A comprehensive model for x-ray projection imaging system efficiency and image quality characterization in the presence of scattered radiation

The analytical characterization of the spatial distribution of scatter via different scatter point spread functions (PSFs) for medical x-ray imaging systems was initiated by Seibert et al (1984) and Boone et al (1986). The spatial distribution of x-ray radiation for a (primary) point x-ray source incident on the object can be modelled in the image plane as a beam point spread function (PSF_b). Considering a homogeneous (shift invariant) x-ray beam, the beam PSF is independent of the position in the image plane, and the x-ray beam can be computed as the sum of the beam PSF of each point. Functions fitted to scatter PSFs obtained by Monte Carlo simulations were used to assess the PSF of beams with scatter and grid out (equation (1) and figure 2). Due to radial symmetry, the 2D PSF_b were considered in polar coordinates

$$\begin{align} \displaystyle {\rm PS{{F}}_{b}}(r)={{K}_{1}}\cdot \left(1-{\rm SF} \right)\cdot \frac{\delta (r)}{r}+{{K}_{2}}\cdot {\rm SF}\cdot \left(\frac{a}{{{\left(1+{{\left({r}/{{{k}_{1}}}\; \right)}^{2}} \right)}^{3/2}}}+\frac{1-a}{{{\left(1+{{\left({r}/{{{k}_{2}}}\; \right)}^{2}} \right)}^{3/2}}} \right).\nonumber \end{align} \tag{ 1 }$$

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Where r is the radial distance from the origin, K₁ and K₂ are normalization constants, the Dirac function δ(r)/r represents the primary beam while the second term is the scatter PSF, SF is the scatter fraction in the beam, and the coefficients a, k₁ and k₂ determine the shape of the scatter distribution. The optical transfer function of the x-ray beam (OTF_b) is the Fourier transform of PSF_b. We define the x-ray beam transfer function (BTF) as the modulus of OTF_b normalized to 1.0 at the zero frequency. The derivation of OTF_b and BTF is given in appendix A.

$$\begin{align} \displaystyle {\rm BTF}(\,f )=\left(1-{\rm SF} \right)+{\rm SF}\cdot \frac{ak_{1}^{2}\exp \left(-2\pi {{k}_{1}}f \right)+(1-a )k_{2}^{2}\exp \left(-2\pi {{k}_{2}}f \right)}{ak_{1}^{2}+(1-a )k_{2}^{2}}\nonumber \end{align} \tag{ 2 }$$

The system MTF (MTF_sys) measured with scatter in the image plane is the product of the presampling MTF (MTF), i.e. the detector and focal spot blurs, and the BTF at the ASD output (BTF_out).

$$\begin{align} \displaystyle \begin{array}{@{}l@{}}{\rm MT}{{{\rm F}}_{{\rm sys}}}(\,f )={\rm BT}{{{\rm F}}_{{\rm out}}}(\,f )\cdot {\rm MTF}(\,f ): \nonumber \\ {\rm MT}{{{\rm F}}_{{\rm sys}}}(\,f )=\left(1-{\rm S}{{{\rm F}}_{{\rm out}}}+{\rm S}{{{\rm F}}_{{\rm out}}}\cdot \frac{ak_{1}^{2}\exp \left(-2\pi {{k}_{1}}f \right)+(1-a )k_{2}^{2}\exp \left(-2\pi {{k}_{2}}f \right)}{ak_{1}^{2}+(1-a )k_{2}^{2}} \right)\cdot {\rm MTF}(\,f ) \nonumber \\ \end{array}\nonumber \end{align} \tag{ 3a }$$

$$\begin{align} \displaystyle {\rm MT}{{{\rm F}}_{{\rm sys}}}(\,f )\cong \left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\cdot {\rm MTF}(\,f )\,{\rm for}\,f>1/k=\max \left(1/{{k}_{1}};\,1/{{k}_{2}} \right)\nonumber \end{align} \tag{ 3b }$$

where ${\rm S}{{{\rm F}}_{{\rm out}}}={{{S}_{{\rm out}}}}/{\left({{P}_{{\rm out}}}+{{S}_{{\rm out}}} \right)}\;$ is the scatter fraction in the output image (or at the ASD output). The scatter term of BTF_out drops quickly to zero at very low frequency, between 0 and 1/k. As already shown in previous studies (Salvagnini et al 2012, 2013), the multiplication of presampling MTF by (1  −  SF_out) gives the system MTF for all frequencies above 1/k. Scattered radiation therefore decreases the signal by (1  −  SF_out) for all spatial frequencies important for object/target detection within an image and acts practically as a factor that reduces contrast without modifying the intrinsic sharpness of the imaging system. For $f>{1}/{k}\;$, the signal in the output image contains only primary photons:

$$\begin{align} \displaystyle \left({{S}_{{\rm out}}}+{{P}_{{\rm out}}} \right)\cdot {\rm MT}{{{\rm F}}_{{\rm sys}}}(\,f )\cong \left({{S}_{{\rm out}}}+{{P}_{{\rm out}}} \right)\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\cdot {\rm MTF}(\,f )={{P}_{{\rm out}}}\cdot {\rm MTF}(\,f )\nonumber \end{align} \tag{ 4 }$$

where the relation $\frac{{{P}_{{\rm out}}}}{{{P}_{{\rm out}}}+~{{S}_{{\rm out}}}}=1-{\rm S}{{{\rm F}}_{{\rm out}}}~$ has been used. Equation (4) shows that scatter contributes to very low frequency signal only within a narrow frequency bandwidth, decreasingly between 0 and 1/k.

The input noise equivalent quanta (NEQ_in) describes the signal-to-noise ratio squared (image quality) at the ASD entrance.

$$\begin{align} \displaystyle {\rm NE}{{{\rm Q}}_{{\rm in}}}(\,f )=\frac{{{\left({{P}_{{\rm in}}}+{{S}_{{\rm in}}} \right)}^{2}}\cdot {\rm BTF}_{{\rm in}}^{2}(\,f )}{{\rm NP}{{{\rm S}}_{{\rm in}}}}=\frac{{\rm BTF}_{{\rm in}}^{2}(\,f )}{{\rm NNP}{{{\rm S}}_{{\rm in}}}}=\left({{{\tilde{P}}}_{{\rm in}}}+{{{\tilde{S}}}_{{\rm in}}} \right)\cdot {\rm BTF}_{{\rm in}}^{2}(\,f )\nonumber \end{align} \tag{ 5 }$$

${{P}_{{\rm in}}}+{{S}_{{\rm in}}}$ is the input air kerma (in µGy) at the ASD entrance. Quantum noise in the x-ray beam is made of primary and scattered photons. NPS_in is the quantum noise power spectrum at the ASD input expressed in µGy² · mm² (IEC 2015), and NNPS_in is the NPS_in normalized by the mean signal squared: ${\rm NNP}{{{\rm S}}_{{\rm in}}}={{\rm NP}{{{\rm S}}_{{\rm in}}}}/{{{\left({{P}_{{\rm in}}}+{{S}_{{\rm in}}} \right)}^{2}}}\;$. Quantum NNPS_in is expressed in mm² and follows Poissonian statistics. It is therefore a white noise whose amplitude is directly linked to ${{\tilde{P}}_{{\rm in}}}+{{\tilde{S}}_{{\rm in}}}$, the photon fluence (photons mm⁻²) at the ASD input: ${\rm NNP}{{{\rm S}}_{{\rm in}}}={1}/{\left({{{\tilde{P}}}_{{\rm in}}}+{{{\tilde{S}}}_{{\rm in}}} \right)}\;$.

The model we propose considers the ASD as a photon selector characterized by primary and scatter transmissions, without introducing correlations in noise (Chan et al 1990). Hence, the NPS at the ASD output (NPS_out) is white and determined by the photon fluence (${{\tilde{P}}_{{\rm out}}}+{{\tilde{S}}_{{\rm out}}}$ photons mm⁻²) at the ASD output (and in the output image). The NEQ at the ASD output is given by equation (6):

$$\begin{align} \displaystyle {\rm NE}{{{\rm Q}}_{{\rm out}}}(\,f )=\frac{{{\left({{P}_{{\rm out}}}+{{S}_{{\rm out}}} \right)}^{2}}\cdot {\rm BTF}_{{\rm out}}^{2}(\,f )}{{\rm NP}{{{\rm S}}_{{\rm out}}}}=\frac{{\rm BTF}_{{\rm out}}^{2}(\,f )}{{\rm NNP}{{{\rm S}}_{{\rm out}}}}=\left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)\cdot {\rm BTF}_{{\rm out}}^{2}(\,f )\nonumber \end{align} \tag{ 6 }$$

The MTF and NPS shapes are modified by the detector, and the NEQ in the image is spatial-frequency dependent:

$$\begin{align} \displaystyle {\rm NEQ}(\,f )=\frac{{{\left({{P}_{{\rm out}}}+{{S}_{{\rm out}}} \right)}^{2}}\cdot {\rm MTF}_{{\rm sys}}^{2}(\,f )}{{\rm NPS}(\,f )}=\frac{{\rm MTF}_{{\rm sys}}^{2}(\,f )}{{\rm NNPS}(\,f )}\nonumber \end{align} \tag{ 7 }$$

These general expressions for NEQs can be simplified for all spatial frequencies important for object/target detection ($f>{1}/{k}\;$).

