth‐dependent PSF

The PSF associated with an arbitrary ionization point is now calculated. From Fig. 2, the $v_1$‐coordinate of the ionization point (I) is (l − z) tan θ, where θ is the incidence angle. The PSF is

$$P_{\mathrm{I}}=\delta \left[v_1^{\prime }-(l-z)tan\theta \right]\delta (v_2^{\prime }),$$ (7)

where the subscript “I” is an abbreviation for “Ionization”. The angle θ can be determined from Fig. 1.

$$tan\theta =\frac{r}{d-l}$$ (8)

Combining Eqs. (5), (6), (7), (8) yields

$$P_{\mathrm{I}}=\delta \left(\frac{{\xi }_1}{r}x+\frac{{\xi }_2}{r}y+\frac{r}{d-l}z-\frac{\mathit{dr}}{d-l}\right)\delta \left(\frac{-{\xi }_2}{r}x+\frac{{\xi }_1}{r}y\right).$$ (9)

In a digital system, a TFT array bins the signal produced by the x‐ray converter into dels. We model the PSF of the TFT array using

$$P_{\mathrm{TFT}}=\frac{1}{a_x{a}_y}\text{rect}\left(\frac{x}{a_x}\right)·\text{rect}\left(\frac{y}{a_y}\right),$$ (10)

where a _x and a _y denote the del size in the x and y directions, respectively, and P _TFT denotes the PSF of the TFT array. If the x and y subscripts are removed, it is assumed that the del is square (a _x  = a _y  = a). Combining the effects of ionization and TFT sampling, the PSF can be written as the two‐dimensional (2D) convolution

$$P_z=P_{\mathrm{I}}{\ast }_2{P}_{\mathrm{TFT}},$$ (11)

where the subscript z emphasizes that this PSF is associated with a specific depth.

2.C. Net PSF

One can integrate P _z over the thickness (l) to determine the net PSF of the system. This function is denoted P _Net.

$$P_{\mathit{Net}}={\int }_0^{l}N·P_z·dz$$ (12)

To convert Eq. (12) into a form with infinite integration limits, one can introduce a rectangle function to match the interval of integration, so that

$$P_{\mathit{Net}}={\int }_{-\infty }^{\infty }N·P_z·\mathit{rect}\left(\frac{z-l/2}{l}\right)dz,$$ (13)

where

$$\text{rect}(v)\equiv \left\{\begin{array}{cc}1,\hfill & |v|\le 1/2\hfill \\ 0,\hfill & |v|>1/2\hfill \end{array}\right..$$ (14)

In Eq. (12), the integrand is weighted by N, the relative number of x‐ray quanta at each depth z. The function N is determined from the linear attenuation coefficient (μ) of selenium

$$N=C_{\mathrm{I}}{e}^{-\mu (l-z)sec\theta },$$ (15)

where

$$sec\theta =\sqrt{1+{\left(\frac{r}{d-l}\right)}^2}.$$ (16)

In Eq. (15), C _I is a normalization term, which can be determined from the normalization convention used by Swank¹⁵

$${\int }_0^{l}N·dz=1$$ (17)

giving

$$C_{\mathrm{I}}=\frac{\mu \sqrt{1+{\left(\frac{r}{d-l}\right)}^2}}{1-e^{-\mu l\sqrt{1+{\left(\frac{r}{d-l}\right)}^2}}}.$$ (18)

Combining Eqs. (9), (10), (11), (13), (15), (16) yields

$$\begin{array}{cc}\hfill P_{\mathrm{Net}}& ={\int }_{-\infty }^{\infty }{C}_{\mathrm{I}}{e}^{-\mu \sqrt{1+{(\frac{r}{d-l})}^2}(l-z)}\hfill \\ & ·\left[{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(\begin{array}{c}\delta \left(\frac{{\xi }_1}{r}{\omega }_1+\frac{{\xi }_2}{r}{\omega }_2+\frac{r}{d-l}z-\frac{\mathit{dr}}{d-l}\right)\\ ·\delta \left(\frac{-{\xi }_2}{r}{\omega }_1+\frac{{\xi }_1}{r}{\omega }_2\right)\frac{1}{a_x{a}_y}\mathit{rect}\left(\frac{x-{\omega }_1}{a_x}\right)\mathit{rect}\left(\frac{y-{\omega }_2}{a_y}\right)\end{array}\right)d{\omega }_{1}d{\omega }_2\right]\mathit{rect}\left(\frac{z-l/2}{l}\right)dz.\hfill \end{array}$$ (19)

