### Extension of the 2ABD model to DBT

To evaluate 2ABD for DBT, we followed the same approach developed for DM procedures [13].

Four mammographic devices were utilised: two Selenia Dimensions (Hologic, Bedford, Mass, USA), devices A and B and two Amulet Innovality (Fujifilm Medical System Inc., USA), devices C and D. Device A was used to develop and validate our method through experimental measurements. Therefore, this device was chosen as reference. Additionally, in order to further test our method, a set of measurements was performed on the other three devices.

The Selenia Dimensions model performs both DM and DBT and offers three anode-filter combinations (W-Rh or W-Ag for DM, W-Al for DBT). The DBT angular range of the x-ray tube is ± 7.5°. The Amulet Innovality model employs the W-Rh anode-filter combination for DM and the W-Al anode-filter combination for DBT; two acquisition modes can be selected: the standard mode with a ± 7.5° x-ray tube angular range and the high-resolution mode with a ± 20° x-ray tube angular range.

The 2ABD calculation method for DBT was developed starting from two main approximations: a homogeneous phantom (polystyrene, C_{8}H_{8}, with admixture of 2.1 ± 0.2% of TiO_{2}) with planar dimensions of 16 × 16 cm^{2}, and variable thickness was employed to simulate the breast in experimental measurements; the beam attenuation was expressed as a function of the phantom depth following the exponential decay model.

Under these assumptions, the 2ABD was defined as follows:

$$ 2\mathrm{ABD}\approx \frac{1}{T}\cdot {\int}_0^T{k}_{a,i}\cdot C\cdot \exp \left(-m\cdot x\right)\kern0.5em dx $$

(1)

where *T* is the breast thickness, *k*_{a,i} is the incident air kerma on the breast/phantom surface, *m* is a parameter related to the beam attenuation in the phantom, and *C* is a conversion factor from the *k*_{a,i} to dose in the phantom. The factor *C* accounts also for the backscatter contribution to the *k*_{a,i}. Specifically, the C factor is given by: \( C=B\bullet {\left(\frac{\overline{\mu_{\mathrm{en}}}}{\rho}\right)}_{air}^{ph} \), where *B* is the backscatter factor and \( {\left(\frac{\overline{\mu_{\mathrm{en}}}}{\rho}\right)}_{air}^{ph} \) is the ratio between the mass energy absorption coefficient of the phantom and the mass energy absorption coefficient of air, averaged over the x-ray energy spectrum. The *B* factor was evaluated from experimental measurements, and a value of 1.1 was adopted in this work. The *B* value was obtained by performing two air kerma measurements, with and without the phantom, and taking the ratio of the two detector readings respectively. No appreciable variations were observed in the range 2–9 cm of the phantom thickness. A value of ~ 0.7 was adopted for \( {\left(\frac{\overline{\mu_{\mathrm{en}}}}{\rho}\right)}_{\mathrm{air}}^{\mathrm{ph}} \) (no significant variations were observed by varying the tube voltage, for all the anode/filter combination employed in our study). The \( {\left(\frac{\overline{\mu_{\mathrm{en}}}}{\rho}\right)}_{\mathrm{air}}^{\mathrm{ph}} \) value was computed by considering the weight average value of the \( {\left(\frac{\mu_{\mathrm{en}}}{\rho}\right)}_{\mathrm{air}}^{\mathrm{ph}} \) on the x-ray energy spectrum of the mammographic device. The \( {\left(\frac{\mu_{\mathrm{en}}}{\rho}\right)}_{\mathrm{air}}^{\mathrm{ph}} \) values were obtained from the National Institute of Standards and Technology website (https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients). The above mentioned quantities (*k*_{a, i}, *C*, *m*) are required in order to calculate 2ABD in any clinical condition. Therefore, a simple model for estimating *k*_{a,i} and *m* was developed.

### Evaluation of *k*_{a,i}

A set of air kerma measurements were performed on the Selenia Dimensions device in DBT modality through a flat 60-cm^{3} ionisation chamber coupled to an electrometer (20X60E chamber model, 2026C Radcal Corporation®, Monrovia, CA, USA) setting different kVp and mAs values. In order to better simulate the clinical settings, the *k*_{a,i} was measured by adopting the closest exposure parameters to the automatic exposure control (AEC) conditions. Additionally, the x-ray tube was free to rotate as in clinical examinations.