Particular case for $f>{1}/{k}\;$:

Using ${\rm BTF}\cong 1-{\rm SF}$ (equation (3b)) and $\frac{{\tilde{P}}}{\tilde{P}+\tilde{S}}=1-{\rm SF}$, the NEQs can be simplified to:

$$\begin{align} \displaystyle {\rm NE}{{{\rm Q}}_{{\rm in}}}\cong \left({{{\tilde{P}}}_{{\rm in}}}+{{{\tilde{S}}}_{{\rm in}}} \right){{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{2}}={{\tilde{P}}_{{\rm in}}}\cdot \left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)\nonumber \end{align} \tag{ 8 }$$

$$\begin{align} \displaystyle {\rm NE}{{{\rm Q}}_{{\rm out}}}\cong \left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right){{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}={{\tilde{P}}_{{\rm out}}}\cdot \left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\nonumber \end{align} \tag{ 9 }$$

$$\begin{align} \displaystyle {\rm NEQ}(\,f )\cong {{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}\cdot \frac{{\rm MT}{{{\rm F}}^{2}}(\,f )}{{\rm NNPS}(\,f )}\nonumber \end{align} \tag{ 10 }$$

The fraction of scattered radiation in the output image (SF_out) reduces the NEQ by ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}$. This loss in NEQ due to scatter can be offset by an increase in DAK of ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-2}}$, termed the scatter compensation factor in Siewerdsen and Jaffray (2000). The NEQ in the absence of scatter is a particular case where SF  =  0. The notation NEQ in our study is therefore used irrespective of the presence or absence of scatter.

2.3.1. System DQE (DQE_sys).

Cascaded linear analysis has been successfully used to provide comprehensive descriptions of detector DQE by modelling the signal and noise transfer as successive gain and blurring stages (Rabbani et al 1987, Siewerdsen et al 1997, Zhao and Rowlands 1997, Cunningham and Shaw 1999, Cunningham et al 2004, Kim 2006, Hunt et al 2007, Monnin et al 2016). In this study, cascaded system analysis is applied to a system composed of detector and ASD (grid, slit or air gap) (figure 1). The DQE describes the ability of the imaging device to preserve the SNR present in the radiation field in the resulting image (Shaw 1978, IEC 2015). We therefore define the DQE for the different elements of the imaging system as the efficiency of SNR² (NEQ) transfer through these elements. For the ASD, we define T_p, T_s and T_t as the primary, scatter and total transmissions of the ASD, respectively:

$$\begin{align} \displaystyle {{T}_{{\rm p}}}=\frac{{{{\tilde{P}}}_{{\rm out}}}}{{{{\tilde{P}}}_{{\rm in}}}}=\frac{{{P}_{{\rm out}}}}{{{P}_{{\rm in}}}}\nonumber \end{align} \tag{ 11a }$$

$$\begin{align} \displaystyle {{T}_{{\rm s}}}=\frac{{{{\tilde{S}}}_{{\rm out}}}}{{{{\tilde{S}}}_{{\rm in}}}}=\frac{{{S}_{{\rm out}}}}{{{S}_{{\rm in}}}}\nonumber \end{align} \tag{ 11b }$$

$$\begin{align} \displaystyle {{T}_{{\rm t}}}=\frac{{{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}}}{{{{\tilde{P}}}_{{\rm in}}}+{{{\tilde{S}}}_{{\rm in}}}}=\frac{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}{{{P}_{{\rm in}}}+{{S}_{{\rm in}}}}={{T}_{{\rm p}}}-\left({{T}_{{\rm p}}}-{{T}_{{\rm s}}} \right)\cdot {\rm S}{{{\rm F}}_{{\rm in}}}\nonumber \end{align} \tag{ 11c }$$

The ASD DQE (DQE_ASD) is defined as:

$$\begin{align} \displaystyle {\rm DQ}{{{\rm E}}_{{\rm ASD}}}(\,f )=\frac{{\rm NE}{{{\rm Q}}_{{\rm out}}}(\,f )}{{\rm NE}{{{\rm Q}}_{{\rm in}}}(\,f )}=\frac{{{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}}}{{{{\tilde{P}}}_{{\rm in}}}+{{{\tilde{S}}}_{{\rm in}}}}\cdot \frac{{\rm BTF}_{{\rm out}}^{2}(\,f )}{{\rm BTF}_{{\rm in}}^{2}(\,f )}={{T}_{{\rm t}}}\cdot \frac{{\rm BTF}_{{\rm out}}^{2}(\,f )}{{\rm BTF}_{{\rm in}}^{2}(\,f )}\nonumber \end{align} \tag{ 12 }$$

DQE_ASD describes the variation in NEQ due to the use of the ASD, without changing patient dose. It depends on the SNR balance between the primary absorption and scatter rejection of the ASD; this is greater than 1.0 if the ASD increases the SNR and lower than 1.0 otherwise. In the absence of an ASD, DQE_ASD is neutral, i.e. ${\rm DQ}{{{\rm E}}_{{\rm ASD}}}=1$.

The detector DQE (DQE_d) is defined as the ratio between NEQ(f) in the output image and NEQ_out(f) at the detector entrance (equations (6) and (7)):

$$\begin{align} \displaystyle {\rm DQ}{{{\rm E}}_{{\rm d}}}(\,f )=\frac{{\rm NEQ}(\,f )}{{\rm NE}{{{\rm Q}}_{{\rm out}}}(\,f )}=\frac{{\rm MTF}_{{\rm sys}}^{2}(\,f )}{{\rm BTF}_{{\rm out}}^{2}(\,f )\cdot {\rm NNPS}(\,f )\cdot \left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}\nonumber \end{align} \tag{ 13 }$$

We define system DQE (DQE_sys) as the efficiency of SNR² (NEQ) transfer through the imaging system, given in the cascaded model by the product of DQE_ASD and DQE_d:

$$\begin{align} \displaystyle {\rm DQ}{{{\rm E}}_{{\rm sys}}}(\,f )={\rm DQ}{{{\rm E}}_{{\rm ASD}}}(\,f )\cdot {\rm DQ}{{{\rm E}}_{{\rm d}}}(\,f )=\frac{{\rm MTF}_{{\rm sys}}^{2}(\,f )}{{\rm BTF}_{{\rm in}}^{2}(\,f )\cdot {\rm NNPS}(\,f )\cdot \left({{{\tilde{P}}}_{{\rm in}}}+{{{\tilde{S}}}_{{\rm in}}} \right)}\nonumber \end{align} \tag{ 14 }$$

These general expressions for DQEs can be simplified for all spatial frequencies important for object/target detection ($f>{1}/{k}\;$).

Particular case for $f>{1}/{k}\;$:

Using equations (8), (9), (11a) and (11c) and again the relation $\frac{{\tilde{P}}}{\tilde{P}+\tilde{S}}=1-{\rm SF}$, DQE_ASD can be expressed as a function of T_p and T_t:

$$\begin{align} \displaystyle {\rm DQ}{{{\rm E}}_{{\rm ASD}}}=\frac{{\rm NE}{{{\rm Q}}_{{\rm out}}}}{{\rm NE}{{{\rm Q}}_{{\rm in}}}}=\frac{{{{\tilde{P}}}_{{\rm out}}}}{{{{\tilde{P}}}_{{\rm in}}}}\cdot \frac{1-{\rm S}{{{\rm F}}_{{\rm out}}}}{1-{\rm S}{{{\rm F}}_{{\rm in}}}}=\frac{T_{{\rm p}}^{2}}{{{T}_{{\rm t}}}}\nonumber \end{align} \tag{ 15a }$$

The ASD DQE can also be expressed as a function of the ASD selectivity ($\Sigma ={{{T}_{{\rm p}}}}/{{{T}_{{\rm s}}}}\;$):

$$\begin{align} \displaystyle {\rm DQ{{E}_{ASD}}}=\frac{{{T}_{p}}\cdot \Sigma }{\Sigma -(\Sigma -1 )\cdot {\rm S{{F}_{in}}}}\nonumber \end{align} \tag{ 15b }$$

The primary transmission (numerator) is linked to the change in (primary) signal and the denominator to the noise level. The square root of this factor was introduced by Neitzel (1992) as the ratio of SNR of images acquired with and without grid, named the SNR improvement factor (K_SNR). This is equivalent to the factors K_SDNR or SIF used in later studies (Shen et al 2006, Carton et al 2009, Chen et al 2015) and used in IEC 60627 (IEC 2013) to characterize the ability of the grid to improve SDNR or SNR. A perfect ASD with (T_p;T_s)  =  (1;0) would lead to the highest ${\rm DQ}{{{\rm E}}_{{\rm ASD}}}={{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{-1}}$.