We can re‐order the integration so that the integral over z is calculated first.

$$\begin{array}{cc}\hfill P_{\mathrm{Net}}=& \frac{C_{\mathrm{I}}}{a_x{a}_y}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\delta \left(\frac{-{\xi }_2}{r}{\omega }_1+\frac{{\xi }_1}{r}{\omega }_2\right)\hfill \\ & ·\mathrm{rect}\left(\frac{x-{\omega }_1}{a_x}\right)\mathrm{rect}\left(\frac{y-{\omega }_2}{a_y}\right)\hfill \\ & ·{\int }_{-\infty }^{\infty }\left(\begin{array}{c}{e}^{-\mu \sqrt{1+{\left(\frac{r}{d-l}\right)}^2}\left(l-z\right)}\mathrm{rect}\left(\frac{z-l/2}{l}\right)\\ ·\delta \left(\frac{{\xi }_1}{r}{\omega }_1+\frac{{\xi }_2}{r}{\omega }_2+\frac{r}{d-l}z-\frac{\mathit{dr}}{d-l}\right)\end{array}\right)dzd{\omega }_{1}d{\omega }_2\hfill \end{array}$$ (20)

The integral over z can be evaluated using the identity for the delta function of a composition.^{16, 17} The identity presumes that the argument of the delta function has a finite number of zeros and that there are no repeated zeros.

$$\delta [g(z)]=\sum _k\frac{\delta (z-z_k)}{|g^{\prime }(z_k)|}$$ (21)

Each term z _k denotes the kth zero of the function g(z). In Eq. (20), the argument of the delta function has only one zero; namely,

$$z=\frac{-{\xi }_1(d-l)}{r^2}{\omega }_1-\frac{{\xi }_2(d-l)}{r^2}{\omega }_2+d$$ (22)

Thus

$$P_{\mathrm{Net}}=C_{\mathrm{Net}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\delta \left(\frac{-{\xi }_2}{r}{\omega }_1+\frac{{\xi }_1}{r}{\omega }_2\right)\text{rect}\left(\frac{x-{\omega }_1}{a_x}\right)\text{rect}\left(\frac{y-{\omega }_2}{a_y}\right){\int }_{-\infty }^{\infty }\left(\begin{array}{c}{e}^{-\mu \sqrt{1+{\left(\frac{r}{d-l}\right)}^2}\left(l-z\right)}\text{rect}\left(\frac{z-l/2}{l}\right)\\ ·\delta \left[z-\left(\frac{-{\xi }_1\left(d-l\right)}{r^2}{\omega }_1-\frac{{\xi }_2\left(d-l\right)}{r^2}{\omega }_2+d\right)\right]\end{array}\right)dzd{\omega }_{1}d{\omega }_2$$ (23)

where

$$C_{\mathrm{Net}}=\frac{C_{\mathrm{I}}(d-l)}{a_x{a}_{y}r}.$$ (24)

The integral can be evaluated using the definition of the delta function

$${\int }_{-\infty }^{\infty }\varphi (z)·\delta (z-\beta )·dz\equiv \varphi (\beta ),$$ (25)

where ϕ is an arbitrary function of z and $\beta \in \mathbb{R}$. This gives

$$P_{\mathrm{Net}}=C_{\mathrm{Net}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(\begin{array}{c}{e}^{\frac{-\mu }{r}\sqrt{1+{\left(\frac{d-l}{r}\right)}^2}\left({\xi }_1{\omega }_1+{\xi }_2{\omega }_2-r^2\right)}\text{rect}\left(\frac{-{\xi }_1\left(d-l\right)}{l{r}^2}{\omega }_1-\frac{{\xi }_2\left(d-l\right)}{l{r}^2}{\omega }_2+\frac{2d-l}{2l}\right)\\ ·\delta \left(\frac{-{\xi }_2}{r}{\omega }_1+\frac{{\xi }_1}{r}{\omega }_2\right)\text{rect}\left(\frac{x-{\omega }_1}{a_x}\right)\text{rect}\left(\frac{y-{\omega }_2}{a_y}\right)\end{array}\right)d{\omega }_{1}d{\omega }_2.$$ (26)

Eq. (26) can be evaluated in closed form. There are three cases for the values of ξ ₁ and ξ ₂ that are now considered separately.