The *k*_{a,i} at the breast surface depends on tube-voltage, tube load, anode-filter combination, breast thickness, and distance between the x-ray source and the upper surface of the breast. The following relationship was employed:

$$ {k}_{a,i}=\eta \cdot \left(\alpha \cdot {\mathrm{kVp}}^2+\beta \cdot k\mathrm{Vp}+\gamma \right)\cdot \mathrm{mAs}\cdot {\left(\frac{\mathrm{FSD}}{\mathrm{FSD}-T}\right)}^2 $$

(2)

where FSD is the focus-to-support distance (67.5 cm for the Selenia Dimensions model, 65 cm for the Amulet Innovality model), *T* is the breast thickness, *η* is a correction factor which takes into account differences in the x-ray tube yield (air kerma to tube load ratio) of different mammographic devices. It can be defined as:

$$ \eta =\frac{Y_{tb}(FSD)}{Y_0\left({FSD}_0\right)}\cdot {\left(\frac{\left({FSD}_0\right)}{(FSD)}\right)}^2 $$

(3)

where *Y*_{tb} represents the yield (mGy/mAs) of the x-ray tube used, and *Y*_{0} is the reference tube yield (*i.e.*, the tube yield of the device A). FSD and FSD_{0} are the distances from the x-ray source at which *Y*_{tb} and *Y*_{0} are evaluated (a reference distance of FSD_{0} = 67.5 cm was chosen in our case for the device A). Both *Y*_{0} and *Y*_{tb} must be evaluated at the same tube voltage (32 kVp in our case). *α*, *β*, and γ are fitting parameters derived from the experimental measurements for a fixed anode-filter combination (W-Al in our case). The choice of 32 kVp as reference tube voltage was due to two main reasons: it is one of the most used voltages at our centre, and it lies in the middle of the tube voltage range used in DBT.

The accuracy of our method was evaluated in a number of exposure settings by comparing the air kerma measured through the ionisation chamber and the *k*_{a,i} calculated through Eq. (2).

Uncertainties in measured air kerma were referred to the accuracy of the detector, while uncertainties associated to the parameters involved in Eq. (2) were employed to estimate the final uncertainties of the computed air kerma values.

### Evaluation of *m*

The following exponential relationship was employed to derive *m* as a function of kVp:

$$ I(d)={I}_0\cdot \exp \left(-m\cdot d\right) $$

(4)

To simulate the breast, a homogeneous phantom with a density of 1.04 ± 0.04 g/cm^{3} (mean ± standard deviation), composed of polystyrene (C_{8}H_{8}) with admixture of 2.1 ± 0.2 % TiO_{2}, consisting of many squared plates (16 × 16 cm^{2}) was used. The thickness of each plate was 0.5 or 1.0 cm. The ionisation chamber was placed between the phantom plates to measure the beam intensities *I* at different depths *d*.

Measurements were performed on the Selenia Dimensions equipment (device A) in a range of 26–48 kVp and 40 mAs. The inverse square law was adopted to account for the variations in source-to-chamber distance due to the different depth of the phantom.

The *m* dependence from kVp was expressed as:

$$ m=\frac{a}{k{\mathrm{Vp}}^b} $$

(5)

where *a* and *b* are fitting parameters.

### Calculation of 2ABD

Once *k*_{a,i} from Eq. (2) and *m* from Eq. (5) were evaluated, 2ABD was computed from Eq. (1) in a number of clinical settings. Only the tube voltage, the tube load, the breast thickness, the value of the *η* parameter, and the FSD are required to estimate *k*_{a,i} and *m* for a given anode-filter combination and, therefore, to calculate 2ABD. The overall uncertainties in 2ABD calculations were estimated by applying the uncertainty propagation formula for *α*, *β*, *γ*, *η*, kVp, mAs, *T*, and FSD.

The 2ABD calculated by applying Eq. (1)—*i.e.*, by employing only the abovementioned input parameters of the model—was compared to AGD for different breast glandularity. AGD was computed by considering a breast of the same phantom thickness with different breast glandularity and adopting the same exposure parameters of the phantom. Uncertainties in 2ABD were obtained by considering a 4% of accuracy in the ionisation chamber (as reported in the Model 2026C, Radiation Monitor Controller Manual) and by propagating uncertainties of all quantities involved in the calculation. The total uncertainty in AGD calculation was considered to be 20% [14].

### Comparison of 2ABD in DM and DBT procedures

The homogeneous phantom was also employed for comparing the 2ABD values for DM and DBT in AEC conditions on device A. A number of automatic exposures for different phantom thicknesses (from 2.5 to 7.5 cm) were executed in both modalities, and the corresponding input parameters required for the 2ABD calculation were recorded. The 2ABD values were computed according to Eq. (1) for both modalities.

The AGD was calculated as the reference dosimetry index by following the approach by Dance et al. [8] for the same AEC conditions of the 2ABD calculation.