Using equation (3b), the detector DQE and system DQE can be expressed as follows:

$$\begin{align} \displaystyle {\rm DQ}{{{\rm E}}_{{\rm d}}}(\,f )=\frac{{\rm MTF}_{{\rm sys}}^{2}(\,f )}{{\rm NNPS}(\,f )\cdot {{{\tilde{P}}}_{{\rm out}}}\cdot \left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}=\frac{{\rm MT}{{{\rm F}}^{2}}(\,f )}{{\rm NNPS}(\,f )\cdot \left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}\nonumber \end{align} \tag{ 16 }$$

$$\begin{align} \displaystyle {\rm DQ}{{{\rm E}}_{{\rm sys}}}(\,f )=\frac{{\rm MTF}_{{\rm sys}}^{2}(\,f )}{{\rm NNPS}(\,f )\cdot {{{\tilde{P}}}_{{\rm in}}}\cdot \left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}=\frac{T_{{\rm p}}^{2}}{{{T}_{{\rm t}}}}\cdot \frac{{\rm MT}{{{\rm F}}^{2}}(\,f )}{{\rm NNPS}(\,f )\cdot \left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}\nonumber \end{align} \tag{ 17 }$$

In the absence of an ASD, DQE_sys reverts to DQE_d. Therefore, DQE_sys is higher or lower than DQE_d, depending on the ability of the ASD to modify the SNR.

2.3.2. Effective DQE (eDQE).

Following from its definition (Samei et al 2004, 2009), the eDQE for a frequency f at the detector plane is given by:

$$\begin{align} \displaystyle {\rm eDQE}(\,f )=\frac{{{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}\cdot {\rm MT}{{{\rm F}}^{2}}(\,f )}{{\rm NNPS}(\,f )\cdot {{{\tilde{P}}}_{{\rm in}}}}={{T}_{{\rm p}}}\cdot \left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\cdot {\rm DQ}{{{\rm E}}_{{\rm d}}}(\,f )\nonumber \end{align} \tag{ 18a }$$

The eDQE is the NEQ normalized by ${{\tilde{P}}_{{\rm in}}}$, the ${\rm NE}{{{\rm Q}}_{{\rm in}}}$ for an ideal beam with only primary photons:

$$\begin{align} \displaystyle {\rm eDQE}(\,f )=\frac{{\rm NEQ}(\,f )}{{{{\tilde{P}}}_{{\rm in}}}}\nonumber \end{align} \tag{ 18b }$$

This gives eDQE equal to 1.0 for a perfect system composed of a perfect detector (that does not add noise) and a perfect ASD defined by (T_p; T_s)  =  (1; 0). The eDQE can be only lower than DQE_d for a real ASD.

2.3.3. Generalized DQE (gDQE).

Turning to the gDQE proposed by Kyprianou et al (2004, 2005a), signal is taken to be the gMTF, a parameter that characterizes blurring from the detector, focal spot and scatter. The gMTF is equivalent to the system MTF (MTF_sys), and using equation (3b) gives:

$$\begin{align} \displaystyle {\rm gDQE}(\,f )=\frac{{\rm gMT}{{{\rm F}}^{2}}(\,f )}{{\rm NNPS}(\,f )\cdot \left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}\cong \frac{{{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}\cdot {\rm MT}{{{\rm F}}^{2}}(\,f )}{{\rm NNPS}(\,f )\cdot \left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}={{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}\cdot {\rm DQ}{{{\rm E}}_{{\rm d}}}(\,f )\nonumber \end{align} \tag{ 19a }$$

The gDQE is the NEQ normalized by the total photon fluence at the detector, and is therefore different from the eDQE:

$$\begin{align} \displaystyle {\rm gDQE}(\,f )\cong \frac{{\rm NEQ}(\,f )}{{{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}}}\nonumber \end{align} \tag{ 19b }$$

Like eDQE, gDQE reverts to DQE_d for a perfect ASD and is equal to 1.0 for a perfect system, but scales differently compared to eDQE.

2.3.4. Links between DQE_sys, eDQE and gDQE (table 1 and figure 3).

Useful links between the different efficiency metrics are given in table 1 for the case where $f>{1}/{k}\;$ (validity range of equation (3b)). Compared to the ideal input beam without scatter considered in Wagner et al (1980) and Samei et al (2009) for eDQE, DQE_sys uses the real scatter fraction at the ASD entrance (SF_in). The eDQE and gDQE are proportional to the detector and ASD DQEs, but decrease with SF_in, mixing efficiency (DQE) with image quality (NEQ). The system DQE depends only on the ASD and detector DQEs, and is therefore a pure imaging system efficiency metric. The DQE_sys follows the ASD efficiency (DQE_ASD) and increases with SF_in. The eDQE and gDQE can be only lower than DQE_d, whereas DQE_sys can be higher or lower than DQE_d, depending on DQE_ASD. This is clearly illustrated when plotting the metrics as a function of SF_in for a theoretical ASD with T_p  =  0.7 and T_s  =  0.2 (figure 3).

Table 1. Conversion factors between the efficiency metrics.

To convert
To	${\rm DQ}{{{\rm E}}_{{\rm d}}}$	${\rm DQ}{{{\rm E}}_{{\rm sys}}}$	${\rm eDQE}$	${\rm gDQE}$
From
${\rm NEQ}$	$\tilde{P}_{{\rm out}}^{-1}\cdot {{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-1}}$	$\tilde{P}_{{\rm in}}^{-1}\cdot {{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{-1}}$	$\tilde{P}_{{\rm in}}^{-1}$	${{\left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}^{-1}}$
${\rm DQ}{{{\rm E}}_{{\rm d}}}$	1	${T_{{\rm p}}^{2}}/{{{T}_{{\rm t}}}}\;$	${T_{{\rm p}}^{2}\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}/{{{T}_{{\rm t}}}}\;$	${T_{{\rm p}}^{2}{{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{2}}}/{T_{{\rm t}}^{2}}\;$
${\rm DQ}{{{\rm E}}_{{\rm sys}}}$	${{{T}_{{\rm t}}}}/{T_{{\rm p}}^{2}}\;$	1	$\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)$	${{{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{2}}}/{{{T}_{{\rm t}}}}\;$
${\rm eDQE}$	${{{T}_{{\rm t}}}}/{\left(T_{{\rm p}}^{2}\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right) \right)}\;$	${{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{-1}}$	1	${\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}/{{{T}_{{\rm t}}}}\;$
${\rm gDQE}$	${T_{{\rm t}}^{2}}/{\left(T_{{\rm p}}^{2}{{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{2}} \right)}\;$	${{{T}_{{\rm t}}}}/{{{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}^{2}}}\;$	${{{T}_{{\rm t}}}}/{\left(1-{\rm S}{{{\rm F}}_{{\rm in}}} \right)}\;$	1

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Considering a thin object of thickness T and linear attenuation coefficient µ in a homogeneous material, the object contrast in the image ΔP_out, expressed as a difference in primary signal between the object and the background, will be proportional to the primary DAK in the image background (P_out).

$$\begin{align} \displaystyle \Delta {{P}_{{\rm out}}}={{P}_{{\rm out}}}\left(1-\exp (-\mu T ) \right)\cong \mu T{{P}_{{\rm out}}}\nonumber \end{align} \tag{ 20 }$$

For a quantum noise limited system, the standard deviation of pixel values is proportional to the square root of the DAK, $\sigma ={{k}_{q}}\sqrt{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}$, where k_q is a gain factor. Under these assumptions, SDNR increases with the square root of the DAK but decreases with the scatter fraction in the image by the factor (1  −  SF_out).