2.C.1. Case 1: ray incidence in ±x direction

First, ray incidence in the ±x direction is considered; that is, ξ ₁ ≠ 0 and ξ ₂ = 0. From Eq. (4), it follows that r = |ξ ₁|.

$$P_{\mathrm{Net},1}=C_{\mathrm{Net}}{\int }_{-\infty }^{\infty }\delta \left(\frac{{\xi }_1}{|{\xi }_1|}{\omega }_2\right)\text{rect}\left(\frac{y-{\omega }_2}{a_y}\right)d{\omega }_2{\int }_{-\infty }^{\infty }{e}^{\frac{-{\xi }_1}{|{\xi }_1|}\mu \left({\omega }_1-{\xi }_1\right)\sqrt{1+{\left(\frac{d-l}{{\xi }_1}\right)}^2}}\text{rect}\left(\frac{-\left(d-l\right)}{l{\xi }_1}{\omega }_1+\frac{2d-l}{2l}\right)\text{rect}\left(\frac{x-{\omega }_1}{a_x}\right)d{\omega }_1$$ (27)

The delta function can be simplified using the sign function (“sgn”)

$$\delta \left(\frac{{\xi }_1}{|{\xi }_1|}{\omega }_2\right)=\delta \left[\text{sgn}\left({\xi }_1\right)·{\omega }_2\right]$$ (28)

$$=\delta ({\omega }_2),$$ (29)

where

$$\text{sgn}(v)\equiv \left\{\begin{array}{c}-1,v<0\hfill \\ 0,v=0\hfill \\ 1,v>0.\hfill \end{array}\right.$$ (30)

The exponential in Eq. (27) can also be simplified with the sign function. Additionally, it is useful to flip the signs of the arguments of the two rect functions in Eq. (27); this is justified since the rect function [Eq. (14)] is an even function.

$$P_{\mathrm{Net},1}=C_{\mathrm{Net}}\text{rect}\left(\frac{y}{a_y}\right){\int }_{-\infty }^{\infty }{e}^{-\text{sgn}\left({\xi }_1\right)·\mu \left({\omega }_1-{\xi }_1\right)\sqrt{1+{\left(\frac{d-l}{{\xi }_1}\right)}^2}}\text{rect}\left(\frac{d-l}{l{\xi }_1}{\omega }_1+\frac{l-2d}{2l}\right)\text{rect}\left(\frac{{\omega }_1-x}{a_x}\right)d{\omega }_1$$ (31)

To determine the interval of overlap for the rect functions in Eq. (31), the integrand needs to be rewritten in a form that shows the centroid and width of each rect function.

$$P_{\mathrm{Net},1}=C_{\mathrm{Net}}\text{rect}\left(\frac{y}{a_y}\right){\int }_{-\infty }^{\infty }{e}^{-\text{sgn}\left({\xi }_1\right)·\mu \left({\omega }_1-{\xi }_1\right)\sqrt{1+{\left(\frac{d-l}{{\xi }_1}\right)}^2}}\text{rect}\left(\frac{{\omega }_1-\frac{(2d-l){\xi }_1}{2(d-l)}}{\frac{{\xi }_{1}l}{d-l}}\right)\text{rect}\left(\frac{{\omega }_1-x}{a_x}\right)d{\omega }_1$$ (32)

For each rect function, the centroid is given by the offset term in the numerator, while the width is given by the denominator. The left‐hand endpoints of the two rect functions are thus

$$h_{1,1}=min\left\{{\xi }_1,\frac{d{\xi }_1}{d-l}\right\}$$ (33)

$$h_{1,2}=x-\frac{a_x}{2}.$$ (34)

The first subscript identifies the case number, while the second subscript identifies the order of the rect function when reading Eq. (32) from left‐to‐right. Similarly, the right‐hand endpoints are

$$j_{1,1}=max\left\{{\xi }_1,\frac{d{\xi }_1}{d-l}\right\}$$ (35)

$$j_{1,2}=x+\frac{a_x}{2}.$$ (36)

It is useful to re‐number the rect functions based on the positioning of the left‐hand endpoints.

