2ABD
2ABD can be expressed as:
$$ 2 ABD=\frac{\int_0^d{k}_{a,i}{e}^{-{\mu}_{en}x} dx}{d}=\frac{k_{a,i}}{\mu_{en}d}\left( 1-{e}^{-{\mu}_{en}d}\right) $$
(3)
where μ
en
is the energy absorption coefficient in a water-equivalent soft tissue (density 1.0 ± 0.1 g/cm3) and d is the breast thickness. Note that μ
en
depends on kVp and anode-filter combination while k
a,i
depends on tube-voltage kVp, tube current-exposure time product mAs, anode-filter combination, d, and the distance between the x-ray tube focus and the upper surface of the breast (FID-d). kVp, mAs, anode-filter combination, and d are provided by the mammography unit and reported in the DICOM-header file stored for each procedure and FID is the (fixed) focus-to-image distance, known for the mammography device used.
X-ray beam characterisation: evaluation of k
a,i
In the range of kVp usually employed in mammography (22–34 kVp), the incident air kerma on the central axis of the x-ray beam at a distance (FID-d) from the tube focus can be linearly related to kVp:
$$ {k}_{a,i}=\frac{Y_{tb}}{Y_0}\left(\alpha \cdot kVp+\beta \right)\cdot mAs\cdot {\left(\frac{FID}{FID-d}\right)}^2 $$
(4)
Y
0
and Y
tb
are the yields of the reference x-ray tube and the actual x-ray tube. The reference x-ray tube is that used for the experimental measurements needed for α and β calculation.
In order to determine the parameters α and β, a set of measurements of k
a,i
was done for various kVp (range 22–34) and mAs (range 10–100) by using a reference x-ray mammography tube (i.e. the mammography unit used for the experimental measurements) whose yield was Y
0
for five different anode-filter combinations: Mo-Mo, Mo-Rh and Rh-Rh (Senograph DS General Electric Medical Systems, Waukesha, WI, USA); and W-Rh and W-Ag (Selenia Dimensions, Hologic, Bedford, MA, USA). k
a,i
, kVp and mAs were measured by using a solid-state detector coupled to a multimeter (Piranha®, RTI-Electronics AB, Molndal, Sweden) placed 6 cm from the chest wall edge at the centre of the mammography flat support plate (d = 0) with the compression paddle between the x-ray-tube focus and the detector.
Each measurement of k
a,i
in the same position, with the same kVp and mAs, was repeated five times. The average values of k
a,i
were fitted (least squares method) to Equation 4 to determine α and β for each anode-filter combination. The fit uncertainties associated to α and β were used in the 2ABD error evaluation.
A comparison of k
a,i
calculated by Equation 4 and k
a,i
directly measured for different anode-filter combinations and different values of d has been done (using four different mammography devices whose yield was Y
tb
, located in different hospitals) to test the accuracy of Equation 4. Note that both Y
0
and Y
tb
must be calculated at the same value of kVp (28 kVp in this work).
Exponential attenuation: evaluation of μ
en
The energy absorption coefficient μ
en
(kVp, anode-filter combination) in a material can be evaluated by the equation:
$$ I(x)={I}_0\cdot {e}^{-{\mu}_{en}x} $$
(5)
where x is the thickness of the attenuating beam-crossed material, I
0
is the incident beam intensity, and I is the attenuated beam intensity.
A set of experimental measurements was done varying the kVp (range 22–34) for each anode-filter combination in order to assess the curves represented by Equation 5. I
0
and I(x) were evaluated by using a 60-cm3 ionisation chamber coupled to a 2026C Radcal Corporation® (Monrovia, CA, USA) electrometer placed under increasing depths x (range 0.0–5.5 cm) of solid water (density 1.0 ± 0.1 g/cm3). The measured I
0
and I(x) were fitted to Equation 5 (least squares method) to evaluate μ
en
.
Once evaluated, α and β and the energy absorption coefficient μ
en
for the five anode-filter combination considered, the 2ABD was calculated by Eq. 3 for each different mammographic device used.
Comparison between AGD and 2ABD
Twenty different mammograms were selected from the picture archiving and communication system (PACS) of our university hospital in order to compare AGD calculated by the Dance method (Equation 1) [11,12,13] and the Wu method (Equation 2) [14, 15] to 2ABD calculated by Equation 3. The three most used kVp values were chosen for each anode-filter combination and the most frequent thicknesses were considered for each kVp.
c, g, and s used for the AGD calculation with the Dance method (Equation 1) are reported in the literature [11,12,13]; D
gN
factors needed for the AGD calculation with the Wu method (Equation 2) were also taken from the literature [14,15,16,17].
For each mammogram, it was possible to obtain (from the PACS) the anode-filter combination used, the kVp and mAs set, the breast thickness d, and the age of the patients (needed to choose from the Dance’s table the right value of c to be used in Equation 1). Y
tb
was known from the quality assurance measurements performed on the mammography device under consideration (in this case coincident with Y
0
) while HVL was measured for each anode-filter combination and kVp set in these procedures.
Uncertainties in AGD (using Dance or Wu methods) were estimated considering an overall 20% error [18]. The uncertainty in 2ABD was estimated considering the error propagation for α, β, Y0, Y
tb
, kVp, d, FID, and μ
en
. In particular, the error of thickness d influences both 2ABD and AGD. The uncertainty in thickness may affect the AGD up to 10% [18]. This error component is included in the 20% of total percentage error associated to AGD evaluation. For the calculation of 2ABD, the uncertainty on breast thickness provided by the equipment was considered equal to ± 0.5 cm as reported in the device technical manuals.
The comparison between AGD and 2ABD took into account the overlap between data within their uncertainties.