$$\begin{align} \displaystyle {\rm SDNR}=\frac{\Delta {{P}_{{\rm out}}}}{\sigma }\cong \frac{\mu T{{P}_{{\rm out}}}}{{{k}_{q}}\sqrt{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}}=\frac{\mu T}{{{k}_{q}}}\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\sqrt{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}\nonumber \end{align} \tag{ 21 }$$

The deleterious effect of scatter on the visibility of details such as microcalcifications in mammography can be addressed through the threshold contrast thickness of details. The CDMAM phantom is often used for this purpose. Detectability can be predicted with a good precision with a non-prewhitened model observer with eye filter (NPWE) (Monnin et al 2011, Liu et al 2014). The detectability index d', known to be a predictive value of object visibility, is calculated from the following parameters: object contrast (signal difference ΔP_out) and NPS (both measured with scatter), presampling MTF (measured without scatter), object Fourier signal function (S_obj), and the visual transfer function (VTF) (equation (22a)).

$$\begin{align} \displaystyle {{d}^{\prime }}=\frac{\Delta {{P}_{{\rm out}}}}{\sigma }\underbrace{\frac{\sqrt{2\pi }\displaystyle\int_{0}^{{{f}_{{\rm c}}}}{S_{{\rm obj}}^{2}(\,f )\cdot {\rm MT}{{{\rm F}}^{2}}(\,f )\cdot {\rm VT}{{{\rm F}}^{2}}(\,f )\,f\,{\rm d}f}}{\sqrt{\displaystyle\int_{0}^{{{f}_{{\rm c}}}}{S_{obj}^{2}(\,f )\cdot {\rm MT}{{{\rm F}}^{2}}(\,f )\cdot {\rm VT}{{{\rm F}}^{4}}(\,f )\cdot \frac{{\rm NPS}(\,f )}{{{\sigma }^{2}}}f\,{\rm d}f}}}}_{\cong \kappa {{D}^{\alpha }}}\cong \kappa \frac{\Delta {{P}_{{\rm out}}}}{\sigma }{{D}^{\alpha }}\nonumber \end{align} \tag{ 22a }$$

Equation (4) shows that the primary signal difference ΔP_out should be used with the presampling MTF in equation (22a). Using MTF_sys instead of MTF without modification of equation (22a) would hence underestimate the object detectability. MTF_sys can be used along with the normalized object contrast (ΔP_out /P_out) and the NNPS (equation (22b)).

$$\begin{align} \displaystyle {{d}^{\prime }}=\frac{{\Delta {{P}_{{\rm out}}}}/{{{P}_{{\rm out}}}}\;}{\sigma }\frac{\sqrt{2\pi }\displaystyle\int_{0}^{{{f}_{{\rm c}}}}{S_{{\rm obj}}^{2}(\,f )\cdot {\rm MTF}_{{\rm sys}}^{2}(\,f )\cdot {\rm VT}{{{\rm F}}^{2}}(\,f )\,f\,{\rm d}f}}{\sqrt{\displaystyle\int_{0}^{{{f}_{{\rm c}}}}{S_{{\rm obj}}^{2}(\,f )\cdot {\rm MTF}_{{\rm sys}}^{2}(\,f )\cdot }{\rm VT}{{{\rm F}}^{4}}(\,f )\cdot \frac{{\rm NNPS}(\,f )}{{{\sigma }^{2}}}f\,{\rm d}f}}\nonumber \end{align} \tag{ 22b }$$

Equation (22a) can be approximated for gold discs of different diameters D as the product between the SDNR (ΔP_out/σ) and the factor κD^α, where κ depends on the frequency content of signal and NPS, whose shapes are practically independent of detector air kerma in a quantum limited dose range: α is a power parameter depending on the object shape that has to be adjusted to the data.

Using equations (20) and (22a), the c-d curves of a given imaging system for gold discs of diameters D and attenuation coefficient µ_Au will satisfy:

$$\begin{align} \displaystyle {{\mu }_{{\rm Au}}}\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\sqrt{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}\cdot {{D}^{\alpha }}T={\rm const}\nonumber \end{align} \tag{ 23a }$$

or

$$\begin{align} \displaystyle \log \left({{\mu }_{{\rm Au}}}\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\sqrt{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}\cdot T \right)={\rm const}-\alpha \log D\nonumber \end{align} \tag{ 23b }$$

The c-d curves given in equations (23a) or (23b) show the threshold gold thickness of CDMAM (T) grows by ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-1}}$ compared to the same DAK without scatter. This loss in detectability due to scatter can be offset by an increase in DAK of ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-2}}$, the scatter compensation factor introduced in Siewerdsen and Jaffray (2000).

For the practical measurements, two mammography systems were used in this study, a Siemens Inspiration and a Hologic Selenia Dimensions. Both are flat panel (FP) type detectors that employ an amorphous selenium (a-Se) x-ray converter layer coupled to TFT array with a pixel spacing of 70 µm and 85 µm for the Hologic and Siemens units, respectively. The Siemens system has a standard linear grid with a ratio of 5:1 and strip density of 31 lines cm⁻¹ while the hologic has a high transmission cellular grid (HTC) with a ratio of 4:1 spaced at 23 lines cm⁻¹. The source-to-image receptor (SID) for the Siemens is 650 mm, while for the Hologic this distance is 700 mm. Source to breast support table distances (STD) are 633 mm and 675 mm for the Siemens and Hologic, respectively. Images used for measurements were DICOM 'For Processing' type.

First, in comparing these metrics, the x-ray beams were chosen to keep the change in effective energy to a minimum; this holds detector DQE as close to constant as feasible and thus highlights changes in image quality/system efficiency due to changing scatter content or system composition (e.g. adding a component such as a grid). Two types of x-ray beams were used: 'primary' beams (P) with a low scattered x-ray photon content and 'primary  +  scatter' beams (PS) that reflect the range of SF typically seen in mammography. The experiments using the PS beams were performed for grid-in and grid-out geometries while the P beams used only the grid-in position. The PS beams used the following PMMA thicknesses positioned on the breast support table: 20 mm, 40 mm, 60 mm and 70 mm. These beams are referred to as PS2, PS4, PS6 and PS7—see table 2. The CDMAM phantom was always positioned on top of 20 mm PMMA; any additional PMMA was placed on top of the CDMAM. At typical mammography energies, the CDMAM transmission is equivalent to 10 mm PMMA or 0.5 mm Al (CDMAM manual, Artinis). Hence the CDMAM phantom was replaced in the PMMA stack by a homogeneous (uniform) 0.5 mm Al sheet (>99% purity) of dimension 180 mm  ×  240 mm for the contrast, MTF and NPS calculations. For P beams, the CDMAM or homogeneous (0.5 mm Al) phantom was supported at a height of 20 mm above the breast support table using small plastic blocks.

Table 2. Characteristics of the x-ray beams.

Beam	Filter (mm)		Effective energy (keV)	HVL (mm Al)	${\left({{{\tilde{P}}}_{{\rm out}}}+{{{\tilde{S}}}_{{\rm out}}} \right)}/{{\rm DAK}}\;$ (mm−2 µGy−1)	µ of gold (cm−1)	µ of aluminium (cm−1)
	PMMA	Al
P2	—	1.5	20.75	0.76	6010	0.831	3.138
P4	—	2.5	21.33	0.83	6438	0.766	3.107
P6	—	3.5	21.78	0.90	6763	0.721	3.084
P7	—	4.0	21.98	0.92	6903	0.702	3.073
PS2	20	0.5	20.77	0.75	6031	0.830	3.137
PS4	40	0.5	21.36	0.83	6460	0.765	3.105
PS6	60	0.5	21.81	0.89	6786	0.720	3.082
PS7	70	0.5	22.01	0.92	6926	0.702	3.072

Following the method in the IEC standard for DQE measurement, the P beams were generated using Al sheets (>99% purity) positioned at the x-ray tube exit port, with the compression paddle removed, and matched to the PS beams via measured HVLs (table 2). Thicknesses of 20 mm, 40 mm, 60 mm and 70 mm PMMA were simulated with 1 mm, 2 mm, 3 mm and 3.5 mm Al, respectively, referred to as P2, P4, P6 and P7—see table 2.