$$h_{1,1}^{\prime }=min\left\{h_{1,1},h_{1,2}\right\}$$ (37)

$$h_{1,2}^{\prime }=max\left\{h_{1,1},h_{1,2}\right\}$$ (38)

The primed superscript emphasizes that the ordering is now based on position in space. Under this ordering of the rect functions, the right‐hand endpoints are as follows.

$$j_{1,1}^{\prime }=\left\{\begin{array}{cc}{j}_{1,1},\hfill & h_{1,1}\le h_{1,2}\hfill \\ j_{1,2},\hfill & h_{1,1}>h_{1,2}\hfill \end{array}\right.$$ (39)

$$j_{1,2}^{\prime }=\left\{\begin{array}{cc}{j}_{1,2},\hfill & h_{1,1}\le h_{1,2}\hfill \\ j_{1,1},\hfill & h_{1,1}>h_{1,2}\hfill \end{array}\right.$$ (40)

Two mutually exclusive possibilities for the positioning of the two rect functions are now considered.

If the two rect functions do not overlap, as would occur if the right‐hand endpoint of one rect function is less than or equal to the left‐hand endpoint of the other rect function, then $j_{1,1}^{\prime }\le h_{1,2}^{\prime }$, and the integral in Eq. (32) is zero.

If the two rect functions overlap, the lower limit of overlap is the larger of the two left‐hand endpoints.

$$t_{1,1}=h_{1,2}^{\prime }$$ (41)

Similarly, the upper limit of overlap is the smaller of the two right‐hand endpoints.

$$t_{1,2}=min\{j_{1,1}^{\prime },j_{1,2}^{\prime }\}$$ (42)

The limits of overlap (t _1,1 and t _1,2) are arranged by position based on the second subscript. The integral in Eq. (32) can now be evaluated

$$P_{\mathrm{Net},1}=\left\{\begin{array}{cc}0\hfill & j_{1,1}^{\prime }\le h_{1,2}^{\prime }\hfill \\ C_{\mathrm{Net}}\text{rect}\left(\frac{y}{a_y}\right)·{\int }_{t_{1,1}}^{t_{1,2}}{e}^{-\text{sgn}\left({\xi }_1\right)·\mu \left({\omega }_1-{\xi }_1\right)\sqrt{1+{\left(\frac{d-l}{{\xi }_1}\right)}^2}}d{\omega }_1,\hfill & \text{otherwise}\hfill \end{array}\right.$$ (43)

$$=\left\{\begin{array}{cc}0,\hfill & j_{1,1}^{\prime }\le h_{1,2}^{\prime }\hfill \\ \frac{-\text{sgn}({\xi }_1)·\text{rect}\left(\frac{y}{a_y}\right)}{a_x{a}_y\left(1-e^{-\mu l\sqrt{1+{\left(\frac{{\xi }_1}{d-l}\right)}^2}}\right)}\left[e^{-\text{sgn}({\xi }_1)·\mu (t_{1,2}-{\xi }_1)\sqrt{1+{(\frac{d-l}{{\xi }_1})}^2}}-e^{-\text{sgn}({\xi }_1)·\mu (t_{1,1}-{\xi }_1)\sqrt{1+{(\frac{d-l}{{\xi }_1})}^2}}\right],\hfill & \text{otherwise}\hfill \end{array}\right.$$ (44)

which is a closed‐form solution for the PSF.

2.C.2. Case 2: ray incidence in ±y direction

In the case of ray incidence in the ±y direction. Equation (26) can be evaluated with the substitutions: ξ ₁ = 0 and ξ ₂ ≠ 0. Following the reasoning used in the transition between Eqs. (27) and (32), it can be shown that

$$\begin{array}{cc}& P_{\mathrm{Net},2}=C_{\mathrm{Net}}·\text{rect}\left(\frac{x}{a_x}\right)\hfill \\ & ·{\int }_{-\infty }^{\infty }{e}^{-\text{sgn}\left({\xi }_2\right)·\mu \left({\omega }_2-{\xi }_2\right)\sqrt{1+{\left(\frac{d-l}{{\xi }_2}\right)}^2}}\hfill \\ & ·\text{rect}\left(\frac{{\omega }_2-\frac{(2d-l){\xi }_2}{2(d-l)}}{\frac{{\xi }_{2}l}{d-l}}\right)\text{rect}\left(\frac{{\omega }_2-y}{a_y}\right)d{\omega }_2\hfill \end{array}$$ (45)

Equation (45) is in a form analogous to Eq. (32). This integral can be evaluated with the approach described in Eqs. (33), (34), (35), (36), (37), (38), (39), (40), (41), (42), (43), (44). This solution is given explicitly in the Data S1.