The W/Rh A/F setting was selected as this is used for a broad range of breast thicknesses on both systems. Tube voltage was set to 29 kV for all acquisitions as this is approximately at the centre of the clinical range (typically 25 kV–32 kV) and enabled selection of tube current-time products (mAs) that gave the chosen target (fixed) standard deviation (σ_T). In this study, standard deviation in the non-linearized images was held constant with changing phantom thickness, as this mimics the behaviour of most AEC devices, where mean pixel value (and standard deviation) is held constant as object thickness is changed (Salvagnini et al 2015). The standard deviation obtained for the PS4 beam under automatic exposure control (AEC) was taken as σ_T for the four PS beams with grid out and four P beams for the two mammography systems. For the PS beams with grid in, the mAs settings obtained for the grid out measurements were used. The standard deviation was measured in a 5 mm  ×  5 mm region of interest (ROI) at the standard position (60 mm from the chest wall edge, centred left-right (EC 2006)).

Detector response for both systems was measured using a beam quality of 29 kV, W/Rh with grid out. The Al filters were positioned at the tube exit position and a calibrated dosemeter positioned at the standard position (EC 2006) used to estimate detector air kerma (DAK). Four response functions were measured, using filter thicknesses of 1.5 mm, 2.5 mm, 3.5 mm and 4.0 mm Al. Detailed information on the measurements is given in Monnin et al (2014). The linearized pixel values expressed in DAK values (DAK  =  P_out  +  S_out) were used for contrast, SDNR, MTF and NPS calculations.

A version of the angled edge method was used to measure both the MTF and MTF_sys (Samei et al 1998). For P beams, the edge was supported at a height of 20 mm above the breast support table using small plastic blocks. Measuring the MTF at 20 mm above the table gave an estimate of sharpness at the position of the CDMAM phantom. A 0.8 mm thick steel edge of dimension 120 mm  ×  60 mm was used for the Siemens data while a square tungsten edge of side 50 mm and thickness 0.5 mm was used with the Hologic system. The MTFs_sys were measured using the scatter beams (PS2, PS4, PS6 and PS7), with the edge always positioned on 20 mm PMMA  +  0.5 mm Al. A 1.0 mm thick copper square of side 200 mm was used for the Siemens data while a 1.0 mm thick copper edge of dimension 120 mm  ×  80 mm was used with the Hologic system. For all the P and PS beams, the mAs was adjusted to give a DAK of ~200 µGy. Two images were acquired for each condition studied, and hence the final MTF or MTF_sys curves were averaged from two measurements. An ROI side of 40 mm was used to generate the ESF for the P beams, while an ROI of side 100 mm was used for the ESF for the PS beams. Calculation steps for the MTF are described in Monnin et al (2016).

Three images of the homogenous phantom were acquired with the mAs values established to give σ_T for the P beams (P2, P4, P6 and P7) with grid in and for the PS beams (PS2, PS4, PS6 and PS7) with grid out. For the PS beams with grid in, the mAs settings derived for the grid out measurements were used. From these homogeneous images, the NPS was calculated using half-overlapping 256  ×  256 pixel ROIs. Details of the NPS implementation are given in Monnin et al (2014). The final NPS curves were a radial average of the NPS, excluding the 0° and 90° axial values.

The contrast was assessed with a 5  ×  5  ×  0.2 mm Al plate (purity of 99.5%) positioned at the reference point at a height of 20 mm above the breast support table, on top of the homogeneous 0.5 mm Al phantom for the P beams and on top of 20 mm PMMA for the PS beams. The contrast was calculated as the signal difference (ΔP_out in equation (20)) between pixel value in the small square of aluminium and the background pixel values in linearized images, determined from a 2  ×  2 mm² ROI at the centre of the aluminium and four identical ROIs on the four sides of the aluminium square. SDNR was calculated as the contrast divided by the average standard deviation of pixel values of the four background ROIs.

The CDMAM test object was imaged at 29 kV, W/Rh with the mAs settings determined as described in section 3.2. Eight images of the CDMAM were acquired for each beam and grid condition (section 3.2). The CDMAM images were scored using the CDCOM module and c-d curves were generated using the standard processing method described by Young et al (2006). Uncertainty on the threshold gold thickness was estimated using a bootstrap method.

The scatter fractions in the output image (SF_out) were measured for the four PS beams with grid out.

3.8.1. Beam stop method.

The scatter fractions in the output image (SF_out) were determined experimentally with the beam stop method (Carton et al 2009, Salvagnini et al 2012). A series of lead discs with radii R between 3 and 10 mm were imaged on top of the PMMA, at the centre of the plate, followed by an image without a lead disc. The mean pixel value PV_Q (linearized pixel values) in a circular ROI (2 mm in diameter) positioned on each image at the disc centre gave the scatter and glare (scattering of signal within the detector), noted ${\rm PV}_{{\rm Q}}^{{\rm disc}}={{S}_{{\rm out}}}+{{G}_{{\rm S}}}$, as a function of the disc radius. The mean PV_Q at the same location but without disc gave the primary, scatter and glare: ${\rm P}{{{\rm V}}_{{\rm Q}}}={{P}_{{\rm out}}}+{{S}_{{\rm out}}}+{{G}_{{\rm P}+{\rm S}}}$. With the reasonable assumption that glare is proportional to signal, i.e. ${{G}_{{\rm S}}}=\gamma {{S}_{{\rm out}}}$ and ${{G}_{{\rm P}+{\rm S}}}=\gamma \left({{P}_{{\rm out}}}+{{S}_{{\rm out}}} \right)$, SF_out was calculated with equation (24):

$$\begin{align} \displaystyle {\rm S}{{{\rm F}}_{{\rm out}}}=\frac{{{S}_{{\rm out}}}+{{G}_{{\rm S}}}}{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}+{{G}_{{\rm P}+{\rm S}}}}=\frac{{{S}_{{\rm out}}}}{{{P}_{{\rm out}}}+{{S}_{{\rm out}}}}=\frac{\underset{R\to 0}{\mathop{\lim }}\,{\rm PV}_{{\rm Q}}^{{\rm disc}}(R )}{{\rm P}{{{\rm V}}_{{\rm Q}}}}\nonumber \end{align} \tag{ 24 }$$

The numerator in equation (24) was obtained by extrapolating ${\rm PV}_{{\rm Q}}^{{\rm disc}}(R )$ plotted as a function of the disc radius to the radius zero. Linear or logarithmic functions are usually used for extrapolation (Aslund et al 2006, Shen et al 2006, Carton et al 2009, Salvagnini et al 2012). In our study a fitting function derived from the beam PSF given in equation (1) was used (equation (25)), with the constraints ${\mathop{\lim }}_{R\to 0}\,{\rm PV}_{{\rm Q}}^{{\rm disc}}(R )={{S}_{{\rm out}}}+{{G}_{{\rm S}}}$ and ${\mathop{\lim }}_{R\to \infty }\,{\rm PV}_{{\rm Q}}^{{\rm disc}}(R )=0$:

$$\begin{align} \displaystyle \begin{array}{@{}l@{}} {\rm PV}_{{\rm Q}}^{{\rm disc}}(R )={\rm DAK}\cdot \displaystyle\int_{0}^{2\pi }{\displaystyle\int_{R}^{\infty }{{\rm PS}{{{\rm F}}_{{\rm b}}}(r)r\,{\rm d}r{\rm d}\theta }} \nonumber \\ \quad\quad\quad\quad \,\,= \frac{{{S}_{{\rm out}}}+{{G}_{S}}}{ak_{1}^{2}+(1-a )k_{2}^{2}}\cdot \left(\frac{ak_{1}^{2}}{{{\left(1+{{\left({R}/{{{k}_{1}}}\; \right)}^{2}} \right)}^{1/2}}}+\frac{(1-a )k_{2}^{2}}{{{\left(1+{{\left({R}/{{{k}_{2}}}\; \right)}^{2}} \right)}^{1/2}}} \right) \end{array}\nonumber \end{align} \tag{ 25 }$$

3.8.2. Low-frequency drop of MTF_sys.

3.8.3. Monte Carlo simulation.

Grid transmissions and beam components were calculated from images with linearized pixel values (PV_Q). The total (T_t) and primary (T_p) grid transmissions were determined for the four P beams from images of circular pinholes of different diameters. A series of circular lead collimators with radii R between 2 and 10 mm were fixed at the tube exit and imaged at a fixed mAs, followed by an image without a collimator. This procedure was repeated with the grid in place and with the grid removed. With the same circular ROI and calculation assumptions as those used for the beam stop method described in section 3.8.1, T_t and T_p were calculated with equations (26a) and (26b), respectively:

$$\begin{align} \displaystyle {{T}_{{\rm t}}}=\frac{{{P}_{{\rm out}}}+S_{{\rm out}}}{{{P}_{{\rm in}}}+{{S}_{{\rm in}}}}=\frac{^{+}{\rm P}{{{\rm V}}_{{\rm Q}}}}{^{-}{\rm P}{{{\rm V}}_{{\rm Q}}}}\nonumber \end{align} \tag{ 26a }$$

$$\begin{align} \displaystyle {{T}_{{\rm p}}}=\frac{{{P}_{{\rm out}}}}{{{P}_{{\rm in}}}}=\underset{R\to 0}{\mathop{\lim }}\,\frac{^{+}{\rm PV}_{{\rm Q}}^{{\rm ph}}(R )}{^{-}{\rm PV}_{{\rm Q}}^{{\rm ph}}(R )}\nonumber \end{align} \tag{ 26b }$$