2.C.3. Case 3: ray incidence in an oblique direction

In the case of ray incidence in an oblique direction, Eq. (26) can be evaluated with the substitutions: ξ ₁ ≠ 0 and ξ ₂ ≠ 0. Rewriting the delta function in Eq. (26)

$$\delta \left(\frac{-{\xi }_2}{r}{\omega }_1+\frac{{\xi }_1}{r}{\omega }_2\right)=\frac{r}{|{\xi }_1|}\delta \left({\omega }_2-\frac{{\xi }_2}{{\xi }_1}{\omega }_1\right),$$ (46)

the ω ₂‐integral can be evaluated.

$$\begin{array}{cc}\hfill P_{\mathrm{Net},3}=& \frac{C_{\mathrm{Net}}r}{|{\xi }_1|}{\int }_{-\infty }^{\infty }{e}^{\frac{-\mu r\left({\omega }_1-{\xi }_1\right)}{{\xi }_1}\sqrt{1+{\left(\frac{d-l}{r}\right)}^2}}\hfill \\ & ·\text{rect}\left(\frac{{\omega }_1-\frac{(2d-l){\xi }_1}{2(d-l)}}{\frac{{\xi }_{1}l}{d-l}}\right)\text{rect}\left(\frac{{\omega }_1-x}{a_x}\right)\hfill \\ & ·\text{rect}\left(\frac{{\omega }_1-\frac{y{\xi }_1}{{\xi }_2}}{\frac{a_y{\xi }_1}{{\xi }_2}}\right)d{\omega }_1\hfill \end{array}$$ (47)

To evaluate Eq. (47) in closed form, it is necessary to determine the interval of overlap for the three rect functions in the integrand. This solution is detailed in the Data S1.

2.D. Coordinate transformation

Figure 1 shows how these equations can be applied to a breast imaging system by transforming the coordinate system. In Fig. 1, point M is a reference point in the plane z = 0; that is, the midpoint of the chest wall side of the detector. In our previous work, the focal spot coordinates (x _FS and y _FS) were measured relative to this point.^{18, 19} In DM, x _FS and y _FS should be roughly aligned with point M. The exact positioning in a physical system is sensitive to geometric imprecisions.

The equations for transforming the coordinate system between reference points O and M are as follows:

$${\xi }_1=v_1-x_{\mathrm{FS}}$$ (48)

$${\xi }_2=v_2-y_{\mathrm{FS}},$$ (49)

where v ₁ and v ₂ are the coordinates of point B (the entrance point of the ray) relative to point M as shown in Fig. 1.

3. Results

3.A. Modeling parameters

We model a next‐generation tomosynthesis (NGT) system that was constructed at the University of Pennsylvania for research use.^{20, 21, 22, 23, 24, 25} This system, which is discussed in more detail in Part 2, is capable of source motion in two directions (x and y). For the purpose of Part 1, only a 2D DM image is simulated with the acquisition parameters given in Table 1.

Table 1.

The acquisition parameters for the simulation of a DM system are summarized

Acquisition parameter	Value (mm)
Del size (a)	0.085
d	650.0
Se thickness (l)	0.200
x FS	0.5a (0.0425)
y FS	0

Similar to previous work^{18, 19}, the focal spot is treated as a point in the plane of the chest wall (y _FS = 0), as shown in Fig. 3 by the positioning of the origin (O) along this side of the detector. This differs from the positioning of the origin in Fig. 1, which is more generalized for an arbitrary coordinate for the focal spot.

Figure 3.

(a) For simulation of a DM system, the focal spot is treated as a point in the plane of the chest wall, and hence the origin (O) is positioned along this side of the detector. Each point on the circle centered on point O is characterized by the same incidence angle (θ), but by a different azimuthal angle (Γ) for the ray; here, Γ = 90°. (b) Four examples are given (0°, 45°, 90°, and 135°) to illustrate the polar angle, α.