The mean pixel values measured without a pinhole (PV_Q) were used in equation (26a). The plus and minus signs indicate the grid position, i.e. in place and removed. Equation (26b) was evaluated by extrapolating the mean pixel values ${\rm PV}_{{\rm Q}}^{{\rm ph}}(R )$ measured in a circular ROI (2 mm in diameter) positioned on each image at the pinhole centre and plotted as a function of the pinhole radius to the radius zero. A fitting function derived from the beam PSF given in equation (1) was used (equation (27)), with the constraints ${\mathop{\lim }}_{R\to 0}\,{\rm PV}_{{\rm Q}}^{{\rm ph}}(R )={{P}_{{\rm out}}}+{{G}_{{\rm P}}}$ and ${\mathop{\lim }}_{R\to \infty }\,{\rm PV}_{{\rm Q}}^{{\rm ph}}(R )={{P}_{{\rm out}}}+{{G}_{{\rm P}}}+{{S}_{{\rm out}}}+{{G}_{{\rm S}}}$:

$$\begin{align} \displaystyle \begin{array}{@{}l@{}} {\rm PV}_{{\rm Q}}^{{\rm ph}}(R )={\rm DAK}\cdot \displaystyle\int_{0}^{2\pi }{\displaystyle\int_{0}^{R}{{\rm PS}{{{\rm F}}_{{\rm b}}}(r)}}r{\rm d}r{\rm d}\theta \, \nonumber \\ ={{P}_{out}}+{{G}_{P}}+\frac{{{S}_{out}}+{{G}_{S}}}{ak_{1}^{2}+(1-a )k_{2}^{2}}\cdot \left(ak_{1}^{2}\cdot \left(1-\frac{1}{{{\left(1+{{\left({R}/{{{k}_{1}}}\; \right)}^{2}} \right)}^{1/2}}} \right)+(1-a )k_{2}^{2}\cdot \left(1-\frac{1}{{{\left(1+{{\left({R}/{{{k}_{2}}}\; \right)}^{2}} \right)}^{1/2}}} \right) \right) \end{array}\nonumber \end{align} \tag{ 27 }$$

The differences between the primary grid transmissions measured for the four PS and corresponding P beams were below 2%, thus average values were used for T_p. The PS beam components were then calculated as shown in table 3, using DAK, T_p and T_t. The scatter grid transmissions (T_s) were calculated with equation (11b), using the calculated values S_out and S_in, and assumed the same for the corresponding P beams (very close HVLs). The parameters for the P beams were finally calculated with DAK, T_p, T_t and T_s, as shown in table 3.

Table 3. Determination of the x-ray beam components.

Beam type	Grid	Pout	Sout	Pin	Sin	SFout	SFin
PS	Out	$\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)\cdot {\rm DAK}$	${\rm S}{{{\rm F}}_{{\rm out}}}\cdot {\rm DAK}$	${{P}_{{\rm out}}}$	${{S}_{{\rm out}}}$	Measured (beam stop)	${\rm S}{{{\rm F}}_{{\rm out}}}$
PS	In	${{P}_{{\rm in}}}\cdot {{T}_{{\rm p}}}$	${\rm DAK}-{{P}_{{\rm in}}}\cdot {{T}_{{\rm p}}}$	Known (=PS grid out)	Known (=PS grid out)	$\frac{{\rm DAK}-{{P}_{{\rm in}}}\cdot {{T}_{{\rm p}}}}{{\rm DAK}}$	Known (=PS grid out)
P	In	$\frac{{{T}_{{\rm p}}}}{{{T}_{{\rm t}}}}\cdot \frac{{{T}_{{\rm t}}}-{{T}_{{\rm s}}}}{{{T}_{{\rm p}}}-{{T}_{{\rm s}}}}\cdot {\rm DAK}$	$\frac{{{T}_{{\rm s}}}}{{{T}_{{\rm t}}}}\cdot \frac{{{T}_{{\rm p}}}-{{T}_{{\rm t}}}}{{{T}_{{\rm p}}}-{{T}_{{\rm s}}}}\cdot {\rm DAK}$	$\frac{{{T}_{{\rm t}}}-{{T}_{{\rm s}}}}{{{T}_{{\rm t}}}\left({{T}_{{\rm p}}}-{{T}_{{\rm s}}} \right)}\cdot {\rm DAK}$	$\frac{{{T}_{{\rm p}}}-{{T}_{{\rm t}}}}{{{T}_{{\rm t}}}\left({{T}_{{\rm p}}}-{{T}_{{\rm s}}} \right)}\cdot {\rm DAK}$	$\frac{{{T}_{{\rm s}}}}{{{T}_{{\rm t}}}}\cdot \frac{{{T}_{{\rm p}}}-{{T}_{{\rm t}}}}{{{T}_{{\rm p}}}-{{T}_{{\rm s}}}}$	$\frac{{{T}_{{\rm p}}}-{{T}_{{\rm t}}}}{{{T}_{{\rm p}}}-{{T}_{{\rm s}}}}$

All DQE metrics start from the detector DQE (DQE_d) (equation (16)). DQE_sys was calculated from equation (17), using T_p and T_t. The eDQE was calculated via equation (18a) using T_p and SF_out; this is a different approach to computing eDQE compared to that of Samei et al (2009), which involves the measurement of the primary beam transmission through the phantom to estimate P_in. The gDQE was calculated using equation (19a). Finally, NEQ was calculated using equation (10).

The calculated beam parameters are given in table 5. The DAK for a constant image noise (target standard deviation of pixel values σ_T) decreases as beam energy increases. This is consistent with results of an earlier study (Marshall 2009) and is partly a result of the increase in detector gain with mean photon energy. Although not shown, the S_out/P_out ratios increase linearly with the PMMA thickness, as expected (Boone et al 2000). With differences below 3%, SF_out measured with the beam stop technique (illustration in figure 4 for the Selenia Dimensions, data in table 4) and average SF_out calculated from Monte Carlo (MC) simulations for the PS beams with grid out are in good agreement. Agreement was slightly poorer using the low-frequency drop of MTF_sys where an underestimate of SF_out compared to the two other methods was seen, ranging from  −6% (20 mm PMMA) to  −10% (70 mm PMMA). In using MTF_sys to estimate SF_out, the hypotheses of signal and SF_out stationarity within the different ROIs have to be made and this is not perfectly met in practice. The MC data showed that SF_out decreases from image centre to edge and that average SF_out is lower than the central value by 6–8%.

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The primary grid transmission (T_p), which varied between 0.75 and 0.77 for the two systems, increases with the beam energy whereas the total grid transmission (T_t) for PS beams falls from 0.60 to 0.43 with increasing SF_in (table 5), consistent with previous work (Rezentes et al 1999, Carton et al 2009, Salvagnini et al 2012). The T_s values measured in this study are within the range expected for mammography grids (Shen et al 2006, Aichinger et al 2012, p 126). The utility of the grid is given by the grid DQE (DQE_ASD in equation (15a) and table 5), which increases with SF_in, T_p and the grid selectivity ($\Sigma ={{{T}_{{\rm p}}}}/{{{T}_{{\rm s}}}}\;$). DQE_ASD grows with SF_in from T_p (SF_in  =  0) to ${{T}_{{\rm p}}}\cdot \Sigma$ (SF_in  =  1). DQE_ASD is mainly determined by T_p for low SF_in, and is increasingly weighted by the grid selectivity as SF_in increases. The cellular grid of the Selenia has a higher selectivity than the classical linear grid of the Siemens, and hence a DQE_ASD increasing faster with SF_in (figure 5). In mammography, SF_in typically lies between 0.2 and 0.5 and consequently the higher selectivity of the cellular grid is of limited value in terms of DQE_ASD.