The NGT system was built with the AXS‐2430 detector described previously. The del dimensions are 0.085 mm × 0.085 mm, and the active area is 304.64 mm × 239.36 mm. Each del is labeled with integer indices m _x and m _y . The coordinates of the centroid of an arbitrary del are

$$v_1=\left(m_x-1792-1/2\right)a_x$$ (50)

$$v_2=\left(m_y-1/2\right)a_y,$$ (51)

where m _x varies from 1 to 3584 and m _y varies from 1 to 2816.

Since there is an even number of dels along the chest wall, point M is at the interface between two dels. For the purpose of this simulation, we assume that x _FS is aligned with the centroid of the del with an x index of 1793; that is, x _FS is displaced from point M by half of a pixel (Table 1). This positioning is chosen for convenience so that the centroids of the dels with this x index are aligned with the azimuthal angle Γ = 90° (Fig. 3), which is considered in Section 3.B.1.

3.B. Calculation of the net PSF

3.B.1. Ray Incidence along 90° azimuthal angle

In Fig. 4(b), the net PSF is calculated at 20 keV at a position with 15° incidence; that is, at the coordinates m _x  = 1793 and m _y  = 2049. This position is aligned with the azimuthal angle Γ = 90° (Fig. 3). The origin of the surface plot is matched to this position. Due to the obliquity effect, the surface is not a perfect 2D rect function [Fig. 4(a)], like the PSF for detector sampling (P _TFT).

Figure 4.

The PSF is illustrated by surface plots at 20 keV. (a) The PSF for del sampling (P _TFT) is a 2D rect function with dimensions 0.085 mm × 0.085 mm. (b) The net PSF is shown at a position with 15° incidence and a 90° azimuthal angle. The cross sections are rect functions along the x direction (perpendicular to the ray incidence direction), but are blurred along the y direction (parallel with the ray incidence direction). [Color figure can be viewed at wileyonlinelibrary.com]

In Fig. 5, cross sections through this surface plot are calculated along various polar angles (α) using the transformation

$$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right)+\left(\begin{array}{cc}cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{array}\right)\left(\begin{array}{c}{x}^{″}\\ y^{″}\end{array}\right).$$ (52)

Figure 5.

Cross sections of the net PSF are shown at a position with 15° incidence and a 90° azimuthal angle. (a) The signal is a rect function along the 0° polar angle (perpendicular to the ray incidence direction). (b) Along the 45° or 135° polar angle, the net PSF is blurred over a 0.120 mm length, matching the diagonal of the del. (c) The net PSF is broader than the width of the del (0.085 mm) along a 90° polar angle (the ray incidence direction). [Color figure can be viewed at wileyonlinelibrary.com]

Figure 3(a) illustrates the rotated coordinates x′′ and y′′. Also, Fig. 3(b) shows four polar angles (α = 0°, 45°, 90°, and 135°), which are used as examples throughout this paper. In Fig. 5, the cross sections align with the centroid of the del; thus, y′′ = 0.

If the polar angle is perpendicular to the ray incidence direction (α = 0°), Fig. 5(a) shows that the cross section is a 2D rect function with 0.085 mm width. This width matches the del size. Conversely, if the polar angle is aligned with the ray incidence direction (α = 90°), the width of the net PSF is broadened by 63.2% [Fig. 5(c)]. This broadening is an illustration of the obliquity effect.

Measurements can also be made along 45° or 135° polar angles, which lie along the diagonal of the del. These two cross sections are equivalent, as shown in Fig. 5(b). The width of this cross section is 0.120 mm; that is, $\sqrt{2}$ times larger than 0.085 mm.

3.B.2. Ray incidence along 45° azimuthal angle

In Fig. 3, a circle centered on point O can be used to identify positions with the same incidence angle but different azimuthal angles for the ray incidence direction. The position corresponding to a 45° azimuthal angle (Γ) is illustrated in Fig. 6 (point C). At 15° incidence, the del that is most closely aligned with this azimuthal angle has the coordinates m _x  = 3241 and m _y  = 1449. Figure 7 shows cross sections of the net PSF along various polar angles, similar to Fig. 5.

Figure 6.

Varying the position around the circle centered on point O changes the azimuthal angle (Γ); here, Γ = 45°.

Figure 7.

Cross sections of the net PSF are shown at a position with the same incidence angle as Fig. 5 but a different azimuthal angle (Γ = 45°). The cross sections differ from those in Fig. 5 at similar polar angles; this illustrates the non‐stationarity of the net PSF. [Color figure can be viewed at wileyonlinelibrary.com]

In Section 3.B.1., we found that the net PSF is a rect function along the polar angle perpendicular to the ray incidence direction. This result is no longer the case, as shown in Fig. 7(c) at a 135° polar angle. This finding illustrates the non‐stationarity of the net PSF.

Figure 7 indicates that the net PSF is 0.174 mm wide along the ray incidence direction [α = 45°, Fig. 7(b)], and is 0.123 mm wide along a 45° angle relative to the ray incidence direction [α = 0° or 90°, Fig. 7(a)]. These cross sections are broader than the reference rect functions. Also, they do not match the cross sections in Section 3.B.1. at similar polar angles, further demonstrating the non‐stationarity of the net PSF.

3.B.3. Effect of x‐ray energy

Although low energies are typically used in breast imaging to maximize subject contrast,^{26, 27} there are applications that require higher energy; for example, contrast‐enhanced digital mammography (CE‐DM) and contrast‐enhanced digital breast tomosynthesis (CE‐DBT).^{28, 29, 30, 31, 32, 33, 34, 35, 36, 37} In CE‐DM and CE‐DBT, an iodinated contrast agent is used to quantify blood flow to a tumor. X‐ray images are acquired at energies above and below the K edge of iodine (33.2 keV), and a weighted logarithmic subtraction is used to measure perfusion.

For simulation of CE‐DM, the PSF at 40 keV is also shown in Figs. 5 and 7. The PSF deviates more strongly from the reference rect function at 40 keV (μ = 3.02 mm⁻¹) than at 20 keV (μ = 20.2 mm⁻¹). These results illustrate that there is loss of resolution at high energy; this is expected, since the x‐ray beam is attenuated less quickly as it passes through the x‐ray converter.

3.B.4. Effect of incidence angle

Figure 8 considers four incidence angles (θ = 0°, 7.5°, 15.0°, and 22.5°) to illustrate the effect on the PSF. For the purpose of this figure, the 90° azimuthal angle (Γ) is considered (Fig. 3), and the polar angle (α) is aligned with this direction (Γ = α = 90°). The x‐ray energy is taken to be 20 keV.

Figure 8.

The net PSF is broadened with increasing obliquity. This figure assumes that the x‐ray energy is 20 keV, and that the PSF is measured along the polar angle aligned with the ray incidence direction (Γ = α = 90°). [Color figure can be viewed at wileyonlinelibrary.com]

Figure 8 demonstrates that increasing the incidence angle gives rise to a broader PSF. This result is expected; the x‐ray beam projects onto the exit surface with the length l tan θ (line segment $\overline{\text{QC}}$ in Fig. 2). Since tan θ is an increasing function, the signal is spread across a broader length at higher obliquity.

4. Discussion

In previous work, the PSF of an a‐Se detector was calculated from first principles, assuming that the incidence angle (θ) was given and that the ray incidence direction was the x direction.¹¹ By contrast, this paper determines how the incidence angle and the ray incidence direction vary with position based on the divergence of the x‐ray beam. This allows the non‐stationarity of the PSF to be demonstrated. In addition, this paper includes digitization in the model, unlike previous work.

We analyze cross sections through the PSF at various polar angles. Oblique incidence blurs the shape of the PSF relative to a reference rect function matching the del size. The resolution loss is more pronounced at high energy, since there is less attenuation in the x‐ray converter. In addition, the PSF is blurred more strongly at positions with higher obliquity; for example, at positions with increasing distance (y) from the chest wall in DM. Although these results were illustrated with the AXS‐2430 detector, similar results are expected in other a‐Se detectors.

This paper has applications in simulating a‐Se detectors in virtual clinical trials (VCTs), which can be used to evaluate new imaging technologies at low cost without requiring human subjects.³⁸ Since P _Net was evaluated with a closed‐form solution, this paper ensures that the detector signal calculation in a VCT is computationally efficient.

Some of the limitations of this paper and directions for future modeling are now discussed. In future work, the three‐dimensional (3D) PSF of a DBT reconstruction should be calculated from the 2D PSFs of the projection images. In some DBT systems, the detector rotates during the scan.^{18, 19, 39, 40, 41} Detector rotation can be modeled with a different coordinate transformation for point M (Section 2.D.). In addition, the 3D PSF should model the reconstruction filter.⁴²

In this paper, it is assumed that the electron and hole migrate in perfect orthogonal paths to opposite sides of the a‐Se x‐ray converter (Fig. 2). However, future work should consider additional x‐ray interactions such as K fluorescence and Compton scatter, which can act as sources of blurring according to the work of Que and Rowlands.¹¹ Also, the focal spot can introduce blurring⁴³ that was not modeled in this paper, since the x‐ray source was assumed to be point‐like.