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DQE_ASD becomes greater than 1.0 for SF_in  ⩾  0.29 and  ⩾  0.31 for the Hologic and Siemens, respectively, corresponding to 50% glandular breast thicknesses of ~38 mm and ~42 mm (Boone et al 2000), as shown in figure 5. For lower SF_in, the benefit of scatter rejection does not compensate sufficiently for the loss in primary photons; only a grid with T_p  =  1 improves SNR for all thicknesses. For the conditions used in this study, the grids of the Hologic and Siemens only improve SNR for breasts thicker than 38 and 42 mm, respectively, and hence should be removed below. This outcome in terms of DQE_ASD is consistent with K_SDNR results (${{K}_{{\rm SDNR}}}=\sqrt{{\rm DQ}{{{\rm E}}_{{\rm ASD}}}}={{{T}_{{\rm p}}}}/{\sqrt{{{T}_{{\rm t}}}}}\;$) in previous studies (Chakraborty 1999, Veldkamp et al 2003, Shen et al 2006). Cunha et al (2010) showed K_SDNR becomes greater than 1.0 for breasts thicker than 3 cm for grids used at 28 kV (Mo/Mo), while Carton et al (2009) measured a SDNR improvement for S_out/P_out ratios higher than 0.4 for 35 kV (Rh/Rh), that corresponds approximately to SF_in  =  0.29, both close to our results. More recently, Chen et al (2015) similarly showed that grid-less imaging in digital mammography gave higher SDNR for PMMAs thinner than 4 cm. Gennaro et al (2007) determined a grid break-even at 6.5 cm PMMA (32 kV—Rh/Rh), a thickness much higher than determined in our study. The characteristics of the grid were however not given and the origin of the difference is therefore difficult to establish (most probably due to a lower T_p). It is interesting to note that the grid modifies the SDNR by the factor ${{{T}_{{\rm p}}}}/{\sqrt{{{T}_{{\rm t}}}}}\;$ without any change in patient dose. The AEC of most radiology systems will however keep the DAK constant and increase the patient dose by ${1}/{{{T}_{{\rm t}}}}\;$, changing SDNR by a factor ${{{T}_{{\rm p}}}}/{{{T}_{{\rm t}}}}\;$.

A scan-slot system was not tested in our study but we can estimate a theoretical DQE_ASD from scatter rejection data published for the Philips MicroDose multi-slit system (Aslund et al 2006). For this system T_p  =  1 because the slit width (0.7 mm) of the post-collimator is wider than the input primary beam. An approximate value T_s  ≈  0.04 was used, which is close to the value of 0.03 reported in older studies for multi-slit mammography systems (Yester et al 1981, Barnes et al 1993), and corresponds to the SDQE  ≈  0.96 given in Aslund et al (2006). High primary transmission and excellent scatter rejection hence give a DQE_ASD close to that for an ideal ASD (figure 5).

Figure 6 plots the presampling MTF curves (simply denoted MTF in our study) measured without scatter but including focus and detector blurring. Also plotted are measured MTFs_sys (presampling MTFs measured in the presence of scattered radiation), along with the MTF_sys determined from equation (3a). It can be seen that calculated MTFs_sys closely matches measured MTFs_sys, an indication that cascading the individual transfer functions of the separate processes is a valid step. This in turn, is consistent with the linear transfer of independent and uncorrelated physical processes, as described by Kyprianou et al (2004, 2005a) who multiplied the linear combination of the scatter MTF (MTF_S), focus MTF (MTF_F) and detector presampling MTF (MTF_D). MTF_sys incorporates these three effects of signal blurring. This result supports the derivation of the system transfer function as the product of the presampling MTF and the BTF (convolution of the impulse responses (PSFs) of the different sources of image degradation). The FWHM of scatter PSFs ranged between 4.9 mm (beam PS2) and 6.4 mm (beam PS7) for the four PS beams with grid out (figure 2), suggesting considerable low spatial frequency content in wide-angle scattering. Furthermore this is evidence that the BTF drops at very low frequency to a value (1  −  SF_out), and can be practically considered as a constant equal to (1  −  SF_out) for spatial frequencies higher than 1/k  ≈  0.1 mm⁻¹ (equation (3b)), the frequency range useful for radiology.

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Two methods are thus available when determining MTF_sys. A direct measure can be made, using a large, knife-edge embedded in PMMA, as described by Salvagnini et al (2012). MTF_sys is however difficult to assess precisely for a number of reasons. First, the sharp edge must be large enough to enable an accurate estimate of the long LSF tails of the wide-angle scatter distribution. Truncation of the LSF tails within a finite ROI causes a spectral truncation at very low spatial frequency. This incorrectly inflates the MTF at all spatial frequencies (Illers et al 2005, Samei et al 2006, Friedman and Cunningham 2008). Secondly, the value of SF_out obtained from MTF_sys contains the contribution of glare. The glare fraction in the image can be estimated from the low-frequency drop of the presampling MTF, and then subtracted from the measured low-frequency drop of MTF_sys to obtain SF_out (Salvagnini et al 2012). Thirdly, the calculation of MTF_sys assumes the spatial stationarity of the detector and beam PSFs. Wide-angle scattering within the finite size of the x-ray field results in a slowly varying SF_out across the image plane. The assumption of spatial stationarity is thus not fully met and this will somewhat decrease the precision of the MTF_sys low-frequency drop estimate.

The alternative is to perform separate measures of the presampling MTF (without scatter) and of SF_out, typically using a beam stop method. The beam stop method gives a direct and local measure of SF_out, and is therefore expected to give a more accurate estimate of low-frequency drop. This also quantifies the spatial spread of the scatter PSF (factors k₁ and k₂ in equation (1)). In this work, average SF_out in the PMMA calculated from Monte Carlo simulations and by beam blocks were consistent, whereas the low-frequency drop of MTF_sys underestimates SF_out by ~6%. This may be partially explained by differences in SF_out between the centre and the edge of the ROI used to compute MTF_sys. The assumptions of spatial invariance are not fully met. A small contribution to the difference may also come from the scattered radiation originating from the edge device that tends to inflate the MTF (Neitzel et al 2004), and hence can reduce the low-frequency drop.

Pairwise comparison of NEQs between equivalent beams without and with scatter (P beams versus PS beams without grid) shows that NEQ varies proportionally to (1  −  SF_out)² for a constant image noise (equation (10) and figure 7). The PS beams with grid have the same mAs and SF_in as the corresponding PS beams without grid, but reduced DAK and SF_out. In terms of NEQ, $P_{{\rm out}}^{2}$ is reduced by the primary grid transmission ($T_{{\rm p}}^{2}$) whereas the NPS varies with the total grid transmission (T_t). Hence, the variation in NEQ due to the grid follows the ratio ${T_{{\rm p}}^{2}}/{{{T}_{{\rm t}}}}\;$, i.e. an increase or a decrease if DQE_ASD is greater or less than 1.0. The grid is seen to improve the NEQ for the beams PS6 and PS7, slightly for PS4, but decreases the NEQ for PS2.

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The contrast of the 0.2 mm aluminium foil measured on the images (ΔP_out) was normalized by the product µ_AlT, where µ_Al is the attenuation coefficient of aluminium given in table 2 and T  =  0.2 mm. The values ${\Delta {{P}_{{\rm out}}}}/{\left({{\mu }_{{\rm Al}}}T \right)}\;$ were then compared to primary DAK in the image (P_out) (figure 8). Good equivalence between ${\Delta {{P}_{{\rm out}}}}/{\left({{\mu }_{{\rm Al}}}T \right)}\;$ and P_out is seen, consistent with the expectation that object contrast is made of primary photons only (equation (20)), regardless the number of scattered photons in the image, whereas the noise increases with the total number of detected photons, without distinction between primary and scatter. Hence, as shown in figure 8, working at constant number of photons at the detector (typical of an AEC) will give a constant quantum noise but will not ensure a constant contrast or SDNR, both varying with (1 – SF_out) (equation (21)). The factor ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}$ represents the degree to which increased DAK can be used to compensate the SDNR for x-ray scatter degradation. This was shown by Siewerdsen and Jaffray (2000), and has been verified in a recent study into AEC set-up for a mammography system (Salvagnini et al 2015).