This paper presumes that the collection efficiency of selenium is 100%. However, future work should model how the collection efficiency is dependent on the depth (z) of the interaction.⁸ Also, while this paper presumes that the entire area of each del is sensitive to x‐rays, future work should model the fill factor.

5. Conclusion

This paper is an extension of previous work modeling the PSF for oblique x‐ray incidence in a‐Se detectors.¹¹ There are two differences between this paper and previous work. First, this paper develops a model of the non‐stationarity of the PSF based on the divergent x‐ray beam geometry. Second, while previous work calculated the PSF for the x‐ray converter, this paper provides a complementary derivation for a digital system.

In this paper, the solution for the PSF is determined in closed form. This formula is a tool that can be used in future simulations of a‐Se detectors (e.g., in VCTs). One advantage of a closed‐form solution is that computation time is minimized relative to numerical methods.

Conflicts of interest

Andrew D. A. Maidment is a scientific advisor to RTT, and his spouse is an employee and shareholder of RTT.

Appendix: Nomenclature

Symbol	Meaning
*2	Two‐dimensional convolution operator
α	Polar angle (defined in Figs. 3 and 6)
Γ	Azimuthal angle of ray (Fig. 1)
δ	Delta function
θ	Incidence angle (Figs. 1, 2).
μ	Linear attenuation coefficient of selenium
ξ 1, ξ 2	Coordinates of point B relative to point S (Fig. 1)
ω 1, ω 2	Dummy variables in the evaluation of a convolution [Eq. (19)].
a x , a y	Del dimensions; if a x  = a y , the dimension is abbreviated a
C I	Normalization term for P I [Eq. (18)]
C Net	Normalization term for P Net [Eq. (24)]
CE‐DBT	Contrast‐enhanced digital breast tomosynthesis
CE‐DM	Contrast‐enhanced digital mammography.
d	Distance between focal spot and origin in Fig. 1
DBT	Digital breast tomosynthesis
Del	Detector element
DM	Digital mammography
FS	Focal spot
h i,k	Left‐hand endpoint of rect function [i = case number, k = ordering based on Eq. (32)]
$h_{i,k}^{\prime }$	Left‐hand endpoint of rect function [i = case number, k = ordering by positioning of left‐hand endpoints]
j i,k	Right‐hand endpoint of rect function [i = case number, k = ordering based on Eq. (32)]
$j_{i,k}^{\prime }$	Right‐hand endpoint of rect function [i = case number, k = ordering by positioning of left‐hand endpoints]
l	Selenium thickness
m x , m y	Del indices [Eqs. (50), (51)]
MTF	Modulation transfer function
N	Relative number of x‐ray quanta at depth z [Eq. (15)]
P I	PSF associated with arbitrary ionization point in x‐ray converter
P Net	Net PSF (additional subscripts denote case number)
P TFT	PSF of TFT array
P z	Depth‐dependent PSF (combined effect of P I and P TFT)
PSF	Point spread function
r	Distance between points B and S in Fig. 1
t i,k	Lower (k = 1) or upper (k = 2) limit of overlap of rect functions (i = case number)
TFT	Thin‐film transistor
v 1, v 2	Coordinate transformation for point B [Eqs. (48), (49)]
$v_1^{\prime }$	Position (relative to point Q) measured along x ′ direction in Fig. 2
$v_2^{\prime }$	Position (relative to point Q) measured along y ′ direction
VCT	Virtual clinical trial
x ′	Position along ray incidence direction
x ′′	Position measured along polar angle α in Figs. 3 and 6
x FS, y FS	Focal spot coordinates (Fig. 1)
y ′	Position perpendicular to ray incidence direction
y ′′	Position measured perpendicular to polar angle α in Figs. 3 and 6
Z	Atomic number

Supporting information

Data S1. Closed‐form solutions for the net PSF in Case 2 (Section 2.C.2.) and Case 3 (Section 2.C.3.) are derived.