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Turning to the CDMAM data, figures 9(a) and (b) present the c-d curves for the Hologic and Siemens systems, respectively. The threshold gold thickness (T) in the upper graph in each figure reveals the negative influence of increasing SF_out on detection. Equations (23a) and (23b) predict the value ${{{P}_{{\rm out}}}{{\mu }_{{\rm Au}}}T}/{\sigma }\;$ will remain constant (µ_Au values in table 2). Considering ${{{P}_{{\rm out}}}{{\mu }_{{\rm Au}}}T}/{\sigma }\;$ instead of T alone in the c-d curves cancels the differences in detectability levels (lower graphs in figures 9(a) and (b)). The pairwise variations of threshold thicknesses between equivalent beams without and with scatter (P beams versus PS beams without grid) show that T is systematically increased by ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-1}}$ (at constant noise σ_T) compared to the conditions without scatter. This result validates the link between MTF and MTF_sys of equation (4) that shows the signal scales with the number of primary photons (for frequencies higher than 1/k), whereas the noise increases with the total amount of photons. The SF in the output image (SF_out) reduces the NEQ by ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{2}}$ and increases T by ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-1}}$, that can be offset by an increase in DAK of ${{\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)}^{-2}}$, called 'scatter compensation factor' in Siewerdsen and Jaffray (2000).

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The experimental variations in T with SF_out behave as predicted by the NPWE observer, that makes use of the image noise (composed of primary and scattered photons without distinction) and the primary object signal (contrast). The NPWE model can thus also be used for image quality characterization with scatter. The effect of scatter in the observer model may be taken into account in two ways: (1) using the presampling MTF (including the detector and focal spot blurs) along to the absolute object contrast (signal difference ΔP_out) and NPS (both measured on the images with scatter) (equation (23a)) or (2) using the system MTF (MTF_sys) along to the normalized object contrast (ΔP_out/P_out) and NNPS, all three measured with scatter (equation (23b)). Both methods have been used in the literature: Monnin et al (2011) used the presampling MTF, while Liu et al (2014) used generalized MTF and NNPS in their observer model. The latter study showed the equivalence of the two approaches.

Figures 10(a) and (b) plot DQE_sys, the average detector DQE (IEC 2015), eDQE (Samei et al 2009) and gDQE (Kyprianou et al 2004, 2005a, for the Selenia Dimensions and Inspiration, respectively. All the images were made with the detector carbon cover in place, and all these detection efficiency metrics—even the detector DQE—include the loss in detection efficiency due to x-ray absorption within the detector cover. Although not shown, maximum variation in the individual DQE curves for a given detector for the P beams was 2%, consistent with the beams being designed to have only small changes in effective energy. This small change in detector DQE with energy is consistent with results for an earlier a-Se detector (Marshall 2009) and hence we use an average curve to represent detector DQE.

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For the conditions with scatter (PS beams) and grid out, figure 10 shows how eDQE and gDQE decrease with SF_in. If no grid is used, DQE_sys reverts to DQE_d and is constant, whereas image quality reflected in the NEQ is degraded as SF_in increases. The system DQE varies only if there is a change in SNR (NEQ) transfer through the system, but image quality (~NEQ) can vary independently because of variations in the quantity of input primary photons (SF_in). Without the grid, the only element in the imaging system is the detector: if detector DQE is ~constant (energy is kept ~constant) then a change in system efficiency is not expected. The system DQE cascades detector and ASD DQEs, and quantifies SNR changes as an output/input balance for the whole imaging system (detector  +  ASD). It differs from detector DQE only when an ASD is used, and is therefore a parameter suitable for the imaging efficiency assessment of a detector/ASD pair, as a function of SF_in. The eDQE compares the real system to a perfect detector with a perfect grid (DQE_d  =  1 and SF_out  =  0), and scales as (1  −  SF_in) for the case without grid. The gDQE scales as (1  −  SF_out)², as does NEQ. When imaging without grid, the ability of the imaging system to select primary photons or reject scattered photons does not vary between the different beams and hence we expect no change in system efficiency, yet eDQE and gDQE decrease with increasing PMMA thickness implying a reduction in system efficiency. This result suggests that eDQE and gDQE are mixing image quality and system efficiency. This property comes from the use of a grid DQE as defined by Wagner et al (1980) and implicitly taken up for eDQE and gDQE that ranks the real grid to an ideal device with (T_p; T_s)  =  (1; 0). As a consequence, both eDQE and gDQE, for detectors coupled to real ASDs, behave as if the grid were the source of the scatter incident on the detector and can therefore only be lower than the detector DQE (for real grids) or equal to the detector DQE for the case of an ideal grid. We suggest that metrics of system efficiency and metrics of image quality should be kept separate. 'Image quality' can then be defined and calculated separately, for example using a detectability index for some task in combination with the NEQ (Siewerdsen and Jaffray 2000) or even using a task generic parameter such as SDNR.

The introduction of the grid into the system increases DQE_sys and eDQE by ${T_{{\rm p}}^{{\rm 2}}}/{{{T}_{{\rm t}}}}\;$ (the grid DQE), and gDQE by the factor ${T_{{\rm p}}^{2}}/{T_{{\rm t}}^{2}}\;$. These variations can be tracked for image quality (NEQ with and without grid in figure 7) and for DQE_sys, eDQE and gDQE in figure 10. Variations in SNR due to the grid can be basically quantified with SDNR measurements, and related to the dose (mAs) for a given beam quality. Figure 11 compares SDNR for a 0.2 mm Al target for grid in and grid out using the four PS beams. The SDNR with grid out was measured for different tube loads (mAs) and compared to the SDNR with grid in with the mAs chosen by the AEC. For the beam PS7, the AEC mAs of the Selenia Dimensions was above the maximum allowed for 29 kV—W/Rh and thus 400 mAs (i.e. maximum) was chosen. As expected from equation (21), the SDNR increases as a power function of the DAK and hence of mAs for fixed x-ray beam and phantom. SDNR therefore scales with $\left(1-{\rm S}{{{\rm F}}_{{\rm out}}} \right)$ i.e. decreases with the PMMA thickness. For a given mAs, the grid improves the SDNR for the beams PS6 and PS7, but reduces SDNR for the beam PS2, and marginally modifies SDNR for PS4. The introduction of the grid into the system modifies the SDNR by a factor close to the square root of the measured grid DQEs (${{{T}_{{\rm p}}}}/{\sqrt{{{T}_{{\rm t}}}}}\;$) for the different thicknesses (SF_in). In our study, DQE_sys is higher than the detector DQE for 50% glandular breast thicker than approximately 38 and 42 mm for the Hologic and Siemens, respectively. Below these break-even thicknesses the negative effect of the grid (loss of SDNR due to a loss in primary photons) overcomes the positive effect (increase of SDNR due to rejection of scatter) and degrades the image SDNR.

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System DQE considers the imaging system as a cascaded chain of elements. The BTF and presampling MTF are independent, and are multiplied as cascaded processes to describe the signal transfer through the imaging chain. DQE_sys is simply the product between the detector and ASD DQEs, where DQE_ASD ranks the real SNR² transfer through the ASD without considering a hypothetical ideal case. DQE_sys reverts to the detector DQE in the absence of an ASD, and will be higher or lower than the detector DQE, depending on how the ASD modifies image SNR. System DQE can be seen as a natural extension of the classical detector DQE and can be used to study the detective efficiency of complete imaging systems for beams with scatter. The newly proposed metric DQE_sys is an extension of current detector characterization for imaging systems with ASD and beams with scatter. Ideal (primary) and real (primary  +  scatter) mammography beams were matched in this study using effective energy (HVL) and the assumed equivalence of 10 mm PMMA ~0.5 mm Al. This is similar to the use of Al filters at the tube exit in the IEC recommendations (2015) for detector DQE assessments in order to create a primary beam with the same effective energy or half value layer (HVL) as typical exit spectra from patients. Characterizing a system using DQE_sys could begin with standard (primary beam) detector DQE assessment for the chosen kV and A/F. The additional Al filter thickness at the tube exit should match the HVL of the beam with PMMA. The effect of scatter on image quality (NEQ) and system efficiency (DQE_sys) can then be established separately by measuring SF_out and DQE_ASD for chosen scatter conditions (different SF_in). DQE_ASD is multiplied by detector DQE to obtain DQE_sys. The NEQ and DQE_sys assessment can be repeated for different beam energies (different kV and/or A/F) as required. The measurement of MTF_sys, which can be a challenge to assess precisely in scatter beams, is not required for NEQ and DQE_sys. Just three supplementary variables are needed: the primary and total ASD transmissions and the SF at the detector (SF_out). Different methodologies have already been used to quantify SF_out, and the beam stop method described in this study gave consistent results. T_t is easily obtained through measurements of signal ratios with grid in/out on linearized images, while an extrapolation of the signal ratio grid in/out to a point source beam was used for T_p measurements. The procedure followed in this study to determine T_p and T_t is however not practicable for slit-scanning systems because the beam components P_out and S_out are not accessible; an alternative method is proposed in appendix B for such systems.